1 00:00:00,125 --> 00:00:01,780 The following content is provided 2 00:00:01,780 --> 00:00:04,019 under a creative commons license. 3 00:00:04,019 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,730 continue to offer high quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,217 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,217 --> 00:00:17,842 at ocw.mit.edu. 8 00:00:20,976 --> 00:00:23,160 PROFESSOR: So today the basic idea 9 00:00:23,160 --> 00:00:27,050 is to try to understand the toggle switch. 10 00:00:27,050 --> 00:00:30,260 How such a thing can be made, why it represents a memory 11 00:00:30,260 --> 00:00:31,656 module. 12 00:00:31,656 --> 00:00:33,030 But then we'll relatively quickly 13 00:00:33,030 --> 00:00:35,209 get into two themes that are going 14 00:00:35,209 --> 00:00:37,250 to be useful throughout the rest of the semester. 15 00:00:37,250 --> 00:00:41,960 First is these dimensionless equations 16 00:00:41,960 --> 00:00:44,780 that often pop up in the analysis of these gene 17 00:00:44,780 --> 00:00:46,130 circuits. 18 00:00:46,130 --> 00:00:49,880 And it's absolutely essential that you understand 19 00:00:49,880 --> 00:00:52,220 how to get to the dimensionless equations, 20 00:00:52,220 --> 00:00:58,140 and also what these parameters end up meaning. 21 00:00:58,140 --> 00:01:00,850 And then we'll, at the end, talk about stability analysis. 22 00:01:00,850 --> 00:01:05,010 How is it that you can determine whether a particular set 23 00:01:05,010 --> 00:01:09,060 of interacting pieces in a cell or in an ecosystem 24 00:01:09,060 --> 00:01:11,600 whatnot-- once you get it in the form of an equation, how's 25 00:01:11,600 --> 00:01:13,480 is it you can determine whether a particular fixed point 26 00:01:13,480 --> 00:01:15,150 or a particular location is going 27 00:01:15,150 --> 00:01:17,696 to be stable to perturbations? 28 00:01:17,696 --> 00:01:19,070 This is going to be useful for us 29 00:01:19,070 --> 00:01:22,540 to determine where the gene network is 30 00:01:22,540 --> 00:01:24,850 going to kind of move towards. 31 00:01:24,850 --> 00:01:27,600 Also it'll be useful for us to determine whether a gene 32 00:01:27,600 --> 00:01:29,140 network is going to oscillate. 33 00:01:29,140 --> 00:01:32,710 And later, it'll be relevant in the context of predator prey 34 00:01:32,710 --> 00:01:34,280 oscillations, and a bunch of things. 35 00:01:39,820 --> 00:01:44,450 So can somebody maybe explain briefly 36 00:01:44,450 --> 00:01:45,850 the idea of the toggle switch? 37 00:01:49,150 --> 00:01:49,720 Yes, please. 38 00:01:49,720 --> 00:01:53,154 AUDIENCE: You have these two genes that repress expression 39 00:01:53,154 --> 00:01:54,360 to each other. 40 00:01:54,360 --> 00:01:57,216 So when one of them is large, the other one 41 00:01:57,216 --> 00:01:58,494 is not expressed. [INAUDIBLE]. 42 00:01:58,494 --> 00:02:00,320 PROFESSOR: Perfect. 43 00:02:00,320 --> 00:02:03,111 So you'll have two genes that are going to mutually repress 44 00:02:03,111 --> 00:02:03,610 each other. 45 00:02:03,610 --> 00:02:08,190 So in this case, each of them will be a transcription factor 46 00:02:08,190 --> 00:02:11,080 of some sort that will bind to the other promoter 47 00:02:11,080 --> 00:02:11,910 and repress it. 48 00:02:11,910 --> 00:02:14,985 So there are various levels of abstraction 49 00:02:14,985 --> 00:02:17,030 that we might use to describe such things. 50 00:02:17,030 --> 00:02:20,370 So we might, for example, just say A repressing B, 51 00:02:20,370 --> 00:02:23,109 B repressing A. Of course when you write it that way, 52 00:02:23,109 --> 00:02:25,400 it doesn't have to be in the context of a gene network. 53 00:02:25,400 --> 00:02:29,030 This A's and B's could just be chemicals, 54 00:02:29,030 --> 00:02:31,780 they could be species eating each other. 55 00:02:31,780 --> 00:02:34,130 It could be almost anything. 56 00:02:34,130 --> 00:02:36,940 Now in this framework, to get a basic sense 57 00:02:36,940 --> 00:02:41,560 of why this thing might have two alternative stable states, 58 00:02:41,560 --> 00:02:45,360 is that often we like to take the Boolean approximation. 59 00:02:45,360 --> 00:02:48,560 This thing will not always work, but it's a useful thing 60 00:02:48,560 --> 00:02:52,120 to do to just kind of first get a sense of what might possibly 61 00:02:52,120 --> 00:02:52,710 be happening. 62 00:02:52,710 --> 00:02:56,420 So we might say, 0 corresponds to some sort of low. 63 00:02:56,420 --> 00:02:58,630 1 might correspond to high. 64 00:02:58,630 --> 00:03:00,559 And of course, these things have to be put it 65 00:03:00,559 --> 00:03:03,100 in quotes, because we haven't specified what we mean by this. 66 00:03:03,100 --> 00:03:04,840 But it's useful to just make sure 67 00:03:04,840 --> 00:03:07,140 that we're all thinking about the same things. 68 00:03:07,140 --> 00:03:08,610 And then in the context of A and B, 69 00:03:08,610 --> 00:03:11,190 we can just say, well, there's a number of different states 70 00:03:11,190 --> 00:03:13,690 it could possibly be in. 71 00:03:13,690 --> 00:03:19,650 And you can ask whether this assignment of logic values 72 00:03:19,650 --> 00:03:23,050 keeps everybody happy. 73 00:03:23,050 --> 00:03:30,274 And so you might ask, well, is 0, 0 a mutually happy state? 74 00:03:30,274 --> 00:03:31,690 And of course then you have to say 75 00:03:31,690 --> 00:03:35,630 well, if you're in the 0 state, you're 76 00:03:35,630 --> 00:03:39,252 not repressing the other guy, but maybe 0 77 00:03:39,252 --> 00:03:40,710 is sort of your equilibrium anyway. 78 00:03:40,710 --> 00:03:43,330 So what we have to do is we have to make the assumption 79 00:03:43,330 --> 00:03:46,670 that when you're not being repressed, in that case 80 00:03:46,670 --> 00:03:49,820 the promoter will be actively making that protein. 81 00:03:49,820 --> 00:03:52,845 So then you'll go to some sort of high or 1 state. 82 00:03:52,845 --> 00:03:55,470 So in that case, you say, if you start out with both repressed, 83 00:03:55,470 --> 00:03:57,910 maybe both of them should start trying 84 00:03:57,910 --> 00:04:00,430 to increase their levels. 85 00:04:00,430 --> 00:04:03,610 So this, in some ways, is not a stable state. 86 00:04:03,610 --> 00:04:05,730 Similarly here, this is also not a stable state. 87 00:04:05,730 --> 00:04:07,460 Because in this case, they're both 88 00:04:07,460 --> 00:04:09,550 going to be trying to repress one another. 89 00:04:09,550 --> 00:04:11,133 So then they'll both start coming down 90 00:04:11,133 --> 00:04:14,080 and then the situation may resolve into one of these two. 91 00:04:14,080 --> 00:04:18,019 So this is just where either A or B is on, 92 00:04:18,019 --> 00:04:19,390 and repressing the other one. 93 00:04:19,390 --> 00:04:22,010 And so for example, in context of the repressilator, 94 00:04:22,010 --> 00:04:24,730 on Thursday, this is just a useful way 95 00:04:24,730 --> 00:04:28,350 to start imagining how this A repressing B, 96 00:04:28,350 --> 00:04:30,790 repressing C repressing A-- how such a loop can 97 00:04:30,790 --> 00:04:32,780 lead to oscillations. 98 00:04:32,780 --> 00:04:34,840 This kind of analysis does not at all 99 00:04:34,840 --> 00:04:38,334 prove that there are oscillations in any given 100 00:04:38,334 --> 00:04:39,500 manifestation of this thing. 101 00:04:39,500 --> 00:04:40,874 But it's useful to just make sure 102 00:04:40,874 --> 00:04:44,480 that you're roughly getting the idea of what the system might 103 00:04:44,480 --> 00:04:46,440 be doing. 104 00:04:46,440 --> 00:04:50,250 Now in many cases, we'll want to be a little bit more explicit, 105 00:04:50,250 --> 00:04:53,740 and draw the gene network in more detail. 106 00:04:53,740 --> 00:04:55,946 And there are multiple manifestations 107 00:04:55,946 --> 00:04:56,820 of the Thomas Switch. 108 00:04:56,820 --> 00:04:58,270 There are many of them that have been made. 109 00:04:58,270 --> 00:05:00,170 So the important thing is not too necessarily 110 00:05:00,170 --> 00:05:03,990 keep track of exactly what the components are, 111 00:05:03,990 --> 00:05:08,440 but in one case for example, we might have A corresponding 112 00:05:08,440 --> 00:05:12,140 to something here. 113 00:05:12,140 --> 00:05:16,030 It's coming back and repressing again. 114 00:05:16,030 --> 00:05:20,720 Expression of this B. And this might all 115 00:05:20,720 --> 00:05:23,790 be on one piece of DNA. 116 00:05:23,790 --> 00:05:29,110 Whereas this B here will come back and repress A. 117 00:05:29,110 --> 00:05:33,260 Now one thing that is-- and I just want to mention here, 118 00:05:33,260 --> 00:05:34,970 they also have a GFP. 119 00:05:38,080 --> 00:05:40,440 And this actually is a case where 120 00:05:40,440 --> 00:05:44,480 those two can be expressed off of a single promoter. 121 00:05:44,480 --> 00:05:46,460 So this is often done in bacteria 122 00:05:46,460 --> 00:05:49,100 where there's a single promoter-- 123 00:05:49,100 --> 00:05:53,380 so RNA polymerase actually will transcribe both of these genes. 124 00:05:53,380 --> 00:05:57,060 This repressor B, as well as this fluorescent protein. 125 00:05:57,060 --> 00:05:59,630 So there are going to be alternative loading sites 126 00:05:59,630 --> 00:06:01,600 for the ribosome in that case. 127 00:06:01,600 --> 00:06:03,880 And eukaryotes typically do not do this. 128 00:06:06,657 --> 00:06:07,156 Yeah? 129 00:06:07,156 --> 00:06:08,092 AUDIENCE: [INAUDIBLE]. 130 00:06:12,011 --> 00:06:13,010 PROFESSOR: That's right. 131 00:06:13,010 --> 00:06:13,280 So-- 132 00:06:13,280 --> 00:06:15,405 AUDIENCE: And it will transcribe everything until-- 133 00:06:15,405 --> 00:06:16,500 PROFESSOR: Exactly. 134 00:06:16,500 --> 00:06:21,030 So here comes-- so an RNA polymerase down here 135 00:06:21,030 --> 00:06:22,210 made this whole thing. 136 00:06:22,210 --> 00:06:24,410 And now you might have two separate locations where 137 00:06:24,410 --> 00:06:28,840 the ribosome loads and makes this protein B. 138 00:06:28,840 --> 00:06:31,257 And then a different ribosome would make this GFP. 139 00:06:31,257 --> 00:06:32,590 AUDIENCE: And when does it stop? 140 00:06:32,590 --> 00:06:33,631 It just keeps going down? 141 00:06:33,631 --> 00:06:34,590 PROFESSOR: Yes. 142 00:06:34,590 --> 00:06:36,398 So there's a termination sequence. 143 00:06:36,398 --> 00:06:38,314 AUDIENCE: No, no, but it would B and then GFP, 144 00:06:38,314 --> 00:06:39,170 and then it would just keep going? 145 00:06:39,170 --> 00:06:39,880 PROFESSOR: No. 146 00:06:39,880 --> 00:06:43,550 The ribosome is told to basically start here and end 147 00:06:43,550 --> 00:06:44,150 here. 148 00:06:44,150 --> 00:06:46,100 So then it just makes the B protein. 149 00:06:46,100 --> 00:06:48,926 And then another ribosome binds here. 150 00:06:48,926 --> 00:06:52,610 AUDIENCE: If you had more proteins on the same strand 151 00:06:52,610 --> 00:06:53,540 after GFP-- 152 00:06:53,540 --> 00:06:54,350 PROFESSOR: And when you say protein, 153 00:06:54,350 --> 00:06:55,210 you're referring to the ribosome. 154 00:06:55,210 --> 00:06:56,527 [INTERPOSING VOICES] 155 00:06:56,527 --> 00:06:57,837 AUDIENCE: I mean G's right? 156 00:06:57,837 --> 00:06:58,420 PROFESSOR: Oh! 157 00:06:58,420 --> 00:07:00,294 OK, you're saying if there were another gene? 158 00:07:00,294 --> 00:07:02,660 AUDIENCE: No, if you have more genes coded, 159 00:07:02,660 --> 00:07:04,950 if you're coding for more proteins after GFP-- 160 00:07:04,950 --> 00:07:06,600 if you have more genes on a 161 00:07:06,600 --> 00:07:08,226 [INTERPOSING VOICES] 162 00:07:08,226 --> 00:07:12,816 --it will just keep going for an arbitrarily long-- 163 00:07:12,816 --> 00:07:14,991 [INAUDIBLE] 164 00:07:14,991 --> 00:07:16,990 PROFESSOR: Arbitrary is always a dangerous word. 165 00:07:16,990 --> 00:07:18,595 But they can be more than two. 166 00:07:18,595 --> 00:07:21,650 And actually at the biophysics retreat that some of you guys 167 00:07:21,650 --> 00:07:24,320 were at just last two days, there was a great talk 168 00:07:24,320 --> 00:07:27,140 by Gene-Wei Li, who's going to be a new incoming biology 169 00:07:27,140 --> 00:07:27,890 faculty member. 170 00:07:27,890 --> 00:07:32,310 And he was talking about the FO F1 ATP synthase. 171 00:07:32,310 --> 00:07:34,635 So it's the thing responsible for making ATP. 172 00:07:37,560 --> 00:07:38,710 He analyzes process work. 173 00:07:38,710 --> 00:07:39,950 And there are many sub units. 174 00:07:39,950 --> 00:07:41,570 So there are half a dozen or so. 175 00:07:41,570 --> 00:07:44,190 And so it's a very long transcript. 176 00:07:44,190 --> 00:07:46,330 And then what he showed is that actually you 177 00:07:46,330 --> 00:07:48,180 have different rates of synthesis 178 00:07:48,180 --> 00:07:51,620 of the different genes on this one transcript. 179 00:07:51,620 --> 00:07:55,590 And in some cases you want actually more copies of one 180 00:07:55,590 --> 00:07:57,340 of the subunits than another one. 181 00:07:57,340 --> 00:08:01,030 And so then actually if the final protein, 182 00:08:01,030 --> 00:08:03,980 if it needs 12 of these, only one of these, then 183 00:08:03,980 --> 00:08:05,990 actually you can make 12 times as much of this, 184 00:08:05,990 --> 00:08:08,460 because you just have more translation 185 00:08:08,460 --> 00:08:09,460 here than you did here. 186 00:08:09,460 --> 00:08:10,640 And then it's great, because then you 187 00:08:10,640 --> 00:08:12,514 have all the right ratios, all the components 188 00:08:12,514 --> 00:08:14,260 to make the protein. 189 00:08:14,260 --> 00:08:16,220 So you can actually have additional regulation 190 00:08:16,220 --> 00:08:17,510 even at that stage. 191 00:08:21,262 --> 00:08:23,470 It's possible not everybody followed that discussion, 192 00:08:23,470 --> 00:08:24,810 and my apologies. 193 00:08:24,810 --> 00:08:27,180 But feel free to just erase it from your brain 194 00:08:27,180 --> 00:08:31,520 if you're too confused. 195 00:08:31,520 --> 00:08:36,500 But what you need to keep track of here is the level of GFP 196 00:08:36,500 --> 00:08:41,730 is going to be perhaps proportional to the level of B. 197 00:08:41,730 --> 00:08:46,160 Because they're being expressed at the same time. 198 00:08:46,160 --> 00:08:50,460 Now in order for this thing to be a memory module, 199 00:08:50,460 --> 00:08:52,970 you also want to be able to reset the state. 200 00:08:52,970 --> 00:08:54,759 So if you were in this state, you'd 201 00:08:54,759 --> 00:08:57,050 like to be able to somehow get it to move to this state 202 00:08:57,050 --> 00:08:58,440 instead. 203 00:08:58,440 --> 00:09:00,190 Does anybody remember what the inputs 204 00:09:00,190 --> 00:09:04,030 were in the context of the sample toggle switch, 205 00:09:04,030 --> 00:09:07,330 that it was in that review? 206 00:09:07,330 --> 00:09:10,240 AUDIENCE: [INAUDIBLE]. 207 00:09:10,240 --> 00:09:11,310 PROFESSOR: Right. 208 00:09:11,310 --> 00:09:14,530 OK, so there are multiple versions of toggle switch. 209 00:09:14,530 --> 00:09:19,120 And indeed in one case this was just a small molecule, IPTG, 210 00:09:19,120 --> 00:09:25,570 and that's because this was in that case, the lac I. 211 00:09:25,570 --> 00:09:28,950 And this then represses the repression. 212 00:09:28,950 --> 00:09:30,610 So in many contexts this class, you'll 213 00:09:30,610 --> 00:09:34,890 have to remember that a minus, minus is equal to a plus. 214 00:09:34,890 --> 00:09:38,850 And in different toggle switches you indeed 215 00:09:38,850 --> 00:09:42,600 have different ways of inhibiting this repression. 216 00:09:42,600 --> 00:09:44,600 So this could be another small molecule, ATCN. 217 00:09:44,600 --> 00:09:48,970 In the example that they had in this review, 218 00:09:48,970 --> 00:09:52,310 it was actually heat that did this. 219 00:09:52,310 --> 00:09:54,909 But that's just because this transcription factor was 220 00:09:54,909 --> 00:09:56,200 a temperature sensitive mutant. 221 00:09:56,200 --> 00:09:58,450 So above some temperature, it could no longer repress. 222 00:10:01,740 --> 00:10:06,370 So for example, if you start out in high 223 00:10:06,370 --> 00:10:10,050 GFP-- so in this case, where they-- all right. 224 00:10:12,810 --> 00:10:16,740 If the cells are in high GFP, and you 225 00:10:16,740 --> 00:10:21,640 want to switch it into the alternative state, 226 00:10:21,640 --> 00:10:25,660 what stimulus do you want to apply here? 227 00:10:25,660 --> 00:10:27,169 So I'll give you a guess. 228 00:10:27,169 --> 00:10:28,710 It's going to be either heat or IPTG. 229 00:10:32,700 --> 00:10:35,290 I just want to make sure that we can all read these diagrams. 230 00:10:38,660 --> 00:10:42,990 This is to switch from the high GFP, 231 00:10:42,990 --> 00:10:46,210 and we want to go to the low GFP. 232 00:10:46,210 --> 00:10:50,815 Give you 15 seconds to just try to read off this diagram. 233 00:10:59,900 --> 00:11:01,103 Do you need more time? 234 00:11:01,103 --> 00:11:02,020 No. 235 00:11:02,020 --> 00:11:02,520 Ready. 236 00:11:02,520 --> 00:11:06,730 Three, two, one. 237 00:11:06,730 --> 00:11:10,170 So we have a majority are B. So a majority of people 238 00:11:10,170 --> 00:11:15,120 are saying, well, in this case, you have a lot of GFP. 239 00:11:15,120 --> 00:11:16,620 Means you have a lot of this protein 240 00:11:16,620 --> 00:11:19,810 B. In this case, the lac I. 241 00:11:19,810 --> 00:11:24,330 And we don't have very much of this other protein. 242 00:11:24,330 --> 00:11:26,840 So that means that if we want to switch the state, 243 00:11:26,840 --> 00:11:31,160 and get a lot of A, we have to stop this repression. 244 00:11:31,160 --> 00:11:34,440 So we have to add IPTG. 245 00:11:34,440 --> 00:11:37,740 Any questions about what we mean by the various symbols 246 00:11:37,740 --> 00:11:38,810 up on this board? 247 00:11:47,445 --> 00:11:55,962 So after we had IPTG, then indeed the GFP should go down. 248 00:11:55,962 --> 00:11:57,920 How long is it going to take for it to go down? 249 00:11:57,920 --> 00:11:59,505 Does this switch go down immediately? 250 00:12:02,100 --> 00:12:04,570 After you add the IPTG, how long do you 251 00:12:04,570 --> 00:12:07,210 think it's going to take for this repressor, lac 252 00:12:07,210 --> 00:12:10,410 I to fall off of that promoter? 253 00:12:10,410 --> 00:12:16,600 Do you think it's going to be seconds or hours? 254 00:12:16,600 --> 00:12:18,480 It's actually seconds, and that's 255 00:12:18,480 --> 00:12:22,470 because this IPTG rapidly can go across the membrane, 256 00:12:22,470 --> 00:12:25,660 it'll rapidly bind to the inhibitor lac I, 257 00:12:25,660 --> 00:12:28,730 and then lac I will fall off. 258 00:12:28,730 --> 00:12:31,720 So in this case, lac I is actually still present, 259 00:12:31,720 --> 00:12:35,600 so it's going to take hours, actually for lac I to go away. 260 00:12:35,600 --> 00:12:40,670 But it takes seconds for the lac I to become inactive, 261 00:12:40,670 --> 00:12:42,490 ineffective. 262 00:12:42,490 --> 00:12:45,640 So this is the separation of time scales idea. 263 00:12:45,640 --> 00:12:48,070 But how long is it going to take for the concentration 264 00:12:48,070 --> 00:12:52,070 of-- well, how long is it going to take for GFP to go away 265 00:12:52,070 --> 00:12:52,859 maybe? 266 00:12:52,859 --> 00:12:54,400 Is that going to be seconds or hours? 267 00:12:54,400 --> 00:12:55,346 AUDIENCE: [INAUDIBLE]. 268 00:12:55,346 --> 00:12:59,130 PROFESSOR: Yeah, In this environment the bacterium-- 269 00:12:59,130 --> 00:13:03,300 they might be dividing every half hour. 270 00:13:03,300 --> 00:13:05,637 That's the characteristic time-- if these are stable, 271 00:13:05,637 --> 00:13:07,970 that would be the characteristic time scale for example, 272 00:13:07,970 --> 00:13:10,070 for GFP to go down. 273 00:13:10,070 --> 00:13:13,270 And of course, that's even after it's inhibited. 274 00:13:13,270 --> 00:13:15,080 After you stop making GFP. 275 00:13:15,080 --> 00:13:18,374 In this case, once you had the IPTG, the lac 276 00:13:18,374 --> 00:13:20,040 I is going to fall off of this promoter, 277 00:13:20,040 --> 00:13:22,530 and then we have to first make A. 278 00:13:22,530 --> 00:13:25,490 And only after we make A will we start repressing 279 00:13:25,490 --> 00:13:27,430 expression of the B and GFP. 280 00:13:31,580 --> 00:13:36,660 That's the process that you expect to take, hours, 281 00:13:36,660 --> 00:13:40,440 based on the generation times. 282 00:13:40,440 --> 00:13:43,210 But this is a memory module because after we've 283 00:13:43,210 --> 00:13:50,340 added the IPTG, now we can, in principle, take the IPTG away, 284 00:13:50,340 --> 00:13:51,270 and it'll stay low. 285 00:13:55,410 --> 00:13:57,750 So in principle, this IPTG signal 286 00:13:57,750 --> 00:13:59,700 can be a transient signal. 287 00:13:59,700 --> 00:14:02,000 And the cells will remember that they encountered IPTG. 288 00:14:04,960 --> 00:14:07,380 That's what makes this a memory. 289 00:14:07,380 --> 00:14:10,720 This input can be transient. 290 00:14:10,720 --> 00:14:12,250 Transient signal is remembered. 291 00:14:21,460 --> 00:14:25,080 Now a big part of why this why this paper was important 292 00:14:25,080 --> 00:14:27,970 is because this toggle switch was constructed out 293 00:14:27,970 --> 00:14:32,910 of components that previously in principle, had not even ever 294 00:14:32,910 --> 00:14:34,410 seen each other before. 295 00:14:34,410 --> 00:14:39,320 So maybe that lac I and this promoter had been put together, 296 00:14:39,320 --> 00:14:42,920 but this whole system, this whole gene network 297 00:14:42,920 --> 00:14:46,490 was composed of individual components 298 00:14:46,490 --> 00:14:48,809 that they were synthetic. 299 00:14:48,809 --> 00:14:50,600 And were put together because they thought, 300 00:14:50,600 --> 00:14:51,770 oh this should kind of work. 301 00:14:55,000 --> 00:14:58,260 And it's led to, I think a real flowering of, 302 00:14:58,260 --> 00:15:00,930 this intersection between modeling and experiment. 303 00:15:00,930 --> 00:15:04,830 This thing was built based on a model that told them 304 00:15:04,830 --> 00:15:07,720 that maybe if we do this, these things 305 00:15:07,720 --> 00:15:12,104 they multiplarize in order to repress and so forth. 306 00:15:12,104 --> 00:15:14,520 So there was a real sense in which the modeling-- and then 307 00:15:14,520 --> 00:15:16,940 we're going to talk more about the modeling-- on how 308 00:15:16,940 --> 00:15:21,560 it was essential to get an idea of how we should construct 309 00:15:21,560 --> 00:15:23,290 this thing, and to guide our work. 310 00:15:23,290 --> 00:15:25,780 Because if any of you do work in the lab, 311 00:15:25,780 --> 00:15:27,290 you'll know that things are hard. 312 00:15:27,290 --> 00:15:30,039 And often components don't behave 313 00:15:30,039 --> 00:15:31,830 the way you think they should and so forth. 314 00:15:31,830 --> 00:15:34,134 Now modeling can't save you from all that pain, 315 00:15:34,134 --> 00:15:36,300 but at least it can guide you in the right direction 316 00:15:36,300 --> 00:15:38,770 so that it limits the number of things you have to try. 317 00:15:41,810 --> 00:15:45,160 Are there any questions about this element, before we switch 318 00:15:45,160 --> 00:15:47,956 over to some of the dimensionless 319 00:15:47,956 --> 00:15:49,580 equations that we're going to be using? 320 00:15:54,920 --> 00:15:58,180 So the reading for today was composed 321 00:15:58,180 --> 00:16:01,200 of a review that these two pieces, 322 00:16:01,200 --> 00:16:03,940 and one of which I think might be hard for some people, 323 00:16:03,940 --> 00:16:06,590 and the other one might be hard for other people. 324 00:16:06,590 --> 00:16:10,540 For those of you with maybe more limited experimental experience 325 00:16:10,540 --> 00:16:13,130 in biology, maybe the review was a challenge 326 00:16:13,130 --> 00:16:15,380 to try to understand all of the nomenclature 327 00:16:15,380 --> 00:16:18,764 and the words, whereas those of you that have not 328 00:16:18,764 --> 00:16:20,930 played with differential equations as much recently, 329 00:16:20,930 --> 00:16:26,040 may have found the reading on modeling the toggle switch 330 00:16:26,040 --> 00:16:29,090 and stability analysis to be more challenging. 331 00:16:29,090 --> 00:16:30,760 We will, in some cases, do this where 332 00:16:30,760 --> 00:16:34,660 we have two different hopefully shorter kinds readings. 333 00:16:34,660 --> 00:16:37,280 And hopefully they're not both hard for you, 334 00:16:37,280 --> 00:16:39,280 because then you'll spend a lot of time reading. 335 00:16:39,280 --> 00:16:40,840 But then it will be good for you. 336 00:16:40,840 --> 00:16:44,480 So don't run away quite yet. 337 00:16:44,480 --> 00:16:47,250 But from my standpoint, it's essential 338 00:16:47,250 --> 00:16:51,020 that you develop intuition behind these ideas. 339 00:16:51,020 --> 00:16:54,736 So once you have this equation that somebody gives you, 340 00:16:54,736 --> 00:16:56,110 you have to be able to figure out 341 00:16:56,110 --> 00:17:00,620 what assumptions have they made, and what should the behavior be 342 00:17:00,620 --> 00:17:03,290 roughly, before you go off and you do simulations 343 00:17:03,290 --> 00:17:05,589 and full mathematical analysis. 344 00:17:05,589 --> 00:17:07,630 So a lot of what we're going to do in this class, 345 00:17:07,630 --> 00:17:09,650 and in particular during the lectures, 346 00:17:09,650 --> 00:17:12,234 is to try to work on that intuition. 347 00:17:12,234 --> 00:17:13,650 And the first thing you need to do 348 00:17:13,650 --> 00:17:16,490 is make sure that you know if you don't understand something. 349 00:17:16,490 --> 00:17:18,430 Sometimes things look really simple, 350 00:17:18,430 --> 00:17:20,240 especially these dimensionless equations. 351 00:17:20,240 --> 00:17:23,319 Part of what's attractive about them is that they are simpler. 352 00:17:23,319 --> 00:17:26,460 But the connection to experiments 353 00:17:26,460 --> 00:17:29,350 can be quite challenging. 354 00:17:29,350 --> 00:17:31,290 And we'll kind of see some of this. 355 00:17:31,290 --> 00:17:33,520 So these dimensionless equations-- 356 00:17:33,520 --> 00:17:38,130 dimensionless equations-- I think 357 00:17:38,130 --> 00:17:43,600 that they're both good and bad. 358 00:17:43,600 --> 00:17:46,590 The good is that you can figure out 359 00:17:46,590 --> 00:17:50,130 what are the essential features of the model. 360 00:17:50,130 --> 00:17:54,131 So you can focus your attention on the essential mathematical 361 00:17:54,131 --> 00:17:54,630 features. 362 00:17:58,960 --> 00:18:02,780 The disadvantage is that you might not even 363 00:18:02,780 --> 00:18:06,300 know which of the parameters change when you do something 364 00:18:06,300 --> 00:18:08,630 experimentally. 365 00:18:08,630 --> 00:18:11,200 So the problem is that the connection 366 00:18:11,200 --> 00:18:18,130 to the biology or experiments is obscured in some cases. 367 00:18:20,567 --> 00:18:21,650 Connection to experiments. 368 00:18:26,450 --> 00:18:28,470 And as an example of this, what we want to do 369 00:18:28,470 --> 00:18:31,759 is look at these equations for the toggle switch. 370 00:18:31,759 --> 00:18:33,300 I know that you guys just did reading 371 00:18:33,300 --> 00:18:36,830 on how to get to these final pair of equations. 372 00:18:36,830 --> 00:18:39,330 It's important to remember that in the context of the paper, 373 00:18:39,330 --> 00:18:40,870 they had a model. 374 00:18:40,870 --> 00:18:42,370 Here are the dimensionless equations 375 00:18:42,370 --> 00:18:44,010 that describe our system. 376 00:18:44,010 --> 00:18:46,470 And we use them to design the toggle switch. 377 00:18:46,470 --> 00:18:50,110 But just from that, you don't necessarily 378 00:18:50,110 --> 00:18:52,880 realize what's been done. 379 00:18:52,880 --> 00:18:54,990 So we want to make sure we understand it. 380 00:18:54,990 --> 00:18:58,400 So the equations that we want to be comfortable with 381 00:18:58,400 --> 00:18:59,650 are the following. 382 00:18:59,650 --> 00:19:04,970 So it's u dot du dt. 383 00:19:04,970 --> 00:19:09,310 Now here they use alpha as the rate of expression. 384 00:19:09,310 --> 00:19:11,450 So beware, in the past, we sometimes 385 00:19:11,450 --> 00:19:13,620 used alpha as a rate of degradation. 386 00:19:13,620 --> 00:19:16,240 So fair warning. 387 00:19:16,240 --> 00:19:27,150 Alpha 1 1 plus that's v beta. v to the beta minus u v dot 388 00:19:27,150 --> 00:19:35,080 is equal to-- here is an alpha 2 divided by 1 plus u 389 00:19:35,080 --> 00:19:41,630 to the gamma minus v. Now in many, many cases, 390 00:19:41,630 --> 00:19:43,980 we're going to get-- equations that 391 00:19:43,980 --> 00:19:46,330 look like this are going to pop up time and time again 392 00:19:46,330 --> 00:19:48,163 over the rest of this semester, and you just 393 00:19:48,163 --> 00:19:52,500 have to be intimately familiar with them. 394 00:19:52,500 --> 00:19:54,060 So first of all, can somebody say 395 00:19:54,060 --> 00:19:57,740 why this might be capturing the dynamics of a toggle switch? 396 00:20:08,080 --> 00:20:09,380 I can say something. 397 00:20:12,770 --> 00:20:13,480 Yes, please. 398 00:20:13,480 --> 00:20:17,384 AUDIENCE: There's some more of each u or [INAUDIBLE]. 399 00:20:20,800 --> 00:20:22,160 PROFESSOR: That's right. 400 00:20:22,160 --> 00:20:26,070 So the more v you have, the less production you're 401 00:20:26,070 --> 00:20:30,000 going to have of u, and vice versa. 402 00:20:30,000 --> 00:20:32,020 Now in many cases, beta and gamma 403 00:20:32,020 --> 00:20:34,160 are going to be something that's larger than 1. 404 00:20:34,160 --> 00:20:36,880 This is capturing some element of cooperativity 405 00:20:36,880 --> 00:20:39,970 in the repression on each side. 406 00:20:39,970 --> 00:20:43,150 So this thing is indeed just a u repressing a v, and vice versa. 407 00:20:46,920 --> 00:20:50,850 Now these things are wonderfully simple equations. 408 00:20:50,850 --> 00:20:54,880 See that there are four parameters that are completely 409 00:20:54,880 --> 00:20:57,810 specifying the dynamics. 410 00:20:57,810 --> 00:20:59,640 Now you'll notice that the world is always 411 00:20:59,640 --> 00:21:02,220 to be more complicated than the four parameters. 412 00:21:02,220 --> 00:21:05,750 Now there are two ways in which these complications went away. 413 00:21:05,750 --> 00:21:09,420 One is that we are modeling a simple-- 414 00:21:09,420 --> 00:21:11,709 it's a simple model of a complex system. 415 00:21:11,709 --> 00:21:13,500 But the other is that even the simple model 416 00:21:13,500 --> 00:21:17,950 has been simplified by going to this dimensionless version. 417 00:21:17,950 --> 00:21:23,520 So I want to make sure that you understood the reading 418 00:21:23,520 --> 00:21:26,310 to the point where at least you understand 419 00:21:26,310 --> 00:21:30,660 what units of concentrations, times, and so forth 420 00:21:30,660 --> 00:21:33,230 are in this model. 421 00:21:33,230 --> 00:21:39,460 So first I want to ask about the effective lifetimes of proteins 422 00:21:39,460 --> 00:21:59,410 u and v. Of u verses v. So we'll maybe call this tau u 423 00:21:59,410 --> 00:22:03,530 and tau v. And I want to know which one-- 424 00:22:03,530 --> 00:22:06,260 and how these things are related to each other. 425 00:22:09,710 --> 00:22:12,074 Tau u greater than [INAUDIBLE]. 426 00:22:40,440 --> 00:22:41,573 DK is again, don't know. 427 00:22:55,560 --> 00:23:00,220 Now of course in principle, the lifetimes of u and v 428 00:23:00,220 --> 00:23:01,589 can be anything. 429 00:23:01,589 --> 00:23:03,880 The question is, once we've written down that equation, 430 00:23:03,880 --> 00:23:05,710 have we actually said anything about that? 431 00:23:05,710 --> 00:23:07,460 Have we already made in assumption or not? 432 00:23:19,895 --> 00:23:23,470 Do you need more time? 433 00:23:23,470 --> 00:23:31,060 I see a fair number of quizzical faces, which 434 00:23:31,060 --> 00:23:33,920 may mean that even with extra time, the quizzical faces 435 00:23:33,920 --> 00:23:35,061 would not go away. 436 00:23:35,061 --> 00:23:35,560 Question. 437 00:23:35,560 --> 00:23:36,050 Yes? 438 00:23:36,050 --> 00:23:37,841 AUDIENCE: Are you asking about the lifetime 439 00:23:37,841 --> 00:23:40,181 of individual [INAUDIBLE]? 440 00:23:40,181 --> 00:23:41,040 PROFESSOR: OK. 441 00:23:41,040 --> 00:23:43,010 All right. 442 00:23:43,010 --> 00:23:43,510 Yeah. 443 00:23:43,510 --> 00:23:46,280 So when I say effective lifetime, 444 00:23:46,280 --> 00:23:50,330 that's because if it's a stable protein, 445 00:23:50,330 --> 00:23:53,250 then the lifetime of that individual protein 446 00:23:53,250 --> 00:23:56,800 is maybe infinite, but you get an effective lifetime because 447 00:23:56,800 --> 00:23:58,177 of this dilution effect. 448 00:23:58,177 --> 00:24:00,010 So when I say effective lifetime in general, 449 00:24:00,010 --> 00:24:01,510 in this class, what I'm referring to 450 00:24:01,510 --> 00:24:06,250 is the sum of two effects of dilution to the cell growth, 451 00:24:06,250 --> 00:24:09,888 as well as actual degradation. 452 00:24:09,888 --> 00:24:11,776 AUDIENCE: And you're saying [INAUDIBLE]. 453 00:24:14,608 --> 00:24:15,500 PROFESSOR: No. 454 00:24:15,500 --> 00:24:19,690 No, I'm saying given that the Collins Lab-- in their paper, 455 00:24:19,690 --> 00:24:22,010 wrote down those set of equations. 456 00:24:22,010 --> 00:24:25,410 I'm asking have they already specified anything 457 00:24:25,410 --> 00:24:28,090 about the effective lifetimes of these two proteins? 458 00:24:30,815 --> 00:24:33,270 AUDIENCE: So in those equations, [INAUDIBLE]. 459 00:24:39,180 --> 00:24:42,907 Your question asks in units of that dimensionless time? 460 00:24:42,907 --> 00:24:44,490 PROFESSOR: Yeah, I mean we can compare 461 00:24:44,490 --> 00:24:46,170 two times in some dimensionless-- yeah, 462 00:24:46,170 --> 00:24:47,669 we're going to also talk about that. 463 00:24:47,669 --> 00:24:49,804 Maybe I should have done that first. 464 00:24:49,804 --> 00:24:51,470 But this is a nice question because it's 465 00:24:51,470 --> 00:24:53,650 highlighting that you get a pair of equations, 466 00:24:53,650 --> 00:24:58,110 they look obvious, but then some of those basic things 467 00:24:58,110 --> 00:25:01,350 about the system are some how not quite clear. 468 00:25:01,350 --> 00:25:03,670 So I'm going talk about in this units of whatever time 469 00:25:03,670 --> 00:25:05,600 is being-- however time-- we're going 470 00:25:05,600 --> 00:25:08,450 to discuss how time is being measured in a moment as well. 471 00:25:08,450 --> 00:25:11,930 But however time is being measured, 472 00:25:11,930 --> 00:25:15,150 is there some relationship here or not? 473 00:25:18,690 --> 00:25:20,400 Let's go ahead and vote. 474 00:25:20,400 --> 00:25:23,530 And it's fine, if you really do not 475 00:25:23,530 --> 00:25:28,040 know what I'm talking about, you can say E and that's fine. 476 00:25:28,040 --> 00:25:29,440 Let's see where we are. 477 00:25:29,440 --> 00:25:31,670 Ready, three, two, one. 478 00:25:35,130 --> 00:25:38,460 So I would say that it's split between B's and D's. 479 00:25:41,866 --> 00:25:43,490 And that's great, because that means we 480 00:25:43,490 --> 00:25:45,490 have something to talk about. 481 00:25:45,490 --> 00:25:47,914 There's broad agreement that it's one of these two. 482 00:25:47,914 --> 00:25:48,830 So turn your neighbor. 483 00:25:48,830 --> 00:25:50,330 You should be able to find somebody 484 00:25:50,330 --> 00:25:51,897 that disagrees with you. 485 00:25:51,897 --> 00:25:53,980 If you can't find anybody that disagrees with you, 486 00:25:53,980 --> 00:25:55,814 you could think about how parameters 487 00:25:55,814 --> 00:25:58,230 are going to change as you vary other experimental things. 488 00:26:01,526 --> 00:26:04,016 [CLASSROOM CHATTER] 489 00:27:21,442 --> 00:27:23,400 PROFESSOR: Why don't we go ahead and reconvene? 490 00:27:23,400 --> 00:27:25,320 I just want to see where we are. 491 00:27:25,320 --> 00:27:28,120 So let's get our cards ready. 492 00:27:28,120 --> 00:27:31,930 And we're still working on this one, so you can ignore this. 493 00:27:31,930 --> 00:27:32,501 Is it B or D? 494 00:27:32,501 --> 00:27:33,000 Ready? 495 00:27:33,000 --> 00:27:34,125 Three, two, one. 496 00:27:36,680 --> 00:27:41,640 So I'd say there's some migration towards B, 497 00:27:41,640 --> 00:27:43,190 but not 100% still. 498 00:27:43,190 --> 00:27:46,790 So I'm going to side with this here. 499 00:27:46,790 --> 00:27:52,142 Can somebody volunteer why they're saying that? 500 00:27:52,142 --> 00:27:59,990 AUDIENCE: Well, if I remember correctly then they [INAUDIBLE] 501 00:27:59,990 --> 00:28:03,328 time by multiplying time by degradation rate. 502 00:28:03,328 --> 00:28:05,718 And they do that with the same degradation 503 00:28:05,718 --> 00:28:08,522 rate for both [INAUDIBLE]. 504 00:28:08,522 --> 00:28:09,230 PROFESSOR: Right. 505 00:28:09,230 --> 00:28:11,820 So the answer here is yeah-- so the way that we got to this non 506 00:28:11,820 --> 00:28:14,236 dimensional time, that we're going to discuss in a moment, 507 00:28:14,236 --> 00:28:17,820 is multiplying by-- or divided by some degradation rate. 508 00:28:17,820 --> 00:28:19,830 And you remember the derivation. 509 00:28:19,830 --> 00:28:22,400 You remember that it was [INAUDIBLE]. 510 00:28:22,400 --> 00:28:27,310 But in many cases you don't get to read the derivation 511 00:28:27,310 --> 00:28:29,530 before you have to answer it. 512 00:28:29,530 --> 00:28:31,620 In many cases you just get these equations, 513 00:28:31,620 --> 00:28:34,500 and you have to figure out what the authors have assumed. 514 00:28:34,500 --> 00:28:37,360 So I think your answer is very much correct, 515 00:28:37,360 --> 00:28:39,570 but it might be-- I'm glad that you're 516 00:28:39,570 --> 00:28:43,282 using the pre-class reading, but at the same time 517 00:28:43,282 --> 00:28:45,240 we have to be able to answer this question just 518 00:28:45,240 --> 00:28:47,150 from these equations. 519 00:28:47,150 --> 00:28:48,776 Because it is contained there. 520 00:28:48,776 --> 00:28:49,276 Yeah? 521 00:28:49,276 --> 00:28:52,657 AUDIENCE: So if you remove production [INAUDIBLE]. 522 00:28:52,657 --> 00:28:53,460 PROFESSOR: Perfect. 523 00:28:53,460 --> 00:28:55,001 AUDIENCE: If you solve that equation, 524 00:28:55,001 --> 00:28:57,316 it's exponential to E, and the [INAUDIBLE]. 525 00:29:00,648 --> 00:29:01,820 PROFESSOR: Yes. 526 00:29:01,820 --> 00:29:02,430 That's right. 527 00:29:02,430 --> 00:29:05,050 So the statement here is that it's 528 00:29:05,050 --> 00:29:07,300 nice to just imagine that we shut down production. 529 00:29:07,300 --> 00:29:09,260 So these first terms go away. 530 00:29:09,260 --> 00:29:12,330 All right now we just have u dot is equal to minus u. 531 00:29:12,330 --> 00:29:16,145 So the concentration of u and B will both fall exponentially. 532 00:29:16,145 --> 00:29:18,520 And they're going to fall the same rate in whatever units 533 00:29:18,520 --> 00:29:20,490 of time-- and we'll discuss this in a moment-- 534 00:29:20,490 --> 00:29:23,270 but however time is being measured, 535 00:29:23,270 --> 00:29:24,860 they're going to fall at the same rate 536 00:29:24,860 --> 00:29:28,310 because it's whatever appears in front of these two 537 00:29:28,310 --> 00:29:29,830 that determines that rate. 538 00:29:29,830 --> 00:29:32,630 So we've already assumed that the effective lifetime 539 00:29:32,630 --> 00:29:34,862 of u and v are the same once we've 540 00:29:34,862 --> 00:29:36,070 written down these equations. 541 00:29:36,070 --> 00:29:37,444 Now that doesn't have to be true. 542 00:29:37,444 --> 00:29:39,300 That means that if experimentally, you 543 00:29:39,300 --> 00:29:41,640 want to make a toggle switch with two proteins, 544 00:29:41,640 --> 00:29:43,570 with different st abilities, then 545 00:29:43,570 --> 00:29:44,820 you can't use these equations. 546 00:29:44,820 --> 00:29:47,412 You have to add a delta on one of the two equations 547 00:29:47,412 --> 00:29:49,870 to capture that dynamic that they have different lifetimes. 548 00:29:52,900 --> 00:29:55,860 So these equations are simple because we've already 549 00:29:55,860 --> 00:29:59,010 combine things, but they've also already assumed some things 550 00:29:59,010 --> 00:30:01,890 to make it look more simple and more symmetric. 551 00:30:01,890 --> 00:30:04,100 So for example, they're allowing for 552 00:30:04,100 --> 00:30:07,490 different effective cooperativities, beta gamma, 553 00:30:07,490 --> 00:30:09,190 for the two repressors. 554 00:30:09,190 --> 00:30:11,446 They didn't have to do that if they didn't want to. 555 00:30:11,446 --> 00:30:13,070 If they wanted to, they could have just 556 00:30:13,070 --> 00:30:16,060 had beta in both equations. 557 00:30:16,060 --> 00:30:18,200 And that case they would be assuming 558 00:30:18,200 --> 00:30:20,620 that the effective cooperativity of the repression 559 00:30:20,620 --> 00:30:21,930 is the same for the two. 560 00:30:21,930 --> 00:30:23,820 So depending on what you write down, 561 00:30:23,820 --> 00:30:26,272 you're making different assumptions about the system. 562 00:30:26,272 --> 00:30:28,480 And so you have to be able to look at these equations 563 00:30:28,480 --> 00:30:30,526 and figure out what assumptions have been made. 564 00:30:33,220 --> 00:30:33,920 Yes, question. 565 00:30:33,920 --> 00:30:35,396 AUDIENCE: I'm having trouble seeing 566 00:30:35,396 --> 00:30:37,364 what the first term is doing. 567 00:30:37,364 --> 00:30:41,300 So the second one is [INAUDIBLE]. 568 00:30:41,300 --> 00:30:44,780 PROFESSOR: So broadly, the first term in these equations 569 00:30:44,780 --> 00:30:46,620 is the production rate of the protein. 570 00:30:46,620 --> 00:30:49,260 And the second term is some sort of effective degradation, 571 00:30:49,260 --> 00:30:50,593 but it could be due to dilution. 572 00:30:52,780 --> 00:30:55,570 The nice thing about just ignoring the production term 573 00:30:55,570 --> 00:30:58,250 is that it's a way of focusing only on the effective lifetime 574 00:30:58,250 --> 00:30:58,750 portion. 575 00:31:02,730 --> 00:31:06,820 This effective lifetime is it's captured here, 576 00:31:06,820 --> 00:31:09,475 irrespective of whether there's production or not. 577 00:31:09,475 --> 00:31:10,850 Of course the production is going 578 00:31:10,850 --> 00:31:14,410 to affect what the equilibrium is that we go to and so forth, 579 00:31:14,410 --> 00:31:18,759 but remember for example, that this effective lifetime that's 580 00:31:18,759 --> 00:31:21,050 set for time scale both to come up to some equilibrium, 581 00:31:21,050 --> 00:31:23,070 as well as come down to some equilibrium. 582 00:31:23,070 --> 00:31:25,500 That's telling us about this effective lifetime 583 00:31:25,500 --> 00:31:30,110 is essential to tell us about the rate 584 00:31:30,110 --> 00:31:32,520 that concentration is going to change within the cell, 585 00:31:32,520 --> 00:31:35,390 whether you're going up or down. 586 00:31:35,390 --> 00:31:38,370 Did that answer your question a little bit? 587 00:31:38,370 --> 00:31:39,148 Yes? 588 00:31:39,148 --> 00:31:41,770 AUDIENCE: That last thing you said 589 00:31:41,770 --> 00:31:43,710 is not quite true because-- 590 00:31:43,710 --> 00:31:45,350 PROFESSOR: Because of the toggle. 591 00:31:45,350 --> 00:31:47,510 I agree. 592 00:31:47,510 --> 00:31:50,500 What I was really referring to there was if we get rid of one, 593 00:31:50,500 --> 00:31:52,450 and we're just talking about where we just 594 00:31:52,450 --> 00:31:55,030 manually set the production rate to one thing or another. 595 00:31:55,030 --> 00:31:57,780 The actual dynamics of this are more complicated, certainly. 596 00:32:01,960 --> 00:32:03,470 So is everybody happy with this idea 597 00:32:03,470 --> 00:32:05,678 that just by writing down those equations, we've made 598 00:32:05,678 --> 00:32:08,440 some assumptions constraining relationships between u and v 599 00:32:08,440 --> 00:32:10,430 in some ways, but not in other ways. 600 00:32:10,430 --> 00:32:11,652 Yeah? 601 00:32:11,652 --> 00:32:14,110 So we've kind of already danced around this other question, 602 00:32:14,110 --> 00:32:17,730 but I want to make sure that we address it head on. 603 00:32:20,300 --> 00:32:22,730 We want to know what is this unit of time. 604 00:32:22,730 --> 00:32:25,550 If I say oh, at time T [INAUDIBLE] 1 versus time 605 00:32:25,550 --> 00:32:31,080 [INAUDIBLE] to 2-- so if delta t-- what is 1 in this case? 606 00:32:31,080 --> 00:32:33,630 Are we referring to one second? 607 00:32:33,630 --> 00:32:34,660 CGS. 608 00:32:34,660 --> 00:32:38,680 Or is it one hour corresponding to another unit? 609 00:32:38,680 --> 00:32:41,256 Cell generation time, effective lifetime, or don't know? 610 00:32:44,904 --> 00:32:48,430 Ten seconds, since we've already kind of said this, 611 00:32:48,430 --> 00:32:52,040 but it's important enough to make sure. 612 00:33:02,041 --> 00:33:02,540 Ready? 613 00:33:08,600 --> 00:33:09,100 No? 614 00:33:09,100 --> 00:33:09,599 Yes? 615 00:33:12,362 --> 00:33:14,570 I'll give you another ten seconds, just to make sure. 616 00:33:34,760 --> 00:33:37,010 Do you need more time? 617 00:33:37,010 --> 00:33:37,510 Let's vote. 618 00:33:37,510 --> 00:33:38,070 Ready? 619 00:33:38,070 --> 00:33:41,797 Three, two, one. 620 00:33:41,797 --> 00:33:42,296 OK. 621 00:33:45,850 --> 00:33:47,970 A majority of the group is agreeing in this case 622 00:33:47,970 --> 00:33:49,136 it's the effective lifetime. 623 00:33:53,360 --> 00:33:55,590 And it's not the cell generation time necessarily 624 00:33:55,590 --> 00:33:58,720 because if we actually have a degradation 625 00:33:58,720 --> 00:34:02,046 tag on this protein, if it's not a stable protein then 626 00:34:02,046 --> 00:34:03,920 the effective lifetime is going to be shorter 627 00:34:03,920 --> 00:34:06,980 than the cell generation time. 628 00:34:06,980 --> 00:34:09,780 So we've normalized time by dividing out 629 00:34:09,780 --> 00:34:16,320 that-- whatever that delta thing would have been on the right. 630 00:34:16,320 --> 00:34:22,360 Now this is great because it makes the equations simple. 631 00:34:22,360 --> 00:34:25,870 But remember what that means is that if we change 632 00:34:25,870 --> 00:34:28,510 the degradation rate, then all of a sudden 633 00:34:28,510 --> 00:34:32,122 it's not obvious what's going to happen. 634 00:34:32,122 --> 00:34:33,580 Experimentally we're always allowed 635 00:34:33,580 --> 00:34:37,060 to affect the degradation rate. 636 00:34:37,060 --> 00:34:41,620 But the question is, which parameter or parameters 637 00:34:41,620 --> 00:34:47,194 change if you add a degradation tag? 638 00:34:47,194 --> 00:34:48,860 So if you increase the degradation rate. 639 00:34:55,400 --> 00:34:58,332 That's what I want to do. 640 00:34:58,332 --> 00:34:59,790 We'll do that in just moment there. 641 00:34:59,790 --> 00:35:01,623 I have another question that I wanted to do. 642 00:35:01,623 --> 00:35:03,647 Do you guys remember the equations up there? 643 00:35:03,647 --> 00:35:05,230 Maybe I'll come over to the other side 644 00:35:05,230 --> 00:35:08,080 just so that I-- it's useful to be able to stare at those 645 00:35:08,080 --> 00:35:10,344 equations as we discuss. 646 00:35:21,660 --> 00:35:24,070 So the question is, what is the unit 647 00:35:24,070 --> 00:35:25,720 of concentration of protein u? 648 00:35:30,240 --> 00:35:32,620 And of course, these are dimensionless. 649 00:35:32,620 --> 00:35:36,820 What I really mean is what does u equal to 1 650 00:35:36,820 --> 00:35:37,990 mean in real units? 651 00:35:41,540 --> 00:35:44,450 I'll just write that. 652 00:35:44,450 --> 00:35:48,000 What does u equal to 1 mean? 653 00:35:48,000 --> 00:35:50,130 What does u equal to 1 correspond to? 654 00:35:50,130 --> 00:35:53,800 In something that is recognizable to 655 00:35:53,800 --> 00:35:55,470 an experimentalist in the lab? 656 00:36:34,667 --> 00:36:36,000 You guys understand what I mean? 657 00:36:38,740 --> 00:36:41,190 So if in this model I say, u is equal to 1, 658 00:36:41,190 --> 00:36:42,340 or u is equal to 10. 659 00:36:42,340 --> 00:36:45,360 What do those numbers-- what do they mean? 660 00:36:54,450 --> 00:36:57,258 So are there any questions about my question first of all? 661 00:36:57,258 --> 00:36:57,758 Yes? 662 00:36:57,758 --> 00:37:01,704 AUDIENCE: Are we [INAUDIBLE] solely on seeing those 663 00:37:01,704 --> 00:37:02,204 equations? 664 00:37:02,204 --> 00:37:06,650 Or based on the reading of [INAUDIBLE]? 665 00:37:06,650 --> 00:37:09,890 PROFESSOR: I would say that the reading actually 666 00:37:09,890 --> 00:37:11,660 has a somewhat more complicated model, 667 00:37:11,660 --> 00:37:13,790 and involves multiple steps. 668 00:37:13,790 --> 00:37:16,500 So this is really just looking at those equations, 669 00:37:16,500 --> 00:37:20,310 given that we can see that there is 670 00:37:20,310 --> 00:37:25,370 some statement about how u and v are repressing each other. 671 00:37:25,370 --> 00:37:28,710 So we're saying that there is some function that describes-- 672 00:37:28,710 --> 00:37:31,170 it's a phenomenological function-- describes the input 673 00:37:31,170 --> 00:37:33,711 output relationship in terms of some concentration of u leads 674 00:37:33,711 --> 00:37:36,540 to some repression of v and vice versa. 675 00:37:36,540 --> 00:37:39,464 From that actually, we should be able to say something 676 00:37:39,464 --> 00:37:41,630 about what we've already assumed in these equations. 677 00:38:05,914 --> 00:38:06,830 Do you need more time? 678 00:38:06,830 --> 00:38:07,455 Yeah, question. 679 00:38:07,455 --> 00:38:12,420 AUDIENCE: So the k of u promoter is-- that's defined as the-- 680 00:38:12,420 --> 00:38:15,870 PROFESSOR: That's the binding of v to the u promoter-- the piece 681 00:38:15,870 --> 00:38:18,135 of DNA in front of the u gene. 682 00:38:24,090 --> 00:38:28,190 Do we need more time, or shall we give it a go? 683 00:38:28,190 --> 00:38:28,760 Ready? 684 00:38:28,760 --> 00:38:33,000 Three, two, one. 685 00:38:33,000 --> 00:38:33,500 All right. 686 00:38:33,500 --> 00:38:37,203 So we got some A, B, C, D's. 687 00:38:39,760 --> 00:38:41,660 No E's. 688 00:38:41,660 --> 00:38:44,500 At least it means that your neighbor has an opinion. 689 00:38:44,500 --> 00:38:49,790 He or she cannot say that-- Turn to your neighbor and discuss. 690 00:38:49,790 --> 00:38:52,778 [CLASSROOM CHATTER] 691 00:40:40,150 --> 00:40:43,210 Did you guys all decide you're comfortable? 692 00:40:43,210 --> 00:40:45,500 Why don't we go ahead and reconvene. 693 00:40:45,500 --> 00:40:48,450 It seems like it may be we're coming to consensus here. 694 00:40:48,450 --> 00:40:49,320 Ready? 695 00:40:49,320 --> 00:40:52,790 Three, two, one. 696 00:40:52,790 --> 00:41:04,830 So now we got a clear majority agreeing that it should be D. 697 00:41:04,830 --> 00:41:09,690 So what happened was that over here in the original equations 698 00:41:09,690 --> 00:41:14,720 we had a real concentration of u divided by some k. 699 00:41:14,720 --> 00:41:18,540 And that was the k for repressing v. Because v 700 00:41:18,540 --> 00:41:21,500 is this guy that is over here. 701 00:41:21,500 --> 00:41:25,540 And what we did is we then just turned u divided by this k 702 00:41:25,540 --> 00:41:28,920 for binding this v promoter into just u. 703 00:41:28,920 --> 00:41:33,400 So in particular, when that concentration of u 704 00:41:33,400 --> 00:41:36,330 is equal to its associated k, that 705 00:41:36,330 --> 00:41:38,430 corresponds to half repression. 706 00:41:38,430 --> 00:41:39,540 Same thing here. 707 00:41:39,540 --> 00:41:42,736 One way to think about this is just that when u is equal to 1, 708 00:41:42,736 --> 00:41:47,760 we're getting half of maximum possible repression. 709 00:41:47,760 --> 00:41:51,100 Similarly, that's what v equal to 1 means. 710 00:41:51,100 --> 00:41:55,770 U and v, do they have to have the same-- I mean, 711 00:41:55,770 --> 00:41:57,670 does u equal to 1 and v equal to 1 712 00:41:57,670 --> 00:41:59,420 mean the same thing in terms of the number 713 00:41:59,420 --> 00:42:01,350 of the proteins in the cell? 714 00:42:01,350 --> 00:42:01,850 No. 715 00:42:01,850 --> 00:42:03,850 Not necessarily. 716 00:42:03,850 --> 00:42:07,170 So we've allowed for the possibility 717 00:42:07,170 --> 00:42:11,660 that those things are measured in different real units. 718 00:42:11,660 --> 00:42:13,960 But in both cases, it's telling us 719 00:42:13,960 --> 00:42:16,740 about how the strength of repression. 720 00:42:16,740 --> 00:42:19,840 And u and v equal to 1 tells us about that crossing point 721 00:42:19,840 --> 00:42:22,530 where the other promoter is half repressed. 722 00:42:25,530 --> 00:42:29,190 Are there any questions about what happened there? 723 00:42:29,190 --> 00:42:32,673 This is especially confusing because it doesn't enter 724 00:42:32,673 --> 00:42:35,080 into the equations at all. 725 00:42:35,080 --> 00:42:37,910 Things just went away. 726 00:42:37,910 --> 00:42:40,110 But you know that this is the dimensionless versions 727 00:42:40,110 --> 00:42:42,550 of the equations because we have u here, 728 00:42:42,550 --> 00:42:44,420 and we're adding 1 to it. 729 00:42:44,420 --> 00:42:48,128 Are we allowed to add things with different units? 730 00:42:48,128 --> 00:42:50,700 No. 731 00:42:50,700 --> 00:42:53,420 Never. 732 00:42:53,420 --> 00:42:56,736 What that means is that-- since they're being added, 733 00:42:56,736 --> 00:42:58,360 that means that we already know that we 734 00:42:58,360 --> 00:43:01,677 made-- this is the dimensionless version of the equation. 735 00:43:01,677 --> 00:43:03,260 And it actually then immediately tells 736 00:43:03,260 --> 00:43:05,190 us what u equal to 1 means. 737 00:43:09,630 --> 00:43:11,130 So now what we want to do is we want 738 00:43:11,130 --> 00:43:14,940 to make sure that our intuition on this is tip top shape. 739 00:43:14,940 --> 00:43:17,130 In particular, we want to know if I 740 00:43:17,130 --> 00:43:19,530 want to change the dynamics of the system-- 741 00:43:19,530 --> 00:43:22,707 so let's say that we go and we spend lots of time 742 00:43:22,707 --> 00:43:24,290 calculating the fixed point stability, 743 00:43:24,290 --> 00:43:26,540 and we do everything on these equations. 744 00:43:26,540 --> 00:43:28,120 And then we know what we need to do 745 00:43:28,120 --> 00:43:29,953 is we need to change some parameter in order 746 00:43:29,953 --> 00:43:32,020 to get say, a toggle switch. 747 00:43:32,020 --> 00:43:34,100 We need to know how we do that. 748 00:43:34,100 --> 00:43:38,820 We need to know how the parameters in real life, 749 00:43:38,820 --> 00:43:41,390 or even in the context of the model, how is it 750 00:43:41,390 --> 00:43:44,420 that the parameters you can actually change experimentally, 751 00:43:44,420 --> 00:43:50,400 how is it that the affect or not the parameters in this model? 752 00:43:50,400 --> 00:43:55,680 So the question is, let's say we increase the degradation 753 00:43:55,680 --> 00:43:58,267 rate of these two transcription factors u 754 00:43:58,267 --> 00:44:01,375 and v, which of the parameters are going to change? 755 00:44:08,770 --> 00:44:12,029 So I'll give you 30 seconds to think about what 756 00:44:12,029 --> 00:44:13,070 should be happening here. 757 00:44:46,260 --> 00:44:47,180 Do you need more time? 758 00:44:49,850 --> 00:44:51,372 Let's see where we are. 759 00:44:51,372 --> 00:44:52,030 Ready? 760 00:44:52,030 --> 00:44:54,675 Three, two, one. 761 00:44:54,675 --> 00:44:56,050 And this is the possibility where 762 00:44:56,050 --> 00:44:57,270 you can put up two things. 763 00:44:57,270 --> 00:45:01,320 Remember our fabulous card system? 764 00:45:01,320 --> 00:45:03,575 So I think most people are saying it's going to be A 765 00:45:03,575 --> 00:45:06,860 and B are going to change. 766 00:45:06,860 --> 00:45:10,210 So beta and gamma are capturing how cooperative that transition 767 00:45:10,210 --> 00:45:10,710 is. 768 00:45:10,710 --> 00:45:14,180 And the cooperativity is not affected 769 00:45:14,180 --> 00:45:20,089 by the questions of the exact concentration, or time scale 770 00:45:20,089 --> 00:45:20,630 and so forth. 771 00:45:20,630 --> 00:45:23,090 Because that has to do with the molecular nature 772 00:45:23,090 --> 00:45:26,480 of the interactions with the promoter. 773 00:45:26,480 --> 00:45:29,520 So in some ways beta gamma are the simplest things 774 00:45:29,520 --> 00:45:32,620 in this system. 775 00:45:32,620 --> 00:45:37,122 Now the question is, do alpha 1-- do they go up 776 00:45:37,122 --> 00:45:38,010 or do they go down? 777 00:45:38,010 --> 00:45:39,676 Now that I've told you that they change. 778 00:45:42,750 --> 00:45:46,075 If degradation rate goes up, alpha 1 and alpha 2, 779 00:45:46,075 --> 00:45:47,450 do they go up or do they go down? 780 00:45:51,900 --> 00:45:54,060 I'll give you 10 seconds to think about it. 781 00:46:03,744 --> 00:46:04,660 Do you need more time? 782 00:46:10,630 --> 00:46:11,340 Let's see it. 783 00:46:11,340 --> 00:46:11,840 Ready? 784 00:46:11,840 --> 00:46:13,215 Three, two, one. 785 00:46:17,540 --> 00:46:20,380 It's a majority are saying it's going go down. 786 00:46:20,380 --> 00:46:22,440 Can somebody offer up an intuitive explanation 787 00:46:22,440 --> 00:46:23,356 for why this might be? 788 00:46:28,460 --> 00:46:28,960 [INAUDIBLE] 789 00:46:28,960 --> 00:46:31,852 AUDIENCE: The degradation rate goes up, 790 00:46:31,852 --> 00:46:34,262 it means the times scale goes down. 791 00:46:34,262 --> 00:46:37,154 As you produce a fixed number of per unit time, 792 00:46:37,154 --> 00:46:39,256 the unit time gets smaller and produces less. 793 00:46:39,256 --> 00:46:41,860 PROFESSOR: Right, so if the degradation rate goes up, 794 00:46:41,860 --> 00:46:44,039 that's kind of reducing this unit of time. 795 00:46:44,039 --> 00:46:46,080 So you just are not going to make as much protein 796 00:46:46,080 --> 00:46:48,610 in that unit of time. 797 00:46:48,610 --> 00:46:50,290 The way that I like to think about this 798 00:46:50,290 --> 00:46:52,370 maybe is that if the degradation rate goes up, 799 00:46:52,370 --> 00:46:57,180 then the real concentration of the protein should go down. 800 00:46:57,180 --> 00:47:00,260 And that means that the repression 801 00:47:00,260 --> 00:47:02,894 should be less effective. 802 00:47:02,894 --> 00:47:04,310 And then wait, is this helping me? 803 00:47:04,310 --> 00:47:13,635 No, now that I'm saying this-- I don't like 804 00:47:13,635 --> 00:47:14,760 to think about it that way. 805 00:47:22,080 --> 00:47:24,179 [INTERPOSING VOICES] 806 00:47:24,179 --> 00:47:24,720 That's right. 807 00:47:24,720 --> 00:47:26,650 Yeah, so-- that's right. 808 00:47:29,980 --> 00:47:33,135 You should decrease the sort of-- So 809 00:47:33,135 --> 00:47:35,480 if the degradation rate goes up, you decrease 810 00:47:35,480 --> 00:47:38,160 the real concentration, which it means indeed 811 00:47:38,160 --> 00:47:39,570 that you are at steady state. 812 00:47:39,570 --> 00:47:41,940 Say a less effective repressor. 813 00:47:41,940 --> 00:47:43,900 That means the concentration goes down 814 00:47:43,900 --> 00:47:48,260 in these units of how repressive are you. 815 00:47:48,260 --> 00:47:50,910 I think the way of thinking about it was correct, 816 00:47:50,910 --> 00:47:54,880 just the words were not. 817 00:47:54,880 --> 00:47:59,180 You can also then go-- and it's useful in the context of when 818 00:47:59,180 --> 00:48:01,600 you actually go and you do the math of removing all 819 00:48:01,600 --> 00:48:04,080 these parameters, just to make sure that-- because it's 820 00:48:04,080 --> 00:48:07,040 easy to do this thing where you just divide everything out, 821 00:48:07,040 --> 00:48:09,120 and you're happy, and whistling and so forth. 822 00:48:09,120 --> 00:48:10,870 But at the end of the day, you really just 823 00:48:10,870 --> 00:48:13,690 have no sense of what happened. 824 00:48:13,690 --> 00:48:17,430 Of how the parameters that come out of the model 825 00:48:17,430 --> 00:48:21,700 are affected by the real things that you can change. 826 00:48:21,700 --> 00:48:23,900 And in some ways what's funny about these equations 827 00:48:23,900 --> 00:48:27,550 is that essentially everything is in the alphas. 828 00:48:27,550 --> 00:48:30,940 Because beta and gamma, that's this cooperativity parameter. 829 00:48:30,940 --> 00:48:32,420 That's what it is. 830 00:48:32,420 --> 00:48:33,320 That's all it is. 831 00:48:33,320 --> 00:48:35,745 So that means that everything else ends up in the alphas. 832 00:48:39,290 --> 00:48:44,340 So the strength of the promoter, the concentration 833 00:48:44,340 --> 00:48:46,880 you need to repress, the lifetime, everything 834 00:48:46,880 --> 00:48:48,380 rolls up in these alphas. 835 00:48:48,380 --> 00:48:51,130 And this is the beauty of the dimensionless equations. 836 00:48:51,130 --> 00:48:53,630 Is it's telling you that you don't have all these different, 837 00:48:53,630 --> 00:48:55,110 separate knobs. 838 00:48:55,110 --> 00:48:57,360 You can't change one and change another, 839 00:48:57,360 --> 00:48:59,280 and go into some funny regime. 840 00:48:59,280 --> 00:49:02,010 Because it's all rolled into one fundamental parameter, 841 00:49:02,010 --> 00:49:03,250 these alphas. 842 00:49:03,250 --> 00:49:05,370 That's telling you that once you understand 843 00:49:05,370 --> 00:49:08,577 how these equations behave, then you in principle understand 844 00:49:08,577 --> 00:49:11,035 everything that could possibly happen in that simple model. 845 00:49:11,035 --> 00:49:12,350 But that's what's wonderful about 846 00:49:12,350 --> 00:49:13,516 the dimensionless equations. 847 00:49:13,516 --> 00:49:15,150 But the problem is that you sometimes 848 00:49:15,150 --> 00:49:17,610 lose track of how it's related to the real experimental 849 00:49:17,610 --> 00:49:18,110 things. 850 00:49:18,110 --> 00:49:21,770 So I would say that I very much like these dimensionless 851 00:49:21,770 --> 00:49:22,360 equations. 852 00:49:22,360 --> 00:49:24,894 They clarify things for you. 853 00:49:24,894 --> 00:49:26,685 But you have to spend the time to make sure 854 00:49:26,685 --> 00:49:32,080 that you understand where all of reality went. 855 00:49:32,080 --> 00:49:36,070 Because it all ends up in this mathematical equation, 856 00:49:36,070 --> 00:49:37,680 and that simplifies things. 857 00:49:37,680 --> 00:49:43,700 You're not floating in a sea of symbols anymore, 858 00:49:43,700 --> 00:49:47,220 but it's really easy to lose track of the connection 859 00:49:47,220 --> 00:49:49,450 to real measurements. 860 00:49:49,450 --> 00:49:51,620 And that's why we want to do modeling 861 00:49:51,620 --> 00:49:54,090 so we can make that connection. 862 00:49:54,090 --> 00:49:57,580 So do the dimensionless equations, but make sure 863 00:49:57,580 --> 00:50:00,380 that you play with them a bit so you know what 864 00:50:00,380 --> 00:50:02,190 changes when you change what. 865 00:50:02,190 --> 00:50:05,680 And we'll actually see a bit later, some other cases 866 00:50:05,680 --> 00:50:10,880 where it's actually quite tricky to figure out what's happening. 867 00:50:10,880 --> 00:50:13,890 On exams I always want to just ask a equation like this, 868 00:50:13,890 --> 00:50:16,690 if this parameter goes off-- and then the TAs always say, 869 00:50:16,690 --> 00:50:20,050 it's too hard of an question. 870 00:50:20,050 --> 00:50:22,760 I feel like it's like the most basic thing you would want 871 00:50:22,760 --> 00:50:25,050 from an equation like this. 872 00:50:25,050 --> 00:50:28,330 Is that if you increase the strength of this-- 873 00:50:28,330 --> 00:50:31,170 But it actually is-- it's surprisingly difficult. 874 00:50:31,170 --> 00:50:33,780 So maybe this year I'll convince the TAs 875 00:50:33,780 --> 00:50:37,150 that it's an OK question. 876 00:50:37,150 --> 00:50:39,830 Are there any questions about where we are right now? 877 00:50:44,209 --> 00:50:46,000 So what I want to do for the last half hour 878 00:50:46,000 --> 00:50:48,125 is talk about stability analysis. 879 00:50:50,970 --> 00:50:53,220 The first context in which stability comes up 880 00:50:53,220 --> 00:50:56,240 in this class is indeed in this toggle switch. 881 00:50:56,240 --> 00:50:59,449 But it's not-- I would say-- the most satisfying application 882 00:50:59,449 --> 00:51:00,240 of it in some ways. 883 00:51:00,240 --> 00:51:04,070 Just because you end up with equations that all you can do 884 00:51:04,070 --> 00:51:04,630 is plot them. 885 00:51:04,630 --> 00:51:06,500 And it's useful to be able to recapitulate, 886 00:51:06,500 --> 00:51:09,250 to understand how the figures from that paper come about. 887 00:51:09,250 --> 00:51:14,110 But at the same time it's not always-- 888 00:51:14,110 --> 00:51:19,851 after you find the solution, you don't feel so happy about it 889 00:51:19,851 --> 00:51:20,350 either. 890 00:51:20,350 --> 00:51:23,510 Because in terms of complexity as a function of time, 891 00:51:23,510 --> 00:51:25,020 things always start out simple. 892 00:51:25,020 --> 00:51:27,160 And then in the course of the calculations 893 00:51:27,160 --> 00:51:28,470 things get complicated. 894 00:51:28,470 --> 00:51:30,400 And the problems that are fun to solve 895 00:51:30,400 --> 00:51:32,150 are the cases where it comes simple again. 896 00:51:32,150 --> 00:51:36,624 This one-- it never quite converges. 897 00:51:36,624 --> 00:51:38,540 I do want to talk about the stability analysis 898 00:51:38,540 --> 00:51:41,232 so that you can understand the calculation that 899 00:51:41,232 --> 00:51:41,940 was in the notes. 900 00:51:41,940 --> 00:51:44,169 But also so that you can just get some more intuition 901 00:51:44,169 --> 00:51:45,960 about some of the other problems that we're 902 00:51:45,960 --> 00:51:47,710 going to be solving in the next few weeks. 903 00:51:51,170 --> 00:51:56,040 Just to make sure that we're all talking about the same thing, 904 00:51:56,040 --> 00:51:58,930 it's useful to start by just making sure 905 00:51:58,930 --> 00:52:02,269 that in a one dimensional problem, 906 00:52:02,269 --> 00:52:03,935 we understand what we mean by stability. 907 00:52:14,520 --> 00:52:15,800 We're going to be fast. 908 00:52:15,800 --> 00:52:20,760 X equals 0 is stable if and only if what? 909 00:52:36,640 --> 00:52:37,710 I'll give you 10 seconds. 910 00:52:47,910 --> 00:52:48,710 Ready? 911 00:52:48,710 --> 00:52:50,295 Three, two, one. 912 00:52:53,321 --> 00:52:53,820 All right. 913 00:52:56,340 --> 00:53:01,530 So this is always-- So there's actually 914 00:53:01,530 --> 00:53:05,060 a fair number of answers here. 915 00:53:05,060 --> 00:53:10,990 And this is tricky because the temptation is always 916 00:53:10,990 --> 00:53:12,880 to jump into the two dimensional stability 917 00:53:12,880 --> 00:53:15,350 analysis, or the n dimensional stability analysis. 918 00:53:15,350 --> 00:53:19,150 And the thing is that we have to make sure 919 00:53:19,150 --> 00:53:21,640 that we are completely comfortable with a one-- talked 920 00:53:21,640 --> 00:53:24,223 about one dimension before you talk about multiple dimensions, 921 00:53:24,223 --> 00:53:25,630 because then everything is lost. 922 00:53:30,880 --> 00:53:32,690 I'm not going to have you guys discuss, 923 00:53:32,690 --> 00:53:37,380 the but it's going to be C. Many people are 924 00:53:37,380 --> 00:53:41,980 saying A or other things. 925 00:53:41,980 --> 00:53:45,060 There is a context in which this guy comes in. 926 00:53:45,060 --> 00:53:46,120 But let's just make sure. 927 00:53:50,310 --> 00:53:51,440 So x equal to 0. 928 00:53:51,440 --> 00:53:54,520 Now first of all, is this thing-- is x equals 0, 929 00:53:54,520 --> 00:53:58,250 is it always a fixed point of the system? 930 00:53:58,250 --> 00:53:59,060 Yes. 931 00:53:59,060 --> 00:54:01,280 So fixed point means that if you go right there, then 932 00:54:01,280 --> 00:54:03,015 in principle you don't move off it. 933 00:54:03,015 --> 00:54:04,890 So it doesn't say anything about whether it's 934 00:54:04,890 --> 00:54:05,750 stable or unstable. 935 00:54:05,750 --> 00:54:08,960 But if x equals zero, then indeed x dot is equal to 0. 936 00:54:08,960 --> 00:54:11,240 So that's a fixed point. 937 00:54:11,240 --> 00:54:16,380 But if you have positive x-- in order for that to be stable, 938 00:54:16,380 --> 00:54:18,220 you have to have a negative change in x. 939 00:54:23,917 --> 00:54:26,250 So if you talk about the behavior as a function of time, 940 00:54:26,250 --> 00:54:28,160 this function of x here. 941 00:54:28,160 --> 00:54:31,190 If you start out above 0, the definition of stable 942 00:54:31,190 --> 00:54:33,470 is if you go a little bit away, you should come back. 943 00:54:38,350 --> 00:54:41,780 And that means that x dot has to be negative. 944 00:54:41,780 --> 00:54:44,710 So if positive x-- and similarly if x is negative, 945 00:54:44,710 --> 00:54:49,380 you want it to be stable, then you 946 00:54:49,380 --> 00:54:50,630 need the x dot to be positive. 947 00:54:50,630 --> 00:54:54,130 So this is-- A less than 0. 948 00:54:54,130 --> 00:54:57,220 Now there is a context in which a stability condition 949 00:54:57,220 --> 00:54:59,570 looks something a little bit more like this. 950 00:54:59,570 --> 00:55:05,080 And can somebody say when it is that you get something 951 00:55:05,080 --> 00:55:06,505 to look-- condition around 1? 952 00:55:11,430 --> 00:55:11,930 Yes? 953 00:55:11,930 --> 00:55:12,870 AUDIENCE: Discreet. 954 00:55:12,870 --> 00:55:15,270 PROFESSOR: Yeah, in discrete maps then indeed 955 00:55:15,270 --> 00:55:17,890 the condition for stability looks something like this. 956 00:55:17,890 --> 00:55:21,590 If you have something that looks the x of t plus 1, 957 00:55:21,590 --> 00:55:32,030 is equal to axt, then if the condition for x 958 00:55:32,030 --> 00:55:35,857 equal to 0 being stable is for indeed a to be less than 1. 959 00:55:35,857 --> 00:55:38,440 Or it's in this case it's really the magnitude of a being less 960 00:55:38,440 --> 00:55:38,940 than 1. 961 00:55:38,940 --> 00:55:41,190 Because in that case you might get hopping. 962 00:55:41,190 --> 00:55:43,560 But then the condition is around the 1 thing, 963 00:55:43,560 --> 00:55:46,110 whereas here 0 [INAUDIBLE] indeed. 964 00:55:46,110 --> 00:55:50,300 Solution is different-- it's going to go exponentially 965 00:55:50,300 --> 00:55:54,650 to-- so x is a function of time is just 966 00:55:54,650 --> 00:55:58,580 going to be some x naught e to the-- and is it at, 967 00:55:58,580 --> 00:56:02,900 or a is less than 0. 968 00:56:02,900 --> 00:56:03,800 It's just at, right? 969 00:56:09,250 --> 00:56:12,510 And in this case, we just have one dimension. 970 00:56:12,510 --> 00:56:16,520 So the eigenvector [INAUDIBLE] is really just x. 971 00:56:16,520 --> 00:56:19,270 There's just one eigenvalue, and a is indeed the eigenvalue. 972 00:56:19,270 --> 00:56:22,710 So the stability for x equals 0 to be stable 973 00:56:22,710 --> 00:56:26,290 is that all-- and in general, for n dimensions, 974 00:56:26,290 --> 00:56:28,965 is that you need all the eigenvalues to be less than 0. 975 00:56:28,965 --> 00:56:30,840 And in this case, a is indeed the eigenvalue. 976 00:56:30,840 --> 00:56:33,260 And you need that to be less than 0. 977 00:56:33,260 --> 00:56:35,480 So if you're not comfortable with this, 978 00:56:35,480 --> 00:56:38,340 or you got something other than C, 979 00:56:38,340 --> 00:56:43,960 then I think it's essential that you take the time now 980 00:56:43,960 --> 00:56:47,192 to go over all of these ideas. 981 00:56:47,192 --> 00:56:48,150 First in one dimension. 982 00:56:48,150 --> 00:56:49,941 Make sure you're all comfortable with that. 983 00:56:49,941 --> 00:56:52,160 But then to look again at these two dimensional, 984 00:56:52,160 --> 00:56:53,800 high dimensional stability analyses. 985 00:56:53,800 --> 00:56:57,402 Because if these things you're finding tricky 986 00:56:57,402 --> 00:56:59,360 just because it's been a while since you looked 987 00:56:59,360 --> 00:57:00,680 at this stuff, that's fine. 988 00:57:00,680 --> 00:57:02,930 But if you don't spend the time to iron out now, 989 00:57:02,930 --> 00:57:06,890 then everything gets much more painful later. 990 00:57:06,890 --> 00:57:11,970 I highly recommend Strogatz's book on dynamical systems. 991 00:57:11,970 --> 00:57:15,190 It's a beautiful book with just as 992 00:57:15,190 --> 00:57:17,300 clear as a textbook could be. 993 00:57:17,300 --> 00:57:20,150 So that book is available at various libraries and reading 994 00:57:20,150 --> 00:57:21,100 rooms and so forth. 995 00:57:21,100 --> 00:57:22,840 So it's a great reference. 996 00:57:25,570 --> 00:57:28,820 Now we want to do is generalize this idea 997 00:57:28,820 --> 00:57:30,930 to think about in n dimensions. 998 00:57:30,930 --> 00:57:36,470 So now what we have a vector x. 999 00:57:36,470 --> 00:57:41,650 And there's going to be some matrix A that-- oh, and x dot. 1000 00:57:44,640 --> 00:57:47,430 So the change in this vector x is 1001 00:57:47,430 --> 00:57:51,660 going to be described by a linear set of equations, 1002 00:57:51,660 --> 00:57:53,535 so that they can be specified by some matrix. 1003 00:57:57,050 --> 00:57:58,540 To determine the stability, we're 1004 00:57:58,540 --> 00:58:04,500 going to then look at the eigenvalues of this matrix. 1005 00:58:04,500 --> 00:58:07,770 Now in many cases in this class, what we're going to do 1006 00:58:07,770 --> 00:58:11,259 is we're going to find that some set of non-linear equations 1007 00:58:11,259 --> 00:58:13,050 has a fixed point somewhere, and then we're 1008 00:58:13,050 --> 00:58:15,760 going to linearize around that fixed point 1009 00:58:15,760 --> 00:58:18,570 to convert into a linear problem that looks like this. 1010 00:58:21,810 --> 00:58:24,552 But for now what we're going to do-- 1011 00:58:24,552 --> 00:58:26,385 the general statement is for n n dimensions. 1012 00:58:29,070 --> 00:58:33,060 You want all the eigenvalues-- lambda I in particular-- 1013 00:58:33,060 --> 00:58:37,740 to be the real part to be less than 0. 1014 00:58:37,740 --> 00:58:40,110 And that has to be true for all the eigenvalues. 1015 00:58:40,110 --> 00:58:43,210 We're going to be talking about the conditions in a two 1016 00:58:43,210 --> 00:58:44,695 dimensional system where there are 1017 00:58:44,695 --> 00:58:47,070 in principles and shortcuts, but it's all the same thing. 1018 00:58:47,070 --> 00:58:48,036 It's all that you need. 1019 00:58:48,036 --> 00:58:50,160 The real part of all the eigenvalues be less than 0 1020 00:58:50,160 --> 00:58:51,409 for the fixed point be stable. 1021 00:59:10,920 --> 00:59:12,670 So for now what we're going to do is we're 1022 00:59:12,670 --> 00:59:14,930 just going to imagine that we have-- assume we 1023 00:59:14,930 --> 00:59:19,010 have two dimensional problem. 1024 00:59:19,010 --> 00:59:21,200 This vector x, we're going to write as x and y. 1025 00:59:28,370 --> 00:59:31,050 So we can write the dynamics like this. 1026 00:59:34,180 --> 00:59:34,750 This is x. 1027 00:59:34,750 --> 00:59:36,035 This is x dot, y dot. 1028 00:59:36,035 --> 00:59:37,410 And this is really the same thing 1029 00:59:37,410 --> 00:59:43,190 as saying that x dot can be described by some ax plus by, 1030 00:59:43,190 --> 00:59:47,675 and y dot is some cx plus dy. 1031 00:59:52,730 --> 00:59:55,360 And so we want to be as comfortable as we can 1032 00:59:55,360 --> 00:59:57,255 with all the possible things that 1033 00:59:57,255 --> 01:00:03,710 could happen in these two linear and differential equations. 1034 01:00:03,710 --> 01:00:06,470 And particularly we want to know what the condition for 0, 1035 01:00:06,470 --> 01:00:09,090 0 being stable? 1036 01:00:09,090 --> 01:00:15,760 0, 0 is stable if and only if-- now there's 1037 01:00:15,760 --> 01:00:18,021 a rule that you found in your reading 1038 01:00:18,021 --> 01:00:20,020 having to do with the trace and the determinant. 1039 01:00:20,020 --> 01:00:21,980 So trace is the sum of these two. 1040 01:00:21,980 --> 01:00:25,280 Determinant is product minus the product here. 1041 01:00:25,280 --> 01:00:27,800 Now this is not the kind of thing 1042 01:00:27,800 --> 01:00:29,020 that you have to memorize. 1043 01:00:29,020 --> 01:00:31,270 But it's useful to know that there is a simple rule, 1044 01:00:31,270 --> 01:00:33,770 because it allows you to quickly determine whether something 1045 01:00:33,770 --> 01:00:35,900 could possibly be stable. 1046 01:00:35,900 --> 01:00:37,400 And you don't have to memorize it, 1047 01:00:37,400 --> 01:00:43,460 but you should be able to figure it back out later. 1048 01:00:43,460 --> 01:00:48,747 So what we have is the trace of this thing being less than 0-- 1049 01:00:48,747 --> 01:00:50,330 so I'm going to give you some options. 1050 01:00:53,930 --> 01:00:56,560 Now of course, you can always look at your notes 1051 01:00:56,560 --> 01:00:58,437 and that is not going to help you. 1052 01:00:58,437 --> 01:01:00,520 You're not being graded on your answers right now. 1053 01:01:00,520 --> 01:01:04,164 I'm going to encourage you to try and think about it 1054 01:01:04,164 --> 01:01:06,580 and see if you can recapitulate what this thing should be. 1055 01:01:06,580 --> 01:01:09,390 So it's less than, greater than-- 1056 01:01:22,380 --> 01:01:24,230 So for example you can start to think 1057 01:01:24,230 --> 01:01:29,500 about-- you should be able to write down some equation 1058 01:01:29,500 --> 01:01:32,120 that you know is stable. 1059 01:01:32,120 --> 01:01:35,010 And figure out, OK, what conditions would it satisfy. 1060 01:01:35,010 --> 01:01:39,500 That's a common, useful trick to be able to do in life. 1061 01:01:39,500 --> 01:01:43,019 To dredge these things out of memory. 1062 01:01:43,019 --> 01:01:44,310 Do you understand the question? 1063 01:01:48,740 --> 01:01:51,830 I'm going to give you 30 seconds because it's 1064 01:01:51,830 --> 01:01:56,751 well worth trying to figure out what this rule should be. 1065 01:01:56,751 --> 01:01:58,250 So from the reading you know there's 1066 01:01:58,250 --> 01:02:00,940 some rule about the trace and the determinant. 1067 01:02:00,940 --> 01:02:02,970 And indeed they both have to be true 1068 01:02:02,970 --> 01:02:07,280 in order-- this is an and sign. 1069 01:02:07,280 --> 01:02:08,544 May be an and sign. 1070 01:02:46,582 --> 01:02:48,190 Do you need more time? 1071 01:02:48,190 --> 01:02:53,340 It's OK if you haven't actually been able to recapitulate this. 1072 01:02:53,340 --> 01:02:58,454 But it's useful to think about it. 1073 01:02:58,454 --> 01:03:00,120 Should we go ahead and vote just to see? 1074 01:03:00,120 --> 01:03:02,200 I'm just curious where we are. 1075 01:03:02,200 --> 01:03:02,830 Ready? 1076 01:03:02,830 --> 01:03:07,630 Three, two, one. 1077 01:03:07,630 --> 01:03:10,770 So we have a lots of B's. 1078 01:03:10,770 --> 01:03:15,670 And can somebody give me an example of a matrix A 1079 01:03:15,670 --> 01:03:18,506 that really ought to be stable? 1080 01:03:18,506 --> 01:03:19,006 Yeah? 1081 01:03:19,006 --> 01:03:20,440 AUDIENCE: Negative identity. 1082 01:03:20,440 --> 01:03:22,010 PROFESSOR: Negative identity, OK. 1083 01:03:22,010 --> 01:03:26,750 So if the matrix A is something that's minus 1 here. 1084 01:03:26,750 --> 01:03:30,800 0, 0 minus 1. 1085 01:03:30,800 --> 01:03:33,360 Then x and y are uncoupled. 1086 01:03:33,360 --> 01:03:37,512 x decays exponentially, y decays exponentially. 1087 01:03:37,512 --> 01:03:39,220 Now we can just directly say, well, this, 1088 01:03:39,220 --> 01:03:41,780 the trace is definitely negative, 1089 01:03:41,780 --> 01:03:44,610 and the determinant's positive. 1090 01:03:44,610 --> 01:03:46,460 It's 1 minus 0. 1091 01:03:46,460 --> 01:03:48,939 So that gets us here. 1092 01:03:48,939 --> 01:03:50,980 Because I think there are many situations in life 1093 01:03:50,980 --> 01:03:53,660 that are like this, where you know that there's some rule 1094 01:03:53,660 --> 01:03:55,480 and you can't remember what it is. 1095 01:03:55,480 --> 01:03:57,211 You don't need to actually remember, 1096 01:03:57,211 --> 01:03:58,710 once you know that there's the rule, 1097 01:03:58,710 --> 01:04:02,130 then you can figure out what it had to have been. 1098 01:04:02,130 --> 01:04:03,480 This is not a proof. 1099 01:04:03,480 --> 01:04:05,070 The proof is only a few lines. 1100 01:04:05,070 --> 01:04:07,940 You can do it, but the point is that this kind of situation 1101 01:04:07,940 --> 01:04:09,090 comes up a lot. 1102 01:04:09,090 --> 01:04:11,274 It's useful to just have simple things 1103 01:04:11,274 --> 01:04:12,940 that you know what the answer has to be, 1104 01:04:12,940 --> 01:04:15,530 and then that allows you to figure out where things were. 1105 01:04:18,470 --> 01:04:20,560 And indeed what you'll find is that for a two 1106 01:04:20,560 --> 01:04:22,889 dimensional system, this condition 1107 01:04:22,889 --> 01:04:24,680 that the trace of the matrix is less than 0 1108 01:04:24,680 --> 01:04:26,830 and the determinant is larger, that 1109 01:04:26,830 --> 01:04:29,760 is equivalent to the statement that the real part 1110 01:04:29,760 --> 01:04:31,470 of both eigenvalues is less than 0. 1111 01:04:34,600 --> 01:04:38,784 And there's the derivation is simple, and it's in your notes. 1112 01:04:38,784 --> 01:04:41,434 Are there any questions about where we are right now? 1113 01:04:47,070 --> 01:04:50,200 So what I want to just say a few more things 1114 01:04:50,200 --> 01:04:53,610 about the trajectories here. 1115 01:04:53,610 --> 01:04:56,920 Can somebody explain the notion of what's 1116 01:04:56,920 --> 01:04:58,860 an intuitive statement about the eigenvectors 1117 01:04:58,860 --> 01:05:00,309 that you get out here? 1118 01:05:08,570 --> 01:05:10,430 Why do we like eigenvectors? 1119 01:05:10,430 --> 01:05:12,340 What are they useful for? 1120 01:05:19,052 --> 01:05:21,052 AUDIENCE: Decoupling the differential equations. 1121 01:05:21,052 --> 01:05:23,300 PROFESSOR: Right, so they're decoupling, 1122 01:05:23,300 --> 01:05:25,730 and there's a sense that-- I guess 1123 01:05:25,730 --> 01:05:29,580 the way I like to think about this is that you have 1124 01:05:29,580 --> 01:05:32,830 a fixed point say, like this. 1125 01:05:32,830 --> 01:05:37,250 Now if these things are stable, that 1126 01:05:37,250 --> 01:05:39,300 means the arrows are all say coming in. 1127 01:05:50,040 --> 01:05:56,140 Are these eigenvalues-- are they purely real, 1128 01:05:56,140 --> 01:05:57,199 or are they complex? 1129 01:05:57,199 --> 01:05:57,990 Reminder, somebody? 1130 01:06:01,800 --> 01:06:03,480 Real and negative. 1131 01:06:03,480 --> 01:06:05,550 There's no imaginary component. 1132 01:06:05,550 --> 01:06:08,970 This is a stable fixed point with real negative eigenvalues. 1133 01:06:08,970 --> 01:06:11,270 Now the idea of these eigenvectors 1134 01:06:11,270 --> 01:06:12,810 is that these are the two directions 1135 01:06:12,810 --> 01:06:15,393 in which if you start out on one of them, you'll stay on them. 1136 01:06:18,260 --> 01:06:21,040 And in general then you can-- any other trajectory 1137 01:06:21,040 --> 01:06:24,180 you can decompose as a combination of the pads 1138 01:06:24,180 --> 01:06:24,720 on the two. 1139 01:06:29,530 --> 01:06:33,620 The position as function of time can always 1140 01:06:33,620 --> 01:06:42,150 described as the sum of the eigenvectors 1141 01:06:42,150 --> 01:06:44,340 where you grow or you shrink along 1142 01:06:44,340 --> 01:06:47,419 each eigenvector exponentially. 1143 01:06:47,419 --> 01:06:49,460 Now for this thing to be stable, all these lambda 1144 01:06:49,460 --> 01:06:50,668 I's are going to be negative. 1145 01:06:53,400 --> 01:07:02,550 We have this property that-- that the dynamics 1146 01:07:02,550 --> 01:07:06,680 of this matrix kind of keep you along the direction 1147 01:07:06,680 --> 01:07:09,950 of the eigenvector. 1148 01:07:09,950 --> 01:07:13,456 What this is saying is that if you start somewhere random, 1149 01:07:13,456 --> 01:07:14,830 that is off one the eigenvectors, 1150 01:07:14,830 --> 01:07:18,530 then you can decompose the trajectory along each. 1151 01:07:18,530 --> 01:07:23,110 But I just want to highlight that it's often 1152 01:07:23,110 --> 01:07:25,610 useful to draw what these things end up looking like. 1153 01:07:31,372 --> 01:07:33,330 Now I want to make sure I get this one correct. 1154 01:07:39,295 --> 01:07:41,670 I'm a little bit worried I'm going to do something funny. 1155 01:07:44,430 --> 01:07:48,440 So here-- we're going to say this is v1 and this is v2. 1156 01:07:48,440 --> 01:07:50,854 So this direction is one eigenvector, 1157 01:07:50,854 --> 01:07:52,020 this direction is the other. 1158 01:08:03,311 --> 01:08:05,560 So let's imagine that the trajectories look like this. 1159 01:08:05,560 --> 01:08:08,500 The question is which eigenvalue is closer to 0? 1160 01:08:31,149 --> 01:08:42,840 Is it A eigenvector v1, or B eigenvector-- So this is really 1161 01:08:42,840 --> 01:08:45,620 lambda 1 verses lambda 2. 1162 01:08:45,620 --> 01:08:47,410 1 is closer to 0 than is 2. 1163 01:08:51,149 --> 01:08:57,220 Which of the eigenvalues is closer to 0? 1164 01:08:57,220 --> 01:09:01,359 I'll give you 20 seconds to think about this because it's 1165 01:09:01,359 --> 01:09:04,029 useful to be able to extract these things 1166 01:09:04,029 --> 01:09:05,220 from the trajectories. 1167 01:09:29,594 --> 01:09:30,510 Do you need more time? 1168 01:09:33,819 --> 01:09:37,350 And it's fine if you don't really 1169 01:09:37,350 --> 01:09:42,640 understand how you can get to this question from this figure, 1170 01:09:42,640 --> 01:09:51,069 then go ahead and flash C, D, or E, just so I know where we are. 1171 01:09:51,069 --> 01:09:51,670 Let's vote. 1172 01:09:51,670 --> 01:09:52,170 Ready? 1173 01:09:52,170 --> 01:09:54,335 Three, two, one. 1174 01:09:57,280 --> 01:10:00,350 All right, so there's at least a majority 1175 01:10:00,350 --> 01:10:04,470 are saying that it's going to be lambda 2. 1176 01:10:04,470 --> 01:10:07,651 Can somebody offer why that is? 1177 01:10:07,651 --> 01:10:08,150 Yes? 1178 01:10:08,150 --> 01:10:12,632 AUDIENCE: I sort of see it as whichever 1179 01:10:12,632 --> 01:10:14,877 one is bigger is going to be more 1180 01:10:14,877 --> 01:10:17,053 effective at squeezing things toward the origin 1181 01:10:17,053 --> 01:10:17,955 along that axis. 1182 01:10:17,955 --> 01:10:21,100 PROFESSOR: OK and bigger-- and you're saying more negative, 1183 01:10:21,100 --> 01:10:24,070 in this case, you're saying? 1184 01:10:24,070 --> 01:10:24,840 Yes, right. 1185 01:10:24,840 --> 01:10:26,940 So yeah that's right. 1186 01:10:26,940 --> 01:10:32,400 So there's some notion that lambda 1 has 1187 01:10:32,400 --> 01:10:33,810 to be more negative than lambda 2 1188 01:10:33,810 --> 01:10:39,840 because you first collapse along the direction of eigenvector 1, 1189 01:10:39,840 --> 01:10:42,255 and then you slowly come in along the direction 1190 01:10:42,255 --> 01:10:44,980 of eigenvector 2. 1191 01:10:44,980 --> 01:10:49,470 That's saying that you have these two exponential decays, 1192 01:10:49,470 --> 01:10:51,180 and the directional on the eigenvector 1 1193 01:10:51,180 --> 01:10:55,820 is more rapid than the directional on eigenvector 2. 1194 01:10:55,820 --> 01:10:58,760 I think in many of these cases in dynamical systems, 1195 01:10:58,760 --> 01:11:01,040 differential equations, it's hugely valuable 1196 01:11:01,040 --> 01:11:03,490 to be able to draw the trajectories. 1197 01:11:03,490 --> 01:11:04,907 So in many cases, what we're going 1198 01:11:04,907 --> 01:11:06,573 to do over the course of the semester is 1199 01:11:06,573 --> 01:11:09,140 we're going to have a simple pair of differential equations 1200 01:11:09,140 --> 01:11:11,090 that are going to be two proteins, 1201 01:11:11,090 --> 01:11:15,690 or they're going to be rabbits and foxes or whatever. 1202 01:11:15,690 --> 01:11:18,324 So we're going to locate where the fixed points are. 1203 01:11:18,324 --> 01:11:19,740 And then we're going to figure out 1204 01:11:19,740 --> 01:11:21,740 the stabilities and the eigenvectors. 1205 01:11:21,740 --> 01:11:24,520 And then we can really understand the entire dynamics 1206 01:11:24,520 --> 01:11:27,700 of the system without solving the full thing, 1207 01:11:27,700 --> 01:11:29,064 without using a computer. 1208 01:11:29,064 --> 01:11:31,230 But just by figuring out where the fixed points are, 1209 01:11:31,230 --> 01:11:32,810 and then the dynamics around there. 1210 01:11:32,810 --> 01:11:35,200 Then you can basically understand 1211 01:11:35,200 --> 01:11:37,590 all the dynamics of the system. 1212 01:11:37,590 --> 01:11:40,080 But you have to make sure that you 1213 01:11:40,080 --> 01:11:43,690 develop intuition of how systems behave near their fixed points. 1214 01:11:46,680 --> 01:11:51,050 Now this is a case where both of the eigenvalues were real. 1215 01:11:51,050 --> 01:11:56,094 But of course, if you have complex eigenvalues, 1216 01:11:56,094 --> 01:11:57,510 if you do the calculations, you'll 1217 01:11:57,510 --> 01:11:59,017 see that the eigenvalues are going 1218 01:11:59,017 --> 01:12:00,930 to be a complex conjugates of each other. 1219 01:12:00,930 --> 01:12:02,470 And there you get spirals. 1220 01:12:08,200 --> 01:12:13,670 So trajectories might look like so. 1221 01:12:13,670 --> 01:12:17,160 So there are two qualitatively different ways that the fixed 1222 01:12:17,160 --> 01:12:18,500 point can be stable. 1223 01:12:18,500 --> 01:12:23,730 You can have spiraling through a state of the fixed point, 1224 01:12:23,730 --> 01:12:29,080 or you can come in via these straight lines. 1225 01:12:29,080 --> 01:12:31,420 Of course, the specific trajectories in some cases, 1226 01:12:31,420 --> 01:12:32,090 can be curved. 1227 01:12:32,090 --> 01:12:34,140 And so if you look at a particular trajectory, 1228 01:12:34,140 --> 01:12:37,170 sometimes even these can look a little bit spirally, 1229 01:12:37,170 --> 01:12:40,580 so be careful. 1230 01:12:40,580 --> 01:12:44,130 Those are the two basic ways it works. 1231 01:12:44,130 --> 01:12:46,950 There's a simple way of getting a sense 1232 01:12:46,950 --> 01:12:50,220 of the dynamics of a system, as a function of the trace 1233 01:12:50,220 --> 01:12:52,180 and the determinant. 1234 01:12:52,180 --> 01:12:55,180 So what we have-- something here. 1235 01:12:59,296 --> 01:13:04,150 If you go ahead and you do the calculation, what you find 1236 01:13:04,150 --> 01:13:14,860 is that the two eigenvalues are going to be described by this. 1237 01:13:30,574 --> 01:13:32,990 And then the basic dynamics are going to be the following. 1238 01:13:32,990 --> 01:13:41,050 So down here, you have a case where the eigenvalues are real, 1239 01:13:41,050 --> 01:13:44,231 and they're of opposite sign. 1240 01:13:44,231 --> 01:13:46,105 And in this case, is that stable or unstable? 1241 01:13:51,050 --> 01:13:52,660 Unstable. 1242 01:13:52,660 --> 01:13:55,020 So in order for it to be stable, all the eigenvalues 1243 01:13:55,020 --> 01:13:56,890 have to have negative real components. 1244 01:13:56,890 --> 01:13:58,840 So if they have opposite sign, one of them 1245 01:13:58,840 --> 01:13:59,890 is going to be positive. 1246 01:13:59,890 --> 01:14:01,430 So this is all unstable down here. 1247 01:14:04,130 --> 01:14:07,300 Over here you have a case where the eigenvalues are real, 1248 01:14:07,300 --> 01:14:08,410 greater than 0. 1249 01:14:08,410 --> 01:14:10,870 So in this case, the trajectories 1250 01:14:10,870 --> 01:14:14,300 are also coming out somehow. 1251 01:14:14,300 --> 01:14:17,910 All right, here, this is the case where they are real, 1252 01:14:17,910 --> 01:14:19,540 but now both less than 0. 1253 01:14:19,540 --> 01:14:21,890 So this is again, this is like what we drew here, where 1254 01:14:21,890 --> 01:14:25,200 everything is stable coming in. 1255 01:14:25,200 --> 01:14:29,410 And up here is where we get the spirals. 1256 01:14:29,410 --> 01:14:32,970 But over here, it's the real part is less than 0. 1257 01:14:32,970 --> 01:14:36,547 So this is where we have the spirals coming in, 1258 01:14:36,547 --> 01:14:38,630 whereas over here, the real part is greater than 0 1259 01:14:38,630 --> 01:14:39,980 and we have spirals coming out. 1260 01:14:44,340 --> 01:14:45,747 Oh, I'm sorry. 1261 01:14:45,747 --> 01:14:47,080 Uh, that would have been useful. 1262 01:14:47,080 --> 01:14:50,775 So this is in the trace of A and this is in the determinant. 1263 01:14:55,400 --> 01:15:01,460 So these are the variety of possible outcomes 1264 01:15:01,460 --> 01:15:05,030 when you have a two dimensional system that's 1265 01:15:05,030 --> 01:15:07,403 already a linear two dimensional system. 1266 01:15:15,117 --> 01:15:16,700 And this is indeed consistent with it, 1267 01:15:16,700 --> 01:15:18,075 to have a stable fixed point, you 1268 01:15:18,075 --> 01:15:19,600 need to have the trace less than 0, 1269 01:15:19,600 --> 01:15:21,113 and the determinant greater than 0. 1270 01:15:21,113 --> 01:15:22,154 So it's in this quadrant. 1271 01:15:30,390 --> 01:15:48,700 Now it's-- finally it's useful to-- so let's just for now, 1272 01:15:48,700 --> 01:15:50,910 stick with the linear system. 1273 01:15:50,910 --> 01:16:00,680 And just imagine a case where we have in this A-- so 1274 01:16:00,680 --> 01:16:05,080 remember it's A, B, C, and D. So we 1275 01:16:05,080 --> 01:16:08,290 saw one way in which this thing could be stable. 1276 01:16:08,290 --> 01:16:10,850 Was that if b and c were both equal 1277 01:16:10,850 --> 01:16:13,610 to 0, so there's no cross interaction, 1278 01:16:13,610 --> 01:16:15,880 but a and d were both less than 0, then 1279 01:16:15,880 --> 01:16:17,220 it's all trivially stable. 1280 01:16:17,220 --> 01:16:18,690 Now the question is, if something 1281 01:16:18,690 --> 01:16:22,430 that looks like this, a is equal to minus 2 and d 1282 01:16:22,430 --> 01:16:25,720 is equal to plus 1. 1283 01:16:25,720 --> 01:16:28,304 Questions is, could such a system be stable? 1284 01:16:37,570 --> 01:16:39,367 So I'm seeing some nods. 1285 01:16:39,367 --> 01:16:41,200 And on the face of this, you say, oh, that's 1286 01:16:41,200 --> 01:16:43,030 a little bit surprising. 1287 01:16:43,030 --> 01:16:46,790 Because d being plus 1, what that's saying 1288 01:16:46,790 --> 01:16:50,740 is that y on its own is unstable. 1289 01:16:50,740 --> 01:16:54,967 So if you start out with no x and no y, 1290 01:16:54,967 --> 01:16:57,550 and you add a little bit of y, y starts growing exponentially. 1291 01:17:00,550 --> 01:17:06,160 So y on its own is unstable, but x is stable its own. 1292 01:17:06,160 --> 01:17:09,010 And the trace being less than 0-- 1293 01:17:09,010 --> 01:17:14,130 that's saying that there's some sense in which if y 1294 01:17:14,130 --> 01:17:16,920 is unstable on its own, then x has to somehow be 1295 01:17:16,920 --> 01:17:20,390 more stable than y is unstable. 1296 01:17:20,390 --> 01:17:24,520 Because the sum of those things still has to be negative. 1297 01:17:24,520 --> 01:17:26,335 Now that was necessary, but not sufficient, 1298 01:17:26,335 --> 01:17:27,460 in order to have stability. 1299 01:17:27,460 --> 01:17:31,520 Because we also need to have a condition on the determinant. 1300 01:17:31,520 --> 01:17:37,119 Now so the trace of a here-- that's minus 1. 1301 01:17:37,119 --> 01:17:37,910 That's less than 0. 1302 01:17:37,910 --> 01:17:38,840 OK, that's great. 1303 01:17:38,840 --> 01:17:44,020 Now the determinant-- now it's going to be a times d. 1304 01:17:44,020 --> 01:17:47,440 That's minus 2. 1305 01:17:47,440 --> 01:17:49,960 But then we also have to say minus b times c, right? 1306 01:17:52,470 --> 01:17:55,500 And this thing has to be greater than 0. 1307 01:17:55,500 --> 01:17:58,280 So what you see here is that b and c-- they 1308 01:17:58,280 --> 01:18:03,880 have to have opposite signs order for this thing to work, 1309 01:18:03,880 --> 01:18:07,310 in order for the origin to be stable. 1310 01:18:07,310 --> 01:18:09,785 And the product has to somehow be strong. 1311 01:18:13,070 --> 01:18:17,240 Now this makes sense, because of course, if b and c, 1312 01:18:17,240 --> 01:18:20,050 or even for matter, just one of them were 0, 1313 01:18:20,050 --> 01:18:23,530 then it would be impossible to get the stability. 1314 01:18:23,530 --> 01:18:28,480 But for example, if we have some situation where 1315 01:18:28,480 --> 01:18:35,020 we have some x that's inhibiting itself-- that's a here-- 1316 01:18:35,020 --> 01:18:37,310 but then y is activating itself. 1317 01:18:37,310 --> 01:18:42,750 So y here is somehow on its own, unstable. 1318 01:18:42,750 --> 01:18:47,190 Then what you need is you need maybe something that 1319 01:18:47,190 --> 01:18:49,280 looks like this. 1320 01:18:49,280 --> 01:18:52,240 Some cross activation and or repression. 1321 01:18:52,240 --> 01:18:55,710 And what's interesting is that it doesn't matter which 1322 01:18:55,710 --> 01:18:58,560 of the two, b or c is negative. 1323 01:18:58,560 --> 01:19:03,200 You can get actually, the origin to be stable in either way. 1324 01:19:03,200 --> 01:19:08,710 But in this situation, you need the direction of the regulation 1325 01:19:08,710 --> 01:19:10,720 to be in opposite directions. 1326 01:19:15,650 --> 01:19:17,070 And if you'd like, you could then 1327 01:19:17,070 --> 01:19:23,540 play with-- you could think for example 1328 01:19:23,540 --> 01:19:27,090 about the directions of these trajectories around the origin, 1329 01:19:27,090 --> 01:19:27,970 and so forth. 1330 01:19:27,970 --> 01:19:31,390 But it's useful, I think, to play with the simplest toy 1331 01:19:31,390 --> 01:19:33,797 systems that you can imagine, just so that you can get 1332 01:19:33,797 --> 01:19:35,880 a sense of what are the basic ingredients that you 1333 01:19:35,880 --> 01:19:38,530 need in order to get stability in something like this. 1334 01:19:38,530 --> 01:19:42,027 Because once you start doing the whole linearization 1335 01:19:42,027 --> 01:19:43,860 around the fixed point, and then calculating 1336 01:19:43,860 --> 01:19:46,430 traces and determinants, you're not going to have it. 1337 01:19:46,430 --> 01:19:47,880 You're going to lose all your intuition about that 1338 01:19:47,880 --> 01:19:49,004 about things at that stage. 1339 01:19:49,004 --> 01:19:51,550 So it's useful to make sure that you nail things 1340 01:19:51,550 --> 01:19:53,320 down in this context. 1341 01:19:53,320 --> 01:19:56,660 We are out of time, so I'll let you go.