WEBVTT 00:00:00.070 --> 00:00:02.500 The following content is provided under a Creative 00:00:02.500 --> 00:00:04.019 Commons license. 00:00:04.019 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.730 continue to offer high-quality educational resources for free. 00:00:10.730 --> 00:00:13.340 To make a donation or view additional materials 00:00:13.340 --> 00:00:17.217 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.217 --> 00:00:17.842 at ocw.mit.edu. 00:00:20.152 --> 00:00:22.360 PROFESSOR: All right, so today what we're going to do 00:00:22.360 --> 00:00:25.170 is we're just start with a short review problem 00:00:25.170 --> 00:00:27.070 on rugged landscapes, just so that you 00:00:27.070 --> 00:00:28.770 get some sense of the kind of thing 00:00:28.770 --> 00:00:32.330 that I would expect you to be able to do a week from today. 00:00:32.330 --> 00:00:36.680 And then we'll get into the core topic of the class, which 00:00:36.680 --> 00:00:39.150 is evolutionary game theory. 00:00:39.150 --> 00:00:41.180 And we'll discuss why it is that you 00:00:41.180 --> 00:00:43.519 don't need to invoke any notion of rationality, which 00:00:43.519 --> 00:00:45.560 is kind of the traditional thing we do when we're 00:00:45.560 --> 00:00:47.685 talking about game theory applied to human decision 00:00:47.685 --> 00:00:48.820 making. 00:00:48.820 --> 00:00:50.806 Then we'll try to understand this difference 00:00:50.806 --> 00:00:53.180 to know what a Nash equilibrium is in the context of game 00:00:53.180 --> 00:00:56.120 theory versus an evolutionary stable strategy 00:00:56.120 --> 00:00:57.924 in this context. 00:00:57.924 --> 00:01:00.340 And we'll say something about the evolution of cooperation 00:01:00.340 --> 00:01:04.260 and experiments that one can do with microbial populations 00:01:04.260 --> 00:01:05.010 in the laboratory. 00:01:07.600 --> 00:01:11.045 Are there any questions before I get started? 00:01:19.312 --> 00:01:25.330 All right, so just on this question of evolutionary paths, 00:01:25.330 --> 00:01:27.780 on Tuesday we discussed the Weinreich paper 00:01:27.780 --> 00:01:31.510 where he talked about sort of different models 00:01:31.510 --> 00:01:34.350 that you might use to try to make estimates of the path 00:01:34.350 --> 00:01:36.480 that evolution might take on that fitness landscape 00:01:36.480 --> 00:01:37.930 that he measured. 00:01:37.930 --> 00:01:41.560 So he measured this MIC, the minimum inhibitory 00:01:41.560 --> 00:01:45.097 concentration, on all 2 to the 5 or 32 different states, 00:01:45.097 --> 00:01:47.305 and then tried to say something about the probability 00:01:47.305 --> 00:01:49.370 that different paths will be taken. 00:01:49.370 --> 00:01:51.870 So I just want to explore this question 00:01:51.870 --> 00:01:55.162 about paths in a simpler landscape, where 00:01:55.162 --> 00:01:57.620 by construction here, I'm to going to give you some fitness 00:01:57.620 --> 00:02:00.130 values just so that we can be clear about why 00:02:00.130 --> 00:02:02.194 it is that there might be different paths, 00:02:02.194 --> 00:02:03.610 or what determines the probability 00:02:03.610 --> 00:02:05.146 that different paths are taken. 00:02:05.146 --> 00:02:06.770 So what we want to do is assume that we 00:02:06.770 --> 00:02:12.445 are in a population that is experiencing this Moran process 00:02:12.445 --> 00:02:18.660 or Moran model, constant population size N equal to, 00:02:18.660 --> 00:02:21.780 in this case, we'll say 1,000. 00:02:21.780 --> 00:02:27.740 And let's say that the mutation rate is 10 to the minus 6. 00:02:27.740 --> 00:02:30.204 So each time that an individual divides, it has a 1 00:02:30.204 --> 00:02:32.610 in a million probability of mutating. 00:02:32.610 --> 00:02:36.785 And that's a per base pair mutation rate. 00:02:36.785 --> 00:02:39.494 And I'll show you what I mean by that. 00:02:39.494 --> 00:02:41.535 And in particular, we're going to have genotypes. 00:02:44.890 --> 00:02:46.360 Originally when we discussed this, 00:02:46.360 --> 00:02:48.845 we were talking about just mutations, maybe A's and B's. 00:02:48.845 --> 00:02:50.220 But now, what we're going to have 00:02:50.220 --> 00:02:54.920 is just a short genome that's string length 2. 00:02:54.920 --> 00:03:02.890 So we might have 0, 0, which has relative fitness 1, 0, 1. 00:03:14.880 --> 00:03:17.810 So we're assuming that this is relative fitness 00:03:17.810 --> 00:03:19.380 as compared to the 0, 0 state. 00:03:19.380 --> 00:03:25.350 We're going to start in the 0, 0 state with 1,000 00:03:25.350 --> 00:03:28.390 isogenic individuals, all 0, 0. 00:03:28.390 --> 00:03:32.020 And the question is, what's going to happen eventually? 00:03:34.660 --> 00:03:55.980 And in particular, what path will be taken on this landscape 00:03:55.980 --> 00:03:56.480 here? 00:04:03.372 --> 00:04:04.830 In particular, what we want to know 00:04:04.830 --> 00:04:07.000 is the probability of taking this path. 00:04:33.989 --> 00:04:36.030 You can start thinking about it while I write out 00:04:36.030 --> 00:04:38.630 some possibilities that we can vote for, 00:04:38.630 --> 00:04:40.626 and I'll give you a minute to think about it. 00:04:40.626 --> 00:04:45.090 So don't-- 00:05:19.350 --> 00:05:22.644 Are there any questions about what I'm trying to ask here? 00:05:22.644 --> 00:05:28.299 AUDIENCE: So this is the long time? 00:05:28.299 --> 00:05:30.480 So we assume that in the long time, 00:05:30.480 --> 00:05:32.880 it will go from 0, 0 to 1, 1? 00:05:32.880 --> 00:05:36.400 PROFESSOR: That's right, yes, so if we wait long enough, 00:05:36.400 --> 00:05:39.900 the population will get there, and the 1, 1 genotype 00:05:39.900 --> 00:05:42.210 will fix in the population. 00:05:42.210 --> 00:05:44.490 We can talk a bit later about how long it's 00:05:44.490 --> 00:05:46.432 going to take to get there, and so forth. 00:05:46.432 --> 00:05:48.892 AUDIENCE: And we're assuming that from 0, 1, 00:05:48.892 --> 00:05:52.342 it can't go back to 0, 0? 00:05:52.342 --> 00:05:53.050 PROFESSOR: Right. 00:05:53.050 --> 00:05:57.080 Yeah, so we'll discuss the situations 00:05:57.080 --> 00:05:58.480 when we have to worry about that, 00:05:58.480 --> 00:06:00.010 and when we don't, and so forth. 00:06:00.010 --> 00:06:02.320 But for now, if you'd like, we can 00:06:02.320 --> 00:06:05.400 say that this is even just mu sub b, the beneficial mutation 00:06:05.400 --> 00:06:09.200 rate per base pair, assuming that the 0's can only 00:06:09.200 --> 00:06:10.770 turn into 1s. 00:06:10.770 --> 00:06:13.130 Then after we think about this, we 00:06:13.130 --> 00:06:17.029 could figure out if that's important, 00:06:17.029 --> 00:06:18.570 or when it's important, and so forth. 00:06:23.130 --> 00:06:23.880 Yes. 00:06:23.880 --> 00:06:24.796 AUDIENCE: [INAUDIBLE]? 00:06:30.880 --> 00:06:31.540 PROFESSOR: No. 00:06:31.540 --> 00:06:34.720 All right, so we're starting with all 1,000 individuals 00:06:34.720 --> 00:06:36.770 being in the 0, 0 state, because now we're 00:06:36.770 --> 00:06:38.187 allowing some mutation rate. 00:06:38.187 --> 00:06:39.141 AUDIENCE: [INAUDIBLE]. 00:06:45.579 --> 00:06:47.120 PROFESSOR: And you also have to think 00:06:47.120 --> 00:06:51.370 about this first mutation-- will it fix or not? 00:06:55.128 --> 00:06:59.000 AUDIENCE: But it's not [INAUDIBLE] first mutation, 00:06:59.000 --> 00:07:01.904 the mutation of one element of that population 00:07:01.904 --> 00:07:06.141 will go to 0, 1 [INAUDIBLE]? 00:07:06.141 --> 00:07:08.390 PROFESSOR: I'm not sure if I understand your question. 00:07:08.390 --> 00:07:11.785 AUDIENCE: So if we start out with all 0, 0, and then one 00:07:11.785 --> 00:07:12.755 mutation [INAUDIBLE]? 00:07:16.150 --> 00:07:19.545 And if that mutation-- are we assuming 00:07:19.545 --> 00:07:22.470 that that mutation is 0, 1 and then figuring out [INAUDIBLE]? 00:07:22.470 --> 00:07:24.740 PROFESSOR: Well, OK, you're asking kind 00:07:24.740 --> 00:07:26.902 of what I mean by path here. 00:07:26.902 --> 00:07:28.230 AUDIENCE: Yeah, I guess. 00:07:28.230 --> 00:07:31.180 PROFESSOR: Yeah, all right, so I'll say path 00:07:31.180 --> 00:07:37.130 means that that this was the dominant probability 00:07:37.130 --> 00:07:40.180 trajectory of the population through there. 00:07:40.180 --> 00:07:43.140 We'll also discuss whether it somehow 00:07:43.140 --> 00:07:46.690 is very likely is going to kind of have to go through one 00:07:46.690 --> 00:07:48.930 or the other of them. 00:07:48.930 --> 00:07:52.450 The probability of getting both mutations in one generation 00:07:52.450 --> 00:07:55.230 is going to be 10 to the minus 12. 00:07:55.230 --> 00:08:00.040 So that's going to be a very rare thing, at least given 00:08:00.040 --> 00:08:02.157 these parameters, and so forth. 00:08:02.157 --> 00:08:03.990 And then there's another question, which is, 00:08:03.990 --> 00:08:07.780 will 0, 1 actually fix in the population 00:08:07.780 --> 00:08:12.247 before you later fix this population? 00:08:12.247 --> 00:08:14.580 And actually, I think the answers to all these questions 00:08:14.580 --> 00:08:16.163 are in principal already on the board. 00:08:20.330 --> 00:08:22.404 Because there's a question of do we 00:08:22.404 --> 00:08:24.070 have to worry about clonal interference? 00:08:24.070 --> 00:08:27.060 Are these things neutral or not? 00:08:27.060 --> 00:08:30.500 And really, this is in some ways a very simple problem. 00:08:30.500 --> 00:08:32.010 But in another way, you have to keep 00:08:32.010 --> 00:08:34.110 track of lots of different things, 00:08:34.110 --> 00:08:37.080 and which regime we're in and so forth. 00:08:37.080 --> 00:08:40.820 So that's what makes it such a wonderful exam problem. 00:08:40.820 --> 00:08:44.780 If you understand what's going on, 00:08:44.780 --> 00:08:46.477 you can answer it in a minute. 00:08:46.477 --> 00:08:48.310 But if you don't understand what's going on, 00:08:48.310 --> 00:08:49.590 it'll take you an hour. 00:08:52.010 --> 00:08:52.510 Yes? 00:08:52.510 --> 00:08:53.010 No? 00:08:53.010 --> 00:08:53.610 Maybe? 00:08:53.610 --> 00:08:57.269 Well, I'll give you another 20 seconds. 00:08:57.269 --> 00:08:58.935 Hopefully, you've been thinking about it 00:08:58.935 --> 00:08:59.976 while we've been talking. 00:09:19.305 --> 00:09:20.680 All right, do you need more time? 00:10:10.000 --> 00:10:11.470 Why don't we go ahead and vote? 00:10:11.470 --> 00:10:13.820 I think it's very likely that we will not 00:10:13.820 --> 00:10:15.889 be at the kind of 100% mark, in which case 00:10:15.889 --> 00:10:18.222 you'll have a chance to talk about it and think about it 00:10:18.222 --> 00:10:18.722 some more. 00:10:18.722 --> 00:10:19.370 Ready? 00:10:19.370 --> 00:10:21.460 Three, two, one. 00:10:25.476 --> 00:10:29.710 OK, all right, so we do have a fair range of answers. 00:10:29.710 --> 00:10:32.050 I'd say it might be kind of something like 50-50. 00:10:32.050 --> 00:10:33.090 And that's great. 00:10:33.090 --> 00:10:35.880 It means that there should be something to talk about. 00:10:35.880 --> 00:10:37.072 So turn to a neighbor. 00:10:37.072 --> 00:10:38.530 You should be able to find somebody 00:10:38.530 --> 00:10:40.180 that disagrees with you. 00:10:40.180 --> 00:10:44.080 And if everyone around you agrees, you can maybe-- 00:10:44.080 --> 00:10:46.690 all right, so there's a group of D's and a group of B's here, 00:10:46.690 --> 00:10:48.064 which means that everybody-- 00:10:48.064 --> 00:10:48.980 AUDIENCE: Let's fight. 00:10:48.980 --> 00:10:49.680 PROFESSOR: All right, so everybody 00:10:49.680 --> 00:10:50.950 thinks that everybody agrees with them, 00:10:50.950 --> 00:10:53.366 but you just need to look a little bit more long distance. 00:10:53.366 --> 00:10:55.270 So turn to a pseudo-neighbor. 00:10:55.270 --> 00:10:58.910 You should be able to find somebody there. 00:10:58.910 --> 00:11:02.695 It's roughly even here, so you should be able to find someone. 00:11:02.695 --> 00:11:06.118 [INTERPOSING VOICES] 00:11:46.329 --> 00:11:48.495 So I don't see much in the way of vibrant discussion 00:11:48.495 --> 00:11:49.280 and argument. 00:11:49.280 --> 00:11:52.882 You guys should be passionately defending your choice here. 00:11:52.882 --> 00:11:56.354 [INTERPOSING VOICES] 00:12:41.264 --> 00:12:42.680 Yeah, that's a higher order point. 00:12:42.680 --> 00:12:43.846 I wouldn't worry about that. 00:12:46.089 --> 00:12:46.922 [INTERPOSING VOICES] 00:12:59.350 --> 00:13:02.050 All right, it looks like people are having a nice discussion. 00:13:02.050 --> 00:13:04.350 But I might still go ahead and cut it short, 00:13:04.350 --> 00:13:07.842 just so that we can get on to evolutionary game theory. 00:13:07.842 --> 00:13:09.550 But I would like to see where people are. 00:13:09.550 --> 00:13:11.730 And we'll discuss it as a group, so don't 00:13:11.730 --> 00:13:14.960 be too disappointed if you don't get finished there. 00:13:14.960 --> 00:13:17.140 But I do want to see kind of where we are. 00:13:17.140 --> 00:13:18.150 Ready? 00:13:18.150 --> 00:13:19.970 Three, two, one. 00:13:23.748 --> 00:13:30.090 OK, so it still is, maybe, split roughly equally between D 00:13:30.090 --> 00:13:35.200 and maybe a B-ish and some C's. 00:13:35.200 --> 00:13:38.840 All right, does somebody want to volunteer their explanation? 00:13:44.250 --> 00:13:45.468 Yes. 00:13:45.468 --> 00:13:47.963 AUDIENCE: I'm not sure how good it is, 00:13:47.963 --> 00:13:51.955 but I was thinking about what's the probability of going 00:13:51.955 --> 00:13:55.448 to 0, 1 instead of 1, 0. 00:13:55.448 --> 00:14:04.929 And I just took it as the ratio of the extra benefit of 0, 1 00:14:04.929 --> 00:14:07.460 over the benefit of 1, 0. 00:14:07.460 --> 00:14:09.910 PROFESSOR: Sure, OK, and just to start out, 00:14:09.910 --> 00:14:11.510 which answer are you arguing for? 00:14:11.510 --> 00:14:12.100 AUDIENCE: D. 00:14:12.100 --> 00:14:13.130 PROFESSOR: D, OK, all right. 00:14:13.130 --> 00:14:15.180 So you're saying D. And you're saying, all right, 00:14:15.180 --> 00:14:18.500 maybe because of the extra, that 1, 0 is somehow 00:14:18.500 --> 00:14:20.520 more fit than 0, 1. 00:14:20.520 --> 00:14:24.290 And you've taken some relative rates or ratios 00:14:24.290 --> 00:14:27.218 for which reason? 00:14:27.218 --> 00:14:31.130 AUDIENCE: Well, I took 0.02 and then 0.1, which is 1/5. 00:14:31.130 --> 00:14:34.176 And then I decided that that should 00:14:34.176 --> 00:14:36.509 be around what it is, but slightly less, because there's 00:14:36.509 --> 00:14:40.010 also a chance that [INAUDIBLE]. 00:14:40.010 --> 00:14:40.810 PROFESSOR: OK, yes. 00:14:40.810 --> 00:14:43.320 I think the arguments there-- there's 00:14:43.320 --> 00:14:46.730 a lot of truth to the arguments that you're saying. 00:14:46.730 --> 00:14:50.580 Yeah, it's a little-- right and another question 00:14:50.580 --> 00:14:55.080 is exactly why might it be 1/6 instead of 1/5 is, 00:14:55.080 --> 00:14:58.630 I think, a little bit hazy in this here. 00:14:58.630 --> 00:15:00.309 It's OK, but it's close. 00:15:00.309 --> 00:15:02.100 Does somebody want to offer an explanation? 00:15:02.100 --> 00:15:05.580 So here, that was an argument of roughly maybe why it's D-ish. 00:15:05.580 --> 00:15:07.720 Because D is very different from B-- 00:15:07.720 --> 00:15:09.930 order of magnitude different. 00:15:09.930 --> 00:15:13.460 So can somebody offer why their neighbor thought it was B? 00:15:13.460 --> 00:15:14.460 Yeah. 00:15:14.460 --> 00:15:18.039 AUDIENCE: So I knew that it was B, because I considered the two 00:15:18.039 --> 00:15:20.631 paths, both from 0, 0 to 0, 1. 00:15:20.631 --> 00:15:23.310 I first checked S, N and it's non-neutral. 00:15:23.310 --> 00:15:27.766 So probably [INAUDIBLE] S. So the probability 00:15:27.766 --> 00:15:31.378 for that first path would be the S for 0, 1, 00:15:31.378 --> 00:15:34.820 so it's 0.02, which is 1/50, multiplied by the probability 00:15:34.820 --> 00:15:38.800 that the other [INAUDIBLE] 1, 0 would die out [INAUDIBLE]. 00:15:38.800 --> 00:15:41.360 PROFESSOR: Right, so there are two related questions. 00:15:41.360 --> 00:15:43.150 And I think that this explanation here 00:15:43.150 --> 00:15:45.780 is answering a slightly different question. 00:15:45.780 --> 00:15:47.350 OK, so let me try to explain what 00:15:47.350 --> 00:15:50.050 the two questions are here. 00:15:50.050 --> 00:15:51.830 So the question that you're answering 00:15:51.830 --> 00:15:58.410 is, if you have kind of 998 individuals that 00:15:58.410 --> 00:16:01.860 are 0, 0 individuals, and you have one that's 0, 1, 00:16:01.860 --> 00:16:05.200 and you have one individual that is 1, 0. 00:16:05.200 --> 00:16:08.660 So this is like these problems that we did a couple weeks ago, 00:16:08.660 --> 00:16:11.324 where we said, you imagine in the population you have 00:16:11.324 --> 00:16:13.740 a couple different kinds of mutants that are present maybe 00:16:13.740 --> 00:16:15.599 in one copy. 00:16:15.599 --> 00:16:17.140 And then we were asking, well, what's 00:16:17.140 --> 00:16:19.930 the probability that this individual is going to fix? 00:16:19.930 --> 00:16:21.440 And what's the probability this individual is going to fix? 00:16:21.440 --> 00:16:23.380 And what's the probability that these guys are 00:16:23.380 --> 00:16:26.580 going to go extinct, and this one will therefore fix? 00:16:26.580 --> 00:16:30.270 And I think that's the calculation that you're 00:16:30.270 --> 00:16:32.440 describing, where you say, OK, well, 00:16:32.440 --> 00:16:35.710 in order for this individual to fix, 00:16:35.710 --> 00:16:40.840 he has to survive stochastic extinction, which happens 00:16:40.840 --> 00:16:43.820 with the probability of 2%. 00:16:43.820 --> 00:16:49.190 And the 1, 0 individual has to go extinct, 00:16:49.190 --> 00:16:53.220 which happens 90% of the time. 00:16:53.220 --> 00:16:57.340 And so this is, indeed, answering the question 00:16:57.340 --> 00:17:02.300 that if you had one copy of each of these two mutant individuals 00:17:02.300 --> 00:17:06.910 in the population, that's the answer to what 00:17:06.910 --> 00:17:10.460 is the probability that this 0, 1 mutant 00:17:10.460 --> 00:17:11.880 would fix in the population. 00:17:11.880 --> 00:17:13.300 Right? 00:17:13.300 --> 00:17:16.290 But that's a slightly different question than if we ask, 00:17:16.290 --> 00:17:19.236 we're going to start with an entire population at 0, 0, 00:17:19.236 --> 00:17:21.319 and now these mutations will be occurring randomly 00:17:21.319 --> 00:17:22.200 at some rate. 00:17:22.200 --> 00:17:23.970 And then something's going to happen. 00:17:23.970 --> 00:17:25.250 Somehow the population is going to climb up 00:17:25.250 --> 00:17:26.349 this fitness landscape. 00:17:26.349 --> 00:17:29.230 And we're trying to figure out the relative probability 00:17:29.230 --> 00:17:32.010 that it's going to take kind of one path or another. 00:17:32.010 --> 00:17:37.090 Do you see the difference between these two questions? 00:17:37.090 --> 00:17:44.550 So indeed, this is the correct answer to a different question. 00:17:44.550 --> 00:17:47.260 And so it's going to end up being D. 00:17:47.260 --> 00:17:50.990 And now we want to try to figure out how to get there. 00:17:50.990 --> 00:17:54.310 Because I think it is a bit tricky. 00:17:54.310 --> 00:17:56.580 And in order in order to figure out to get there, 00:17:56.580 --> 00:17:59.860 we have to make sure we keep track of which 00:17:59.860 --> 00:18:01.210 parameter regime we're in. 00:18:01.210 --> 00:18:03.390 So there are a couple of questions we have to ask. 00:18:03.390 --> 00:18:05.590 First of all, we have to remember 00:18:05.590 --> 00:18:07.940 that we start out with everybody, all 1,000 00:18:07.940 --> 00:18:09.190 individuals in the 0, 0 state. 00:18:09.190 --> 00:18:12.210 So there are initially no mutants in the population. 00:18:12.210 --> 00:18:14.240 But they're just replicating at some rate. 00:18:14.240 --> 00:18:16.910 And every now and then, mutation's going to occur. 00:18:16.910 --> 00:18:20.480 Now one thing we have to answer, we have to think about, 00:18:20.480 --> 00:18:22.550 is whether these are nearly neutral mutations. 00:18:28.440 --> 00:18:29.350 Verbally yes or no? 00:18:29.350 --> 00:18:29.850 Ready? 00:18:29.850 --> 00:18:31.610 Three, two, one. 00:18:31.610 --> 00:18:32.180 AUDIENCE: No. 00:18:32.180 --> 00:18:33.570 PROFESSOR: No, right. 00:18:33.570 --> 00:18:35.017 And that's because we want to ask 00:18:35.017 --> 00:18:36.850 for, if it's nearly neutral, we want to ask, 00:18:36.850 --> 00:18:40.510 is the magnitude of S times N much greater or much less 00:18:40.510 --> 00:18:42.040 than 1? 00:18:42.040 --> 00:18:44.450 In particular, if they're much greater than 1, 00:18:44.450 --> 00:18:49.559 as is the case here, then we're in a nice, simple regime. 00:18:49.559 --> 00:18:51.600 And it's easy to get paralyzed in this situation, 00:18:51.600 --> 00:18:56.920 because there's more than one S. But in both cases, 00:18:56.920 --> 00:18:58.260 S times N is much larger than 1. 00:18:58.260 --> 00:19:01.580 We can take the smaller S, which is 2 over 100, 00:19:01.580 --> 00:19:03.540 and S times N is 20, right? 00:19:03.540 --> 00:19:10.680 So S for the 0, 1 state times N is 20, 00:19:10.680 --> 00:19:14.080 which is much greater than 1. 00:19:14.080 --> 00:19:15.710 What this tells us is that if we do 00:19:15.710 --> 00:19:19.340 get this mutant appearing in the population, 00:19:19.340 --> 00:19:23.810 then he or she will have a probability S of surviving 00:19:23.810 --> 00:19:24.810 stochastic extinction. 00:19:28.050 --> 00:19:34.470 So probability of surviving stochastic extinction 00:19:34.470 --> 00:19:38.220 if the individual appears is equal to S 0, 00:19:38.220 --> 00:19:42.450 1, which is equal to 2%. 00:19:42.450 --> 00:19:57.500 Whereas for the 1, 0 state, that's going to be 0.1. 00:19:57.500 --> 00:20:00.660 Now, this is assuming that the mutation appears 00:20:00.660 --> 00:20:03.350 in the population, that's the probability it will survive 00:20:03.350 --> 00:20:05.300 stochastic extinction. 00:20:05.300 --> 00:20:08.730 Now, just as a reminder, surviving stochastic extinction 00:20:08.730 --> 00:20:12.540 roughly corresponds to this becoming an established idea. 00:20:12.540 --> 00:20:14.600 And becoming established was what again? 00:20:21.980 --> 00:20:22.952 AUDIENCE: [INAUDIBLE]. 00:20:22.952 --> 00:20:23.910 PROFESSOR: What's that? 00:20:23.910 --> 00:20:26.090 AUDIENCE: S1 is [INAUDIBLE]. 00:20:26.090 --> 00:20:27.440 PROFESSOR: It's when S1-- 00:20:27.440 --> 00:20:28.380 AUDIENCE: [INAUDIBLE]. 00:20:28.380 --> 00:20:29.630 PROFESSOR: Yeah, that's right. 00:20:29.630 --> 00:20:32.860 All right, so established-- when we say established, 00:20:32.860 --> 00:20:37.960 what we mean is that this corresponds 00:20:37.960 --> 00:20:42.280 to saying that this probability that we talked about 00:20:42.280 --> 00:20:47.050 before this X sub i is approximately equal to 1. 00:20:47.050 --> 00:20:49.050 So this question is, how many individuals do you 00:20:49.050 --> 00:20:51.910 have to get to in the population before you're 00:20:51.910 --> 00:20:54.030 very likely to fix? 00:20:54.030 --> 00:21:00.130 And what we found is that that number established went as 1 00:21:00.130 --> 00:21:02.570 over the selection coefficient. 00:21:02.570 --> 00:21:06.264 So in this case, you would need to have 50 individuals 00:21:06.264 --> 00:21:08.430 before you were kind of more likely to fix than not. 00:21:08.430 --> 00:21:10.013 So if you want to be much more likely, 00:21:10.013 --> 00:21:13.420 you might need twice that or so. 00:21:13.420 --> 00:21:16.242 Do you guys remember that? 00:21:16.242 --> 00:21:18.450 This is not important for this question, necessarily, 00:21:18.450 --> 00:21:22.284 but it might be important at a later date. 00:21:22.284 --> 00:21:24.260 AUDIENCE: And so for nearly neutral mutations, 00:21:24.260 --> 00:21:25.989 the whole point is that the number needed 00:21:25.989 --> 00:21:29.200 to become established is equal to the population. 00:21:29.200 --> 00:21:32.430 PROFESSOR: Yeah, so everything kind of works, right? 00:21:35.910 --> 00:21:37.710 OK, so the way that we can think about this 00:21:37.710 --> 00:21:41.020 is, now we have this population, 1,000 individuals. 00:21:41.020 --> 00:21:43.590 They're dividing at some rate. 00:21:43.590 --> 00:21:45.060 Mutations are going to appear. 00:21:47.640 --> 00:21:51.270 Now we know if they did appear, the probability they would fix. 00:21:51.270 --> 00:21:53.785 This is assuming there's no clonal interference, right? 00:22:05.375 --> 00:22:07.000 Because if there's clonal interference, 00:22:07.000 --> 00:22:09.450 then surviving stochastic extinction 00:22:09.450 --> 00:22:11.572 is not the same thing as fixing. 00:22:11.572 --> 00:22:13.155 If they both appear in the population, 00:22:13.155 --> 00:22:15.780 and they both survive stochastic extinction, 00:22:15.780 --> 00:22:19.440 then this mutant loses to this mutant. 00:22:19.440 --> 00:22:21.570 That's the clonal interference. 00:22:21.570 --> 00:22:24.236 Do we have to worry about clonal interference in this situation? 00:22:50.330 --> 00:22:54.340 So remember, this was comparing the two time scales. 00:22:54.340 --> 00:22:57.400 This was comparing the time between 00:22:57.400 --> 00:23:06.998 successive establishment events, which went as 1 over mu N S. 00:23:06.998 --> 00:23:11.200 And the other one is the time between the time to fix, 00:23:11.200 --> 00:23:16.130 which went as 1 over S log of NS, right? 00:23:16.130 --> 00:23:17.710 So we can ignore clonal interference 00:23:17.710 --> 00:23:19.180 if this is much larger than that. 00:23:22.730 --> 00:23:24.820 So no clonal interference corresponds 00:23:24.820 --> 00:23:32.960 to mu N log NS much less than 1. 00:23:32.960 --> 00:23:35.860 No clonal interference, same as this statement. 00:23:35.860 --> 00:23:36.480 Is that right? 00:23:40.630 --> 00:23:41.386 Did I do it right? 00:23:41.386 --> 00:23:43.920 OK. 00:23:43.920 --> 00:23:47.650 So and once again, there are multiple S's, and it's 00:23:47.650 --> 00:23:50.220 easy to get kind of upset about this. 00:23:50.220 --> 00:23:53.096 But you can just use whichever S would be-- 00:23:53.096 --> 00:23:54.970 which S would you want to use to be kind of-- 00:23:57.221 --> 00:23:58.762 AUDIENCE: Small or large [INAUDIBLE]? 00:24:01.400 --> 00:24:04.550 PROFESSOR: To be on the safe or conservative side, 00:24:04.550 --> 00:24:06.760 we want to take this to be as big as possible. 00:24:06.760 --> 00:24:11.120 So we take S actually as big as we can, right? 00:24:11.120 --> 00:24:12.250 It's in the log. 00:24:12.250 --> 00:24:13.720 So details, right? 00:24:13.720 --> 00:24:20.900 But we can see we have 10 to the minus 6, 10 to the 3, 00:24:20.900 --> 00:24:28.430 and then this is the log of maybe 100, which 00:24:28.430 --> 00:24:32.012 is, like, 4 or 5. 00:24:32.012 --> 00:24:32.970 Is it closer to 4 or 5? 00:24:32.970 --> 00:24:35.480 I don't know, but it doesn't matter. 00:24:35.480 --> 00:24:36.680 We'll say 5. 00:24:36.680 --> 00:24:38.070 This is indeed much less than 1. 00:24:43.420 --> 00:24:45.010 So indeed, we don't have to worry 00:24:45.010 --> 00:24:46.680 about clonal interference. 00:24:46.680 --> 00:24:49.240 This is a wonderful simplification. 00:24:49.240 --> 00:24:52.440 What it's saying is that the population is dividing. 00:24:52.440 --> 00:24:55.760 Every now and then, a mutation occurs in the population. 00:24:55.760 --> 00:24:58.290 It could be either the 0, 1 or the 1, 0. 00:24:58.290 --> 00:25:01.000 But in either case, the fate of that mutation 00:25:01.000 --> 00:25:04.039 is resolved before the next mutation occurs. 00:25:04.039 --> 00:25:05.580 So you don't need to worry about them 00:25:05.580 --> 00:25:06.900 competing in the population. 00:25:06.900 --> 00:25:10.440 Instead, just at some constant rate they're appearing. 00:25:10.440 --> 00:25:13.290 And given that they appear, there's some probability 00:25:13.290 --> 00:25:14.610 that they're going to fix. 00:25:14.610 --> 00:25:17.420 So that leads to effective rates going to each of those two 00:25:17.420 --> 00:25:20.690 steps-- going to 0, 1 or 1, 0. 00:25:20.690 --> 00:25:24.570 And in particular, this is like a chemical reaction, where we 00:25:24.570 --> 00:25:29.110 have some chemical state here. 00:25:29.110 --> 00:25:30.540 We have two rates. 00:25:30.540 --> 00:25:35.760 There's the k going to 0, 1, the k going to 1, 0. 00:25:39.410 --> 00:25:41.899 And what we know is we know the ratio of those rates. 00:25:41.899 --> 00:25:43.440 And that's everything we need to know 00:25:43.440 --> 00:25:45.481 to calculate the relative probabilities of taking 00:25:45.481 --> 00:25:50.810 those states, because the probability of going through 00:25:50.810 --> 00:25:53.960 to 0, 1-- we want to go that direction-- 0, 1, this 00:25:53.960 --> 00:26:02.330 is going to be given by k 0, 1 divided by k 0, 1 plus k 1, 0. 00:26:02.330 --> 00:26:05.810 So this is how we get 1/6 instead of 1/5. 00:26:05.810 --> 00:26:10.950 Because this thing is 1/5 of that. 00:26:10.950 --> 00:26:12.870 So it's like 1, and then 1, 5. 00:26:21.140 --> 00:26:24.270 So this is actually, in principle, not quite answering 00:26:24.270 --> 00:26:26.320 the question that I asked, because this 00:26:26.320 --> 00:26:30.250 is talking about the relative probability of the first state, 00:26:30.250 --> 00:26:32.120 the first mutant to fix. 00:26:32.120 --> 00:26:35.780 In principle, it is possible that from there, there's 00:26:35.780 --> 00:26:37.040 some rate of coming back. 00:26:37.040 --> 00:26:40.214 Or they might not necessarily move forward on up that hill. 00:26:40.214 --> 00:26:42.130 Do you guys understand what I'm talking about? 00:26:42.130 --> 00:26:43.588 Because it goes from here to there. 00:26:43.588 --> 00:26:47.520 Because we really want to know about this next step, 00:26:47.520 --> 00:26:49.290 going to the 1, 1 state. 00:26:52.980 --> 00:26:57.760 But in this case do we have to worry about going backwards? 00:26:57.760 --> 00:26:58.320 No. 00:26:58.320 --> 00:26:58.990 And why not? 00:27:02.910 --> 00:27:03.720 It's very unlikely. 00:27:03.720 --> 00:27:05.502 And in particular, you could think now 00:27:05.502 --> 00:27:07.710 that you're here you can talk about the rate of going 00:27:07.710 --> 00:27:11.310 to the 1, 1 state as compared to the rate of going to 0, 1. 00:27:11.310 --> 00:27:17.080 And those are going to be exponentially different. 00:27:17.080 --> 00:27:19.950 Because just as this was a non-neutral beneficial 00:27:19.950 --> 00:27:21.820 mutation, that means that going from 0, 00:27:21.820 --> 00:27:24.260 1 back is going to be a non-neutral deleterious 00:27:24.260 --> 00:27:25.685 mutation. 00:27:25.685 --> 00:27:28.310 So the probability of fixing it in the back direction is not 0, 00:27:28.310 --> 00:27:29.726 but it's exponentially suppressed. 00:27:36.060 --> 00:27:38.890 I think it's very important to understand 00:27:38.890 --> 00:27:41.290 all the different pieces of this kind of puzzle, 00:27:41.290 --> 00:27:44.070 because it incorporates many different ideas that we've 00:27:44.070 --> 00:27:46.665 talked about over the last few weeks. 00:27:46.665 --> 00:27:50.480 If there are questions, please ask now. 00:27:50.480 --> 00:27:51.277 Yes. 00:27:51.277 --> 00:27:52.818 AUDIENCE: What about the [INAUDIBLE]? 00:27:58.285 --> 00:28:09.219 [INAUDIBLE] 0, 0 to 1, 0 to 1, 1? 00:28:09.219 --> 00:28:13.195 Then it seems like the benefit of 1, 0 versus [INAUDIBLE]. 00:28:15.700 --> 00:28:19.300 PROFESSOR: All right, so you're wondering about-- 00:28:19.300 --> 00:28:22.810 so the fitness of the 1, 1 state was 1.2. 00:28:22.810 --> 00:28:26.860 So you're pointing out that it's actually easier 00:28:26.860 --> 00:28:29.700 to go from the 0, 1 state to the 1, 1 as 00:28:29.700 --> 00:28:31.645 compared to the 1, 0 to the 1, 1. 00:28:31.645 --> 00:28:34.615 AUDIENCE: Right, which seems like a reason for why 00:28:34.615 --> 00:28:38.486 we wouldn't care about [INAUDIBLE]. 00:28:38.486 --> 00:28:39.610 PROFESSOR: Yeah, OK, right. 00:28:39.610 --> 00:28:43.810 So if anything, in some ways, this 00:28:43.810 --> 00:28:46.590 actually provides a bias going towards the 0, 1 state, 00:28:46.590 --> 00:28:49.456 because it's saying that if we do get to 0, 1, 00:28:49.456 --> 00:28:51.080 it's actually easier to move forward as 00:28:51.080 --> 00:28:52.324 compared to this other path. 00:28:52.324 --> 00:28:53.990 In practice, it doesn't actually matter, 00:28:53.990 --> 00:28:56.890 because this acts as a ratchet. 00:28:56.890 --> 00:28:59.220 Because all these mutations are non-neutral, 00:28:59.220 --> 00:29:03.660 once you fix this state or this one, you can't go back. 00:29:03.660 --> 00:29:05.980 So the population will move forward 00:29:05.980 --> 00:29:07.835 once it gets to one of those two states. 00:29:07.835 --> 00:29:09.960 Now I mean, it would be a very interesting question 00:29:09.960 --> 00:29:14.920 to ask if we instead did a different arrangement. 00:29:14.920 --> 00:29:18.060 What would the rate of evolution be, and so forth? 00:29:18.060 --> 00:29:21.690 Yeah, but what you're saying is certainly true, 00:29:21.690 --> 00:29:24.360 that if this took up all of the benefit going here, 00:29:24.360 --> 00:29:25.957 then it may not actually be somehow 00:29:25.957 --> 00:29:28.540 an optimal path in terms of the rate of evolution or something 00:29:28.540 --> 00:29:29.039 like that. 00:29:32.420 --> 00:29:34.510 I'll think about that when designing. 00:29:34.510 --> 00:29:36.848 Problems. 00:29:36.848 --> 00:29:42.560 AUDIENCE: In this system, 0, 0 eventually becomes 1, 1. 00:29:42.560 --> 00:29:43.560 PROFESSOR: That's right. 00:29:43.560 --> 00:29:46.590 AUDIENCE: So the probability is 1. 00:29:46.590 --> 00:29:50.310 PROFESSOR: That's right, so we are guaranteed 00:29:50.310 --> 00:29:53.470 that we will eventually evolve to this peak in the fitness 00:29:53.470 --> 00:29:53.970 landscape. 00:29:53.970 --> 00:29:57.430 And so what we're asking here is which of these two paths 00:29:57.430 --> 00:29:58.621 is going to be taken. 00:29:58.621 --> 00:30:02.058 AUDIENCE: Yeah, so how to mathematically prove 00:30:02.058 --> 00:30:06.995 that the system will go from 0, 0 to 1, 1? 00:30:06.995 --> 00:30:09.120 PROFESSOR: I mean, I feel like I kind of proved it, 00:30:09.120 --> 00:30:11.411 although I understand that nothing I said was rigorous. 00:30:15.580 --> 00:30:18.670 And of course, there are non-zero probabilities 00:30:18.670 --> 00:30:19.620 of going backwards. 00:30:19.620 --> 00:30:22.390 It's just that they are reduced. 00:30:22.390 --> 00:30:24.510 And actually, you can prove, for those of you 00:30:24.510 --> 00:30:26.580 who are interested in such things, 00:30:26.580 --> 00:30:31.510 that over long time scales, there's 00:30:31.510 --> 00:30:34.260 going to be an equilibrium that distribution 00:30:34.260 --> 00:30:40.610 over all these states, where the probability of being 00:30:40.610 --> 00:30:47.490 in a particular state will-- it goes as the fitness. 00:30:47.490 --> 00:30:52.100 It scales as the relative fitness to the Nth power. 00:30:54.620 --> 00:30:57.940 So we talk about these fitness landscapes 00:30:57.940 --> 00:30:59.670 as energy landscapes. 00:30:59.670 --> 00:31:04.760 And indeed, in this regime where you have small mutation rates, 00:31:04.760 --> 00:31:06.550 then it's going to be a detailed balance. 00:31:06.550 --> 00:31:08.570 And it's actually a thermodynamic system. 00:31:08.570 --> 00:31:11.930 So then in that case, you can make a correspondence 00:31:11.930 --> 00:31:14.210 between everything that we normally talk about, 00:31:14.210 --> 00:31:18.020 where fitness is like energy and population 00:31:18.020 --> 00:31:20.720 size is like temperature. 00:31:20.720 --> 00:31:24.610 So the relative amplitude of being in this peak as compared 00:31:24.610 --> 00:31:27.670 to the other states is going to be, in this case, 00:31:27.670 --> 00:31:30.150 the ratios of those things is, indeed, described 00:31:30.150 --> 00:31:33.160 by the ratios of the fitnesses. 00:31:33.160 --> 00:31:38.550 And it's going to go as kind of like 1.1 to the 1,000th power, 00:31:38.550 --> 00:31:40.690 which is big. 00:31:40.690 --> 00:31:42.760 Which means that the population has really 00:31:42.760 --> 00:31:47.500 cohered at this peak in the finished landscape. 00:31:47.500 --> 00:31:48.726 Yeah. 00:31:48.726 --> 00:31:52.386 AUDIENCE: So if you want to calculate a problem going 00:31:52.386 --> 00:31:57.022 from 0, 1 to 0, 0, then [INAUDIBLE] 00:31:57.022 --> 00:32:01.440 that would just be-- I guess I'm not sure. 00:32:01.440 --> 00:32:03.225 PROFESSOR: OK, you want to know the rate 00:32:03.225 --> 00:32:04.080 that that's going to happen. 00:32:04.080 --> 00:32:04.500 AUDIENCE: Yeah. 00:32:04.500 --> 00:32:04.990 PROFESSOR: No, that's fine. 00:32:04.990 --> 00:32:05.690 Let's do that. 00:32:10.170 --> 00:32:13.110 So for example, let's imagine that we 00:32:13.110 --> 00:32:23.650 don't have-- so let's imagine that we just have the 0, 0 00:32:23.650 --> 00:32:28.190 and the 0, 1 states, just so we don't have to worry 00:32:28.190 --> 00:32:29.980 about going up the landscape. 00:32:29.980 --> 00:32:38.090 And so what we have is we have r is relative fitness 1 and 1.02. 00:32:38.090 --> 00:32:41.200 Now what we want to do is we want to ask, well, 00:32:41.200 --> 00:32:44.390 what is the rate of going back and forth? 00:32:44.390 --> 00:32:51.620 Well, so the rate of going forward, 00:32:51.620 --> 00:32:55.760 well, we sample mutations at a rate mu. 00:32:55.760 --> 00:32:57.980 And this is mu only for this one state, 00:32:57.980 --> 00:33:02.270 because pretend that we're not going to mutate this other one. 00:33:02.270 --> 00:33:07.580 So rate mu N, you have mutations appearing. 00:33:07.580 --> 00:33:10.789 And times this s, 0.02, is the probability 00:33:10.789 --> 00:33:12.830 that it'll actually fix in the forward direction. 00:33:15.580 --> 00:33:19.700 And now what we want to know is the rate of coming back. 00:33:19.700 --> 00:33:21.720 Well, the beginning part's the same, 00:33:21.720 --> 00:33:25.640 because we have mu N is the rate that you 00:33:25.640 --> 00:33:27.970 get this deleterious mutant in the population. 00:33:27.970 --> 00:33:30.595 But then we need to multiply it by the probability of fixation. 00:33:33.720 --> 00:33:36.630 And the probability of fixation is-- 00:33:36.630 --> 00:33:43.290 there was this thing X1, which was 1 minus-- now this is r, 00:33:43.290 --> 00:33:45.670 but it's r in the other direction, so be careful. 00:33:45.670 --> 00:33:47.550 Because the general equation was 1 over. 00:33:52.620 --> 00:33:55.750 But now r, instead of being 1.02, is 1/1.02. 00:34:02.360 --> 00:34:05.000 So which of these terms is going to be dominant? 00:34:05.000 --> 00:34:07.540 This thing gets up to be some really big number 00:34:07.540 --> 00:34:09.090 is our problem. 00:34:09.090 --> 00:34:11.780 So we should be able to figure this out, though. 00:34:11.780 --> 00:34:14.389 Because this new r is 1/1.02. 00:34:14.389 --> 00:34:21.300 So we for example, have 1 minus 1.02, 1 minus 1.02 00:34:21.300 --> 00:34:24.754 to the 1,000. 00:34:24.754 --> 00:34:26.420 All right, so this is a negative number, 00:34:26.420 --> 00:34:27.878 but this is a negative number, too. 00:34:27.878 --> 00:34:37.925 So we end up with 0.02-- 00:34:37.925 --> 00:34:39.280 AUDIENCE: 200. 00:34:39.280 --> 00:34:41.550 PROFESSOR: Is it 200? 00:34:41.550 --> 00:34:44.139 Yeah, you're keeping only the first, 00:34:44.139 --> 00:34:47.000 which, since it's much larger than 1, it's bigger than 200. 00:34:47.000 --> 00:34:47.500 Right? 00:34:47.500 --> 00:34:50.480 I mean do you guys understand what I'm saying? 00:34:50.480 --> 00:34:52.550 You can't keep just the first term in a series. 00:34:52.550 --> 00:34:57.070 If the terms grow with number. 00:34:57.070 --> 00:35:02.340 AUDIENCE: [INAUDIBLE] squared, 3.98 or something. 00:35:02.340 --> 00:35:03.948 PROFESSOR: Wait, which one? 00:35:03.948 --> 00:35:07.760 AUDIENCE: 1.02 to the 1,000. 00:35:07.760 --> 00:35:08.800 PROFESSOR: It's 4? 00:35:08.800 --> 00:35:11.440 OK, all right. 00:35:11.440 --> 00:35:15.040 OK, so it's 1 minus 4. 00:35:15.040 --> 00:35:21.200 So it's 2/300. 00:35:21.200 --> 00:35:26.430 OK, so this is teamwork, right? 00:35:26.430 --> 00:35:30.859 OK, so there's less than a 1% probability of it fixing. 00:35:30.859 --> 00:35:31.650 Is this believable? 00:35:37.592 --> 00:35:39.972 AUDIENCE: It's about right. 00:35:39.972 --> 00:35:47.055 PROFESSOR: 2, 1,000, 50-- I think 00:35:47.055 --> 00:35:52.465 that you did it 1.02 to the 100 rather than 1.02 to the 1,000. 00:35:52.465 --> 00:35:54.900 AUDIENCE: OK. 00:35:54.900 --> 00:35:56.330 PROFESSOR: No? 00:35:56.330 --> 00:35:57.987 Do you not have it in front of you? 00:35:57.987 --> 00:35:59.448 AUDIENCE: No, it's 3-- [INAUDIBLE]. 00:36:02.857 --> 00:36:05.407 AUDIENCE: 4 times 10 to the 8. 00:36:05.407 --> 00:36:07.240 AUDIENCE: I never thought that my calculator 00:36:07.240 --> 00:36:08.710 would become so controversial. 00:36:08.710 --> 00:36:10.126 AUDIENCE: Oh, 4 times 10 to the 8. 00:36:18.440 --> 00:36:21.080 PROFESSOR: Yes, sorry, I was just saying this 00:36:21.080 --> 00:36:23.580 doesn't-- so this is why I'm saying you always check to make 00:36:23.580 --> 00:36:26.900 sure that your calculation makes any sense at all. 00:36:26.900 --> 00:36:29.780 So it's not this. 00:36:29.780 --> 00:36:31.817 But it's tiny, right? 00:36:31.817 --> 00:36:34.620 AUDIENCE: Yes, [INAUDIBLE]. 00:36:34.620 --> 00:36:36.620 PROFESSOR: Yeah, because this didn't make sense, 00:36:36.620 --> 00:36:39.520 because this was of the same order as-- well, 00:36:39.520 --> 00:36:43.772 this would be larger than 1 over N, so it's totally nonsensical. 00:36:43.772 --> 00:36:45.980 Because 1 over N would be the probability of fixation 00:36:45.980 --> 00:36:47.050 of a neutral mutation. 00:36:47.050 --> 00:36:49.182 This is a deleterious mutation. 00:36:49.182 --> 00:36:50.390 It's not even nearly neutral. 00:36:50.390 --> 00:36:54.481 So it has to be much less than 1 over N, right? 00:36:54.481 --> 00:36:58.317 So this whole thing is 10 to the minus 10, 00:36:58.317 --> 00:36:59.275 or something like that? 00:37:03.300 --> 00:37:08.309 OK, 4 times 10 to the minus 8. 00:37:08.309 --> 00:37:09.100 Well, OK, whatever. 00:37:11.620 --> 00:37:12.630 It's 10 to the minus 9. 00:37:12.630 --> 00:37:14.440 It's something small. 00:37:14.440 --> 00:37:18.080 So this times the probability of fixation, which is 10 minus 9-- 00:37:18.080 --> 00:37:20.607 this is how you would calculate the rate of going backwards. 00:37:20.607 --> 00:37:22.440 There's some rate that the mutation appears, 00:37:22.440 --> 00:37:24.450 and you multiply by the probability that it would fix. 00:37:24.450 --> 00:37:25.120 And it's tiny. 00:37:29.520 --> 00:37:30.020 OK? 00:37:32.960 --> 00:37:34.950 All right, any other questions about how 00:37:34.950 --> 00:37:37.120 to think about these sorts of evolutionary dynamics 00:37:37.120 --> 00:37:40.594 with presence of mutation, fixation, everything? 00:37:40.594 --> 00:37:41.814 Yeah. 00:37:41.814 --> 00:37:43.272 AUDIENCE: Can we handle a situation 00:37:43.272 --> 00:37:48.350 where [INAUDIBLE] interference is important at this point? 00:37:48.350 --> 00:37:51.150 PROFESSOR: Yeah, so this is what you do in your problem set 00:37:51.150 --> 00:37:54.090 with simulations. 00:37:54.090 --> 00:37:54.590 Yeah. 00:37:54.590 --> 00:37:55.923 AUDIENCE: [INAUDIBLE] numerical. 00:37:55.923 --> 00:37:59.780 PROFESSOR: You know, I think that it 00:37:59.780 --> 00:38:01.988 gets really messy with clonal interference, I'll say. 00:38:01.988 --> 00:38:05.059 AUDIENCE: But, like, with basic-- 00:38:05.059 --> 00:38:07.308 I guess I was thinking about it and you could probably 00:38:07.308 --> 00:38:10.284 imagine that [INAUDIBLE] calculate the probability 00:38:10.284 --> 00:38:14.760 that 1, 0 doesn't arise first. 00:38:14.760 --> 00:38:17.500 PROFESSOR: Right, yeah, OK, this is an important statement. 00:38:17.500 --> 00:38:20.850 In the limit, as you get more and more mutations, when 00:38:20.850 --> 00:38:22.600 clonal interference is really significant, 00:38:22.600 --> 00:38:24.224 then you're pretty much just guaranteed 00:38:24.224 --> 00:38:25.600 to take the 1, 0 path. 00:38:25.600 --> 00:38:29.037 Because if you have many mutants, 00:38:29.037 --> 00:38:30.870 the definition of clonal interference is you 00:38:30.870 --> 00:38:33.332 have multiple mutations that have established. 00:38:33.332 --> 00:38:35.040 And once you have multiple mutations that 00:38:35.040 --> 00:38:37.350 have established, then it's likely that one of them 00:38:37.350 --> 00:38:38.520 is going to be this. 00:38:38.520 --> 00:38:41.957 And if it's established, it's going to win. 00:38:41.957 --> 00:38:44.540 But the other thing is that as you go up in the mutation rate, 00:38:44.540 --> 00:38:47.260 you don't even do successive fixations. 00:38:47.260 --> 00:38:50.785 So it may be that neither state ever actually fixes, 00:38:50.785 --> 00:38:52.410 because it could be that the 1, 0 state 00:38:52.410 --> 00:38:55.100 is growing exponentially, but is a minority of the population. 00:38:55.100 --> 00:38:57.740 And it gets another mutation that allows it to go to 1, 1. 00:38:57.740 --> 00:39:00.000 So as you increase the mutation rate, 00:39:00.000 --> 00:39:02.450 you don't have to actually take single steps. 00:39:02.450 --> 00:39:04.730 You can kind of move through states. 00:39:04.730 --> 00:39:08.760 And there's a whole literature of the rate at which you 00:39:08.760 --> 00:39:10.410 cross fitness valleys. 00:39:10.410 --> 00:39:12.650 So this is like tunneling in quantum mechanics or so. 00:39:12.650 --> 00:39:14.275 And it has a lot of the same behaviors, 00:39:14.275 --> 00:39:17.029 in the sense of exponential suppression of probabilities 00:39:17.029 --> 00:39:19.570 as a function of the depth and the width of the valley you're 00:39:19.570 --> 00:39:20.390 trying to traverse. 00:39:20.390 --> 00:39:22.870 And there's some very nice papers, 00:39:22.870 --> 00:39:25.420 if you're interested in looking at this stuff. 00:39:25.420 --> 00:39:28.775 And one of them is actually in the syllabus that I mentioned. 00:39:28.775 --> 00:39:29.900 I'm trying to remember who. 00:39:29.900 --> 00:39:32.020 It was Journal of Theoretical Biology, 00:39:32.020 --> 00:39:34.629 but I put it on as optional reading for those of you 00:39:34.629 --> 00:39:35.420 who are interested. 00:39:40.470 --> 00:39:41.505 All right, OK. 00:39:41.505 --> 00:39:43.630 So what I want to do now is I want to switch gears, 00:39:43.630 --> 00:39:45.330 so we can think about this evolutionary game theory 00:39:45.330 --> 00:39:45.829 business. 00:39:49.950 --> 00:39:51.830 And I think the most important thing 00:39:51.830 --> 00:39:54.200 to stress when thinking about evolutionary game theory 00:39:54.200 --> 00:39:57.721 is just that this point that we don't need to assume anything 00:39:57.721 --> 00:39:58.470 about rationality. 00:40:02.750 --> 00:40:10.270 Because the puzzles that we like to give each other in your dorm 00:40:10.270 --> 00:40:13.600 rooms Friday night, you give these logic 00:40:13.600 --> 00:40:15.850 puzzles to each other. 00:40:15.850 --> 00:40:17.140 Is that-- I don't know. 00:40:17.140 --> 00:40:18.330 [LAUGHTER] 00:40:18.330 --> 00:40:21.330 OK, let me just say, back when I was in college, that was, like, 00:40:21.330 --> 00:40:23.930 all the cool kids were doing it. 00:40:23.930 --> 00:40:27.600 But in these puzzles, you assume that there's hyper-rationality. 00:40:27.600 --> 00:40:32.130 You assume that if this guy knows 00:40:32.130 --> 00:40:33.896 that I did this, and that, and if I 00:40:33.896 --> 00:40:35.020 did that, he would do that. 00:40:35.020 --> 00:40:36.645 And then you end up, and then you 00:40:36.645 --> 00:40:38.270 have the villagers that are jumping off 00:40:38.270 --> 00:40:40.625 cliffs on the seventh day. 00:40:40.625 --> 00:40:42.000 Have you guys done these puzzles? 00:40:42.000 --> 00:40:42.360 No? 00:40:42.360 --> 00:40:42.720 OK. 00:40:42.720 --> 00:40:42.980 All right. 00:40:42.980 --> 00:40:43.480 Well. 00:40:43.480 --> 00:40:45.960 [LAUGHTER] 00:40:45.960 --> 00:40:49.620 So the point was that people assume 00:40:49.620 --> 00:40:51.740 that when we're talking about game theory, 00:40:51.740 --> 00:40:55.320 you have to invoke this hyper-rationality even 00:40:55.320 --> 00:40:57.060 humans don't engage in. 00:40:57.060 --> 00:40:59.130 And I think that it's just very important 00:40:59.130 --> 00:41:01.379 to remember that we're talking about evolutionary game 00:41:01.379 --> 00:41:04.990 theory in the case of, well, biological evolution. 00:41:04.990 --> 00:41:06.840 You don't assume anything about rationality. 00:41:06.840 --> 00:41:11.656 Instead, you simply have mutations that sample 00:41:11.656 --> 00:41:12.530 different strategies. 00:41:17.550 --> 00:41:26.050 And then you have differences in fitness 00:41:26.050 --> 00:41:29.470 that just lead to evolution towards the same solutions 00:41:29.470 --> 00:41:31.080 of the game. 00:41:31.080 --> 00:41:36.710 So it's evolution to the game solutions, 00:41:36.710 --> 00:41:38.710 so the Nash equilibrium, for example. 00:41:43.020 --> 00:41:45.840 So it's not that we think that the cells are 00:41:45.840 --> 00:41:50.210 engaging in any sort of weird puzzle solving. 00:41:50.210 --> 00:41:52.290 Instead, they're just mutations. 00:41:52.290 --> 00:41:56.190 And the more fit individuals spread in the population. 00:41:56.190 --> 00:41:59.400 And somehow, you evolve to the same or similar solutions, 00:41:59.400 --> 00:42:02.360 to these Nash equilibria in the context of game theory. 00:42:06.190 --> 00:42:10.040 And we'll see how this plays out in a few concrete examples. 00:42:15.130 --> 00:42:20.640 Now, there are always different ways of looking at these games. 00:42:20.640 --> 00:42:23.579 One thing I want to stress, though, is that all 00:42:23.579 --> 00:42:25.120 the selection that we've been talking 00:42:25.120 --> 00:42:29.210 about in the last few weeks, that all 00:42:29.210 --> 00:42:31.250 is consistent with game theory in the sense 00:42:31.250 --> 00:42:33.500 that the idea of the game theory is 00:42:33.500 --> 00:42:36.200 that we allow for the possibility 00:42:36.200 --> 00:42:38.240 that the fitness of individuals depends 00:42:38.240 --> 00:42:41.060 upon the rest of the population. 00:42:41.060 --> 00:42:43.200 Whereas in all these calculations we've been doing, 00:42:43.200 --> 00:42:47.210 I told you, all right, I just gave you some fitnesses. 00:42:47.210 --> 00:42:50.550 So I said, here we have a 0, 0 state that has some fitness. 00:42:50.550 --> 00:42:53.090 0, 1 has a higher fitness, and so forth. 00:42:53.090 --> 00:42:56.050 But in general, these fitness values 00:42:56.050 --> 00:43:00.990 may depend upon what the population composition is. 00:43:00.990 --> 00:43:03.590 And in that situation, then you want to use evolutionary game 00:43:03.590 --> 00:43:05.910 theory. 00:43:05.910 --> 00:43:08.870 In many cases, people just assume 00:43:08.870 --> 00:43:12.950 that you can do something like this-- that you can describe it 00:43:12.950 --> 00:43:14.290 as some fitness landscape. 00:43:14.290 --> 00:43:17.950 But you can't do that if there's this frequency-dependent 00:43:17.950 --> 00:43:20.260 selection-- if there's any sort of evolutionary game 00:43:20.260 --> 00:43:22.340 interactions going on. 00:43:22.340 --> 00:43:27.410 So it's just important. 00:43:27.410 --> 00:43:32.280 If the fitnesses depend on composition-- 00:43:32.280 --> 00:43:34.990 this is the population composition-- 00:43:34.990 --> 00:43:39.610 then you cannot even define a fitness landscape. 00:43:43.250 --> 00:43:44.430 Then no fitness landscape. 00:43:47.800 --> 00:43:51.120 For example, you can have situations where the population 00:43:51.120 --> 00:43:54.700 evolves to lower fitness. 00:43:54.700 --> 00:43:56.510 So you can have a situation where, 00:43:56.510 --> 00:44:00.040 if I tell you individual 0, you measure its growth 00:44:00.040 --> 00:44:05.760 rate, whatnot, its fitness might be 1. 00:44:05.760 --> 00:44:10.680 So this is genome, and this is fitness. 00:44:10.680 --> 00:44:13.090 Now, if I go and I measure the fitness 00:44:13.090 --> 00:44:17.380 of some other individual, different genome-- so 00:44:17.380 --> 00:44:20.480 another strain of bacteria or yeast or whatever-- 00:44:20.480 --> 00:44:23.650 and you say, oh, well, its fitness is 1.2. 00:44:23.650 --> 00:44:26.730 So this strain has higher fitness than this strain. 00:44:26.730 --> 00:44:32.160 Now, it would be very natural to assume that this strain will 00:44:32.160 --> 00:44:34.380 out-compete this strain. 00:44:34.380 --> 00:44:37.080 And indeed, that's been the assumption in everything 00:44:37.080 --> 00:44:38.690 we've been talking about. 00:44:38.690 --> 00:44:41.580 But it's not necessarily true. 00:44:41.580 --> 00:44:45.840 And that's the basic insight of evolutionary game theory, 00:44:45.840 --> 00:44:50.130 is that just knowing the fitness of a pure population 00:44:50.130 --> 00:44:53.110 is not actually enough information to know that it's 00:44:53.110 --> 00:44:54.600 going to be selected for. 00:44:54.600 --> 00:44:57.800 Because it's still possible that in a mixed population, 00:44:57.800 --> 00:45:00.880 the genome 0 may actually have higher fitness 00:45:00.880 --> 00:45:01.892 than the genome 1. 00:45:06.050 --> 00:45:08.760 And once you kind of study these things, 00:45:08.760 --> 00:45:10.430 it's kind of clear that it can happen. 00:45:10.430 --> 00:45:12.800 But then it's easy to then go back to the lab 00:45:12.800 --> 00:45:13.900 and forget that it's true. 00:45:17.930 --> 00:45:21.933 And so we'll see how this plays out. 00:45:21.933 --> 00:45:25.790 AUDIENCE: On this game theory, [INAUDIBLE]? 00:45:25.790 --> 00:45:27.530 PROFESSOR: No. 00:45:27.530 --> 00:45:29.710 That's the other thing, is that I 00:45:29.710 --> 00:45:31.842 like to just draw these things as graphs, because I 00:45:31.842 --> 00:45:33.800 think it's much easier to see what's happening. 00:45:33.800 --> 00:45:35.800 And it's clear that things can be non-linear. 00:45:35.800 --> 00:45:39.940 But the basic insights are all intact. 00:45:39.940 --> 00:45:42.800 From my standpoint as kind of an experimentalist-- 00:45:42.800 --> 00:45:47.090 don't forget about the exam-- I think 00:45:47.090 --> 00:45:50.689 that the more formal evolutionary game theory 00:45:50.689 --> 00:45:52.730 thing-- these two-player games that you guys just 00:45:52.730 --> 00:45:56.200 read about-- I think they're important because they tell you 00:45:56.200 --> 00:46:00.054 what are the possible outcomes of measurements or of systems, 00:46:00.054 --> 00:46:02.220 even in the most simple situation where everything's 00:46:02.220 --> 00:46:02.950 linear. 00:46:02.950 --> 00:46:04.700 Now, when things are not linear, of course 00:46:04.700 --> 00:46:06.530 you can get even richer dynamics. 00:46:06.530 --> 00:46:10.391 But in practice, you basically get the categories of outcomes 00:46:10.391 --> 00:46:11.140 that we saw there. 00:46:22.230 --> 00:46:27.350 So maybe what I'll do is-- so what we're going to do 00:46:27.350 --> 00:46:30.380 is think about competition between two individuals 00:46:30.380 --> 00:46:38.260 A and B. And often, we talk about these things 00:46:38.260 --> 00:46:41.866 in the context of the two-player games, where we have A, B, C, 00:46:41.866 --> 00:46:47.390 D. And because this is really importing the kind of approach, 00:46:47.390 --> 00:46:51.230 or the nomenclature, from conventional game theory, 00:46:51.230 --> 00:46:54.020 and then immediately applying it to populations where you just 00:46:54.020 --> 00:46:55.990 assume that all the individuals have 00:46:55.990 --> 00:46:58.280 equal probability of interacting with everybody 00:46:58.280 --> 00:46:59.317 in the population. 00:46:59.317 --> 00:47:00.900 So it's what you would get if you just 00:47:00.900 --> 00:47:04.850 had some two-player game like they study in game theory, 00:47:04.850 --> 00:47:07.290 but in a population of 1,000 or whatnot. 00:47:07.290 --> 00:47:09.790 You just made a bunch of random pairwise interactions. 00:47:09.790 --> 00:47:10.930 You had them play the game. 00:47:10.930 --> 00:47:12.846 And then you had them do that again over time. 00:47:12.846 --> 00:47:18.550 And then this is the payouts that you read about 00:47:18.550 --> 00:47:20.577 in Chapter Four are kind of what would happen 00:47:20.577 --> 00:47:22.410 in that sort of situation, where everybody's 00:47:22.410 --> 00:47:26.020 interacting with everybody else with equal probability. 00:47:26.020 --> 00:47:28.450 Now, remember the way that you read 00:47:28.450 --> 00:47:33.340 this is that, depending upon the strategy that these guys are 00:47:33.340 --> 00:47:37.590 following, you get different payouts. 00:47:37.590 --> 00:47:41.990 And normally what we say is that if this could be, 00:47:41.990 --> 00:47:45.350 for example, strategy one and two, strategy one and two. 00:47:45.350 --> 00:47:47.880 And this is telling us about the payout 00:47:47.880 --> 00:47:50.670 that the A individual gets depending on what he does, 00:47:50.670 --> 00:47:52.986 and depending upon what his opponent does. 00:47:52.986 --> 00:47:55.360 Now, we're not explicitly saying what the payout to the B 00:47:55.360 --> 00:47:56.785 individual is, but we're assuming 00:47:56.785 --> 00:47:59.160 that this is a symmetric game, so you could figure it out 00:47:59.160 --> 00:48:02.290 by looking at the opposite entry. 00:48:02.290 --> 00:48:05.710 So if A follows strategy one, B follows strategy one, 00:48:05.710 --> 00:48:10.420 then individual A gets little a fitness, 00:48:10.420 --> 00:48:13.030 whereas B also gets little a fitness, 00:48:13.030 --> 00:48:15.840 because it's a symmetric game. 00:48:15.840 --> 00:48:18.930 So the case it's different is when we're in the diagonals. 00:48:22.410 --> 00:48:26.240 And from this framework, you can see 00:48:26.240 --> 00:48:28.470 that there are going to be already 00:48:28.470 --> 00:48:30.550 a bunch of kind of non-trivial things 00:48:30.550 --> 00:48:33.470 that can happen, even in this regime where everything's 00:48:33.470 --> 00:48:35.750 linear. 00:48:35.750 --> 00:48:42.070 And the probably best well-known of these 00:48:42.070 --> 00:48:50.500 is this Prisoner's Dilemma, which 00:48:50.500 --> 00:48:53.770 is the standard model of cooperation 00:48:53.770 --> 00:48:54.949 in the field of game theory. 00:48:54.949 --> 00:48:56.740 So there's a story that goes along with it. 00:48:56.740 --> 00:49:02.120 It's this idea that-- I'm sure you guys watch these cop shows, 00:49:02.120 --> 00:49:05.110 where you have the cops bring in the two accomplices. 00:49:05.110 --> 00:49:07.300 And then they put them in separate rooms. 00:49:07.300 --> 00:49:09.220 And they tell them that they have 00:49:09.220 --> 00:49:11.520 to confess to committing the crime, 00:49:11.520 --> 00:49:13.800 because the guy in the other room is confessing, 00:49:13.800 --> 00:49:15.300 and if he doesn't confess, then he's 00:49:15.300 --> 00:49:16.880 going to be in trouble, et cetera. 00:49:16.880 --> 00:49:20.470 You've seen these cop shows? 00:49:20.470 --> 00:49:23.900 And incidentally, in these questions, 00:49:23.900 --> 00:49:25.510 when cops are questioning witnesses, 00:49:25.510 --> 00:49:31.730 they're actually allowed to lie to the person being questioned, 00:49:31.730 --> 00:49:34.250 which feels a little bit weird, actually. 00:49:34.250 --> 00:49:35.670 Doesn't it? 00:49:35.670 --> 00:49:37.630 I know, I know, this is not relevant. 00:49:40.459 --> 00:49:42.000 So the idea of the prisoner's dilemma 00:49:42.000 --> 00:49:46.787 is that if you set up these jail sentences in the right way, 00:49:46.787 --> 00:49:48.870 then it could be the case that each individual has 00:49:48.870 --> 00:49:52.260 the incentive to confess, even though both individuals would 00:49:52.260 --> 00:49:57.200 be better off if they cooperated. 00:49:57.200 --> 00:50:01.132 And you can come up with some reasonable payout structure 00:50:01.132 --> 00:50:02.090 that has that property. 00:50:18.000 --> 00:50:24.515 And we'll call this-- so this is for individual one, 00:50:24.515 --> 00:50:26.010 say and individual two. 00:50:26.010 --> 00:50:29.550 So there are different strategies you can follow. 00:50:29.550 --> 00:50:32.920 And do you guys remember from the reading 00:50:32.920 --> 00:50:35.800 slash my explanation how to read these charts? 00:50:39.460 --> 00:50:44.510 All right, now the question is, just to remind ourselves, 00:50:44.510 --> 00:50:47.370 what is the Nash equilibrium of this game? 00:50:51.290 --> 00:50:54.300 And I know that you read about it last night. 00:50:54.300 --> 00:50:55.640 Well, use your cards. 00:50:55.640 --> 00:50:59.190 Is it C or is it D? 00:50:59.190 --> 00:51:02.910 Or is there no Nash equilibrium, you can flash something else. 00:51:10.590 --> 00:51:12.520 AUDIENCE: Are those negative or positive? 00:51:12.520 --> 00:51:14.103 PROFESSOR: These are positive fitness. 00:51:16.670 --> 00:51:19.970 I kind of don't like the Prisoner's Dilemma as a story, 00:51:19.970 --> 00:51:22.164 because it's not very intuitive, because you 00:51:22.164 --> 00:51:23.830 have to actually specify the jail terms, 00:51:23.830 --> 00:51:25.990 and you have to remember that jail terms are bad, not good. 00:51:25.990 --> 00:51:27.370 So these are good things, OK? 00:51:27.370 --> 00:51:30.960 These are years off that you get as a result 00:51:30.960 --> 00:51:32.210 of doing one thing or another. 00:51:35.930 --> 00:51:37.560 You want to get big numbers. 00:51:37.560 --> 00:51:38.360 Ready? 00:51:38.360 --> 00:51:40.220 Three, two, one. 00:51:44.640 --> 00:51:46.980 So at least we have a majority that are D, 00:51:46.980 --> 00:51:49.270 but it's not all of them. 00:51:49.270 --> 00:51:51.300 And I think this is basically a reflection-- 00:51:51.300 --> 00:51:53.300 and D is indeed the Nash equilibrium. 00:51:56.550 --> 00:51:58.690 It's to do this strategy D that we're saying here. 00:51:58.690 --> 00:52:01.060 All right, now the question is why? 00:52:01.060 --> 00:52:03.320 And part of the challenge here is just understanding 00:52:03.320 --> 00:52:05.540 how to read these charts. 00:52:05.540 --> 00:52:09.670 Now, first of all, the payout that everybody gets if everyone 00:52:09.670 --> 00:52:12.130 follows strategy D is what? 00:52:12.130 --> 00:52:14.130 Verbally, three, two, one. 00:52:14.130 --> 00:52:15.170 AUDIENCE: One. 00:52:15.170 --> 00:52:17.440 PROFESSOR: So everybody gets payout one. 00:52:17.440 --> 00:52:20.100 Now, if you look at this chart, you 00:52:20.100 --> 00:52:23.410 say, well, gee, that is a shame. 00:52:23.410 --> 00:52:28.900 Because 1 is just not the biggest number you see here. 00:52:28.900 --> 00:52:30.820 And indeed, the important point to note 00:52:30.820 --> 00:52:34.670 here is that if both players had followed this strategy 00:52:34.670 --> 00:52:39.930 C for cooperate, D for defect, then both individuals would be 00:52:39.930 --> 00:52:43.840 getting fitness 3, or payout 3. 00:52:43.840 --> 00:52:46.550 So the idea here is that both individuals would 00:52:46.550 --> 00:52:49.140 do better if they both played strategy C. 00:52:49.140 --> 00:52:51.660 But the problem is that that's not evolutionarily stable. 00:52:51.660 --> 00:52:53.590 Or in the context of game theory, 00:52:53.590 --> 00:52:56.770 that is cheatable in some ways. 00:52:56.770 --> 00:52:59.020 And so the reason that this is a Nash equilibrium 00:52:59.020 --> 00:53:01.890 is that you ask-- so a Nash equilibrium, what it means, 00:53:01.890 --> 00:53:05.070 if you recall, is that if everyone's 00:53:05.070 --> 00:53:07.640 playing that strategy, then nobody has the incentive 00:53:07.640 --> 00:53:09.770 to change strategy. 00:53:09.770 --> 00:53:11.825 So no incentive to change strategy. 00:53:17.902 --> 00:53:19.360 So now you just imagine, let's say, 00:53:19.360 --> 00:53:20.390 that you're playing against somebody else, 00:53:20.390 --> 00:53:22.100 or in the context of biology, it's 00:53:22.100 --> 00:53:24.746 a population of individuals following the D strategy. 00:53:24.746 --> 00:53:27.740 The question is whether you as an individual 00:53:27.740 --> 00:53:32.040 would have the incentive to switch to the other strategy? 00:53:32.040 --> 00:53:34.830 And the answer is no, because what you have control over 00:53:34.830 --> 00:53:38.510 is this rows. 00:53:38.510 --> 00:53:41.450 The column is specified by the rest of the population. 00:53:41.450 --> 00:53:45.095 So if you're in this state, what you have a choice of 00:53:45.095 --> 00:53:47.612 is to switch to the cooperate strategy, which 00:53:47.612 --> 00:53:48.570 would be to go up here. 00:53:48.570 --> 00:53:53.890 So you have a choice to move up to this 0 payout, 00:53:53.890 --> 00:53:55.732 but that's not to your advantage. 00:53:55.732 --> 00:53:57.940 Now, it's true that your opponent would get payout 5. 00:53:57.940 --> 00:54:01.170 So you opponent would actually do wonderfully. 00:54:01.170 --> 00:54:02.370 But you would do poorly. 00:54:02.370 --> 00:54:05.890 So you'd be selected against, if you imagine 00:54:05.890 --> 00:54:07.980 this being in the context of biology-- 00:54:07.980 --> 00:54:09.990 that you have a genotype that are playing 00:54:09.990 --> 00:54:13.120 D. If you're a mutant that starts following this strategy 00:54:13.120 --> 00:54:17.627 C, you have lower fitness, so you're selected against. 00:54:17.627 --> 00:54:19.835 So that's saying that the strategy D is noninvadable. 00:54:23.050 --> 00:54:25.040 We can also think about what happens if we're 00:54:25.040 --> 00:54:28.710 a population of cooperators. 00:54:28.710 --> 00:54:31.251 Now everybody has high fitness-- fitness 3. 00:54:31.251 --> 00:54:32.750 Question is, what happens if there's 00:54:32.750 --> 00:54:35.070 a mutation that leads to one individual following the D 00:54:35.070 --> 00:54:35.570 strategy? 00:54:41.560 --> 00:54:45.426 Is he selected four or not? 00:54:45.426 --> 00:54:46.320 AUDIENCE: Yes. 00:54:46.320 --> 00:54:48.570 PROFESSOR: Yes, so the point here 00:54:48.570 --> 00:54:53.410 is that you always will have higher fitness, 00:54:53.410 --> 00:54:57.010 regardless of what your opponent does in the context of a game 00:54:57.010 --> 00:54:59.260 theory situation, or regardless of the distribution 00:54:59.260 --> 00:55:00.910 of cooperation and defection in the population. 00:55:00.910 --> 00:55:02.410 It's always better to be a defector. 00:55:05.280 --> 00:55:11.308 So the problem here is, it's always better to play D. 00:55:21.280 --> 00:55:28.911 Now, I really like drawing the graphs of these things, 00:55:28.911 --> 00:55:30.660 because I think it's just much more clear. 00:55:34.770 --> 00:55:37.940 And you can either draw the fitness 00:55:37.940 --> 00:55:41.860 of the two types minus each other, or just the raw fitness. 00:55:41.860 --> 00:55:43.086 Yes. 00:55:43.086 --> 00:55:45.537 AUDIENCE: So what if instead of 5, you have 7? 00:55:45.537 --> 00:55:49.830 Because then the population as a whole's fitness 00:55:49.830 --> 00:55:54.123 decreases when you [INAUDIBLE]. 00:55:54.123 --> 00:55:56.680 So how does that-- 00:55:56.680 --> 00:55:59.236 PROFESSOR: So you're saying if this 5 were a 7 instead? 00:55:59.236 --> 00:55:59.930 AUDIENCE: Yeah. 00:55:59.930 --> 00:56:01.950 PROFESSOR: Right, then what you're saying 00:56:01.950 --> 00:56:04.020 is that-- so it doesn't change. 00:56:04.020 --> 00:56:07.940 The Nash equilibrium is still defect. 00:56:07.940 --> 00:56:13.600 The subtle thing here is that, in general, 00:56:13.600 --> 00:56:17.180 in terms of game theory, we like it when the mean of these two 00:56:17.180 --> 00:56:19.437 is smaller than this one. 00:56:19.437 --> 00:56:20.770 That's why you're asking, right? 00:56:20.770 --> 00:56:23.120 Exactly, because that's right. 00:56:23.120 --> 00:56:24.130 Exactly, right. 00:56:24.130 --> 00:56:27.630 So yeah, so that's a slightly more complicated situation, 00:56:27.630 --> 00:56:31.490 because in that situation, then, if you had two rational agents, 00:56:31.490 --> 00:56:34.820 say, playing this game, then you could alternate cooperation 00:56:34.820 --> 00:56:35.940 and defection. 00:56:35.940 --> 00:56:39.060 And that would actually be the ultimate form of cooperation 00:56:39.060 --> 00:56:42.030 in such a game, because you could actually get a higher 00:56:42.030 --> 00:56:45.180 payout by alternating. 00:56:45.180 --> 00:56:47.390 Right, so we've chosen the numbers 00:56:47.390 --> 00:56:54.480 as they are so that this is subtlety is not an issue. 00:56:54.480 --> 00:56:56.280 Does everybody understand the issue there? 00:57:00.030 --> 00:57:02.850 So in the context of evolutionary game theory, what 00:57:02.850 --> 00:57:07.030 we can do is we can plot as a function of the fraction 00:57:07.030 --> 00:57:10.610 of the population that's cooperator between 0 and 1, 00:57:10.610 --> 00:57:12.770 say. 00:57:12.770 --> 00:57:16.420 And we can plot the payout for the cooperator 00:57:16.420 --> 00:57:17.350 and for the defector. 00:57:23.020 --> 00:57:29.510 For example, I'm going to draw a solid line for the cooperator, 00:57:29.510 --> 00:57:32.450 dashed line for the defector. 00:57:32.450 --> 00:57:39.150 Now the question is, what should be the y-axes on either ends 00:57:39.150 --> 00:57:39.730 and so forth? 00:57:39.730 --> 00:57:40.479 Do you understand? 00:57:43.030 --> 00:57:45.850 So what should these things look like? 00:57:45.850 --> 00:57:47.780 I'd like to encourage you to-- I'll 00:57:47.780 --> 00:57:51.055 give you 30 seconds to try to draw 00:57:51.055 --> 00:57:52.180 what this should look like. 00:57:52.180 --> 00:57:59.610 So this is the payout or the expected payout. 00:57:59.610 --> 00:58:03.610 So we're assuming that you're going to interact randomly 00:58:03.610 --> 00:58:05.860 with the other members of the population as a function 00:58:05.860 --> 00:58:07.290 of the fraction cooperator. 00:58:07.290 --> 00:58:09.570 So then 1 minus that will be the fraction defector. 00:58:13.170 --> 00:58:15.321 Do you understand what I'm trying to ask you to do? 00:58:15.321 --> 00:58:17.966 AUDIENCE: So the scale on the right-hand side supposed 00:58:17.966 --> 00:58:20.140 to be for the defectors? 00:58:20.140 --> 00:58:21.759 [INAUDIBLE]? 00:58:21.759 --> 00:58:24.050 PROFESSOR: This is just a legend, or key, or something. 00:58:24.050 --> 00:58:26.630 So I want you to draw something over here that's 00:58:26.630 --> 00:58:28.030 a solid line and a dashed line. 00:58:28.030 --> 00:58:29.380 AUDIENCE: All right, so it's just one scale. 00:58:29.380 --> 00:58:30.005 And you don't-- 00:58:30.005 --> 00:58:31.088 PROFESSOR: It's one scale. 00:58:31.088 --> 00:58:32.230 AUDIENCE: [INAUDIBLE]. 00:58:32.230 --> 00:58:34.614 PROFESSOR: Oh, yeah, sorry. 00:58:34.614 --> 00:58:36.780 I'm just telling you what's going to be a solid line 00:58:36.780 --> 00:58:39.540 and what's going to be a dashed line. 00:58:39.540 --> 00:58:42.290 And I'll give you a hint, that up here is number 5. 00:59:23.450 --> 00:59:27.775 This is going to be the expected payout for a lone individual 00:59:27.775 --> 00:59:29.650 given the rest of the population is following 00:59:29.650 --> 00:59:31.070 some fraction of cooperator. 01:00:05.630 --> 01:00:08.136 Do you guys understand what I'm asking you to do? 01:00:08.136 --> 01:00:10.010 Because I'm a little bit concerned that there 01:00:10.010 --> 01:00:14.706 are very few plots in front. 01:00:14.706 --> 01:00:16.020 AUDIENCE: What is fc? 01:00:16.020 --> 01:00:17.660 PROFESSOR: So this is the fraction of the population 01:00:17.660 --> 01:00:18.409 that's cooperator. 01:00:40.700 --> 01:00:44.530 Well, I was giving you a chance to think about it. 01:00:44.530 --> 01:00:46.930 But from looking around, I think that maybe you're 01:00:46.930 --> 01:00:48.960 not quite sure what I'm trying to ask you to do. 01:00:48.960 --> 01:00:50.964 So I'm trying to plot the expected payout 01:00:50.964 --> 01:00:53.380 for an individual that is either cooperating or defecting, 01:00:53.380 --> 01:00:55.421 based on the fact that the rest of the population 01:00:55.421 --> 01:01:00.590 has some composition between all cooperate or all defect. 01:01:00.590 --> 01:01:03.170 So it's the evolution game theory extension 01:01:03.170 --> 01:01:06.460 of this simple model. 01:01:06.460 --> 01:01:11.360 So first, we can ask, well, if the entire population is 01:01:11.360 --> 01:01:15.460 cooperating, we want to know the fitness for a cooperator 01:01:15.460 --> 01:01:16.000 or defector. 01:01:16.000 --> 01:01:20.710 Well, this is really just saying that we're 01:01:20.710 --> 01:01:25.190 all the way over on here, and we just choose between the two. 01:01:25.190 --> 01:01:27.250 And the defector is the 5 one. 01:01:27.250 --> 01:01:30.390 So this is going to be a dashed line that's 01:01:30.390 --> 01:01:31.880 going to start from here. 01:01:31.880 --> 01:01:33.530 And then this is 2 and 1/2, 3. 01:01:37.080 --> 01:01:40.435 So this is cooperator starts here, and defector starts here. 01:01:40.435 --> 01:01:43.390 Do you understand what I'm? 01:01:43.390 --> 01:01:45.290 Now, OK, let's see. 01:01:45.290 --> 01:01:48.740 So now this is the one where if everybody else is defecting, 01:01:48.740 --> 01:01:51.250 well, now, the cooperator line goes to what? 01:01:51.250 --> 01:01:54.160 Verbally, three, two, one. 01:01:54.160 --> 01:01:54.740 AUDIENCE: 0. 01:01:54.740 --> 01:01:56.900 PROFESSOR: 0. 01:01:56.900 --> 01:01:58.410 The defector line goes to 1. 01:02:09.310 --> 01:02:11.640 OK, that line, I started going the wrong direction, 01:02:11.640 --> 01:02:13.370 but that's supposed to be a line. 01:02:15.940 --> 01:02:18.020 So this is an example of what this 01:02:18.020 --> 01:02:20.914 looks like for the Prisoner's Dilemma. 01:02:20.914 --> 01:02:22.580 And what you see is the defector fitness 01:02:22.580 --> 01:02:25.180 is always above the cooperator fitness. 01:02:25.180 --> 01:02:27.502 So for any population composition, 01:02:27.502 --> 01:02:29.460 defectors have higher fitness than cooperators. 01:02:29.460 --> 01:02:34.630 So evolution brings you to the pure defecting state, 01:02:34.630 --> 01:02:36.170 where you have fitness 1. 01:02:40.001 --> 01:02:41.500 And if you want, you could calculate 01:02:41.500 --> 01:02:45.230 what the mean fitness of the population is, for example. 01:02:45.230 --> 01:02:48.400 And the mean fitness starts out over here, 01:02:48.400 --> 01:02:49.830 and ends up over here. 01:02:49.830 --> 01:02:51.930 So the mean fitness decreases over time. 01:02:54.610 --> 01:03:01.240 Now, you can imagine that in the simple, two-player models, 01:03:01.240 --> 01:03:02.590 all these are lines. 01:03:02.590 --> 01:03:04.970 But you can imagine that the only thing that's important 01:03:04.970 --> 01:03:09.340 are how these lines cross each other. 01:03:09.340 --> 01:03:12.090 So for example, there are only a few different things 01:03:12.090 --> 01:03:14.160 that can happen. 01:03:14.160 --> 01:03:24.260 You can have one strategy that dominates, 01:03:24.260 --> 01:03:26.660 which is what occurred here. 01:03:26.660 --> 01:03:29.240 And surprisingly, that does not mean 01:03:29.240 --> 01:03:32.290 that that strategy is higher fitness, 01:03:32.290 --> 01:03:36.070 in the sense that you may evolve to a state of low fitness. 01:03:36.070 --> 01:03:37.166 That's what's weird. 01:03:37.166 --> 01:03:43.930 You can have coexistence, or you can have bi-stability. 01:03:48.880 --> 01:03:53.175 So I'll give you another example of this. 01:03:59.420 --> 01:04:01.990 So now we're just going to have two strategies. 01:04:01.990 --> 01:04:07.516 The strategies-- we'll just call them A and B. 01:04:11.460 --> 01:04:16.045 And the question is, what is the Nash equilibrium? 01:04:22.620 --> 01:04:34.500 Is it A, B, or C should be neither, D is both. 01:04:34.500 --> 01:04:35.950 Do you understand? 01:04:35.950 --> 01:04:38.950 I'm going to ask, because if this is the game 01:04:38.950 --> 01:04:41.670 and this is the interaction, is the Nash equilibrium A, 01:04:41.670 --> 01:04:43.400 Nash equilibrium B? 01:04:43.400 --> 01:04:47.240 If you vote C, it means neither, D means both. 01:04:47.240 --> 01:04:50.590 Do you understand the question? 01:04:50.590 --> 01:04:53.474 I'll give you 30 seconds to think about it. 01:05:38.660 --> 01:05:44.670 All right, are we ready to vote? 01:05:44.670 --> 01:05:47.935 Ready, three, two, one. 01:05:53.621 --> 01:05:55.370 All right, so we have a fair distribution. 01:05:55.370 --> 01:06:01.250 I may not have us vote, but yeah, in this case, 01:06:01.250 --> 01:06:04.970 they're actually both Nash equilibria. 01:06:04.970 --> 01:06:06.500 So let's see this. 01:06:06.500 --> 01:06:09.290 If both individuals, or an entire population, 01:06:09.290 --> 01:06:13.000 say, is playing A, they're getting fitness 5. 01:06:13.000 --> 01:06:14.790 Question is, as a lone individual, 01:06:14.790 --> 01:06:17.720 you can choose to switch over and get fitness 3. 01:06:17.720 --> 01:06:19.610 Do you want to do that? 01:06:19.610 --> 01:06:21.150 No. 01:06:21.150 --> 01:06:23.510 So that means that A is going to be a Nash equilibrium. 01:06:23.510 --> 01:06:25.010 Incidentally, the difference between 01:06:25.010 --> 01:06:27.780 the so-called regular Nash equilibrium and the strict Nash 01:06:27.780 --> 01:06:32.050 equilibrium is that Nash equilibrium 01:06:32.050 --> 01:06:35.630 means that no individual has the incentive to change strategy. 01:06:35.630 --> 01:06:40.860 A strict Nash equilibrium means that any change in strategy 01:06:40.860 --> 01:06:42.800 leads to an actual decrease in fitness. 01:06:42.800 --> 01:06:45.920 So it's a question of whether you can make neutral changes 01:06:45.920 --> 01:06:47.690 in strategy or not. 01:06:47.690 --> 01:06:48.440 Do you understand? 01:06:51.800 --> 01:06:54.540 So A is a Nash equilibrium. 01:06:54.540 --> 01:06:55.570 What about B? 01:06:55.570 --> 01:06:59.530 Well, in that case, everybody's getting to fitness 1. 01:06:59.530 --> 01:07:02.980 Now, as a lone individual, what can you do? 01:07:02.980 --> 01:07:04.850 All you can do is switch. 01:07:04.850 --> 01:07:07.050 As an individual, you can only choose rows. 01:07:07.050 --> 01:07:08.260 So you go up to 0. 01:07:08.260 --> 01:07:09.950 That's a decrease of fitness. 01:07:09.950 --> 01:07:14.530 So that means that strategy B is also a Nash equilibrium. 01:07:17.410 --> 01:07:19.450 So there are two Nash equilibria in this game. 01:07:19.450 --> 01:07:21.850 And what does that mean about which regime 01:07:21.850 --> 01:07:25.830 you're in here, if you convert this into an evolutionary game 01:07:25.830 --> 01:07:27.075 theory scenario? 01:07:37.460 --> 01:07:41.700 Ready, three, two, one. 01:07:41.700 --> 01:07:43.550 OK, so a majority is saying yes. 01:07:43.550 --> 01:07:47.130 This is indeed a situation in which you have bi-stability. 01:07:50.030 --> 01:07:53.417 So what does that mean in terms of these lines if we draw them? 01:07:57.730 --> 01:08:07.130 So this is payout as a function of the fraction that 01:08:07.130 --> 01:08:08.555 is playing the A strategy. 01:08:12.000 --> 01:08:13.100 Should the lines cross? 01:08:13.100 --> 01:08:13.620 Yes or no? 01:08:13.620 --> 01:08:15.997 Ready, three, two, one. 01:08:15.997 --> 01:08:16.580 AUDIENCE: Yes. 01:08:16.580 --> 01:08:20.898 PROFESSOR: Yes, and indeed, in principle, 01:08:20.898 --> 01:08:22.689 the math that we do in all these situations 01:08:22.689 --> 01:08:25.319 is kind of super simple. 01:08:25.319 --> 01:08:28.540 Yet it's easy to get confused about what's going on 01:08:28.540 --> 01:08:29.770 in all these situations. 01:08:29.770 --> 01:08:33.930 So the idea here is that if the population is A, that 01:08:33.930 --> 01:08:37.044 means that the A here is at 5. 01:08:41.390 --> 01:08:42.694 But then it goes down to 0. 01:08:47.140 --> 01:08:50.448 Whereas over here, B here is 3. 01:08:50.448 --> 01:08:51.364 And then it goes to 1. 01:08:59.200 --> 01:09:01.960 Because these two lines cross, does that 01:09:01.960 --> 01:09:06.970 mean that you have bi-stability? 01:09:06.970 --> 01:09:10.278 Ready, yes or no, three, two, one, 01:09:10.278 --> 01:09:10.819 AUDIENCE: No. 01:09:10.819 --> 01:09:13.780 PROFESSOR: No, and why not? 01:09:13.780 --> 01:09:15.210 AUDIENCE: [INAUDIBLE]. 01:09:15.210 --> 01:09:16.120 PROFESSOR: That's right, because you can also 01:09:16.120 --> 01:09:18.411 do the other thing, and then that leads to coexistence. 01:09:22.310 --> 01:09:26.270 Now, in some ways coexistence is the most subtle 01:09:26.270 --> 01:09:27.580 of the situations. 01:09:27.580 --> 01:09:30.955 And that's for an interesting reason. 01:09:30.955 --> 01:09:32.746 AUDIENCE: Sorry, sir, you said you can also 01:09:32.746 --> 01:09:33.537 do the other thing. 01:09:33.537 --> 01:09:34.830 What is the other thing here? 01:09:37.398 --> 01:09:39.189 PROFESSOR: I'm saying that these things can 01:09:39.189 --> 01:09:42.250 cross in the other orientation. 01:09:42.250 --> 01:09:52.734 Let me put a matrix out there, and then-- 01:09:52.734 --> 01:09:56.130 so this is something that, for example, is what's 01:09:56.130 --> 01:09:57.480 known as a Hawk/Dove game. 01:09:57.480 --> 01:09:58.755 Or it has many other names. 01:10:12.341 --> 01:10:14.840 And we can maybe figure out what would be the Hawk strategy, 01:10:14.840 --> 01:10:16.048 and what's the Dove strategy. 01:10:19.710 --> 01:10:23.080 Now, we want to ask the same question-- is A a Nash 01:10:23.080 --> 01:10:23.630 equilibrium? 01:10:23.630 --> 01:10:25.480 Is B a Nash equilibrium? 01:10:25.480 --> 01:10:26.440 Is it neither? 01:10:26.440 --> 01:10:27.300 Or is it both? 01:10:27.300 --> 01:10:30.104 And maybe I shouldn't have covered this up, 01:10:30.104 --> 01:10:32.020 so you're not influenced, in case you actually 01:10:32.020 --> 01:10:34.290 did do the reading. 01:10:34.290 --> 01:10:37.090 Then I don't want to you to be influenced by this. 01:10:40.960 --> 01:10:42.590 So think about it for 30 seconds. 01:11:09.430 --> 01:11:11.047 Do you need more time? 01:11:11.047 --> 01:11:12.130 Let's go see where we are. 01:11:12.130 --> 01:11:16.650 Ready, three, two, one. 01:11:16.650 --> 01:11:19.200 All right, so most of the group is agreeing in this case, 01:11:19.200 --> 01:11:21.320 neither are the Nash equilibrium. 01:11:24.590 --> 01:11:29.320 So neither are a Nash equilibrium. 01:11:29.320 --> 01:11:32.810 Does that mean that this game has no Nash equilibrium? 01:11:32.810 --> 01:11:36.556 Yes or no, verbally-- ready, three, two, one. 01:11:36.556 --> 01:11:37.270 AUDIENCE: No. 01:11:37.270 --> 01:11:39.140 PROFESSOR: No, it does not mean that. 01:11:39.140 --> 01:11:40.920 This game has a Nash equilibrium. 01:11:40.920 --> 01:11:43.505 And indeed, all games like this have Nash equilibria. 01:11:47.200 --> 01:11:49.270 And this is what Nash won the Nobel Prize for, 01:11:49.270 --> 01:11:53.660 so this is the famous one-page paper published in PNAS. 01:11:53.660 --> 01:11:56.500 If you look at it, I have no idea what it says. 01:12:00.510 --> 01:12:03.930 I mean, he basically just pointed out 01:12:03.930 --> 01:12:07.950 that this theorem implies this, implies that-- done. 01:12:07.950 --> 01:12:14.390 And so it's good that somebody knew what he was saying, 01:12:14.390 --> 01:12:17.050 otherwise we'd be in trouble, all of us. 01:12:17.050 --> 01:12:20.760 So what he proved is that such games, 01:12:20.760 --> 01:12:26.050 even with more players, more options, and so forth, 01:12:26.050 --> 01:12:30.860 they always have such a solution in this sense. 01:12:30.860 --> 01:12:32.870 There exists some strategy such as if everybody 01:12:32.870 --> 01:12:34.870 were playing in, nobody would have the incentive 01:12:34.870 --> 01:12:36.290 to change strategy. 01:12:36.290 --> 01:12:39.330 But you have to include so-called probabilistic or 01:12:39.330 --> 01:12:40.180 mixed strategies. 01:12:46.400 --> 01:12:49.350 And we can draw what this thing is. 01:12:49.350 --> 01:12:53.590 So just like always, so everyone else is following A, 01:12:53.590 --> 01:12:55.045 then A starts here at 3. 01:12:58.170 --> 01:12:59.660 And then it goes to 1. 01:13:04.010 --> 01:13:07.830 Whereas the B individuals start at 5, and they go to 0. 01:13:14.220 --> 01:13:17.530 So this looks very similar to that, 01:13:17.530 --> 01:13:19.720 but they're rather different, in the sense 01:13:19.720 --> 01:13:26.120 that in this situation, we had bi-stability. 01:13:26.120 --> 01:13:31.410 So if you look at the direction of evolution, 01:13:31.410 --> 01:13:32.970 depending upon where you start, you 01:13:32.970 --> 01:13:36.330 go to either all B or all A. Whereas 01:13:36.330 --> 01:13:41.210 in this situation over here, we have coexistence. 01:13:41.210 --> 01:13:42.840 Does not matter where you start. 01:13:42.840 --> 01:13:44.940 So long as you have some members of both A and B 01:13:44.940 --> 01:13:47.810 in the population, you'll always evolve to the same equilibrium. 01:13:47.810 --> 01:13:50.860 Now, the important thing here that's, I think, 01:13:50.860 --> 01:13:54.220 interesting is that in a population, 01:13:54.220 --> 01:13:57.430 if you have genetic A's and genetic B's that are each 01:13:57.430 --> 01:14:00.840 giving birth to their own type, then 01:14:00.840 --> 01:14:05.020 you evolve to some coexistence of genotypes. 01:14:05.020 --> 01:14:08.020 So here, this is some fraction. 01:14:08.020 --> 01:14:17.677 f a star is the equilibrium fraction, fraction 01:14:17.677 --> 01:14:18.635 of A in the population. 01:14:21.240 --> 01:14:25.320 So this is a case where you have genetic diversity that 01:14:25.320 --> 01:14:28.590 leads to phenotypic diversity in the population. 01:14:28.590 --> 01:14:34.430 Whereas the mixed Nash equilibrium-- this 01:14:34.430 --> 01:14:37.560 is a situation where you have, in principle, 01:14:37.560 --> 01:14:41.020 genetic homogeneity. 01:14:41.020 --> 01:14:48.849 So this is a single genotype that is implementing 01:14:48.849 --> 01:14:49.890 phenotypic heterogeneity. 01:14:58.174 --> 01:14:59.840 And indeed, one of the things that we've 01:14:59.840 --> 01:15:01.860 been excited about exploring in my group 01:15:01.860 --> 01:15:09.160 is this distinction here, where it's known that in many cases, 01:15:09.160 --> 01:15:12.360 isogenic populations of microbes can exhibit 01:15:12.360 --> 01:15:15.100 a diversity of phenotypes as a result of, for example, 01:15:15.100 --> 01:15:17.555 stochastic gene expression and bi-stability. 01:15:21.150 --> 01:15:23.230 So that's a molecular mechanism for how 01:15:23.230 --> 01:15:25.400 you might get heterogeneity. 01:15:25.400 --> 01:15:28.280 Another question is, what is the evolution explanation 01:15:28.280 --> 01:15:31.140 for why that behavior might have evolved? 01:15:31.140 --> 01:15:33.820 Now in general, we cannot prove why something evolved, 01:15:33.820 --> 01:15:37.390 but we can make educated guesses that make experimentally 01:15:37.390 --> 01:15:38.684 testable hypotheses. 01:15:38.684 --> 01:15:40.100 And for example, in the experiment 01:15:40.100 --> 01:15:44.360 that we've been doing, we've been looking at bi-modality 01:15:44.360 --> 01:15:48.230 in expression of the galactose genes in yeast. 01:15:48.230 --> 01:15:53.815 And that was still a problem set in early-- oh, 01:15:53.815 --> 01:15:55.700 no, we removed that one this year. 01:15:55.700 --> 01:16:02.880 Well, so experimentally yeast, in some environments, 01:16:02.880 --> 01:16:04.840 bimodally or stochastically activate 01:16:04.840 --> 01:16:07.470 the genes required to break down the sugar galactose. 01:16:07.470 --> 01:16:09.600 And what we've demonstrated is that if you 01:16:09.600 --> 01:16:13.590 make the mutants that always turn on or always don't 01:16:13.590 --> 01:16:16.160 turn on these genes, then they're actually playing game 01:16:16.160 --> 01:16:20.050 where you actually get this exact thing-- where 01:16:20.050 --> 01:16:22.640 you get evolution towards coexistence of those two 01:16:22.640 --> 01:16:23.740 strategies. 01:16:23.740 --> 01:16:26.070 So that's saying that maybe the wild type that 01:16:26.070 --> 01:16:29.270 follows this stochastic, mixed strategy-- it 01:16:29.270 --> 01:16:32.210 may be implementing the solution of some game that 01:16:32.210 --> 01:16:34.640 is a result of such frequency dependence. 01:16:34.640 --> 01:16:36.720 There are other possible explanations to this. 01:16:36.720 --> 01:16:38.760 In the coming weeks, we'll talk about this idea 01:16:38.760 --> 01:16:41.801 of bet hedging-- that given uncertain or fluctuating 01:16:41.801 --> 01:16:43.300 environments, it may be advantageous 01:16:43.300 --> 01:16:46.520 for clonal populations to have a variety of different strategies 01:16:46.520 --> 01:16:47.890 to cope with that uncertainty. 01:16:47.890 --> 01:16:50.225 So we'll talk about those models later. 01:16:50.225 --> 01:16:52.350 But since we're talking about mixed strategies now, 01:16:52.350 --> 01:16:53.647 I wanted to mention that. 01:16:53.647 --> 01:16:54.810 Yeah. 01:16:54.810 --> 01:16:56.658 AUDIENCE: So f a star, just to be sure, 01:16:56.658 --> 01:16:59.440 is going to converge to this probability [INAUDIBLE]? 01:16:59.440 --> 01:17:04.380 PROFESSOR: Exactly, so the Nash equilibrium mixed strategy 01:17:04.380 --> 01:17:08.680 plays A with probability-- so it's p should 01:17:08.680 --> 01:17:10.020 be equal to f a star. 01:17:13.260 --> 01:17:16.710 Exactly, so indeed, the heterogeneity there 01:17:16.710 --> 01:17:19.980 can be implemented either way. 01:17:19.980 --> 01:17:22.516 It's either coexistence of genotypes following 01:17:22.516 --> 01:17:23.890 different strategies, or it could 01:17:23.890 --> 01:17:25.587 be one genotype implementing both, 01:17:25.587 --> 01:17:27.420 or it could be a mixture of those, actually. 01:17:27.420 --> 01:17:30.670 Indeed, a characteristic of these situations 01:17:30.670 --> 01:17:38.070 is that, let's say that you have a genotype-- a population has 01:17:38.070 --> 01:17:40.810 this genotype that is implementing the mixed Nash 01:17:40.810 --> 01:17:44.680 equilibrium choosing strategy A with probability p. 01:17:44.680 --> 01:17:46.400 That's that equilibrium fraction. 01:17:46.400 --> 01:17:49.530 What's interesting is that any individual 01:17:49.530 --> 01:17:54.850 in the population following any strategy has the same fitness. 01:17:54.850 --> 01:17:58.280 And of course, that's kind of why this was in equilibrium. 01:17:58.280 --> 01:18:01.295 This equilibrium is when the two strategies have equal fitness. 01:18:01.295 --> 01:18:02.920 But the funny thing is, what that means 01:18:02.920 --> 01:18:05.137 is, it doesn't matter what you do at the equilibrium. 01:18:05.137 --> 01:18:06.970 Depending on how you look at it, it's either 01:18:06.970 --> 01:18:09.030 super deep or super trivial. 01:18:09.030 --> 01:18:11.630 But it's a weird thing that if your at the equilibrium, 01:18:11.630 --> 01:18:13.520 or if the population or the opponent 01:18:13.520 --> 01:18:15.942 is playing this Nash equilibrium in these games, 01:18:15.942 --> 01:18:17.650 then it just does not matter what you do. 01:18:17.650 --> 01:18:21.920 You can do A. You can do B, actually, in any fraction. 01:18:21.920 --> 01:18:23.810 So since A and B have the same fitness, 01:18:23.810 --> 01:18:26.480 you can choose between them at any frequency you want, 01:18:26.480 --> 01:18:29.270 and you have the same fitness, if the rest of the population 01:18:29.270 --> 01:18:33.330 is playing this mixed Nash equilibrium. 01:18:33.330 --> 01:18:37.490 And indeed, in this there are nice conditions 01:18:37.490 --> 01:18:42.070 for what makes it this Nash equilibrium. 01:18:42.070 --> 01:18:46.770 And I'm going to just highlight that you should make sense 01:18:46.770 --> 01:18:49.910 of why it means what it is. 01:18:49.910 --> 01:18:54.690 So if the payout, the expected fitness or payout-- 01:18:54.690 --> 01:18:56.820 if you're following the Nash equilibrium 01:18:56.820 --> 01:19:03.890 against Nash equilibrium-- is equal to this guy. 01:19:03.890 --> 01:19:07.230 So that's what I just said-- that it 01:19:07.230 --> 01:19:09.700 doesn't matter what you do. 01:19:09.700 --> 01:19:11.490 If everyone else is doing p star, 01:19:11.490 --> 01:19:14.370 you have the same fitness. 01:19:14.370 --> 01:19:16.360 So that's saying it's a Nash equilibrium. 01:19:19.430 --> 01:19:22.980 Whereas there's another interesting kind of statement 01:19:22.980 --> 01:19:25.010 here, that-- 01:19:25.010 --> 01:19:27.435 AUDIENCE: [INAUDIBLE]? 01:19:27.435 --> 01:19:30.587 That you can't unilaterally increase your fitness 01:19:30.587 --> 01:19:32.770 by switching. 01:19:32.770 --> 01:19:34.900 PROFESSOR: Right, it's an equality, which 01:19:34.900 --> 01:19:36.250 means it is a Nash equilibrium. 01:19:36.250 --> 01:19:38.599 Because it's saying that you don't have the incentive 01:19:38.599 --> 01:19:39.390 to change strategy. 01:19:39.390 --> 01:19:42.080 It's true that you're not dis-incentivized. 01:19:42.080 --> 01:19:43.810 So it's a Nash equilibrium. 01:19:43.810 --> 01:19:45.914 It's not a strict Nash equilibrium. 01:19:45.914 --> 01:19:46.830 AUDIENCE: [INAUDIBLE]. 01:19:50.754 --> 01:19:52.920 PROFESSOR: Well, it has to be greater than/equal to, 01:19:52.920 --> 01:19:55.084 and it's actually equal to. 01:19:55.084 --> 01:19:57.250 The condition it has to be greater than or equal to, 01:19:57.250 --> 01:20:00.239 but it's equal to, which means it is a Nash equilibrium. 01:20:00.239 --> 01:20:03.017 AUDIENCE: [INAUDIBLE] Nash equilibrium for that situation, 01:20:03.017 --> 01:20:04.410 [INAUDIBLE] strategy. 01:20:04.410 --> 01:20:04.710 PROFESSOR: That's right. 01:20:04.710 --> 01:20:06.293 So this is not the definition of that. 01:20:06.293 --> 01:20:08.070 But this thing is true, which means 01:20:08.070 --> 01:20:09.278 that it's a Nash equilibrium. 01:20:12.500 --> 01:20:18.690 And this other thing that's interesting is that-- 01:20:18.690 --> 01:20:21.690 so this tells us that it's actually one of these ESS's. 01:20:21.690 --> 01:20:28.340 And if you have questions about this, I'm happy to answer it. 01:20:28.340 --> 01:20:30.640 It's explained in the book as well. 01:20:30.640 --> 01:20:32.700 We are out of time, so I should let you go. 01:20:32.700 --> 01:20:35.645 But good luck on the exam next Thursday. 01:20:35.645 --> 01:20:37.770 If you have questions and you want to meet with me, 01:20:37.770 --> 01:20:38.811 I'm available on Tuesday. 01:20:38.811 --> 01:20:40.940 So please let me know.