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all right today we're going to talk  about protein folding  and its relation to linkage folding  we're going to look at sort of a  mechanical  model of proteins this is an example of  a protein from lecture one  there's a ton out there in this place  called the protein data bank  all freely available it's really hard to  get pictures  like this but you get some idea that  there's a linkage  embedded in here you see various little  uh spheres and edges that's of course  not reality those spheres are actually  atoms and they're the kind of amorphous  blobs  the edges are chemical bonds and those  are  connections we don't know whether  they're it's not really matter but it's  a  force this is a rather messy picture  this is what a protein folds into some  3d shape most proteins fold consistently  into one shape we don't really know how  that happens we can't  watch it happen so  we the big challenge is to know how  proteins fold  given a protein what does it fold into  that's the protein folding problem  major unsolved problem in biology  biochemistry  the protein design problem is i want to  make a particular 3d shape so that it  docks into something binds to a virus  whatever  what protein should i synthesize in  order for it to fold into that shape  that is potentially an easier question  algorithmically it's the  really useful one from a drug design  standpoint you come up some some new  virus comes along  you design a drug to attack it and only  it you build it  usually you would manufacture some  synthetic dna you feed it into a cell  dna goes to the rna goes to the mrna  goes to the protein  you all remember biology 101 hopefully  we don't need to know much about it if  you look  at what's called the backbone of the  protein protein is basically a chain  and attached to the chain are various  amino acids today i'm going to ignore  the amino acids which is a little crazy  and just think about the backbone chain  backbone chain looks something like this  um one of the challenges of video  recording a class is i can only use  copyright free or creative commons  images this one i couldn't get one so i  had to draw it  there's various measurements here  certain numbers of angstroms those are  the chemical bonds various uh  atoms here nitrogen carbon so on  hydrogen but basically it's a chain  zigzags back and forth  you can also see the angles here they're  not quite all the same but they're very  similar  all the lengths and the angles are close  so uh it zigzags this is really in three  dimensions  and try to draw the sphere so you can  see the three dimensionality but it's a  little tricky  and then attached on the sides are the  amino acids i'm going to focus just on  the backbone  the way this thing is allowed to fold  these lengths as far as we know are are  pretty static they probably  jiggle a little bit but you can think of  them as edges so you can think of this  as a linkage  the catch is also the angles are fixed  because the way this atom wants to bind  to other things has  very fixed angle patterns if you ever  played with a chemistry construction set  that's that's how they work to have  holes at just particular angles  so if you think of like a robotic arm  normally  uh like here i have a two edge robotic  arm let's say  normally you have two degrees of freedom  in three dimensions you can change the  angle  and you can spin around this edge  now i'm saying the angle is fixed for  example here it's say at 90 degrees all  i can do is spin  i'm not allowed to flex my muscle in  this way  okay so that is the model all of in this  case we have a tree  all of the angles here are fixed but you  can still for example take this entire  sub chain  and spin it around this edge that'll  preserve all the angles and all the  lengths  that's all you're allowed to do you take  an edge you spin it spin once half of  the edge relative to the other half  these are called fixed angle linkages  and they've been studied quite a lot  because of their connection to protein  folding  so embedded in the term linkage we  assume that the edge lengths are fixed  and then we add  the constraint that the angles are fixed  and the motivation is the backbone is  something like put  the background of a protein is something  like a fixed angle tree  of course it's not much of a tree most  of it is a chain there's just  small objects hanging off and if you add  the amino acids there are bigger things  hanging off but still constant size  they'll have some cycles they're not  trees but  it's slightly more approximately maybe i  should draw  wavier lines uh it's a chain  usually an open chain although  occasionally closed chain  so we think a lot about fixed angle  chains and sometimes about fixed angle  trees  now fixed angle linkages are harder to  think about than  universal joints that's the usual kind  of linkage um  so and we know 3d linkages are kind of  tough nonetheless we found lots of  really interesting mathematical problems  to solve here  and that is the topic of today at some  level  we are thinking about the mechanics of  protein folding  we're throwing away energy we're  throwing away the  the actuators in real life that make  proteins fold  we're just imagining given this  mechanical model of how a protein might  fold  what's possible so it's some sense it's  broader than reality  and the hope is you find an interesting  algorithm for  how to fold these protein chains maybe  that's the algorithm that nature is  implementing  that's the kind of general picture we're  not constrained by  reality and by how nature actually folds  things  so i'm going to talk today about four  main  problems here  first one's called span second one's  called flattening  third one is flat state connectivity  and the fourth one is locked  our good friend locked chains  and of course there are lock chains  because we're constraining linkages even  more than before so if you take knitting  needles it'll still be locked  because you add extra constraints makes  it harder to fold  but there are actually some interesting  positive results we can give of chains  that are not locked in some sense  and flat state connectivity is about the  same kind of thing instead of worrying  about getting from anywhere to anywhere  we just worry about getting from one  flat state to another flat state flat  means  lying in a plane flattening is about is  there such a configuration  and span is about given  robotic arm like a more complicated one  like with multiple  edges how far apart can the endpoints  get and how close can the endpoints get  the universal chain is not very exciting  the farthest thing is when it's straight  and the least far they can get is when  it's closed and you can always do that  i think well no i guess you can't always  close it up  that's a little non-trivial but for  fixed angle linkages you can't  straighten out because you have to  preserve the angles  so it's kind of what is the straightest  like configuration given that the angles  are fixed  so let's start with span  so the span of a configuration is the  distance  between the endpoints  and  in general you'll find the max span and  the min span  this search was begun by a guy named  mike sauce  who was a phd student at mcgill  and he proved that if you want to find  for example  flat state that lives in two dimensions  with the minimum  or the maximum span  this is np hard  this is in his phd thesis  question  oh here i'm assuming open chain i should  say that  which most proteins are yeah i've been  talking about trees and stuff i  hear i mean chain otherwise there aren't  two end points to think about  good uh so here are his np hardness  proofs  in fact the problems are mp complete  they're pretty  simple the problem here we're reducing  from is  partition i give you a bunch of integers  i want to divide them into two halves of  equal sum  and the top example is minimum flat span  problem so you have you make an  orthogonal chain  where the horizontal edges are long and  they're proportional to the integers  you're given  and the vertical edges are really tiny  and so what you'd like to do  what all you can do is sort of flip in  the because you have to stay in the  plane  you can flip one of the vertical edges  say and make any of these edges go left  or right  you get that freedom but so  each integer you get to choose do i go  right by that amount and go or do i go  left by that amount and if the amount  you go left is equal to the amount you  go right in other words there's a  partition into two equal sums  then those endpoints will be aligned and  then their distance will be very tiny  otherwise it will be quite large because  the horizontal distances are all big  so that's kind of an e very easy np  hardness proof  to maximize your flat span  instead of mapping your integers onto  lengths you map them onto angles or turn  angles i won't  specify that too precisely but again if  you if you make your total  counterclockwise turn equal to your  total clockwise turn  then the two end edges which are super  super long  will be parallel and to maximize the  distance between the endpoints you want  them to be parallel  if you make them go some other angle  they're closer  now both of these proofs rely on the  requirement that you want a flat  configuration with minimum and maximum  span  now there's a claim that flat  configurations matter for proteins  so it's a natural constraint but what  about the general problem what about  i have something in three dimensions i  want to maximize i have a fixed angle  chain in 3d  maximize or minimize the span  both of those problems are open can you  solve them in polynomial time  for 3d  max span  so the non-flat version just for  maximization  there's been a lot of work and there are  two papers on the subject one of them is  by nadia  and joe o'rourke another one is by  borsea and stranu  and i just want to quickly summarize  that because  there's a lot of stuff there but  essentially they find  what the structure of those spans look  like i have  an early figure that's in our book  before all this work was done  of a simple chain this black guy one  two three four bars open chain and in  that  black three-dimensional state it  maximizes the span the  the green span there and if you look  from above which is this picture  of course the endpoints look much closer  in in projection  and the red configuration is the max  span if you restrict to flat  configurations  so here of course 3d buys you something  in general  it always will an interesting thing is  that this max span  the green line passes through another  vertex  that seems kind of weird and in fact  there's a general  theorem there sort of characterizing the  structure  of these chains it's still not known  whether we can solve this problem in  polynomial time but  for orthogonal chains where all the  angles  are 90 degrees  uh we can solve that in linear time i  guess  and here's what it looks like  suppose you have some orthogonal chain  orthogonal chains are nice because you  can draw them in a plane  as a staircase so there's a nice  canonical configuration  what one way to think about  how to find the  maxspan configuration i'm just going to  give a high level overview here this  won't be a complete algorithm  is you triangulate that staircase in  this sort of obvious way of connecting  every endpoint to the one to a head  this you can think about  this is like a body that's hinging  around here because i can  spin if i spin the left part of this  chain around this edge it's like hinging  that triangle around that hinge  same thing you can think of these  triangles as just being hinged together  like  in rigid origami it's the same  class of motions and now  you can uh what i'm going to do is  compute a  shortest path in this surface from here  to here confusingly this is called a  geodesic shortest path although it's not  really  related to geodesics from polyhedral  surfaces  but if i compute a shortest path it's  going to go like to this vertex and then  probably to that vertex but i'm  constrained to say  stay inside the unit of those triangles  i want to go from one endpoint to  another  then i claim that okay the  these two edges will stay planar of  course they form a triangle  uh i claim these four edges will stay  planar  and in the orthogonal case they'll stay  zigzag and then also these two guys will  stay in their own plane  and then uh i claim that actually this  wiggly line which is not straight  because it bends here and it bends here  the total length of that wiggly line is  the max span  and you achieve that by folding this  planar part with respect to this planar  part with respect to this planar part  so that the wiggly lines become aligned  and straight  and that's very hard to draw but it can  be done and that's what you do in the  orthogonal case and that gives you  the answer in linear time with enough  work  for non-orthogonal though it's open  whether you can do this in polynomial  time  maybe it's np hard actually i don't know  all right  that's all i want to say about span next  we go  to flattening  i guess the first question about  flattening and and the main one we'll  we'll talk about here until we get to  flat state connectivity is  does a fixed angle chain have a flat  state at all  can you even draw it in the plane  without crossings  so we're restricted here to have no no  self intersections  we want a flat state  no self intersection  then there would be a question of given  some configuration can actually  continuously get to a flat state but  sort of the simplest question is ignore  about getting  ignore getting there just is there a  flat state  and this problem is np-hard  again mike sauce and his advisor godfrey  toussaint  it's a little more complicated but it's  basically the same idea as that very  simple proof  which was just to map integers to a  little zigzag  staircase here so the goal is to force  x to end up being the two endpoints of  the green curve to be aligned with each  other that will exist if and only if  there is a partition of the given  integers  and there's all this infrastructure this  sort of  uh there's a little lock here and a key  and and some structure on the left  basically forces the picture  to look like that um so the first claim  is that the black stuff  is basically unique i think there's one  global reflection you can do  that doesn't affect anything but you try  any of the other flips again we're  restricted to flat states here so  there's only sort of a  bounded number of things you can do  finite number of things you can do  you try all of them they self-intersect  so the black thing is basically forced  and it forces the end point this  endpoint x from the black side  to be aligned with this very narrow  spike  and because the angles are preserved  that red guy is going to be vertical it  can't go down so it must go up  and so only if this thing is aligned in  the center aligned with x  in other words this problem has a  partition  will this have a flat state so it's not  the most exciting example this is only a  weak np hardness proof  lots of interesting questions still open  here like if all the lengths are  the same if they're all equal  then we don't know or if all the lengths  are even polynomially bounded this needs  really really long lengths versus really  really tiny lengths exponentially  exponential and ratio all these problems  are open  and that's flattening so we're going  very quick uh because there  isn't uh well partly because i'm more  excited about this but there's more work  in these two  parts so i'm gonna focus on that next  topic  is flat state connectivity  so the idea is to think about the  configuration space of these fixed angle  chains let's say and we kind of know  that it's going to be disconnected  because there are knitting needles there  are nasty things  so there's maybe various connected  components  but let's say that we really care about  flat states  and the question is are they connected  to each other  so in other words do all the flat states  mark them with x's  do they all appear there's only finitely  many so configurations there's  it's this continuum and there are these  messy blobs semi-algebraic sets  but flat states those are discrete  things because we have fixed  angles you can flip or not flip every  edge so most exponentially many of them  so finite are they all in one component  so i can get for if i pick two of my  favorite  flat states there's a path between them  or are some of them  in multiple components so in this case  we call it flat state disconnected  and if they're all like this we call it  flat state connected  and we just like to know which chains  which  fixed angle trees whatever are flat  state connected  versus flat state disconnected i would  say  the big the big open problem here is  our is every flat as every fixed angle  chain open chain flat state connected  that is still open we have lots of  results in  in that direction so the top four  results are about open chains but they  have an extra constraint  example open chains that have a monotone  configuration like the staircase  those are flat state connected and so  in fact whenever the angles between the  edges are  either orthogonal or obtuse then  their flats stay connected when the  angles are acute  we're not really sure if all the angles  are equal and acute  then we can do it but if they're  different and acute we don't know  unless the edges are all unit length  and the angles are in this funny range  then we can do it so there's all these  special cases we can solve  the most relevant to proteins is  actually obtuse  chains so we've solved sort of the main  problem  with this second result but there's a  natural  theoretical question here is are all  open chains flat state connected  or do we get disconnectivity i will show  you that i'll show you the orthogonal  case in a little bit  we can do some stuff we have multiple  chains that are attached to some  blob like a cell closed chains is a  little bit  for disconnected uh we don't have  very interesting examples i would say  this is funny because locked examples  are easy to come by  but flat state disconnected examples are  a little trickier because flat is so  constrained  so let me just show you these examples  this is what we call a partially rigid  fixed angle tree so not only are the  angles fixed  but also the black edges  are not in fact only the blue edges here  are allowed to spin  everything else is held rigid so these  arms are somehow forced to be in exactly  that geometry  i can spin it around this edge just  spin it up into 3d for example these are  two different flat states of the same  linkage  right all i all the only difference  between these two i haven't rotated or  anything is i've taken each of these  arms  and flipped it around a blue axis  if i do all four of them i would get  this picture but the claim is you cannot  do that without self-intersection  the intuition is when there aren't very  oh one other thing  this makes this slightly more  interesting it's weird to say well why  did you force some of the edges to be  rigid and not others  one way to force that is to use a  general graph  if you add some extra edges to sort of  brace this and all these angles are  fixed  then this linkage will behave exactly  like that one  so that at least is somewhat more  natural although what we really care  about  are chains maybe trees  but we don't know whether there's a we  also don't know whether all  true fixed angle trees are flat state  connected  these are the the worst examples we know  let me give you an idea of why that  it doesn't work this is a little  animation of  just a couple of moves attempted it's  just going to cycle  through that and these are some  static images of same kind of thing  so the intuition is the following you  have four arms you have two  sides to the plane there's up and down  for four  arms and two sides at least two of them  are going to have to go to the same side  the best you can do is two and two or  three and one but in either case you  have two sides  go to the two arms to go on the same  side  now it could be like in this image that  they're  opposite arms so there's this arm here  and there's this arm here so they're  connected by 180 degree angle  and those guys when they fold up  actually these edges will just hit each  other  dead on so that's uh  kind of obvious from a geometric  standpoint maybe you call it cheating  for them to hit dead on you can twiddle  the edge length so that they will  properly intersect without dead-on  collision  without being degenerate basically the  alternative is the  and this is a little harder to see  geometrically and that's why we drew  that  animation is that you have  you have one arm and you have an  adjacent arm connected by a 90 degree  angle  now here there's clearly some collision  going on and if you happen to fold it up  90 degrees like that and then fold the  other guy obviously you get stuck  but maybe you fold it a little bit and  the other guy goes a little bit more and  there could be some dance between those  two degrees of freedom those two arms  that somehow gets them both to pass over  to the other side it's obviously not  possible  how do you prove it well you can prove  it with topology  not not theory or link theory  so it's very cute proof uh you start  with  so here's the the full example but i've  highlighted the two arms in red  that are going to move and i imagine  connecting the endpoints  of each arm with these little blue ropes  underneath the plane so all right they  they're both going on the same side  let's say that somehow pass through each  other on the top side  then i'm free to connect stuff on the  bottom and i shouldn't collide with that  so if somehow both of these guys flip  over  so the arm on the left a3  flips over a3 stays where it is but now  the arm is on the top the north side  instead of the south side  and the other guy from b  to b3 used to go like this and now it  goes like this  if that happened somehow then these  ropes could remain intact during that  whole motion and on the top  you have two closed loops that are not  interlocked  on the bottom you have two closed loops  that are interlocked so there's no way  to get from there to there without  colliding somewhere  the blue stuff didn't move so the red  stuff must have collided  so even just topologically you are  screwed  that is our only negative example lots  of interesting open questions here  on the positive side let me show you  for orthogonal chains and the same  algorithm works for obtuse chains  all the angles are obtuse uh how they  are flat state connected  so in order to show its flat stay  connected i take i want to think about  two flat states  and show that i can fold from one to the  other  via some intermediate 3d stuff  let's start with one of the flat states  so it's orthogonal so in two dimensions  they really  all the edges will be horizontal or  vertical in 3d they can kind of be in  many  many different angles many different  dihedral angles in 2d it's pretty simple  and all i need to do is sort of pick up  that chain and i'm going to try to pick  it up into a staircase  because there's only one staircase if i  can make it a staircase  i make configuration flat configuration  a staircase flat configuration b  staircase i just fed x in the middle  right  once they're both staircases i play one  motion and the other one backwards  get from anywhere to anywhere so here's  all you do you take the first edge  and you just rotate it up to the red  line a  uh and then you take the next edge  and you take both of those edges and you  just rotate them  like this so you get that little  two-step  staircase now i'd really like to pick up  this edge but  i want to first get these two edges in a  plane with that edge  so i rotate this flag over to the left i  get  those two guys and now they're in a  plane with this and i just lift that up  then i'm going to flip then rotate up  flip rotate flip rotate  here's some more examples so if at this  point  i have this staircase um  sorry i guess originally i have from v3  to d  up there i it's not in plane with this  guy  so i just rotate it like that i'm  spinning around this edge  so now i have from v3 to e and then i  rotate it up along that green arc  and i get a bigger staircase above  the chain and because everything's  staying above it will never penetrate  the plane it'll never hit anybody else  and i'm building a staircase by design i  always rotate this  there's actually two ways i could be in  plane but i always rotate it so that  when i pick some an edge up it'll be in  a staircase  so this is actually really easy and  slight generalization is to  obtuse chains then instead of making a  staircase we make a  monotone get this right  yeah some z monotone state so it goes  monotone and z  out of the plane and that's enough to  avoid collision  and you get a canonical configuration  also if you have acute angles  but all the angles are equal then  there's a natural  canonical state which is just like a  compressed staircase and that will work  here too although that takes more effort  that was in a separate paper  but big open question is chains with  arbitrary angles we have no idea  it seems very hard to do an operation  like this  wow we are burning through this is fun  so the next topic is about  locked chains  now as i said you can take a knitting  needles example  which has five edges  and that will still be locked if you  force the angles to be fixed because it  was locked without the angles being  fixed  now it required a length ratio of uh  three to one i think yeah this edge had  to be longer than the sum of those three  uh so let me put down  uh some open problems so you may recall  in the case of universal chains  universal joints i should say  the big open question was can you lock  a universal joint 3d chain with  unit edge lengths so equilateral  every edge is the same length  is there a lock chain like the knitting  needles where all the edge lengths are  the same and one of the motivations for  that is in proteins  the edge lengths are all within like 50  of each other  so it's pretty natural of course  we don't have universal  joints with proteins we have fixed angle  joints  so the big open problem for fixed angle  joints  i guess we'll do this in parts  is there  a locked  3d fixed angle chain  that's equilateral okay i'm going to add  some conditions here  so that's first natural question  knitting needles doesn't suffice we need  a three to one length ratio as far as we  know  uh yeah  turns out that question is not very  interesting i need to do slightly  non-linear  editing here so you take a knitting  needles example  and you just subdivide the edges into  lots of little tiny bars it doesn't have  to be this extreme  you could not subdivide these edges at  all and make these guys subdivide them  into like three or four parts  because the angles are fixed these guys  act as a single bar there's really no  difference  maybe you make slight curve there and  then they can wiggle they can bend a  little bit  but really not much so if you just  say oh i want it to be unit length i  don't constrain what the angles are  but i fix them then it's trivial to come  up with locked examples  so that's not very interesting  what if i make it not only equilateral  the lengths are the same  if i make it equally angular  because again in proteins all the angles  are  similar they're around 110 108  something like that they're all pretty  close within if i think within  10 20 of each other well here there's  also a locked example  and just to show you how research was  done  back at the turn of the century this is  pre  web 2.0 pre-ajax and all that fancy  stuff  we used ascii art this was email was the  tool of choice  i know it's hard to imagine a time 2002  so long ago and i i tracked this down  this is the original  claim where it looks we call this the  crossed legs example because it's  like two legs crossed around each other  and uh  this is the the first time we thought oh  maybe it can be done  unit length this is stefan langerman and  uh and here for the first time ever  this is not the first model but this is  the first photograph of any model i'm  aware of  this is the cross legs example um this  is made with a  construction toy that used to be sold  around here but is no longer in  production it's hard to  so they're pretty hard to get is straws  nicely colored straws  and the cool part are these uh these  connectors  so the connectors force particular  angles in this case every angle is 45  degrees  so this is equiangular and equilateral  because all the straws  i'm told are the same length that's how  they're sold  and you can do edge spins so  whoops that's called cheating  it's not totally obvious that this is  locked  the problem with the model is that the  edges can bend  but if you treat it properly and  only spin around edges  then you're you're stuck now  there is one thing you can do let me see  if i  uh  yeah like this so here i'm almost in a  plane i've got the purple edge right  against the  pink one is it easier to see from that  angle i don't know  so here this guy can come out  and this guy can barely go  along the edge so actually this doesn't  quite work for equilateral it works for  one plus epsilon that's why i added  these little  nubs at the ends so if it if  they're all exactly equal length and you  allow just a  abrasion of the endpoint then this could  go around  like that and then you'd be unlocked if  you just add slightly  either you change the angles to be not  quite equal so ben  make this a little smaller or you make  the lengths a little bit longer at the  ends  then the claim is it's locked we don't  actually have a formal proof of this  we're just remembering hey we should  probably write this up i was talking to  stefan last night  and uh yeah so someday we will prove  that this is locked  but certainly looks like it so this  isn't open yet  i mean modulo the details of that proof  equilateral and equiangular  seems easy to lock with fixed angle  chains  in fact even easier that this example  only has  four edges so even less than the  knitting needles fixed angles  make for complicated motions i guess  make it hard to unlock things so i need  to add one more constraint  and the constraint is obtuse  so again all of these properties are  enjoyed by proteins protein  protein backbones have all these  properties even if you looked at fixed  angle trees  is there something like this that's  locked and now we don't know  and this seems quite tricky i guess the  intuition is that obtuse or and usually  we think about orthogonal just because  it's easier to draw the pictures but  reality is more like 108 degrees  uh conjectures of two chains obtuse  fixed angle chains behave  kind of like universal joints and with  universal joints we don't know whether  equal  equilateral is enough so  it's tricky  what if you instead had like  made your ribbon lengths basically like  a bunch of little unit obtuse angle  connectors  and then when you hit the the big turns  it's just  you know up to yeah you can definitely  you can make this example be entirely  obtuse you can make every angle obtuse  here you could arc a little bit  here you could arc some more but not too  sharp and because here  we actually know that this part can be  made a string we don't really care what  it looks like  so you can make it fairly obtuse it's  just that these guys should not bend  much they have to  be long no matter how you fold them so  if you want  equilateral and obtuse that's also easy  but to make all the angles actually be  equal as far as we know you cannot take  that knitting needle subdivided  make all the lengths equal and all the  angles equal and make them obtuse  that's open but any two out of the three  it's easy  uh of course in reality they're not  quite equilateral they're not quite  equal angular  but it's still open for the if you have  like a small range for the lengths and a  small range for the angles  this is open we pose it this way because  it's the cleanest geometrically  but the the real question you care about  is  when these are fuzzy constraints obtuse  is real but these guys are  fuzzier so  uh if you think about proteins uh which  fold very well  in nature there are a couple of reasons  they might fold well  we know as far as fixed angle chains go  it's actually  quite easy to find locked examples and  this is somewhat intuitive but bear with  me  because there are locked examples in  this configuration space we believe  these configuration spaces are really  ugly nasty  so it would be very hard even if you  know oh i only need to fold something in  my component  if these guys are highly disconnected  and flat states are all over the place  it's probably even within this connected  component it looks really ugly  and so it's very hard to find a path  from one state to another  probably p-space complete although we  don't know that for sure  but that's the intuition locked equals  messy  when there are no locked configurations  like carpenter's roles  we get really nice algorithms it's super  easy to get from state a to state b  now if you're nature or you're designing  nature let's say  or you're building your own virtual  world second life  and you want to design proteins you  would like to design them in such a way  that they fold easily because it happens  all the time  every thing that is being acted on by  our body every living thing that we know  has tons of little proteins that are  doing all the work they are  fold into their shape and they do  something that's proteins plus  rna but mostly proteins so  uh to understand life we should  understand proteins now how do proteins  fold so well when we know there are all  these locked configurations  one possible answer is that proteins  have extra structure namely these three  things  which somehow make it very easy to  algorithmically go from a to b  notice i'm not even i'm not assuming  anything about how  proteins fold in terms of  what is the mechanism that drives them  because  we don't really understand those  mechanisms there's hydrophobia which we  don't really know how it works  it's all these little forces um that we  don't fully understand  lots of we understand lots of parts of  the story but not the whole story  and what's convenient about these kinds  of problems is you don't need to assume  anything about how it actually happens  all we're assuming is the mechanical  behavior of proteins and how they could  possibly fold  and the the idea is if there's a lot  configurations that's probably the wrong  model because then everything's messy  now there's also evolution coming into  play and maybe some proteins are easy to  fold some  proteins are hard to fold that's an  interesting question which should be  experimented with  but let's hope that there's a model  things are mutating randomly you really  like everything to fold nicely  maybe it's because you have all three of  these properties approximately  in real proteins  so the general idea is that nature has  some extra constraints that make  protein folding easy we just have to  figure out  what they are and why it makes them easy  unfortunately this is still an open  problem if this had an algorithm  that would be a natural candidate for  what nature is doing using its  mechanical  or using its for energies and forces and  so on  this would be a rather unsatisfactory  ending if this was  if the climax was an open problem  we have a theorem too  and this is what i'll cover in most  detail  and it's paper called producible protein  chains  protein chains just means fixed angle  chains open chains  and the idea is well yeah there are  these constraints  uh or there are these extra features we  don't know how to exploit them so  let's not even worry about them suppose  they don't even exist  maybe i'm going to assume obtuse but  none of the others  there's another constraint in how  proteins fold or really how proteins are  created  they're created by a machine a molecular  machine made up of a whole bunch of  proteins and rna  called the ribosome you may have heard  of it translates  messenger rna into proteins so there's  some mrna  around here maybe don't know exactly how  this machine works  but there are actually very accurate  three-dimensional reconstructions of the  ribosome  with no copyright free images you're  going to have to there's a link on this  slide that goes to  the cool amp 3d models of the ribosome  with a slice away so you can see this  there's a tunnel down here  and the prot the protein gets sort of  created here the backbone gets created  here  and starts going through this tunnel  there's a bend in the tunnel around here  where it's conjectured an amino acid  gets attached  and then that it goes out the tunnel and  the protein starts spewing out here and  presumably folding at the same time we  don't really know  so this is how proteins are created  um the birds and bees i guess of  proteins  so what's interesting about this is it's  not like a protein  exists and then folds which is how a lot  of  people might think about it at first  glance  that's natural way to model protein  folding you start with a protein say in  just  zigzag configuration if it's obtuse  there's a nice zigzag monotone  configuration then you see what is  the best configuration i could fold into  for some notion of best  and that's sort of what this  configuration space picture is about  it's  if i already have a protein what  configurations can i reach by motions  and that is interesting that's important  because you're still going to have to  reach by emotion but it's actually more  flexible than that  because protein could just be partially  built  the the rest of the protein hasn't been  built and it could start folding already  and it might be easier to fold when you  don't have the obstacles  of your existing protein so that's both  a worry  but it's also a convenient structure  because this ribosome is a giant  obstacle  it's bigger than most proteins there if  your protein is really long maybe it  could go over here  but most of the time it's going to stay  on one side of this plane  because locally this thing is basically  flat if you look at the real  3d pictures not the schematic  now this is good news for a geometer  because there's this giant obstacle  think of it as a half space which the  protein cannot penetrate  while it's being produced over here  that's it that half half space  constraint  is enough to get really good algorithms  for  folding your chain it's weird because  we've made our problem both harder  because the protein is only partially  produced at any time and it can fold  which is part of it  but we've also made our life easier  because there's this big obstacle  yeah right we're out of this we're going  to get that the angles in the protein  are constrained  and in particular for this angle  uh it depends i mean you could  in this picture because it's  perpendicular here yeah uh  you can the the sharpest angle you can  make is 90 degrees  more or less that's a good point  so it's a convenient match between the  chemistry which also forces the angles  to be obtuse i guess  i don't know a ton of chemistry but also  the ribosome just geometrically forces  we're going to use a property like that  our model is going to be a little bit  more  both more general and simpler  we're going to imagine that the ribosome  is a cone  it's part of the upper cone here this is  like a  mirror image and  in reality that cone is actually a plane  and everything above the plane  but to be more general we're going to  allow some angle alpha here it's also  just easier to think about when alpha is  smaller than  90 but it's work everything i say will  work when alpha equals 90 and that is  sort of the reality  case so so the model is  so the ribosome is a cone who we call  this the half angle of the cone  from the vertical axis to the edge of  the cone is alpha  so if you're going from one axis to the  other would be two alpha  the the model is you start with one  link of your chain which is inside the  cone it spews out  uh through the apex okay that's that's  the exit of the tunnel  here we're allowing the tunnel to be  actually quite free it doesn't have to  be perpendicular  to the to the apex or the plane of the  apex  so the edge comes out and as soon as the  vert  the end point of the chain reaches here  then a new link is created  this is like a very simple model for how  a chain can come out of a cone without  worrying about what's happening inside  the cone  imagine everything's totally free this  is like you can allow self-intersection  the cone who knows what  but once you come outside the cone  you're not allowed to self-intersect  and you're not allowed to intersect the  cone once you come out you can't go back  in  so that is a model of producing protein  chains  and if you have a cone of angle alpha we  call this an alpha  producible chain  for whatever reason we often call it a  beta producible chain  just change the variables  so if you think of the ribosome as a  cone with half angle beta  you can produce it like this that is  beta producible now this is a pretty  powerful model  because you only have to worry about it  link by link you don't have to worry  about the rest of the chain until it  gets spews outside of the cone  but it's restrictive in that you cannot  penetrate the [ __ ]  all right one thing we can talk about  is angles so  i'm gonna write call a chain a less than  or equal to alpha chain  if all the turn angles are less than or  equal to alpha  i don't know if i've used turn angles in  this class probably  if i have two edges the angle would be  this  the turn angle would be this supplement  yeah i guess we use trit angles way back  in origami land kawasaki's theorem and  so on  it's just if you're going straight how  much do you have to turn to  get to the next edge so we'd like uh  fairly obtuse things so alpha's going to  be small there isn't a ton of turn  but in general less than or equal to  alpha chain for some alpha  now there's a relation as jason was  mentioning there's a relation between  alpha and beta  in the ribosome because you always  exited orthogonally to the  the plane that was your cone the  sharpest angle you could get was 90  degree turn angle  here we're a little freer because this  edge  can wiggle around as long as it touches  the apex  so if you're up against the cone  you have to slide out into the  complementary cone that was the  previous picture and as soon as you get  there  you could create a new edge which is  like this so the sharpest angle you can  get  is actually twice beta in general  we're going to have alpha over 2  is less than or equal to beta that is  you can get up to  beta equals 2 alpha get that right  and also in the obtuse case this is not  too exciting  but it's true uh  there's actually some open problems here  when you have that full flexibility and  you set  um sorry alpha to two beta not the other  way around  uh i'm going to assume here  that alpha equals beta  this will be convenient and it's the  interesting case because in reality the  cone  has a half angle of 90 degrees so beta  is 90.  and the sharpest angle we're going to  make is always obtuse so  saying that you have a less than or  equal to 90 chain is just fine  but in on the mathematical side i think  we solve the case when alpha is less  than or equal to beta um  but not when alpha over 2 is less than  or equal to beta that's a weaker  constraint  so there is a range where it's not so  easy  all right now what do i claim about  these chains  other than their angles are not so sharp  i claim they're good algorithms for  folding them  what could i possibly mean there are  still locked configurations  is that true well  yeah i mean presumably this is acute but  you take the obtuse versions of this guy  because i didn't constrain the edge  links or anything i just said that  they're that the angles are obtuse  so i could just sort of round these  corners make it obtuse you know add lots  of dots just at the corners  that will be obtuse and a chain  like this will be producible a chain  with these angles and these edge links  can be produced from a cone but  this configuration of this chain cannot  be produced  i claim i claim anything that can be  produced  is in one connected component so while i  can make a linkage that is locked in  that there are  bad configurations you can't get out of  the things you can actually make  you can always get out of so there's  going to be  the space of producible configurations  uh maybe there's some stuff that's  unproducible but still connected to it i  don't  i don't know doesn't matter too much i  won't worry about this stuff  there's other bad locked configurations  that you cannot  cannot reach here but everything that's  producible is in one connected component  of the configuration space  that's property one that's kind of nice  also all the flat states are going to be  in here  this is actually pretty easy i just need  to prove that flat states are producible  which we'll worry about later so in  particular these guys are flat state  connected  all the producible protein chains are  flat state connected  that's interesting because we don't even  know that all chains are flat state  connected  but here i guess we know that obtuse  chains are flat stay connected  so maybe it's not so surprising  but what's important is not only are the  flat states connected to each other and  the producible states are connected to  each other  but producibles connected to flat flat  states everything is together here  i might have more properties but that's  already some good news and there's  algorithms to do all of this  how do we prove it well  as usual we use the fedex method  and some since one of the challenges is  what is the natural canonical state for  protein chains in fact we're just going  to assume that our chain  is in reality we're going to assume that  it's an orthogonal chain  an obtuse chain i should say but in  general  for any lesser equal to alpha chain for  whatever alpha you like  and it will be the half angle of the  cone so alpha equals beta  we will define a canonical configuration  i think we called it the alpha ccc  so it's going to be kind of like a helix  i think i have an example an actual  computed example  that's not the best picture because you  can't see everything that's going on but  this will  this is an actual canonical  configuration of a particular  chain  let me tell you how it works in general  so in general we have some  chain v1 v2 sorry starting at v0 v1 v2  um i want to define a canonic and  there's defined lengths between the two  and there's defined angles between every  triple  in sequence so i'm going to start with  v0 somewhere  it doesn't really matter by translation  say the origin of space  and  what i'm going to do is draw a cone  whose apex is at v0 and the half angle  here is going to be alpha over 2 not  alpha  this is a smaller cone than by a factor  of 2  than the ribosome that's important  so there's this uh vertical line  and to pick v1 i'm just going to use  this  the right edge which is let's say this  is the x direction this  is the z direction maximum x coordinate  it's going to lie on the cone maximum  export it that's v1  okay now v1 let me  redraw this picture a little lower so  there was  v0 v1  now i want to draw v2 i want to draw it  above  so what i'll do is draw a vertical cone  whose half angle here is alpha over two  and i want to draw v2 on the cone here  of course the the height of the cone is  the length of the edge  not the height but you know this you can  think of the cone as infinite  and then i just clip this to when it has  the right length  so again i'm at this might be a  different height cone i  clip it to whatever the length v1 v2 is  i want it to be somewhere on this cone  but now i'm constrained to have the  correct angle at v1  i can't just put it over here because  then the angle here would be 180  presumably i don't want to make 180  degree angle  so in reality what happens is that  there's a cone  which is um  who's whose axis is the edge v0v1  so i extend v0v1 out here which in this  case happens to lie here  and i make a cone like that  let me draw it slightly more accurately  okay in this case the center  axis of the cone would go right here  whatever the extension of v0v1 was  and to be to have the right angle at v1  v2 must be on that cone conveniently  there are two intersection points  between those two cones  i can choose either one of them to be v2  and i will choose the counterclockwise  most  one which is this one  this is going to be v2 so draw that edge  now i repeat so from v2  i'm going to draw a vertical cone whose  half angle here the half angle was  always alpha over 2.  if angle is alpha over 2. this cone  had angle  or half angle whatever the angle v0 v1  v2  was is that the half angle or the angle  the  half angle so i think this is right  okay and then i take the intersection of  those two cones  and that will give me where v3 is so i  do the same thing for v2 for v3 and so  on  this i have a unique choice at every  moment  yeah basically and the only exception  was at the beginning here when i had a  vertical line  these cones two cones could actually be  equal and then the intersection is  the entire cone in that case i guess i  choose the maximum x-coordinate one  again  and then i'll i have a canonical choice  of everything  along the way it will always go up and  it sort of spirals around because of the  counterclockwise  most choice and the result is a picture  like this  anything else i need to say here all  right  i claim this canonical configuration  lies in an  alpha over two half angle cone  that's true by construction the  challenge is does the construction  really work so i start  obviously with one cone and i can think  of this as actually an infinite cone  that goes  out to infinity here and i claim the  entire construction will lie inside that  cone  and it's kind of obvious because v2 v1 i  chose v2 to lie  in in this co the same  cone just translated up to start at v1  instead of starting at v0  of course this cone is contained in this  bigger one  and by induction in fact the entire rest  of the chain will lie  in this smaller cone therefore it lies  in the big one also  okay fine  so by construction it will lie in alpha  over two cone  the worry is that these two cones don't  intersect  here we have an angle the half angle of  the cone is whatever angle  the angle is at v1 sorry this should not  be the angle this should be the turn  angle  that that angle is how much you turn  from v0 v1  now we know that we're assuming that the  turn angles are all at most alpha  this cone so this the the turn angle  cone could actually be twice as big  as the vertical cone that we were always  using we always use a vertical cone  half angle alpha over two but because  it's okay because we always keep these  edges like v0 v1  was on the edge of the cone uh  and so if when we extend it it lies on  the edge of this vertical cone  so its angle in in the most  extreme case its half angle is alpha  which would look like this would go all  the way over  from the right side of the cone to the  left side of the cone in general it's  not going to be right and left but it's  going to be some side and the  antipodal point and because the  double angle of the cone is alpha it's  still okay  you will intersect somewhere on the cone  this is a subtle detail but it's really  crucial because we start with a chain  that has relatively large angles alpha  and we get it  into we squeeze it into a cone that  still has double  twice its angle is alpha but we kind of  compress it  into something of half angle alpha over  two you might think oh i'm just  changing the definition calling a half  angle therefore i could stand alpha over  two but it's  a little tricky to actually get it to  fit in a vertical alpha over two cone  once we have this it's really easy to  canonicalize a chain  a producible chain so let me tell you  how to do that  so we're going to use the fedex method  of taking some configuration  and canonicalizing it and then  uncanonicalizing to something else  we're also going to use a new method  which i just came up with the term  it's called the momento method which is  you play  the movie in reverse  so i guess also the merlin method uh  that's more complicated so  we have a movie here in mind  which is how was the chain produced so  what i want to show  is that if i want to start with a  producible  configuration and chain  somehow it got produced so  uh you had your cone and the thing  starts spewing out and folding and doing  whatever  uh that's that's an animation in some  sense of of one edge coming out and  stuff is folding at the same time then  an edge is created then another edge  comes out and so on  what i want to do is play that movie  backwards this is pretty intuitive idea  i just want to start feeding the edges  back into the cone  and just keep stuffing them in now what  happens out here  is easy because we know it doesn't  penetrate the cone that's the assumption  and we know whatever was created here  could be uncreated as long as you  can't afford to erase edges one by one  that's the tricky part how do i erase an  edge usually i can't  but it's not i don't have to erase any  edge like if i had to erase this one  that would be hard because some motion  here might penetrate where that edge  ought to have been and erasing the edge  will not  will make it  adding the edge makes it harder to fold  so i can't erase it  but the edges i have to erase are the  ones that have been fully inserted into  the cone so if i can  somehow do something inside the cone  i would be okay all this work  of defining a canonical configuration  was about forcing a chain to stay inside  a cone and not only an alpha cone which  is what we're going to have  as the ribosome but an alpha over two  cone this is smaller than my ribosome  cone  by a factor of two and i need that  why do i need that because this thing  i have no control over the outside chain  so the way that it approaches the cone  could be as sharp as like this  where i have it the first edge that is  or the the current edge that is inside  the cone  as far as the movie is concerned there's  only one edge there's nothing up here  i want to put something up here in a  cone naturally  but i don't get to control the first  angle  because that is controlled by this  motion maybe it really needed to go  sharp like that so they could make a  sharper angle or whatever  so you know you might say well of course  i can put it  inside an alpha cone which is the same  as  as this alpha sorry  bad picture this is alpha  this would be a problem because i have  this weird angle coming in  and now suddenly i have to bend back  like that maybe the turn angle here is  not so sharp as alpha maybe it's  one degree so i really can't  force the rest of my chain to lie in  this cone ah  but i claim i can force it  to lie in this cone  with half angle alpha over two  this is a little more subtle  oh right i wanted to mention that  helices appear in nature  they appear in proteins but they also  appear in this crazy  climber plant marty have you seen this  in costa rica  yes yeah there's some really incredible  wildlife in costa rica i've never been  there  but i've seen lots of pictures and this  is uh  spirals in practice also  um proteins tend to form these things  they call them alpha helices because  they spin like an alpha i guess  so this it's kind of neat that the  canonical configuration which is totally  geometrically motivated  also appears in biology not that kind of  biology though  so here is the picture i have  the big cone that is my ribosome and  here i'm going to write beta  for the beta for big i guess and then  we know that we can canonicalize a chain  or there at least exists a canonical  configuration of the chain  where everything lies inside a cone  of half angle alpha over two now the  problem is that  cone we want it to be at a funny angle  let me draw a real picture  here's a ribosome has a big angle  here called alpha  now i'm going to do the not extreme case  try to be a little more general  there's some edge that is currently  entering and we have no control over  that edge or the rest of the chain  that's plate that's determined by the  movie which we're trying to play  backwards  what we what we have to control and what  we're free to control  is the rest of the chain because as far  as the movie is concerned that hasn't  been  created yet or it hasn't been just it's  already been destroyed depending on  where you're playing forwards or  backwards  uh we need to say what happens to it and  i what i want to happen  is so that by the time this edge is  inside  that edge plus the rest is in the  canonical configuration  if i can achieve that then as edges come  in they become canonicalized and then  everything will be canonical and  inside the cone we're done that's our  goal  canonicalization so uh  what's the deal well in reality  there's some cone uh  yeah it can penetrate like that could  penetrate the outside cone  and this is getting messy  so if i extend this line  there's a cone of half angle here which  is equal to whatever the turn angle is  at that vertex which is again specified  we're not free to set it to whatever we  want we know the next edge  must lie on this cone  what i do uh  for the now i have on the other hand off  to the side i have in mind a vertical  cone  whose half angle is alpha over two so  it's quite small  and i know um how do we do it initially  the first edge was along the maximum x  direction and then it you know it  spirals up from there  okay here's what i'm going to do  there are sort of two situations i what  i'd like to do  is put a cone here that is  vertical something like that  just like i have here the trouble is the  right side of this cone  does not intersect the boundary of this  cone and i need it to in order to form  the right angle here  it's going to it might go through the  middle of the cone like it does here  so then what i do is i take this  canonical configuration and i rotate it  so that this is hard to draw  it'll be something like this  i think there's no hope of seeing this  so it's on the surface of this cone it's  not on the far right edge it's going to  be some  some intermediate point and that's  that's exactly where it intersects this  cone  it's just like the previous picture just  harder to see i'm taking the  intersection of these two cones  if i just rotate the picture then it  will lie on the intersection  if they intersect but they might not  intersect  so let me go here  try it again  so the easy case is when i can draw a  vertical cone  and it intersects the the cone that i  need to intersect  the harder case is  maybe it's more extreme  maybe it's a very tight angle here  so there's a very small turn angle i  have to intersect this cone  because my the cone that i'm working  with is actually smaller  it might it's half half of this  angle it might not intersect this cone  in that case  i'm going to rotate the cone to fall  over a little bit  so instead of being like that it's going  to be like this  okay so i want the intersection of these  two cones which i've conveniently made  this this edge here so the first edge  will be  lie along here and then it's going to  spiral it from there so i still have the  canonical configuration i've just tilted  it  tilting is going to be necessary because  i have this angle to match up  the convenient thing about the canonical  this the alpha ccc canonical  configuration  is that i have this half angle of alpha  over two  so i can afford to tilt it by up to  alpha over two  it will still stay within the ribosome  which is of  half angle alpha and all i need to show  here and  sort of once you think about it for a  while it's obvious that  you ideally i don't tilt it at all got  tons of rooms  huge amount of room here but sometimes  i'll have to tilt it  but by at most alpha over two so i will  stay inside  the ribosome because this apex was  inside the cone  and this smaller cone even if i tilt it  all the way  to meet this edge it will stay inside  the cone in fact  it'll stay inside the big cone in fact  this cone that i tilt  the one that contains the rest of the  canonical configuration will always  contain  the up direction  that's how to see it if it always  contains the up direction  then at most it's that big  and that will uh because that is  ha that's an angle of alpha because that  was the half angle of the big cone  and so that would be a half angle of  alpha over two as long as you contain  the up direction at all times  you will not fall outside the big cone  so the rest is just memento so  you play this movie backwards as things  come in here  uh you this cone is going to wiggle back  and forth depending on how this angle  changes  once the edge gets all the way in you  absorb it into the canonical  configuration it's a little bit of work  there but you just sort of twist the  cone around  until that algorithm that we described  for producing canonical configuration  would actually produce what's inside the  cone  then the next edge comes in the cone  wiggles around until the edge gets all  the way in then you  canonicalize what's inside the cone and  repeat  so if you had a way to get it out you  can put it back in  and keep track of all the stuff that  happens on the inside  that's what this theorem says and then  there's just  slightly more to say  if you have a flat state  we want to prove these things are flat  state connected so  take some flat state i ideally should  have only obtuse angles  i claim this flat state can be produced  using a cone  of the appropriate angle this is the  sharpest turn angle here is alpha then  you need an alpha cone  and the way to think about that is to  think of the cone moving  instead of as the chain moving  it's a lot easier so you want the chain  to lie around here  so you start by moving the cone  i guess like this so that it just barely  touches the plane  and this edge spews out  is that the right way to think about it  no no no it should be like this  so you you take the cone this this is  like putting  frosting on a cake okay so you move your  cone like here  you squirt out some this this edge  okay then uh you do it so that when  you're here and the new edge is created  uh it's still in the plane and then you  just sort of move around there i'm just  going to leave it as a sketch like that  once you know that you can produce any  flat state of course you can  reorient yourself by relativity so that  the chain is moving instead of the cone  so you can produce this thing with a  ribosome  once you know all flat states can be  produced and you know all  producible configurations can be  canonicalized then you know it's flat  stay connected and you know all  canonical things are flattenable and  vice versa  by continuous motions without  self-intersection and all of this is  algorithmic  can tell you how to go from one place to  another  and so this is a candidate algorithm i  would say for how nature folds proteins  just thinking about the mechanics not  worrying about how it's implemented  maybe you take this model and then you  try to make it  physical physical forces  and you get a way to fold proteins that  of course remains a mystery but next  time we will talk about  uh some very simple models that are  motivated  more closely by biology of how proteins  might actually fold  and talk about the complexities that you  get there

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all right today we're going to talk  about protein folding  and its relation to linkage folding  we're going to look at sort of a  mechanical  model of proteins this is an example of  a protein from lecture one  there's a ton out there in this place  called the protein data bank  all freely available it's really hard to  get pictures  like this but you get some idea that  there's a linkage  embedded in here you see various little  uh spheres and edges that's of course  not reality those spheres are actually  atoms and they're the kind of amorphous  blobs  the edges are chemical bonds and those  are  connections we don't know whether  they're it's not really matter but it's  a  force this is a rather messy picture  this is what a protein folds into some  3d shape most proteins fold consistently  into one shape we don't really know how  that happens we can't  watch it happen so  we the big challenge is to know how  proteins fold  given a protein what does it fold into  that's the protein folding problem  major unsolved problem in biology  biochemistry  the protein design problem is i want to  make a particular 3d shape so that it  docks into something binds to a virus  whatever  what protein should i synthesize in  order for it to fold into that shape  that is potentially an easier question  algorithmically it's the  really useful one from a drug design  standpoint you come up some some new  virus comes along  you design a drug to attack it and only  it you build it  usually you would manufacture some  synthetic dna you feed it into a cell  dna goes to the rna goes to the mrna  goes to the protein  you all remember biology 101 hopefully  we don't need to know much about it if  you look  at what's called the backbone of the  protein protein is basically a chain  and attached to the chain are various  amino acids today i'm going to ignore  the amino acids which is a little crazy  and just think about the backbone chain  backbone chain looks something like this  um one of the challenges of video  recording a class is i can only use  copyright free or creative commons  images this one i couldn't get one so i  had to draw it  there's various measurements here  certain numbers of angstroms those are  the chemical bonds various uh  atoms here nitrogen carbon so on  hydrogen but basically it's a chain  zigzags back and forth  you can also see the angles here they're  not quite all the same but they're very  similar  all the lengths and the angles are close  so uh it zigzags this is really in three  dimensions  and try to draw the sphere so you can  see the three dimensionality but it's a  little tricky  and then attached on the sides are the  amino acids i'm going to focus just on  the backbone  the way this thing is allowed to fold  these lengths as far as we know are are  pretty static they probably  jiggle a little bit but you can think of  them as edges so you can think of this  as a linkage  the catch is also the angles are fixed  because the way this atom wants to bind  to other things has  very fixed angle patterns if you ever  played with a chemistry construction set  that's that's how they work to have  holes at just particular angles  so if you think of like a robotic arm  normally  uh like here i have a two edge robotic  arm let's say  normally you have two degrees of freedom  in three dimensions you can change the  angle  and you can spin around this edge  now i'm saying the angle is fixed for  example here it's say at 90 degrees all  i can do is spin  i'm not allowed to flex my muscle in  this way  okay so that is the model all of in this  case we have a tree  all of the angles here are fixed but you  can still for example take this entire  sub chain  and spin it around this edge that'll  preserve all the angles and all the  lengths  that's all you're allowed to do you take  an edge you spin it spin once half of  the edge relative to the other half  these are called fixed angle linkages  and they've been studied quite a lot  because of their connection to protein  folding  so embedded in the term linkage we  assume that the edge lengths are fixed  and then we add  the constraint that the angles are fixed  and the motivation is the backbone is  something like put  the background of a protein is something  like a fixed angle tree  of course it's not much of a tree most  of it is a chain there's just  small objects hanging off and if you add  the amino acids there are bigger things  hanging off but still constant size  they'll have some cycles they're not  trees but  it's slightly more approximately maybe i  should draw  wavier lines uh it's a chain  usually an open chain although  occasionally closed chain  so we think a lot about fixed angle  chains and sometimes about fixed angle  trees  now fixed angle linkages are harder to  think about than  universal joints that's the usual kind  of linkage um  so and we know 3d linkages are kind of  tough nonetheless we found lots of  really interesting mathematical problems  to solve here  and that is the topic of today at some  level  we are thinking about the mechanics of  protein folding  we're throwing away energy we're  throwing away the  the actuators in real life that make  proteins fold  we're just imagining given this  mechanical model of how a protein might  fold  what's possible so it's some sense it's  broader than reality  and the hope is you find an interesting  algorithm for  how to fold these protein chains maybe  that's the algorithm that nature is  implementing  that's the kind of general picture we're  not constrained by  reality and by how nature actually folds  things  so i'm going to talk today about four  main  problems here  first one's called span second one's  called flattening  third one is flat state connectivity  and the fourth one is locked  our good friend locked chains  and of course there are lock chains  because we're constraining linkages even  more than before so if you take knitting  needles it'll still be locked  because you add extra constraints makes  it harder to fold  but there are actually some interesting  positive results we can give of chains  that are not locked in some sense  and flat state connectivity is about the  same kind of thing instead of worrying  about getting from anywhere to anywhere  we just worry about getting from one  flat state to another flat state flat  means  lying in a plane flattening is about is  there such a configuration  and span is about given  robotic arm like a more complicated one  like with multiple  edges how far apart can the endpoints  get and how close can the endpoints get  the universal chain is not very exciting  the farthest thing is when it's straight  and the least far they can get is when  it's closed and you can always do that  i think well no i guess you can't always  close it up  that's a little non-trivial but for  fixed angle linkages you can't  straighten out because you have to  preserve the angles  so it's kind of what is the straightest  like configuration given that the angles  are fixed  so let's start with span  so the span of a configuration is the  distance  between the endpoints  and  in general you'll find the max span and  the min span  this search was begun by a guy named  mike sauce  who was a phd student at mcgill  and he proved that if you want to find  for example  flat state that lives in two dimensions  with the minimum  or the maximum span  this is np hard  this is in his phd thesis  question  oh here i'm assuming open chain i should  say that  which most proteins are yeah i've been  talking about trees and stuff i  hear i mean chain otherwise there aren't  two end points to think about  good uh so here are his np hardness  proofs  in fact the problems are mp complete  they're pretty  simple the problem here we're reducing  from is  partition i give you a bunch of integers  i want to divide them into two halves of  equal sum  and the top example is minimum flat span  problem so you have you make an  orthogonal chain  where the horizontal edges are long and  they're proportional to the integers  you're given  and the vertical edges are really tiny  and so what you'd like to do  what all you can do is sort of flip in  the because you have to stay in the  plane  you can flip one of the vertical edges  say and make any of these edges go left  or right  you get that freedom but so  each integer you get to choose do i go  right by that amount and go or do i go  left by that amount and if the amount  you go left is equal to the amount you  go right in other words there's a  partition into two equal sums  then those endpoints will be aligned and  then their distance will be very tiny  otherwise it will be quite large because  the horizontal distances are all big  so that's kind of an e very easy np  hardness proof  to maximize your flat span  instead of mapping your integers onto  lengths you map them onto angles or turn  angles i won't  specify that too precisely but again if  you if you make your total  counterclockwise turn equal to your  total clockwise turn  then the two end edges which are super  super long  will be parallel and to maximize the  distance between the endpoints you want  them to be parallel  if you make them go some other angle  they're closer  now both of these proofs rely on the  requirement that you want a flat  configuration with minimum and maximum  span  now there's a claim that flat  configurations matter for proteins  so it's a natural constraint but what  about the general problem what about  i have something in three dimensions i  want to maximize i have a fixed angle  chain in 3d  maximize or minimize the span  both of those problems are open can you  solve them in polynomial time  for 3d  max span  so the non-flat version just for  maximization  there's been a lot of work and there are  two papers on the subject one of them is  by nadia  and joe o'rourke another one is by  borsea and stranu  and i just want to quickly summarize  that because  there's a lot of stuff there but  essentially they find  what the structure of those spans look  like i have  an early figure that's in our book  before all this work was done  of a simple chain this black guy one  two three four bars open chain and in  that  black three-dimensional state it  maximizes the span the  the green span there and if you look  from above which is this picture  of course the endpoints look much closer  in in projection  and the red configuration is the max  span if you restrict to flat  configurations  so here of course 3d buys you something  in general  it always will an interesting thing is  that this max span  the green line passes through another  vertex  that seems kind of weird and in fact  there's a general  theorem there sort of characterizing the  structure  of these chains it's still not known  whether we can solve this problem in  polynomial time but  for orthogonal chains where all the  angles  are 90 degrees  uh we can solve that in linear time i  guess  and here's what it looks like  suppose you have some orthogonal chain  orthogonal chains are nice because you  can draw them in a plane  as a staircase so there's a nice  canonical configuration  what one way to think about  how to find the  maxspan configuration i'm just going to  give a high level overview here this  won't be a complete algorithm  is you triangulate that staircase in  this sort of obvious way of connecting  every endpoint to the one to a head  this you can think about  this is like a body that's hinging  around here because i can  spin if i spin the left part of this  chain around this edge it's like hinging  that triangle around that hinge  same thing you can think of these  triangles as just being hinged together  like  in rigid origami it's the same  class of motions and now  you can uh what i'm going to do is  compute a  shortest path in this surface from here  to here confusingly this is called a  geodesic shortest path although it's not  really  related to geodesics from polyhedral  surfaces  but if i compute a shortest path it's  going to go like to this vertex and then  probably to that vertex but i'm  constrained to say  stay inside the unit of those triangles  i want to go from one endpoint to  another  then i claim that okay the  these two edges will stay planar of  course they form a triangle  uh i claim these four edges will stay  planar  and in the orthogonal case they'll stay  zigzag and then also these two guys will  stay in their own plane  and then uh i claim that actually this  wiggly line which is not straight  because it bends here and it bends here  the total length of that wiggly line is  the max span  and you achieve that by folding this  planar part with respect to this planar  part with respect to this planar part  so that the wiggly lines become aligned  and straight  and that's very hard to draw but it can  be done and that's what you do in the  orthogonal case and that gives you  the answer in linear time with enough  work  for non-orthogonal though it's open  whether you can do this in polynomial  time  maybe it's np hard actually i don't know  all right  that's all i want to say about span next  we go  to flattening  i guess the first question about  flattening and and the main one we'll  we'll talk about here until we get to  flat state connectivity is  does a fixed angle chain have a flat  state at all  can you even draw it in the plane  without crossings  so we're restricted here to have no no  self intersections  we want a flat state  no self intersection  then there would be a question of given  some configuration can actually  continuously get to a flat state but  sort of the simplest question is ignore  about getting  ignore getting there just is there a  flat state  and this problem is np-hard  again mike sauce and his advisor godfrey  toussaint  it's a little more complicated but it's  basically the same idea as that very  simple proof  which was just to map integers to a  little zigzag  staircase here so the goal is to force  x to end up being the two endpoints of  the green curve to be aligned with each  other that will exist if and only if  there is a partition of the given  integers  and there's all this infrastructure this  sort of  uh there's a little lock here and a key  and and some structure on the left  basically forces the picture  to look like that um so the first claim  is that the black stuff  is basically unique i think there's one  global reflection you can do  that doesn't affect anything but you try  any of the other flips again we're  restricted to flat states here so  there's only sort of a  bounded number of things you can do  finite number of things you can do  you try all of them they self-intersect  so the black thing is basically forced  and it forces the end point this  endpoint x from the black side  to be aligned with this very narrow  spike  and because the angles are preserved  that red guy is going to be vertical it  can't go down so it must go up  and so only if this thing is aligned in  the center aligned with x  in other words this problem has a  partition  will this have a flat state so it's not  the most exciting example this is only a  weak np hardness proof  lots of interesting questions still open  here like if all the lengths are  the same if they're all equal  then we don't know or if all the lengths  are even polynomially bounded this needs  really really long lengths versus really  really tiny lengths exponentially  exponential and ratio all these problems  are open  and that's flattening so we're going  very quick uh because there  isn't uh well partly because i'm more  excited about this but there's more work  in these two  parts so i'm gonna focus on that next  topic  is flat state connectivity  so the idea is to think about the  configuration space of these fixed angle  chains let's say and we kind of know  that it's going to be disconnected  because there are knitting needles there  are nasty things  so there's maybe various connected  components  but let's say that we really care about  flat states  and the question is are they connected  to each other  so in other words do all the flat states  mark them with x's  do they all appear there's only finitely  many so configurations there's  it's this continuum and there are these  messy blobs semi-algebraic sets  but flat states those are discrete  things because we have fixed  angles you can flip or not flip every  edge so most exponentially many of them  so finite are they all in one component  so i can get for if i pick two of my  favorite  flat states there's a path between them  or are some of them  in multiple components so in this case  we call it flat state disconnected  and if they're all like this we call it  flat state connected  and we just like to know which chains  which  fixed angle trees whatever are flat  state connected  versus flat state disconnected i would  say  the big the big open problem here is  our is every flat as every fixed angle  chain open chain flat state connected  that is still open we have lots of  results in  in that direction so the top four  results are about open chains but they  have an extra constraint  example open chains that have a monotone  configuration like the staircase  those are flat state connected and so  in fact whenever the angles between the  edges are  either orthogonal or obtuse then  their flats stay connected when the  angles are acute  we're not really sure if all the angles  are equal and acute  then we can do it but if they're  different and acute we don't know  unless the edges are all unit length  and the angles are in this funny range  then we can do it so there's all these  special cases we can solve  the most relevant to proteins is  actually obtuse  chains so we've solved sort of the main  problem  with this second result but there's a  natural  theoretical question here is are all  open chains flat state connected  or do we get disconnectivity i will show  you that i'll show you the orthogonal  case in a little bit  we can do some stuff we have multiple  chains that are attached to some  blob like a cell closed chains is a  little bit  for disconnected uh we don't have  very interesting examples i would say  this is funny because locked examples  are easy to come by  but flat state disconnected examples are  a little trickier because flat is so  constrained  so let me just show you these examples  this is what we call a partially rigid  fixed angle tree so not only are the  angles fixed  but also the black edges  are not in fact only the blue edges here  are allowed to spin  everything else is held rigid so these  arms are somehow forced to be in exactly  that geometry  i can spin it around this edge just  spin it up into 3d for example these are  two different flat states of the same  linkage  right all i all the only difference  between these two i haven't rotated or  anything is i've taken each of these  arms  and flipped it around a blue axis  if i do all four of them i would get  this picture but the claim is you cannot  do that without self-intersection  the intuition is when there aren't very  oh one other thing  this makes this slightly more  interesting it's weird to say well why  did you force some of the edges to be  rigid and not others  one way to force that is to use a  general graph  if you add some extra edges to sort of  brace this and all these angles are  fixed  then this linkage will behave exactly  like that one  so that at least is somewhat more  natural although what we really care  about  are chains maybe trees  but we don't know whether there's a we  also don't know whether all  true fixed angle trees are flat state  connected  these are the the worst examples we know  let me give you an idea of why that  it doesn't work this is a little  animation of  just a couple of moves attempted it's  just going to cycle  through that and these are some  static images of same kind of thing  so the intuition is the following you  have four arms you have two  sides to the plane there's up and down  for four  arms and two sides at least two of them  are going to have to go to the same side  the best you can do is two and two or  three and one but in either case you  have two sides  go to the two arms to go on the same  side  now it could be like in this image that  they're  opposite arms so there's this arm here  and there's this arm here so they're  connected by 180 degree angle  and those guys when they fold up  actually these edges will just hit each  other  dead on so that's uh  kind of obvious from a geometric  standpoint maybe you call it cheating  for them to hit dead on you can twiddle  the edge length so that they will  properly intersect without dead-on  collision  without being degenerate basically the  alternative is the  and this is a little harder to see  geometrically and that's why we drew  that  animation is that you have  you have one arm and you have an  adjacent arm connected by a 90 degree  angle  now here there's clearly some collision  going on and if you happen to fold it up  90 degrees like that and then fold the  other guy obviously you get stuck  but maybe you fold it a little bit and  the other guy goes a little bit more and  there could be some dance between those  two degrees of freedom those two arms  that somehow gets them both to pass over  to the other side it's obviously not  possible  how do you prove it well you can prove  it with topology  not not theory or link theory  so it's very cute proof uh you start  with  so here's the the full example but i've  highlighted the two arms in red  that are going to move and i imagine  connecting the endpoints  of each arm with these little blue ropes  underneath the plane so all right they  they're both going on the same side  let's say that somehow pass through each  other on the top side  then i'm free to connect stuff on the  bottom and i shouldn't collide with that  so if somehow both of these guys flip  over  so the arm on the left a3  flips over a3 stays where it is but now  the arm is on the top the north side  instead of the south side  and the other guy from b  to b3 used to go like this and now it  goes like this  if that happened somehow then these  ropes could remain intact during that  whole motion and on the top  you have two closed loops that are not  interlocked  on the bottom you have two closed loops  that are interlocked so there's no way  to get from there to there without  colliding somewhere  the blue stuff didn't move so the red  stuff must have collided  so even just topologically you are  screwed  that is our only negative example lots  of interesting open questions here  on the positive side let me show you  for orthogonal chains and the same  algorithm works for obtuse chains  all the angles are obtuse uh how they  are flat state connected  so in order to show its flat stay  connected i take i want to think about  two flat states  and show that i can fold from one to the  other  via some intermediate 3d stuff  let's start with one of the flat states  so it's orthogonal so in two dimensions  they really  all the edges will be horizontal or  vertical in 3d they can kind of be in  many  many different angles many different  dihedral angles in 2d it's pretty simple  and all i need to do is sort of pick up  that chain and i'm going to try to pick  it up into a staircase  because there's only one staircase if i  can make it a staircase  i make configuration flat configuration  a staircase flat configuration b  staircase i just fed x in the middle  right  once they're both staircases i play one  motion and the other one backwards  get from anywhere to anywhere so here's  all you do you take the first edge  and you just rotate it up to the red  line a  uh and then you take the next edge  and you take both of those edges and you  just rotate them  like this so you get that little  two-step  staircase now i'd really like to pick up  this edge but  i want to first get these two edges in a  plane with that edge  so i rotate this flag over to the left i  get  those two guys and now they're in a  plane with this and i just lift that up  then i'm going to flip then rotate up  flip rotate flip rotate  here's some more examples so if at this  point  i have this staircase um  sorry i guess originally i have from v3  to d  up there i it's not in plane with this  guy  so i just rotate it like that i'm  spinning around this edge  so now i have from v3 to e and then i  rotate it up along that green arc  and i get a bigger staircase above  the chain and because everything's  staying above it will never penetrate  the plane it'll never hit anybody else  and i'm building a staircase by design i  always rotate this  there's actually two ways i could be in  plane but i always rotate it so that  when i pick some an edge up it'll be in  a staircase  so this is actually really easy and  slight generalization is to  obtuse chains then instead of making a  staircase we make a  monotone get this right  yeah some z monotone state so it goes  monotone and z  out of the plane and that's enough to  avoid collision  and you get a canonical configuration  also if you have acute angles  but all the angles are equal then  there's a natural  canonical state which is just like a  compressed staircase and that will work  here too although that takes more effort  that was in a separate paper  but big open question is chains with  arbitrary angles we have no idea  it seems very hard to do an operation  like this  wow we are burning through this is fun  so the next topic is about  locked chains  now as i said you can take a knitting  needles example  which has five edges  and that will still be locked if you  force the angles to be fixed because it  was locked without the angles being  fixed  now it required a length ratio of uh  three to one i think yeah this edge had  to be longer than the sum of those three  uh so let me put down  uh some open problems so you may recall  in the case of universal chains  universal joints i should say  the big open question was can you lock  a universal joint 3d chain with  unit edge lengths so equilateral  every edge is the same length  is there a lock chain like the knitting  needles where all the edge lengths are  the same and one of the motivations for  that is in proteins  the edge lengths are all within like 50  of each other  so it's pretty natural of course  we don't have universal  joints with proteins we have fixed angle  joints  so the big open problem for fixed angle  joints  i guess we'll do this in parts  is there  a locked  3d fixed angle chain  that's equilateral okay i'm going to add  some conditions here  so that's first natural question  knitting needles doesn't suffice we need  a three to one length ratio as far as we  know  uh yeah  turns out that question is not very  interesting i need to do slightly  non-linear  editing here so you take a knitting  needles example  and you just subdivide the edges into  lots of little tiny bars it doesn't have  to be this extreme  you could not subdivide these edges at  all and make these guys subdivide them  into like three or four parts  because the angles are fixed these guys  act as a single bar there's really no  difference  maybe you make slight curve there and  then they can wiggle they can bend a  little bit  but really not much so if you just  say oh i want it to be unit length i  don't constrain what the angles are  but i fix them then it's trivial to come  up with locked examples  so that's not very interesting  what if i make it not only equilateral  the lengths are the same  if i make it equally angular  because again in proteins all the angles  are  similar they're around 110 108  something like that they're all pretty  close within if i think within  10 20 of each other well here there's  also a locked example  and just to show you how research was  done  back at the turn of the century this is  pre  web 2.0 pre-ajax and all that fancy  stuff  we used ascii art this was email was the  tool of choice  i know it's hard to imagine a time 2002  so long ago and i i tracked this down  this is the original  claim where it looks we call this the  crossed legs example because it's  like two legs crossed around each other  and uh  this is the the first time we thought oh  maybe it can be done  unit length this is stefan langerman and  uh and here for the first time ever  this is not the first model but this is  the first photograph of any model i'm  aware of  this is the cross legs example um this  is made with a  construction toy that used to be sold  around here but is no longer in  production it's hard to  so they're pretty hard to get is straws  nicely colored straws  and the cool part are these uh these  connectors  so the connectors force particular  angles in this case every angle is 45  degrees  so this is equiangular and equilateral  because all the straws  i'm told are the same length that's how  they're sold  and you can do edge spins so  whoops that's called cheating  it's not totally obvious that this is  locked  the problem with the model is that the  edges can bend  but if you treat it properly and  only spin around edges  then you're you're stuck now  there is one thing you can do let me see  if i  uh  yeah like this so here i'm almost in a  plane i've got the purple edge right  against the  pink one is it easier to see from that  angle i don't know  so here this guy can come out  and this guy can barely go  along the edge so actually this doesn't  quite work for equilateral it works for  one plus epsilon that's why i added  these little  nubs at the ends so if it if  they're all exactly equal length and you  allow just a  abrasion of the endpoint then this could  go around  like that and then you'd be unlocked if  you just add slightly  either you change the angles to be not  quite equal so ben  make this a little smaller or you make  the lengths a little bit longer at the  ends  then the claim is it's locked we don't  actually have a formal proof of this  we're just remembering hey we should  probably write this up i was talking to  stefan last night  and uh yeah so someday we will prove  that this is locked  but certainly looks like it so this  isn't open yet  i mean modulo the details of that proof  equilateral and equiangular  seems easy to lock with fixed angle  chains  in fact even easier that this example  only has  four edges so even less than the  knitting needles fixed angles  make for complicated motions i guess  make it hard to unlock things so i need  to add one more constraint  and the constraint is obtuse  so again all of these properties are  enjoyed by proteins protein  protein backbones have all these  properties even if you looked at fixed  angle trees  is there something like this that's  locked and now we don't know  and this seems quite tricky i guess the  intuition is that obtuse or and usually  we think about orthogonal just because  it's easier to draw the pictures but  reality is more like 108 degrees  uh conjectures of two chains obtuse  fixed angle chains behave  kind of like universal joints and with  universal joints we don't know whether  equal  equilateral is enough so  it's tricky  what if you instead had like  made your ribbon lengths basically like  a bunch of little unit obtuse angle  connectors  and then when you hit the the big turns  it's just  you know up to yeah you can definitely  you can make this example be entirely  obtuse you can make every angle obtuse  here you could arc a little bit  here you could arc some more but not too  sharp and because here  we actually know that this part can be  made a string we don't really care what  it looks like  so you can make it fairly obtuse it's  just that these guys should not bend  much they have to  be long no matter how you fold them so  if you want  equilateral and obtuse that's also easy  but to make all the angles actually be  equal as far as we know you cannot take  that knitting needle subdivided  make all the lengths equal and all the  angles equal and make them obtuse  that's open but any two out of the three  it's easy  uh of course in reality they're not  quite equilateral they're not quite  equal angular  but it's still open for the if you have  like a small range for the lengths and a  small range for the angles  this is open we pose it this way because  it's the cleanest geometrically  but the the real question you care about  is  when these are fuzzy constraints obtuse  is real but these guys are  fuzzier so  uh if you think about proteins uh which  fold very well  in nature there are a couple of reasons  they might fold well  we know as far as fixed angle chains go  it's actually  quite easy to find locked examples and  this is somewhat intuitive but bear with  me  because there are locked examples in  this configuration space we believe  these configuration spaces are really  ugly nasty  so it would be very hard even if you  know oh i only need to fold something in  my component  if these guys are highly disconnected  and flat states are all over the place  it's probably even within this connected  component it looks really ugly  and so it's very hard to find a path  from one state to another  probably p-space complete although we  don't know that for sure  but that's the intuition locked equals  messy  when there are no locked configurations  like carpenter's roles  we get really nice algorithms it's super  easy to get from state a to state b  now if you're nature or you're designing  nature let's say  or you're building your own virtual  world second life  and you want to design proteins you  would like to design them in such a way  that they fold easily because it happens  all the time  every thing that is being acted on by  our body every living thing that we know  has tons of little proteins that are  doing all the work they are  fold into their shape and they do  something that's proteins plus  rna but mostly proteins so  uh to understand life we should  understand proteins now how do proteins  fold so well when we know there are all  these locked configurations  one possible answer is that proteins  have extra structure namely these three  things  which somehow make it very easy to  algorithmically go from a to b  notice i'm not even i'm not assuming  anything about how  proteins fold in terms of  what is the mechanism that drives them  because  we don't really understand those  mechanisms there's hydrophobia which we  don't really know how it works  it's all these little forces um that we  don't fully understand  lots of we understand lots of parts of  the story but not the whole story  and what's convenient about these kinds  of problems is you don't need to assume  anything about how it actually happens  all we're assuming is the mechanical  behavior of proteins and how they could  possibly fold  and the the idea is if there's a lot  configurations that's probably the wrong  model because then everything's messy  now there's also evolution coming into  play and maybe some proteins are easy to  fold some  proteins are hard to fold that's an  interesting question which should be  experimented with  but let's hope that there's a model  things are mutating randomly you really  like everything to fold nicely  maybe it's because you have all three of  these properties approximately  in real proteins  so the general idea is that nature has  some extra constraints that make  protein folding easy we just have to  figure out  what they are and why it makes them easy  unfortunately this is still an open  problem if this had an algorithm  that would be a natural candidate for  what nature is doing using its  mechanical  or using its for energies and forces and  so on  this would be a rather unsatisfactory  ending if this was  if the climax was an open problem  we have a theorem too  and this is what i'll cover in most  detail  and it's paper called producible protein  chains  protein chains just means fixed angle  chains open chains  and the idea is well yeah there are  these constraints  uh or there are these extra features we  don't know how to exploit them so  let's not even worry about them suppose  they don't even exist  maybe i'm going to assume obtuse but  none of the others  there's another constraint in how  proteins fold or really how proteins are  created  they're created by a machine a molecular  machine made up of a whole bunch of  proteins and rna  called the ribosome you may have heard  of it translates  messenger rna into proteins so there's  some mrna  around here maybe don't know exactly how  this machine works  but there are actually very accurate  three-dimensional reconstructions of the  ribosome  with no copyright free images you're  going to have to there's a link on this  slide that goes to  the cool amp 3d models of the ribosome  with a slice away so you can see this  there's a tunnel down here  and the prot the protein gets sort of  created here the backbone gets created  here  and starts going through this tunnel  there's a bend in the tunnel around here  where it's conjectured an amino acid  gets attached  and then that it goes out the tunnel and  the protein starts spewing out here and  presumably folding at the same time we  don't really know  so this is how proteins are created  um the birds and bees i guess of  proteins  so what's interesting about this is it's  not like a protein  exists and then folds which is how a lot  of  people might think about it at first  glance  that's natural way to model protein  folding you start with a protein say in  just  zigzag configuration if it's obtuse  there's a nice zigzag monotone  configuration then you see what is  the best configuration i could fold into  for some notion of best  and that's sort of what this  configuration space picture is about  it's  if i already have a protein what  configurations can i reach by motions  and that is interesting that's important  because you're still going to have to  reach by emotion but it's actually more  flexible than that  because protein could just be partially  built  the the rest of the protein hasn't been  built and it could start folding already  and it might be easier to fold when you  don't have the obstacles  of your existing protein so that's both  a worry  but it's also a convenient structure  because this ribosome is a giant  obstacle  it's bigger than most proteins there if  your protein is really long maybe it  could go over here  but most of the time it's going to stay  on one side of this plane  because locally this thing is basically  flat if you look at the real  3d pictures not the schematic  now this is good news for a geometer  because there's this giant obstacle  think of it as a half space which the  protein cannot penetrate  while it's being produced over here  that's it that half half space  constraint  is enough to get really good algorithms  for  folding your chain it's weird because  we've made our problem both harder  because the protein is only partially  produced at any time and it can fold  which is part of it  but we've also made our life easier  because there's this big obstacle  yeah right we're out of this we're going  to get that the angles in the protein  are constrained  and in particular for this angle  uh it depends i mean you could  in this picture because it's  perpendicular here yeah uh  you can the the sharpest angle you can  make is 90 degrees  more or less that's a good point  so it's a convenient match between the  chemistry which also forces the angles  to be obtuse i guess  i don't know a ton of chemistry but also  the ribosome just geometrically forces  we're going to use a property like that  our model is going to be a little bit  more  both more general and simpler  we're going to imagine that the ribosome  is a cone  it's part of the upper cone here this is  like a  mirror image and  in reality that cone is actually a plane  and everything above the plane  but to be more general we're going to  allow some angle alpha here it's also  just easier to think about when alpha is  smaller than  90 but it's work everything i say will  work when alpha equals 90 and that is  sort of the reality  case so so the model is  so the ribosome is a cone who we call  this the half angle of the cone  from the vertical axis to the edge of  the cone is alpha  so if you're going from one axis to the  other would be two alpha  the the model is you start with one  link of your chain which is inside the  cone it spews out  uh through the apex okay that's that's  the exit of the tunnel  here we're allowing the tunnel to be  actually quite free it doesn't have to  be perpendicular  to the to the apex or the plane of the  apex  so the edge comes out and as soon as the  vert  the end point of the chain reaches here  then a new link is created  this is like a very simple model for how  a chain can come out of a cone without  worrying about what's happening inside  the cone  imagine everything's totally free this  is like you can allow self-intersection  the cone who knows what  but once you come outside the cone  you're not allowed to self-intersect  and you're not allowed to intersect the  cone once you come out you can't go back  in  so that is a model of producing protein  chains  and if you have a cone of angle alpha we  call this an alpha  producible chain  for whatever reason we often call it a  beta producible chain  just change the variables  so if you think of the ribosome as a  cone with half angle beta  you can produce it like this that is  beta producible now this is a pretty  powerful model  because you only have to worry about it  link by link you don't have to worry  about the rest of the chain until it  gets spews outside of the cone  but it's restrictive in that you cannot  penetrate the [ __ ]  all right one thing we can talk about  is angles so  i'm gonna write call a chain a less than  or equal to alpha chain  if all the turn angles are less than or  equal to alpha  i don't know if i've used turn angles in  this class probably  if i have two edges the angle would be  this  the turn angle would be this supplement  yeah i guess we use trit angles way back  in origami land kawasaki's theorem and  so on  it's just if you're going straight how  much do you have to turn to  get to the next edge so we'd like uh  fairly obtuse things so alpha's going to  be small there isn't a ton of turn  but in general less than or equal to  alpha chain for some alpha  now there's a relation as jason was  mentioning there's a relation between  alpha and beta  in the ribosome because you always  exited orthogonally to the  the plane that was your cone the  sharpest angle you could get was 90  degree turn angle  here we're a little freer because this  edge  can wiggle around as long as it touches  the apex  so if you're up against the cone  you have to slide out into the  complementary cone that was the  previous picture and as soon as you get  there  you could create a new edge which is  like this so the sharpest angle you can  get  is actually twice beta in general  we're going to have alpha over 2  is less than or equal to beta that is  you can get up to  beta equals 2 alpha get that right  and also in the obtuse case this is not  too exciting  but it's true uh  there's actually some open problems here  when you have that full flexibility and  you set  um sorry alpha to two beta not the other  way around  uh i'm going to assume here  that alpha equals beta  this will be convenient and it's the  interesting case because in reality the  cone  has a half angle of 90 degrees so beta  is 90.  and the sharpest angle we're going to  make is always obtuse so  saying that you have a less than or  equal to 90 chain is just fine  but in on the mathematical side i think  we solve the case when alpha is less  than or equal to beta um  but not when alpha over 2 is less than  or equal to beta that's a weaker  constraint  so there is a range where it's not so  easy  all right now what do i claim about  these chains  other than their angles are not so sharp  i claim they're good algorithms for  folding them  what could i possibly mean there are  still locked configurations  is that true well  yeah i mean presumably this is acute but  you take the obtuse versions of this guy  because i didn't constrain the edge  links or anything i just said that  they're that the angles are obtuse  so i could just sort of round these  corners make it obtuse you know add lots  of dots just at the corners  that will be obtuse and a chain  like this will be producible a chain  with these angles and these edge links  can be produced from a cone but  this configuration of this chain cannot  be produced  i claim i claim anything that can be  produced  is in one connected component so while i  can make a linkage that is locked in  that there are  bad configurations you can't get out of  the things you can actually make  you can always get out of so there's  going to be  the space of producible configurations  uh maybe there's some stuff that's  unproducible but still connected to it i  don't  i don't know doesn't matter too much i  won't worry about this stuff  there's other bad locked configurations  that you cannot  cannot reach here but everything that's  producible is in one connected component  of the configuration space  that's property one that's kind of nice  also all the flat states are going to be  in here  this is actually pretty easy i just need  to prove that flat states are producible  which we'll worry about later so in  particular these guys are flat state  connected  all the producible protein chains are  flat state connected  that's interesting because we don't even  know that all chains are flat state  connected  but here i guess we know that obtuse  chains are flat stay connected  so maybe it's not so surprising  but what's important is not only are the  flat states connected to each other and  the producible states are connected to  each other  but producibles connected to flat flat  states everything is together here  i might have more properties but that's  already some good news and there's  algorithms to do all of this  how do we prove it well  as usual we use the fedex method  and some since one of the challenges is  what is the natural canonical state for  protein chains in fact we're just going  to assume that our chain  is in reality we're going to assume that  it's an orthogonal chain  an obtuse chain i should say but in  general  for any lesser equal to alpha chain for  whatever alpha you like  and it will be the half angle of the  cone so alpha equals beta  we will define a canonical configuration  i think we called it the alpha ccc  so it's going to be kind of like a helix  i think i have an example an actual  computed example  that's not the best picture because you  can't see everything that's going on but  this will  this is an actual canonical  configuration of a particular  chain  let me tell you how it works in general  so in general we have some  chain v1 v2 sorry starting at v0 v1 v2  um i want to define a canonic and  there's defined lengths between the two  and there's defined angles between every  triple  in sequence so i'm going to start with  v0 somewhere  it doesn't really matter by translation  say the origin of space  and  what i'm going to do is draw a cone  whose apex is at v0 and the half angle  here is going to be alpha over 2 not  alpha  this is a smaller cone than by a factor  of 2  than the ribosome that's important  so there's this uh vertical line  and to pick v1 i'm just going to use  this  the right edge which is let's say this  is the x direction this  is the z direction maximum x coordinate  it's going to lie on the cone maximum  export it that's v1  okay now v1 let me  redraw this picture a little lower so  there was  v0 v1  now i want to draw v2 i want to draw it  above  so what i'll do is draw a vertical cone  whose half angle here is alpha over two  and i want to draw v2 on the cone here  of course the the height of the cone is  the length of the edge  not the height but you know this you can  think of the cone as infinite  and then i just clip this to when it has  the right length  so again i'm at this might be a  different height cone i  clip it to whatever the length v1 v2 is  i want it to be somewhere on this cone  but now i'm constrained to have the  correct angle at v1  i can't just put it over here because  then the angle here would be 180  presumably i don't want to make 180  degree angle  so in reality what happens is that  there's a cone  which is um  who's whose axis is the edge v0v1  so i extend v0v1 out here which in this  case happens to lie here  and i make a cone like that  let me draw it slightly more accurately  okay in this case the center  axis of the cone would go right here  whatever the extension of v0v1 was  and to be to have the right angle at v1  v2 must be on that cone conveniently  there are two intersection points  between those two cones  i can choose either one of them to be v2  and i will choose the counterclockwise  most  one which is this one  this is going to be v2 so draw that edge  now i repeat so from v2  i'm going to draw a vertical cone whose  half angle here the half angle was  always alpha over 2.  if angle is alpha over 2. this cone  had angle  or half angle whatever the angle v0 v1  v2  was is that the half angle or the angle  the  half angle so i think this is right  okay and then i take the intersection of  those two cones  and that will give me where v3 is so i  do the same thing for v2 for v3 and so  on  this i have a unique choice at every  moment  yeah basically and the only exception  was at the beginning here when i had a  vertical line  these cones two cones could actually be  equal and then the intersection is  the entire cone in that case i guess i  choose the maximum x-coordinate one  again  and then i'll i have a canonical choice  of everything  along the way it will always go up and  it sort of spirals around because of the  counterclockwise  most choice and the result is a picture  like this  anything else i need to say here all  right  i claim this canonical configuration  lies in an  alpha over two half angle cone  that's true by construction the  challenge is does the construction  really work so i start  obviously with one cone and i can think  of this as actually an infinite cone  that goes  out to infinity here and i claim the  entire construction will lie inside that  cone  and it's kind of obvious because v2 v1 i  chose v2 to lie  in in this co the same  cone just translated up to start at v1  instead of starting at v0  of course this cone is contained in this  bigger one  and by induction in fact the entire rest  of the chain will lie  in this smaller cone therefore it lies  in the big one also  okay fine  so by construction it will lie in alpha  over two cone  the worry is that these two cones don't  intersect  here we have an angle the half angle of  the cone is whatever angle  the angle is at v1 sorry this should not  be the angle this should be the turn  angle  that that angle is how much you turn  from v0 v1  now we know that we're assuming that the  turn angles are all at most alpha  this cone so this the the turn angle  cone could actually be twice as big  as the vertical cone that we were always  using we always use a vertical cone  half angle alpha over two but because  it's okay because we always keep these  edges like v0 v1  was on the edge of the cone uh  and so if when we extend it it lies on  the edge of this vertical cone  so its angle in in the most  extreme case its half angle is alpha  which would look like this would go all  the way over  from the right side of the cone to the  left side of the cone in general it's  not going to be right and left but it's  going to be some side and the  antipodal point and because the  double angle of the cone is alpha it's  still okay  you will intersect somewhere on the cone  this is a subtle detail but it's really  crucial because we start with a chain  that has relatively large angles alpha  and we get it  into we squeeze it into a cone that  still has double  twice its angle is alpha but we kind of  compress it  into something of half angle alpha over  two you might think oh i'm just  changing the definition calling a half  angle therefore i could stand alpha over  two but it's  a little tricky to actually get it to  fit in a vertical alpha over two cone  once we have this it's really easy to  canonicalize a chain  a producible chain so let me tell you  how to do that  so we're going to use the fedex method  of taking some configuration  and canonicalizing it and then  uncanonicalizing to something else  we're also going to use a new method  which i just came up with the term  it's called the momento method which is  you play  the movie in reverse  so i guess also the merlin method uh  that's more complicated so  we have a movie here in mind  which is how was the chain produced so  what i want to show  is that if i want to start with a  producible  configuration and chain  somehow it got produced so  uh you had your cone and the thing  starts spewing out and folding and doing  whatever  uh that's that's an animation in some  sense of of one edge coming out and  stuff is folding at the same time then  an edge is created then another edge  comes out and so on  what i want to do is play that movie  backwards this is pretty intuitive idea  i just want to start feeding the edges  back into the cone  and just keep stuffing them in now what  happens out here  is easy because we know it doesn't  penetrate the cone that's the assumption  and we know whatever was created here  could be uncreated as long as you  can't afford to erase edges one by one  that's the tricky part how do i erase an  edge usually i can't  but it's not i don't have to erase any  edge like if i had to erase this one  that would be hard because some motion  here might penetrate where that edge  ought to have been and erasing the edge  will not  will make it  adding the edge makes it harder to fold  so i can't erase it  but the edges i have to erase are the  ones that have been fully inserted into  the cone so if i can  somehow do something inside the cone  i would be okay all this work  of defining a canonical configuration  was about forcing a chain to stay inside  a cone and not only an alpha cone which  is what we're going to have  as the ribosome but an alpha over two  cone this is smaller than my ribosome  cone  by a factor of two and i need that  why do i need that because this thing  i have no control over the outside chain  so the way that it approaches the cone  could be as sharp as like this  where i have it the first edge that is  or the the current edge that is inside  the cone  as far as the movie is concerned there's  only one edge there's nothing up here  i want to put something up here in a  cone naturally  but i don't get to control the first  angle  because that is controlled by this  motion maybe it really needed to go  sharp like that so they could make a  sharper angle or whatever  so you know you might say well of course  i can put it  inside an alpha cone which is the same  as  as this alpha sorry  bad picture this is alpha  this would be a problem because i have  this weird angle coming in  and now suddenly i have to bend back  like that maybe the turn angle here is  not so sharp as alpha maybe it's  one degree so i really can't  force the rest of my chain to lie in  this cone ah  but i claim i can force it  to lie in this cone  with half angle alpha over two  this is a little more subtle  oh right i wanted to mention that  helices appear in nature  they appear in proteins but they also  appear in this crazy  climber plant marty have you seen this  in costa rica  yes yeah there's some really incredible  wildlife in costa rica i've never been  there  but i've seen lots of pictures and this  is uh  spirals in practice also  um proteins tend to form these things  they call them alpha helices because  they spin like an alpha i guess  so this it's kind of neat that the  canonical configuration which is totally  geometrically motivated  also appears in biology not that kind of  biology though  so here is the picture i have  the big cone that is my ribosome and  here i'm going to write beta  for the beta for big i guess and then  we know that we can canonicalize a chain  or there at least exists a canonical  configuration of the chain  where everything lies inside a cone  of half angle alpha over two now the  problem is that  cone we want it to be at a funny angle  let me draw a real picture  here's a ribosome has a big angle  here called alpha  now i'm going to do the not extreme case  try to be a little more general  there's some edge that is currently  entering and we have no control over  that edge or the rest of the chain  that's plate that's determined by the  movie which we're trying to play  backwards  what we what we have to control and what  we're free to control  is the rest of the chain because as far  as the movie is concerned that hasn't  been  created yet or it hasn't been just it's  already been destroyed depending on  where you're playing forwards or  backwards  uh we need to say what happens to it and  i what i want to happen  is so that by the time this edge is  inside  that edge plus the rest is in the  canonical configuration  if i can achieve that then as edges come  in they become canonicalized and then  everything will be canonical and  inside the cone we're done that's our  goal  canonicalization so uh  what's the deal well in reality  there's some cone uh  yeah it can penetrate like that could  penetrate the outside cone  and this is getting messy  so if i extend this line  there's a cone of half angle here which  is equal to whatever the turn angle is  at that vertex which is again specified  we're not free to set it to whatever we  want we know the next edge  must lie on this cone  what i do uh  for the now i have on the other hand off  to the side i have in mind a vertical  cone  whose half angle is alpha over two so  it's quite small  and i know um how do we do it initially  the first edge was along the maximum x  direction and then it you know it  spirals up from there  okay here's what i'm going to do  there are sort of two situations i what  i'd like to do  is put a cone here that is  vertical something like that  just like i have here the trouble is the  right side of this cone  does not intersect the boundary of this  cone and i need it to in order to form  the right angle here  it's going to it might go through the  middle of the cone like it does here  so then what i do is i take this  canonical configuration and i rotate it  so that this is hard to draw  it'll be something like this  i think there's no hope of seeing this  so it's on the surface of this cone it's  not on the far right edge it's going to  be some  some intermediate point and that's  that's exactly where it intersects this  cone  it's just like the previous picture just  harder to see i'm taking the  intersection of these two cones  if i just rotate the picture then it  will lie on the intersection  if they intersect but they might not  intersect  so let me go here  try it again  so the easy case is when i can draw a  vertical cone  and it intersects the the cone that i  need to intersect  the harder case is  maybe it's more extreme  maybe it's a very tight angle here  so there's a very small turn angle i  have to intersect this cone  because my the cone that i'm working  with is actually smaller  it might it's half half of this  angle it might not intersect this cone  in that case  i'm going to rotate the cone to fall  over a little bit  so instead of being like that it's going  to be like this  okay so i want the intersection of these  two cones which i've conveniently made  this this edge here so the first edge  will be  lie along here and then it's going to  spiral it from there so i still have the  canonical configuration i've just tilted  it  tilting is going to be necessary because  i have this angle to match up  the convenient thing about the canonical  this the alpha ccc canonical  configuration  is that i have this half angle of alpha  over two  so i can afford to tilt it by up to  alpha over two  it will still stay within the ribosome  which is of  half angle alpha and all i need to show  here and  sort of once you think about it for a  while it's obvious that  you ideally i don't tilt it at all got  tons of rooms  huge amount of room here but sometimes  i'll have to tilt it  but by at most alpha over two so i will  stay inside  the ribosome because this apex was  inside the cone  and this smaller cone even if i tilt it  all the way  to meet this edge it will stay inside  the cone in fact  it'll stay inside the big cone in fact  this cone that i tilt  the one that contains the rest of the  canonical configuration will always  contain  the up direction  that's how to see it if it always  contains the up direction  then at most it's that big  and that will uh because that is  ha that's an angle of alpha because that  was the half angle of the big cone  and so that would be a half angle of  alpha over two as long as you contain  the up direction at all times  you will not fall outside the big cone  so the rest is just memento so  you play this movie backwards as things  come in here  uh you this cone is going to wiggle back  and forth depending on how this angle  changes  once the edge gets all the way in you  absorb it into the canonical  configuration it's a little bit of work  there but you just sort of twist the  cone around  until that algorithm that we described  for producing canonical configuration  would actually produce what's inside the  cone  then the next edge comes in the cone  wiggles around until the edge gets all  the way in then you  canonicalize what's inside the cone and  repeat  so if you had a way to get it out you  can put it back in  and keep track of all the stuff that  happens on the inside  that's what this theorem says and then  there's just  slightly more to say  if you have a flat state  we want to prove these things are flat  state connected so  take some flat state i ideally should  have only obtuse angles  i claim this flat state can be produced  using a cone  of the appropriate angle this is the  sharpest turn angle here is alpha then  you need an alpha cone  and the way to think about that is to  think of the cone moving  instead of as the chain moving  it's a lot easier so you want the chain  to lie around here  so you start by moving the cone  i guess like this so that it just barely  touches the plane  and this edge spews out  is that the right way to think about it  no no no it should be like this  so you you take the cone this this is  like putting  frosting on a cake okay so you move your  cone like here  you squirt out some this this edge  okay then uh you do it so that when  you're here and the new edge is created  uh it's still in the plane and then you  just sort of move around there i'm just  going to leave it as a sketch like that  once you know that you can produce any  flat state of course you can  reorient yourself by relativity so that  the chain is moving instead of the cone  so you can produce this thing with a  ribosome  once you know all flat states can be  produced and you know all  producible configurations can be  canonicalized then you know it's flat  stay connected and you know all  canonical things are flattenable and  vice versa  by continuous motions without  self-intersection and all of this is  algorithmic  can tell you how to go from one place to  another  and so this is a candidate algorithm i  would say for how nature folds proteins  just thinking about the mechanics not  worrying about how it's implemented  maybe you take this model and then you  try to make it  physical physical forces  and you get a way to fold proteins that  of course remains a mystery but next  time we will talk about  uh some very simple models that are  motivated  more closely by biology of how proteins  might actually fold  and talk about the complexities that you  get there
 

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248
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249
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250
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251
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252
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253
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255
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256
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257
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258
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259
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260
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261
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262
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263
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264
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265
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266
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267
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268
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269
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270
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271
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272
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273
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274
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275
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276
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277
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278
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279
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280
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281
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282
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283
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284
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285
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286
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287
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288
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289
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290
00:10:06,160 --> 00:10:06,790


291
00:10:06,790 --> 00:10:08,870

 

292
00:10:08,870 --> 00:10:10,870

 

293
00:10:10,870 --> 00:10:11,509

 

294
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295
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296
00:10:15,990 --> 00:10:17,990

 

297
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298
00:10:20,150 --> 00:10:21,509

 

299
00:10:21,509 --> 00:10:22,470

 

300
00:10:22,470 --> 00:10:24,389

 

301
00:10:24,389 --> 00:10:26,150

 

302
00:10:26,150 --> 00:10:27,910

 

303
00:10:27,910 --> 00:10:29,269

 

304
00:10:29,269 --> 00:10:31,670

 

305
00:10:31,670 --> 00:10:33,829

 

306
00:10:33,829 --> 00:10:36,069

 

307
00:10:36,069 --> 00:10:39,430

 

308
00:10:39,430 --> 00:10:41,110

 

309
00:10:41,110 --> 00:10:43,110

 

310
00:10:43,110 --> 00:10:44,069

 

311
00:10:44,069 --> 00:10:46,550

 

312
00:10:46,550 --> 00:10:48,470

 

313
00:10:48,470 --> 00:10:49,910

 

314
00:10:49,910 --> 00:10:51,910

 

315
00:10:51,910 --> 00:10:54,069

 

316
00:10:54,069 --> 00:10:55,110

 

317
00:10:55,110 --> 00:10:58,470

 

318
00:10:58,470 --> 00:10:59,670

 

319
00:10:59,670 --> 00:11:01,030

 

320
00:11:01,030 --> 00:11:03,829

 

321
00:11:03,829 --> 00:11:06,550

 

322
00:11:06,550 --> 00:11:10,790

 

323
00:11:10,790 --> 00:11:12,790

 

324
00:11:12,790 --> 00:11:14,630

 

325
00:11:14,630 --> 00:11:14,640

 

326
00:11:14,640 --> 00:11:15,269


327
00:11:15,269 --> 00:11:16,630

 

328
00:11:16,630 --> 00:11:20,230

 

329
00:11:20,230 --> 00:11:22,550

 

330
00:11:22,550 --> 00:11:24,389

 

331
00:11:24,389 --> 00:11:25,910

 

332
00:11:25,910 --> 00:11:27,670

 

333
00:11:27,670 --> 00:11:28,949

 

334
00:11:28,949 --> 00:11:34,389

 

335
00:11:34,389 --> 00:11:36,470

 

336
00:11:36,470 --> 00:11:39,110

 

337
00:11:39,110 --> 00:11:42,550

 

338
00:11:42,550 --> 00:11:49,030

 

339
00:11:49,030 --> 00:11:51,350

 

340
00:11:51,350 --> 00:11:51,360

 

341
00:11:51,360 --> 00:11:52,550


342
00:11:52,550 --> 00:11:54,230

 

343
00:11:54,230 --> 00:11:56,629

 

344
00:11:56,629 --> 00:11:57,670

 

345
00:11:57,670 --> 00:12:00,310

 

346
00:12:00,310 --> 00:12:02,790

 

347
00:12:02,790 --> 00:12:06,310

 

348
00:12:06,310 --> 00:12:07,110

 

349
00:12:07,110 --> 00:12:08,949

 

350
00:12:08,949 --> 00:12:11,269

 

351
00:12:11,269 --> 00:12:13,590

 

352
00:12:13,590 --> 00:12:14,949

 

353
00:12:14,949 --> 00:12:17,590

 

354
00:12:17,590 --> 00:12:19,509

 

355
00:12:19,509 --> 00:12:22,790

 

356
00:12:22,790 --> 00:12:25,590

 

357
00:12:25,590 --> 00:12:25,600

 

358
00:12:25,600 --> 00:12:26,150


359
00:12:26,150 --> 00:12:27,990

 

360
00:12:27,990 --> 00:12:29,670

 

361
00:12:29,670 --> 00:12:33,030

 

362
00:12:33,030 --> 00:12:35,190

 

363
00:12:35,190 --> 00:12:37,350

 

364
00:12:37,350 --> 00:12:39,430

 

365
00:12:39,430 --> 00:12:42,470

 

366
00:12:42,470 --> 00:12:43,750

 

367
00:12:43,750 --> 00:12:43,760

 

368
00:12:43,760 --> 00:12:45,030


369
00:12:45,030 --> 00:12:47,670

 

370
00:12:47,670 --> 00:12:48,790

 

371
00:12:48,790 --> 00:12:50,870

 

372
00:12:50,870 --> 00:12:52,710

 

373
00:12:52,710 --> 00:12:55,110

 

374
00:12:55,110 --> 00:12:55,120

 

375
00:12:55,120 --> 00:12:56,470


376
00:12:56,470 --> 00:12:58,710

 

377
00:12:58,710 --> 00:13:01,030

 

378
00:13:01,030 --> 00:13:03,670

 

379
00:13:03,670 --> 00:13:03,680

 

380
00:13:03,680 --> 00:13:04,389


381
00:13:04,389 --> 00:13:06,629

 

382
00:13:06,629 --> 00:13:07,829

 

383
00:13:07,829 --> 00:13:09,670

 

384
00:13:09,670 --> 00:13:12,629

 

385
00:13:12,629 --> 00:13:12,639

 

386
00:13:12,639 --> 00:13:15,509


387
00:13:15,509 --> 00:13:23,430

 

388
00:13:23,430 --> 00:13:27,190

 

389
00:13:27,190 --> 00:13:27,200

 

390
00:13:27,200 --> 00:13:31,430


391
00:13:31,430 --> 00:13:39,430

 

392
00:13:39,430 --> 00:13:40,870

 

393
00:13:40,870 --> 00:13:42,389

 

394
00:13:42,389 --> 00:13:43,509

 

395
00:13:43,509 --> 00:13:45,750

 

396
00:13:45,750 --> 00:13:48,790

 

397
00:13:48,790 --> 00:13:51,829

 

398
00:13:51,829 --> 00:13:56,150

 

399
00:13:56,150 --> 00:13:59,189

 

400
00:13:59,189 --> 00:14:00,629

 

401
00:14:00,629 --> 00:14:02,310

 

402
00:14:02,310 --> 00:14:05,110

 

403
00:14:05,110 --> 00:14:06,949

 

404
00:14:06,949 --> 00:14:10,150

 

405
00:14:10,150 --> 00:14:13,430

 

406
00:14:13,430 --> 00:14:15,590

 

407
00:14:15,590 --> 00:14:17,030

 

408
00:14:17,030 --> 00:14:19,110

 

409
00:14:19,110 --> 00:14:20,870

 

410
00:14:20,870 --> 00:14:22,710

 

411
00:14:22,710 --> 00:14:24,389

 

412
00:14:24,389 --> 00:14:26,069

 

413
00:14:26,069 --> 00:14:26,079

 

414
00:14:26,079 --> 00:14:26,470


415
00:14:26,470 --> 00:14:30,710

 

416
00:14:30,710 --> 00:14:33,750

 

417
00:14:33,750 --> 00:14:36,150

 

418
00:14:36,150 --> 00:14:36,790

 

419
00:14:36,790 --> 00:14:40,310

 

420
00:14:40,310 --> 00:14:42,069

 

421
00:14:42,069 --> 00:14:43,750

 

422
00:14:43,750 --> 00:14:43,760

 

423
00:14:43,760 --> 00:14:44,069


424
00:14:44,069 --> 00:14:46,710

 

425
00:14:46,710 --> 00:14:46,720

 

426
00:14:46,720 --> 00:14:47,670


427
00:14:47,670 --> 00:14:49,590

 

428
00:14:49,590 --> 00:14:51,590

 

429
00:14:51,590 --> 00:14:53,829

 

430
00:14:53,829 --> 00:14:54,790

 

431
00:14:54,790 --> 00:14:56,710

 

432
00:14:56,710 --> 00:14:57,990

 

433
00:14:57,990 --> 00:14:58,000

 

434
00:14:58,000 --> 00:14:59,269


435
00:14:59,269 --> 00:15:02,790

 

436
00:15:02,790 --> 00:15:04,389

 

437
00:15:04,389 --> 00:15:06,389

 

438
00:15:06,389 --> 00:15:09,590

 

439
00:15:09,590 --> 00:15:09,600

 

440
00:15:09,600 --> 00:15:10,150


441
00:15:10,150 --> 00:15:11,509

 

442
00:15:11,509 --> 00:15:14,550

 

443
00:15:14,550 --> 00:15:16,310

 

444
00:15:16,310 --> 00:15:18,790

 

445
00:15:18,790 --> 00:15:20,870

 

446
00:15:20,870 --> 00:15:23,110

 

447
00:15:23,110 --> 00:15:24,710

 

448
00:15:24,710 --> 00:15:26,069

 

449
00:15:26,069 --> 00:15:28,310

 

450
00:15:28,310 --> 00:15:30,069

 

451
00:15:30,069 --> 00:15:32,150

 

452
00:15:32,150 --> 00:15:35,030

 

453
00:15:35,030 --> 00:15:36,470

 

454
00:15:36,470 --> 00:15:38,389

 

455
00:15:38,389 --> 00:15:39,990

 

456
00:15:39,990 --> 00:15:41,990

 

457
00:15:41,990 --> 00:15:45,269

 

458
00:15:45,269 --> 00:15:45,279

 

459
00:15:45,279 --> 00:15:48,230


460
00:15:48,230 --> 00:15:50,629

 

461
00:15:50,629 --> 00:15:51,990

 

462
00:15:51,990 --> 00:15:52,000

 

463
00:15:52,000 --> 00:15:52,710


464
00:15:52,710 --> 00:15:58,389

 

465
00:15:58,389 --> 00:16:05,670

 

466
00:16:05,670 --> 00:16:08,710

 

467
00:16:08,710 --> 00:16:09,189

 

468
00:16:09,189 --> 00:16:20,150

 

469
00:16:20,150 --> 00:16:21,749

 

470
00:16:21,749 --> 00:16:23,749

 

471
00:16:23,749 --> 00:16:25,430

 

472
00:16:25,430 --> 00:16:27,269

 

473
00:16:27,269 --> 00:16:30,870

 

474
00:16:30,870 --> 00:16:32,470

 

475
00:16:32,470 --> 00:16:34,870

 

476
00:16:34,870 --> 00:16:35,990

 

477
00:16:35,990 --> 00:16:39,110

 

478
00:16:39,110 --> 00:16:40,310

 

479
00:16:40,310 --> 00:16:46,310

 

480
00:16:46,310 --> 00:16:50,870

 

481
00:16:50,870 --> 00:16:52,389

 

482
00:16:52,389 --> 00:16:53,829

 

483
00:16:53,829 --> 00:16:55,430

 

484
00:16:55,430 --> 00:16:56,870

 

485
00:16:56,870 --> 00:16:57,749

 

486
00:16:57,749 --> 00:16:59,590

 

487
00:16:59,590 --> 00:17:01,030

 

488
00:17:01,030 --> 00:17:04,150

 

489
00:17:04,150 --> 00:17:06,390

 

490
00:17:06,390 --> 00:17:06,400

 

491
00:17:06,400 --> 00:17:09,829


492
00:17:09,829 --> 00:17:11,270

 

493
00:17:11,270 --> 00:17:12,870

 

494
00:17:12,870 --> 00:17:14,470

 

495
00:17:14,470 --> 00:17:17,350

 

496
00:17:17,350 --> 00:17:18,470

 

497
00:17:18,470 --> 00:17:22,390

 

498
00:17:22,390 --> 00:17:24,630

 

499
00:17:24,630 --> 00:17:26,230

 

500
00:17:26,230 --> 00:17:27,669

 

501
00:17:27,669 --> 00:17:29,270

 

502
00:17:29,270 --> 00:17:29,280

 

503
00:17:29,280 --> 00:17:30,789


504
00:17:30,789 --> 00:17:32,950

 

505
00:17:32,950 --> 00:17:33,830

 

506
00:17:33,830 --> 00:17:37,029

 

507
00:17:37,029 --> 00:17:39,430

 

508
00:17:39,430 --> 00:17:41,190

 

509
00:17:41,190 --> 00:17:44,549

 

510
00:17:44,549 --> 00:17:46,150

 

511
00:17:46,150 --> 00:17:48,150

 

512
00:17:48,150 --> 00:17:49,750

 

513
00:17:49,750 --> 00:17:51,590

 

514
00:17:51,590 --> 00:17:53,590

 

515
00:17:53,590 --> 00:17:55,190

 

516
00:17:55,190 --> 00:17:56,549

 

517
00:17:56,549 --> 00:17:57,830

 

518
00:17:57,830 --> 00:17:59,590

 

519
00:17:59,590 --> 00:18:01,750

 

520
00:18:01,750 --> 00:18:03,909

 

521
00:18:03,909 --> 00:18:05,590

 

522
00:18:05,590 --> 00:18:07,510

 

523
00:18:07,510 --> 00:18:09,510

 

524
00:18:09,510 --> 00:18:09,520

 

525
00:18:09,520 --> 00:18:10,870


526
00:18:10,870 --> 00:18:12,630

 

527
00:18:12,630 --> 00:18:14,150

 

528
00:18:14,150 --> 00:18:16,230

 

529
00:18:16,230 --> 00:18:18,549

 

530
00:18:18,549 --> 00:18:20,470

 

531
00:18:20,470 --> 00:18:22,070

 

532
00:18:22,070 --> 00:18:22,080

 

533
00:18:22,080 --> 00:18:23,590


534
00:18:23,590 --> 00:18:27,669

 

535
00:18:27,669 --> 00:18:29,270

 

536
00:18:29,270 --> 00:18:31,510

 

537
00:18:31,510 --> 00:18:33,590

 

538
00:18:33,590 --> 00:18:36,630

 

539
00:18:36,630 --> 00:18:39,669

 

540
00:18:39,669 --> 00:18:41,909

 

541
00:18:41,909 --> 00:18:44,150

 

542
00:18:44,150 --> 00:18:45,990

 

543
00:18:45,990 --> 00:18:48,549

 

544
00:18:48,549 --> 00:18:51,190

 

545
00:18:51,190 --> 00:18:52,870

 

546
00:18:52,870 --> 00:18:55,029

 

547
00:18:55,029 --> 00:18:56,630

 

548
00:18:56,630 --> 00:18:59,669

 

549
00:18:59,669 --> 00:19:01,110

 

550
00:19:01,110 --> 00:19:02,150

 

551
00:19:02,150 --> 00:19:05,350

 

552
00:19:05,350 --> 00:19:05,360

 

553
00:19:05,360 --> 00:19:05,909


554
00:19:05,909 --> 00:19:21,590

 

555
00:19:21,590 --> 00:19:24,070

 

556
00:19:24,070 --> 00:19:26,390

 

557
00:19:26,390 --> 00:19:29,190

 

558
00:19:29,190 --> 00:19:30,470

 

559
00:19:30,470 --> 00:19:31,830

 

560
00:19:31,830 --> 00:19:33,270

 

561
00:19:33,270 --> 00:19:36,830

 

562
00:19:36,830 --> 00:19:36,840

 

563
00:19:36,840 --> 00:19:40,310


564
00:19:40,310 --> 00:19:43,669

 

565
00:19:43,669 --> 00:19:47,430

 

566
00:19:47,430 --> 00:19:49,990

 

567
00:19:49,990 --> 00:19:50,870

 

568
00:19:50,870 --> 00:19:53,190

 

569
00:19:53,190 --> 00:19:54,630

 

570
00:19:54,630 --> 00:19:56,710

 

571
00:19:56,710 --> 00:19:58,549

 

572
00:19:58,549 --> 00:19:59,990

 

573
00:19:59,990 --> 00:20:02,230

 

574
00:20:02,230 --> 00:20:04,149

 

575
00:20:04,149 --> 00:20:05,750

 

576
00:20:05,750 --> 00:20:07,350

 

577
00:20:07,350 --> 00:20:09,750

 

578
00:20:09,750 --> 00:20:12,789

 

579
00:20:12,789 --> 00:20:14,870

 

580
00:20:14,870 --> 00:20:14,880

 

581
00:20:14,880 --> 00:20:15,669


582
00:20:15,669 --> 00:20:17,990

 

583
00:20:17,990 --> 00:20:19,350

 

584
00:20:19,350 --> 00:20:22,230

 

585
00:20:22,230 --> 00:20:27,669

 

586
00:20:27,669 --> 00:20:29,590

 

587
00:20:29,590 --> 00:20:32,310

 

588
00:20:32,310 --> 00:20:35,029

 

589
00:20:35,029 --> 00:20:35,039

 

590
00:20:35,039 --> 00:20:35,350


591
00:20:35,350 --> 00:20:37,990

 

592
00:20:37,990 --> 00:20:38,870

 

593
00:20:38,870 --> 00:20:41,590

 

594
00:20:41,590 --> 00:20:41,600

 

595
00:20:41,600 --> 00:20:42,710


596
00:20:42,710 --> 00:20:45,990

 

597
00:20:45,990 --> 00:20:49,510

 

598
00:20:49,510 --> 00:20:53,430

 

599
00:20:53,430 --> 00:20:55,590

 

600
00:20:55,590 --> 00:20:59,350

 

601
00:20:59,350 --> 00:21:02,070

 

602
00:21:02,070 --> 00:21:03,750

 

603
00:21:03,750 --> 00:21:05,669

 

604
00:21:05,669 --> 00:21:07,990

 

605
00:21:07,990 --> 00:21:10,390

 

606
00:21:10,390 --> 00:21:14,310

 

607
00:21:14,310 --> 00:21:16,630

 

608
00:21:16,630 --> 00:21:17,830

 

609
00:21:17,830 --> 00:21:21,909

 

610
00:21:21,909 --> 00:21:24,149

 

611
00:21:24,149 --> 00:21:25,110

 

612
00:21:25,110 --> 00:21:27,029

 

613
00:21:27,029 --> 00:21:28,470

 

614
00:21:28,470 --> 00:21:29,990

 

615
00:21:29,990 --> 00:21:33,029

 

616
00:21:33,029 --> 00:21:36,149

 

617
00:21:36,149 --> 00:21:38,230

 

618
00:21:38,230 --> 00:21:39,669

 

619
00:21:39,669 --> 00:21:41,990

 

620
00:21:41,990 --> 00:21:43,669

 

621
00:21:43,669 --> 00:21:45,029

 

622
00:21:45,029 --> 00:21:47,270

 

623
00:21:47,270 --> 00:21:47,280

 

624
00:21:47,280 --> 00:21:48,310


625
00:21:48,310 --> 00:21:51,190

 

626
00:21:51,190 --> 00:21:51,200

 

627
00:21:51,200 --> 00:21:51,750


628
00:21:51,750 --> 00:21:53,830

 

629
00:21:53,830 --> 00:21:55,909

 

630
00:21:55,909 --> 00:21:57,909

 

631
00:21:57,909 --> 00:21:59,750

 

632
00:21:59,750 --> 00:22:01,830

 

633
00:22:01,830 --> 00:22:03,270

 

634
00:22:03,270 --> 00:22:05,110

 

635
00:22:05,110 --> 00:22:07,990

 

636
00:22:07,990 --> 00:22:10,789

 

637
00:22:10,789 --> 00:22:15,029

 

638
00:22:15,029 --> 00:22:16,870

 

639
00:22:16,870 --> 00:22:18,710

 

640
00:22:18,710 --> 00:22:20,310

 

641
00:22:20,310 --> 00:22:22,230

 

642
00:22:22,230 --> 00:22:23,750

 

643
00:22:23,750 --> 00:22:23,760

 

644
00:22:23,760 --> 00:22:25,430


645
00:22:25,430 --> 00:22:29,270

 

646
00:22:29,270 --> 00:22:32,630

 

647
00:22:32,630 --> 00:22:34,710

 

648
00:22:34,710 --> 00:22:36,390

 

649
00:22:36,390 --> 00:22:40,070

 

650
00:22:40,070 --> 00:22:42,310

 

651
00:22:42,310 --> 00:22:43,510

 

652
00:22:43,510 --> 00:22:45,909

 

653
00:22:45,909 --> 00:22:47,990

 

654
00:22:47,990 --> 00:22:49,110

 

655
00:22:49,110 --> 00:22:52,149

 

656
00:22:52,149 --> 00:22:54,789

 

657
00:22:54,789 --> 00:22:56,470

 

658
00:22:56,470 --> 00:22:56,480

 

659
00:22:56,480 --> 00:22:58,230


660
00:22:58,230 --> 00:22:59,990

 

661
00:22:59,990 --> 00:23:02,149

 

662
00:23:02,149 --> 00:23:04,070

 

663
00:23:04,070 --> 00:23:04,080

 

664
00:23:04,080 --> 00:23:05,029


665
00:23:05,029 --> 00:23:08,549

 

666
00:23:08,549 --> 00:23:10,230

 

667
00:23:10,230 --> 00:23:11,909

 

668
00:23:11,909 --> 00:23:14,390

 

669
00:23:14,390 --> 00:23:16,630

 

670
00:23:16,630 --> 00:23:18,710

 

671
00:23:18,710 --> 00:23:19,830

 

672
00:23:19,830 --> 00:23:21,270

 

673
00:23:21,270 --> 00:23:22,710

 

674
00:23:22,710 --> 00:23:24,870

 

675
00:23:24,870 --> 00:23:26,789

 

676
00:23:26,789 --> 00:23:27,990

 

677
00:23:27,990 --> 00:23:29,750

 

678
00:23:29,750 --> 00:23:31,190

 

679
00:23:31,190 --> 00:23:31,200

 

680
00:23:31,200 --> 00:23:31,830


681
00:23:31,830 --> 00:23:33,669

 

682
00:23:33,669 --> 00:23:35,110

 

683
00:23:35,110 --> 00:23:38,070

 

684
00:23:38,070 --> 00:23:39,510

 

685
00:23:39,510 --> 00:23:39,520

 

686
00:23:39,520 --> 00:23:39,909


687
00:23:39,909 --> 00:23:43,590

 

688
00:23:43,590 --> 00:23:45,590

 

689
00:23:45,590 --> 00:23:46,789

 

690
00:23:46,789 --> 00:23:49,430

 

691
00:23:49,430 --> 00:23:49,440

 

692
00:23:49,440 --> 00:23:50,310


693
00:23:50,310 --> 00:23:53,909

 

694
00:23:53,909 --> 00:23:57,029

 

695
00:23:57,029 --> 00:23:58,310

 

696
00:23:58,310 --> 00:24:00,310

 

697
00:24:00,310 --> 00:24:02,470

 

698
00:24:02,470 --> 00:24:03,750

 

699
00:24:03,750 --> 00:24:09,269

 

700
00:24:09,269 --> 00:24:12,710

 

701
00:24:12,710 --> 00:24:14,310

 

702
00:24:14,310 --> 00:24:16,310

 

703
00:24:16,310 --> 00:24:19,110

 

704
00:24:19,110 --> 00:24:19,669

 

705
00:24:19,669 --> 00:24:22,870

 

706
00:24:22,870 --> 00:24:24,710

 

707
00:24:24,710 --> 00:24:26,230

 

708
00:24:26,230 --> 00:24:28,070

 

709
00:24:28,070 --> 00:24:29,510

 

710
00:24:29,510 --> 00:24:31,750

 

711
00:24:31,750 --> 00:24:31,760

 

712
00:24:31,760 --> 00:24:32,549


713
00:24:32,549 --> 00:24:35,430

 

714
00:24:35,430 --> 00:24:35,440

 

715
00:24:35,440 --> 00:24:35,830


716
00:24:35,830 --> 00:24:38,950

 

717
00:24:38,950 --> 00:24:41,430

 

718
00:24:41,430 --> 00:24:44,149

 

719
00:24:44,149 --> 00:24:46,390

 

720
00:24:46,390 --> 00:24:48,230

 

721
00:24:48,230 --> 00:24:48,240

 

722
00:24:48,240 --> 00:24:48,710


723
00:24:48,710 --> 00:24:51,909

 

724
00:24:51,909 --> 00:24:53,269

 

725
00:24:53,269 --> 00:24:54,549

 

726
00:24:54,549 --> 00:24:56,070

 

727
00:24:56,070 --> 00:24:57,430

 

728
00:24:57,430 --> 00:25:00,230

 

729
00:25:00,230 --> 00:25:00,240

 

730
00:25:00,240 --> 00:25:00,950


731
00:25:00,950 --> 00:25:04,149

 

732
00:25:04,149 --> 00:25:05,190

 

733
00:25:05,190 --> 00:25:06,549

 

734
00:25:06,549 --> 00:25:08,310

 

735
00:25:08,310 --> 00:25:08,320

 

736
00:25:08,320 --> 00:25:08,950


737
00:25:08,950 --> 00:25:13,029

 

738
00:25:13,029 --> 00:25:14,310

 

739
00:25:14,310 --> 00:25:16,310

 

740
00:25:16,310 --> 00:25:16,320

 

741
00:25:16,320 --> 00:25:17,190


742
00:25:17,190 --> 00:25:18,789

 

743
00:25:18,789 --> 00:25:20,789

 

744
00:25:20,789 --> 00:25:22,310

 

745
00:25:22,310 --> 00:25:23,990

 

746
00:25:23,990 --> 00:25:25,590

 

747
00:25:25,590 --> 00:25:27,350

 

748
00:25:27,350 --> 00:25:29,110

 

749
00:25:29,110 --> 00:25:31,510

 

750
00:25:31,510 --> 00:25:34,310

 

751
00:25:34,310 --> 00:25:35,590

 

752
00:25:35,590 --> 00:25:35,600

 

753
00:25:35,600 --> 00:25:36,230


754
00:25:36,230 --> 00:25:38,950

 

755
00:25:38,950 --> 00:25:40,230

 

756
00:25:40,230 --> 00:25:43,269

 

757
00:25:43,269 --> 00:25:45,990

 

758
00:25:45,990 --> 00:25:46,000

 

759
00:25:46,000 --> 00:25:47,110


760
00:25:47,110 --> 00:25:48,789

 

761
00:25:48,789 --> 00:25:50,870

 

762
00:25:50,870 --> 00:25:53,669

 

763
00:25:53,669 --> 00:25:55,590

 

764
00:25:55,590 --> 00:25:59,669

 

765
00:25:59,669 --> 00:26:01,990

 

766
00:26:01,990 --> 00:26:03,269

 

767
00:26:03,269 --> 00:26:04,710

 

768
00:26:04,710 --> 00:26:06,149

 

769
00:26:06,149 --> 00:26:07,669

 

770
00:26:07,669 --> 00:26:10,310

 

771
00:26:10,310 --> 00:26:13,110

 

772
00:26:13,110 --> 00:26:13,120

 

773
00:26:13,120 --> 00:26:14,070


774
00:26:14,070 --> 00:26:17,190

 

775
00:26:17,190 --> 00:26:20,549

 

776
00:26:20,549 --> 00:26:22,710

 

777
00:26:22,710 --> 00:26:24,549

 

778
00:26:24,549 --> 00:26:27,909

 

779
00:26:27,909 --> 00:26:30,789

 

780
00:26:30,789 --> 00:26:31,669

 

781
00:26:31,669 --> 00:26:34,070

 

782
00:26:34,070 --> 00:26:35,669

 

783
00:26:35,669 --> 00:26:37,430

 

784
00:26:37,430 --> 00:26:40,149

 

785
00:26:40,149 --> 00:26:40,159

 

786
00:26:40,159 --> 00:26:41,190


787
00:26:41,190 --> 00:26:42,789

 

788
00:26:42,789 --> 00:26:44,630

 

789
00:26:44,630 --> 00:26:45,750

 

790
00:26:45,750 --> 00:26:46,870

 

791
00:26:46,870 --> 00:26:48,630

 

792
00:26:48,630 --> 00:26:50,470

 

793
00:26:50,470 --> 00:26:52,230

 

794
00:26:52,230 --> 00:26:52,240

 

795
00:26:52,240 --> 00:26:56,230


796
00:26:56,230 --> 00:26:59,269

 

797
00:26:59,269 --> 00:27:02,390

 

798
00:27:02,390 --> 00:27:06,710

 

799
00:27:06,710 --> 00:27:08,710

 

800
00:27:08,710 --> 00:27:10,710

 

801
00:27:10,710 --> 00:27:13,430

 

802
00:27:13,430 --> 00:27:15,190

 

803
00:27:15,190 --> 00:27:16,390

 

804
00:27:16,390 --> 00:27:18,549

 

805
00:27:18,549 --> 00:27:20,149

 

806
00:27:20,149 --> 00:27:22,710

 

807
00:27:22,710 --> 00:27:22,720

 

808
00:27:22,720 --> 00:27:23,510


809
00:27:23,510 --> 00:27:27,510

 

810
00:27:27,510 --> 00:27:30,549

 

811
00:27:30,549 --> 00:27:33,590

 

812
00:27:33,590 --> 00:27:34,310

 

813
00:27:34,310 --> 00:27:35,750

 

814
00:27:35,750 --> 00:27:37,510

 

815
00:27:37,510 --> 00:27:37,520

 

816
00:27:37,520 --> 00:27:38,470


817
00:27:38,470 --> 00:27:40,549

 

818
00:27:40,549 --> 00:27:43,510

 

819
00:27:43,510 --> 00:27:45,269

 

820
00:27:45,269 --> 00:27:46,870

 

821
00:27:46,870 --> 00:27:47,990

 

822
00:27:47,990 --> 00:27:49,830

 

823
00:27:49,830 --> 00:27:51,350

 

824
00:27:51,350 --> 00:27:53,350

 

825
00:27:53,350 --> 00:27:55,029

 

826
00:27:55,029 --> 00:27:57,750

 

827
00:27:57,750 --> 00:27:57,760

 

828
00:27:57,760 --> 00:27:58,070


829
00:27:58,070 --> 00:27:59,909

 

830
00:27:59,909 --> 00:28:01,990

 

831
00:28:01,990 --> 00:28:03,669

 

832
00:28:03,669 --> 00:28:05,510

 

833
00:28:05,510 --> 00:28:07,350

 

834
00:28:07,350 --> 00:28:09,830

 

835
00:28:09,830 --> 00:28:13,029

 

836
00:28:13,029 --> 00:28:14,630

 

837
00:28:14,630 --> 00:28:16,070

 

838
00:28:16,070 --> 00:28:18,389

 

839
00:28:18,389 --> 00:28:18,399

 

840
00:28:18,399 --> 00:28:19,110


841
00:28:19,110 --> 00:28:22,230

 

842
00:28:22,230 --> 00:28:23,110

 

843
00:28:23,110 --> 00:28:25,350

 

844
00:28:25,350 --> 00:28:26,710

 

845
00:28:26,710 --> 00:28:30,070

 

846
00:28:30,070 --> 00:28:30,080

 

847
00:28:30,080 --> 00:28:30,549


848
00:28:30,549 --> 00:28:33,269

 

849
00:28:33,269 --> 00:28:35,510

 

850
00:28:35,510 --> 00:28:38,070

 

851
00:28:38,070 --> 00:28:40,470

 

852
00:28:40,470 --> 00:28:43,510

 

853
00:28:43,510 --> 00:28:43,520

 

854
00:28:43,520 --> 00:28:44,149


855
00:28:44,149 --> 00:28:47,909

 

856
00:28:47,909 --> 00:28:50,470

 

857
00:28:50,470 --> 00:28:51,110

 

858
00:28:51,110 --> 00:28:53,830

 

859
00:28:53,830 --> 00:28:53,840

 

860
00:28:53,840 --> 00:28:54,389


861
00:28:54,389 --> 00:28:56,310

 

862
00:28:56,310 --> 00:28:58,389

 

863
00:28:58,389 --> 00:29:01,669

 

864
00:29:01,669 --> 00:29:04,389

 

865
00:29:04,389 --> 00:29:07,510

 

866
00:29:07,510 --> 00:29:08,789

 

867
00:29:08,789 --> 00:29:10,470

 

868
00:29:10,470 --> 00:29:12,549

 

869
00:29:12,549 --> 00:29:14,789

 

870
00:29:14,789 --> 00:29:15,990

 

871
00:29:15,990 --> 00:29:17,350

 

872
00:29:17,350 --> 00:29:19,029

 

873
00:29:19,029 --> 00:29:20,789

 

874
00:29:20,789 --> 00:29:22,149

 

875
00:29:22,149 --> 00:29:24,070

 

876
00:29:24,070 --> 00:29:26,310

 

877
00:29:26,310 --> 00:29:29,669

 

878
00:29:29,669 --> 00:29:32,070

 

879
00:29:32,070 --> 00:29:38,549

 

880
00:29:38,549 --> 00:29:40,710

 

881
00:29:40,710 --> 00:29:41,909

 

882
00:29:41,909 --> 00:29:43,830

 

883
00:29:43,830 --> 00:29:44,950

 

884
00:29:44,950 --> 00:29:47,029

 

885
00:29:47,029 --> 00:29:49,269

 

886
00:29:49,269 --> 00:29:51,590

 

887
00:29:51,590 --> 00:29:53,269

 

888
00:29:53,269 --> 00:29:54,789

 

889
00:29:54,789 --> 00:29:56,950

 

890
00:29:56,950 --> 00:29:58,710

 

891
00:29:58,710 --> 00:30:02,789

 

892
00:30:02,789 --> 00:30:05,110

 

893
00:30:05,110 --> 00:30:07,350

 

894
00:30:07,350 --> 00:30:10,230

 

895
00:30:10,230 --> 00:30:11,190

 

896
00:30:11,190 --> 00:30:15,669

 

897
00:30:15,669 --> 00:30:19,110

 

898
00:30:19,110 --> 00:30:23,110

 

899
00:30:23,110 --> 00:30:24,470

 

900
00:30:24,470 --> 00:30:26,789

 

901
00:30:26,789 --> 00:30:34,070

 

902
00:30:34,070 --> 00:30:35,430

 

903
00:30:35,430 --> 00:30:37,269

 

904
00:30:37,269 --> 00:30:38,630

 

905
00:30:38,630 --> 00:30:38,640

 

906
00:30:38,640 --> 00:30:39,750


907
00:30:39,750 --> 00:30:42,950

 

908
00:30:42,950 --> 00:30:45,110

 

909
00:30:45,110 --> 00:30:49,269

 

910
00:30:49,269 --> 00:30:52,310

 

911
00:30:52,310 --> 00:30:55,990

 

912
00:30:55,990 --> 00:31:00,070

 

913
00:31:00,070 --> 00:31:10,070

 

914
00:31:10,070 --> 00:31:13,830

 

915
00:31:13,830 --> 00:31:16,870

 

916
00:31:16,870 --> 00:31:21,590

 

917
00:31:21,590 --> 00:31:24,789

 

918
00:31:24,789 --> 00:31:26,310

 

919
00:31:26,310 --> 00:31:27,909

 

920
00:31:27,909 --> 00:31:29,430

 

921
00:31:29,430 --> 00:31:31,190

 

922
00:31:31,190 --> 00:31:33,669

 

923
00:31:33,669 --> 00:31:35,350

 

924
00:31:35,350 --> 00:31:38,870

 

925
00:31:38,870 --> 00:31:42,950

 

926
00:31:42,950 --> 00:31:45,110

 

927
00:31:45,110 --> 00:31:45,120

 

928
00:31:45,120 --> 00:31:48,470


929
00:31:48,470 --> 00:31:50,310

 

930
00:31:50,310 --> 00:31:50,320

 

931
00:31:50,320 --> 00:31:52,389


932
00:31:52,389 --> 00:31:55,669

 

933
00:31:55,669 --> 00:31:59,990

 

934
00:31:59,990 --> 00:32:03,350

 

935
00:32:03,350 --> 00:32:23,029

 

936
00:32:23,029 --> 00:32:25,190

 

937
00:32:25,190 --> 00:32:30,870

 

938
00:32:30,870 --> 00:32:32,470

 

939
00:32:32,470 --> 00:32:34,870

 

940
00:32:34,870 --> 00:32:36,549

 

941
00:32:36,549 --> 00:32:36,559

 

942
00:32:36,559 --> 00:32:40,830


943
00:32:40,830 --> 00:32:42,950

 

944
00:32:42,950 --> 00:32:44,149

 

945
00:32:44,149 --> 00:32:46,389

 

946
00:32:46,389 --> 00:32:46,399

 

947
00:32:46,399 --> 00:32:47,430


948
00:32:47,430 --> 00:32:49,190

 

949
00:32:49,190 --> 00:32:50,710

 

950
00:32:50,710 --> 00:32:52,310

 

951
00:32:52,310 --> 00:32:54,310

 

952
00:32:54,310 --> 00:32:55,430

 

953
00:32:55,430 --> 00:32:57,269

 

954
00:32:57,269 --> 00:32:59,269

 

955
00:32:59,269 --> 00:33:01,190

 

956
00:33:01,190 --> 00:33:04,230

 

957
00:33:04,230 --> 00:33:07,110

 

958
00:33:07,110 --> 00:33:07,120

 

959
00:33:07,120 --> 00:33:07,909


960
00:33:07,909 --> 00:33:09,830

 

961
00:33:09,830 --> 00:33:11,350

 

962
00:33:11,350 --> 00:33:11,990

 

963
00:33:11,990 --> 00:33:15,029

 

964
00:33:15,029 --> 00:33:17,029

 

965
00:33:17,029 --> 00:33:18,789

 

966
00:33:18,789 --> 00:33:22,149

 

967
00:33:22,149 --> 00:33:23,509

 

968
00:33:23,509 --> 00:33:26,870

 

969
00:33:26,870 --> 00:33:29,190

 

970
00:33:29,190 --> 00:33:30,549

 

971
00:33:30,549 --> 00:33:37,750

 

972
00:33:37,750 --> 00:33:40,230

 

973
00:33:40,230 --> 00:33:40,240

 

974
00:33:40,240 --> 00:33:41,110


975
00:33:41,110 --> 00:33:44,310

 

976
00:33:44,310 --> 00:33:45,830

 

977
00:33:45,830 --> 00:33:47,830

 

978
00:33:47,830 --> 00:33:51,029

 

979
00:33:51,029 --> 00:33:52,470

 

980
00:33:52,470 --> 00:33:55,350

 

981
00:33:55,350 --> 00:33:55,360

 

982
00:33:55,360 --> 00:33:55,750


983
00:33:55,750 --> 00:33:57,990

 

984
00:33:57,990 --> 00:33:58,000

 

985
00:33:58,000 --> 00:33:58,789


986
00:33:58,789 --> 00:34:01,669

 

987
00:34:01,669 --> 00:34:01,679

 

988
00:34:01,679 --> 00:34:02,310


989
00:34:02,310 --> 00:34:05,590

 

990
00:34:05,590 --> 00:34:06,789

 

991
00:34:06,789 --> 00:34:10,149

 

992
00:34:10,149 --> 00:34:13,990

 

993
00:34:13,990 --> 00:34:15,669

 

994
00:34:15,669 --> 00:34:17,349

 

995
00:34:17,349 --> 00:34:19,190

 

996
00:34:19,190 --> 00:34:21,270

 

997
00:34:21,270 --> 00:34:22,470

 

998
00:34:22,470 --> 00:34:24,310

 

999
00:34:24,310 --> 00:34:25,510

 

1000
00:34:25,510 --> 00:34:28,550

 

1001
00:34:28,550 --> 00:34:32,710

 

1002
00:34:32,710 --> 00:34:34,230

 

1003
00:34:34,230 --> 00:34:35,829

 

1004
00:34:35,829 --> 00:34:36,629

 

1005
00:34:36,629 --> 00:34:39,589

 

1006
00:34:39,589 --> 00:34:40,629

 

1007
00:34:40,629 --> 00:34:42,230

 

1008
00:34:42,230 --> 00:34:43,349

 

1009
00:34:43,349 --> 00:34:44,710

 

1010
00:34:44,710 --> 00:34:47,829

 

1011
00:34:47,829 --> 00:34:49,190

 

1012
00:34:49,190 --> 00:34:51,349

 

1013
00:34:51,349 --> 00:34:51,359

 

1014
00:34:51,359 --> 00:34:52,710


1015
00:34:52,710 --> 00:34:54,790

 

1016
00:34:54,790 --> 00:34:56,790

 

1017
00:34:56,790 --> 00:34:56,800

 

1018
00:34:56,800 --> 00:34:57,510


1019
00:34:57,510 --> 00:35:00,150

 

1020
00:35:00,150 --> 00:35:01,190

 

1021
00:35:01,190 --> 00:35:03,190

 

1022
00:35:03,190 --> 00:35:04,310

 

1023
00:35:04,310 --> 00:35:07,670

 

1024
00:35:07,670 --> 00:35:11,510

 

1025
00:35:11,510 --> 00:35:13,109

 

1026
00:35:13,109 --> 00:35:13,119

 

1027
00:35:13,119 --> 00:35:15,349


1028
00:35:15,349 --> 00:35:16,710

 

1029
00:35:16,710 --> 00:35:18,550

 

1030
00:35:18,550 --> 00:35:21,750

 

1031
00:35:21,750 --> 00:35:25,829

 

1032
00:35:25,829 --> 00:35:29,990

 

1033
00:35:29,990 --> 00:35:32,470

 

1034
00:35:32,470 --> 00:35:34,390

 

1035
00:35:34,390 --> 00:35:34,400

 

1036
00:35:34,400 --> 00:35:37,430


1037
00:35:37,430 --> 00:35:40,470

 

1038
00:35:40,470 --> 00:35:42,310

 

1039
00:35:42,310 --> 00:35:43,910

 

1040
00:35:43,910 --> 00:35:45,750

 

1041
00:35:45,750 --> 00:35:47,190

 

1042
00:35:47,190 --> 00:35:50,230

 

1043
00:35:50,230 --> 00:35:53,910

 

1044
00:35:53,910 --> 00:35:55,270

 

1045
00:35:55,270 --> 00:35:56,710

 

1046
00:35:56,710 --> 00:35:58,310

 

1047
00:35:58,310 --> 00:35:59,270

 

1048
00:35:59,270 --> 00:36:02,390

 

1049
00:36:02,390 --> 00:36:04,310

 

1050
00:36:04,310 --> 00:36:05,910

 

1051
00:36:05,910 --> 00:36:08,310

 

1052
00:36:08,310 --> 00:36:09,030

 

1053
00:36:09,030 --> 00:36:12,069

 

1054
00:36:12,069 --> 00:36:13,670

 

1055
00:36:13,670 --> 00:36:15,430

 

1056
00:36:15,430 --> 00:36:16,950

 

1057
00:36:16,950 --> 00:36:18,310

 

1058
00:36:18,310 --> 00:36:19,750

 

1059
00:36:19,750 --> 00:36:19,760

 

1060
00:36:19,760 --> 00:36:20,630


1061
00:36:20,630 --> 00:36:22,310

 

1062
00:36:22,310 --> 00:36:23,829

 

1063
00:36:23,829 --> 00:36:25,270

 

1064
00:36:25,270 --> 00:36:27,109

 

1065
00:36:27,109 --> 00:36:29,990

 

1066
00:36:29,990 --> 00:36:32,950

 

1067
00:36:32,950 --> 00:36:34,069

 

1068
00:36:34,069 --> 00:36:36,630

 

1069
00:36:36,630 --> 00:36:38,390

 

1070
00:36:38,390 --> 00:36:41,030

 

1071
00:36:41,030 --> 00:36:42,790

 

1072
00:36:42,790 --> 00:36:45,190

 

1073
00:36:45,190 --> 00:36:45,200

 

1074
00:36:45,200 --> 00:36:46,150


1075
00:36:46,150 --> 00:36:48,470

 

1076
00:36:48,470 --> 00:36:49,829

 

1077
00:36:49,829 --> 00:36:52,230

 

1078
00:36:52,230 --> 00:36:56,870

 

1079
00:36:56,870 --> 00:37:01,190

 

1080
00:37:01,190 --> 00:37:03,430

 

1081
00:37:03,430 --> 00:37:05,030

 

1082
00:37:05,030 --> 00:37:08,630

 

1083
00:37:08,630 --> 00:37:11,910

 

1084
00:37:11,910 --> 00:37:15,109

 

1085
00:37:15,109 --> 00:37:17,030

 

1086
00:37:17,030 --> 00:37:18,470

 

1087
00:37:18,470 --> 00:37:20,390

 

1088
00:37:20,390 --> 00:37:21,910

 

1089
00:37:21,910 --> 00:37:24,870

 

1090
00:37:24,870 --> 00:37:27,109

 

1091
00:37:27,109 --> 00:37:29,190

 

1092
00:37:29,190 --> 00:37:30,550

 

1093
00:37:30,550 --> 00:37:31,910

 

1094
00:37:31,910 --> 00:37:34,550

 

1095
00:37:34,550 --> 00:37:37,349

 

1096
00:37:37,349 --> 00:37:38,790

 

1097
00:37:38,790 --> 00:37:41,430

 

1098
00:37:41,430 --> 00:37:42,950

 

1099
00:37:42,950 --> 00:37:42,960

 

1100
00:37:42,960 --> 00:37:43,670


1101
00:37:43,670 --> 00:37:47,430

 

1102
00:37:47,430 --> 00:38:02,230

 

1103
00:38:02,230 --> 00:38:05,589

 

1104
00:38:05,589 --> 00:38:08,790

 

1105
00:38:08,790 --> 00:38:12,310

 

1106
00:38:12,310 --> 00:38:12,320

 

1107
00:38:12,320 --> 00:38:13,349


1108
00:38:13,349 --> 00:38:15,990

 

1109
00:38:15,990 --> 00:38:16,470

 

1110
00:38:16,470 --> 00:38:19,430

 

1111
00:38:19,430 --> 00:38:21,190

 

1112
00:38:21,190 --> 00:38:22,950

 

1113
00:38:22,950 --> 00:38:24,630

 

1114
00:38:24,630 --> 00:38:26,390

 

1115
00:38:26,390 --> 00:38:28,470

 

1116
00:38:28,470 --> 00:38:30,069

 

1117
00:38:30,069 --> 00:38:31,589

 

1118
00:38:31,589 --> 00:38:32,470

 

1119
00:38:32,470 --> 00:38:33,829

 

1120
00:38:33,829 --> 00:38:35,510

 

1121
00:38:35,510 --> 00:38:36,790

 

1122
00:38:36,790 --> 00:38:39,750

 

1123
00:38:39,750 --> 00:38:42,150

 

1124
00:38:42,150 --> 00:38:45,670

 

1125
00:38:45,670 --> 00:38:47,589

 

1126
00:38:47,589 --> 00:38:50,310

 

1127
00:38:50,310 --> 00:38:52,310

 

1128
00:38:52,310 --> 00:38:53,750

 

1129
00:38:53,750 --> 00:38:56,230

 

1130
00:38:56,230 --> 00:38:58,470

 

1131
00:38:58,470 --> 00:39:02,230

 

1132
00:39:02,230 --> 00:39:04,230

 

1133
00:39:04,230 --> 00:39:05,430

 

1134
00:39:05,430 --> 00:39:06,550

 

1135
00:39:06,550 --> 00:39:08,230

 

1136
00:39:08,230 --> 00:39:09,990

 

1137
00:39:09,990 --> 00:39:11,349

 

1138
00:39:11,349 --> 00:39:13,030

 

1139
00:39:13,030 --> 00:39:15,430

 

1140
00:39:15,430 --> 00:39:17,510

 

1141
00:39:17,510 --> 00:39:17,520

 

1142
00:39:17,520 --> 00:39:20,150


1143
00:39:20,150 --> 00:39:22,069

 

1144
00:39:22,069 --> 00:39:23,910

 

1145
00:39:23,910 --> 00:39:27,190

 

1146
00:39:27,190 --> 00:39:30,630

 

1147
00:39:30,630 --> 00:39:31,510

 

1148
00:39:31,510 --> 00:39:34,790

 

1149
00:39:34,790 --> 00:39:36,150

 

1150
00:39:36,150 --> 00:39:39,030

 

1151
00:39:39,030 --> 00:39:39,589

 

1152
00:39:39,589 --> 00:39:42,470

 

1153
00:39:42,470 --> 00:39:44,470

 

1154
00:39:44,470 --> 00:39:44,480

 

1155
00:39:44,480 --> 00:39:45,030


1156
00:39:45,030 --> 00:39:46,390

 

1157
00:39:46,390 --> 00:39:48,230

 

1158
00:39:48,230 --> 00:39:50,150

 

1159
00:39:50,150 --> 00:39:51,829

 

1160
00:39:51,829 --> 00:39:54,069

 

1161
00:39:54,069 --> 00:39:56,230

 

1162
00:39:56,230 --> 00:39:58,470

 

1163
00:39:58,470 --> 00:39:59,990

 

1164
00:39:59,990 --> 00:40:01,910

 

1165
00:40:01,910 --> 00:40:04,470

 

1166
00:40:04,470 --> 00:40:06,630

 

1167
00:40:06,630 --> 00:40:08,710

 

1168
00:40:08,710 --> 00:40:10,630

 

1169
00:40:10,630 --> 00:40:12,630

 

1170
00:40:12,630 --> 00:40:15,670

 

1171
00:40:15,670 --> 00:40:17,829

 

1172
00:40:17,829 --> 00:40:17,839

 

1173
00:40:17,839 --> 00:40:19,270


1174
00:40:19,270 --> 00:40:20,950

 

1175
00:40:20,950 --> 00:40:22,309

 

1176
00:40:22,309 --> 00:40:24,069

 

1177
00:40:24,069 --> 00:40:27,270

 

1178
00:40:27,270 --> 00:40:29,270

 

1179
00:40:29,270 --> 00:40:30,390

 

1180
00:40:30,390 --> 00:40:31,829

 

1181
00:40:31,829 --> 00:40:34,630

 

1182
00:40:34,630 --> 00:40:36,550

 

1183
00:40:36,550 --> 00:40:38,069

 

1184
00:40:38,069 --> 00:40:40,069

 

1185
00:40:40,069 --> 00:40:40,710

 

1186
00:40:40,710 --> 00:40:43,349

 

1187
00:40:43,349 --> 00:40:45,430

 

1188
00:40:45,430 --> 00:40:47,109

 

1189
00:40:47,109 --> 00:40:48,630

 

1190
00:40:48,630 --> 00:40:49,750

 

1191
00:40:49,750 --> 00:40:51,990

 

1192
00:40:51,990 --> 00:40:55,589

 

1193
00:40:55,589 --> 00:40:57,430

 

1194
00:40:57,430 --> 00:40:59,750

 

1195
00:40:59,750 --> 00:41:02,230

 

1196
00:41:02,230 --> 00:41:04,390

 

1197
00:41:04,390 --> 00:41:06,069

 

1198
00:41:06,069 --> 00:41:07,829

 

1199
00:41:07,829 --> 00:41:07,839

 

1200
00:41:07,839 --> 00:41:08,470


1201
00:41:08,470 --> 00:41:10,150

 

1202
00:41:10,150 --> 00:41:13,030

 

1203
00:41:13,030 --> 00:41:14,790

 

1204
00:41:14,790 --> 00:41:16,309

 

1205
00:41:16,309 --> 00:41:19,670

 

1206
00:41:19,670 --> 00:41:22,069

 

1207
00:41:22,069 --> 00:41:22,079

 

1208
00:41:22,079 --> 00:41:23,109


1209
00:41:23,109 --> 00:41:24,550

 

1210
00:41:24,550 --> 00:41:26,870

 

1211
00:41:26,870 --> 00:41:28,390

 

1212
00:41:28,390 --> 00:41:31,829

 

1213
00:41:31,829 --> 00:41:33,829

 

1214
00:41:33,829 --> 00:41:36,230

 

1215
00:41:36,230 --> 00:41:39,109

 

1216
00:41:39,109 --> 00:41:40,710

 

1217
00:41:40,710 --> 00:41:42,069

 

1218
00:41:42,069 --> 00:41:44,710

 

1219
00:41:44,710 --> 00:41:46,230

 

1220
00:41:46,230 --> 00:41:48,470

 

1221
00:41:48,470 --> 00:41:50,069

 

1222
00:41:50,069 --> 00:41:52,790

 

1223
00:41:52,790 --> 00:41:54,550

 

1224
00:41:54,550 --> 00:41:56,390

 

1225
00:41:56,390 --> 00:41:58,150

 

1226
00:41:58,150 --> 00:41:59,670

 

1227
00:41:59,670 --> 00:42:00,390

 

1228
00:42:00,390 --> 00:42:02,230

 

1229
00:42:02,230 --> 00:42:03,829

 

1230
00:42:03,829 --> 00:42:05,430

 

1231
00:42:05,430 --> 00:42:08,630

 

1232
00:42:08,630 --> 00:42:10,230

 

1233
00:42:10,230 --> 00:42:12,470

 

1234
00:42:12,470 --> 00:42:14,470

 

1235
00:42:14,470 --> 00:42:16,309

 

1236
00:42:16,309 --> 00:42:19,829

 

1237
00:42:19,829 --> 00:42:22,950

 

1238
00:42:22,950 --> 00:42:24,950

 

1239
00:42:24,950 --> 00:42:27,430

 

1240
00:42:27,430 --> 00:42:28,470

 

1241
00:42:28,470 --> 00:42:30,710

 

1242
00:42:30,710 --> 00:42:31,829

 

1243
00:42:31,829 --> 00:42:33,430

 

1244
00:42:33,430 --> 00:42:34,710

 

1245
00:42:34,710 --> 00:42:36,150

 

1246
00:42:36,150 --> 00:42:36,160

 

1247
00:42:36,160 --> 00:42:37,190


1248
00:42:37,190 --> 00:42:39,270

 

1249
00:42:39,270 --> 00:42:42,230

 

1250
00:42:42,230 --> 00:42:43,990

 

1251
00:42:43,990 --> 00:42:45,589

 

1252
00:42:45,589 --> 00:42:49,430

 

1253
00:42:49,430 --> 00:42:53,190

 

1254
00:42:53,190 --> 00:42:56,829

 

1255
00:42:56,829 --> 00:42:56,839

 

1256
00:42:56,839 --> 00:43:00,230


1257
00:43:00,230 --> 00:43:02,870

 

1258
00:43:02,870 --> 00:43:02,880

 

1259
00:43:02,880 --> 00:43:04,470


1260
00:43:04,470 --> 00:43:07,030

 

1261
00:43:07,030 --> 00:43:09,510

 

1262
00:43:09,510 --> 00:43:11,510

 

1263
00:43:11,510 --> 00:43:12,710

 

1264
00:43:12,710 --> 00:43:15,190

 

1265
00:43:15,190 --> 00:43:17,109

 

1266
00:43:17,109 --> 00:43:18,870

 

1267
00:43:18,870 --> 00:43:20,309

 

1268
00:43:20,309 --> 00:43:22,710

 

1269
00:43:22,710 --> 00:43:26,069

 

1270
00:43:26,069 --> 00:43:29,349

 

1271
00:43:29,349 --> 00:43:32,069

 

1272
00:43:32,069 --> 00:43:32,079

 

1273
00:43:32,079 --> 00:43:35,510


1274
00:43:35,510 --> 00:43:38,230

 

1275
00:43:38,230 --> 00:43:39,670

 

1276
00:43:39,670 --> 00:43:42,069

 

1277
00:43:42,069 --> 00:43:44,309

 

1278
00:43:44,309 --> 00:43:45,910

 

1279
00:43:45,910 --> 00:43:48,630

 

1280
00:43:48,630 --> 00:43:49,750

 

1281
00:43:49,750 --> 00:43:51,990

 

1282
00:43:51,990 --> 00:43:53,109

 

1283
00:43:53,109 --> 00:43:55,430

 

1284
00:43:55,430 --> 00:43:56,870

 

1285
00:43:56,870 --> 00:43:56,880

 

1286
00:43:56,880 --> 00:43:58,150


1287
00:43:58,150 --> 00:43:59,829

 

1288
00:43:59,829 --> 00:44:01,349

 

1289
00:44:01,349 --> 00:44:03,510

 

1290
00:44:03,510 --> 00:44:07,109

 

1291
00:44:07,109 --> 00:44:08,870

 

1292
00:44:08,870 --> 00:44:10,950

 

1293
00:44:10,950 --> 00:44:12,870

 

1294
00:44:12,870 --> 00:44:15,109

 

1295
00:44:15,109 --> 00:44:15,119

 

1296
00:44:15,119 --> 00:44:15,589


1297
00:44:15,589 --> 00:44:17,349

 

1298
00:44:17,349 --> 00:44:19,910

 

1299
00:44:19,910 --> 00:44:21,910

 

1300
00:44:21,910 --> 00:44:23,270

 

1301
00:44:23,270 --> 00:44:25,270

 

1302
00:44:25,270 --> 00:44:27,349

 

1303
00:44:27,349 --> 00:44:29,190

 

1304
00:44:29,190 --> 00:44:31,670

 

1305
00:44:31,670 --> 00:44:35,349

 

1306
00:44:35,349 --> 00:44:37,990

 

1307
00:44:37,990 --> 00:44:38,000

 

1308
00:44:38,000 --> 00:44:38,870


1309
00:44:38,870 --> 00:44:42,790

 

1310
00:44:42,790 --> 00:44:43,990

 

1311
00:44:43,990 --> 00:44:46,550

 

1312
00:44:46,550 --> 00:44:46,560

 

1313
00:44:46,560 --> 00:44:47,430


1314
00:44:47,430 --> 00:44:49,270

 

1315
00:44:49,270 --> 00:44:49,280

 

1316
00:44:49,280 --> 00:44:50,630


1317
00:44:50,630 --> 00:44:52,150

 

1318
00:44:52,150 --> 00:44:53,589

 

1319
00:44:53,589 --> 00:44:53,599

 

1320
00:44:53,599 --> 00:44:53,910


1321
00:44:53,910 --> 00:44:56,150

 

1322
00:44:56,150 --> 00:44:57,670

 

1323
00:44:57,670 --> 00:44:59,589

 

1324
00:44:59,589 --> 00:45:01,349

 

1325
00:45:01,349 --> 00:45:05,430

 

1326
00:45:05,430 --> 00:45:06,470

 

1327
00:45:06,470 --> 00:45:08,390

 

1328
00:45:08,390 --> 00:45:08,400

 

1329
00:45:08,400 --> 00:45:08,870


1330
00:45:08,870 --> 00:45:10,790

 

1331
00:45:10,790 --> 00:45:12,950

 

1332
00:45:12,950 --> 00:45:14,550

 

1333
00:45:14,550 --> 00:45:15,990

 

1334
00:45:15,990 --> 00:45:18,550

 

1335
00:45:18,550 --> 00:45:20,150

 

1336
00:45:20,150 --> 00:45:22,470

 

1337
00:45:22,470 --> 00:45:22,480

 

1338
00:45:22,480 --> 00:45:23,589


1339
00:45:23,589 --> 00:45:25,349

 

1340
00:45:25,349 --> 00:45:28,630

 

1341
00:45:28,630 --> 00:45:30,230

 

1342
00:45:30,230 --> 00:45:31,990

 

1343
00:45:31,990 --> 00:45:34,870

 

1344
00:45:34,870 --> 00:45:35,829

 

1345
00:45:35,829 --> 00:45:38,950

 

1346
00:45:38,950 --> 00:45:41,430

 

1347
00:45:41,430 --> 00:45:41,440

 

1348
00:45:41,440 --> 00:45:42,550


1349
00:45:42,550 --> 00:45:45,589

 

1350
00:45:45,589 --> 00:45:47,030

 

1351
00:45:47,030 --> 00:45:47,910

 

1352
00:45:47,910 --> 00:45:49,510

 

1353
00:45:49,510 --> 00:45:52,230

 

1354
00:45:52,230 --> 00:45:54,150

 

1355
00:45:54,150 --> 00:45:55,910

 

1356
00:45:55,910 --> 00:45:59,190

 

1357
00:45:59,190 --> 00:46:01,190

 

1358
00:46:01,190 --> 00:46:03,270

 

1359
00:46:03,270 --> 00:46:05,589

 

1360
00:46:05,589 --> 00:46:07,109

 

1361
00:46:07,109 --> 00:46:12,710

 

1362
00:46:12,710 --> 00:46:15,349

 

1363
00:46:15,349 --> 00:46:15,359

 

1364
00:46:15,359 --> 00:46:16,790


1365
00:46:16,790 --> 00:46:19,030

 

1366
00:46:19,030 --> 00:46:19,040

 

1367
00:46:19,040 --> 00:46:20,069


1368
00:46:20,069 --> 00:46:22,069

 

1369
00:46:22,069 --> 00:46:24,069

 

1370
00:46:24,069 --> 00:46:26,150

 

1371
00:46:26,150 --> 00:46:27,829

 

1372
00:46:27,829 --> 00:46:29,349

 

1373
00:46:29,349 --> 00:46:30,630

 

1374
00:46:30,630 --> 00:46:37,670

 

1375
00:46:37,670 --> 00:46:39,589

 

1376
00:46:39,589 --> 00:46:41,190

 

1377
00:46:41,190 --> 00:46:42,230

 

1378
00:46:42,230 --> 00:46:46,390

 

1379
00:46:46,390 --> 00:46:49,589

 

1380
00:46:49,589 --> 00:46:50,710

 

1381
00:46:50,710 --> 00:46:52,950

 

1382
00:46:52,950 --> 00:46:55,589

 

1383
00:46:55,589 --> 00:46:57,030

 

1384
00:46:57,030 --> 00:47:00,550

 

1385
00:47:00,550 --> 00:47:03,510

 

1386
00:47:03,510 --> 00:47:05,270

 

1387
00:47:05,270 --> 00:47:07,109

 

1388
00:47:07,109 --> 00:47:09,270

 

1389
00:47:09,270 --> 00:47:11,190

 

1390
00:47:11,190 --> 00:47:13,109

 

1391
00:47:13,109 --> 00:47:14,710

 

1392
00:47:14,710 --> 00:47:14,720

 

1393
00:47:14,720 --> 00:47:16,550


1394
00:47:16,550 --> 00:47:20,069

 

1395
00:47:20,069 --> 00:47:21,990

 

1396
00:47:21,990 --> 00:47:23,670

 

1397
00:47:23,670 --> 00:47:26,069

 

1398
00:47:26,069 --> 00:47:27,270

 

1399
00:47:27,270 --> 00:47:30,309

 

1400
00:47:30,309 --> 00:47:32,870

 

1401
00:47:32,870 --> 00:47:34,230

 

1402
00:47:34,230 --> 00:47:35,670

 

1403
00:47:35,670 --> 00:47:38,309

 

1404
00:47:38,309 --> 00:47:39,829

 

1405
00:47:39,829 --> 00:47:40,630

 

1406
00:47:40,630 --> 00:47:42,710

 

1407
00:47:42,710 --> 00:47:44,230

 

1408
00:47:44,230 --> 00:47:45,589

 

1409
00:47:45,589 --> 00:47:49,910

 

1410
00:47:49,910 --> 00:47:52,790

 

1411
00:47:52,790 --> 00:47:54,630

 

1412
00:47:54,630 --> 00:47:57,349

 

1413
00:47:57,349 --> 00:47:58,630

 

1414
00:47:58,630 --> 00:48:00,150

 

1415
00:48:00,150 --> 00:48:03,109

 

1416
00:48:03,109 --> 00:48:07,270

 

1417
00:48:07,270 --> 00:48:09,190

 

1418
00:48:09,190 --> 00:48:12,150

 

1419
00:48:12,150 --> 00:48:15,430

 

1420
00:48:15,430 --> 00:48:16,710

 

1421
00:48:16,710 --> 00:48:18,230

 

1422
00:48:18,230 --> 00:48:20,309

 

1423
00:48:20,309 --> 00:48:21,670

 

1424
00:48:21,670 --> 00:48:24,870

 

1425
00:48:24,870 --> 00:48:24,880

 

1426
00:48:24,880 --> 00:48:25,750


1427
00:48:25,750 --> 00:48:28,950

 

1428
00:48:28,950 --> 00:48:28,960

 

1429
00:48:28,960 --> 00:48:29,349


1430
00:48:29,349 --> 00:48:31,589

 

1431
00:48:31,589 --> 00:48:33,270

 

1432
00:48:33,270 --> 00:48:36,790

 

1433
00:48:36,790 --> 00:48:39,270

 

1434
00:48:39,270 --> 00:48:40,630

 

1435
00:48:40,630 --> 00:48:41,750

 

1436
00:48:41,750 --> 00:48:44,470

 

1437
00:48:44,470 --> 00:48:45,990

 

1438
00:48:45,990 --> 00:48:47,349

 

1439
00:48:47,349 --> 00:48:48,790

 

1440
00:48:48,790 --> 00:48:50,470

 

1441
00:48:50,470 --> 00:48:52,069

 

1442
00:48:52,069 --> 00:48:54,390

 

1443
00:48:54,390 --> 00:48:54,400

 

1444
00:48:54,400 --> 00:48:56,470


1445
00:48:56,470 --> 00:48:59,510

 

1446
00:48:59,510 --> 00:48:59,520

 

1447
00:48:59,520 --> 00:49:00,790


1448
00:49:00,790 --> 00:49:04,309

 

1449
00:49:04,309 --> 00:49:05,829

 

1450
00:49:05,829 --> 00:49:09,990

 

1451
00:49:09,990 --> 00:49:11,829

 

1452
00:49:11,829 --> 00:49:14,069

 

1453
00:49:14,069 --> 00:49:40,549

 

1454
00:49:40,549 --> 00:49:41,990

 

1455
00:49:41,990 --> 00:49:43,750

 

1456
00:49:43,750 --> 00:49:45,750

 

1457
00:49:45,750 --> 00:49:47,190

 

1458
00:49:47,190 --> 00:49:48,470

 

1459
00:49:48,470 --> 00:49:50,790

 

1460
00:49:50,790 --> 00:49:52,150

 

1461
00:49:52,150 --> 00:49:53,589

 

1462
00:49:53,589 --> 00:49:55,589

 

1463
00:49:55,589 --> 00:49:57,030

 

1464
00:49:57,030 --> 00:50:01,430

 

1465
00:50:01,430 --> 00:50:04,549

 

1466
00:50:04,549 --> 00:50:08,390

 

1467
00:50:08,390 --> 00:50:12,150

 

1468
00:50:12,150 --> 00:50:14,790

 

1469
00:50:14,790 --> 00:50:17,270

 

1470
00:50:17,270 --> 00:50:22,230

 

1471
00:50:22,230 --> 00:50:24,150

 

1472
00:50:24,150 --> 00:50:26,790

 

1473
00:50:26,790 --> 00:50:30,069

 

1474
00:50:30,069 --> 00:50:30,079

 

1475
00:50:30,079 --> 00:50:31,109


1476
00:50:31,109 --> 00:50:38,309

 

1477
00:50:38,309 --> 00:50:39,670

 

1478
00:50:39,670 --> 00:50:42,069

 

1479
00:50:42,069 --> 00:50:42,630

 

1480
00:50:42,630 --> 00:50:43,910

 

1481
00:50:43,910 --> 00:50:45,750

 

1482
00:50:45,750 --> 00:50:49,190

 

1483
00:50:49,190 --> 00:50:51,109

 

1484
00:50:51,109 --> 00:50:53,510

 

1485
00:50:53,510 --> 00:50:55,109

 

1486
00:50:55,109 --> 00:50:57,270

 

1487
00:50:57,270 --> 00:50:59,109

 

1488
00:50:59,109 --> 00:51:00,230

 

1489
00:51:00,230 --> 00:51:01,990

 

1490
00:51:01,990 --> 00:51:03,750

 

1491
00:51:03,750 --> 00:51:05,910

 

1492
00:51:05,910 --> 00:51:09,109

 

1493
00:51:09,109 --> 00:51:10,870

 

1494
00:51:10,870 --> 00:51:14,230

 

1495
00:51:14,230 --> 00:51:16,870

 

1496
00:51:16,870 --> 00:51:16,880

 

1497
00:51:16,880 --> 00:51:17,270


1498
00:51:17,270 --> 00:51:19,270

 

1499
00:51:19,270 --> 00:51:20,870

 

1500
00:51:20,870 --> 00:51:24,230

 

1501
00:51:24,230 --> 00:51:25,430

 

1502
00:51:25,430 --> 00:51:27,270

 

1503
00:51:27,270 --> 00:51:30,549

 

1504
00:51:30,549 --> 00:51:30,559

 

1505
00:51:30,559 --> 00:51:31,349


1506
00:51:31,349 --> 00:51:32,630

 

1507
00:51:32,630 --> 00:51:35,510

 

1508
00:51:35,510 --> 00:51:35,520

 

1509
00:51:35,520 --> 00:51:36,230


1510
00:51:36,230 --> 00:51:40,230

 

1511
00:51:40,230 --> 00:51:44,549

 

1512
00:51:44,549 --> 00:51:46,069

 

1513
00:51:46,069 --> 00:51:47,829

 

1514
00:51:47,829 --> 00:51:56,870

 

1515
00:51:56,870 --> 00:51:58,870

 

1516
00:51:58,870 --> 00:52:01,190

 

1517
00:52:01,190 --> 00:52:05,589

 

1518
00:52:05,589 --> 00:52:07,030

 

1519
00:52:07,030 --> 00:52:08,950

 

1520
00:52:08,950 --> 00:52:09,990

 

1521
00:52:09,990 --> 00:52:13,270

 

1522
00:52:13,270 --> 00:52:14,549

 

1523
00:52:14,549 --> 00:52:18,230

 

1524
00:52:18,230 --> 00:52:22,230

 

1525
00:52:22,230 --> 00:52:24,950

 

1526
00:52:24,950 --> 00:52:27,190

 

1527
00:52:27,190 --> 00:52:27,200

 

1528
00:52:27,200 --> 00:52:28,069


1529
00:52:28,069 --> 00:52:30,309

 

1530
00:52:30,309 --> 00:52:32,069

 

1531
00:52:32,069 --> 00:52:33,510

 

1532
00:52:33,510 --> 00:52:35,670

 

1533
00:52:35,670 --> 00:52:36,790

 

1534
00:52:36,790 --> 00:52:39,349

 

1535
00:52:39,349 --> 00:52:41,510

 

1536
00:52:41,510 --> 00:52:43,190

 

1537
00:52:43,190 --> 00:52:45,990

 

1538
00:52:45,990 --> 00:52:48,470

 

1539
00:52:48,470 --> 00:52:49,750

 

1540
00:52:49,750 --> 00:52:49,760

 

1541
00:52:49,760 --> 00:52:51,430


1542
00:52:51,430 --> 00:52:53,910

 

1543
00:52:53,910 --> 00:52:53,920

 

1544
00:52:53,920 --> 00:52:56,710


1545
00:52:56,710 --> 00:52:59,829

 

1546
00:52:59,829 --> 00:53:00,950

 

1547
00:53:00,950 --> 00:53:05,109

 

1548
00:53:05,109 --> 00:53:07,670

 

1549
00:53:07,670 --> 00:53:10,470

 

1550
00:53:10,470 --> 00:53:12,549

 

1551
00:53:12,549 --> 00:53:14,790

 

1552
00:53:14,790 --> 00:53:19,109

 

1553
00:53:19,109 --> 00:53:22,470

 

1554
00:53:22,470 --> 00:53:25,750

 

1555
00:53:25,750 --> 00:53:26,870

 

1556
00:53:26,870 --> 00:53:28,069

 

1557
00:53:28,069 --> 00:53:30,470

 

1558
00:53:30,470 --> 00:53:31,670

 

1559
00:53:31,670 --> 00:53:33,990

 

1560
00:53:33,990 --> 00:53:35,829

 

1561
00:53:35,829 --> 00:53:39,270

 

1562
00:53:39,270 --> 00:53:41,109

 

1563
00:53:41,109 --> 00:53:43,109

 

1564
00:53:43,109 --> 00:53:46,150

 

1565
00:53:46,150 --> 00:53:48,549

 

1566
00:53:48,549 --> 00:53:49,589

 

1567
00:53:49,589 --> 00:53:52,069

 

1568
00:53:52,069 --> 00:53:52,079

 

1569
00:53:52,079 --> 00:53:53,670


1570
00:53:53,670 --> 00:53:57,030

 

1571
00:53:57,030 --> 00:53:58,870

 

1572
00:53:58,870 --> 00:53:59,430

 

1573
00:53:59,430 --> 00:54:01,510

 

1574
00:54:01,510 --> 00:54:03,030

 

1575
00:54:03,030 --> 00:54:05,430

 

1576
00:54:05,430 --> 00:54:06,870

 

1577
00:54:06,870 --> 00:54:12,470

 

1578
00:54:12,470 --> 00:54:14,470

 

1579
00:54:14,470 --> 00:54:16,309

 

1580
00:54:16,309 --> 00:54:16,319

 

1581
00:54:16,319 --> 00:54:16,710


1582
00:54:16,710 --> 00:54:18,470

 

1583
00:54:18,470 --> 00:54:20,309

 

1584
00:54:20,309 --> 00:54:23,190

 

1585
00:54:23,190 --> 00:54:23,990

 

1586
00:54:23,990 --> 00:54:25,990

 

1587
00:54:25,990 --> 00:54:27,910

 

1588
00:54:27,910 --> 00:54:29,349

 

1589
00:54:29,349 --> 00:54:32,390

 

1590
00:54:32,390 --> 00:54:34,950

 

1591
00:54:34,950 --> 00:54:36,870

 

1592
00:54:36,870 --> 00:54:38,309

 

1593
00:54:38,309 --> 00:54:40,390

 

1594
00:54:40,390 --> 00:54:42,630

 

1595
00:54:42,630 --> 00:54:44,309

 

1596
00:54:44,309 --> 00:54:44,319

 

1597
00:54:44,319 --> 00:54:45,829


1598
00:54:45,829 --> 00:54:47,750

 

1599
00:54:47,750 --> 00:54:49,270

 

1600
00:54:49,270 --> 00:54:50,549

 

1601
00:54:50,549 --> 00:54:51,990

 

1602
00:54:51,990 --> 00:54:52,000

 

1603
00:54:52,000 --> 00:54:52,870


1604
00:54:52,870 --> 00:54:55,270

 

1605
00:54:55,270 --> 00:54:57,510

 

1606
00:54:57,510 --> 00:55:01,270

 

1607
00:55:01,270 --> 00:55:03,109

 

1608
00:55:03,109 --> 00:55:04,549

 

1609
00:55:04,549 --> 00:55:05,990

 

1610
00:55:05,990 --> 00:55:06,630

 

1611
00:55:06,630 --> 00:55:08,710

 

1612
00:55:08,710 --> 00:55:12,789

 

1613
00:55:12,789 --> 00:55:14,309

 

1614
00:55:14,309 --> 00:55:15,349

 

1615
00:55:15,349 --> 00:55:19,109

 

1616
00:55:19,109 --> 00:55:28,390

 

1617
00:55:28,390 --> 00:55:31,829

 

1618
00:55:31,829 --> 00:55:38,789

 

1619
00:55:38,789 --> 00:55:43,030

 

1620
00:55:43,030 --> 00:55:46,789

 

1621
00:55:46,789 --> 00:55:48,390

 

1622
00:55:48,390 --> 00:55:51,990

 

1623
00:55:51,990 --> 00:55:53,270

 

1624
00:55:53,270 --> 00:55:56,230

 

1625
00:55:56,230 --> 00:55:56,240

 

1626
00:55:56,240 --> 00:55:57,190


1627
00:55:57,190 --> 00:55:59,750

 

1628
00:55:59,750 --> 00:56:00,950

 

1629
00:56:00,950 --> 00:56:02,630

 

1630
00:56:02,630 --> 00:56:07,109

 

1631
00:56:07,109 --> 00:56:21,829

 

1632
00:56:21,829 --> 00:56:28,829

 

1633
00:56:28,829 --> 00:56:32,309

 

1634
00:56:32,309 --> 00:56:35,109

 

1635
00:56:35,109 --> 00:56:36,710

 

1636
00:56:36,710 --> 00:56:38,150

 

1637
00:56:38,150 --> 00:56:39,589

 

1638
00:56:39,589 --> 00:56:40,069

 

1639
00:56:40,069 --> 00:56:41,829

 

1640
00:56:41,829 --> 00:56:43,589

 

1641
00:56:43,589 --> 00:56:43,599

 

1642
00:56:43,599 --> 00:56:47,829


1643
00:56:47,829 --> 00:56:50,150

 

1644
00:56:50,150 --> 00:56:52,470

 

1645
00:56:52,470 --> 00:56:57,510

 

1646
00:56:57,510 --> 00:57:00,069

 

1647
00:57:00,069 --> 00:57:02,150

 

1648
00:57:02,150 --> 00:57:04,230

 

1649
00:57:04,230 --> 00:57:04,240

 

1650
00:57:04,240 --> 00:57:05,430


1651
00:57:05,430 --> 00:57:09,190

 

1652
00:57:09,190 --> 00:57:10,950

 

1653
00:57:10,950 --> 00:57:12,549

 

1654
00:57:12,549 --> 00:57:15,829

 

1655
00:57:15,829 --> 00:57:15,839

 

1656
00:57:15,839 --> 00:57:18,950


1657
00:57:18,950 --> 00:57:24,630

 

1658
00:57:24,630 --> 00:57:30,950

 

1659
00:57:30,950 --> 00:57:33,589

 

1660
00:57:33,589 --> 00:57:33,599

 

1661
00:57:33,599 --> 00:57:34,549


1662
00:57:34,549 --> 00:57:36,950

 

1663
00:57:36,950 --> 00:57:37,910

 

1664
00:57:37,910 --> 00:57:42,309

 

1665
00:57:42,309 --> 00:57:45,670

 

1666
00:57:45,670 --> 00:57:48,470

 

1667
00:57:48,470 --> 00:57:48,480

 

1668
00:57:48,480 --> 00:57:48,870


1669
00:57:48,870 --> 00:57:51,589

 

1670
00:57:51,589 --> 00:57:53,030

 

1671
00:57:53,030 --> 00:57:56,789

 

1672
00:57:56,789 --> 00:57:58,309

 

1673
00:57:58,309 --> 00:57:59,990

 

1674
00:57:59,990 --> 00:58:03,510

 

1675
00:58:03,510 --> 00:58:05,670

 

1676
00:58:05,670 --> 00:58:07,430

 

1677
00:58:07,430 --> 00:58:10,470

 

1678
00:58:10,470 --> 00:58:12,390

 

1679
00:58:12,390 --> 00:58:12,400

 

1680
00:58:12,400 --> 00:58:15,190


1681
00:58:15,190 --> 00:58:20,789

 

1682
00:58:20,789 --> 00:58:27,030

 

1683
00:58:27,030 --> 00:58:29,190

 

1684
00:58:29,190 --> 00:58:30,789

 

1685
00:58:30,789 --> 00:58:32,230

 

1686
00:58:32,230 --> 00:58:35,349

 

1687
00:58:35,349 --> 00:58:36,789

 

1688
00:58:36,789 --> 00:58:38,789

 

1689
00:58:38,789 --> 00:58:40,069

 

1690
00:58:40,069 --> 00:58:41,589

 

1691
00:58:41,589 --> 00:58:43,109

 

1692
00:58:43,109 --> 00:58:45,349

 

1693
00:58:45,349 --> 00:58:47,910

 

1694
00:58:47,910 --> 00:58:49,349

 

1695
00:58:49,349 --> 00:58:50,950

 

1696
00:58:50,950 --> 00:58:52,789

 

1697
00:58:52,789 --> 00:58:54,309

 

1698
00:58:54,309 --> 00:58:56,069

 

1699
00:58:56,069 --> 00:58:57,510

 

1700
00:58:57,510 --> 00:58:59,270

 

1701
00:58:59,270 --> 00:59:00,549

 

1702
00:59:00,549 --> 00:59:05,910

 

1703
00:59:05,910 --> 00:59:09,430

 

1704
00:59:09,430 --> 00:59:13,030

 

1705
00:59:13,030 --> 00:59:15,349

 

1706
00:59:15,349 --> 00:59:19,030

 

1707
00:59:19,030 --> 00:59:26,549

 

1708
00:59:26,549 --> 00:59:30,789

 

1709
00:59:30,789 --> 00:59:33,270

 

1710
00:59:33,270 --> 00:59:37,910

 

1711
00:59:37,910 --> 00:59:42,309

 

1712
00:59:42,309 --> 00:59:45,829

 

1713
00:59:45,829 --> 00:59:47,349

 

1714
00:59:47,349 --> 00:59:49,030

 

1715
00:59:49,030 --> 00:59:52,069

 

1716
00:59:52,069 --> 00:59:54,950

 

1717
00:59:54,950 --> 00:59:54,960

 

1718
00:59:54,960 --> 00:59:55,349


1719
00:59:55,349 --> 00:59:58,390

 

1720
00:59:58,390 --> 01:00:03,829

 

1721
01:00:03,829 --> 01:00:07,109

 

1722
01:00:07,109 --> 01:00:09,030

 

1723
01:00:09,030 --> 01:00:10,789

 

1724
01:00:10,789 --> 01:00:12,710

 

1725
01:00:12,710 --> 01:00:17,829

 

1726
01:00:17,829 --> 01:00:20,950

 

1727
01:00:20,950 --> 01:00:24,630

 

1728
01:00:24,630 --> 01:00:24,640

 

1729
01:00:24,640 --> 01:00:26,069


1730
01:00:26,069 --> 01:00:28,950

 

1731
01:00:28,950 --> 01:00:28,960

 

1732
01:00:28,960 --> 01:00:29,349


1733
01:00:29,349 --> 01:00:33,910

 

1734
01:00:33,910 --> 01:00:37,190

 

1735
01:00:37,190 --> 01:00:38,870

 

1736
01:00:38,870 --> 01:00:41,829

 

1737
01:00:41,829 --> 01:00:43,990

 

1738
01:00:43,990 --> 01:00:44,000

 

1739
01:00:44,000 --> 01:00:44,630


1740
01:00:44,630 --> 01:00:46,230

 

1741
01:00:46,230 --> 01:00:46,240

 

1742
01:00:46,240 --> 01:00:50,309


1743
01:00:50,309 --> 01:00:53,990

 

1744
01:00:53,990 --> 01:00:55,349

 

1745
01:00:55,349 --> 01:00:57,190

 

1746
01:00:57,190 --> 01:00:58,630

 

1747
01:00:58,630 --> 01:01:00,390

 

1748
01:01:00,390 --> 01:01:02,710

 

1749
01:01:02,710 --> 01:01:04,309

 

1750
01:01:04,309 --> 01:01:04,319

 

1751
01:01:04,319 --> 01:01:05,270


1752
01:01:05,270 --> 01:01:07,190

 

1753
01:01:07,190 --> 01:01:08,390

 

1754
01:01:08,390 --> 01:01:09,990

 

1755
01:01:09,990 --> 01:01:11,670

 

1756
01:01:11,670 --> 01:01:11,680

 

1757
01:01:11,680 --> 01:01:12,870


1758
01:01:12,870 --> 01:01:16,710

 

1759
01:01:16,710 --> 01:01:22,150

 

1760
01:01:22,150 --> 01:01:24,710

 

1761
01:01:24,710 --> 01:01:24,720

 

1762
01:01:24,720 --> 01:01:26,230


1763
01:01:26,230 --> 01:01:31,109

 

1764
01:01:31,109 --> 01:01:34,150

 

1765
01:01:34,150 --> 01:01:41,190

 

1766
01:01:41,190 --> 01:01:44,549

 

1767
01:01:44,549 --> 01:01:45,910

 

1768
01:01:45,910 --> 01:01:48,150

 

1769
01:01:48,150 --> 01:01:50,230

 

1770
01:01:50,230 --> 01:01:51,589

 

1771
01:01:51,589 --> 01:01:51,990

 

1772
01:01:51,990 --> 01:01:54,710

 

1773
01:01:54,710 --> 01:01:56,470

 

1774
01:01:56,470 --> 01:01:56,480

 

1775
01:01:56,480 --> 01:01:57,910


1776
01:01:57,910 --> 01:02:01,109

 

1777
01:02:01,109 --> 01:02:02,549

 

1778
01:02:02,549 --> 01:02:06,230

 

1779
01:02:06,230 --> 01:02:08,470

 

1780
01:02:08,470 --> 01:02:10,710

 

1781
01:02:10,710 --> 01:02:12,309

 

1782
01:02:12,309 --> 01:02:13,750

 

1783
01:02:13,750 --> 01:02:15,829

 

1784
01:02:15,829 --> 01:02:17,029

 

1785
01:02:17,029 --> 01:02:19,589

 

1786
01:02:19,589 --> 01:02:22,309

 

1787
01:02:22,309 --> 01:02:25,349

 

1788
01:02:25,349 --> 01:02:27,430

 

1789
01:02:27,430 --> 01:02:28,549

 

1790
01:02:28,549 --> 01:02:30,390

 

1791
01:02:30,390 --> 01:02:30,400

 

1792
01:02:30,400 --> 01:02:33,270


1793
01:02:33,270 --> 01:02:35,829

 

1794
01:02:35,829 --> 01:02:37,750

 

1795
01:02:37,750 --> 01:02:40,470

 

1796
01:02:40,470 --> 01:02:46,829

 

1797
01:02:46,829 --> 01:02:46,839

 

1798
01:02:46,839 --> 01:02:48,390


1799
01:02:48,390 --> 01:02:50,230

 

1800
01:02:50,230 --> 01:02:52,150

 

1801
01:02:52,150 --> 01:02:54,230

 

1802
01:02:54,230 --> 01:02:57,270

 

1803
01:02:57,270 --> 01:03:00,309

 

1804
01:03:00,309 --> 01:03:02,630

 

1805
01:03:02,630 --> 01:03:04,230

 

1806
01:03:04,230 --> 01:03:06,230

 

1807
01:03:06,230 --> 01:03:10,390

 

1808
01:03:10,390 --> 01:03:12,309

 

1809
01:03:12,309 --> 01:03:14,150

 

1810
01:03:14,150 --> 01:03:17,589

 

1811
01:03:17,589 --> 01:03:19,670

 

1812
01:03:19,670 --> 01:03:22,390

 

1813
01:03:22,390 --> 01:03:25,589

 

1814
01:03:25,589 --> 01:03:29,029

 

1815
01:03:29,029 --> 01:03:31,990

 

1816
01:03:31,990 --> 01:03:32,630

 

1817
01:03:32,630 --> 01:03:33,829

 

1818
01:03:33,829 --> 01:03:35,270

 

1819
01:03:35,270 --> 01:03:36,390

 

1820
01:03:36,390 --> 01:03:37,990

 

1821
01:03:37,990 --> 01:03:41,910

 

1822
01:03:41,910 --> 01:03:44,630

 

1823
01:03:44,630 --> 01:03:45,510

 

1824
01:03:45,510 --> 01:03:49,430

 

1825
01:03:49,430 --> 01:03:51,589

 

1826
01:03:51,589 --> 01:03:53,750

 

1827
01:03:53,750 --> 01:03:56,710

 

1828
01:03:56,710 --> 01:03:57,510

 

1829
01:03:57,510 --> 01:03:59,990

 

1830
01:03:59,990 --> 01:04:01,349

 

1831
01:04:01,349 --> 01:04:04,230

 

1832
01:04:04,230 --> 01:04:05,029

 

1833
01:04:05,029 --> 01:04:07,109

 

1834
01:04:07,109 --> 01:04:09,349

 

1835
01:04:09,349 --> 01:04:11,029

 

1836
01:04:11,029 --> 01:04:12,870

 

1837
01:04:12,870 --> 01:04:13,829

 

1838
01:04:13,829 --> 01:04:15,349

 

1839
01:04:15,349 --> 01:04:18,390

 

1840
01:04:18,390 --> 01:04:21,029

 

1841
01:04:21,029 --> 01:04:22,390

 

1842
01:04:22,390 --> 01:04:24,710

 

1843
01:04:24,710 --> 01:04:52,150

 

1844
01:04:52,150 --> 01:04:54,390

 

1845
01:04:54,390 --> 01:04:56,069

 

1846
01:04:56,069 --> 01:04:57,430

 

1847
01:04:57,430 --> 01:04:59,750

 

1848
01:04:59,750 --> 01:05:01,589

 

1849
01:05:01,589 --> 01:05:03,270

 

1850
01:05:03,270 --> 01:05:05,990

 

1851
01:05:05,990 --> 01:05:06,789

 

1852
01:05:06,789 --> 01:05:10,230

 

1853
01:05:10,230 --> 01:05:13,910

 

1854
01:05:13,910 --> 01:05:17,109

 

1855
01:05:17,109 --> 01:05:21,270

 

1856
01:05:21,270 --> 01:05:24,230

 

1857
01:05:24,230 --> 01:05:25,430

 

1858
01:05:25,430 --> 01:05:28,150

 

1859
01:05:28,150 --> 01:05:28,160

 

1860
01:05:28,160 --> 01:05:29,829


1861
01:05:29,829 --> 01:05:34,710

 

1862
01:05:34,710 --> 01:05:38,150

 

1863
01:05:38,150 --> 01:05:41,349

 

1864
01:05:41,349 --> 01:05:43,190

 

1865
01:05:43,190 --> 01:05:43,200

 

1866
01:05:43,200 --> 01:05:44,789


1867
01:05:44,789 --> 01:05:46,950

 

1868
01:05:46,950 --> 01:05:48,950

 

1869
01:05:48,950 --> 01:05:50,470

 

1870
01:05:50,470 --> 01:05:51,910

 

1871
01:05:51,910 --> 01:05:53,109

 

1872
01:05:53,109 --> 01:05:54,789

 

1873
01:05:54,789 --> 01:05:57,510

 

1874
01:05:57,510 --> 01:05:59,029

 

1875
01:05:59,029 --> 01:06:01,190

 

1876
01:06:01,190 --> 01:06:03,270

 

1877
01:06:03,270 --> 01:06:04,390

 

1878
01:06:04,390 --> 01:06:06,549

 

1879
01:06:06,549 --> 01:06:08,870

 

1880
01:06:08,870 --> 01:06:10,789

 

1881
01:06:10,789 --> 01:06:13,029

 

1882
01:06:13,029 --> 01:06:16,710

 

1883
01:06:16,710 --> 01:06:18,950

 

1884
01:06:18,950 --> 01:06:21,510

 

1885
01:06:21,510 --> 01:06:23,510

 

1886
01:06:23,510 --> 01:06:25,589

 

1887
01:06:25,589 --> 01:06:27,589

 

1888
01:06:27,589 --> 01:06:29,510

 

1889
01:06:29,510 --> 01:06:32,390

 

1890
01:06:32,390 --> 01:06:33,029

 

1891
01:06:33,029 --> 01:06:37,109

 

1892
01:06:37,109 --> 01:06:39,109

 

1893
01:06:39,109 --> 01:06:41,829

 

1894
01:06:41,829 --> 01:06:43,349

 

1895
01:06:43,349 --> 01:06:45,029

 

1896
01:06:45,029 --> 01:06:45,990

 

1897
01:06:45,990 --> 01:06:49,029

 

1898
01:06:49,029 --> 01:06:53,270

 

1899
01:06:53,270 --> 01:06:55,510

 

1900
01:06:55,510 --> 01:06:57,670

 

1901
01:06:57,670 --> 01:06:59,670

 

1902
01:06:59,670 --> 01:07:01,349

 

1903
01:07:01,349 --> 01:07:03,349

 

1904
01:07:03,349 --> 01:07:05,750

 

1905
01:07:05,750 --> 01:07:05,760

 

1906
01:07:05,760 --> 01:07:06,870


1907
01:07:06,870 --> 01:07:10,710

 

1908
01:07:10,710 --> 01:07:14,470

 

1909
01:07:14,470 --> 01:07:17,910

 

1910
01:07:17,910 --> 01:07:19,510

 

1911
01:07:19,510 --> 01:07:22,470

 

1912
01:07:22,470 --> 01:07:25,349

 

1913
01:07:25,349 --> 01:07:27,029

 

1914
01:07:27,029 --> 01:07:27,589

 

1915
01:07:27,589 --> 01:07:28,950

 

1916
01:07:28,950 --> 01:07:31,910

 

1917
01:07:31,910 --> 01:07:34,230

 

1918
01:07:34,230 --> 01:07:37,829

 

1919
01:07:37,829 --> 01:07:40,309

 

1920
01:07:40,309 --> 01:07:40,319

 

1921
01:07:40,319 --> 01:07:40,950


1922
01:07:40,950 --> 01:07:42,230

 

1923
01:07:42,230 --> 01:07:44,710

 

1924
01:07:44,710 --> 01:07:45,990

 

1925
01:07:45,990 --> 01:07:52,710

 

1926
01:07:52,710 --> 01:07:56,230

 

1927
01:07:56,230 --> 01:07:56,950

 

1928
01:07:56,950 --> 01:08:00,230

 

1929
01:08:00,230 --> 01:08:00,240

 

1930
01:08:00,240 --> 01:08:00,789


1931
01:08:00,789 --> 01:08:03,829

 

1932
01:08:03,829 --> 01:08:07,750

 

1933
01:08:07,750 --> 01:08:09,670

 

1934
01:08:09,670 --> 01:08:11,589

 

1935
01:08:11,589 --> 01:08:13,829

 

1936
01:08:13,829 --> 01:08:15,510

 

1937
01:08:15,510 --> 01:08:18,070

 

1938
01:08:18,070 --> 01:08:21,349

 

1939
01:08:21,349 --> 01:08:23,110

 

1940
01:08:23,110 --> 01:08:25,110

 

1941
01:08:25,110 --> 01:08:28,229

 

1942
01:08:28,229 --> 01:08:31,749

 

1943
01:08:31,749 --> 01:08:35,030

 

1944
01:08:35,030 --> 01:08:38,709

 

1945
01:08:38,709 --> 01:08:40,709

 

1946
01:08:40,709 --> 01:08:43,669

 

1947
01:08:43,669 --> 01:08:45,749

 

1948
01:08:45,749 --> 01:08:47,189

 

1949
01:08:47,189 --> 01:08:49,910

 

1950
01:08:49,910 --> 01:08:50,950

 

1951
01:08:50,950 --> 01:08:53,430

 

1952
01:08:53,430 --> 01:08:54,870

 

1953
01:08:54,870 --> 01:08:54,880

 

1954
01:08:54,880 --> 01:08:55,269


1955
01:08:55,269 --> 01:08:56,789

 

1956
01:08:56,789 --> 01:08:58,630

 

1957
01:08:58,630 --> 01:09:01,910

 

1958
01:09:01,910 --> 01:09:03,910

 

1959
01:09:03,910 --> 01:09:05,189

 

1960
01:09:05,189 --> 01:09:08,870

 

1961
01:09:08,870 --> 01:09:10,309

 

1962
01:09:10,309 --> 01:09:12,149

 

1963
01:09:12,149 --> 01:09:13,749

 

1964
01:09:13,749 --> 01:09:17,269

 

1965
01:09:17,269 --> 01:09:19,590

 

1966
01:09:19,590 --> 01:09:23,269

 

1967
01:09:23,269 --> 01:09:25,510

 

1968
01:09:25,510 --> 01:09:27,430

 

1969
01:09:27,430 --> 01:09:30,950

 

1970
01:09:30,950 --> 01:09:34,070

 

1971
01:09:34,070 --> 01:09:35,829

 

1972
01:09:35,829 --> 01:09:37,829

 

1973
01:09:37,829 --> 01:09:40,870

 

1974
01:09:40,870 --> 01:09:44,070

 

1975
01:09:44,070 --> 01:09:45,829

 

1976
01:09:45,829 --> 01:09:51,189

 

1977
01:09:51,189 --> 01:09:59,510

 

1978
01:09:59,510 --> 01:10:04,070

 

1979
01:10:04,070 --> 01:10:08,070

 

1980
01:10:08,070 --> 01:10:10,950

 

1981
01:10:10,950 --> 01:10:13,110

 

1982
01:10:13,110 --> 01:10:14,870

 

1983
01:10:14,870 --> 01:10:16,630

 

1984
01:10:16,630 --> 01:10:19,270

 

1985
01:10:19,270 --> 01:10:21,189

 

1986
01:10:21,189 --> 01:10:22,229

 

1987
01:10:22,229 --> 01:10:22,239

 

1988
01:10:22,239 --> 01:10:24,390


1989
01:10:24,390 --> 01:10:26,950

 

1990
01:10:26,950 --> 01:10:28,070

 

1991
01:10:28,070 --> 01:10:29,590

 

1992
01:10:29,590 --> 01:10:30,790

 

1993
01:10:30,790 --> 01:10:30,800

 

1994
01:10:30,800 --> 01:10:31,110


1995
01:10:31,110 --> 01:10:32,870

 

1996
01:10:32,870 --> 01:10:34,229

 

1997
01:10:34,229 --> 01:10:35,189

 

1998
01:10:35,189 --> 01:10:35,199

 

1999
01:10:35,199 --> 01:10:36,550


2000
01:10:36,550 --> 01:10:38,229

 

2001
01:10:38,229 --> 01:10:40,070

 

2002
01:10:40,070 --> 01:10:41,669

 

2003
01:10:41,669 --> 01:10:41,679

 

2004
01:10:41,679 --> 01:10:43,430


2005
01:10:43,430 --> 01:10:45,270

 

2006
01:10:45,270 --> 01:10:47,110

 

2007
01:10:47,110 --> 01:10:49,189

 

2008
01:10:49,189 --> 01:10:51,110

 

2009
01:10:51,110 --> 01:10:52,470

 

2010
01:10:52,470 --> 01:10:54,550

 

2011
01:10:54,550 --> 01:10:54,560

 

2012
01:10:54,560 --> 01:10:55,990


2013
01:10:55,990 --> 01:10:59,270

 

2014
01:10:59,270 --> 01:11:02,790

 

2015
01:11:02,790 --> 01:11:06,870

 

2016
01:11:06,870 --> 01:11:09,350

 

2017
01:11:09,350 --> 01:11:10,950

 

2018
01:11:10,950 --> 01:11:15,510

 

2019
01:11:15,510 --> 01:11:19,590

 

2020
01:11:19,590 --> 01:11:21,270

 

2021
01:11:21,270 --> 01:11:25,430

 

2022
01:11:25,430 --> 01:11:28,630

 

2023
01:11:28,630 --> 01:11:30,470

 

2024
01:11:30,470 --> 01:11:32,070

 

2025
01:11:32,070 --> 01:11:36,390

 

2026
01:11:36,390 --> 01:11:40,790

 

2027
01:11:40,790 --> 01:11:43,750

 

2028
01:11:43,750 --> 01:11:45,750

 

2029
01:11:45,750 --> 01:11:45,760

 

2030
01:11:45,760 --> 01:11:46,550


2031
01:11:46,550 --> 01:11:48,950

 

2032
01:11:48,950 --> 01:11:51,189

 

2033
01:11:51,189 --> 01:11:55,110

 

2034
01:11:55,110 --> 01:11:58,310

 

2035
01:11:58,310 --> 01:11:59,669

 

2036
01:11:59,669 --> 01:12:03,189

 

2037
01:12:03,189 --> 01:12:07,669

 

2038
01:12:07,669 --> 01:12:10,070

 

2039
01:12:10,070 --> 01:12:11,350

 

2040
01:12:11,350 --> 01:12:14,950

 

2041
01:12:14,950 --> 01:12:19,910

 

2042
01:12:19,910 --> 01:12:22,470

 

2043
01:12:22,470 --> 01:12:24,630

 

2044
01:12:24,630 --> 01:12:26,470

 

2045
01:12:26,470 --> 01:12:28,070

 

2046
01:12:28,070 --> 01:12:28,950

 

2047
01:12:28,950 --> 01:12:30,310

 

2048
01:12:30,310 --> 01:12:32,390

 

2049
01:12:32,390 --> 01:12:33,510

 

2050
01:12:33,510 --> 01:12:36,470

 

2051
01:12:36,470 --> 01:12:40,870

 

2052
01:12:40,870 --> 01:12:44,709

 

2053
01:12:44,709 --> 01:12:47,510

 

2054
01:12:47,510 --> 01:12:49,430

 

2055
01:12:49,430 --> 01:12:51,030

 

2056
01:12:51,030 --> 01:12:51,669

 

2057
01:12:51,669 --> 01:12:53,830

 

2058
01:12:53,830 --> 01:12:55,350

 

2059
01:12:55,350 --> 01:12:55,360

 

2060
01:12:55,360 --> 01:12:55,830


2061
01:12:55,830 --> 01:12:57,189

 

2062
01:12:57,189 --> 01:12:59,270

 

2063
01:12:59,270 --> 01:13:00,950

 

2064
01:13:00,950 --> 01:13:02,950

 

2065
01:13:02,950 --> 01:13:04,229

 

2066
01:13:04,229 --> 01:13:07,350

 

2067
01:13:07,350 --> 01:13:07,360

 

2068
01:13:07,360 --> 01:13:10,790


2069
01:13:10,790 --> 01:13:14,149

 

2070
01:13:14,149 --> 01:13:19,030

 

2071
01:13:19,030 --> 01:13:21,590

 

2072
01:13:21,590 --> 01:13:22,630

 

2073
01:13:22,630 --> 01:13:24,709

 

2074
01:13:24,709 --> 01:13:27,350

 

2075
01:13:27,350 --> 01:13:30,709

 

2076
01:13:30,709 --> 01:13:33,990

 

2077
01:13:33,990 --> 01:13:38,630

 

2078
01:13:38,630 --> 01:13:41,350

 

2079
01:13:41,350 --> 01:13:42,630

 

2080
01:13:42,630 --> 01:13:44,229

 

2081
01:13:44,229 --> 01:13:46,310

 

2082
01:13:46,310 --> 01:13:49,669

 

2083
01:13:49,669 --> 01:13:51,750

 

2084
01:13:51,750 --> 01:13:52,950

 

2085
01:13:52,950 --> 01:13:55,590

 

2086
01:13:55,590 --> 01:13:59,510

 

2087
01:13:59,510 --> 01:14:01,030

 

2088
01:14:01,030 --> 01:14:07,990

 

2089
01:14:07,990 --> 01:14:10,070

 

2090
01:14:10,070 --> 01:14:12,229

 

2091
01:14:12,229 --> 01:14:15,830

 

2092
01:14:15,830 --> 01:14:16,149

 

2093
01:14:16,149 --> 01:14:17,270

 

2094
01:14:17,270 --> 01:14:19,430

 

2095
01:14:19,430 --> 01:14:21,110

 

2096
01:14:21,110 --> 01:14:21,120

 

2097
01:14:21,120 --> 01:14:22,790


2098
01:14:22,790 --> 01:14:24,870

 

2099
01:14:24,870 --> 01:14:26,870

 

2100
01:14:26,870 --> 01:14:30,390

 

2101
01:14:30,390 --> 01:14:32,790

 

2102
01:14:32,790 --> 01:14:32,800

 

2103
01:14:32,800 --> 01:14:34,149


2104
01:14:34,149 --> 01:14:37,350

 

2105
01:14:37,350 --> 01:14:38,790

 

2106
01:14:38,790 --> 01:14:41,510

 

2107
01:14:41,510 --> 01:14:43,669

 

2108
01:14:43,669 --> 01:14:45,910

 

2109
01:14:45,910 --> 01:14:47,270

 

2110
01:14:47,270 --> 01:14:51,110

 

2111
01:14:51,110 --> 01:14:51,669

 

2112
01:14:51,669 --> 01:14:54,310

 

2113
01:14:54,310 --> 01:14:56,070

 

2114
01:14:56,070 --> 01:14:58,950

 

2115
01:14:58,950 --> 01:14:59,750

 

2116
01:14:59,750 --> 01:15:02,070

 

2117
01:15:02,070 --> 01:15:03,189

 

2118
01:15:03,189 --> 01:15:05,189

 

2119
01:15:05,189 --> 01:15:06,550

 

2120
01:15:06,550 --> 01:15:08,390

 

2121
01:15:08,390 --> 01:15:09,669

 

2122
01:15:09,669 --> 01:15:12,149

 

2123
01:15:12,149 --> 01:15:13,270

 

2124
01:15:13,270 --> 01:15:16,709

 

2125
01:15:16,709 --> 01:15:18,149

 

2126
01:15:18,149 --> 01:15:19,669

 

2127
01:15:19,669 --> 01:15:21,669

 

2128
01:15:21,669 --> 01:15:23,110

 

2129
01:15:23,110 --> 01:15:25,030

 

2130
01:15:25,030 --> 01:15:25,040

 

2131
01:15:25,040 --> 01:15:26,229


2132
01:15:26,229 --> 01:15:29,750

 

2133
01:15:29,750 --> 01:15:32,149

 

2134
01:15:32,149 --> 01:15:33,510

 

2135
01:15:33,510 --> 01:15:37,030

 

2136
01:15:37,030 --> 01:15:40,149

 

2137
01:15:40,149 --> 01:15:42,229

 

2138
01:15:42,229 --> 01:15:44,070

 

2139
01:15:44,070 --> 01:15:45,590

 

2140
01:15:45,590 --> 01:15:47,669

 

2141
01:15:47,669 --> 01:15:49,430

 

2142
01:15:49,430 --> 01:15:53,189

 

2143
01:15:53,189 --> 01:15:57,189

 

2144
01:15:57,189 --> 01:15:59,910

 

2145
01:15:59,910 --> 01:16:01,189

 

2146
01:16:01,189 --> 01:16:03,590

 

2147
01:16:03,590 --> 01:16:05,270

 

2148
01:16:05,270 --> 01:16:05,280

 

2149
01:16:05,280 --> 01:16:06,870


2150
01:16:06,870 --> 01:16:09,350

 

2151
01:16:09,350 --> 01:16:10,709

 

2152
01:16:10,709 --> 01:16:12,310

 

2153
01:16:12,310 --> 01:16:13,590

 

2154
01:16:13,590 --> 01:16:14,790

 

2155
01:16:14,790 --> 01:16:17,350

 

2156
01:16:17,350 --> 01:16:18,950

 

2157
01:16:18,950 --> 01:16:20,470

 

2158
01:16:20,470 --> 01:16:20,480

 

2159
01:16:20,480 --> 01:16:21,189


2160
01:16:21,189 --> 01:16:22,950

 

2161
01:16:22,950 --> 01:16:24,790

 

2162
01:16:24,790 --> 01:16:25,830

 

2163
01:16:25,830 --> 01:16:27,910

 

2164
01:16:27,910 --> 01:16:27,920

 

2165
01:16:27,920 --> 01:16:29,270


2166
01:16:29,270 --> 01:16:31,430

 

2167
01:16:31,430 --> 01:16:32,390

 

2168
01:16:32,390 --> 01:16:33,910

 

2169
01:16:33,910 --> 01:16:35,750

 

2170
01:16:35,750 --> 01:16:38,470

 

2171
01:16:38,470 --> 01:16:39,189

 

2172
01:16:39,189 --> 01:16:43,189

 

2173
01:16:43,189 --> 01:16:49,669

 

2174
01:16:49,669 --> 01:16:51,110

 

2175
01:16:51,110 --> 01:16:52,870

 

2176
01:16:52,870 --> 01:16:55,990

 

2177
01:16:55,990 --> 01:16:59,030

 

2178
01:16:59,030 --> 01:17:01,350

 

2179
01:17:01,350 --> 01:17:03,110

 

2180
01:17:03,110 --> 01:17:05,669

 

2181
01:17:05,669 --> 01:17:07,430

 

2182
01:17:07,430 --> 01:17:09,990

 

2183
01:17:09,990 --> 01:17:11,510

 

2184
01:17:11,510 --> 01:17:13,189

 

2185
01:17:13,189 --> 01:17:16,790

 

2186
01:17:16,790 --> 01:17:18,790

 

2187
01:17:18,790 --> 01:17:20,310

 

2188
01:17:20,310 --> 01:17:25,510

 

2189
01:17:25,510 --> 01:17:28,229

 

2190
01:17:28,229 --> 01:17:29,669

 

2191
01:17:29,669 --> 01:17:33,189

 

2192
01:17:33,189 --> 01:17:34,550

 

2193
01:17:34,550 --> 01:17:36,950

 

2194
01:17:36,950 --> 01:17:39,189

 

2195
01:17:39,189 --> 01:17:40,070

 

2196
01:17:40,070 --> 01:17:42,470

 

2197
01:17:42,470 --> 01:17:43,350

 

2198
01:17:43,350 --> 01:17:47,430

 

2199
01:17:47,430 --> 01:17:51,270

 

2200
01:17:51,270 --> 01:17:53,590

 

2201
01:17:53,590 --> 01:17:56,709

 

2202
01:17:56,709 --> 01:17:58,870

 

2203
01:17:58,870 --> 01:18:01,830

 

2204
01:18:01,830 --> 01:18:03,350

 

2205
01:18:03,350 --> 01:18:05,669

 

2206
01:18:05,669 --> 01:18:08,870

 

2207
01:18:08,870 --> 01:18:10,790

 

2208
01:18:10,790 --> 01:18:12,070

 

2209
01:18:12,070 --> 01:18:12,080

 

2210
01:18:12,080 --> 01:18:13,270


2211
01:18:13,270 --> 01:18:15,510

 

2212
01:18:15,510 --> 01:18:17,189

 

2213
01:18:17,189 --> 01:18:18,709

 

2214
01:18:18,709 --> 01:18:20,630

 

2215
01:18:20,630 --> 01:18:21,830

 

2216
01:18:21,830 --> 01:18:23,590

 

2217
01:18:23,590 --> 01:18:24,870

 

2218
01:18:24,870 --> 01:18:26,149

 

2219
01:18:26,149 --> 01:18:28,550

 

2220
01:18:28,550 --> 01:18:28,560

 

2221
01:18:28,560 --> 01:18:29,910


2222
01:18:29,910 --> 01:18:32,070

 

2223
01:18:32,070 --> 01:18:32,080

 

2224
01:18:32,080 --> 01:18:33,350


2225
01:18:33,350 --> 01:18:36,310

 

2226
01:18:36,310 --> 01:18:38,630

 

2227
01:18:38,630 --> 01:18:40,470

 

2228
01:18:40,470 --> 01:18:42,310

 

2229
01:18:42,310 --> 01:18:45,110

 

2230
01:18:45,110 --> 01:18:46,229

 

2231
01:18:46,229 --> 01:18:50,310

 

2232
01:18:50,310 --> 01:18:52,310

 

2233
01:18:52,310 --> 01:18:54,390

 

2234
01:18:54,390 --> 01:18:55,910

 

2235
01:18:55,910 --> 01:18:58,790

 

2236
01:18:58,790 --> 01:18:58,800

 

2237
01:18:58,800 --> 01:18:59,590


2238
01:18:59,590 --> 01:19:02,550

 

2239
01:19:02,550 --> 01:19:03,910

 

2240
01:19:03,910 --> 01:19:05,750

 

2241
01:19:05,750 --> 01:19:09,840