1 00:00:00,070 --> 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 --> 00:00:17,236 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,236 --> 00:00:17,861 at ocw.mit.edu. 8 00:00:21,320 --> 00:00:24,850 PROFESSOR: So today we're going to take 9 00:00:24,850 --> 00:00:29,270 a little bit of a lightning tour through some basic topics 10 00:00:29,270 --> 00:00:34,060 on chemical enzyme kinetics, before thinking a little bit 11 00:00:34,060 --> 00:00:37,110 about some, what you might call, simple input-output 12 00:00:37,110 --> 00:00:40,814 relationships, in terms of gene expression. 13 00:00:40,814 --> 00:00:42,230 I've got separation of time scale, 14 00:00:42,230 --> 00:00:45,910 and also this basic notion that for a stable protein 15 00:00:45,910 --> 00:00:47,809 there's a natural time scale over which 16 00:00:47,809 --> 00:00:49,350 the concentration will go up or down, 17 00:00:49,350 --> 00:00:51,640 and that's dictated by the cell generation time. 18 00:00:54,450 --> 00:00:57,610 At the end, we'll then talk about different ways you get 19 00:00:57,610 --> 00:00:59,800 this thing of ultrasensitivity. 20 00:00:59,800 --> 00:01:02,080 So, how is it that you can make it 21 00:01:02,080 --> 00:01:05,090 so that a small change in the input concentration, 22 00:01:05,090 --> 00:01:07,090 say the concentration of a transcription factor, 23 00:01:07,090 --> 00:01:10,010 might be able to lead to a large change in the output, 24 00:01:10,010 --> 00:01:11,930 or the gene expression of its target? 25 00:01:11,930 --> 00:01:13,620 We'll talk about how you can have, 26 00:01:13,620 --> 00:01:15,760 for example, cooperative binding at the promoter, 27 00:01:15,760 --> 00:01:18,520 or you can have multimerization, kind of leads to similar things 28 00:01:18,520 --> 00:01:19,200 here. 29 00:01:19,200 --> 00:01:21,420 But also, we're going to talk about this idea 30 00:01:21,420 --> 00:01:23,860 of molecular titration. 31 00:01:23,860 --> 00:01:26,980 So if you have another protein that acts as kind of a sponge, 32 00:01:26,980 --> 00:01:30,780 then this connect can lead to a similar effect. 33 00:01:30,780 --> 00:01:34,450 And this is indeed observed in various natural contexts. 34 00:01:34,450 --> 00:01:43,010 So this is motivated by a work by Nick Buchler, B-U-C-H-L-E-R. 35 00:01:43,010 --> 00:01:45,680 Turns out I was down at Princeton, no sorry, 36 00:01:45,680 --> 00:01:48,920 I was out at Duke yesterday, and so I got to hang out with Nick 37 00:01:48,920 --> 00:01:51,580 and talk about this work. 38 00:01:51,580 --> 00:01:54,669 I had previously told him that in my first lecture 39 00:01:54,669 --> 00:01:56,210 in the systems biology class, we like 40 00:01:56,210 --> 00:01:59,440 to discuss this molecular titration effect. 41 00:01:59,440 --> 00:02:04,880 He was instrumental in elucidating how it worked. 42 00:02:04,880 --> 00:02:07,910 All right so let's go ahead and get started. 43 00:02:07,910 --> 00:02:11,904 So hopefully you all have these cards. 44 00:02:11,904 --> 00:02:14,070 We're going to start out with some simple questions, 45 00:02:14,070 --> 00:02:15,610 just to make sure that you know how 46 00:02:15,610 --> 00:02:18,040 to use the complicated devices that 47 00:02:18,040 --> 00:02:19,320 are sitting in front of you. 48 00:02:19,320 --> 00:02:19,819 OK? 49 00:02:22,630 --> 00:02:24,250 So what we want to do is to start 50 00:02:24,250 --> 00:02:27,360 by thinking about a situation where you have two molecules. 51 00:02:27,360 --> 00:02:30,300 We're going to call them E and S, 52 00:02:30,300 --> 00:02:33,620 and of course, you can imagine what these might possibly stand 53 00:02:33,620 --> 00:02:37,140 for in one context or another. 54 00:02:37,140 --> 00:02:40,840 There are two rates here that are describing 55 00:02:40,840 --> 00:02:44,140 the rate that these molecules, E and S, find each other, Kf. 56 00:02:44,140 --> 00:02:47,780 And Kr is defining the rate at which this complex is 57 00:02:47,780 --> 00:02:49,390 going to fall apart. 58 00:02:49,390 --> 00:02:56,820 So there's this forward rate that 59 00:02:56,820 --> 00:03:01,870 is sum Kf, times concentration of E, 60 00:03:01,870 --> 00:03:03,350 times the concentration of S. Now 61 00:03:03,350 --> 00:03:07,640 I'd say for the first part of this lecture, 62 00:03:07,640 --> 00:03:10,360 we will indeed use the chemistry convention of concentrations 63 00:03:10,360 --> 00:03:13,680 here, although we will quickly get tired of these brackets 64 00:03:13,680 --> 00:03:16,930 and we'll just start writing the letters. 65 00:03:16,930 --> 00:03:19,397 And hopefully it is self evident in the context 66 00:03:19,397 --> 00:03:21,230 that I'm referring to a concentration rather 67 00:03:21,230 --> 00:03:23,980 than something else, but if you're ever confused, please 68 00:03:23,980 --> 00:03:25,440 ask. 69 00:03:25,440 --> 00:03:30,630 So this is forward rate, and then the reverse rate 70 00:03:30,630 --> 00:03:31,890 is something similar. 71 00:03:31,890 --> 00:03:34,024 So we have this Kr. 72 00:03:34,024 --> 00:03:35,440 Now this is just the concentration 73 00:03:35,440 --> 00:03:36,490 of the complex ES. 74 00:03:39,270 --> 00:03:42,360 Many of you have spent a lot of time thinking 75 00:03:42,360 --> 00:03:44,770 about how we often define things. 76 00:03:44,770 --> 00:03:47,670 This Kd, dissociation constant, is defined 77 00:03:47,670 --> 00:03:52,760 as the ratio Kr over Kf. 78 00:03:55,780 --> 00:03:58,682 Now, just so we can practice using our cards, what 79 00:03:58,682 --> 00:04:03,615 we're going to ask first is what the units of this Kd thing is. 80 00:04:06,530 --> 00:04:09,260 Now, in general, when I ask such a question, 81 00:04:09,260 --> 00:04:11,875 I will give you some A, B, C, D options. 82 00:04:14,242 --> 00:04:16,200 You can start thinking even before I write down 83 00:04:16,200 --> 00:04:17,839 the options. 84 00:04:17,839 --> 00:04:20,630 Yes? 85 00:04:20,630 --> 00:04:22,130 I'll encourage you to think before I 86 00:04:22,130 --> 00:04:23,255 start writing down options. 87 00:04:25,670 --> 00:04:34,070 So it's either dimensionless, units of concentration, 88 00:04:34,070 --> 00:04:39,790 1 over concentration, 1 over time, 89 00:04:39,790 --> 00:04:42,740 and I will often include at the bottom something that 90 00:04:42,740 --> 00:04:45,080 simply is, don't know. 91 00:04:45,080 --> 00:04:48,744 And that is if you're really confused 92 00:04:48,744 --> 00:04:50,160 about what I'm talking about, then 93 00:04:50,160 --> 00:04:54,430 feel free to just flash me that, and that at least tells me 94 00:04:54,430 --> 00:04:58,480 that I'm gibbering nonsense. 95 00:04:58,480 --> 00:05:01,440 So there's going to be a very strict set of rules 96 00:05:01,440 --> 00:05:03,620 for how we do these flash cards, all right? 97 00:05:03,620 --> 00:05:06,400 You don't get to vote before I tell you to vote. 98 00:05:06,400 --> 00:05:08,226 You have to keep on thinking. 99 00:05:08,226 --> 00:05:09,850 If you think you know the right answer, 100 00:05:09,850 --> 00:05:14,460 check limits, just do whatever it is to keep on thinking. 101 00:05:14,460 --> 00:05:17,630 And then we vote simultaneously. 102 00:05:17,630 --> 00:05:20,240 That way, it builds up the tension, everyone gets excited, 103 00:05:20,240 --> 00:05:22,000 and then you vote. 104 00:05:22,000 --> 00:05:24,312 It also provides me an opportunity 105 00:05:24,312 --> 00:05:26,770 to make sure that I can see that everybody's participating. 106 00:05:26,770 --> 00:05:30,210 So if you don't vote, then you have the opportunity 107 00:05:30,210 --> 00:05:33,190 to tell the group what you think the answer should be and why. 108 00:05:37,860 --> 00:05:41,010 And the cards they're both colored, 109 00:05:41,010 --> 00:05:44,190 and they have letters on there, so the letters correspond 110 00:05:44,190 --> 00:05:45,689 to the answer. 111 00:05:45,689 --> 00:05:46,730 We're all on top of this? 112 00:05:49,799 --> 00:05:51,090 Have you had a chance to think? 113 00:05:51,090 --> 00:05:54,831 Or has my talking bothered you? 114 00:05:54,831 --> 00:05:55,330 Both. 115 00:05:55,330 --> 00:05:56,490 OK. 116 00:05:56,490 --> 00:05:59,980 So what we do is I'm going to ask, do you need more time? 117 00:05:59,980 --> 00:06:02,680 If you need more time, just nod or something like that, 118 00:06:02,680 --> 00:06:06,690 and if more than a few people nod, I'll give you more time. 119 00:06:06,690 --> 00:06:10,580 But you guys are totally all right. 120 00:06:10,580 --> 00:06:12,390 OK let's see how we are. 121 00:06:12,390 --> 00:06:14,650 So then I'll say, OK, we're ready. 122 00:06:14,650 --> 00:06:17,060 And then we're going to go three, two, one, and then 123 00:06:17,060 --> 00:06:18,860 I want a vote by your chest. 124 00:06:18,860 --> 00:06:21,250 You don't need to display it to the group. 125 00:06:21,250 --> 00:06:23,542 It's just here, and then tell me what you think. 126 00:06:23,542 --> 00:06:24,250 All right, ready. 127 00:06:24,250 --> 00:06:26,210 Three, two, one. 128 00:06:29,540 --> 00:06:32,870 Broadly, people know how to use the cards. 129 00:06:32,870 --> 00:06:35,650 And there's a clear majority of the group, 130 00:06:35,650 --> 00:06:38,414 although it's not 100%, that are saying that this thing is 131 00:06:38,414 --> 00:06:39,080 a concentration. 132 00:06:42,420 --> 00:06:46,730 I'd say that if the group is maybe between 25, 133 00:06:46,730 --> 00:06:48,930 75% correct on these sorts of things, 134 00:06:48,930 --> 00:06:53,330 then I will often have you pair off in, well, in pairs. 135 00:06:53,330 --> 00:06:55,050 And the goal there would be to try 136 00:06:55,050 --> 00:06:56,883 to convince your neighbor that you're right. 137 00:06:56,883 --> 00:06:59,680 In this case we're a bit above 75%, 138 00:06:59,680 --> 00:07:04,700 so I've already indicated what the answer is. 139 00:07:04,700 --> 00:07:09,800 Can somebody just quickly say, why is this a concentration? 140 00:07:09,800 --> 00:07:10,840 Maybe in the back. 141 00:07:10,840 --> 00:07:12,963 AUDIENCE: So, we know that both rates need 142 00:07:12,963 --> 00:07:14,270 to have the same dimensions-- 143 00:07:14,270 --> 00:07:16,769 PROFESSOR: Yeah, and what are the dimensions of these rates? 144 00:07:16,769 --> 00:07:18,870 AUDIENCE: [INAUDIBLE]. 145 00:07:18,870 --> 00:07:21,230 PROFESSOR: OK. 146 00:07:21,230 --> 00:07:26,320 So, there are actually different conventions, in principle, 147 00:07:26,320 --> 00:07:31,960 but we will often be working in numbers 148 00:07:31,960 --> 00:07:33,654 in most of this class, in which case 149 00:07:33,654 --> 00:07:35,320 it would actually just be a 1 over time. 150 00:07:35,320 --> 00:07:38,670 But depending on whether you're doing chemistry-- So 151 00:07:38,670 --> 00:07:41,100 the numerators may be ambiguous, depending, 152 00:07:41,100 --> 00:07:45,280 but the important thing is that they're definitely the same. 153 00:07:45,280 --> 00:07:48,140 These are definitely going to be the same. 154 00:07:48,140 --> 00:07:50,514 But it is true that in an awful lot of this class, 155 00:07:50,514 --> 00:07:52,180 we're going to be thinking about numbers 156 00:07:52,180 --> 00:07:53,140 rather than the concentrations. 157 00:07:53,140 --> 00:07:54,681 Because, for a lot of the class we'll 158 00:07:54,681 --> 00:07:58,930 be thinking about finite number fluctuations 159 00:07:58,930 --> 00:08:03,020 of stochastic dynamics, in which case, concentration, who knows 160 00:08:03,020 --> 00:08:04,420 what's going to happen? 161 00:08:04,420 --> 00:08:07,730 But these have to have the same units, right? 162 00:08:07,730 --> 00:08:10,270 And the important thing here is if you look at the right, 163 00:08:10,270 --> 00:08:14,720 you see this guy has an extra concentration up here. 164 00:08:14,720 --> 00:08:18,690 So, I think this is, on the one hand, a trivial point, 165 00:08:18,690 --> 00:08:21,097 but it's just really easy to forget about 166 00:08:21,097 --> 00:08:21,930 as you move forward. 167 00:08:21,930 --> 00:08:24,640 Because these things, they look awfully similar, right? 168 00:08:24,640 --> 00:08:27,430 There's a K, little subscript something, right? 169 00:08:27,430 --> 00:08:29,620 So just be careful about this kind of thing. 170 00:08:34,929 --> 00:08:38,419 Are there any questions about what I've said so far? 171 00:08:38,419 --> 00:08:39,610 So in these cases. 172 00:08:39,610 --> 00:08:41,640 I think that it's really very useful 173 00:08:41,640 --> 00:08:44,890 to try to get some intuition for what's going on. 174 00:08:44,890 --> 00:08:46,820 These are all just definitions, but you 175 00:08:46,820 --> 00:08:48,861 want to ask, well, what happens if concentrations 176 00:08:48,861 --> 00:08:50,415 of various things move around? 177 00:08:50,415 --> 00:08:52,840 What we want to think about is just the fraction. 178 00:08:52,840 --> 00:08:56,420 And the reason we call this E is because, for now, we 179 00:08:56,420 --> 00:09:03,550 might be calling this an enzyme and, over here, a substrate, 180 00:09:03,550 --> 00:09:06,370 something that the enzyme is acting on. 181 00:09:06,370 --> 00:09:08,499 But we'll see that, in many cases, 182 00:09:08,499 --> 00:09:10,540 we might be thinking about one of these as being, 183 00:09:10,540 --> 00:09:12,319 let's say, the piece of DNA, and then 184 00:09:12,319 --> 00:09:13,610 this is not even the substrate. 185 00:09:13,610 --> 00:09:15,885 Then maybe this is the RNA polymerase that 186 00:09:15,885 --> 00:09:17,010 will lead to transcription. 187 00:09:17,010 --> 00:09:20,560 So in various contexts, we'll think about these things having 188 00:09:20,560 --> 00:09:22,510 different molecular identities. 189 00:09:22,510 --> 00:09:26,360 But for now, E and S, possibly enzyme substrate. 190 00:09:26,360 --> 00:09:29,520 So the question is, if for now we just think, 191 00:09:29,520 --> 00:09:32,620 these are just two molecules of whatever sort, 192 00:09:32,620 --> 00:09:35,070 at some concentration, and we just want to make sure 193 00:09:35,070 --> 00:09:37,040 that we are on top of what's going 194 00:09:37,040 --> 00:09:39,000 to happen if the concentrations of each 195 00:09:39,000 --> 00:09:41,390 of these molecular components goes 196 00:09:41,390 --> 00:09:43,830 either to 0 or to infinity. 197 00:09:43,830 --> 00:09:47,170 I think that before you do any math in life, 198 00:09:47,170 --> 00:09:49,510 it's good to just think about these sorts of limits, 199 00:09:49,510 --> 00:09:53,690 because it helps to make sure that your intuition is correct. 200 00:09:53,690 --> 00:09:57,972 In many, many cases, if you think about the problem 201 00:09:57,972 --> 00:10:00,180 before you do any math, then when you go do the math, 202 00:10:00,180 --> 00:10:01,221 you'll get some solution. 203 00:10:01,221 --> 00:10:03,330 You can check to see whether your solution is 204 00:10:03,330 --> 00:10:05,665 consistent with what your intuition said. 205 00:10:05,665 --> 00:10:07,040 And if they disagree it means you 206 00:10:07,040 --> 00:10:10,290 have to update either your intuition, or the solution, 207 00:10:10,290 --> 00:10:11,610 or maybe both. 208 00:10:11,610 --> 00:10:13,110 It's possible. 209 00:10:13,110 --> 00:10:15,130 But at least one of them has to be updated, 210 00:10:15,130 --> 00:10:18,830 and that's a way of both getting better scores on your exams, 211 00:10:18,830 --> 00:10:22,660 but also improving your scientific intuition. 212 00:10:22,660 --> 00:10:25,810 So, in particular, we just want to do some limits. 213 00:10:29,840 --> 00:10:33,040 We'll think in the context, for example, 214 00:10:33,040 --> 00:10:36,240 if the total concentration-- your adding of the small S-- 215 00:10:36,240 --> 00:10:38,150 if it goes to zero, what we're going 216 00:10:38,150 --> 00:10:43,570 to try to get intuition about is this fraction of E-bound. 217 00:10:43,570 --> 00:10:44,590 It might be the enzyme. 218 00:10:44,590 --> 00:10:46,160 So the fraction of this thing bound, 219 00:10:46,160 --> 00:10:50,260 it's defined by the concentration of the complex, 220 00:10:50,260 --> 00:10:53,290 divided by the concentration of the enzyme, 221 00:10:53,290 --> 00:10:56,490 plus the concentration of the enzyme in the complex, 222 00:10:56,490 --> 00:11:00,360 assuming that this is the only two places that the enzyme can 223 00:11:00,360 --> 00:11:02,310 be located. 224 00:11:02,310 --> 00:11:09,890 Now, these three arrows, that, in general, means a definition. 225 00:11:09,890 --> 00:11:13,070 So the question is, if we come here, 226 00:11:13,070 --> 00:11:18,090 what happens to the fraction of this enzyme that's bound? 227 00:11:18,090 --> 00:11:34,180 And, again, can't be determined-- which 228 00:11:34,180 --> 00:11:35,650 is different from don't know. 229 00:11:40,729 --> 00:11:42,770 And we're going to just do a few different limits 230 00:11:42,770 --> 00:11:45,430 so we want to maybe go through these quickly. 231 00:11:45,430 --> 00:11:47,990 I'll give you 10 seconds to prepare your card. 232 00:11:58,460 --> 00:11:59,500 All right, ready? 233 00:11:59,500 --> 00:12:01,315 Three, two, one. 234 00:12:04,260 --> 00:12:10,300 So we're pretty good. 235 00:12:10,300 --> 00:12:12,570 So I'd say a majority, at least, of the group 236 00:12:12,570 --> 00:12:16,630 is saying that in this case, the fraction bound should go to 0. 237 00:12:16,630 --> 00:12:18,974 Intuitively, why should that be? 238 00:12:18,974 --> 00:12:21,340 AUDIENCE: [INAUDIBLE] 239 00:12:21,340 --> 00:12:23,662 PROFESSOR: A little louder. 240 00:12:23,662 --> 00:12:25,120 AUDIENCE: You have nothing to bind. 241 00:12:25,120 --> 00:12:26,078 PROFESSOR: Yeah, right. 242 00:12:26,078 --> 00:12:30,120 So if there's no S around at all, 243 00:12:30,120 --> 00:12:33,130 then you shouldn't have much of this complex, right? 244 00:12:33,130 --> 00:12:35,220 But you still have some enzymes. 245 00:12:35,220 --> 00:12:38,750 This thing should go to 0, and that kind of makes sense. 246 00:12:38,750 --> 00:12:46,470 And, similarly, if you add a lot, a lot of this substrate? 247 00:12:46,470 --> 00:12:49,478 I'll give you eight seconds. 248 00:12:49,478 --> 00:12:52,160 AUDIENCE: So, you're moving the substrate [INAUDIBLE]. 249 00:12:52,160 --> 00:12:54,900 PROFESSOR: Yes, so S total, this is the total. 250 00:12:54,900 --> 00:12:56,800 This is if you have a test tube, and this 251 00:12:56,800 --> 00:13:03,040 is the amount of this sugar that you add in there. 252 00:13:03,040 --> 00:13:09,170 So S total, then, is the sum of that and S. 253 00:13:09,170 --> 00:13:12,450 Do you need more time? 254 00:13:12,450 --> 00:13:12,950 Ready. 255 00:13:12,950 --> 00:13:16,640 Three, two, one. 256 00:13:16,640 --> 00:13:21,760 So we have maybe an island of people that disagree. 257 00:13:25,370 --> 00:13:28,070 At least a majority are saying, in this case, 258 00:13:28,070 --> 00:13:31,120 it should have to go to 1. 259 00:13:31,120 --> 00:13:35,454 Of course, this is a limit and a limit. 260 00:13:35,454 --> 00:13:37,870 It's always going to be between 0 and 1, but in the limit, 261 00:13:37,870 --> 00:13:38,890 it does go to 1. 262 00:13:38,890 --> 00:13:41,560 So if you just add so much of the substrate 263 00:13:41,560 --> 00:13:44,240 then you should be able to saturate that binding 264 00:13:44,240 --> 00:13:47,210 and drive all of that enzyme into the bound state. 265 00:13:50,900 --> 00:13:54,460 So this one is actually, maybe, a little bit more subtle. 266 00:13:54,460 --> 00:14:02,627 So what if we take this limit? 267 00:14:02,627 --> 00:14:04,960 I'm going to give you 20 seconds to think about it, just 268 00:14:04,960 --> 00:14:08,446 because it's roughly three times as hard as the last one. 269 00:14:34,454 --> 00:14:35,370 Do you need more time? 270 00:14:35,370 --> 00:14:37,660 Or do you think that you have something you believe in, 271 00:14:37,660 --> 00:14:39,493 that you're willing to turn to your neighbor 272 00:14:39,493 --> 00:14:41,990 and-- Let's see where we are. 273 00:14:41,990 --> 00:14:42,840 Ready. 274 00:14:42,840 --> 00:14:44,445 Three, two, one. 275 00:14:47,426 --> 00:14:48,550 All right, so this is good. 276 00:14:48,550 --> 00:14:54,170 So we have a fair distribution, and this is one 277 00:14:54,170 --> 00:14:57,970 that reasonable people might be able to argue about. 278 00:14:57,970 --> 00:15:00,890 It's worth having the argument. 279 00:15:00,890 --> 00:15:03,040 Give yourself 30 seconds, turn your neighbor, 280 00:15:03,040 --> 00:15:05,280 preferably a neighbor that disagrees with you, 281 00:15:05,280 --> 00:15:07,780 and tell them why you said what you said. 282 00:16:08,890 --> 00:16:10,640 If in a pair, you've already convinced 283 00:16:10,640 --> 00:16:12,890 each other of something, then go ahead and look around 284 00:16:12,890 --> 00:16:15,000 to see if there's another pair that has maybe 285 00:16:15,000 --> 00:16:16,380 settled on a different answer. 286 00:16:43,990 --> 00:16:47,380 I think that you guys are still kind of passionately arguing, 287 00:16:47,380 --> 00:16:50,800 but maybe we'll go ahead and convene, 288 00:16:50,800 --> 00:16:55,379 and try to get a sense of-- I think, from the sound of it 289 00:16:55,379 --> 00:16:57,420 at least, there's some disagreement about the way 290 00:16:57,420 --> 00:17:00,390 to think about this. 291 00:17:00,390 --> 00:17:01,890 In general, if you want to volunteer 292 00:17:01,890 --> 00:17:04,780 an opinion or an explanation, what I like to do is, 293 00:17:04,780 --> 00:17:08,440 I like to tell the group what your neighbor thought. 294 00:17:08,440 --> 00:17:13,276 So go ahead, anybody, it could be a neighbor in quotes. 295 00:17:13,276 --> 00:17:15,359 Anybody want to volunteer one possible explanation 296 00:17:15,359 --> 00:17:18,109 of how to think about this? 297 00:17:18,109 --> 00:17:21,109 AUDIENCE: Well, what my neighbors thought 298 00:17:21,109 --> 00:17:27,440 was if E total is defined as E plus ES, 299 00:17:27,440 --> 00:17:33,450 then as E total goes to 0, then this goes to 0 over 0 300 00:17:33,450 --> 00:17:37,930 at some vague, unclear-- 301 00:17:37,930 --> 00:17:39,700 PROFESSOR: Yeah, although, right. 302 00:17:39,700 --> 00:17:41,750 So you're saying, maybe it's all going to 0, 303 00:17:41,750 --> 00:17:43,580 and then this is just philosophy. 304 00:17:45,984 --> 00:17:48,150 Not that I'm putting words in your neighbor's mouth. 305 00:17:52,040 --> 00:17:55,885 So there's a sense in which this is true, but mathematically, 306 00:17:55,885 --> 00:17:59,630 and actually also physically, there are well-defined ways 307 00:17:59,630 --> 00:18:01,430 of taking such a limit, right? 308 00:18:01,430 --> 00:18:05,350 So if you get 0 over 0, then you can use L'Hopital's Rule, 309 00:18:05,350 --> 00:18:07,840 which we'll have the opportunity to pull out sometime 310 00:18:07,840 --> 00:18:08,710 within the class. 311 00:18:08,710 --> 00:18:10,395 Of course, the most difficult part of L'Hopital's Rule 312 00:18:10,395 --> 00:18:11,760 is knowing how to spell it. 313 00:18:21,214 --> 00:18:23,130 So that's one answer, the mathematical answer, 314 00:18:23,130 --> 00:18:25,860 that you should be able to just take this limit and so forth. 315 00:18:25,860 --> 00:18:28,110 But I think there is another physical answer, which is 316 00:18:28,110 --> 00:18:30,870 that this could happen, right? 317 00:18:30,870 --> 00:18:36,206 And something is going to occur, right? 318 00:18:36,206 --> 00:18:37,580 You could, in principle, measure. 319 00:18:37,580 --> 00:18:39,980 And even if you just had a single enzyme there, 320 00:18:39,980 --> 00:18:45,170 this Fb would be the fraction of time that that enzyme is bound. 321 00:18:45,170 --> 00:18:49,820 So this is a well-defined experimental question 322 00:18:49,820 --> 00:18:52,430 and the answer should arise from these interactions. 323 00:18:56,150 --> 00:19:00,290 But then how do we edit it-- what does it all mean? 324 00:19:00,290 --> 00:19:02,850 How do we figure out the answer? 325 00:19:02,850 --> 00:19:06,930 What's another possible view on this? 326 00:19:06,930 --> 00:19:08,242 Maybe in the back. 327 00:19:08,242 --> 00:19:09,700 AUDIENCE: My neighbor thought that, 328 00:19:09,700 --> 00:19:12,694 if there's a non-zero concentration of S, 329 00:19:12,694 --> 00:19:15,480 and the concentration of E goes to 0, 330 00:19:15,480 --> 00:19:18,047 then all E will be bound [? at some point ?]. 331 00:19:18,047 --> 00:19:19,630 PROFESSOR: This is interesting, right? 332 00:19:19,630 --> 00:19:22,800 So, if there's a finite concentration of S, 333 00:19:22,800 --> 00:19:24,870 if E goes to 0, then you say well there's 334 00:19:24,870 --> 00:19:28,934 plenty of S to go around, so I should always get bound. 335 00:19:28,934 --> 00:19:35,326 Is that what the neighbor-- that's another option. 336 00:19:35,326 --> 00:19:36,700 So, so far, we've had an argument 337 00:19:36,700 --> 00:19:40,630 for can't be determined, it's philosophy. 338 00:19:40,630 --> 00:19:44,800 We've had an argument for 1. 339 00:19:44,800 --> 00:19:45,870 Other possibilities? 340 00:19:45,870 --> 00:19:49,694 This is an interesting question, because depending 341 00:19:49,694 --> 00:19:51,860 on how you think about it, you can convince yourself 342 00:19:51,860 --> 00:19:54,670 that it's anything, right? 343 00:19:54,670 --> 00:19:55,726 Other possible answers? 344 00:19:59,270 --> 00:20:00,695 No. 345 00:20:00,695 --> 00:20:02,595 AUDIENCE: So, I don't hear the good answer. 346 00:20:05,868 --> 00:20:10,206 I said E because, when there isn't very much E, 347 00:20:10,206 --> 00:20:19,035 then it's true that a lot of people go into yes, but yeah. 348 00:20:19,035 --> 00:20:20,520 I don't know what to say. 349 00:20:24,490 --> 00:20:26,870 PROFESSOR: Now, that's OK. 350 00:20:26,870 --> 00:20:29,312 Another take on that answer, or a different one? 351 00:20:29,312 --> 00:20:30,270 AUDIENCE: I don't know. 352 00:20:30,270 --> 00:20:31,644 If I had to complete that answer, 353 00:20:31,644 --> 00:20:34,210 maybe something like, but there still might not 354 00:20:34,210 --> 00:20:36,480 be a total large concentration of either, 355 00:20:36,480 --> 00:20:38,736 so there might still be a decent chance for E 356 00:20:38,736 --> 00:20:41,434 to be around for a while without encountering some of S. 357 00:20:41,434 --> 00:20:43,100 AUDIENCE 2: So the forward reaction rate 358 00:20:43,100 --> 00:20:45,400 will also go to 0. 359 00:20:45,400 --> 00:20:47,650 Even though S is very large, the forward reaction rate 360 00:20:47,650 --> 00:20:50,290 will also go to 0, as the concentration [INAUDIBLE]. 361 00:20:50,290 --> 00:20:51,940 PROFESSOR: By the forward rate, it's 362 00:20:51,940 --> 00:20:53,481 not necessarily a variable. it's just 363 00:20:53,481 --> 00:20:56,197 that it depends on the substrate concentration in this case. 364 00:21:00,830 --> 00:21:01,765 Other takes on it? 365 00:21:07,060 --> 00:21:08,810 This is interesting. 366 00:21:08,810 --> 00:21:12,750 So I'm going to argue that the most reasonable way to view 367 00:21:12,750 --> 00:21:14,970 this would give you-- as long as there 368 00:21:14,970 --> 00:21:17,620 is some finite concentration of that substrate around, 369 00:21:17,620 --> 00:21:20,840 and I think that there's a very well-defined sense in which 370 00:21:20,840 --> 00:21:24,170 it's going to go to some finite fraction. 371 00:21:24,170 --> 00:21:26,990 And I think that when you're thinking about this 372 00:21:26,990 --> 00:21:32,250 in the context of molecular kinetics, the chemistry of you, 373 00:21:32,250 --> 00:21:34,660 it still is all well-defined, but the way 374 00:21:34,660 --> 00:21:37,130 that I think that I get the most clear intuition is just 375 00:21:37,130 --> 00:21:41,480 to imagine myself as being that one and only enzyme in the test 376 00:21:41,480 --> 00:21:42,870 tube. 377 00:21:42,870 --> 00:21:44,600 Now there's going to be some rate that I 378 00:21:44,600 --> 00:21:48,000 bind to the substrates, right? 379 00:21:48,000 --> 00:21:49,910 And what's going to determine that rate? 380 00:21:53,776 --> 00:21:55,740 AUDIENCE: The concentration of the substrate? 381 00:21:55,740 --> 00:21:57,781 PROFESSOR: The concentration of substrate, right. 382 00:21:57,781 --> 00:21:59,690 So if I double the substrate concentration, 383 00:21:59,690 --> 00:22:02,690 what should that do to the rate of me binding? 384 00:22:02,690 --> 00:22:03,875 It should double it, yeah. 385 00:22:03,875 --> 00:22:05,250 And then of course there's always 386 00:22:05,250 --> 00:22:07,700 this Kf somewhere in there, and some units, right? 387 00:22:07,700 --> 00:22:10,070 But there's going to be some rate that I bind. 388 00:22:10,070 --> 00:22:11,550 And then when I bind again? 389 00:22:11,550 --> 00:22:14,180 Now I'm just an enzyme substrate. 390 00:22:14,180 --> 00:22:16,739 Now, instead thinking about this in the context of chemical 391 00:22:16,739 --> 00:22:18,280 kinetics, I can just think about this 392 00:22:18,280 --> 00:22:20,920 from the standpoint of an individual molecule, where 393 00:22:20,920 --> 00:22:24,080 I'm a complex, S-bound, and there's some rate 394 00:22:24,080 --> 00:22:25,680 that I fall apart. 395 00:22:25,680 --> 00:22:27,640 And it's just the balance of those two rates 396 00:22:27,640 --> 00:22:30,050 of finding a substrate and falling part 397 00:22:30,050 --> 00:22:32,430 that's going to lead to this fraction bound. 398 00:22:35,150 --> 00:22:36,858 AUDIENCE: Taking what you just explained, 399 00:22:36,858 --> 00:22:41,425 couldn't you think about it as E always either 0 or 1, 400 00:22:41,425 --> 00:22:42,300 because when you're-- 401 00:22:42,300 --> 00:22:42,625 PROFESSOR: Yeah 402 00:22:42,625 --> 00:22:45,041 AUDIENCE: --in the case of having one [? and the same-- ?] 403 00:22:45,041 --> 00:22:46,020 PROFESSOR: Yes. 404 00:22:46,020 --> 00:22:51,290 So, if you'd like, we could put an average sign here, 405 00:22:51,290 --> 00:22:52,821 to say an average over time. 406 00:22:52,821 --> 00:22:55,320 Because in many, many cases, what we're really interested in 407 00:22:55,320 --> 00:22:58,872 is the fraction of time that, say, the promoter is bound, 408 00:22:58,872 --> 00:23:00,330 or the enzyme is bound, or whatnot. 409 00:23:00,330 --> 00:23:04,550 And if you have many, many, many molecules, then 410 00:23:04,550 --> 00:23:07,911 at any moment in time, the average is the time average. 411 00:23:07,911 --> 00:23:09,660 But once you're down to a single molecule, 412 00:23:09,660 --> 00:23:11,493 then you really want to take a time average. 413 00:23:17,445 --> 00:23:20,967 We'll revisit this in just a moment using the math, 414 00:23:20,967 --> 00:23:23,300 but what's interesting is the math also can mislead you. 415 00:23:26,680 --> 00:23:29,890 Other questions? 416 00:23:29,890 --> 00:23:32,582 If somebody wants to argue forcefully 417 00:23:32,582 --> 00:23:34,040 what their neighbor said was right, 418 00:23:34,040 --> 00:23:42,420 then I'm happy to-- we will come back to this in a little bit. 419 00:23:42,420 --> 00:23:44,910 If I want to do the fourth possibility, which 420 00:23:44,910 --> 00:23:47,697 is the total concentration of the enzyme, 421 00:23:47,697 --> 00:23:48,905 it's going to go to infinity. 422 00:23:52,100 --> 00:23:55,270 I'll give you, again, eight seconds to think and then 423 00:23:55,270 --> 00:23:56,860 get your card ready. 424 00:24:10,450 --> 00:24:12,380 Are you ready? 425 00:24:12,380 --> 00:24:13,870 Three, two, one. 426 00:24:16,461 --> 00:24:16,960 OK. 427 00:24:16,960 --> 00:24:21,100 So we actually have a fair amount of disagreement, 428 00:24:21,100 --> 00:24:25,260 between As and Bs it seems. 429 00:24:25,260 --> 00:24:27,890 Go ahead and, again, turn to your neighbor, 430 00:24:27,890 --> 00:24:31,760 but maybe find somebody that disagrees with you. 431 00:24:31,760 --> 00:24:33,565 Sometimes there are pockets of people that 432 00:24:33,565 --> 00:24:34,690 agree one way or the other. 433 00:24:34,690 --> 00:24:38,140 So try to find each other. 434 00:25:10,560 --> 00:25:11,570 Yeah, I know. 435 00:25:11,570 --> 00:25:12,162 I understand. 436 00:25:14,740 --> 00:25:18,280 You can try to figure out the expression for the fraction 437 00:25:18,280 --> 00:25:23,409 bound, and how that behaves. 438 00:25:23,409 --> 00:25:25,534 And that's actually kind of weird as well, frankly. 439 00:25:32,450 --> 00:25:35,707 Why don't we go ahead and reconvene, just so I can see. 440 00:25:35,707 --> 00:25:37,290 And maybe, let's go ahead and re-vote, 441 00:25:37,290 --> 00:25:38,870 so I can see if anybody convinced 442 00:25:38,870 --> 00:25:41,050 anybody of anything else. 443 00:25:41,050 --> 00:25:41,640 Ready. 444 00:25:41,640 --> 00:25:44,570 Three, two, one. 445 00:25:44,570 --> 00:25:47,035 So it seems like now there's pretty good agreement. 446 00:25:47,035 --> 00:25:48,410 The answer to this is going to be 447 00:25:48,410 --> 00:25:52,880 A. There were a fair number of people that said B before. 448 00:25:52,880 --> 00:25:54,724 So here comes our curvy lines. 449 00:25:57,830 --> 00:26:00,890 And so the idea here is that in the limit 450 00:26:00,890 --> 00:26:03,410 of the enzyme concentration going to infinity, 451 00:26:03,410 --> 00:26:05,740 in that limit the fraction bound has to go to 0, 452 00:26:05,740 --> 00:26:09,170 because you just don't have any substrate to bind. 453 00:26:09,170 --> 00:26:12,016 And even if all the substrate's bound-- and indeed 454 00:26:12,016 --> 00:26:13,890 in this limit, what fraction of the substrate 455 00:26:13,890 --> 00:26:16,051 ends up being bound? 456 00:26:16,051 --> 00:26:17,110 AUDIENCE: All of it. 457 00:26:17,110 --> 00:26:19,090 PROFESSOR: All of it, right. 458 00:26:19,090 --> 00:26:20,760 So then, indeed, the fraction bound 459 00:26:20,760 --> 00:26:21,780 is just going to be the concentration 460 00:26:21,780 --> 00:26:24,279 of the substrate divided by the concentration of the enzyme. 461 00:26:27,895 --> 00:26:29,270 In the limit of the concentration 462 00:26:29,270 --> 00:26:31,353 of the enzyme going to infinity, then the fraction 463 00:26:31,353 --> 00:26:33,470 of the enzyme bound is going to go to 0. 464 00:26:33,470 --> 00:26:34,550 You're unhappy. 465 00:26:34,550 --> 00:26:38,090 AUDIENCE: I think having made an extremely eloquent argument, 466 00:26:38,090 --> 00:26:42,560 number 3 for B, I actually think that that answer is wrong. 467 00:26:42,560 --> 00:26:44,020 PROFESSOR: OK. 468 00:26:44,020 --> 00:26:45,664 So this one, right? 469 00:26:45,664 --> 00:26:47,080 Yes, I mean, you had convinced me. 470 00:26:51,450 --> 00:26:56,160 AUDIENCE: So if you take E total to 0 at fixed S-- 471 00:26:56,160 --> 00:26:57,160 PROFESSOR: Fixed S, yes. 472 00:26:57,160 --> 00:27:01,228 AUDIENCE: --then the reaction rate is-- 473 00:27:04,306 --> 00:27:06,722 PROFESSOR: You have to take the limits carefully, I think. 474 00:27:06,722 --> 00:27:10,658 AUDIENCE: --is a ratio of the forward propensity 475 00:27:10,658 --> 00:27:12,134 and the backwards propensity. 476 00:27:14,894 --> 00:27:15,560 PROFESSOR: Yeah. 477 00:27:18,242 --> 00:27:20,200 I think that the clearest way to think about it 478 00:27:20,200 --> 00:27:22,460 is as just that single enzyme. 479 00:27:22,460 --> 00:27:25,632 Because, it's certainly going to have some rate of binding, 480 00:27:25,632 --> 00:27:27,090 and then once it's bound it's going 481 00:27:27,090 --> 00:27:28,548 to have some rate of falling apart. 482 00:27:30,565 --> 00:27:32,690 And that ratio is not a function of whether there's 483 00:27:32,690 --> 00:27:35,230 one enzyme-- I mean, that ratio is well-defined. 484 00:27:38,350 --> 00:27:40,890 The time average of the probability 485 00:27:40,890 --> 00:27:42,710 of that enzyme being bound is, indeed, 486 00:27:42,710 --> 00:27:43,710 a well-defined quantity. 487 00:27:46,320 --> 00:27:50,080 AUDIENCE: So can Ethan say D and it depends on the Kd? 488 00:27:50,080 --> 00:27:54,270 PROFESSOR: Well, people always like to argue Ds. 489 00:27:58,170 --> 00:28:00,590 We can write down what the expression is, actually, now. 490 00:28:00,590 --> 00:28:05,560 And then you can decide whether you think D is justified. 491 00:28:05,560 --> 00:28:07,310 We can also just try to figure out, 492 00:28:07,310 --> 00:28:09,250 what's the equilibrium of this thing? 493 00:28:09,250 --> 00:28:14,972 So the change in the complex concentration 494 00:28:14,972 --> 00:28:16,430 as a function of time is just going 495 00:28:16,430 --> 00:28:19,020 to be the rate of creation minus the rate of destruction. 496 00:28:19,020 --> 00:28:21,640 So there's just going to be this K forward, 497 00:28:21,640 --> 00:28:35,470 ES-- oh sorry, these brackets are awful-- Kr, ES. 498 00:28:35,470 --> 00:28:37,320 And we just want to set this equal to 0, 499 00:28:37,320 --> 00:28:40,950 if we want to figure out what the equilibrium there is. 500 00:28:40,950 --> 00:28:46,840 And in one line, you can find that the fraction bound can 501 00:28:46,840 --> 00:28:50,860 be written as a concentration of the substrate here, 502 00:28:50,860 --> 00:28:56,996 divided by Kd, plus S. Is this correct from standpoint 503 00:28:56,996 --> 00:28:57,495 of units? 504 00:29:04,440 --> 00:29:09,300 So there's something that you might find troubling 505 00:29:09,300 --> 00:29:11,114 about this expression, though. 506 00:29:15,650 --> 00:29:17,090 Is anybody troubled by it? 507 00:29:19,850 --> 00:29:22,317 AUDIENCE: But it equals 0. 508 00:29:22,317 --> 00:29:24,275 PROFESSOR: You could say, what if E total is 0? 509 00:29:27,820 --> 00:29:31,760 I'd say that if E total is actually zero, now 510 00:29:31,760 --> 00:29:33,430 I think that's an ill-defined quantity. 511 00:29:33,430 --> 00:29:36,620 So, at some point, I'll side with the philosophers here. 512 00:29:36,620 --> 00:29:41,160 But if there is an enzyme to talk about fraction bound, 513 00:29:41,160 --> 00:29:44,840 then-- So it's related to this, but it's-- 514 00:29:47,756 --> 00:29:52,032 AUDIENCE: But whenever you have [INAUDIBLE] E [INAUDIBLE]. 515 00:29:52,032 --> 00:29:52,740 PROFESSOR: Right. 516 00:29:52,740 --> 00:29:55,730 So we've already discussed from our intuition 517 00:29:55,730 --> 00:29:59,390 that as E total goes to infinity, 518 00:29:59,390 --> 00:30:03,260 then the fraction bound is supposed to go what? 519 00:30:03,260 --> 00:30:05,360 To 0, we decided, right? 520 00:30:05,360 --> 00:30:08,436 And does this expression do that? 521 00:30:08,436 --> 00:30:10,266 AUDIENCE: Well this is S, not S total. 522 00:30:10,266 --> 00:30:10,890 PROFESSOR: Yes! 523 00:30:10,890 --> 00:30:12,360 So this is S, not S total. 524 00:30:12,360 --> 00:30:16,940 And once again, really easy to screw this up. 525 00:30:16,940 --> 00:30:22,790 Because, in many contexts, S and S total are the same thing. 526 00:30:22,790 --> 00:30:24,850 Especially, in context of enzyme kinetics, 527 00:30:24,850 --> 00:30:27,880 it's often the case that the concentration enzyme is really 528 00:30:27,880 --> 00:30:30,680 rather low, and then the substrate concentration 529 00:30:30,680 --> 00:30:33,630 is huge, so then S and S total we can really 530 00:30:33,630 --> 00:30:37,090 treat as being interchangeable. 531 00:30:37,090 --> 00:30:41,100 But here, in the general context, we can't. 532 00:30:41,100 --> 00:30:44,210 And this concentration of S, this thing 533 00:30:44,210 --> 00:30:49,470 is a function of the concentration of the enzyme. 534 00:30:49,470 --> 00:30:52,590 So this guy here, it's a function 535 00:30:52,590 --> 00:30:56,100 of the total amount of substrate you have 536 00:30:56,100 --> 00:31:00,695 and also E total, and for that matter, Kd. 537 00:31:06,960 --> 00:31:11,180 So really this thing is a true statement, 538 00:31:11,180 --> 00:31:14,337 but it's very misleading if you're not keeping track 539 00:31:14,337 --> 00:31:15,420 of what these things mean. 540 00:31:15,420 --> 00:31:19,594 Because, this thing is very simple, except for that 541 00:31:19,594 --> 00:31:22,010 it's really actually complicated because S section depends 542 00:31:22,010 --> 00:31:24,720 on everything right? 543 00:31:24,720 --> 00:31:26,220 Now there's one context in which you 544 00:31:26,220 --> 00:31:29,470 can be safe in assuming that S and S total are the same, 545 00:31:29,470 --> 00:31:32,540 and that's in the limit of, for example, E total going to 0. 546 00:31:32,540 --> 00:31:38,080 So if you just have 1 enzyme, then this expression 547 00:31:38,080 --> 00:31:41,400 is, essentially, always valid. 548 00:31:41,400 --> 00:31:43,344 Every now and then, you might be using up 549 00:31:43,344 --> 00:31:45,260 one of your substrate molecules, occasionally. 550 00:31:45,260 --> 00:31:46,240 Right? 551 00:31:46,240 --> 00:31:49,050 But it's pretty safe to say that in the limit of E total 552 00:31:49,050 --> 00:31:51,600 going 0, when it's just in the limit of one enzyme, 553 00:31:51,600 --> 00:31:54,130 then it's really just described by a curve that 554 00:31:54,130 --> 00:31:56,110 looks like this. 555 00:31:56,110 --> 00:31:59,220 And this curve is something that you see over, and over, 556 00:31:59,220 --> 00:32:02,080 and over again. 557 00:32:02,080 --> 00:32:03,720 And this is the fundamental reason 558 00:32:03,720 --> 00:32:08,020 that Michaelis Menten kinetics looks the way it does. 559 00:32:08,020 --> 00:32:10,390 But this is going to be very useful for us 560 00:32:10,390 --> 00:32:13,770 because, in many contexts that we're interested in, 561 00:32:13,770 --> 00:32:15,810 we want to think about, for example, 562 00:32:15,810 --> 00:32:19,610 the rate expression of some gene. 563 00:32:19,610 --> 00:32:22,180 And what we want to know about is the fraction of time 564 00:32:22,180 --> 00:32:25,890 that it's going to be bound by, say, a transcription factor. 565 00:32:25,890 --> 00:32:27,530 So then, in the simplest case, we 566 00:32:27,530 --> 00:32:31,010 get an input-output relationship that is just given by this. 567 00:32:31,010 --> 00:32:34,120 Because, there's just one, or few copies, of that DNA, 568 00:32:34,120 --> 00:32:36,740 so it doesn't really sequester the transcription factor 569 00:32:36,740 --> 00:32:37,990 that's going to be binding it. 570 00:32:43,810 --> 00:32:48,640 Do you guys understand why this is weird? 571 00:32:48,640 --> 00:32:50,240 A thing you have to be careful of? 572 00:32:54,040 --> 00:32:56,770 And more generally, I strongly recommend 573 00:32:56,770 --> 00:32:59,390 that, in all of these sorts of problems, 574 00:32:59,390 --> 00:33:02,884 it's good to just plot some things. 575 00:33:02,884 --> 00:33:04,300 For example, the fraction bound is 576 00:33:04,300 --> 00:33:08,850 a function of if you vary the substrate concentration. 577 00:33:08,850 --> 00:33:12,420 Because, often you think that you know what's going on, 578 00:33:12,420 --> 00:33:15,460 and then when you just go and sit down to draw some curve, 579 00:33:15,460 --> 00:33:17,070 just get your intuition, you realize 580 00:33:17,070 --> 00:33:18,860 you don't know where it starts, you don't where it ends, 581 00:33:18,860 --> 00:33:20,630 you don't know what it does in between. 582 00:33:20,630 --> 00:33:22,010 It's embarrassing, but it's only when 583 00:33:22,010 --> 00:33:24,301 you sit down and try to do something like that that you 584 00:33:24,301 --> 00:33:26,450 realize that it's not obvious. 585 00:33:26,450 --> 00:33:29,700 So just, for example, it's useful to imagine a situation, 586 00:33:29,700 --> 00:33:32,280 just between a similar E and S, where 587 00:33:32,280 --> 00:33:36,390 we, for the sake of argument, say 588 00:33:36,390 --> 00:33:39,690 that S total is around this Kd. 589 00:33:43,380 --> 00:33:47,460 Now what we want to do is ask, what's the fraction 590 00:33:47,460 --> 00:34:00,000 bound as a function of E total? 591 00:34:00,000 --> 00:34:02,530 So we're going to fix total substrate, 592 00:34:02,530 --> 00:34:05,555 vary the enzyme concentration. 593 00:34:09,270 --> 00:34:11,770 So we already know what the limit here should be. 594 00:34:11,770 --> 00:34:13,560 We go to infinity, what should this go to? 595 00:34:17,429 --> 00:34:18,651 0. 596 00:34:18,651 --> 00:34:20,400 We know that eventually it should go to 0, 597 00:34:20,400 --> 00:34:24,360 and we already figured that out for any finite substrate 598 00:34:24,360 --> 00:34:25,330 concentration. 599 00:34:25,330 --> 00:34:28,739 Incidentally, on many of the exams, 600 00:34:28,739 --> 00:34:33,210 I will ask for plots of curves like this. 601 00:34:33,210 --> 00:34:36,940 So basically you want to indicate 602 00:34:36,940 --> 00:34:38,941 where it goes on one end, where it starts-- 603 00:34:38,941 --> 00:34:41,440 and where is it going to start in the limit of E total going 604 00:34:41,440 --> 00:34:44,239 to 0? 605 00:34:44,239 --> 00:34:46,460 Half. 606 00:34:46,460 --> 00:34:50,489 Then it's actually accurately described by that. 607 00:34:50,489 --> 00:34:51,250 So we start here. 608 00:34:51,250 --> 00:34:52,210 This is 1. 609 00:34:52,210 --> 00:34:55,199 So we start at 1/2. 610 00:34:55,199 --> 00:34:59,590 And it's going to have to go in between those two, right? 611 00:34:59,590 --> 00:35:02,480 So what's the characteristic concentration here 612 00:35:02,480 --> 00:35:05,360 where something-- where it's changing a lot? 613 00:35:14,162 --> 00:35:15,107 AUDIENCE: Kd? 614 00:35:15,107 --> 00:35:15,940 PROFESSOR: Yeah, Kd. 615 00:35:15,940 --> 00:35:18,470 Kd's actually the only concentration in the problem. 616 00:35:18,470 --> 00:35:20,260 So that means that's what sets scale. 617 00:35:20,260 --> 00:35:22,340 And I don't know, Kd, exactly where it should 618 00:35:22,340 --> 00:35:24,680 be, but something in there. 619 00:35:24,680 --> 00:35:26,980 So in these sorts of situations, you 620 00:35:26,980 --> 00:35:30,910 want to get the limits and what is it that sets the scale? 621 00:35:30,910 --> 00:35:32,960 If there's a peak, is where is it? 622 00:35:37,390 --> 00:35:39,639 I encourage for a few toy examples, 623 00:35:39,639 --> 00:35:40,930 just draw some of these things. 624 00:35:40,930 --> 00:35:45,460 It's a fun way to spend a Saturday afternoon. 625 00:35:45,460 --> 00:35:49,215 Are there any questions about what we've said so far? 626 00:35:49,215 --> 00:35:49,715 Yes. 627 00:35:49,715 --> 00:35:53,352 AUDIENCE: What's the Fb when E total equals Kd? 628 00:35:53,352 --> 00:35:54,810 PROFESSOR: So the question is, what 629 00:35:54,810 --> 00:35:57,790 is the fraction of the enzyme that's bound 630 00:35:57,790 --> 00:36:03,270 when E total is equal to Kd? 631 00:36:03,270 --> 00:36:06,790 I think we could figure it out, but it might actually 632 00:36:06,790 --> 00:36:09,770 be a little bit of math. 633 00:36:09,770 --> 00:36:10,270 Let me see. 634 00:36:14,980 --> 00:36:17,070 I would have to think about it, but it's 635 00:36:17,070 --> 00:36:21,070 going to be around a third or a fifth, somewhere in there. 636 00:36:24,117 --> 00:36:25,950 If somebody gets bored with what I'm saying, 637 00:36:25,950 --> 00:36:27,590 they can do the calculation and report to us 638 00:36:27,590 --> 00:36:28,423 at the end of class. 639 00:36:34,830 --> 00:36:41,120 So what we've just done, it feels like a lot of time 640 00:36:41,120 --> 00:36:44,000 to spend on two molecules binding to each other, 641 00:36:44,000 --> 00:36:46,020 but I think that it's good to just make sure 642 00:36:46,020 --> 00:36:48,629 that you're comfortable with the simplest kind of process 643 00:36:48,629 --> 00:36:50,420 before you start thinking about things that 644 00:36:50,420 --> 00:36:53,650 are super complicated, for example, Michaelis Menten 645 00:36:53,650 --> 00:36:55,030 kinetics. 646 00:36:55,030 --> 00:36:57,380 So it's not super complicated. 647 00:36:57,380 --> 00:37:01,260 E plus S. So now it's the same thing here, 648 00:37:01,260 --> 00:37:05,760 where we have K forward, K reverse, to this complex. 649 00:37:05,760 --> 00:37:13,290 But here, at some rate, Kcat, enzyme does something, 650 00:37:13,290 --> 00:37:15,710 turns it into a product. 651 00:37:15,710 --> 00:37:19,410 Now this is a model of how an enzyme works. 652 00:37:19,410 --> 00:37:26,170 It is not a perfect description of reality in any given case, 653 00:37:26,170 --> 00:37:27,740 or in general. 654 00:37:27,740 --> 00:37:30,560 What's the most obvious possible point of concern? 655 00:37:34,402 --> 00:37:35,860 AUDIENCE: There's no way to go back 656 00:37:35,860 --> 00:37:37,160 PROFESSOR: No way to go back, well that's OK. 657 00:37:37,160 --> 00:37:38,368 What's that matter with that? 658 00:37:38,368 --> 00:37:39,870 AUDIENCE: Well, sometimes there is. 659 00:37:39,870 --> 00:37:40,980 PROFESSOR: Well sometimes there is. 660 00:37:40,980 --> 00:37:41,479 OK. 661 00:37:44,089 --> 00:37:45,630 I'd say the problem is, in some ways, 662 00:37:45,630 --> 00:37:47,691 more fundamental than just, sometimes there is. 663 00:37:47,691 --> 00:37:48,190 Right? 664 00:37:53,239 --> 00:37:55,530 It's true that sometimes-- but sometimes lots of things 665 00:37:55,530 --> 00:37:56,029 happen. 666 00:37:56,029 --> 00:38:01,430 Sometimes the enzyme binds 2 substrates. 667 00:38:01,430 --> 00:38:03,660 On any specific case, there are many ways 668 00:38:03,660 --> 00:38:08,050 that this thing can fail, but there's 669 00:38:08,050 --> 00:38:10,285 a more fundamental sense which is a problem. 670 00:38:10,285 --> 00:38:15,430 AUDIENCE: The rate at which it produces the-- P doesn't depend 671 00:38:15,430 --> 00:38:17,020 on any other small molecules. 672 00:38:17,020 --> 00:38:18,520 It depends on other concentrations-- 673 00:38:18,520 --> 00:38:19,990 PROFESSOR: OK, right. 674 00:38:19,990 --> 00:38:22,124 So what Sam is saying is, well this Kcat 675 00:38:22,124 --> 00:38:23,540 is not a function of other things. 676 00:38:23,540 --> 00:38:25,359 AUDIENCE: Yeah. 677 00:38:25,359 --> 00:38:27,400 PROFESSOR: It's true, and in many cases in might, 678 00:38:27,400 --> 00:38:37,200 but there's a real sense in which this thing is failing 679 00:38:37,200 --> 00:38:39,140 fundamentally for any enzyme. 680 00:38:39,140 --> 00:38:41,848 And I just want to make sure that we're all-- 681 00:38:41,848 --> 00:38:43,560 AUDIENCE: Dissociation of P. 682 00:38:43,560 --> 00:38:45,370 PROFESSOR: Dissociation of P, and what-- 683 00:38:45,370 --> 00:38:47,320 AUDIENCE: From the enzyme. 684 00:38:47,320 --> 00:38:50,500 PROFESSOR: So, you don't like the dissociation? 685 00:38:50,500 --> 00:38:52,374 AUDIENCE: We don't have an association there. 686 00:38:52,374 --> 00:38:54,248 You're assuming that the Kf of disocciation-- 687 00:38:54,248 --> 00:38:55,940 PROFESSOR: Oh, right. 688 00:38:55,940 --> 00:39:00,690 Although, I could argue, Kcat is some kind of bulk parameter 689 00:39:00,690 --> 00:39:03,340 that tells you about the rate of breaking some bond 690 00:39:03,340 --> 00:39:04,570 and dissociating. 691 00:39:04,570 --> 00:39:08,150 And it's just a simple model. 692 00:39:08,150 --> 00:39:10,595 We don't want to ask too much of it. 693 00:39:14,554 --> 00:39:16,842 AUDIENCE: K reverse is huge. 694 00:39:16,842 --> 00:39:18,050 PROFESSOR: K reverse is huge? 695 00:39:18,050 --> 00:39:19,841 Well, I haven't told you what K reverse is. 696 00:39:19,841 --> 00:39:23,100 So, it's not huge. 697 00:39:23,100 --> 00:39:24,020 I mean, it could be. 698 00:39:24,020 --> 00:39:26,061 So far we haven't said anything about what it is. 699 00:39:29,230 --> 00:39:32,540 What are the fundamental properties of an enzyme? 700 00:39:32,540 --> 00:39:34,220 Or a catalyst, for that matter? 701 00:39:38,570 --> 00:39:40,460 When you go home for Thanksgiving, 702 00:39:40,460 --> 00:39:44,030 your grandmother asks you, honey, tell me, 703 00:39:44,030 --> 00:39:46,094 what's a catalyst? 704 00:39:46,094 --> 00:39:48,260 AUDIENCE: It's doesn't get used during the reaction. 705 00:39:48,260 --> 00:39:48,710 PROFESSOR: What's that? 706 00:39:48,710 --> 00:39:50,310 AUDIENCE: It doesn't get used up during the reaction. 707 00:39:50,310 --> 00:39:52,393 PROFESSOR: It doesn't get used up in the reaction. 708 00:39:52,393 --> 00:39:55,110 OK, perfect, not used up. 709 00:39:55,110 --> 00:39:58,084 And does this model violate that? 710 00:39:58,084 --> 00:39:58,890 AUDIENCE: No. 711 00:39:58,890 --> 00:40:00,630 PROFESSOR: So all right. 712 00:40:00,630 --> 00:40:01,500 Grandma's happy. 713 00:40:06,750 --> 00:40:10,145 AUDIENCE: You can deactivate or activate these catalysts? 714 00:40:10,145 --> 00:40:11,610 I don't know. 715 00:40:11,610 --> 00:40:13,350 PROFESSOR: Right, so it's true, there's 716 00:40:13,350 --> 00:40:15,230 some enzymes you can activate, deactivate. 717 00:40:15,230 --> 00:40:20,702 How you deactivate a protein enzyme, if you wanted to? 718 00:40:20,702 --> 00:40:21,670 AUDIENCE: Denature. 719 00:40:21,670 --> 00:40:24,600 PROFESSOR: You could denature it from heat or salt. 720 00:40:24,600 --> 00:40:31,620 But that's maybe not one of the most fundamental. 721 00:40:31,620 --> 00:40:34,766 AUDIENCE: You should have a rate from S to P without the enzyme. 722 00:40:34,766 --> 00:40:37,140 PROFESSOR: Right so maybe there should be a rate, S to P, 723 00:40:37,140 --> 00:40:39,230 without the enzyme. 724 00:40:39,230 --> 00:40:40,970 Although, I'm just trying to tell you 725 00:40:40,970 --> 00:40:43,200 about the rate of what the enzyme is doing, 726 00:40:43,200 --> 00:40:47,620 so you could write a difference equation. 727 00:40:47,620 --> 00:40:51,132 If I just let this model go to infinity, then what happens? 728 00:40:51,132 --> 00:40:52,980 AUDIENCE: You get P. 729 00:40:52,980 --> 00:40:55,800 PROFESSOR: You get P. And how much P is it? 730 00:40:55,800 --> 00:40:56,340 A lot of P? 731 00:40:56,340 --> 00:40:57,090 A little bit of P? 732 00:40:57,090 --> 00:40:58,805 AUDIENCE: As much as it can make. 733 00:40:58,805 --> 00:41:00,370 PROFESSOR: It's all pee. 734 00:41:00,370 --> 00:41:01,170 Right? 735 00:41:01,170 --> 00:41:03,390 And how much substrate? 736 00:41:03,390 --> 00:41:05,250 None, right? 737 00:41:05,250 --> 00:41:08,600 So if I just let this go, you have 0 substrate, all product. 738 00:41:11,210 --> 00:41:11,990 Is that OK? 739 00:41:11,990 --> 00:41:18,160 I mean is that, in general-- I like product. 740 00:41:23,490 --> 00:41:25,330 AUDIENCE: Time. 741 00:41:25,330 --> 00:41:26,661 PROFESSOR: Time? 742 00:41:26,661 --> 00:41:29,455 AUDIENCE: How much time it would make too. 743 00:41:29,455 --> 00:41:30,830 PROFESSOR: Well, we can calculate 744 00:41:30,830 --> 00:41:32,690 what the V is in this model, and then we 745 00:41:32,690 --> 00:41:33,981 could figure out what the time. 746 00:41:36,650 --> 00:41:40,310 But there's something wrong with that. 747 00:41:40,310 --> 00:41:42,880 And normally I wouldn't want to belabor the point, 748 00:41:42,880 --> 00:41:45,105 but it's worth belaboring maybe. 749 00:41:45,105 --> 00:41:45,730 AUDIENCE: Yeah. 750 00:41:45,730 --> 00:41:48,705 If we had a back reaction, then that wouldn't happen. 751 00:41:48,705 --> 00:41:50,330 PROFESSOR: Right, OK, so this gets back 752 00:41:50,330 --> 00:41:52,490 to your back reaction, right? 753 00:41:52,490 --> 00:41:54,260 And I like the back rate. 754 00:41:54,260 --> 00:41:56,790 It's just there was something a little more 755 00:41:56,790 --> 00:42:00,540 fundamental than the way you phrased it, was my concern. 756 00:42:00,540 --> 00:42:04,250 Because what you said is, there might be some back rate. 757 00:42:04,250 --> 00:42:05,020 Right? 758 00:42:05,020 --> 00:42:09,960 And I guess what I would say is that there's kind of always 759 00:42:09,960 --> 00:42:13,850 some back rate, or that the equilibrium-- this is 760 00:42:13,850 --> 00:42:17,160 fundamental-- the equilibrium ratio between S and P, 761 00:42:17,160 --> 00:42:18,775 how does the enzyme change it? 762 00:42:18,775 --> 00:42:19,650 AUDIENCE: Not at all. 763 00:42:19,650 --> 00:42:21,510 PROFESSOR: It doesn't change it, right? 764 00:42:21,510 --> 00:42:24,950 So if you take the enzyme, invertase, 765 00:42:24,950 --> 00:42:28,050 and you put it in a test tube with sucrose, 766 00:42:28,050 --> 00:42:32,040 it's going to break down almost all that sucrose, really fast. 767 00:42:32,040 --> 00:42:36,530 It's going to speed things up by a factor of 10 to the 5, 768 00:42:36,530 --> 00:42:38,690 or I don't know, by a lot. 769 00:42:38,690 --> 00:42:42,869 But if you leave the test tube for a year, 770 00:42:42,869 --> 00:42:44,660 it comes to an equilibrium with the enzyme. 771 00:42:44,660 --> 00:42:47,490 If you left it in the test you without the enzyme 772 00:42:47,490 --> 00:42:52,010 for a million years, you would get to the same outcome. 773 00:42:52,010 --> 00:42:55,605 You come to some equilibrium between the substrate 774 00:42:55,605 --> 00:42:56,630 and the product. 775 00:42:56,630 --> 00:43:00,200 And that's a function of the kinetics. 776 00:43:00,200 --> 00:43:03,215 There's a delta G and so forth, but the important point 777 00:43:03,215 --> 00:43:06,210 is that the enzyme does not change that equilibrium. 778 00:43:08,739 --> 00:43:10,114 AUDIENCE: I just have a question. 779 00:43:10,114 --> 00:43:13,042 This molecule is effectively [INAUDIBLE]. 780 00:43:13,042 --> 00:43:15,482 If you leave it in a test tube for a million years, 781 00:43:15,482 --> 00:43:16,950 then the ATP will all be consumed. 782 00:43:16,950 --> 00:43:18,300 PROFESSOR: Yes. 783 00:43:18,300 --> 00:43:20,268 AUDIENCE: But if you keep providing 784 00:43:20,268 --> 00:43:22,649 ATP, then you should-- 785 00:43:22,649 --> 00:43:24,940 PROFESSOR: Well, first of all, not all enzymes actually 786 00:43:24,940 --> 00:43:26,005 are coupled to ATP. 787 00:43:26,005 --> 00:43:29,560 So ATP is a way of putting out a big delta G, right? 788 00:43:29,560 --> 00:43:33,180 So that you can really push things far. 789 00:43:33,180 --> 00:43:36,340 And ATP could, in principle, be included as a co-factor, 790 00:43:36,340 --> 00:43:38,670 and then you take the overall delta G of that, 791 00:43:38,670 --> 00:43:40,380 and then calculate it. 792 00:43:40,380 --> 00:43:41,985 But if you want to keep on adding, 793 00:43:41,985 --> 00:43:43,110 then it complicates things. 794 00:43:43,110 --> 00:43:44,545 But I think, for many enzymes, it's more straightforward 795 00:43:44,545 --> 00:43:46,900 just to think about enzymes that don't require 796 00:43:46,900 --> 00:43:48,530 any extra input of energy. 797 00:43:48,530 --> 00:43:50,390 So they're just lowering the energy barrier 798 00:43:50,390 --> 00:43:52,740 and they're just speeding up the rate of reaction. 799 00:43:52,740 --> 00:43:53,900 But the important point there is that they're 800 00:43:53,900 --> 00:43:55,020 speeding up both rates. 801 00:43:57,960 --> 00:44:01,670 So the equilibrium between those two is not going to change. 802 00:44:01,670 --> 00:44:05,550 And that's why, this thing, it's a great model, 803 00:44:05,550 --> 00:44:08,290 but like all models you have to make 804 00:44:08,290 --> 00:44:11,680 sure you keep track of what the assumptions are going into it. 805 00:44:11,680 --> 00:44:14,659 Because this is going to violate the laws of physics 806 00:44:14,659 --> 00:44:16,200 if you take this model too seriously. 807 00:44:18,971 --> 00:44:21,429 AUDIENCE: I don't understand what the fundamental principle 808 00:44:21,429 --> 00:44:22,674 that's being violated is. 809 00:44:22,674 --> 00:44:26,160 Because why is it not that if you have it stable, 810 00:44:26,160 --> 00:44:29,148 everything is product. 811 00:44:29,148 --> 00:44:31,140 You never see as in nature. 812 00:44:31,140 --> 00:44:34,973 I mean, how is that not a physical situation? 813 00:44:34,973 --> 00:44:37,014 AUDIENCE 2: So if you can get a really small test 814 00:44:37,014 --> 00:44:41,365 tube with one G and one S, isn't it just like-- 815 00:44:41,365 --> 00:44:42,740 PROFESSOR: But the same statement 816 00:44:42,740 --> 00:44:44,260 that we talked about for single, then 817 00:44:44,260 --> 00:44:46,490 you would want-- if you had just a single substrate going 818 00:44:46,490 --> 00:44:48,281 to product-- then you want to look probably 819 00:44:48,281 --> 00:44:49,290 at the time average. 820 00:44:49,290 --> 00:44:53,100 Because, the thing is that the equilibrium is determined 821 00:44:53,100 --> 00:44:57,280 by the delta G of the reaction. 822 00:44:57,280 --> 00:45:00,670 And that's going to determine the equilibrium, whether you 823 00:45:00,670 --> 00:45:05,380 have the enzyme there or not. 824 00:45:05,380 --> 00:45:08,960 So if the delta G is such that it's 825 00:45:08,960 --> 00:45:12,110 at equilibrium-- sort of 90% product, 826 00:45:12,110 --> 00:45:14,812 10% substrate-- then what you can do is go, 827 00:45:14,812 --> 00:45:16,520 well if you start out with all substrate, 828 00:45:16,520 --> 00:45:18,240 this model may work wonderfully. 829 00:45:18,240 --> 00:45:20,490 But then as you're getting closer to that equilibrium, 830 00:45:20,490 --> 00:45:22,281 then this model's going to be breaking down 831 00:45:22,281 --> 00:45:23,930 because this model is not accounting 832 00:45:23,930 --> 00:45:26,780 for the back reaction, as you were saying. 833 00:45:26,780 --> 00:45:29,990 But I just want to stress that it's not just a detailed model, 834 00:45:29,990 --> 00:45:32,124 or it's not just a failure for some enzymes, 835 00:45:32,124 --> 00:45:33,540 this is the way that enzymes work. 836 00:45:37,980 --> 00:45:39,536 Are there other questions about this? 837 00:45:39,536 --> 00:45:41,160 Or different ways of thinking about it? 838 00:45:44,375 --> 00:45:45,250 So, it's not used up. 839 00:45:45,250 --> 00:45:51,990 It speeds up reaction in both directions. 840 00:46:03,655 --> 00:46:05,738 AUDIENCE: I mean, but that's not necessarily true. 841 00:46:05,738 --> 00:46:07,702 You can have an enzyme that is only 842 00:46:07,702 --> 00:46:11,160 really capable of going in one direction. 843 00:46:11,160 --> 00:46:12,140 PROFESSOR: Really? 844 00:46:12,140 --> 00:46:15,510 We should meet after class and you can give me your-- 845 00:46:15,510 --> 00:46:18,456 AUDIENCE: It basically binds in a particular direction. 846 00:46:18,456 --> 00:46:19,830 PROFESSOR: It's just not allowed. 847 00:46:26,280 --> 00:46:28,744 So it's true that enzymes can be-- 848 00:46:28,744 --> 00:46:31,410 and this is getting to the other fundamental point of an enzyme, 849 00:46:31,410 --> 00:46:35,090 which is that they, especially enzymes in biology, 850 00:46:35,090 --> 00:46:39,690 can be exquisitely specific. 851 00:46:39,690 --> 00:46:43,420 What you're saying is that it's really 852 00:46:43,420 --> 00:46:46,010 only catalyzing this one, weird reaction, 853 00:46:46,010 --> 00:46:50,380 going from some funny substrate to some funny product, right? 854 00:46:50,380 --> 00:46:55,360 But that enzyme also speeds up that back reaction, 855 00:46:55,360 --> 00:46:59,590 going from the funny product to the funny substrate. 856 00:46:59,590 --> 00:47:04,560 And that's just like the nature of the beast. 857 00:47:04,560 --> 00:47:09,140 I'm try to think of what I can-- 858 00:47:09,140 --> 00:47:11,575 AUDIENCE: That's where you have one enzyme going one way, 859 00:47:11,575 --> 00:47:15,900 and another going the other way in biology. 860 00:47:15,900 --> 00:47:17,510 PROFESSOR: So it does happen there, 861 00:47:17,510 --> 00:47:20,400 but then what they are often doing 862 00:47:20,400 --> 00:47:24,670 is they're coupling things to ATP hydrolysis or something, 863 00:47:24,670 --> 00:47:31,240 in order to actually make that reaction go in the single way. 864 00:47:31,240 --> 00:47:34,440 Just as kind of like a general statement-- because the way 865 00:47:34,440 --> 00:47:40,010 these things work is that there's some over here 866 00:47:40,010 --> 00:47:43,080 and it's over here somehow, and these enzymes, 867 00:47:43,080 --> 00:47:46,236 they just lower this energy barrier. 868 00:47:46,236 --> 00:47:47,736 AUDIENCE: So the thing that confused 869 00:47:47,736 --> 00:47:49,950 me at first is that I was just thinking of rates, 870 00:47:49,950 --> 00:47:52,019 and I think the thing that's important 871 00:47:52,019 --> 00:47:55,931 is to just realize again that the enzyme doesn't change 872 00:47:55,931 --> 00:47:59,090 the thermodynamics, it only changes that variable 873 00:47:59,090 --> 00:48:00,180 to change where they are. 874 00:48:00,180 --> 00:48:02,600 So the key thing is that it doesn't 875 00:48:02,600 --> 00:48:06,071 change the ratio of the product through the substrate, 876 00:48:06,071 --> 00:48:08,020 the rates are realatively-- 877 00:48:08,020 --> 00:48:10,580 PROFESSOR: Right, because from a thermodynamic standpoint, 878 00:48:10,580 --> 00:48:13,350 it's not used up, which means there's an enzyme here 879 00:48:13,350 --> 00:48:15,230 and an enzyme here. 880 00:48:15,230 --> 00:48:17,840 So these final states, you can think about only 881 00:48:17,840 --> 00:48:19,620 in terms of the substrate and the product, 882 00:48:19,620 --> 00:48:23,582 because the enzyme was there in both beginning and ending. 883 00:48:23,582 --> 00:48:25,040 So from a thermodynamic standpoint, 884 00:48:25,040 --> 00:48:26,980 it's just you're not allowed to change 885 00:48:26,980 --> 00:48:28,413 one rate without the other. 886 00:48:32,200 --> 00:48:42,970 Now in the reading, you saw the Michaelis Menten kinetics, 887 00:48:42,970 --> 00:48:48,980 where you found that once you reach 888 00:48:48,980 --> 00:48:52,510 this equilibrium between the enzyme substrate complex, 889 00:48:52,510 --> 00:48:54,330 the velocity can be described by something 890 00:48:54,330 --> 00:48:57,140 that is rather simple. 891 00:48:57,140 --> 00:49:03,820 There's some Km plus S, and then there's some Vmax. 892 00:49:06,500 --> 00:49:08,940 And if the substrate concentration, 893 00:49:08,940 --> 00:49:10,540 the total concentration is very large, 894 00:49:10,540 --> 00:49:14,810 then you can just think about this is the S total. 895 00:49:14,810 --> 00:49:18,520 Now in this case, this, once again, 896 00:49:18,520 --> 00:49:21,450 can be thought of in this limit of if the enzyme concentration 897 00:49:21,450 --> 00:49:23,950 is really small, then this is really just the fraction 898 00:49:23,950 --> 00:49:25,350 of the enzyme that's bound. 899 00:49:29,825 --> 00:49:31,700 So we've already spent a lot of time thinking 900 00:49:31,700 --> 00:49:36,830 about how to get at the fraction bound, 901 00:49:36,830 --> 00:49:42,670 and the question is, what should this Km be here? 902 00:49:42,670 --> 00:49:45,000 Now that I've told you that it's the fraction bound, 903 00:49:45,000 --> 00:49:49,480 is it just going to be the same thing that we had before? 904 00:49:49,480 --> 00:49:51,020 Is the Km the same thing is the Kd? 905 00:49:55,490 --> 00:50:00,075 So remember, before, we found that Kd was just Kr over Kf. 906 00:50:03,710 --> 00:50:07,770 But you should, in principle, be able ti just look at that 907 00:50:07,770 --> 00:50:10,420 and say what fraction bound should be. 908 00:50:13,294 --> 00:50:16,647 AUDIENCE: Is it Kr over Kf plus-- other way around, 909 00:50:16,647 --> 00:50:19,050 Kr plus Kcat over Kf 910 00:50:19,050 --> 00:50:21,475 PROFESSOR: Yes, because now, from the standpoint 911 00:50:21,475 --> 00:50:27,330 of the enzyme, there's some rate at which you form the complex. 912 00:50:27,330 --> 00:50:29,650 And now the lifetime of that complex 913 00:50:29,650 --> 00:50:31,496 has been reduced, because now there 914 00:50:31,496 --> 00:50:33,790 are two ways for the complex to fall apart, right? 915 00:50:33,790 --> 00:50:36,680 One, is could just go back where it came from, 916 00:50:36,680 --> 00:50:40,200 but the other is that you can catalyze the reaction. 917 00:50:40,200 --> 00:50:43,350 So, from the standpoint of the enzyme and the fraction bound, 918 00:50:43,350 --> 00:50:45,330 then we can just-- the entire discussion 919 00:50:45,330 --> 00:50:47,130 that we had before-- we can just replace Kd 920 00:50:47,130 --> 00:50:51,680 with this new Michaelis constant, Km. 921 00:50:51,680 --> 00:50:56,130 Where now, we say now it's the Kr up in the numerator still, 922 00:50:56,130 --> 00:50:58,810 but now, instead of just being Kf at the bottom, 923 00:50:58,810 --> 00:51:00,760 we have to add Kcat, because there are just 924 00:51:00,760 --> 00:51:04,510 two ways that that enzyme substrate 925 00:51:04,510 --> 00:51:06,430 complex can fall apart. 926 00:51:13,160 --> 00:51:15,354 Oh I'm sorry, I've already messed up. 927 00:51:18,258 --> 00:51:26,815 Kf over-- So Kcat has to be with Kr. 928 00:51:36,490 --> 00:51:39,265 So it just kind of speeds up the effective rate of dissociation. 929 00:51:42,730 --> 00:51:46,300 And of course, depending whether Kcat is large or small 930 00:51:46,300 --> 00:51:50,860 as compared to Kr, this can be either a large or small effect. 931 00:51:50,860 --> 00:51:55,100 But these rates, they just add. 932 00:51:55,100 --> 00:51:58,630 And we'll spend a lot of time thinking about how rates add 933 00:51:58,630 --> 00:52:02,259 and so forth in a few weeks. 934 00:52:02,259 --> 00:52:04,008 AUDIENCE: For this expression to be valid, 935 00:52:04,008 --> 00:52:06,664 don't you need Kcat to be much longer than the other rates? 936 00:52:06,664 --> 00:52:07,580 PROFESSOR: Right, yes. 937 00:52:07,580 --> 00:52:10,485 So you want Kcat to be-- So there's 938 00:52:10,485 --> 00:52:13,110 various kinds of limits in which you can talk about this thing. 939 00:52:13,110 --> 00:52:15,600 So in general, what you want is Kcat to be small, 940 00:52:15,600 --> 00:52:19,652 and you also want the initial transient to have gone away. 941 00:52:19,652 --> 00:52:21,360 Because when you first add the substrate, 942 00:52:21,360 --> 00:52:24,290 you don't yet have any enzyme substrate complex. 943 00:52:24,290 --> 00:52:26,460 So you have to wait until you've gotten 944 00:52:26,460 --> 00:52:30,454 to this so-called steady state, where the Michaelis Menten 945 00:52:30,454 --> 00:52:31,120 formula applies. 946 00:52:31,120 --> 00:52:33,150 And then you also can't have let it 947 00:52:33,150 --> 00:52:34,890 go too far, because then of course you're 948 00:52:34,890 --> 00:52:36,556 going to start running out of substrate. 949 00:52:49,326 --> 00:52:51,200 In the homework, you're going to get a chance 950 00:52:51,200 --> 00:52:53,366 to play with Michaelis Menten kinetics a little bit, 951 00:52:53,366 --> 00:52:56,420 and think about the dynamics when you have 952 00:52:56,420 --> 00:52:58,270 different kinds of inhibitors. 953 00:52:58,270 --> 00:53:02,320 So you can imagine having inhibitors that inhibit 954 00:53:02,320 --> 00:53:03,390 multiple different ways. 955 00:53:03,390 --> 00:53:08,270 You could have an inhibitor the binds the enzyme, 956 00:53:08,270 --> 00:53:12,230 and prevents the enzyme from providing the substrate. 957 00:53:12,230 --> 00:53:13,875 Now should this effect the Vmax? 958 00:53:21,802 --> 00:53:23,260 We'll think about it for 10 seconds 959 00:53:23,260 --> 00:53:28,050 and we'll vote because it's so much fun. 960 00:53:28,050 --> 00:53:29,760 We have these cards. 961 00:53:29,760 --> 00:53:35,930 Vmax change-- and this is with an inhibitor 962 00:53:35,930 --> 00:53:41,650 that binds here-- and forming an EI complex, reversibly. 963 00:53:41,650 --> 00:53:43,655 The question is, does Vmax change? 964 00:53:55,020 --> 00:53:58,560 A is yes and B is no. 965 00:54:04,320 --> 00:54:06,630 I'll give you 10 seconds to think about this. 966 00:54:06,630 --> 00:54:12,570 So Vmax is, again, defined as this rate of product formation 967 00:54:12,570 --> 00:54:15,370 at saturation, when you have a lot of the substrate. 968 00:54:34,460 --> 00:54:36,290 Do you need time? 969 00:54:36,290 --> 00:54:39,489 Or will time help? 970 00:54:39,489 --> 00:54:40,530 Well who wants more time? 971 00:54:40,530 --> 00:54:41,821 Just nod if you want more time. 972 00:54:45,260 --> 00:54:46,930 OK, well let's see how we feel. 973 00:54:46,930 --> 00:54:48,820 Let's go ahead and vote. 974 00:54:48,820 --> 00:54:53,510 If I add this competitive inhibitor, 975 00:54:53,510 --> 00:54:55,640 the question is, will be Vmax change? 976 00:54:55,640 --> 00:54:56,300 Ready. 977 00:54:56,300 --> 00:54:59,930 Three, two, one. 978 00:54:59,930 --> 00:55:04,830 So we have a majority of Bs, but some As. 979 00:55:04,830 --> 00:55:07,570 Can somebody give the intuition for why the Vmax should not 980 00:55:07,570 --> 00:55:08,070 change? 981 00:55:11,120 --> 00:55:12,156 Yes. 982 00:55:12,156 --> 00:55:18,108 AUDIENCE: Vmax is when substrate is 983 00:55:18,108 --> 00:55:21,580 far excess to the enzyme and, at that time, 984 00:55:21,580 --> 00:55:25,260 all of the enzymes bond to the substrate not to the inhibitor. 985 00:55:25,260 --> 00:55:26,260 PROFESSOR: Right, right. 986 00:55:26,260 --> 00:55:30,415 So Vmax occurs when you have lots and lots of substrate. 987 00:55:30,415 --> 00:55:31,790 And, of course, the condition you 988 00:55:31,790 --> 00:55:33,170 have to be a little bit careful, because it's not just 989 00:55:33,170 --> 00:55:35,120 having more substrate than the enzyme, 990 00:55:35,120 --> 00:55:38,490 but it's when the substrate is saturating. 991 00:55:38,490 --> 00:55:40,470 So if you have lots and lots of substrate, 992 00:55:40,470 --> 00:55:42,210 then the important point there is 993 00:55:42,210 --> 00:55:45,436 that it's when you've pushed this reaction all 994 00:55:45,436 --> 00:55:49,414 the way over here, all the enzyme is bound, 995 00:55:49,414 --> 00:55:51,580 and that's when you get this maximal rate of product 996 00:55:51,580 --> 00:55:52,680 formation. 997 00:55:52,680 --> 00:55:55,328 And that's true, you might need more substrate in order 998 00:55:55,328 --> 00:55:57,036 to get all that enzyme bound, because you 999 00:55:57,036 --> 00:56:00,870 have to pull the enzyme away from this side reaction. 1000 00:56:00,870 --> 00:56:03,330 And, indeed, this kind of inhibitor 1001 00:56:03,330 --> 00:56:07,820 alters the Km, the effect of Km of the reaction. 1002 00:56:07,820 --> 00:56:13,380 But it does not affect this Vmax, 1003 00:56:13,380 --> 00:56:18,300 whereas other inhibitors can bind this complex 1004 00:56:18,300 --> 00:56:22,300 and prevent it from catalyzing the reaction. 1005 00:56:22,300 --> 00:56:26,800 And that will instead affect Vmax, but won't affect Km. 1006 00:56:26,800 --> 00:56:30,340 So this was a powerful way that enzymologists 1007 00:56:30,340 --> 00:56:34,020 have used to try to get at mechanism of inhibitors. 1008 00:56:34,020 --> 00:56:35,550 So if you have some small molecule 1009 00:56:35,550 --> 00:56:38,309 you know somehow inhibits some enzymatic reaction 1010 00:56:38,309 --> 00:56:40,100 and you want to know, how is it doing that? 1011 00:56:40,100 --> 00:56:43,630 One thing you can do is you can titrate in that inhibitor 1012 00:56:43,630 --> 00:56:47,130 and then measure the Michaelis Menten 1013 00:56:47,130 --> 00:56:50,520 curve to get out the Vmax and Km to try 1014 00:56:50,520 --> 00:56:52,820 to get a sense mechanism. 1015 00:56:52,820 --> 00:56:55,340 And I always say we should be drawing these things. 1016 00:56:55,340 --> 00:56:58,240 So V is a function of-- and this is 1017 00:56:58,240 --> 00:56:59,950 in the [? lit ?] for a lot of substrate 1018 00:56:59,950 --> 00:57:02,780 relative to the enzyme-- then we can indeed 1019 00:57:02,780 --> 00:57:10,620 say it's going to plateau in Vmax at concentration Km. 1020 00:57:10,620 --> 00:57:11,150 It's at 1/2. 1021 00:57:15,580 --> 00:57:17,040 And then it plateaus. 1022 00:57:17,040 --> 00:57:20,390 This is a very, very, very common curve. 1023 00:57:20,390 --> 00:57:26,090 Lots of things in biology and life start at 0 and plateau, 1024 00:57:26,090 --> 00:57:28,455 and there are almost only two ways you can do that. 1025 00:57:28,455 --> 00:57:29,830 OK, there are more than two ways, 1026 00:57:29,830 --> 00:57:32,880 but there are a very small number of ways you can do that. 1027 00:57:32,880 --> 00:57:33,919 This is one of them. 1028 00:57:38,050 --> 00:57:42,480 Any questions on these Michaelis and Menten kinetics inhibitors? 1029 00:57:42,480 --> 00:57:46,990 You're going to spend a couple hours over the next few days 1030 00:57:46,990 --> 00:57:48,270 thinking about this. 1031 00:57:54,720 --> 00:57:57,190 So what I want to do for the last 20 minutes 1032 00:57:57,190 --> 00:57:59,250 is switch gears a little bit and to think 1033 00:57:59,250 --> 00:58:03,757 about the simple dynamics of gene expression. 1034 00:58:03,757 --> 00:58:05,590 The ideas that we've just been talking about 1035 00:58:05,590 --> 00:58:10,915 end up being just very relevant for the simple models here. 1036 00:58:10,915 --> 00:58:13,040 So what we want to think about is a situation where 1037 00:58:13,040 --> 00:58:18,740 we have some transcription factor, X, that is activating 1038 00:58:18,740 --> 00:58:26,930 expression of gene Y. So we have X activating Y. Now, 1039 00:58:26,930 --> 00:58:28,810 the way we can think about this, for example, 1040 00:58:28,810 --> 00:58:34,350 is that we may have X, which together with some 1041 00:58:34,350 --> 00:58:42,530 signal S of X, turns into some X star 1042 00:58:42,530 --> 00:58:48,150 It's X star that can bind to the promoter 1043 00:58:48,150 --> 00:58:59,360 and lead to expression of Y. 1044 00:58:59,360 --> 00:59:02,040 Now in Uri's book, he talks about this idea 1045 00:59:02,040 --> 00:59:04,160 of a separation of time scales that 1046 00:59:04,160 --> 00:59:08,360 is often useful to invoke when thinking about gene expression. 1047 00:59:08,360 --> 00:59:11,860 In this context, what was the fast event? 1048 00:59:17,527 --> 00:59:18,610 AUDIENCE: Activation of X? 1049 00:59:18,610 --> 00:59:21,360 PROFESSOR: Activation of X. So in many cases, 1050 00:59:21,360 --> 00:59:26,730 if this is a sugar or a small molecule that is going to be, 1051 00:59:26,730 --> 00:59:31,150 in this case, activating X, that can occur really quite quickly. 1052 00:59:31,150 --> 00:59:34,180 Often maybe less than a second. 1053 00:59:34,180 --> 00:59:36,900 The rate-limiting step would then, in many cases, 1054 00:59:36,900 --> 00:59:38,980 be getting the signal into the cell, so 1055 00:59:38,980 --> 00:59:42,450 depending on how that works. 1056 00:59:42,450 --> 00:59:45,680 So this occurs very rapidly. 1057 00:59:45,680 --> 00:59:49,160 What that means is if we look at a signal Sx, 1058 00:59:49,160 --> 00:59:53,790 as a function of time, where it starts out being absent 1059 00:59:53,790 --> 00:59:58,430 and then, all of a sudden, sugar appears in the environment, 1060 00:59:58,430 --> 01:00:02,100 we can think about the concentration of X and X star 1061 01:00:02,100 --> 01:00:09,400 So X starts out high and then quickly goes down, right? 1062 01:00:09,400 --> 01:00:16,270 Whereas X star will do the reverse here, quickly comes up. 1063 01:00:16,270 --> 01:00:17,345 And this should be flat. 1064 01:00:20,090 --> 01:00:23,180 Now, what is it that Y will do as a function of time? 1065 01:00:27,050 --> 01:00:31,360 So if X is an activator that means that before X star became 1066 01:00:31,360 --> 01:00:36,500 available, there was no expression of Y. 1067 01:00:36,500 --> 01:00:37,490 So it should be low. 1068 01:00:42,130 --> 01:00:44,550 So X is quickly activated, turns into X star. 1069 01:00:44,550 --> 01:00:52,171 So we start expressing Y. So, roughly, 1070 01:00:52,171 --> 01:00:53,462 what does this curve look like? 1071 01:01:03,100 --> 01:01:05,666 Somebody please help me. 1072 01:01:05,666 --> 01:01:08,155 AUDIENCE: It's S-shaped. 1073 01:01:08,155 --> 01:01:09,780 PROFESSOR: OK, so it could be S-shaped. 1074 01:01:15,590 --> 01:01:18,420 The thing that's very fast is activation of X, 1075 01:01:18,420 --> 01:01:21,570 and then what's really still rather fast 1076 01:01:21,570 --> 01:01:24,450 is equilibration of X star on this promoter. 1077 01:01:24,450 --> 01:01:27,270 So that might still be very rapid 1078 01:01:27,270 --> 01:01:31,610 because these things were nearly instantaneous, But. 1079 01:01:31,610 --> 01:01:34,590 Coming to equilibrium here still might happen over 1080 01:01:34,590 --> 01:01:37,580 time scales of seconds. 1081 01:01:37,580 --> 01:01:39,730 So that means that you actually, sort of quickly, 1082 01:01:39,730 --> 01:01:43,710 start getting expression, at least on time scales 1083 01:01:43,710 --> 01:01:46,180 are relevant in terms of hours kind of time scales. 1084 01:01:46,180 --> 01:01:48,750 Of course, it still does take time to express. 1085 01:01:48,750 --> 01:01:54,950 So it takes minutes for the RNA polymerase to transcribe, 1086 01:01:54,950 --> 01:01:56,720 and then of course the ribosome's 1087 01:01:56,720 --> 01:01:57,970 going to have to do something. 1088 01:02:04,024 --> 01:02:04,940 What do I want to ask? 1089 01:02:09,530 --> 01:02:11,500 Let's write down the equation that Uri 1090 01:02:11,500 --> 01:02:15,569 invokes because there's a very real sense in which it 1091 01:02:15,569 --> 01:02:17,360 does, maybe, look a little bit more S-like. 1092 01:02:17,360 --> 01:02:21,010 But at least in terms of Uri's kind of formalism, 1093 01:02:21,010 --> 01:02:23,510 he often would say, the change in the concentration 1094 01:02:23,510 --> 01:02:28,260 of this protein, it's going to be some function of, 1095 01:02:28,260 --> 01:02:29,510 in this case X star. 1096 01:02:32,590 --> 01:02:35,740 And then there's another term here, which was the minus alpha 1097 01:02:35,740 --> 01:02:39,714 Y. What was the minus alpha Y due to? 1098 01:02:39,714 --> 01:02:40,630 AUDIENCE: Degradation. 1099 01:02:40,630 --> 01:02:43,210 PROFESSOR: Right, so there are two terms. 1100 01:02:43,210 --> 01:02:47,150 So there's alpha, and it's going to be the sum of two things. 1101 01:02:47,150 --> 01:02:51,060 There's alpha due to degradation. 1102 01:02:51,060 --> 01:02:55,230 So if the protein is degraded actively in some way. 1103 01:02:55,230 --> 01:02:56,960 If the protein is not degraded, then 1104 01:02:56,960 --> 01:02:59,940 does that mean that alpha is equal to 0? 1105 01:02:59,940 --> 01:03:00,750 No. 1106 01:03:00,750 --> 01:03:04,504 So what is this other term? 1107 01:03:04,504 --> 01:03:05,420 AUDIENCE: Cell growth. 1108 01:03:05,420 --> 01:03:07,544 PROFESSOR: Right, so it's alpha due to some growth. 1109 01:03:07,544 --> 01:03:13,800 And cell growth leads to some dilution effect. 1110 01:03:13,800 --> 01:03:16,260 So if you have the same number of proteins in the cell 1111 01:03:16,260 --> 01:03:17,760 that the cell is growing, that means 1112 01:03:17,760 --> 01:03:19,240 the concentration is shrinking. 1113 01:03:19,240 --> 01:03:19,780 Right? 1114 01:03:19,780 --> 01:03:22,210 Now the reality of this process is 1115 01:03:22,210 --> 01:03:26,380 that it's complicated, because cell growth is not uniform. 1116 01:03:26,380 --> 01:03:29,610 But if you kind of average over things, 1117 01:03:29,610 --> 01:03:32,230 then a reasonable description is just to say, 1118 01:03:32,230 --> 01:03:35,540 just a first order effective dilution rate. 1119 01:03:35,540 --> 01:03:41,960 If you want to, you can write down a more detailed formula, 1120 01:03:41,960 --> 01:03:44,900 or a model where, you say if cell growth does this, 1121 01:03:44,900 --> 01:03:47,164 then-- It's going to kind of wiggle a little bit 1122 01:03:47,164 --> 01:03:48,580 over the course of the cell cycle, 1123 01:03:48,580 --> 01:03:50,860 but this is a reasonable description. 1124 01:03:50,860 --> 01:03:55,700 Now what this is saying is that even if there 1125 01:03:55,700 --> 01:03:57,550 is no active degradation, then there still 1126 01:03:57,550 --> 01:04:00,760 is an effective term due to this dilution. 1127 01:04:00,760 --> 01:04:05,610 And this means that if we immediately activate, 1128 01:04:05,610 --> 01:04:09,070 and if F of X star-- at time T equal to 0 here-- if it just 1129 01:04:09,070 --> 01:04:12,880 goes to some beta, then what is the long time 1130 01:04:12,880 --> 01:04:13,990 solution of this equation? 1131 01:04:17,750 --> 01:04:18,700 AUDIENCE: Beta/alpha. 1132 01:04:18,700 --> 01:04:19,970 PROFESSOR: Beta/alpha, right? 1133 01:04:19,970 --> 01:04:23,140 So we know it should eventually come to beta/alpha. 1134 01:04:27,538 --> 01:04:29,912 What's the characteristic time scale for it to get there? 1135 01:04:36,716 --> 01:04:40,582 AUDIENCE: Cell alpha's rate, it's 1/alpha. 1136 01:04:40,582 --> 01:04:41,290 PROFESSOR: Right. 1137 01:04:41,290 --> 01:04:44,141 So characteristic time is 1/alpha. 1138 01:04:44,141 --> 01:04:45,890 The solution to this differential equation 1139 01:04:45,890 --> 01:04:53,850 is just an exponential where, if extend this line here, 1140 01:04:53,850 --> 01:04:56,580 this is 1/alpha. 1141 01:04:56,580 --> 01:04:58,120 So that's time. 1142 01:04:58,120 --> 01:05:00,060 And then, of course, the T 1/2, the time 1143 01:05:00,060 --> 01:05:03,340 it takes to get to 1/2, is indeed different by log 2, 1144 01:05:03,340 --> 01:05:04,877 and that's the cell division time. 1145 01:05:10,330 --> 01:05:17,220 This point here-- this is at T 1/2-- is cell division. 1146 01:05:17,220 --> 01:05:21,155 This is for a stable protein, assuming that alpha degradation 1147 01:05:21,155 --> 01:05:21,738 is equal to 0. 1148 01:05:25,950 --> 01:05:29,150 So the thing to remember is that this basic differential 1149 01:05:29,150 --> 01:05:34,450 equation of Y dot is equal to a minus alpha Y, is 1150 01:05:34,450 --> 01:05:37,264 an exponential by going to 0. 1151 01:05:37,264 --> 01:05:39,180 Whereas if you have a constant term here, then 1152 01:05:39,180 --> 01:05:41,180 it's an exponential going to some nonzero value. 1153 01:05:43,750 --> 01:05:49,990 So, indeed, if the signal here goes away, 1154 01:05:49,990 --> 01:05:52,250 then we quickly come back here. 1155 01:05:52,250 --> 01:05:53,755 So this comes here. 1156 01:05:53,755 --> 01:05:59,620 This comes here, and then this-- does it go back down to 0? 1157 01:05:59,620 --> 01:06:02,840 Is it more or less rapid returning to 0 1158 01:06:02,840 --> 01:06:04,810 than it took to come up? 1159 01:06:09,160 --> 01:06:09,820 All right. 1160 01:06:09,820 --> 01:06:10,320 OK. 1161 01:06:15,860 --> 01:06:17,585 So, how can I phrase this? 1162 01:06:20,570 --> 01:06:27,330 OK, faster decay, question mark. 1163 01:06:27,330 --> 01:06:32,840 A is yes, and B is no, and you can always 1164 01:06:32,840 --> 01:06:35,500 do C or something if you don't know what I'm asking. 1165 01:06:35,500 --> 01:06:40,460 The question is, we've turned off the signal, 1166 01:06:40,460 --> 01:06:43,580 is it going to go away faster, or slower, or the same? 1167 01:06:47,830 --> 01:06:48,455 This is faster. 1168 01:06:51,440 --> 01:06:54,530 B can even be slower maybe. 1169 01:06:54,530 --> 01:06:55,175 C is same. 1170 01:06:59,640 --> 01:07:01,900 Do you understand the options now? 1171 01:07:01,900 --> 01:07:06,734 So we stopped expressing Y, so concentration of X 1172 01:07:06,734 --> 01:07:07,900 is going to decrease, right? 1173 01:07:07,900 --> 01:07:12,110 Question is, it is going to go away faster, slower, 1174 01:07:12,110 --> 01:07:15,310 or the same as the rate that it came up? 1175 01:07:20,330 --> 01:07:23,150 Do you need more time? 1176 01:07:23,150 --> 01:07:23,980 Ready. 1177 01:07:23,980 --> 01:07:26,060 Three, two, one. 1178 01:07:30,100 --> 01:07:35,000 OK so we have a fair agreement that, this thing, 1179 01:07:35,000 --> 01:07:36,140 it's going to be the same. 1180 01:07:36,140 --> 01:07:38,550 So there's a characteristic time for it to come 1181 01:07:38,550 --> 01:07:45,350 and it's the same characteristic time for it to degrade away. 1182 01:07:45,350 --> 01:07:51,570 So this, I would say, is not a priori obvious, 1183 01:07:51,570 --> 01:07:55,450 but it's really just the nature of when you 1184 01:07:55,450 --> 01:07:58,122 have these sorts of situations. 1185 01:07:58,122 --> 01:07:59,580 This sets the time scale for if you 1186 01:07:59,580 --> 01:08:01,650 want to change the concentration-- doesn't matter 1187 01:08:01,650 --> 01:08:03,441 whether you're going to 0, a finite number, 1188 01:08:03,441 --> 01:08:05,630 or if you go from high to low, but not 0. 1189 01:08:05,630 --> 01:08:07,796 Again, it's going to be exponential in the same time 1190 01:08:07,796 --> 01:08:09,930 scale. 1191 01:08:09,930 --> 01:08:12,722 So if you want that to be faster, 1192 01:08:12,722 --> 01:08:14,680 if you want to be able to respond more rapidly, 1193 01:08:14,680 --> 01:08:21,420 then one solution would be to actively degrade the protein. 1194 01:08:21,420 --> 01:08:21,970 Right? 1195 01:08:21,970 --> 01:08:25,500 Now it's obvious that degrading the protein actively 1196 01:08:25,500 --> 01:08:28,470 will allow it to go way more rapidly. 1197 01:08:28,470 --> 01:08:30,700 What's perhaps less obvious is that there's 1198 01:08:30,700 --> 01:08:34,229 a real sense in which degrading the protein allows 1199 01:08:34,229 --> 01:08:37,930 this response to be more rapid as well. 1200 01:08:37,930 --> 01:08:42,440 But of course, did we keep everything constant? 1201 01:08:42,440 --> 01:08:44,502 If I say, oh I want the curve to look like this, 1202 01:08:44,502 --> 01:08:46,210 can I just increase the degradation rate? 1203 01:08:49,472 --> 01:08:52,279 AUDIENCE: [INAUDIBLE]. 1204 01:08:52,279 --> 01:08:55,500 PROFESSOR: Well let's say that X star is already maximal 1205 01:08:55,500 --> 01:08:56,380 saturating. 1206 01:08:56,380 --> 01:08:59,193 So we're already-- well, OK. 1207 01:08:59,193 --> 01:08:59,859 So I understand. 1208 01:08:59,859 --> 01:09:02,630 OK, now I understand what you're saying. 1209 01:09:02,630 --> 01:09:04,630 You need to increase beta, and that could either 1210 01:09:04,630 --> 01:09:06,609 be by increasing X star or it could 1211 01:09:06,609 --> 01:09:11,060 be just by increasing the strength of that promoter. 1212 01:09:11,060 --> 01:09:15,039 So the idea is that if you want a more rapid on or off, 1213 01:09:15,039 --> 01:09:16,830 you can also increase the degradation rate. 1214 01:09:16,830 --> 01:09:18,640 But there's a cost to that, which 1215 01:09:18,640 --> 01:09:22,109 is that you have to make more protein. 1216 01:09:22,109 --> 01:09:27,640 And, indeed, many transcription factors are actively degraded. 1217 01:09:27,640 --> 01:09:31,520 And that may be because if you are using these transcription 1218 01:09:31,520 --> 01:09:33,069 factors to turn things on and off, 1219 01:09:33,069 --> 01:09:37,350 then you want to get rapid responses. 1220 01:09:37,350 --> 01:09:39,560 And, also, transcription factors are often not 1221 01:09:39,560 --> 01:09:42,314 expressed at the same high levels as structural proteins. 1222 01:09:42,314 --> 01:09:44,189 So that means that the cost of degrading them 1223 01:09:44,189 --> 01:09:46,450 is not going to be as severe. 1224 01:09:46,450 --> 01:09:52,024 If you're actively degrading cell wall type of things, 1225 01:09:52,024 --> 01:09:53,399 that's going to be really costly. 1226 01:09:56,175 --> 01:09:57,550 Are there any questions of what I 1227 01:09:57,550 --> 01:09:59,790 mean by this discussion of active degradation, 1228 01:09:59,790 --> 01:10:03,370 why it might help, costs? 1229 01:10:03,370 --> 01:10:05,540 Because, over the next week or two, 1230 01:10:05,540 --> 01:10:08,560 we're going to see multiple possible solutions 1231 01:10:08,560 --> 01:10:09,320 to this problem. 1232 01:10:09,320 --> 01:10:12,260 If you want to increase the rate that you respond 1233 01:10:12,260 --> 01:10:14,490 to some environmental change, one way you can do it 1234 01:10:14,490 --> 01:10:17,824 is by actively degrading some of the signaling proteins. 1235 01:10:17,824 --> 01:10:20,240 But there are other solutions we're going to come up with, 1236 01:10:20,240 --> 01:10:21,240 such as auto regulation. 1237 01:10:29,167 --> 01:10:30,750 In the last few minutes here, I wanted 1238 01:10:30,750 --> 01:10:36,240 to say something about this question of ultrasensitivity. 1239 01:10:36,240 --> 01:10:37,660 So there are many cases where you 1240 01:10:37,660 --> 01:10:43,350 would like to get very sensitive responses, i.e. 1241 01:10:43,350 --> 01:10:45,570 you'd like to be able to make a modest change 1242 01:10:45,570 --> 01:10:48,080 in the concentration of some, for example, transcription 1243 01:10:48,080 --> 01:10:51,965 factor, and get a significant change in-- I don't know why 1244 01:10:51,965 --> 01:10:55,272 I erased that but-- and you want to be 1245 01:10:55,272 --> 01:10:58,370 able to get a significant change in the output. 1246 01:10:58,370 --> 01:11:00,050 And one way that you can do this is 1247 01:11:00,050 --> 01:11:02,490 by having some sort of cooperative binding. 1248 01:11:02,490 --> 01:11:05,070 If you have dimerization of a transcription factor 1249 01:11:05,070 --> 01:11:06,930 before binding then you, for example, 1250 01:11:06,930 --> 01:11:09,470 can get a more sensitive response. 1251 01:11:09,470 --> 01:11:14,760 So one way to think that this is if you have an X activating Y 1252 01:11:14,760 --> 01:11:21,280 right in this simple case, then the rate 1253 01:11:21,280 --> 01:11:27,440 of Y expression as a function of-- 1254 01:11:27,440 --> 01:11:30,680 and here we're going to write X for now 1255 01:11:30,680 --> 01:11:35,500 and we'll just assume that all Xs are indeed active. 1256 01:11:35,500 --> 01:11:37,380 OK? 1257 01:11:37,380 --> 01:11:39,910 Now if it just is a single X binding Y, 1258 01:11:39,910 --> 01:11:43,580 then this should behave just like this Michaelis Menten 1259 01:11:43,580 --> 01:11:46,280 formula, where there's some maximal rate of expression 1260 01:11:46,280 --> 01:11:47,730 here. 1261 01:11:47,730 --> 01:11:54,100 There's going to be some Kd, which is bound 1/2 the time, 1262 01:11:54,100 --> 01:11:56,454 and then some curve that looks like this. 1263 01:11:56,454 --> 01:11:58,120 So this would be an example of something 1264 01:11:58,120 --> 01:12:01,310 that is not ultrasensitive and that you 1265 01:12:01,310 --> 01:12:06,350 don't get a significant change in the rate of Y expression-- 1266 01:12:06,350 --> 01:12:12,500 or the equilibrium Y value, if you'd like-- 1267 01:12:12,500 --> 01:12:15,110 as a function of changing X. 1268 01:12:15,110 --> 01:12:17,212 As you start having more and more X, 1269 01:12:17,212 --> 01:12:18,670 you would need get more and more Y, 1270 01:12:18,670 --> 01:12:23,080 but that ratio, if you double X, you always 1271 01:12:23,080 --> 01:12:29,260 get less than a doubling of Y. So the question is, 1272 01:12:29,260 --> 01:12:33,230 what can you do to make things somehow more sensitive? 1273 01:12:33,230 --> 01:12:37,580 You'd like something that looks a little bit more-- 1274 01:12:37,580 --> 01:12:44,160 well the ultimate would be 0 and then beta. 1275 01:12:44,160 --> 01:12:49,170 And indeed this would be this logic kind of limit. 1276 01:12:49,170 --> 01:12:53,380 So this is Y expression as a function of X. 1277 01:12:53,380 --> 01:12:56,760 If you didn't get any until some Kd and then all of a sudden you 1278 01:12:56,760 --> 01:12:59,880 had beta, that would be as sensitive as you could 1279 01:12:59,880 --> 01:13:02,550 possibly-- this is ultra-, ultra- sensitive. 1280 01:13:06,450 --> 01:13:09,164 So there's one solution that was talked about in the book 1281 01:13:09,164 --> 01:13:11,330 to get something that's a little bit more like this. 1282 01:13:14,460 --> 01:13:15,710 AUDIENCE: Cooperative binding. 1283 01:13:15,710 --> 01:13:17,251 PROFESSOR: Yeah, cooperative binding. 1284 01:13:17,251 --> 01:13:23,700 So we often describe these functions-- 1285 01:13:23,700 --> 01:13:26,080 this is the rate of expression as a function of X-- 1286 01:13:26,080 --> 01:13:28,420 as via some hill equation. 1287 01:13:28,420 --> 01:13:30,580 So I'm just going to write Xs here. 1288 01:13:30,580 --> 01:13:37,720 So it could be there's X, K plus X here. 1289 01:13:37,720 --> 01:13:41,180 Now if you have cooperative binding 1290 01:13:41,180 --> 01:13:43,560 either at the side of the promoter or dimerization, 1291 01:13:43,560 --> 01:13:45,950 trimerzation, something before binding, 1292 01:13:45,950 --> 01:13:49,810 you can get some effective hill coverage in here, 1293 01:13:49,810 --> 01:13:52,516 where this is going to be up X to the n, X to the n, K 1294 01:13:52,516 --> 01:13:53,240 to then. 1295 01:13:53,240 --> 01:13:55,965 We put K to n here just so that all the units are still 1296 01:13:55,965 --> 01:13:58,260 reasonable. 1297 01:13:58,260 --> 01:14:01,920 And as n increases, this thing becomes more and more sensitive 1298 01:14:01,920 --> 01:14:03,220 or ultrasensitive. 1299 01:14:03,220 --> 01:14:05,950 So this is with n just equal to 1, 1300 01:14:05,950 --> 01:14:09,430 just a monomer binding in a simple way. 1301 01:14:09,430 --> 01:14:11,150 Kd is still the 1/2 mark. 1302 01:14:11,150 --> 01:14:15,590 So things always cross here, but if it's 2 1303 01:14:15,590 --> 01:14:20,610 then it might look like this, now three, four. 1304 01:14:20,610 --> 01:14:24,121 So it gets steeper and steeper as that hill coefficient 1305 01:14:24,121 --> 01:14:24,620 increases. 1306 01:14:34,090 --> 01:14:38,065 As Uri mentions, for many input in many genes, if you go in 1307 01:14:38,065 --> 01:14:39,690 and you measure these things, you often 1308 01:14:39,690 --> 01:14:42,265 get something that's reasonably well-defined by the S 1309 01:14:42,265 --> 01:14:45,060 with a n somewhere between 1 and 4. 1310 01:14:45,060 --> 01:14:47,251 So things are often moderately cooperative. 1311 01:14:57,020 --> 01:15:00,120 Maybe I'll just tell you about this other mechanism 1312 01:15:00,120 --> 01:15:04,620 for ultrasensitivity, this idea of molecular titration. 1313 01:15:04,620 --> 01:15:08,010 I'm going to leave you with the basic model, or the basic idea, 1314 01:15:08,010 --> 01:15:10,820 and then at the beginning of class on Thursday, 1315 01:15:10,820 --> 01:15:12,390 we will try to figure out what are 1316 01:15:12,390 --> 01:15:15,227 the requirements for the various binding affinities 1317 01:15:15,227 --> 01:15:16,560 in order for that model to work. 1318 01:15:22,920 --> 01:15:24,460 So what's neat about this is it's 1319 01:15:24,460 --> 01:15:26,440 a situation where you can get something 1320 01:15:26,440 --> 01:15:29,705 that is ultrasensitive without any cooperativity. 1321 01:15:33,950 --> 01:15:40,530 The idea is that you have some X that is indeed 1322 01:15:40,530 --> 01:15:46,250 binding to the promoter to activate expression of Y, 1323 01:15:46,250 --> 01:15:51,720 but, in addition, you have some other protein, say W, 1324 01:15:51,720 --> 01:15:58,535 that can bind to X and turn it into this complex XW. 1325 01:16:01,710 --> 01:16:06,580 So we can always describe things as there's some Kw here, 1326 01:16:06,580 --> 01:16:11,900 some Kd for X to bind to the promoter, 1327 01:16:11,900 --> 01:16:20,960 and for some relations of Kw, Kd, and W 1328 01:16:20,960 --> 01:16:24,450 total, you can get ultrasensitivity. 1329 01:16:24,450 --> 01:16:28,810 What happens is that if you look at this rate of expression-- 1330 01:16:28,810 --> 01:16:34,250 so this is Y expression-- as a function of X, 1331 01:16:34,250 --> 01:16:39,260 if you don't have any W here, then indeed it just looks 1332 01:16:39,260 --> 01:16:40,540 like our standard thing here. 1333 01:16:43,290 --> 01:16:47,960 Whereas in the this is when you add W, 1334 01:16:47,960 --> 01:16:51,790 when you add this molecular titration phenomenon, 1335 01:16:51,790 --> 01:16:54,710 you can make this curve slide over so you don't 1336 01:16:54,710 --> 01:16:59,310 get significant expression until X-- 1337 01:16:59,310 --> 01:17:00,790 or this is X total if you'd like-- 1338 01:17:00,790 --> 01:17:03,974 is larger than W total and this whole curve just slides over. 1339 01:17:09,170 --> 01:17:12,367 You can see this is ultrasensitive. 1340 01:17:12,367 --> 01:17:14,200 So nothing happens until all of a sudden you 1341 01:17:14,200 --> 01:17:16,280 start getting expression. 1342 01:17:16,280 --> 01:17:18,530 So what we're going to do is on the beginning of class 1343 01:17:18,530 --> 01:17:21,330 on Thursday, we'll try to figure out 1344 01:17:21,330 --> 01:17:26,390 what is the relationship between these different binding 1345 01:17:26,390 --> 01:17:30,240 parameters in order to get something that looks like this. 1346 01:17:34,280 --> 01:17:36,910 Are there any questions about anything 1347 01:17:36,910 --> 01:17:38,527 that we've said so far today? 1348 01:17:42,350 --> 01:17:45,180 With that, why don't we go ahead and quit.