1 00:00:00,080 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 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 --> 00:00:13,340 To make a donation or view additional materials 6 00:00:13,340 --> 00:00:17,215 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,215 --> 00:00:17,840 at ocw.mit.edu. 8 00:00:21,590 --> 00:00:25,230 PROFESSOR: Our goal for today is to basically analyze 9 00:00:25,230 --> 00:00:27,530 this simple model to death. 10 00:00:27,530 --> 00:00:30,520 So we're first going to try to understand 11 00:00:30,520 --> 00:00:34,190 the deterministic behavior of this model gene 12 00:00:34,190 --> 00:00:38,130 expression, where we just get transcription of mRNA, 13 00:00:38,130 --> 00:00:40,100 and then translation of protein. 14 00:00:40,100 --> 00:00:43,272 And after we think we understand the mean behavior, 15 00:00:43,272 --> 00:00:44,730 the deterministic dynamics, then we 16 00:00:44,730 --> 00:00:48,490 will try to understand just stochastic behavior 17 00:00:48,490 --> 00:00:49,400 in this model. 18 00:00:49,400 --> 00:00:51,370 So we're going to try to understand 19 00:00:51,370 --> 00:00:55,450 what's the distribution of mRNA in a cell 20 00:00:55,450 --> 00:00:56,920 in this simple situation. 21 00:00:56,920 --> 00:00:58,770 What's the distribution of protein? 22 00:00:58,770 --> 00:01:01,400 What's going to be the bursting behavior? 23 00:01:01,400 --> 00:01:02,900 Everything you can possibly think of 24 00:01:02,900 --> 00:01:05,070 to ask about this model, we will hopefully 25 00:01:05,070 --> 00:01:08,230 have asked by the end of today's class. 26 00:01:08,230 --> 00:01:10,150 This simple model of gene expression, 27 00:01:10,150 --> 00:01:12,830 as was indicated in the review, is perhaps 28 00:01:12,830 --> 00:01:15,800 a reasonable description of gene expression 29 00:01:15,800 --> 00:01:21,180 in bacteria, when the gene is in some active state. 30 00:01:21,180 --> 00:01:25,312 So there's no repressor, for example, bound. 31 00:01:25,312 --> 00:01:26,770 Although maybe even in the presence 32 00:01:26,770 --> 00:01:28,660 of a repressor, if it's binding and unbinding, 33 00:01:28,660 --> 00:01:31,118 maybe you still end up getting some sort of renormalization 34 00:01:31,118 --> 00:01:32,600 that looks like this. 35 00:01:32,600 --> 00:01:35,220 But this is first order, a reasonable description 36 00:01:35,220 --> 00:01:38,710 of gene expression in bacteria. 37 00:01:38,710 --> 00:01:41,460 And it's the model that was basically 38 00:01:41,460 --> 00:01:44,070 used in the Sunney Xie paper that we talked about 39 00:01:44,070 --> 00:01:44,860 on Tuesday. 40 00:01:44,860 --> 00:01:46,560 And hopefully this model will allow 41 00:01:46,560 --> 00:01:49,880 us to think a little bit more deeply about the data 42 00:01:49,880 --> 00:01:53,640 that they obtained in that paper. 43 00:01:53,640 --> 00:01:57,320 As always, we want to start by understanding 44 00:01:57,320 --> 00:01:58,612 the basic aspects of the model. 45 00:01:58,612 --> 00:02:00,070 So what we're going to do, is we're 46 00:02:00,070 --> 00:02:02,010 going to go through a series of questions 47 00:02:02,010 --> 00:02:03,660 of increasing difficulty. 48 00:02:03,660 --> 00:02:06,580 And in some of them, we are indeed, 49 00:02:06,580 --> 00:02:12,242 the answers will end up being something divided by something. 50 00:02:12,242 --> 00:02:14,200 In which case you take advantage of your cards, 51 00:02:14,200 --> 00:02:16,490 and illustrate that by putting something on top, something 52 00:02:16,490 --> 00:02:16,990 below. 53 00:02:24,410 --> 00:02:28,270 But just first in this model, what is the unit of time? 54 00:02:28,270 --> 00:02:32,072 So if I say t is equal to 1, or delta t is equal to 1, 55 00:02:32,072 --> 00:02:35,460 what am I referring to? 56 00:02:35,460 --> 00:02:36,860 So we're not use the cards. 57 00:02:36,860 --> 00:02:42,070 But in particular, the question is, is delta t equal to 1, 58 00:02:42,070 --> 00:02:46,560 is that a cell cycle necessarily? 59 00:02:46,560 --> 00:02:49,550 Yes or no, ready, three, two, one. 60 00:02:49,550 --> 00:02:54,850 Well I guess now maybe I've complicated things by-- well, 61 00:02:54,850 --> 00:02:57,910 this was really going to be relevant for the later ones. 62 00:02:57,910 --> 00:02:58,910 All right. 63 00:02:58,910 --> 00:03:01,000 Now I've totally confused you. 64 00:03:01,000 --> 00:03:03,590 But can somebody offer why it may or may not 65 00:03:03,590 --> 00:03:08,920 be-- how do we think about the unit of time in this model? 66 00:03:08,920 --> 00:03:10,675 AUDIENCE: Usually the lifetime of one 67 00:03:10,675 --> 00:03:11,800 of the species [INAUDIBLE]. 68 00:03:14,322 --> 00:03:15,030 PROFESSOR: Right. 69 00:03:15,030 --> 00:03:18,640 OK so indeed, what we often do in these non-dimensionalized 70 00:03:18,640 --> 00:03:21,530 models is we set something equal to 1. 71 00:03:21,530 --> 00:03:23,876 Have we set anything equal to 1 here? 72 00:03:23,876 --> 00:03:24,430 No. 73 00:03:24,430 --> 00:03:25,888 So in principle, we've said there's 74 00:03:25,888 --> 00:03:28,440 some degradation rate of the mRNA, some degradation 75 00:03:28,440 --> 00:03:30,600 rate of the protein. 76 00:03:30,600 --> 00:03:34,780 And in general, those will be given in some units involving 77 00:03:34,780 --> 00:03:38,120 seconds or minutes or hours. 78 00:03:38,120 --> 00:03:40,350 So in general, so at this stage, we 79 00:03:40,350 --> 00:03:43,240 have not yet-- we have not actually 80 00:03:43,240 --> 00:03:45,980 gotten to this sort of non-dimensionalized version 81 00:03:45,980 --> 00:03:47,110 of any model. 82 00:03:47,110 --> 00:03:49,490 So in this case this is going to be something 83 00:03:49,490 --> 00:03:53,500 like a seconds, or minutes, or hours, 84 00:03:53,500 --> 00:03:58,230 whatever units we use for those degradation rates. 85 00:03:58,230 --> 00:04:01,488 So we have not done anything where it's the cell generation 86 00:04:01,488 --> 00:04:03,863 time, or the protein lifetime, mRNA lifetime, or anything 87 00:04:03,863 --> 00:04:04,790 like that. 88 00:04:04,790 --> 00:04:07,260 Here everybody happy with this statement so far? 89 00:04:10,200 --> 00:04:12,180 So we'll go ahead and vote here. 90 00:04:12,180 --> 00:04:16,899 So we're going to do some A, B, C, D's. 91 00:04:16,899 --> 00:04:19,890 And you can always combine anything you want. 92 00:04:19,890 --> 00:04:23,630 So we'll go ahead and say this is 93 00:04:23,630 --> 00:04:26,980 the synthesis rate of the mRNA. 94 00:04:26,980 --> 00:04:29,330 This is the degradation rate for the mRNA, the synthesis 95 00:04:29,330 --> 00:04:31,105 rate for the protein, the degradation 96 00:04:31,105 --> 00:04:31,980 rate for the protein. 97 00:04:34,747 --> 00:04:36,830 And if you're just confused, you can just do this. 98 00:04:36,830 --> 00:04:39,288 But in general, for any of the questions we're going to do, 99 00:04:39,288 --> 00:04:40,997 you can do some combination of these guys 100 00:04:40,997 --> 00:04:42,954 by putting things in numerator and denominator. 101 00:04:42,954 --> 00:04:43,576 Yes? 102 00:04:43,576 --> 00:04:45,534 AUDIENCE: Calculate the population of the cells 103 00:04:45,534 --> 00:04:46,650 that were hidden? 104 00:04:46,650 --> 00:04:47,930 PROFESSOR: Yes. 105 00:04:47,930 --> 00:04:51,810 Question is, if you just look at the cell population, 106 00:04:51,810 --> 00:04:53,520 and you find it's growing exponentially, 107 00:04:53,520 --> 00:04:55,100 the question is what is going to be that rate 108 00:04:55,100 --> 00:04:56,016 of exponential growth. 109 00:04:59,570 --> 00:05:01,090 Have I done something wrong already? 110 00:05:04,158 --> 00:05:05,074 AUDIENCE: [INAUDIBLE]? 111 00:05:08,317 --> 00:05:08,900 PROFESSOR: OK. 112 00:05:08,900 --> 00:05:11,450 But I am going to say that now for this we're 113 00:05:11,450 --> 00:05:13,800 going to assume that the protein is stable. 114 00:05:13,800 --> 00:05:15,115 So it's not actually degraded. 115 00:05:22,870 --> 00:05:26,400 This is to remind you of what we read about in chapter one, 116 00:05:26,400 --> 00:05:29,986 maybe of Uri's book, maybe chapter two. 117 00:05:29,986 --> 00:05:31,860 I'll give you 10 seconds to think about this. 118 00:05:40,910 --> 00:05:44,070 Do you need more time? 119 00:05:44,070 --> 00:05:48,590 All right, Ready, three, two, one. 120 00:05:48,590 --> 00:05:50,010 OK. 121 00:05:50,010 --> 00:05:53,230 We got a bunch of C's and a bunch of D's and some E's. 122 00:05:53,230 --> 00:05:53,730 All right. 123 00:05:53,730 --> 00:05:56,250 So the E's are going to argue with me, presumably rather than 124 00:05:56,250 --> 00:05:56,750 a neighbor. 125 00:06:00,770 --> 00:06:01,802 OK. 126 00:06:01,802 --> 00:06:03,510 I think that there are enough people that 127 00:06:03,510 --> 00:06:06,012 are disagreeing on this to maybe go ahead, and turn. 128 00:06:06,012 --> 00:06:07,470 You should be able to find somebody 129 00:06:07,470 --> 00:06:08,580 that disagrees with you. 130 00:06:08,580 --> 00:06:10,980 The distribution was a bit patchy, unfortunately. 131 00:06:10,980 --> 00:06:14,534 Did you guys-- you guys are worried that you're not going 132 00:06:14,534 --> 00:06:15,700 to be able to find somebody. 133 00:06:15,700 --> 00:06:16,199 OK. 134 00:06:16,199 --> 00:06:17,650 Fine, fine. 135 00:06:17,650 --> 00:06:18,650 Yeah? 136 00:06:18,650 --> 00:06:22,580 AUDIENCE: So if the protein is stable, ah, 137 00:06:22,580 --> 00:06:24,512 so the mRNA may not be stable? 138 00:06:24,512 --> 00:06:26,095 PROFESSOR: The mRNA may not be stable. 139 00:06:26,095 --> 00:06:26,926 AUDIENCE: Ah, OK. 140 00:06:26,926 --> 00:06:27,700 That makes sense. 141 00:06:27,700 --> 00:06:29,980 PROFESSOR: And in general which one typically 142 00:06:29,980 --> 00:06:31,899 has a longer lifetime? 143 00:06:31,899 --> 00:06:32,690 AUDIENCE: Proteins. 144 00:06:32,690 --> 00:06:34,931 PROFESSOR: Proteins typically have a longer lifetime. 145 00:06:34,931 --> 00:06:35,430 Right. 146 00:06:35,430 --> 00:06:40,810 So mRNA are actively degraded, typically. 147 00:06:40,810 --> 00:06:44,117 They're also just kind of less stable intrinsically. 148 00:06:44,117 --> 00:06:45,700 But what we're going to assume for now 149 00:06:45,700 --> 00:06:48,890 is that we're working with stable proteins. 150 00:06:48,890 --> 00:06:54,340 In which case the growth rate of the population 151 00:06:54,340 --> 00:06:57,270 will just be this effective degradation 152 00:06:57,270 --> 00:06:59,450 rate of the protein. 153 00:06:59,450 --> 00:07:01,420 So in this model, even if we say there's 154 00:07:01,420 --> 00:07:04,050 no active degradation of the protein, 155 00:07:04,050 --> 00:07:08,940 still there's going to be some effective degradation that's 156 00:07:08,940 --> 00:07:10,930 due to dilution. 157 00:07:10,930 --> 00:07:13,308 So we can say effective, if you like. 158 00:07:16,280 --> 00:07:18,819 So the rate of exponential growth of the population 159 00:07:18,819 --> 00:07:20,610 will be equal to this effective degradation 160 00:07:20,610 --> 00:07:22,151 rate for the protein, if it's stable. 161 00:07:24,590 --> 00:07:26,590 AUDIENCE: So you're talking about the population 162 00:07:26,590 --> 00:07:28,990 of the protein? 163 00:07:28,990 --> 00:07:30,000 PROFESSOR: No. 164 00:07:30,000 --> 00:07:33,240 The growth rate of the cell population. 165 00:07:33,240 --> 00:07:37,620 So this is if we go in there, and you 166 00:07:37,620 --> 00:07:38,980 go into your spectrophotometer. 167 00:07:38,980 --> 00:07:42,190 And you measure population-- numbers in function of time 168 00:07:42,190 --> 00:07:43,280 is growing exponentially. 169 00:07:43,280 --> 00:07:45,300 It'll grow exponentially with this rate. 170 00:07:45,300 --> 00:07:47,710 Because this is what's causing the dilution. 171 00:07:47,710 --> 00:07:52,423 In some ways if you stop making the protein, 172 00:07:52,423 --> 00:07:54,506 and you double the number of cells, and that means 173 00:07:54,506 --> 00:07:57,100 the concentration of the protein in each cell 174 00:07:57,100 --> 00:07:59,420 has to go down by a factor of two. 175 00:07:59,420 --> 00:08:00,420 So that's the statement. 176 00:08:00,420 --> 00:08:05,347 Are there any questions about why I'm making this argument? 177 00:08:05,347 --> 00:08:05,846 Yes? 178 00:08:05,846 --> 00:08:08,280 AUDIENCE: What was the relevance of the protein being stable? 179 00:08:08,280 --> 00:08:08,720 PROFESSOR: All right. 180 00:08:08,720 --> 00:08:10,261 So the relevance of the protein being 181 00:08:10,261 --> 00:08:14,700 stable, because this is in general, this delta, this 182 00:08:14,700 --> 00:08:17,540 is the effective rate. 183 00:08:17,540 --> 00:08:20,970 This is going to be equal to the growth rate of the population. 184 00:08:20,970 --> 00:08:25,740 So you might call it gamma growth 185 00:08:25,740 --> 00:08:30,010 plus the actual degradation. 186 00:08:30,010 --> 00:08:32,599 I don't want to use the same, but I'll just 187 00:08:32,599 --> 00:08:34,770 say plus the degradation rate. 188 00:08:34,770 --> 00:08:40,627 And this is a true physical degradation, true degradation 189 00:08:40,627 --> 00:08:42,750 rate of the protein. 190 00:08:42,750 --> 00:08:47,380 So if it's stable, then we say that this thing is zero. 191 00:08:47,380 --> 00:08:48,820 So when we say stable protein, it 192 00:08:48,820 --> 00:08:50,970 means there's no degradation of the protein. 193 00:08:50,970 --> 00:08:53,700 So this physical degradation rate is zero. 194 00:08:53,700 --> 00:08:56,070 And then the effective degradation rate of the protein 195 00:08:56,070 --> 00:08:59,148 is just equal to the growth rate of the population. 196 00:08:59,148 --> 00:09:02,052 AUDIENCE: OK, so no degradation means stable, basically? 197 00:09:02,052 --> 00:09:05,434 PROFESSOR: Yes, sorry, yeah. 198 00:09:05,434 --> 00:09:07,350 Any other questions about what I mean by this? 199 00:09:11,270 --> 00:09:13,980 So now what we want to do is ask a few other quantities 200 00:09:13,980 --> 00:09:15,240 about this model. 201 00:09:15,240 --> 00:09:21,650 So for example, what will be the number of mRNA per cell? 202 00:09:32,450 --> 00:09:34,455 And this is always going to be the mean. 203 00:09:41,030 --> 00:09:42,570 I'll give you 20 seconds. 204 00:09:42,570 --> 00:09:45,560 In this model what is the mean number of mRNA per cell? 205 00:10:09,160 --> 00:10:10,280 All right. 206 00:10:10,280 --> 00:10:12,580 Ready? 207 00:10:12,580 --> 00:10:14,720 Three, two, one. 208 00:10:21,410 --> 00:10:24,580 And we have let's say a majority of the group 209 00:10:24,580 --> 00:10:28,420 is saying it's A over B, which corresponds to the synthesis 210 00:10:28,420 --> 00:10:31,540 rate of the mRNA divided by the degradation. 211 00:10:31,540 --> 00:10:33,370 Some people are this one? 212 00:10:33,370 --> 00:10:35,750 Yes, so this is indeed, synthesis rate 213 00:10:35,750 --> 00:10:37,320 divided by the degradation rate. 214 00:10:40,120 --> 00:10:45,330 Now this is saying that what happens later 215 00:10:45,330 --> 00:10:50,300 doesn't really matter, for the mean mRNA number. 216 00:10:50,300 --> 00:10:52,860 Because it's just that it's going to be made at some rate. 217 00:10:52,860 --> 00:10:55,290 Its lifetime is given by 1 over delta m. 218 00:10:55,290 --> 00:10:58,509 Now this thing, of course, is again as always, 219 00:10:58,509 --> 00:10:59,800 the effective degradation rate. 220 00:10:59,800 --> 00:11:03,440 So it's the sum of the sort of physical degradation rate, 221 00:11:03,440 --> 00:11:06,570 plus this dilution due to growth. 222 00:11:06,570 --> 00:11:09,170 But in general, the true degradation, 223 00:11:09,170 --> 00:11:11,920 the physical degradation is much faster than the cell division 224 00:11:11,920 --> 00:11:12,980 rate. 225 00:11:12,980 --> 00:11:15,350 So this is very close to actually just 226 00:11:15,350 --> 00:11:18,140 the physical degradation rate. 227 00:11:18,140 --> 00:11:22,640 But in any case, it's just delta m, regardless. 228 00:11:22,640 --> 00:11:26,920 Are there any questions about why this is the way it is? 229 00:11:26,920 --> 00:11:27,820 Yes? 230 00:11:27,820 --> 00:11:30,740 AUDIENCE: Does it matter whether it's only physical? 231 00:11:30,740 --> 00:11:32,610 Because wouldn't it be the same if it were-- 232 00:11:32,610 --> 00:11:34,860 PROFESSOR: It doesn't matter that it's only-- exactly. 233 00:11:34,860 --> 00:11:36,193 That's what I was trying to say. 234 00:11:36,193 --> 00:11:37,680 So the way that this is written, it 235 00:11:37,680 --> 00:11:41,952 doesn't matter whether the-- this is the answer regardless 236 00:11:41,952 --> 00:11:43,410 of whether the physical degradation 237 00:11:43,410 --> 00:11:45,990 rate is much larger than the growth rate or not. 238 00:11:45,990 --> 00:11:46,490 Yeah. 239 00:11:51,240 --> 00:11:52,330 All right. 240 00:11:52,330 --> 00:12:00,150 What is this protein molecules per mRNA? 241 00:12:00,150 --> 00:12:04,690 How many protein molecules are made from each mRNA? 242 00:12:04,690 --> 00:12:40,215 Protein produced-- Do you need more time? 243 00:12:52,040 --> 00:12:52,820 Remember. 244 00:12:52,820 --> 00:12:56,340 This is again, the mean number of proteins 245 00:12:56,340 --> 00:12:58,910 produced from a single mRNA or each mRNA. 246 00:13:02,500 --> 00:13:05,570 Let's go ahead and vote, so I can see where we are. 247 00:13:05,570 --> 00:13:06,330 Ready? 248 00:13:06,330 --> 00:13:08,480 Three, two, one. 249 00:13:14,490 --> 00:13:14,990 OK. 250 00:13:14,990 --> 00:13:20,730 So we have, I'd say, so at least a majority 251 00:13:20,730 --> 00:13:24,961 are saying it's going to be C over B Now. 252 00:13:24,961 --> 00:13:25,460 All right. 253 00:13:25,460 --> 00:13:26,510 So this is interesting. 254 00:13:26,510 --> 00:13:28,660 So this is saying that really what's happening 255 00:13:28,660 --> 00:13:34,190 is that there's a competition once you 256 00:13:34,190 --> 00:13:37,660 make an mRNA that the proteins are going to be getting 257 00:13:37,660 --> 00:13:38,752 fired off at some rate. 258 00:13:38,752 --> 00:13:40,460 But eventually it's going to be degraded. 259 00:13:40,460 --> 00:13:42,305 It's a competition between those two rates 260 00:13:42,305 --> 00:13:45,340 that determines basically how many proteins, 261 00:13:45,340 --> 00:13:48,219 how many times do you fire off a protein before you 262 00:13:48,219 --> 00:13:48,760 get degraded. 263 00:13:52,930 --> 00:13:54,230 Any questions about that logic? 264 00:13:57,767 --> 00:14:00,100 AUDIENCE: Can you please just repeat that one more time? 265 00:14:00,100 --> 00:14:00,810 PROFESSOR: Sure. 266 00:14:00,810 --> 00:14:04,054 Right, so what we're assuming is that OK, an mRNA is produced. 267 00:14:04,054 --> 00:14:05,220 And that's already happened. 268 00:14:05,220 --> 00:14:07,800 So it doesn't matter what Sm is anymore. 269 00:14:07,800 --> 00:14:08,780 So now we have an mRNA. 270 00:14:11,670 --> 00:14:14,580 Eventually this mRNA will be degraded. 271 00:14:14,580 --> 00:14:18,210 But before that happens, we want to know, basically 272 00:14:18,210 --> 00:14:21,710 how many proteins do we expect to be made. 273 00:14:21,710 --> 00:14:24,820 Now if Sp and delta m are the same that means 274 00:14:24,820 --> 00:14:27,800 you kind of expect one protein to be made on average, 275 00:14:27,800 --> 00:14:28,800 before it's degraded. 276 00:14:28,800 --> 00:14:31,380 Or if Sp we're twice delta m, then you 277 00:14:31,380 --> 00:14:34,580 would get two proteins made before it was degraded. 278 00:14:34,580 --> 00:14:36,680 Now this is a mean statement. 279 00:14:36,680 --> 00:14:39,000 We're about to start thinking-- in 10 minutes, 280 00:14:39,000 --> 00:14:41,140 we'll think about this distribution. 281 00:14:41,140 --> 00:14:42,560 And so we have to be careful. 282 00:14:42,560 --> 00:14:46,504 But in terms of mean behavior, this thing is true. 283 00:14:46,504 --> 00:14:48,243 AUDIENCE: So is this different than it 284 00:14:48,243 --> 00:14:53,467 has been for the number of proteins per mRNA in the cell? 285 00:14:53,467 --> 00:14:55,300 PROFESSOR: Is this different from the number 286 00:14:55,300 --> 00:14:57,150 of proteins in the cell? 287 00:14:57,150 --> 00:15:02,551 AUDIENCE: Number of proteins per mRNA in the cell. 288 00:15:02,551 --> 00:15:05,006 Because then you will have to do the protein concentration 289 00:15:05,006 --> 00:15:06,107 over mRNA concentration. 290 00:15:06,107 --> 00:15:06,690 PROFESSOR: OK. 291 00:15:06,690 --> 00:15:07,190 Right. 292 00:15:07,190 --> 00:15:12,140 So this is not the same thing as asking about the ratio 293 00:15:12,140 --> 00:15:13,430 of the number of proteins. 294 00:15:13,430 --> 00:15:16,239 And we can calculate that as well. 295 00:15:16,239 --> 00:15:17,280 Yeah these are different. 296 00:15:19,775 --> 00:15:21,400 This is the number of protein molecules 297 00:15:21,400 --> 00:15:22,632 produced from each mRNA. 298 00:15:22,632 --> 00:15:24,340 So this is just talking about production. 299 00:15:24,340 --> 00:15:25,740 Because indeed, the degradation rates 300 00:15:25,740 --> 00:15:26,823 are going to be different. 301 00:15:26,823 --> 00:15:31,330 So then we can see what that ends up being. 302 00:15:31,330 --> 00:15:34,490 Any other questions about why this one is what it is? 303 00:15:38,930 --> 00:15:43,150 How about the number of mRNA produced per cell cycle? 304 00:15:47,500 --> 00:15:52,380 And for now we're going to ignore factors of log two. 305 00:16:36,840 --> 00:16:37,790 Do you need more time? 306 00:16:41,704 --> 00:16:42,620 So another 10 seconds. 307 00:16:54,596 --> 00:16:56,592 AUDIENCE: Produced but not degraded? 308 00:16:59,110 --> 00:17:00,524 PROFESSOR: Produced, yes. 309 00:17:00,524 --> 00:17:02,024 We're just talking about production. 310 00:17:05,089 --> 00:17:10,980 Because we've already calculated a number of mRNA in the cell. 311 00:17:10,980 --> 00:17:14,670 But now we want to know the mean number produced. 312 00:17:14,670 --> 00:17:17,490 For example, this is the same as the mean number 313 00:17:17,490 --> 00:17:23,910 of protein bursts observed in Sunney Xie's paper. 314 00:17:23,910 --> 00:17:27,089 But this is just the number of mRNA produced per cell cycle. 315 00:17:27,089 --> 00:17:27,589 All right. 316 00:17:27,589 --> 00:17:28,180 Let's see where we are. 317 00:17:28,180 --> 00:17:28,679 Ready? 318 00:17:28,679 --> 00:17:31,811 Three, two, one. 319 00:17:31,811 --> 00:17:32,310 All right. 320 00:17:32,310 --> 00:17:34,720 So we've got lots of A's over D's. 321 00:17:34,720 --> 00:17:36,060 That's sounds nice. 322 00:17:36,060 --> 00:17:38,090 So this is going to be some synthesis rate. 323 00:17:38,090 --> 00:17:40,600 But now the relevant thing is this delta p. 324 00:17:40,600 --> 00:17:43,250 Because that's the cell division rate. 325 00:17:43,250 --> 00:17:46,050 So it's barring issues of log two, 326 00:17:46,050 --> 00:17:47,630 it's approximately the synthesis rate 327 00:17:47,630 --> 00:17:50,770 of the mRNA divided by delta p. 328 00:17:50,770 --> 00:17:54,760 Because this is the growth rate of population. 329 00:17:54,760 --> 00:17:59,595 Cell generation time is log two off of that. 330 00:17:59,595 --> 00:18:01,470 Are there any questions about that statement? 331 00:18:04,810 --> 00:18:06,870 All right, so this is the mean number. 332 00:18:06,870 --> 00:18:09,745 Now from the paper, we know how this thing is distributed. 333 00:18:16,080 --> 00:18:19,270 We should probably-- we're going to use a bunch of distributions 334 00:18:19,270 --> 00:18:21,840 over the next couple. 335 00:18:21,840 --> 00:18:30,540 So we can-- we like exponential distributions. 336 00:18:30,540 --> 00:18:33,810 We like geometric distributions. 337 00:18:36,465 --> 00:18:37,370 We like Poisson. 338 00:18:40,390 --> 00:18:41,500 We like Gaussian. 339 00:18:44,900 --> 00:18:46,160 And we like gamma. 340 00:18:51,672 --> 00:18:53,505 These are various probability distributions. 341 00:18:53,505 --> 00:18:55,887 The question is, how is it that now, not the mean, 342 00:18:55,887 --> 00:18:57,470 but how is the number of mRNA produced 343 00:18:57,470 --> 00:19:00,400 per cell cycle distributed. 344 00:19:00,400 --> 00:19:00,960 Ready? 345 00:19:00,960 --> 00:19:02,800 Three, two, one. 346 00:19:07,580 --> 00:19:08,620 All right. 347 00:19:08,620 --> 00:19:15,290 We've got some-- this side of the rooms a little bit 348 00:19:15,290 --> 00:19:16,530 slower, maybe. 349 00:19:16,530 --> 00:19:18,710 But that's OK. 350 00:19:18,710 --> 00:19:22,370 So maybe some people are not confident of this statement. 351 00:19:22,370 --> 00:19:22,870 OK. 352 00:19:22,870 --> 00:19:24,286 So this one ends up being Poisson. 353 00:19:27,000 --> 00:19:30,630 So this is indeed how the number of-- this 354 00:19:30,630 --> 00:19:36,100 is number mRNA per cycle. 355 00:19:36,100 --> 00:19:38,190 Now this is-- so Poisson, in general, 356 00:19:38,190 --> 00:19:41,080 that's what you get if there's some probability per unit time 357 00:19:41,080 --> 00:19:42,030 that something's going to happen, 358 00:19:42,030 --> 00:19:44,696 and you want to know how many of them happen in some finite time 359 00:19:44,696 --> 00:19:45,240 period. 360 00:19:45,240 --> 00:19:47,115 That's basically the definition of a Poisson. 361 00:19:49,700 --> 00:19:54,110 And this is, if you recall, this is what we talked about 362 00:19:54,110 --> 00:19:54,610 on Tuesday. 363 00:19:54,610 --> 00:20:00,200 The probability observe n, it's given by this mean number. 364 00:20:00,200 --> 00:20:01,820 So if lambda is the mean, then we 365 00:20:01,820 --> 00:20:04,620 get lambda to the n, over n factorial, 366 00:20:04,620 --> 00:20:07,020 e to the minus lambda. 367 00:20:07,020 --> 00:20:10,840 If you go ahead and calculate the mean of this, 368 00:20:10,840 --> 00:20:12,170 you indeed get lambda. 369 00:20:12,170 --> 00:20:17,890 So lambda is equal to the mean, which in this case 370 00:20:17,890 --> 00:20:21,530 was around, well in the case of Sunney's paper, 371 00:20:21,530 --> 00:20:25,821 does anybody remember what that roughly was? 372 00:20:25,821 --> 00:20:26,570 It was around one. 373 00:20:35,490 --> 00:20:37,770 Now what about this other one? 374 00:20:37,770 --> 00:20:40,600 So we also have another mRNA problem, 375 00:20:40,600 --> 00:20:43,900 which is that we calculated the mean number of mRNA per cell. 376 00:20:43,900 --> 00:20:46,790 If you look at a cell, the mean number is this. 377 00:20:46,790 --> 00:20:48,970 But what's the probability distribution 378 00:20:48,970 --> 00:20:51,400 of the number of mRNA per cell? 379 00:20:51,400 --> 00:20:56,940 So we probably-- I'm trying think it-- you probably 380 00:20:56,940 --> 00:20:58,980 don't yet know this answer. 381 00:20:58,980 --> 00:21:00,720 This ends up also being Poisson. 382 00:21:00,720 --> 00:21:02,960 We're going to calculate this in a bit. 383 00:21:02,960 --> 00:21:05,260 But this is very confusing somehow. 384 00:21:05,260 --> 00:21:07,816 That both this thing and this thing, are Poisson. 385 00:21:07,816 --> 00:21:09,190 But they're not the same Poisson, 386 00:21:09,190 --> 00:21:11,410 in the sense they have different lambdas. 387 00:21:11,410 --> 00:21:14,617 Which one is going to be larger? 388 00:21:14,617 --> 00:21:15,492 This one or this one? 389 00:21:18,090 --> 00:21:19,030 The bottom one, right? 390 00:21:19,030 --> 00:21:22,260 And that's because delta m is much larger than delta p, 391 00:21:22,260 --> 00:21:23,910 typically. 392 00:21:23,910 --> 00:21:27,800 So indeed, if you ask, in Sunney's paper, for example, 393 00:21:27,800 --> 00:21:31,460 there was just over one mRNA produced per cell cycle. 394 00:21:31,460 --> 00:21:36,420 But the mean number of mRNA might have been 1/30th of that. 395 00:21:36,420 --> 00:21:39,910 Because the degradation rate was just 1 and 1/2 minutes. 396 00:21:39,910 --> 00:21:43,520 What that's saying is that in a typical situation 397 00:21:43,520 --> 00:21:47,900 you would not see an mRNA in a cell in that condition. 398 00:21:50,800 --> 00:21:52,550 We're going to calculate this in a moment. 399 00:21:52,550 --> 00:21:55,810 So don't worry if you don't see why it's a Poisson. 400 00:21:55,810 --> 00:21:56,859 But don't get confused. 401 00:21:56,859 --> 00:21:58,400 There are two different distributions 402 00:21:58,400 --> 00:22:01,869 that arise from the mRNA in the cell or in the cell cycle. 403 00:22:01,869 --> 00:22:03,160 And they're different Poissons. 404 00:22:06,190 --> 00:22:08,980 And I think that-- I mean I'm sure that in some deep sense 405 00:22:08,980 --> 00:22:10,605 there's a reason that they're the same. 406 00:22:10,605 --> 00:22:13,760 But it's somehow not immediately obvious. 407 00:22:18,456 --> 00:22:20,830 So there was another one that we might have wanted to do, 408 00:22:20,830 --> 00:22:27,070 which is the mean number of proteins in each cell. 409 00:22:42,820 --> 00:22:45,740 Now this one is a bit harder. 410 00:22:45,740 --> 00:22:49,341 And this one is going to take full advantage of the cards 411 00:22:49,341 --> 00:22:50,590 that you have in front of you. 412 00:22:50,590 --> 00:22:52,425 So be prepared. 413 00:22:52,425 --> 00:22:53,800 I'm going to give you 30 seconds. 414 00:22:53,800 --> 00:22:56,201 Because this one you might-- well 415 00:22:56,201 --> 00:22:57,784 you might need a little bit more time. 416 00:23:33,682 --> 00:23:34,640 AUDIENCE: This is hard. 417 00:23:38,719 --> 00:23:39,385 PROFESSOR: Yeah. 418 00:23:39,385 --> 00:23:42,272 Although I think that it's useful to see that 419 00:23:42,272 --> 00:23:43,230 it can be a bit tricky. 420 00:23:43,230 --> 00:23:46,120 Because this really is the simplest possible model. 421 00:23:46,120 --> 00:23:49,370 We're going to talk about some models that 422 00:23:49,370 --> 00:23:50,721 get to be horribly complicated. 423 00:23:50,721 --> 00:23:52,220 And so it's useful to just make sure 424 00:23:52,220 --> 00:23:54,401 you can nail down the intuition on this model. 425 00:24:10,610 --> 00:24:11,150 All right. 426 00:24:11,150 --> 00:24:12,066 Do you need more time? 427 00:24:12,066 --> 00:24:15,179 It's OK if this is escaping you at this moment. 428 00:24:15,179 --> 00:24:16,720 Why don't we go and see where we are? 429 00:24:16,720 --> 00:24:17,350 Ready? 430 00:24:17,350 --> 00:24:19,030 Three, two, one. 431 00:24:21,640 --> 00:24:23,520 All right. 432 00:24:23,520 --> 00:24:26,332 So you know all the naysayers on the cards, 433 00:24:26,332 --> 00:24:27,790 now that you've done this, you feel 434 00:24:27,790 --> 00:24:29,540 like it's an amazing system. 435 00:24:29,540 --> 00:24:31,200 So it's AC over DB. 436 00:24:31,200 --> 00:24:33,540 So the two, the product of the synthesis rates 437 00:24:33,540 --> 00:24:36,060 divided by the product of degradation rates. 438 00:24:36,060 --> 00:24:38,040 So what we have is the synthesis rate 439 00:24:38,040 --> 00:24:41,000 for the mRNA divided by the degradation rate for the mRNA. 440 00:24:41,000 --> 00:24:43,666 Synthesis rate for the proteins divided by the degradation rate 441 00:24:43,666 --> 00:24:45,800 for the proteins. 442 00:24:45,800 --> 00:24:48,210 Can somebody give us a verbal explanation 443 00:24:48,210 --> 00:24:52,014 for why this might have been, or why this is? 444 00:24:52,014 --> 00:24:54,002 Yes? 445 00:24:54,002 --> 00:24:55,990 AUDIENCE: It's the same reasoning 446 00:24:55,990 --> 00:24:58,972 as the number of mRNA per cell. 447 00:24:58,972 --> 00:25:04,439 But instead of just a basel-- like a synthesis rate 448 00:25:04,439 --> 00:25:06,427 doesn't depend on the concentration. 449 00:25:06,427 --> 00:25:08,940 You're just multiplying the synthesis rate 450 00:25:08,940 --> 00:25:11,001 by the number of mRNA. 451 00:25:11,001 --> 00:25:12,250 PROFESSOR: Yeah, that's great. 452 00:25:12,250 --> 00:25:13,160 OK. 453 00:25:13,160 --> 00:25:17,220 So what you're saying is that this thing here was indeed, 454 00:25:17,220 --> 00:25:20,325 we calculate that was the mean number of mRNA in the cell. 455 00:25:20,325 --> 00:25:21,700 If you just start with something, 456 00:25:21,700 --> 00:25:23,120 and you have a production and degradation rate. 457 00:25:23,120 --> 00:25:23,620 OK. 458 00:25:23,620 --> 00:25:26,510 Well that means that if you had one mRNA, then indeed that's 459 00:25:26,510 --> 00:25:30,100 what the concentration would be, is this Sp divided by delta p. 460 00:25:30,100 --> 00:25:33,440 But now we-- well we multiply that by the number of mRNA, 461 00:25:33,440 --> 00:25:34,600 and then we are set. 462 00:25:40,500 --> 00:25:43,000 All right. 463 00:25:43,000 --> 00:25:50,010 Now another question. 464 00:25:50,010 --> 00:25:53,130 We have a distribution, or a mean protein-- 465 00:25:53,130 --> 00:25:58,350 wait sorry, mean number. 466 00:25:58,350 --> 00:26:00,190 We have the mean number of protein produced 467 00:26:00,190 --> 00:26:04,730 from each mRNA, is something. 468 00:26:04,730 --> 00:26:08,720 And the question is, is this the most likely number of proteins 469 00:26:08,720 --> 00:26:10,420 to observe. 470 00:26:10,420 --> 00:26:14,195 Is the distribution here, now this is a mean, 471 00:26:14,195 --> 00:26:15,570 but now we want to start thinking 472 00:26:15,570 --> 00:26:17,960 about the probabilistic stochastic elements. 473 00:26:17,960 --> 00:26:20,590 Is this the most likely, is it like the number of proteins 474 00:26:20,590 --> 00:26:21,932 observed from an mRNA? 475 00:26:25,030 --> 00:26:29,240 The question is, is this most likely. 476 00:26:29,240 --> 00:26:31,380 By which I mean is the probability distribution 477 00:26:31,380 --> 00:26:33,480 peaked here. 478 00:26:33,480 --> 00:26:37,170 So we're going to do an A as a yes, and B is a no. 479 00:26:42,390 --> 00:26:45,520 Does everybody understand the question I'm trying to ask? 480 00:26:45,520 --> 00:26:47,612 So an mRNA is here. 481 00:26:47,612 --> 00:26:49,570 There's going to be some proteins made from it. 482 00:26:49,570 --> 00:26:51,040 This is the mean. 483 00:26:51,040 --> 00:26:54,249 What I want to know is, is that we should somehow expect? 484 00:26:54,249 --> 00:26:56,540 In a sense, is the distribution peaked, the probability 485 00:26:56,540 --> 00:27:00,630 distribution peaked around this value? 486 00:27:00,630 --> 00:27:04,410 And C is-- do I want to do a depends? 487 00:27:04,410 --> 00:27:08,321 Well you can always argue after. 488 00:27:08,321 --> 00:27:11,750 Do you need more time? 489 00:27:11,750 --> 00:27:13,622 AUDIENCE: Is this, you're saying, 490 00:27:13,622 --> 00:27:16,134 is this the most likely number of proteins? 491 00:27:16,134 --> 00:27:16,800 PROFESSOR: Yeah. 492 00:27:16,800 --> 00:27:21,778 What I'm wondering is the mode there? 493 00:27:21,778 --> 00:27:22,444 AUDIENCE: Right. 494 00:27:22,444 --> 00:27:24,560 But only for this quantity? 495 00:27:24,560 --> 00:27:26,360 PROFESSOR: Only yeah. 496 00:27:26,360 --> 00:27:28,200 So now we're not doing means anymore. 497 00:27:28,200 --> 00:27:30,158 We want to know if the probability distribution 498 00:27:30,158 --> 00:27:31,980 of the protein produced from each mRNA 499 00:27:31,980 --> 00:27:33,950 is the mode around this. 500 00:27:33,950 --> 00:27:34,960 Ready? 501 00:27:34,960 --> 00:27:37,810 Three, two, one. 502 00:27:37,810 --> 00:27:38,310 All right. 503 00:27:38,310 --> 00:27:41,144 We got a lot of no's, but some yeses. 504 00:27:41,144 --> 00:27:42,685 So this is actually going to be a no. 505 00:27:42,685 --> 00:27:49,860 And this was because the probability distribution. 506 00:27:49,860 --> 00:27:52,150 The question is, what is the probability distribution 507 00:27:52,150 --> 00:27:55,551 for the number of proteins produced from each mRNA. 508 00:27:55,551 --> 00:27:56,800 It's going to be one of these. 509 00:27:56,800 --> 00:27:57,640 Ready? 510 00:27:57,640 --> 00:27:59,400 Three, two, one. 511 00:28:04,610 --> 00:28:05,110 All right. 512 00:28:05,110 --> 00:28:06,770 So we've got some difference. 513 00:28:06,770 --> 00:28:08,634 But I'd say that most of the group 514 00:28:08,634 --> 00:28:10,050 is saying it's going to be A or B. 515 00:28:10,050 --> 00:28:13,790 And indeed these are almost the same distributions. 516 00:28:13,790 --> 00:28:15,727 What's the difference between them? 517 00:28:15,727 --> 00:28:16,810 AUDIENCE: One's discrete-- 518 00:28:16,810 --> 00:28:16,980 PROFESSOR: Right. 519 00:28:16,980 --> 00:28:17,938 So this guy's discrete. 520 00:28:17,938 --> 00:28:19,680 This guy is continuous. 521 00:28:19,680 --> 00:28:20,180 Right. 522 00:28:20,180 --> 00:28:21,760 Indeed when we're taking about the numbers, 523 00:28:21,760 --> 00:28:23,343 then we should get-- it's a geometric. 524 00:28:23,343 --> 00:28:27,592 But often we're kind of a little bit loose about these things. 525 00:28:27,592 --> 00:28:29,550 So it's not a disaster if you said exponential. 526 00:28:29,550 --> 00:28:32,180 But the key thing is that the distribution 527 00:28:32,180 --> 00:28:34,380 looks something like-- so now I've certainly 528 00:28:34,380 --> 00:28:35,930 drawn it as an exponential. 529 00:28:35,930 --> 00:28:39,250 This is the probability of n as a function of n. 530 00:28:39,250 --> 00:28:41,430 Of course, the geometric thing it looks-- 531 00:28:44,322 --> 00:28:45,286 AUDIENCE: Sorry. 532 00:28:45,286 --> 00:28:49,079 So why should we expect that distribution? 533 00:28:49,079 --> 00:28:51,120 PROFESSOR: Why should we expect the distribution? 534 00:28:51,120 --> 00:28:52,620 So one answer is that because that's 535 00:28:52,620 --> 00:28:53,940 what you read on Tuesday. 536 00:28:53,940 --> 00:29:02,740 But let's go ahead and-- yes, but let's go ahead 537 00:29:02,740 --> 00:29:04,300 and calculate it. 538 00:29:04,300 --> 00:29:05,020 That's useful. 539 00:29:07,530 --> 00:29:09,280 The way to think about this, in some ways, 540 00:29:09,280 --> 00:29:11,060 there's another way to write this perhaps. 541 00:29:11,060 --> 00:29:12,925 Which is that imagine you have an mRNA. 542 00:29:15,820 --> 00:29:20,270 Now at some rate it's going to be degraded. 543 00:29:20,270 --> 00:29:22,460 And maybe we'll keep the degradation rate down, 544 00:29:22,460 --> 00:29:27,320 just for-- so there's a degradation rate. 545 00:29:27,320 --> 00:29:30,090 But then if you'd like, we could draw it like this. 546 00:29:30,090 --> 00:29:33,885 Where this is the synthesis rate for a protein. 547 00:29:33,885 --> 00:29:36,310 And out pops a protein. 548 00:29:39,704 --> 00:29:42,120 And so the idea is that is here we're in some state where, 549 00:29:42,120 --> 00:29:44,540 OK, here we have an mRNA. 550 00:29:44,540 --> 00:29:46,495 Here's the state where we don't have an mRNA. 551 00:29:46,495 --> 00:29:48,620 Now this is the competition between those two rates 552 00:29:48,620 --> 00:29:49,780 that I was telling you about. 553 00:29:49,780 --> 00:29:51,613 There is some degradation rate for the mRNA. 554 00:29:51,613 --> 00:29:54,040 Or there's a synthesis rate where we go around this loop. 555 00:29:54,040 --> 00:29:55,550 If we come around this loop, we come back 556 00:29:55,550 --> 00:29:56,700 to the state with an mRNA. 557 00:29:56,700 --> 00:29:58,090 There still is an mRNA intact. 558 00:29:58,090 --> 00:30:00,776 Just out pops a protein. 559 00:30:00,776 --> 00:30:02,150 So then what we want to do, is we 560 00:30:02,150 --> 00:30:06,310 want to think about what's the number of proteins 561 00:30:06,310 --> 00:30:07,929 that we expect, not just the mean, 562 00:30:07,929 --> 00:30:09,095 but the actual distribution. 563 00:30:13,560 --> 00:30:16,980 So it's useful in these situations 564 00:30:16,980 --> 00:30:25,185 to define some probability rho, which is the probability 565 00:30:25,185 --> 00:30:27,360 you actually, if you're here, it's 566 00:30:27,360 --> 00:30:32,110 the probability that you produce one protein at least. 567 00:30:32,110 --> 00:30:34,930 The question is, which path do you take initially. 568 00:30:34,930 --> 00:30:36,620 Well that's just given by the ratios. 569 00:30:36,620 --> 00:30:39,980 So there's the rate that we take this circular path divided 570 00:30:39,980 --> 00:30:42,120 by the sum of these two other rates. 571 00:30:47,320 --> 00:30:49,710 And then what we can do is we can ask, well 572 00:30:49,710 --> 00:30:55,570 what is the probability that 0 proteins are produced, 573 00:30:55,570 --> 00:30:58,420 probability that we get 0. 574 00:30:58,420 --> 00:31:02,700 Well if we take this path initially, 575 00:31:02,700 --> 00:31:04,019 will we have 0 proteins? 576 00:31:04,019 --> 00:31:04,560 AUDIENCE: No. 577 00:31:04,560 --> 00:31:05,150 PROFESSOR: No. 578 00:31:05,150 --> 00:31:08,110 Right, so this probability is indeed 579 00:31:08,110 --> 00:31:12,010 simply equal to the probability that we do this first, which 580 00:31:12,010 --> 00:31:13,560 is 1 minus rho. 581 00:31:20,394 --> 00:31:22,310 Now what's the probably that we get 1 protein? 582 00:31:22,310 --> 00:31:24,640 Well, only 1? 583 00:31:24,640 --> 00:31:28,740 That's equal to the probability that we first take this path, 584 00:31:28,740 --> 00:31:30,682 and then we take this path. 585 00:31:30,682 --> 00:31:32,390 Well we can multiply those probabilities. 586 00:31:32,390 --> 00:31:35,350 Because we first take the circular path 587 00:31:35,350 --> 00:31:36,770 to make a protein. 588 00:31:36,770 --> 00:31:39,920 And then we take the degradation path. 589 00:31:39,920 --> 00:31:41,690 Well what's the probability we get 2? 590 00:31:41,690 --> 00:31:44,110 Well that's just that we come around here once, 591 00:31:44,110 --> 00:31:45,450 twice, and then degrade. 592 00:31:50,540 --> 00:31:52,830 Now if you're not seeing a pattern here, 593 00:31:52,830 --> 00:31:54,840 then we're in trouble. 594 00:31:57,400 --> 00:31:58,910 So this says the probability of n 595 00:31:58,910 --> 00:32:02,730 will then just be equal to rho to the n, 1 minus rho. 596 00:32:08,850 --> 00:32:11,340 And indeed, it's always useful in order 597 00:32:11,340 --> 00:32:13,370 to warm up your probability muscles, 598 00:32:13,370 --> 00:32:17,880 to check to make sure that this is a normalized probability 599 00:32:17,880 --> 00:32:18,830 distribution. 600 00:32:18,830 --> 00:32:25,820 So sum over all possible ends indeed goes to 1. 601 00:32:25,820 --> 00:32:29,190 And that's just because the sum over a bunch of rho to the n's 602 00:32:29,190 --> 00:32:32,440 is equal to 1 divided by 1 minus rho, which is the term there. 603 00:32:32,440 --> 00:32:34,591 And that goes to 1. 604 00:32:34,591 --> 00:32:37,682 AUDIENCE: So this is making a pretty strong assumption 605 00:32:37,682 --> 00:32:38,890 that they're all independent? 606 00:32:38,890 --> 00:32:41,180 PROFESSOR: Yep, yep. 607 00:32:41,180 --> 00:32:42,540 Yep. 608 00:32:42,540 --> 00:32:47,970 This is assuming that if you've gone around once, you return. 609 00:32:47,970 --> 00:32:50,100 But I've come back to the original state. 610 00:32:50,100 --> 00:32:51,962 AUDIENCE: But do mRNAs like actually 611 00:32:51,962 --> 00:32:55,060 get caught in ribosomes-- 612 00:32:55,060 --> 00:32:58,720 PROFESSOR: There are a lot of things that can be true. 613 00:32:58,720 --> 00:33:03,972 And I would say that in biology and in life, 614 00:33:03,972 --> 00:33:05,430 what you do is you first write down 615 00:33:05,430 --> 00:33:06,790 the simplest possible model. 616 00:33:06,790 --> 00:33:08,540 And then you go and you make measurements. 617 00:33:08,540 --> 00:33:12,300 And you ask whether the simplest possible model can adequately 618 00:33:12,300 --> 00:33:13,380 explain the data. 619 00:33:13,380 --> 00:33:15,086 And if the answer is no, then you're 620 00:33:15,086 --> 00:33:16,960 allowed to start thinking about other things. 621 00:33:16,960 --> 00:33:19,770 Because everything's is in principle true. 622 00:33:19,770 --> 00:33:22,400 In that mRNA, maybe it's this or that. 623 00:33:22,400 --> 00:33:24,130 The question is whether it's significant. 624 00:33:24,130 --> 00:33:26,046 And at least from the data from Sunney's group 625 00:33:26,046 --> 00:33:29,807 would say that in that condition, in those cells, 626 00:33:29,807 --> 00:33:31,390 that those things are not significant, 627 00:33:31,390 --> 00:33:33,950 in the sense that you still get a geometric distribution. 628 00:33:33,950 --> 00:33:37,429 Of course it could also be that those other things actually are 629 00:33:37,429 --> 00:33:38,470 true and are significant. 630 00:33:38,470 --> 00:33:41,120 But then you end up with some new parameters 631 00:33:41,120 --> 00:33:45,260 that describe how things look as a result of all the complexity. 632 00:33:45,260 --> 00:33:48,762 That's also OK in the sense that I'd 633 00:33:48,762 --> 00:33:50,720 say that you can get a quantitative description 634 00:33:50,720 --> 00:33:53,300 of the process by describing it as a geometric with just 635 00:33:53,300 --> 00:33:54,580 a single free parameter. 636 00:33:54,580 --> 00:33:58,750 And they found that the mean was four, or four or five. 637 00:33:58,750 --> 00:34:01,960 The mean number of proteins produced from each mRNA. 638 00:34:01,960 --> 00:34:06,040 But they got this geometric distribution in that paper. 639 00:34:10,809 --> 00:34:11,309 Yes? 640 00:34:15,159 --> 00:34:19,940 And I'll just mention here that the mean of this 641 00:34:19,940 --> 00:34:21,730 is rho divided by 1 minus rho. 642 00:34:24,280 --> 00:34:26,860 So what you see is that as rho goes to 1, 643 00:34:26,860 --> 00:34:29,060 then this thing is going to diverge. 644 00:34:29,060 --> 00:34:30,050 And that makes sense. 645 00:34:30,050 --> 00:34:33,540 because as rho goes to 1, it's saying that you essentially 646 00:34:33,540 --> 00:34:37,750 always synthesize another protein rather than degrading. 647 00:34:37,750 --> 00:34:39,520 And before I move on, I just want 648 00:34:39,520 --> 00:34:41,270 to say one more thing, which is that there 649 00:34:41,270 --> 00:34:44,507 are many different definitions of the geometric distribution, 650 00:34:44,507 --> 00:34:46,382 depending upon whether the probability of rho 651 00:34:46,382 --> 00:34:47,390 is the probability of terminating, 652 00:34:47,390 --> 00:34:49,640 or the probability of going around, and also depending 653 00:34:49,640 --> 00:34:52,300 on whether you're asking what is the-- here 654 00:34:52,300 --> 00:34:53,860 we're talking about the probability 655 00:34:53,860 --> 00:34:56,320 distribution for the number of proteins produced. 656 00:34:56,320 --> 00:34:57,690 Whereas we could have talked about the probability 657 00:34:57,690 --> 00:35:00,148 distribution for the number of times we go around this loop 658 00:35:00,148 --> 00:35:03,150 before, no, no sorry. 659 00:35:03,150 --> 00:35:05,200 That is for the number of proteins produced. 660 00:35:05,200 --> 00:35:09,000 So the other way you could have defined this is the number 661 00:35:09,000 --> 00:35:14,960 of times where it's-- the number of cycles that you had to go 662 00:35:14,960 --> 00:35:17,360 before you went here, in the sense that if you first go 663 00:35:17,360 --> 00:35:19,756 here, you can either call that a 0 or a 1. 664 00:35:19,756 --> 00:35:20,880 Do you see what I'm saying? 665 00:35:20,880 --> 00:35:22,890 And reasonable people can disagree. 666 00:35:22,890 --> 00:35:25,570 But you end up getting distributions that are 667 00:35:25,570 --> 00:35:27,200 just a little bit different. 668 00:35:27,200 --> 00:35:28,250 So watch out. 669 00:35:28,250 --> 00:35:30,060 If you just memorize something, you 670 00:35:30,060 --> 00:35:32,050 might have memorized the equation 671 00:35:32,050 --> 00:35:35,240 for a different definition of this distribution. 672 00:35:35,240 --> 00:35:38,780 Does everyone understand what I tried to say there? 673 00:35:38,780 --> 00:35:39,380 Maybe? 674 00:35:39,380 --> 00:35:40,285 Yeah? 675 00:35:40,285 --> 00:35:41,826 AUDIENCE: When there's no degradation 676 00:35:41,826 --> 00:35:43,492 is it still a Poisson? 677 00:35:43,492 --> 00:35:46,550 PROFESSOR: Ah, if there's no degradation then would this 678 00:35:46,550 --> 00:35:48,040 be a Poisson? 679 00:35:48,040 --> 00:35:52,329 I mean, this would be infinity, right? 680 00:35:52,329 --> 00:35:53,870 AUDIENCE: Right, [INAUDIBLE] protein, 681 00:35:53,870 --> 00:35:57,216 this is done independently, like there's an mRNA. 682 00:35:57,216 --> 00:35:59,620 [INAUDIBLE] proteins independently. 683 00:35:59,620 --> 00:36:01,654 PROFESSOR: OK. 684 00:36:01,654 --> 00:36:02,570 So I want to be clear. 685 00:36:02,570 --> 00:36:08,000 This is, p of n is, this is the probability 686 00:36:08,000 --> 00:36:18,150 distribution for number of proteins n, 687 00:36:18,150 --> 00:36:19,480 produced from a single mRNA. 688 00:36:26,020 --> 00:36:30,059 Now if there's no degradation of the mRNA, 689 00:36:30,059 --> 00:36:31,600 then this thing is not even, I think, 690 00:36:31,600 --> 00:36:35,940 defined in that the number of proteins 691 00:36:35,940 --> 00:36:39,330 produced from that mRNA just really goes to infinity. 692 00:36:39,330 --> 00:36:43,090 If you wanted to ask about the probability distribution 693 00:36:43,090 --> 00:36:45,170 for the number of proteins produced 694 00:36:45,170 --> 00:36:49,470 in some unit, some period of time that would indeed 695 00:36:49,470 --> 00:36:53,410 be a Poisson distribution, assuming 696 00:36:53,410 --> 00:36:56,210 that there's no degradation. 697 00:36:56,210 --> 00:36:58,972 Do you understand what I'm trying to say? 698 00:36:58,972 --> 00:37:02,780 AUDIENCE: So Sp is like 0? 699 00:37:02,780 --> 00:37:06,330 PROFESSOR: If Sp, I'm sorry. 700 00:37:06,330 --> 00:37:07,872 If Sp were 0? 701 00:37:07,872 --> 00:37:09,038 AUDIENCE: Yeah. [INAUDIBLE]. 702 00:37:21,367 --> 00:37:21,950 PROFESSOR: OK. 703 00:37:21,950 --> 00:37:24,325 And you're saying that what would be Poisson distributed? 704 00:37:24,325 --> 00:37:26,225 AUDIENCE: The number of proteins. 705 00:37:36,090 --> 00:37:38,430 PROFESSOR: Yeah, I think that actually-- no I 706 00:37:38,430 --> 00:37:41,940 think that-- I think that you're probably right. 707 00:37:41,940 --> 00:37:45,176 That as Sp goes to 0-- I'm a little bit worried that-- 708 00:37:45,176 --> 00:37:46,050 AUDIENCE: No, no, no. 709 00:37:46,050 --> 00:37:46,670 Zero-th order, sorry. 710 00:37:46,670 --> 00:37:47,436 PROFESSOR: Oh. 711 00:37:47,436 --> 00:37:49,284 AUDIENCE: Not 0. 712 00:37:49,284 --> 00:37:52,220 [INAUDIBLE] 713 00:37:52,220 --> 00:37:53,135 PROFESSOR: OK. 714 00:37:53,135 --> 00:37:55,260 Right, so the mRNA distribution we're about to find 715 00:37:55,260 --> 00:37:58,250 is indeed going to be a Poisson at steady state. 716 00:37:58,250 --> 00:38:01,090 And so if there's some process by which the protein 717 00:38:01,090 --> 00:38:02,980 distribution is really just mirroring 718 00:38:02,980 --> 00:38:07,810 the mRNA distribution, then it will also be Poisson. 719 00:38:07,810 --> 00:38:10,680 Although I think you have to be careful about how you actually 720 00:38:10,680 --> 00:38:12,050 implement that. 721 00:38:12,050 --> 00:38:15,290 Because even in the absence of this geometric bursting, 722 00:38:15,290 --> 00:38:17,390 different things, I think, can happen. 723 00:38:17,390 --> 00:38:20,420 Because for example, if there were exactly 10 proteins 724 00:38:20,420 --> 00:38:23,800 produced from each mRNA, then that probability distribution 725 00:38:23,800 --> 00:38:26,240 is a shift. 726 00:38:26,240 --> 00:38:29,570 But then it's no longer actually going 727 00:38:29,570 --> 00:38:33,150 to be Poisson, because the mean and variance are going to scale 728 00:38:33,150 --> 00:38:35,450 differently if you do that. 729 00:38:35,450 --> 00:38:38,450 Let's maybe do the Poisson distribution for the mRNA 730 00:38:38,450 --> 00:38:38,950 first. 731 00:38:38,950 --> 00:38:42,420 And then we can try to touch back on this. 732 00:38:46,040 --> 00:38:49,790 So this is a plot of kind of geometric distribution 733 00:38:49,790 --> 00:38:51,155 with a mean of 3-4-ish. 734 00:38:56,860 --> 00:39:00,312 Is everybody happy with where we are now? 735 00:39:00,312 --> 00:39:02,060 OK. 736 00:39:02,060 --> 00:39:04,290 Now from this, what we've said so far 737 00:39:04,290 --> 00:39:06,770 is it obvious what the distribution of proteins 738 00:39:06,770 --> 00:39:08,580 will be in a cell? 739 00:39:08,580 --> 00:39:10,080 We can say obvious, yes. 740 00:39:10,080 --> 00:39:12,920 Or not obvious, no. 741 00:39:12,920 --> 00:39:15,330 Ready, just verbal, yes? 742 00:39:15,330 --> 00:39:16,020 Ready or no? 743 00:39:16,020 --> 00:39:16,520 All right. 744 00:39:16,520 --> 00:39:17,020 Ready? 745 00:39:17,020 --> 00:39:17,690 Three, two, one. 746 00:39:17,690 --> 00:39:18,317 AUDIENCE: No. 747 00:39:18,317 --> 00:39:18,900 PROFESSOR: No. 748 00:39:18,900 --> 00:39:19,570 Right. 749 00:39:19,570 --> 00:39:22,560 So we've said that the distribution 750 00:39:22,560 --> 00:39:25,116 of size of protein bursts from single mRNA is geometric. 751 00:39:25,116 --> 00:39:26,490 But that doesn't mean that that's 752 00:39:26,490 --> 00:39:29,919 going to be the distribution of proteins in the cell. 753 00:39:29,919 --> 00:39:31,460 And indeed after, we're going to find 754 00:39:31,460 --> 00:39:33,790 that the distribution of mRNA is going to be Poisson. 755 00:39:33,790 --> 00:39:35,830 But even then it's not obvious what the distribution 756 00:39:35,830 --> 00:39:36,455 of proteins is. 757 00:39:47,410 --> 00:39:47,910 All right. 758 00:39:47,910 --> 00:39:49,410 So what we want to do now is we want 759 00:39:49,410 --> 00:39:51,390 to introduce kind of a simple version of what's 760 00:39:51,390 --> 00:39:52,650 known as the Master Equation. 761 00:39:55,390 --> 00:39:58,010 Now you guys are going to do more reading on this 762 00:39:58,010 --> 00:40:00,330 for the lecture on Tuesday. 763 00:40:00,330 --> 00:40:03,080 Where we're going to talk about the Master Equation, as well 764 00:40:03,080 --> 00:40:06,310 as the Fokker-Planck approximation. 765 00:40:06,310 --> 00:40:08,620 Maybe the Gillespie algorithm, and so forth. 766 00:40:08,620 --> 00:40:13,380 But I want to start by thinking about the this notion 767 00:40:13,380 --> 00:40:15,084 in the simplest possible context. 768 00:40:15,084 --> 00:40:16,500 So what we're going to do is we're 769 00:40:16,500 --> 00:40:18,580 going to think about the world. 770 00:40:18,580 --> 00:40:21,550 So we want to know the steady state, or the equilibrium 771 00:40:21,550 --> 00:40:26,170 distribution of mRNA numbers in the cell, given this process. 772 00:40:26,170 --> 00:40:27,400 So that's great. 773 00:40:27,400 --> 00:40:31,630 We can-- so mRNA distribution, question mark. 774 00:40:40,520 --> 00:40:43,520 Now in this case we don't care about Sp, delta p, 775 00:40:43,520 --> 00:40:47,032 because the only things that are relevant are going to be these. 776 00:40:47,032 --> 00:40:48,490 Now what we're going to do is we're 777 00:40:48,490 --> 00:40:50,830 going to think about the world in which we just 778 00:40:50,830 --> 00:40:54,010 defined states corresponding to the different numbers 779 00:40:54,010 --> 00:40:54,950 of these mRNAs. 780 00:40:57,860 --> 00:41:01,170 So there's a state where there's 0. 781 00:41:01,170 --> 00:41:03,120 We can't go to the left, less than 0, 782 00:41:03,120 --> 00:41:09,490 but we can go to the state where there's 1, or the state where 783 00:41:09,490 --> 00:41:10,620 there's 2, and so forth. 784 00:41:16,340 --> 00:41:18,590 Now the description here is supposed 785 00:41:18,590 --> 00:41:21,680 to be the analog of this over there. 786 00:41:21,680 --> 00:41:25,094 So this is trying to understand the situation where 787 00:41:25,094 --> 00:41:27,010 the deterministic equations would be described 788 00:41:27,010 --> 00:41:32,050 by m dot is equal to this some synthesis rate, 789 00:41:32,050 --> 00:41:35,600 minus a degradation rate that's proportional to the number, 790 00:41:35,600 --> 00:41:39,186 so minus delta m times m. 791 00:41:42,590 --> 00:41:46,880 So what you can see is that the deterministic equations 792 00:41:46,880 --> 00:41:49,010 are very simple. 793 00:41:49,010 --> 00:41:51,296 We already calculated the equilibrium. 794 00:41:51,296 --> 00:41:52,670 So when this thing is equal to 0, 795 00:41:52,670 --> 00:41:55,185 then we get that m equilibrium is just 796 00:41:55,185 --> 00:41:57,810 going to equal to the synthesis rate divided by the degradation 797 00:41:57,810 --> 00:41:58,309 rate. 798 00:42:02,210 --> 00:42:03,830 If we're away from the equilibrium 799 00:42:03,830 --> 00:42:05,836 in this deterministic approximation, 800 00:42:05,836 --> 00:42:07,460 how long is it going to take us to kind 801 00:42:07,460 --> 00:42:10,640 of approach our equilibrium? 802 00:42:10,640 --> 00:42:11,610 Verbal answer, ready? 803 00:42:11,610 --> 00:42:13,100 Three, two, one. 804 00:42:13,100 --> 00:42:14,192 [STUDENTS RESPOND] 805 00:42:14,192 --> 00:42:14,900 PROFESSOR: Right. 806 00:42:14,900 --> 00:42:17,130 So it's going to be 1 over delta m. 807 00:42:17,130 --> 00:42:20,320 So this tells us the characteristic timescale 808 00:42:20,320 --> 00:42:21,160 to come back. 809 00:42:21,160 --> 00:42:28,015 So if we're-- this is the equilibrium Sm over delta m. 810 00:42:28,015 --> 00:42:31,570 This is m as a function of time. 811 00:42:31,570 --> 00:42:32,980 If we're below, we come here. 812 00:42:32,980 --> 00:42:35,070 If we're above, we come here. 813 00:42:35,070 --> 00:42:41,686 And this time is 1 over delta m. 814 00:42:41,686 --> 00:42:43,202 Are the any questions about? 815 00:42:50,920 --> 00:42:53,130 Now this-- so I want to highlight 816 00:42:53,130 --> 00:42:58,640 that this is like the world's simplest dynamical equation, 817 00:42:58,640 --> 00:43:01,654 almost the world's simplest. 818 00:43:01,654 --> 00:43:03,820 Yet what we're going to find is that once we go over 819 00:43:03,820 --> 00:43:05,695 and we try to understand the full probability 820 00:43:05,695 --> 00:43:08,320 distribution of the stochastic system 821 00:43:08,320 --> 00:43:10,470 then it's a little bit more complicated. 822 00:43:10,470 --> 00:43:13,140 In particular, we end up with an infinite set 823 00:43:13,140 --> 00:43:15,710 of differential equations. 824 00:43:15,710 --> 00:43:18,640 So in general the Master Equation format, 825 00:43:18,640 --> 00:43:21,380 where we're going to write differential equations for how 826 00:43:21,380 --> 00:43:25,530 these probabilities change over time. 827 00:43:25,530 --> 00:43:28,350 Now what we've done is we've traded a single differential 828 00:43:28,350 --> 00:43:31,170 equation for an infinite number of differential equations. 829 00:43:31,170 --> 00:43:33,650 So that's a bummer. 830 00:43:33,650 --> 00:43:36,210 But on the other hand, it will allow 831 00:43:36,210 --> 00:43:39,830 us to do the full stochastic treatment. 832 00:43:39,830 --> 00:43:42,341 And it's also a nice, to me, the master equation 833 00:43:42,341 --> 00:43:43,590 is useful in kind of two ways. 834 00:43:43,590 --> 00:43:46,610 One is that it's going to be a tool for us to do 835 00:43:46,610 --> 00:43:48,110 analytic calculations. 836 00:43:48,110 --> 00:43:51,010 But it's also kind of a principled way 837 00:43:51,010 --> 00:43:53,300 of organizing your thoughts so that you can go and do 838 00:43:53,300 --> 00:43:55,860 stochastic simulations, if that's what you want to do. 839 00:43:55,860 --> 00:43:58,240 So it's also just kind of like a weigh station 840 00:43:58,240 --> 00:44:01,150 to kind of help you set up your simulation. 841 00:44:15,754 --> 00:44:17,170 So what we're going to do is we're 842 00:44:17,170 --> 00:44:20,260 going to ask about the general way 843 00:44:20,260 --> 00:44:23,240 that this thing is going to move between different states. 844 00:44:23,240 --> 00:44:26,390 In particular, we are going to have 845 00:44:26,390 --> 00:44:29,950 some general state in here. 846 00:44:29,950 --> 00:44:35,365 Mn, which can go forward or back, Mn plus 1. 847 00:44:47,060 --> 00:44:50,700 Now what we want to do is think about how those probabilities 848 00:44:50,700 --> 00:44:53,970 are going to change over time. 849 00:44:53,970 --> 00:44:57,640 So we typically have fn. 850 00:44:57,640 --> 00:45:04,040 So this is often written as an fn and fn minus 1. 851 00:45:04,040 --> 00:45:06,710 And then this is a g. 852 00:45:06,710 --> 00:45:09,040 I want to make sure I get the n's and n minus ones 853 00:45:09,040 --> 00:45:09,836 correct here. 854 00:45:13,030 --> 00:45:17,890 Typically we write gm plus 1, gn. 855 00:45:17,890 --> 00:45:21,205 So these are telling us about the rates 856 00:45:21,205 --> 00:45:25,350 of being in this state, with say, n mRNAs, as 857 00:45:25,350 --> 00:45:27,190 compared to going here. 858 00:45:27,190 --> 00:45:28,860 We're going here. 859 00:45:28,860 --> 00:45:31,260 So then what we can do is we can write 860 00:45:31,260 --> 00:45:41,560 the change in the probability of mn with respect to time. 861 00:45:41,560 --> 00:45:43,750 Well there are just a few different ways 862 00:45:43,750 --> 00:45:45,890 that the probability can change. 863 00:45:45,890 --> 00:45:49,280 So we can leave the state in two different ways. 864 00:45:49,280 --> 00:45:51,955 fn, gn. 865 00:45:55,210 --> 00:45:58,460 So the way that we lose the probability 866 00:45:58,460 --> 00:46:03,840 is that we have fn plus gn times the probability 867 00:46:03,840 --> 00:46:06,426 that we are in mn. 868 00:46:06,426 --> 00:46:08,285 That's an n there. 869 00:46:08,285 --> 00:46:09,910 And then there are going to be two ways 870 00:46:09,910 --> 00:46:11,660 that we gain probability. 871 00:46:11,660 --> 00:46:14,260 We can gain probably from the mn minus 1. 872 00:46:14,260 --> 00:46:21,450 So this is fn minus 1, plus we can get probability 873 00:46:21,450 --> 00:46:22,960 from the upper state. 874 00:46:22,960 --> 00:46:30,440 That's a gn plus 1 mn plus 1. 875 00:46:30,440 --> 00:46:33,590 So this is just saying that the change in the probability 876 00:46:33,590 --> 00:46:36,470 of being in this state is going to be given by the probability 877 00:46:36,470 --> 00:46:38,061 that we leave the state. 878 00:46:38,061 --> 00:46:38,560 Sorry. 879 00:46:38,560 --> 00:46:40,710 The probability that we enter the state, 880 00:46:40,710 --> 00:46:42,376 minus the probability that we're leaving 881 00:46:42,376 --> 00:46:46,200 the state, kind of the rates. 882 00:46:46,200 --> 00:46:48,570 Now this is going to be true for all n, 883 00:46:48,570 --> 00:46:51,220 except for n equal to zero, we don't have 884 00:46:51,220 --> 00:46:54,330 the terms over on the left. 885 00:46:54,330 --> 00:46:58,170 So this is kind of for all n. 886 00:46:58,170 --> 00:47:00,630 So this is, in particular this is for n, 887 00:47:00,630 --> 00:47:03,250 basically 0 on up to infinity. 888 00:47:03,250 --> 00:47:06,480 So this is a differential equation for the probability 889 00:47:06,480 --> 00:47:07,700 for having an mRNA. 890 00:47:07,700 --> 00:47:11,790 But this is, we have to have a different equation for each n, 891 00:47:11,790 --> 00:47:13,830 0, 1, 2, 3, 4, 5 on up. 892 00:47:16,510 --> 00:47:20,890 So this is what I mean by converting single differential 893 00:47:20,890 --> 00:47:24,990 equation, which is actually an exceedingly simple one, for one 894 00:47:24,990 --> 00:47:26,335 that is for an infinite set. 895 00:47:26,335 --> 00:47:28,460 And each one is even a little bit more complicated. 896 00:47:28,460 --> 00:47:33,400 In general, these f's and n's can be pretty complicated. 897 00:47:33,400 --> 00:47:37,450 In this situation they're not so bad. 898 00:47:37,450 --> 00:47:40,520 But let's make sure. 899 00:47:40,520 --> 00:47:44,470 Can somebody say what fn and gn are equal to? 900 00:47:51,830 --> 00:47:54,144 Any volunteers? 901 00:47:54,144 --> 00:47:56,000 AUDIENCE: [INAUDIBLE] 902 00:47:56,000 --> 00:48:00,670 PROFESSOR: Right so fn, this is rate that we add a new mRNA. 903 00:48:00,670 --> 00:48:04,450 Well that's just synthesis rate for mRNA. 904 00:48:04,450 --> 00:48:06,774 And this guy is what? 905 00:48:06,774 --> 00:48:08,686 AUDIENCE: Delta. 906 00:48:08,686 --> 00:48:10,310 PROFESSOR: So this is degradation rate. 907 00:48:10,310 --> 00:48:14,598 And we actually do have to multiply still by the number n. 908 00:48:18,720 --> 00:48:22,100 And that's because as we go further out here to the right, 909 00:48:22,100 --> 00:48:23,540 then it is true. 910 00:48:23,540 --> 00:48:26,140 The rate at which we come back to the left is increasing. 911 00:48:26,140 --> 00:48:28,670 Because there's just more mRNA that can be degraded. 912 00:48:32,010 --> 00:48:37,540 Now it's worth saying that you can, for example, use 913 00:48:37,540 --> 00:48:39,880 this to simulate the probability distribution 914 00:48:39,880 --> 00:48:41,850 if you start from any distribution you like. 915 00:48:41,850 --> 00:48:44,370 So for example, you could start M0 equal to 1. 916 00:48:44,370 --> 00:48:48,210 And then just simulate how the probability recalibrates 917 00:48:48,210 --> 00:48:50,740 and comes over here. 918 00:48:50,740 --> 00:48:52,340 Similarly, you could do it over here. 919 00:48:52,340 --> 00:48:54,030 You could start with any probability distribution 920 00:48:54,030 --> 00:48:54,590 you want. 921 00:48:54,590 --> 00:48:56,375 And you could use this as a framework 922 00:48:56,375 --> 00:48:58,500 to calculate what the probability distribution will 923 00:48:58,500 --> 00:48:59,620 be at any time later. 924 00:49:03,600 --> 00:49:05,460 But you can also use this just as a way 925 00:49:05,460 --> 00:49:09,950 of figuring out what the equilibrium distribution is 926 00:49:09,950 --> 00:49:10,450 going to be. 927 00:49:10,450 --> 00:49:14,040 Because at equilibrium, we can just ask, 928 00:49:14,040 --> 00:49:15,915 for each one of these arrows, the probability 929 00:49:15,915 --> 00:49:18,289 of moving to the right has to be equal to the probability 930 00:49:18,289 --> 00:49:21,070 of moving to the left, otherwise we wouldn't be at equilibrium. 931 00:49:21,070 --> 00:49:24,202 And that's true for every one of these kinds of pairs of arrows. 932 00:49:27,400 --> 00:49:32,090 And in particular, what we can get, 933 00:49:32,090 --> 00:49:34,555 and I want to make sure that-- so but it's 934 00:49:34,555 --> 00:49:45,360 not that fn is equal to gn-- so it's really going to end up 935 00:49:45,360 --> 00:49:47,600 being that if you see what fn and gn, 936 00:49:47,600 --> 00:49:56,680 so that fn is going to have to be equal to g of n plus 1 937 00:49:56,680 --> 00:49:57,350 for all n. 938 00:49:57,350 --> 00:49:58,320 Yes? 939 00:49:58,320 --> 00:50:00,260 AUDIENCE: Do you also need to multiply 940 00:50:00,260 --> 00:50:01,715 like natural probabilities-- 941 00:50:01,715 --> 00:50:05,860 PROFESSOR: Ah, Yes, yes, indeed. 942 00:50:05,860 --> 00:50:11,100 So that sorry, times mn times m of n plus 1. 943 00:50:14,870 --> 00:50:19,700 So it's the kind of probably flux so we have to equalize. 944 00:50:19,700 --> 00:50:23,780 So this is nice because this gives us a ratio of things. 945 00:50:23,780 --> 00:50:26,440 In particular, this tells us that the probability 946 00:50:26,440 --> 00:50:29,810 of being in the n plus 1 divided by the probability of being n, 947 00:50:29,810 --> 00:50:31,350 and this is at equilibrium. 948 00:50:36,330 --> 00:50:44,130 Is going to be fn divided by gn plus 1. 949 00:50:44,130 --> 00:50:46,830 Which is this synthesis rate. 950 00:50:46,830 --> 00:50:49,710 And then down here is going to be this degradation rate 951 00:50:49,710 --> 00:50:53,300 times, in this case, n plus 1. 952 00:51:00,300 --> 00:51:01,430 So this is useful. 953 00:51:01,430 --> 00:51:04,790 Because for example, if we start at m, 954 00:51:04,790 --> 00:51:11,240 we could say that m1 over m0-- well maybe 955 00:51:11,240 --> 00:51:14,330 we'll even put the m0 over on the right. 956 00:51:17,360 --> 00:51:19,330 So then m1, what is that equal to? 957 00:51:19,330 --> 00:51:21,650 That's going to be synthesis rate divided 958 00:51:21,650 --> 00:51:28,010 degradation rate, times m0. 959 00:51:28,010 --> 00:51:33,420 But then we also know that m2, well that's going to be again, 960 00:51:33,420 --> 00:51:35,596 synthesis rate divided by degradation rate. 961 00:51:35,596 --> 00:51:36,970 And we're going to get a squared. 962 00:51:36,970 --> 00:51:42,550 But then now we have to divide by 1/2 times m0. 963 00:51:42,550 --> 00:51:51,600 Continuing on, m3 we get Sm over delta m cubed, 964 00:51:51,600 --> 00:51:55,750 divided by 1 over 3 times 2 times m0. 965 00:51:58,830 --> 00:52:01,200 So in general, we get the probability 966 00:52:01,200 --> 00:52:05,870 of being in the nth state, is going to be this thing. 967 00:52:05,870 --> 00:52:10,480 We'll call it lambda for now. 968 00:52:10,480 --> 00:52:14,350 Lambda to the n, divided by n factorial, times m0. 969 00:52:19,980 --> 00:52:24,280 Now what's the-- and I'll-- remember lambda here 970 00:52:24,280 --> 00:52:28,426 we've defined it to be the ratio Sm over delta m. 971 00:52:31,120 --> 00:52:33,840 Now if we sum over all these probabilities, 972 00:52:33,840 --> 00:52:35,800 what should we get? 973 00:52:35,800 --> 00:52:36,300 AUDIENCE: 1. 974 00:52:36,300 --> 00:52:37,460 PROFESSOR: 1. 975 00:52:37,460 --> 00:52:42,620 Right, if we sum over this thing, what does that equal to? 976 00:52:45,840 --> 00:52:48,200 It's what? 977 00:52:48,200 --> 00:52:49,640 AUDIENCE: Eta lambda. 978 00:52:49,640 --> 00:52:51,100 PROFESSOR: Eta lambda, right? 979 00:52:51,100 --> 00:52:57,750 So just remember in this world-- the sum over lambda 980 00:52:57,750 --> 00:53:03,490 to the n, n factorial, from n equal to 0 to infinity, 981 00:53:03,490 --> 00:53:05,570 this is indeed the definition of e to the lambda. 982 00:53:08,420 --> 00:53:11,055 So what that means is that the normalization condition is 983 00:53:11,055 --> 00:53:19,951 that m0 has to be equal to e to the minus lambda, which is 984 00:53:19,951 --> 00:53:21,200 indeed a Poisson distribution. 985 00:53:30,548 --> 00:53:32,332 I'll raise it up a little bit. 986 00:53:36,680 --> 00:53:39,160 So this is saying, OK, to back up. 987 00:53:39,160 --> 00:53:42,390 If we just have constant rate of creation of something, 988 00:53:42,390 --> 00:53:44,910 constant rate of degradation of that thing, 989 00:53:44,910 --> 00:53:47,720 on a per item basis, per unit basis, 990 00:53:47,720 --> 00:53:51,050 then you end up getting a Poisson distribution, 991 00:53:51,050 --> 00:53:54,760 at equilibrium for the number of that thing, 992 00:53:54,760 --> 00:53:57,310 in this case, the number of mRNA in the cell. 993 00:54:04,310 --> 00:54:05,940 Questions about why that is? 994 00:54:05,940 --> 00:54:07,700 What happened? 995 00:54:07,700 --> 00:54:08,620 How we calculate it? 996 00:54:11,931 --> 00:54:13,823 AUDIENCE: Could you explain why [INAUDIBLE]? 997 00:54:16,954 --> 00:54:17,620 PROFESSOR: Sure. 998 00:54:24,440 --> 00:54:28,810 So this is basically f of n. 999 00:54:28,810 --> 00:54:30,850 And this is basically this g of n. 1000 00:54:30,850 --> 00:54:35,530 But remember here n is the number of proteins 1001 00:54:35,530 --> 00:54:37,550 or the number of mRNA. 1002 00:54:37,550 --> 00:54:42,480 So then that's in the context of the master equation, 1003 00:54:42,480 --> 00:54:43,900 then m and n are there. 1004 00:54:43,900 --> 00:54:46,416 You get n by the current number of m. 1005 00:54:46,416 --> 00:54:47,290 Does that make sense? 1006 00:54:51,250 --> 00:54:53,270 Yes? 1007 00:54:53,270 --> 00:54:57,546 AUDIENCE: I'm confused how you changed m0 to the e 1008 00:54:57,546 --> 00:54:58,920 to the minus lambda. 1009 00:54:58,920 --> 00:55:00,940 PROFESSOR: OK. 1010 00:55:00,940 --> 00:55:03,130 Well let's just do it. 1011 00:55:03,130 --> 00:55:09,890 So mn, this is the probability that we observe n mRNA. 1012 00:55:09,890 --> 00:55:13,610 And we know that the sum over mn, 1013 00:55:13,610 --> 00:55:18,335 so all these probabilities from n equal to 0 to infinity, 1014 00:55:18,335 --> 00:55:20,410 has to be equal to 1. 1015 00:55:20,410 --> 00:55:22,970 Something has to happen. 1016 00:55:22,970 --> 00:55:24,080 Well let's do this sum. 1017 00:55:24,080 --> 00:55:27,380 This is equal to the sum of lambda to the n, 1018 00:55:27,380 --> 00:55:29,470 over n factorial m0. 1019 00:55:32,890 --> 00:55:36,490 But m0, is this a function of n? 1020 00:55:39,100 --> 00:55:39,690 No. 1021 00:55:39,690 --> 00:55:42,480 m0 is just, this is just the probability at equilibrium 1022 00:55:42,480 --> 00:55:44,410 that you have 0 mRNA. 1023 00:55:44,410 --> 00:55:46,479 So we can just pull this thing out. 1024 00:55:46,479 --> 00:55:48,270 This is just some number, some probability. 1025 00:55:54,560 --> 00:55:56,720 Now the statement is that while this thing, 1026 00:55:56,720 --> 00:55:59,610 this is the definition of e to the lambda. 1027 00:56:03,060 --> 00:56:08,160 So in general, so e to the x we often write is equal to 1, 1028 00:56:08,160 --> 00:56:12,633 plus x, plus x squared over 2, plus dot, dot, dot. 1029 00:56:16,190 --> 00:56:20,820 So this thing is indeed just equal e to the lambda. 1030 00:56:20,820 --> 00:56:24,370 So what we know is that this is still 1. 1031 00:56:24,370 --> 00:56:27,150 So m0 times e to the lambda, is equal to 1, 1032 00:56:27,150 --> 00:56:29,614 so m0 is to the minus lambda. 1033 00:56:35,590 --> 00:56:38,750 Any other questions about how we got here? 1034 00:56:38,750 --> 00:56:40,182 What's going on? 1035 00:56:40,182 --> 00:56:41,598 Yes? 1036 00:56:41,598 --> 00:56:43,570 AUDIENCE: The plot of the solution 1037 00:56:43,570 --> 00:56:48,060 to the adjoining equation, that would 1038 00:56:48,060 --> 00:56:50,370 be like the mean value, that would 1039 00:56:50,370 --> 00:56:52,515 be the behavior of the mean values? 1040 00:56:52,515 --> 00:56:54,930 PROFESSOR: That Is the expected behavior 1041 00:56:54,930 --> 00:56:57,570 of the mean value over time. 1042 00:56:57,570 --> 00:57:00,640 In this case, fn and gn are both linear functions 1043 00:57:00,640 --> 00:57:02,140 of the number of the mRNA. 1044 00:57:02,140 --> 00:57:05,670 Which means that in the context of the master equation, 1045 00:57:05,670 --> 00:57:14,590 if you ask about the expectation of mn, 1046 00:57:14,590 --> 00:57:20,300 this quantity is indeed equal to-- it has the same behavior 1047 00:57:20,300 --> 00:57:28,210 as, over time, as the deterministic equations. 1048 00:57:28,210 --> 00:57:30,890 So if f and g are nonlinear, then 1049 00:57:30,890 --> 00:57:33,420 actually you get a deviation. 1050 00:57:33,420 --> 00:57:35,750 But in this case, it is indeed the same. 1051 00:57:35,750 --> 00:57:41,230 What it means that if you compare 1052 00:57:41,230 --> 00:57:43,680 the stochastic and the deterministic trajectories, 1053 00:57:43,680 --> 00:57:45,900 what you would see is that this thing 1054 00:57:45,900 --> 00:57:51,040 is going to be a little bit jagged, or whatnot. 1055 00:57:51,040 --> 00:57:52,420 And then even at equilibrium it's 1056 00:57:52,420 --> 00:57:54,380 going to come up and down a little bit. 1057 00:57:54,380 --> 00:57:56,213 I'm trying to add a little bit of jaggedness 1058 00:57:56,213 --> 00:57:57,870 because it's discrete. 1059 00:57:57,870 --> 00:58:01,230 But the deterministic equation here 1060 00:58:01,230 --> 00:58:03,630 is what you would get if you average together 1061 00:58:03,630 --> 00:58:07,549 an infinite number of these stochastic trajectories. 1062 00:58:07,549 --> 00:58:09,465 Because another one might have come down here. 1063 00:58:13,200 --> 00:58:15,938 Does that answer? 1064 00:58:15,938 --> 00:58:18,686 AUDIENCE: Is m playing a double role? 1065 00:58:18,686 --> 00:58:20,186 Like in that deterministic equation, 1066 00:58:20,186 --> 00:58:22,210 m is the concentration of mRNA? 1067 00:58:22,210 --> 00:58:25,410 PROFESSOR: I think that I'm-- yeah I think that I should-- 1068 00:58:25,410 --> 00:58:27,320 my nomenclature I think was not very good. 1069 00:58:27,320 --> 00:58:29,360 I've used two different things. 1070 00:58:29,360 --> 00:58:30,880 And now that I'm doing this, I think 1071 00:58:30,880 --> 00:58:36,980 that I should have-- I should have just called it p of n, 1072 00:58:36,980 --> 00:58:38,965 or maybe I should've used n here. 1073 00:58:44,972 --> 00:58:46,930 I think I was trying to be consistent with some 1074 00:58:46,930 --> 00:58:49,340 of the previous, but I think it was a mistake. 1075 00:58:52,460 --> 00:58:53,927 Yes? 1076 00:58:53,927 --> 00:58:55,510 AUDIENCE: Are you plotting stochastic? 1077 00:58:58,820 --> 00:59:05,720 PROFESSOR: I'm plotting-- OK, so no, I'm not. 1078 00:59:05,720 --> 00:59:11,450 So this is if you run an actual stochastic trajectory. 1079 00:59:11,450 --> 00:59:13,180 Then at any moment in time, you just 1080 00:59:13,180 --> 00:59:16,830 have one-- there's some number of mRNA. 1081 00:59:16,830 --> 00:59:19,410 Whereas the sum over the mn's, this 1082 00:59:19,410 --> 00:59:21,290 is talking about the probability distribution 1083 00:59:21,290 --> 00:59:22,430 of the entire thing. 1084 00:59:22,430 --> 00:59:25,580 So really if you started here, the master equation would give 1085 00:59:25,580 --> 00:59:30,550 you some distribution for the n's, some distribution for m's. 1086 00:59:30,550 --> 00:59:33,350 And so if you looked at these over time, 1087 00:59:33,350 --> 00:59:35,510 than the mean of these distributions 1088 00:59:35,510 --> 00:59:40,481 is indeed equal to the deterministic behavior. 1089 00:59:40,481 --> 00:59:40,980 Yes? 1090 00:59:40,980 --> 00:59:43,230 AUDIENCE: Is it possible to recover, like how would we 1091 00:59:43,230 --> 00:59:47,631 recover the differential equation from the master 1092 00:59:47,631 --> 00:59:48,130 equation? 1093 00:59:48,130 --> 00:59:49,280 Is that possible? 1094 00:59:49,280 --> 00:59:50,630 Maybe that would help. 1095 00:59:50,630 --> 00:59:52,760 PROFESSOR: Yeah. 1096 00:59:52,760 --> 00:59:54,814 I think that in the end, there's going 1097 00:59:54,814 --> 00:59:56,730 to be a one-to-one relationship from, I guess, 1098 00:59:56,730 --> 01:00:00,230 this differential equation to the master equation. 1099 01:00:00,230 --> 01:00:01,999 I'm trying to think of any weird case 1100 01:00:01,999 --> 01:00:03,540 or something funny's going to happen. 1101 01:00:03,540 --> 01:00:04,830 Is something funny going to happen? 1102 01:00:04,830 --> 01:00:05,371 AUDIENCE: No. 1103 01:00:05,371 --> 01:00:08,246 But like the easy way is just to write them all in terms 1104 01:00:08,246 --> 01:00:09,222 of the distribution. 1105 01:00:09,222 --> 01:00:13,068 And you can just differentiate the whole sum. 1106 01:00:13,068 --> 01:00:17,460 And in that sum, we express the [INAUDIBLE] 1107 01:00:17,460 --> 01:00:20,388 with your last equation. 1108 01:00:20,388 --> 01:00:23,022 [INAUDIBLE] 1109 01:00:23,022 --> 01:00:23,730 PROFESSOR: Right. 1110 01:00:23,730 --> 01:00:25,966 But I think this is the much more mathematical way. 1111 01:00:25,966 --> 01:00:27,590 I mean because I think that actually, I 1112 01:00:27,590 --> 01:00:31,810 mean, from the differential equation, you 1113 01:00:31,810 --> 01:00:34,440 actually from the terms here, you can actually 1114 01:00:34,440 --> 01:00:36,060 construct the master equation. 1115 01:00:36,060 --> 01:00:39,544 And I think by the same way, you can go from the master 1116 01:00:39,544 --> 01:00:40,960 equation, and I think that there's 1117 01:00:40,960 --> 01:00:42,334 going to be a unique differential 1118 01:00:42,334 --> 01:00:44,950 equation that would have gotten you to that master equation. 1119 01:00:44,950 --> 01:00:46,990 So I think just from the terms you can do it. 1120 01:00:46,990 --> 01:00:49,530 You could also do like moment generating functions 1121 01:00:49,530 --> 01:00:51,620 to get to how things change. 1122 01:00:51,620 --> 01:00:59,540 But I mean I think that it's really from this, for example, 1123 01:00:59,540 --> 01:01:04,050 I think it tells you that that was the differential equation. 1124 01:01:04,050 --> 01:01:06,980 Does that-- 1125 01:01:06,980 --> 01:01:10,100 I mean it's sort of-- the way that we typically 1126 01:01:10,100 --> 01:01:12,350 do things these things, is that we have a differential 1127 01:01:12,350 --> 01:01:14,790 equation, and then we construct the master equation. 1128 01:01:14,790 --> 01:01:17,248 So then we already knew what the differential equation was. 1129 01:01:17,248 --> 01:01:20,100 But I think just from the terms in your master equation, 1130 01:01:20,100 --> 01:01:22,769 you can say, all right. 1131 01:01:22,769 --> 01:01:25,102 This was the differential equation that it started with. 1132 01:01:31,486 --> 01:01:33,360 Any other questions about what happened here? 1133 01:01:40,320 --> 01:01:43,650 So we have, I think, a fair number, a fair knowledge 1134 01:01:43,650 --> 01:01:45,890 of what's going on here now. 1135 01:01:45,890 --> 01:01:49,930 We know that the equilibrium distribution 1136 01:01:49,930 --> 01:01:52,620 of mRNA in the cell is going to be Poisson. 1137 01:01:52,620 --> 01:01:54,600 We also know that the distribution 1138 01:01:54,600 --> 01:01:57,230 of the number of mRNA produced per sell cycle is also Poisson. 1139 01:01:57,230 --> 01:02:00,600 But it's a different Poisson from the first one. 1140 01:02:00,600 --> 01:02:04,097 We know that the number of proteins produced per mRNA 1141 01:02:04,097 --> 01:02:05,805 is going to be geometrically distributed. 1142 01:02:08,775 --> 01:02:10,400 The one thing that we have not yet done 1143 01:02:10,400 --> 01:02:17,650 is to ask about the distribution of protein in the cell. 1144 01:02:17,650 --> 01:02:19,352 So let's say something about that. 1145 01:02:19,352 --> 01:02:21,060 I'm not going to do the whole derivation. 1146 01:02:21,060 --> 01:02:22,560 Because it's harder. 1147 01:02:22,560 --> 01:02:27,180 But I encourage you to-- even the continuous version 1148 01:02:27,180 --> 01:02:30,840 of the derivation is definitely harder than this. 1149 01:02:30,840 --> 01:02:36,620 But then the discrete derivation is even worse. 1150 01:02:36,620 --> 01:02:38,060 So what we're going to talk about, 1151 01:02:38,060 --> 01:02:39,530 and the way we'll typically maybe 1152 01:02:39,530 --> 01:02:41,613 think about this from the standpoint of this class 1153 01:02:41,613 --> 01:02:46,257 is the continuous approximation to-- oh, 1154 01:02:46,257 --> 01:02:47,840 that might have ended up being useful. 1155 01:02:47,840 --> 01:02:49,790 Well it's OK. 1156 01:02:49,790 --> 01:02:54,499 Is the continuous approximation to the real answer. 1157 01:02:54,499 --> 01:02:56,790 And in particular, just the way that the exponential is 1158 01:02:56,790 --> 01:03:00,820 the continuous approximation of the geometric distribution, 1159 01:03:00,820 --> 01:03:04,370 in the same way you can think about the equilibrium 1160 01:03:04,370 --> 01:03:06,370 distribution of protein in the cell. 1161 01:03:06,370 --> 01:03:08,720 In this model is going to be gamma distributed. 1162 01:03:08,720 --> 01:03:11,110 But gamma is a continuous distribution. 1163 01:03:11,110 --> 01:03:17,880 But it's a continuous analog of the negative binomial. 1164 01:03:17,880 --> 01:03:33,401 So let me just make sure I'm-- and Sunney Xie actually has 1165 01:03:33,401 --> 01:03:38,670 a nice PRL paper where he derives the gamma distribution. 1166 01:03:38,670 --> 01:03:40,970 But even earlier actually Paulson 1167 01:03:40,970 --> 01:03:43,700 had derived this negative binomial distribution, 1168 01:03:43,700 --> 01:03:45,690 the discrete version of the solution. 1169 01:03:56,910 --> 01:03:59,379 So this is the number of protein per cell. 1170 01:03:59,379 --> 01:04:00,420 We already know the mean. 1171 01:04:04,870 --> 01:04:09,410 So this is approximately distributed as a gamma. 1172 01:04:09,410 --> 01:04:12,410 A gamma is a distribution that requires two parameters 1173 01:04:12,410 --> 01:04:13,330 to describe. 1174 01:04:13,330 --> 01:04:17,230 So a Poisson can be described by single parameter. 1175 01:04:17,230 --> 01:04:19,150 Gamma is typically described by two. 1176 01:04:23,580 --> 01:04:29,806 And b is going to be the burst size, 1177 01:04:29,806 --> 01:04:43,390 whereas a is the mean number of bursts per cell cycle, which 1178 01:04:43,390 --> 01:04:45,850 is the same as the mean number of mRNA 1179 01:04:45,850 --> 01:04:47,500 produced, so mean number of bursts. 1180 01:04:56,670 --> 01:05:00,080 So the gamma of this a, b. 1181 01:05:18,850 --> 01:05:19,350 All right. 1182 01:05:19,350 --> 01:05:23,920 So the gamma of a is the gamma function. 1183 01:05:23,920 --> 01:05:29,360 It's equal to-- now is it a minus 1 factorial? 1184 01:05:29,360 --> 01:05:31,770 I always get the-- is it a minus 1 or a plus 1 factorial. 1185 01:05:31,770 --> 01:05:33,590 Anybody remember this? 1186 01:05:33,590 --> 01:05:34,370 Yeah, a minus 1. 1187 01:05:43,830 --> 01:05:45,290 I mean it's like a lot of things. 1188 01:05:45,290 --> 01:05:46,740 You look at this equation. 1189 01:05:46,740 --> 01:05:48,980 It doesn't really mean a whole lot. 1190 01:05:48,980 --> 01:05:51,540 But I think that a reasonable way to think about this 1191 01:05:51,540 --> 01:05:55,810 is the gamma is approximately what 1192 01:05:55,810 --> 01:06:02,260 you get when you add together a different exponentials 1193 01:06:02,260 --> 01:06:06,515 with length scale, given by b. 1194 01:06:06,515 --> 01:06:08,140 When you add probability distributions, 1195 01:06:08,140 --> 01:06:09,909 you have to do a convolution. 1196 01:06:09,909 --> 01:06:11,700 So in some ways, the way to think about it, 1197 01:06:11,700 --> 01:06:12,908 and this kind of makes sense. 1198 01:06:12,908 --> 01:06:15,720 Because what is happening is that it 1199 01:06:15,720 --> 01:06:18,631 takes something of order cell division time 1200 01:06:18,631 --> 01:06:19,880 for these proteins to go away. 1201 01:06:19,880 --> 01:06:22,200 Because they're stable. 1202 01:06:22,200 --> 01:06:25,070 Now each-- and so then what you want to know 1203 01:06:25,070 --> 01:06:27,200 is how many proteins are kind of produced 1204 01:06:27,200 --> 01:06:30,200 over the course of a cell cycle. 1205 01:06:30,200 --> 01:06:36,144 Well that actually you can get at by asking how many bursts 1206 01:06:36,144 --> 01:06:37,060 are there going to be. 1207 01:06:37,060 --> 01:06:38,430 And then how big are the bursts? 1208 01:06:41,450 --> 01:06:47,580 So indeed, the mean here is equal to a times b. 1209 01:06:47,580 --> 01:06:50,860 And the variance is equal to a times b squared. 1210 01:06:56,551 --> 01:06:58,550 So for example, if you have a single exponential 1211 01:06:58,550 --> 01:07:06,270 distribution, with burst size b, then this is what you get. 1212 01:07:06,270 --> 01:07:09,820 So this is the probability that you get n proteins. 1213 01:07:09,820 --> 01:07:11,500 And this is this function of n. 1214 01:07:11,500 --> 01:07:15,500 So for a single burst, this is exponentially distributed. 1215 01:07:15,500 --> 01:07:18,990 So this is the continuous version. 1216 01:07:18,990 --> 01:07:22,830 Now if we add together multiple of these bursts, 1217 01:07:22,830 --> 01:07:25,330 this is really saying that we sample from this distribution, 1218 01:07:25,330 --> 01:07:26,240 say twice. 1219 01:07:26,240 --> 01:07:29,120 And then we add the resulting value. 1220 01:07:29,120 --> 01:07:30,790 So this is a convolution. 1221 01:07:30,790 --> 01:07:32,500 You guys will have an opportunity 1222 01:07:32,500 --> 01:07:35,310 to practice this on your problem sets. 1223 01:07:35,310 --> 01:07:36,910 But what happens is that you end up 1224 01:07:36,910 --> 01:07:39,660 getting something that looks like-- it's going to go. 1225 01:07:42,920 --> 01:07:44,380 So it increases linearly. 1226 01:07:44,380 --> 01:07:46,470 If you added three of them together 1227 01:07:46,470 --> 01:07:47,871 this increase is quadratic. 1228 01:07:47,871 --> 01:07:49,120 And it kind of goes like that. 1229 01:07:49,120 --> 01:07:52,415 So this thing becomes kind of-- it 1230 01:07:52,415 --> 01:07:54,652 goes from a distribution where it's 1231 01:07:54,652 --> 01:07:57,110 peaked at 0, to something that's peaked at a nonzero value. 1232 01:08:00,060 --> 01:08:02,660 Now you can ask, for example, what 1233 01:08:02,660 --> 01:08:09,810 happens as for a large a, if you have many bursts, 1234 01:08:09,810 --> 01:08:11,330 what does this thing look like? 1235 01:08:11,330 --> 01:08:14,415 Oh, I wish I hadn't erased my probability distributions. 1236 01:08:17,290 --> 01:08:22,233 So what are the gamma converged to for large a? 1237 01:08:22,233 --> 01:08:23,149 A normal distribution. 1238 01:08:23,149 --> 01:08:23,950 Right? 1239 01:08:23,950 --> 01:08:26,359 So that's the central limit theorem. 1240 01:08:26,359 --> 01:08:28,939 If you take any well-behaved probability distribution, 1241 01:08:28,939 --> 01:08:29,649 you add it. 1242 01:08:29,649 --> 01:08:31,821 You sample from it many times. 1243 01:08:31,821 --> 01:08:33,279 Then you end up getting a Gaussian. 1244 01:08:38,528 --> 01:08:40,069 If you don't remember that very well, 1245 01:08:40,069 --> 01:08:43,580 then this is something to read about over the weekend. 1246 01:08:49,109 --> 01:08:50,580 Just like the Poisson is also going 1247 01:08:50,580 --> 01:08:57,745 to go to-- for large lambda, the Poisson 1248 01:08:57,745 --> 01:08:58,870 also looks like a Gaussian. 1249 01:08:58,870 --> 01:09:00,330 Can somebody give an explanation, 1250 01:09:00,330 --> 01:09:03,500 an intuitive explanation for why that should be? 1251 01:09:03,500 --> 01:09:06,773 Why it-- yes? 1252 01:09:06,773 --> 01:09:08,697 AUDIENCE: Because in a Poisson distribution, 1253 01:09:08,697 --> 01:09:10,307 you can't have anything negative. 1254 01:09:10,307 --> 01:09:10,890 PROFESSOR: OK. 1255 01:09:10,890 --> 01:09:12,848 So a Poisson distribution can't have anything-- 1256 01:09:12,848 --> 01:09:14,970 but now I feel like you're arguing against me. 1257 01:09:14,970 --> 01:09:17,579 Because a Gaussian has negative values, right? 1258 01:09:17,579 --> 01:09:18,245 AUDIENCE: Right. 1259 01:09:18,245 --> 01:09:22,946 So when the mean is really small, only have [INAUDIBLE]. 1260 01:09:22,946 --> 01:09:23,529 PROFESSOR: OK. 1261 01:09:23,529 --> 01:09:24,100 Yeah. 1262 01:09:24,100 --> 01:09:24,990 All right. 1263 01:09:24,990 --> 01:09:28,755 So what you're saying is that Poisson for small lambda 1264 01:09:28,755 --> 01:09:30,035 it can't go negative. 1265 01:09:32,221 --> 01:09:32,720 OK. 1266 01:09:32,720 --> 01:09:35,568 No I think that that's true. 1267 01:09:35,568 --> 01:09:37,609 Yeah, and so somehow the probability distribution 1268 01:09:37,609 --> 01:09:39,974 is somehow piling up, as you say. 1269 01:09:39,974 --> 01:09:41,974 What are some other ways of thinking about this? 1270 01:09:50,463 --> 01:09:51,379 AUDIENCE: [INAUDIBLE]. 1271 01:10:00,289 --> 01:10:04,744 Because if you have a low lambda that means it's a Poisson. 1272 01:10:04,744 --> 01:10:08,704 And then I'm just imagining stretching out. 1273 01:10:08,704 --> 01:10:09,240 [INAUDIBLE] 1274 01:10:12,137 --> 01:10:12,720 PROFESSOR: OK. 1275 01:10:12,720 --> 01:10:14,051 So I think that's fair. 1276 01:10:14,051 --> 01:10:15,550 Another way we can think about this, 1277 01:10:15,550 --> 01:10:17,425 is let's say that we have some process that's 1278 01:10:17,425 --> 01:10:20,600 occurring randomly over some period of time. 1279 01:10:20,600 --> 01:10:23,470 And this could be say, mRNA production. 1280 01:10:23,470 --> 01:10:26,250 And here this is just the number that we observe here, 1281 01:10:26,250 --> 01:10:29,580 this is going to be a Poisson, with some mean lambda. 1282 01:10:29,580 --> 01:10:33,580 Now let's just say that I take another one, same process, 1283 01:10:33,580 --> 01:10:34,520 same period of time. 1284 01:10:34,520 --> 01:10:36,186 How is this guy going to be distributed? 1285 01:10:38,229 --> 01:10:39,520 So this also Poisson of lambda. 1286 01:10:43,430 --> 01:10:47,740 Now let's say I take this probability distribution, 1287 01:10:47,740 --> 01:10:49,970 and I take this probability distribution. 1288 01:10:49,970 --> 01:10:52,259 And I convolve them. 1289 01:10:52,259 --> 01:10:54,050 I'm going to do the calculation of my head. 1290 01:10:54,050 --> 01:10:57,100 I did it. 1291 01:10:57,100 --> 01:11:02,410 So for those of you who haven't done convolutions-- whatever. 1292 01:11:02,410 --> 01:11:04,380 Yes, what's the new distribution going to be? 1293 01:11:11,380 --> 01:11:13,242 Poisson 2 lambda. 1294 01:11:13,242 --> 01:11:14,450 And why does that have to be? 1295 01:11:16,306 --> 01:11:18,306 AUDIENCE: That line was sort of you put it by n. 1296 01:11:21,690 --> 01:11:22,370 PROFESSOR: Yeah. 1297 01:11:22,370 --> 01:11:22,911 That's right. 1298 01:11:22,911 --> 01:11:25,100 This line, I just kind of like I just made it up. 1299 01:11:25,100 --> 01:11:25,990 I could have just said, oh. 1300 01:11:25,990 --> 01:11:27,948 Well it's the same process occurring over here. 1301 01:11:27,948 --> 01:11:29,247 So we have to have the mean. 1302 01:11:29,247 --> 01:11:31,080 It's still is going to be a Poisson process. 1303 01:11:31,080 --> 01:11:35,085 And the mean has to be the-- well we just 1304 01:11:35,085 --> 01:11:35,960 had twice the length. 1305 01:11:35,960 --> 01:11:38,210 And indeed, for independent probability distributions, 1306 01:11:38,210 --> 01:11:39,655 means always add. 1307 01:11:39,655 --> 01:11:41,780 So this all consistent will all the things we know. 1308 01:11:41,780 --> 01:11:43,690 So this has to be a Poisson of 2 lambda. 1309 01:11:43,690 --> 01:11:45,370 If I add another segment on here, 1310 01:11:45,370 --> 01:11:47,600 it has to Poisson of 3 lambda. 1311 01:11:47,600 --> 01:11:51,590 But what you see is that we see that Poisson of n lambda, which 1312 01:11:51,590 --> 01:11:52,840 is the sum over many Poissons. 1313 01:11:55,724 --> 01:11:57,890 Poissons are well-behaved probability distributions. 1314 01:11:57,890 --> 01:11:58,560 You add them together, you're going 1315 01:11:58,560 --> 01:11:59,990 to have to get a Gaussian. 1316 01:11:59,990 --> 01:12:02,840 So you can see that the Poisson has to become 1317 01:12:02,840 --> 01:12:03,940 Gaussian for large lambda. 1318 01:12:03,940 --> 01:12:05,010 And indeed it does. 1319 01:12:09,520 --> 01:12:11,660 So there's a comment about this in the-- 1320 01:12:11,660 --> 01:12:13,950 AUDIENCE: It's a little bit more complicated than this 1321 01:12:13,950 --> 01:12:16,990 because obviously you always just divide 1322 01:12:16,990 --> 01:12:18,430 from lambda [INAUDIBLE]. 1323 01:12:21,895 --> 01:12:24,617 Like you would have to say that Poisson lambda is just 1324 01:12:24,617 --> 01:12:28,340 like a combination of S-- 1325 01:12:28,340 --> 01:12:29,410 PROFESSOR: OK. 1326 01:12:29,410 --> 01:12:31,860 You're saying that if I do this calculation backwards, 1327 01:12:31,860 --> 01:12:32,710 I'm going to get into trouble. 1328 01:12:32,710 --> 01:12:34,043 Because if I try to break them-- 1329 01:12:34,043 --> 01:12:36,689 AUDIENCE: So if you require lambda to be-- you 1330 01:12:36,689 --> 01:12:39,736 have to have like a significant probability of getting at least 1331 01:12:39,736 --> 01:12:40,970 one candidate, right? 1332 01:12:40,970 --> 01:12:45,670 PROFESSOR: So I'd say lambda has to be much, much larger than 1. 1333 01:12:45,670 --> 01:12:50,986 So once you're at lambda of 100, it looks like a Gaussian. 1334 01:12:50,986 --> 01:12:53,110 And in Sunney's paper, he had a comment about this. 1335 01:12:53,110 --> 01:12:54,526 Does anybody remember what it was? 1336 01:12:59,160 --> 01:13:01,290 Was mRNA production really well described 1337 01:13:01,290 --> 01:13:06,395 as-- they mention that actually there 1338 01:13:06,395 --> 01:13:12,105 is some violation of this model in the data. 1339 01:13:12,105 --> 01:13:14,079 AUDIENCE: Does it go into eukaryotes? 1340 01:13:14,079 --> 01:13:16,120 PROFESSOR: Oh, as soon as you go into eukaryotes, 1341 01:13:16,120 --> 01:13:18,160 this is why I stay away from them. 1342 01:13:18,160 --> 01:13:21,350 But even in their data, in E. coli, 1343 01:13:21,350 --> 01:13:23,080 they actually observed a deviation. 1344 01:13:29,955 --> 01:13:32,500 So what they found is that there was a cell cycle dependence 1345 01:13:32,500 --> 01:13:36,810 to this bursting rate, i.e. the mRNA production 1346 01:13:36,810 --> 01:13:39,510 over the course of the cell cycle. 1347 01:13:39,510 --> 01:13:41,450 And presumably their conclusion of this 1348 01:13:41,450 --> 01:13:43,740 was that you have this guy. 1349 01:13:43,740 --> 01:13:47,160 And then he turns into, gets longer. 1350 01:13:47,160 --> 01:13:50,817 And then eventually he septates, and then you get two cells. 1351 01:13:55,540 --> 01:13:59,940 What he found is that these longer cells had actually 1352 01:13:59,940 --> 01:14:03,210 a larger rate of mRNA synthesis than the smaller cells. 1353 01:14:03,210 --> 01:14:04,460 And actually this makes sense. 1354 01:14:04,460 --> 01:14:07,590 Because here you maybe have just one copy of the genome. 1355 01:14:07,590 --> 01:14:12,930 Whereas here you might have-- you're making a second copy. 1356 01:14:12,930 --> 01:14:16,180 So you might have two copies of that gene. 1357 01:14:16,180 --> 01:14:21,117 So it may make sense that this bursting rate should grow. 1358 01:14:21,117 --> 01:14:22,700 But does that mean that you should not 1359 01:14:22,700 --> 01:14:26,130 expect it to be a Poisson distribution for the number 1360 01:14:26,130 --> 01:14:27,642 of bursts per cell cycle? 1361 01:14:33,540 --> 01:14:34,040 No. 1362 01:14:34,040 --> 01:14:37,420 It actually is still-- it still is described by a Poisson. 1363 01:14:37,420 --> 01:14:40,650 Because you can just say, this is the cell cycle. 1364 01:14:40,650 --> 01:14:43,480 And here this is Poisson of sum lambda 1. 1365 01:14:43,480 --> 01:14:45,270 Here is a Poisson of sum lambda 2. 1366 01:14:45,270 --> 01:14:46,970 So there could be a different rate 1367 01:14:46,970 --> 01:14:47,700 over the course of the thing. 1368 01:14:47,700 --> 01:14:49,241 But you still have just two Poissons. 1369 01:14:49,241 --> 01:14:51,230 You still get another Poisson. 1370 01:14:51,230 --> 01:14:54,215 So adding Poissons, gives you backup Poisson. 1371 01:14:54,215 --> 01:14:56,090 They don't have to have the same mean lambda. 1372 01:15:04,230 --> 01:15:07,550 I just want to make one comment about what 1373 01:15:07,550 --> 01:15:09,940 you have to do once you start thinking about eukaryotes. 1374 01:15:09,940 --> 01:15:15,540 And the basic-- so you can see the gamma distribution 1375 01:15:15,540 --> 01:15:19,877 can either be peaked at 0, or it can be peaked at nonzero value. 1376 01:15:19,877 --> 01:15:21,710 So most, for like highly expressed proteins, 1377 01:15:21,710 --> 01:15:24,880 you'll see that it looks something like this. 1378 01:15:24,880 --> 01:15:27,620 Now for eukaryotes, you also have 1379 01:15:27,620 --> 01:15:30,490 to consider there's some rate that you 1380 01:15:30,490 --> 01:15:33,500 go between an active and inactive promoter. 1381 01:15:37,790 --> 01:15:40,470 And this actually makes things much more complicated. 1382 01:15:40,470 --> 01:15:43,220 So there's a rate going to inactive, 1383 01:15:43,220 --> 01:15:45,990 a rate going to active. 1384 01:15:45,990 --> 01:15:48,390 And so now if you look at, for example, 1385 01:15:48,390 --> 01:15:50,980 the mRNA number per cell, you'll see 1386 01:15:50,980 --> 01:15:53,950 that it is no longer a Poisson. 1387 01:15:53,950 --> 01:15:57,230 And I encourage you, if you're curious about such things, 1388 01:15:57,230 --> 01:16:00,340 to come up and look at this. 1389 01:16:00,340 --> 01:16:02,320 The solution for the steady state distribution 1390 01:16:02,320 --> 01:16:03,528 has been solved analytically. 1391 01:16:03,528 --> 01:16:06,630 For example, Arjun Raj, who is the author of the review 1392 01:16:06,630 --> 01:16:11,100 that you guys just read, derived this equation here, 1393 01:16:11,100 --> 01:16:12,756 which I don't know if you can see. 1394 01:16:12,756 --> 01:16:14,130 But even from a distance, you can 1395 01:16:14,130 --> 01:16:15,734 see that this is the solution. 1396 01:16:15,734 --> 01:16:17,525 And this is just for the mRNA distribution. 1397 01:16:17,525 --> 01:16:20,640 This is not even getting to the level of the protein. 1398 01:16:20,640 --> 01:16:24,620 And it involves many gamma functions, as well as 1399 01:16:24,620 --> 01:16:29,210 a confluent hypergeometric function of the first kind, 1400 01:16:29,210 --> 01:16:31,400 which is a disaster. 1401 01:16:31,400 --> 01:16:33,230 But he went to Courant. 1402 01:16:33,230 --> 01:16:34,570 He was an applied mathematician. 1403 01:16:34,570 --> 01:16:36,180 So this is, I guess, this is what 1404 01:16:36,180 --> 01:16:40,030 you can do after doing a PhD in applied mathematics. 1405 01:16:40,030 --> 01:16:43,020 The point though is that it ends up being very complicated. 1406 01:16:43,020 --> 01:16:47,490 And you can get hugely varying distributions for the mRNA. 1407 01:16:47,490 --> 01:16:50,240 And indeed this is seen in individual cells. 1408 01:16:50,240 --> 01:16:53,170 If you look at mammalian cells, just at the mRNA level, 1409 01:16:53,170 --> 01:16:56,260 you can have some cells that have hardly any mRNA, some 1410 01:16:56,260 --> 01:16:57,920 that have a huge number. 1411 01:16:57,920 --> 01:16:59,720 The protein distributions actually 1412 01:16:59,720 --> 01:17:03,890 end up being more regular than the mRNA distributions. 1413 01:17:03,890 --> 01:17:07,290 Because of this difference in lifetime. 1414 01:17:07,290 --> 01:17:09,800 So the mRNA numbers may fluctuate wildly. 1415 01:17:09,800 --> 01:17:12,480 But the protein numbers will fluctuate less, 1416 01:17:12,480 --> 01:17:14,720 because they last longer. 1417 01:17:14,720 --> 01:17:17,360 So then you do some averaging over this crazy mRNA business. 1418 01:17:23,940 --> 01:17:25,680 Now in the last-- yeah, go ahead. 1419 01:17:25,680 --> 01:17:27,862 AUDIENCE: In terms of timescale, like all 1420 01:17:27,862 --> 01:17:31,500 this is switching to the active and inactive promoter, 1421 01:17:31,500 --> 01:17:32,480 like to the other-- 1422 01:17:32,480 --> 01:17:33,390 PROFESSOR: Ah, yes. 1423 01:17:33,390 --> 01:17:36,490 That's a good question. 1424 01:17:36,490 --> 01:17:39,470 I think that people argue very much about this. 1425 01:17:39,470 --> 01:17:41,630 This is kind of minutes. 1426 01:17:41,630 --> 01:17:44,150 This can be hours. 1427 01:17:44,150 --> 01:17:48,370 And this is maybe in between those timescales 1428 01:17:48,370 --> 01:17:51,020 would be typical. 1429 01:17:51,020 --> 01:17:53,540 And when I say hours, especially like in mammalian cells, 1430 01:17:53,540 --> 01:17:55,630 they might only divide once a day or so. 1431 01:17:55,630 --> 01:17:59,460 So then this gets to be many hours. 1432 01:17:59,460 --> 01:18:03,935 And then I'd say minutes is kind of the-- 1433 01:18:10,110 --> 01:18:13,190 So there were many biological examples that 1434 01:18:13,190 --> 01:18:14,690 were discussed in that review. 1435 01:18:14,690 --> 01:18:16,560 And I'm not going to talk about all them. 1436 01:18:16,560 --> 01:18:18,070 But I think that it's a nice review. 1437 01:18:18,070 --> 01:18:20,492 Because it goes over some of the papers that you've read, 1438 01:18:20,492 --> 01:18:22,950 or that we've talked about over the course of the semester. 1439 01:18:22,950 --> 01:18:26,050 It also illustrates some different biological context 1440 01:18:26,050 --> 01:18:28,460 in which noise may play a role. 1441 01:18:28,460 --> 01:18:30,220 But I want to mention one study that 1442 01:18:30,220 --> 01:18:33,450 was done by actually again, Arjun together 1443 01:18:33,450 --> 01:18:37,460 with Hedia Maamar, in collaboration with Dave Dubnau, 1444 01:18:37,460 --> 01:18:40,130 where they were studying this process of competence. 1445 01:18:40,130 --> 01:18:43,060 So in B. subtilis, during sometimes particularly 1446 01:18:43,060 --> 01:18:45,580 of starvation, or other forms of unhappiness, 1447 01:18:45,580 --> 01:18:49,190 they kind of pick up DNA from outside. 1448 01:18:49,190 --> 01:18:51,170 So they'll import DNA. 1449 01:18:51,170 --> 01:18:52,740 Some of it may just be consumed. 1450 01:18:52,740 --> 01:18:56,940 But some of it could actually be incorporated into the genome. 1451 01:18:56,940 --> 01:19:01,130 Now what they found is that this competence process is 1452 01:19:01,130 --> 01:19:02,830 mediated by this protein comK. 1453 01:19:08,000 --> 01:19:10,450 And there was a positive feedback loop, 1454 01:19:10,450 --> 01:19:14,780 where this guy ends up positively activating itself. 1455 01:19:14,780 --> 01:19:19,270 And this helps lead to bistability in this network. 1456 01:19:19,270 --> 01:19:21,270 Only a small fraction of the cells 1457 01:19:21,270 --> 01:19:23,760 kind of get into this high feedback state. 1458 01:19:23,760 --> 01:19:26,660 Only a small fraction of them activate competence and then 1459 01:19:26,660 --> 01:19:28,260 uptake DNA. 1460 01:19:28,260 --> 01:19:30,150 And what they were able to show in that study 1461 01:19:30,150 --> 01:19:33,270 was that it was sort of noise-induced. 1462 01:19:33,270 --> 01:19:36,760 That they were able to vary both the transcription 1463 01:19:36,760 --> 01:19:39,120 rate and the translation rate, in a way 1464 01:19:39,120 --> 01:19:41,840 so as to reduce the noise. 1465 01:19:41,840 --> 01:19:43,580 The mean is the same. 1466 01:19:43,580 --> 01:19:49,650 So if you in the context of this model, what they did 1467 01:19:49,650 --> 01:19:52,640 is they varied transcription rate, 1468 01:19:52,640 --> 01:19:55,640 and they varied translation rate, 1469 01:19:55,640 --> 01:19:58,180 each say, by a factor of 2. 1470 01:19:58,180 --> 01:20:02,550 So they got the same mean, but then different noise. 1471 01:20:02,550 --> 01:20:03,630 And we're out of time. 1472 01:20:03,630 --> 01:20:05,340 I would have you vote. 1473 01:20:05,340 --> 01:20:06,800 But can anybody remember? 1474 01:20:06,800 --> 01:20:10,010 If you want to decrease the noise 1475 01:20:10,010 --> 01:20:12,310 in the number of the proteins, which of these 1476 01:20:12,310 --> 01:20:15,717 do you want to go up, and which do you want to go down? 1477 01:20:15,717 --> 01:20:16,800 And which one is going up? 1478 01:20:16,800 --> 01:20:18,599 Which one's going up let's say? 1479 01:20:18,599 --> 01:20:19,140 AUDIENCE: Sm. 1480 01:20:19,140 --> 01:20:20,270 PROFESSOR: Sm. 1481 01:20:20,270 --> 01:20:23,154 Right, so if you want to reduce the noise, 1482 01:20:23,154 --> 01:20:24,820 but keep the mean constant, you increase 1483 01:20:24,820 --> 01:20:27,120 the rate of transcription, and you decrease 1484 01:20:27,120 --> 01:20:28,120 the rate of translation. 1485 01:20:28,120 --> 01:20:30,460 Because the noise is really driven by this protein 1486 01:20:30,460 --> 01:20:33,250 bursting behavior here. 1487 01:20:33,250 --> 01:20:34,910 And that's precisely what they did. 1488 01:20:34,910 --> 01:20:36,630 They changed those two quantities. 1489 01:20:36,630 --> 01:20:38,480 They got the same mean, lower noise, 1490 01:20:38,480 --> 01:20:41,860 and then they reduced the amount of competence in that sur--