WEBVTT 00:00:00.070 --> 00:00:02.500 The following content is provided under a Creative 00:00:02.500 --> 00:00:04.019 Commons license. 00:00:04.019 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.730 continue to offer high quality, educational resources for free. 00:00:10.730 --> 00:00:13.340 To make a donation, or view additional materials 00:00:13.340 --> 00:00:17.236 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.236 --> 00:00:17.861 at ocw.mit.edu. 00:00:21.320 --> 00:00:24.850 PROFESSOR: So today we're going to take 00:00:24.850 --> 00:00:29.270 a little bit of a lightning tour through some basic topics 00:00:29.270 --> 00:00:34.060 on chemical enzyme kinetics, before thinking a little bit 00:00:34.060 --> 00:00:37.110 about some, what you might call, simple input-output 00:00:37.110 --> 00:00:40.814 relationships, in terms of gene expression. 00:00:40.814 --> 00:00:42.230 I've got separation of time scale, 00:00:42.230 --> 00:00:45.910 and also this basic notion that for a stable protein 00:00:45.910 --> 00:00:47.809 there's a natural time scale over which 00:00:47.809 --> 00:00:49.350 the concentration will go up or down, 00:00:49.350 --> 00:00:51.640 and that's dictated by the cell generation time. 00:00:54.450 --> 00:00:57.610 At the end, we'll then talk about different ways you get 00:00:57.610 --> 00:00:59.800 this thing of ultrasensitivity. 00:00:59.800 --> 00:01:02.080 So, how is it that you can make it 00:01:02.080 --> 00:01:05.090 so that a small change in the input concentration, 00:01:05.090 --> 00:01:07.090 say the concentration of a transcription factor, 00:01:07.090 --> 00:01:10.010 might be able to lead to a large change in the output, 00:01:10.010 --> 00:01:11.930 or the gene expression of its target? 00:01:11.930 --> 00:01:13.620 We'll talk about how you can have, 00:01:13.620 --> 00:01:15.760 for example, cooperative binding at the promoter, 00:01:15.760 --> 00:01:18.520 or you can have multimerization, kind of leads to similar things 00:01:18.520 --> 00:01:19.200 here. 00:01:19.200 --> 00:01:21.420 But also, we're going to talk about this idea 00:01:21.420 --> 00:01:23.860 of molecular titration. 00:01:23.860 --> 00:01:26.980 So if you have another protein that acts as kind of a sponge, 00:01:26.980 --> 00:01:30.780 then this connect can lead to a similar effect. 00:01:30.780 --> 00:01:34.450 And this is indeed observed in various natural contexts. 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. 00:01:43.010 --> 00:01:45.680 Turns out I was down at Princeton, no sorry, 00:01:45.680 --> 00:01:48.920 I was out at Duke yesterday, and so I got to hang out with Nick 00:01:48.920 --> 00:01:51.580 and talk about this work. 00:01:51.580 --> 00:01:54.669 I had previously told him that in my first lecture 00:01:54.669 --> 00:01:56.210 in the systems biology class, we like 00:01:56.210 --> 00:01:59.440 to discuss this molecular titration effect. 00:01:59.440 --> 00:02:04.880 He was instrumental in elucidating how it worked. 00:02:04.880 --> 00:02:07.910 All right so let's go ahead and get started. 00:02:07.910 --> 00:02:11.904 So hopefully you all have these cards. 00:02:11.904 --> 00:02:14.070 We're going to start out with some simple questions, 00:02:14.070 --> 00:02:15.610 just to make sure that you know how 00:02:15.610 --> 00:02:18.040 to use the complicated devices that 00:02:18.040 --> 00:02:19.320 are sitting in front of you. 00:02:19.320 --> 00:02:19.819 OK? 00:02:22.630 --> 00:02:24.250 So what we want to do is to start 00:02:24.250 --> 00:02:27.360 by thinking about a situation where you have two molecules. 00:02:27.360 --> 00:02:30.300 We're going to call them E and S, 00:02:30.300 --> 00:02:33.620 and of course, you can imagine what these might possibly stand 00:02:33.620 --> 00:02:37.140 for in one context or another. 00:02:37.140 --> 00:02:40.840 There are two rates here that are describing 00:02:40.840 --> 00:02:44.140 the rate that these molecules, E and S, find each other, Kf. 00:02:44.140 --> 00:02:47.780 And Kr is defining the rate at which this complex is 00:02:47.780 --> 00:02:49.390 going to fall apart. 00:02:49.390 --> 00:02:56.820 So there's this forward rate that 00:02:56.820 --> 00:03:01.870 is sum Kf, times concentration of E, 00:03:01.870 --> 00:03:03.350 times the concentration of S. Now 00:03:03.350 --> 00:03:07.640 I'd say for the first part of this lecture, 00:03:07.640 --> 00:03:10.360 we will indeed use the chemistry convention of concentrations 00:03:10.360 --> 00:03:13.680 here, although we will quickly get tired of these brackets 00:03:13.680 --> 00:03:16.930 and we'll just start writing the letters. 00:03:16.930 --> 00:03:19.397 And hopefully it is self evident in the context 00:03:19.397 --> 00:03:21.230 that I'm referring to a concentration rather 00:03:21.230 --> 00:03:23.980 than something else, but if you're ever confused, please 00:03:23.980 --> 00:03:25.440 ask. 00:03:25.440 --> 00:03:30.630 So this is forward rate, and then the reverse rate 00:03:30.630 --> 00:03:31.890 is something similar. 00:03:31.890 --> 00:03:34.024 So we have this Kr. 00:03:34.024 --> 00:03:35.440 Now this is just the concentration 00:03:35.440 --> 00:03:36.490 of the complex ES. 00:03:39.270 --> 00:03:42.360 Many of you have spent a lot of time thinking 00:03:42.360 --> 00:03:44.770 about how we often define things. 00:03:44.770 --> 00:03:47.670 This Kd, dissociation constant, is defined 00:03:47.670 --> 00:03:52.760 as the ratio Kr over Kf. 00:03:55.780 --> 00:03:58.682 Now, just so we can practice using our cards, what 00:03:58.682 --> 00:04:03.615 we're going to ask first is what the units of this Kd thing is. 00:04:06.530 --> 00:04:09.260 Now, in general, when I ask such a question, 00:04:09.260 --> 00:04:11.875 I will give you some A, B, C, D options. 00:04:14.242 --> 00:04:16.200 You can start thinking even before I write down 00:04:16.200 --> 00:04:17.839 the options. 00:04:17.839 --> 00:04:20.630 Yes? 00:04:20.630 --> 00:04:22.130 I'll encourage you to think before I 00:04:22.130 --> 00:04:23.255 start writing down options. 00:04:25.670 --> 00:04:34.070 So it's either dimensionless, units of concentration, 00:04:34.070 --> 00:04:39.790 1 over concentration, 1 over time, 00:04:39.790 --> 00:04:42.740 and I will often include at the bottom something that 00:04:42.740 --> 00:04:45.080 simply is, don't know. 00:04:45.080 --> 00:04:48.744 And that is if you're really confused 00:04:48.744 --> 00:04:50.160 about what I'm talking about, then 00:04:50.160 --> 00:04:54.430 feel free to just flash me that, and that at least tells me 00:04:54.430 --> 00:04:58.480 that I'm gibbering nonsense. 00:04:58.480 --> 00:05:01.440 So there's going to be a very strict set of rules 00:05:01.440 --> 00:05:03.620 for how we do these flash cards, all right? 00:05:03.620 --> 00:05:06.400 You don't get to vote before I tell you to vote. 00:05:06.400 --> 00:05:08.226 You have to keep on thinking. 00:05:08.226 --> 00:05:09.850 If you think you know the right answer, 00:05:09.850 --> 00:05:14.460 check limits, just do whatever it is to keep on thinking. 00:05:14.460 --> 00:05:17.630 And then we vote simultaneously. 00:05:17.630 --> 00:05:20.240 That way, it builds up the tension, everyone gets excited, 00:05:20.240 --> 00:05:22.000 and then you vote. 00:05:22.000 --> 00:05:24.312 It also provides me an opportunity 00:05:24.312 --> 00:05:26.770 to make sure that I can see that everybody's participating. 00:05:26.770 --> 00:05:30.210 So if you don't vote, then you have the opportunity 00:05:30.210 --> 00:05:33.190 to tell the group what you think the answer should be and why. 00:05:37.860 --> 00:05:41.010 And the cards they're both colored, 00:05:41.010 --> 00:05:44.190 and they have letters on there, so the letters correspond 00:05:44.190 --> 00:05:45.689 to the answer. 00:05:45.689 --> 00:05:46.730 We're all on top of this? 00:05:49.799 --> 00:05:51.090 Have you had a chance to think? 00:05:51.090 --> 00:05:54.831 Or has my talking bothered you? 00:05:54.831 --> 00:05:55.330 Both. 00:05:55.330 --> 00:05:56.490 OK. 00:05:56.490 --> 00:05:59.980 So what we do is I'm going to ask, do you need more time? 00:05:59.980 --> 00:06:02.680 If you need more time, just nod or something like that, 00:06:02.680 --> 00:06:06.690 and if more than a few people nod, I'll give you more time. 00:06:06.690 --> 00:06:10.580 But you guys are totally all right. 00:06:10.580 --> 00:06:12.390 OK let's see how we are. 00:06:12.390 --> 00:06:14.650 So then I'll say, OK, we're ready. 00:06:14.650 --> 00:06:17.060 And then we're going to go three, two, one, and then 00:06:17.060 --> 00:06:18.860 I want a vote by your chest. 00:06:18.860 --> 00:06:21.250 You don't need to display it to the group. 00:06:21.250 --> 00:06:23.542 It's just here, and then tell me what you think. 00:06:23.542 --> 00:06:24.250 All right, ready. 00:06:24.250 --> 00:06:26.210 Three, two, one. 00:06:29.540 --> 00:06:32.870 Broadly, people know how to use the cards. 00:06:32.870 --> 00:06:35.650 And there's a clear majority of the group, 00:06:35.650 --> 00:06:38.414 although it's not 100%, that are saying that this thing is 00:06:38.414 --> 00:06:39.080 a concentration. 00:06:42.420 --> 00:06:46.730 I'd say that if the group is maybe between 25, 00:06:46.730 --> 00:06:48.930 75% correct on these sorts of things, 00:06:48.930 --> 00:06:53.330 then I will often have you pair off in, well, in pairs. 00:06:53.330 --> 00:06:55.050 And the goal there would be to try 00:06:55.050 --> 00:06:56.883 to convince your neighbor that you're right. 00:06:56.883 --> 00:06:59.680 In this case we're a bit above 75%, 00:06:59.680 --> 00:07:04.700 so I've already indicated what the answer is. 00:07:04.700 --> 00:07:09.800 Can somebody just quickly say, why is this a concentration? 00:07:09.800 --> 00:07:10.840 Maybe in the back. 00:07:10.840 --> 00:07:12.963 AUDIENCE: So, we know that both rates need 00:07:12.963 --> 00:07:14.270 to have the same dimensions-- 00:07:14.270 --> 00:07:16.769 PROFESSOR: Yeah, and what are the dimensions of these rates? 00:07:16.769 --> 00:07:18.870 AUDIENCE: [INAUDIBLE]. 00:07:18.870 --> 00:07:21.230 PROFESSOR: OK. 00:07:21.230 --> 00:07:26.320 So, there are actually different conventions, in principle, 00:07:26.320 --> 00:07:31.960 but we will often be working in numbers 00:07:31.960 --> 00:07:33.654 in most of this class, in which case 00:07:33.654 --> 00:07:35.320 it would actually just be a 1 over time. 00:07:35.320 --> 00:07:38.670 But depending on whether you're doing chemistry-- So 00:07:38.670 --> 00:07:41.100 the numerators may be ambiguous, depending, 00:07:41.100 --> 00:07:45.280 but the important thing is that they're definitely the same. 00:07:45.280 --> 00:07:48.140 These are definitely going to be the same. 00:07:48.140 --> 00:07:50.514 But it is true that in an awful lot of this class, 00:07:50.514 --> 00:07:52.180 we're going to be thinking about numbers 00:07:52.180 --> 00:07:53.140 rather than the concentrations. 00:07:53.140 --> 00:07:54.681 Because, for a lot of the class we'll 00:07:54.681 --> 00:07:58.930 be thinking about finite number fluctuations 00:07:58.930 --> 00:08:03.020 of stochastic dynamics, in which case, concentration, who knows 00:08:03.020 --> 00:08:04.420 what's going to happen? 00:08:04.420 --> 00:08:07.730 But these have to have the same units, right? 00:08:07.730 --> 00:08:10.270 And the important thing here is if you look at the right, 00:08:10.270 --> 00:08:14.720 you see this guy has an extra concentration up here. 00:08:14.720 --> 00:08:18.690 So, I think this is, on the one hand, a trivial point, 00:08:18.690 --> 00:08:21.097 but it's just really easy to forget about 00:08:21.097 --> 00:08:21.930 as you move forward. 00:08:21.930 --> 00:08:24.640 Because these things, they look awfully similar, right? 00:08:24.640 --> 00:08:27.430 There's a K, little subscript something, right? 00:08:27.430 --> 00:08:29.620 So just be careful about this kind of thing. 00:08:34.929 --> 00:08:38.419 Are there any questions about what I've said so far? 00:08:38.419 --> 00:08:39.610 So in these cases. 00:08:39.610 --> 00:08:41.640 I think that it's really very useful 00:08:41.640 --> 00:08:44.890 to try to get some intuition for what's going on. 00:08:44.890 --> 00:08:46.820 These are all just definitions, but you 00:08:46.820 --> 00:08:48.861 want to ask, well, what happens if concentrations 00:08:48.861 --> 00:08:50.415 of various things move around? 00:08:50.415 --> 00:08:52.840 What we want to think about is just the fraction. 00:08:52.840 --> 00:08:56.420 And the reason we call this E is because, for now, we 00:08:56.420 --> 00:09:03.550 might be calling this an enzyme and, over here, a substrate, 00:09:03.550 --> 00:09:06.370 something that the enzyme is acting on. 00:09:06.370 --> 00:09:08.499 But we'll see that, in many cases, 00:09:08.499 --> 00:09:10.540 we might be thinking about one of these as being, 00:09:10.540 --> 00:09:12.319 let's say, the piece of DNA, and then 00:09:12.319 --> 00:09:13.610 this is not even the substrate. 00:09:13.610 --> 00:09:15.885 Then maybe this is the RNA polymerase that 00:09:15.885 --> 00:09:17.010 will lead to transcription. 00:09:17.010 --> 00:09:20.560 So in various contexts, we'll think about these things having 00:09:20.560 --> 00:09:22.510 different molecular identities. 00:09:22.510 --> 00:09:26.360 But for now, E and S, possibly enzyme substrate. 00:09:26.360 --> 00:09:29.520 So the question is, if for now we just think, 00:09:29.520 --> 00:09:32.620 these are just two molecules of whatever sort, 00:09:32.620 --> 00:09:35.070 at some concentration, and we just want to make sure 00:09:35.070 --> 00:09:37.040 that we are on top of what's going 00:09:37.040 --> 00:09:39.000 to happen if the concentrations of each 00:09:39.000 --> 00:09:41.390 of these molecular components goes 00:09:41.390 --> 00:09:43.830 either to 0 or to infinity. 00:09:43.830 --> 00:09:47.170 I think that before you do any math in life, 00:09:47.170 --> 00:09:49.510 it's good to just think about these sorts of limits, 00:09:49.510 --> 00:09:53.690 because it helps to make sure that your intuition is correct. 00:09:53.690 --> 00:09:57.972 In many, many cases, if you think about the problem 00:09:57.972 --> 00:10:00.180 before you do any math, then when you go do the math, 00:10:00.180 --> 00:10:01.221 you'll get some solution. 00:10:01.221 --> 00:10:03.330 You can check to see whether your solution is 00:10:03.330 --> 00:10:05.665 consistent with what your intuition said. 00:10:05.665 --> 00:10:07.040 And if they disagree it means you 00:10:07.040 --> 00:10:10.290 have to update either your intuition, or the solution, 00:10:10.290 --> 00:10:11.610 or maybe both. 00:10:11.610 --> 00:10:13.110 It's possible. 00:10:13.110 --> 00:10:15.130 But at least one of them has to be updated, 00:10:15.130 --> 00:10:18.830 and that's a way of both getting better scores on your exams, 00:10:18.830 --> 00:10:22.660 but also improving your scientific intuition. 00:10:22.660 --> 00:10:25.810 So, in particular, we just want to do some limits. 00:10:29.840 --> 00:10:33.040 We'll think in the context, for example, 00:10:33.040 --> 00:10:36.240 if the total concentration-- your adding of the small S-- 00:10:36.240 --> 00:10:38.150 if it goes to zero, what we're going 00:10:38.150 --> 00:10:43.570 to try to get intuition about is this fraction of E-bound. 00:10:43.570 --> 00:10:44.590 It might be the enzyme. 00:10:44.590 --> 00:10:46.160 So the fraction of this thing bound, 00:10:46.160 --> 00:10:50.260 it's defined by the concentration of the complex, 00:10:50.260 --> 00:10:53.290 divided by the concentration of the enzyme, 00:10:53.290 --> 00:10:56.490 plus the concentration of the enzyme in the complex, 00:10:56.490 --> 00:11:00.360 assuming that this is the only two places that the enzyme can 00:11:00.360 --> 00:11:02.310 be located. 00:11:02.310 --> 00:11:09.890 Now, these three arrows, that, in general, means a definition. 00:11:09.890 --> 00:11:13.070 So the question is, if we come here, 00:11:13.070 --> 00:11:18.090 what happens to the fraction of this enzyme that's bound? 00:11:18.090 --> 00:11:34.180 And, again, can't be determined-- which 00:11:34.180 --> 00:11:35.650 is different from don't know. 00:11:40.729 --> 00:11:42.770 And we're going to just do a few different limits 00:11:42.770 --> 00:11:45.430 so we want to maybe go through these quickly. 00:11:45.430 --> 00:11:47.990 I'll give you 10 seconds to prepare your card. 00:11:58.460 --> 00:11:59.500 All right, ready? 00:11:59.500 --> 00:12:01.315 Three, two, one. 00:12:04.260 --> 00:12:10.300 So we're pretty good. 00:12:10.300 --> 00:12:12.570 So I'd say a majority, at least, of the group 00:12:12.570 --> 00:12:16.630 is saying that in this case, the fraction bound should go to 0. 00:12:16.630 --> 00:12:18.974 Intuitively, why should that be? 00:12:18.974 --> 00:12:21.340 AUDIENCE: [INAUDIBLE] 00:12:21.340 --> 00:12:23.662 PROFESSOR: A little louder. 00:12:23.662 --> 00:12:25.120 AUDIENCE: You have nothing to bind. 00:12:25.120 --> 00:12:26.078 PROFESSOR: Yeah, right. 00:12:26.078 --> 00:12:30.120 So if there's no S around at all, 00:12:30.120 --> 00:12:33.130 then you shouldn't have much of this complex, right? 00:12:33.130 --> 00:12:35.220 But you still have some enzymes. 00:12:35.220 --> 00:12:38.750 This thing should go to 0, and that kind of makes sense. 00:12:38.750 --> 00:12:46.470 And, similarly, if you add a lot, a lot of this substrate? 00:12:46.470 --> 00:12:49.478 I'll give you eight seconds. 00:12:49.478 --> 00:12:52.160 AUDIENCE: So, you're moving the substrate [INAUDIBLE]. 00:12:52.160 --> 00:12:54.900 PROFESSOR: Yes, so S total, this is the total. 00:12:54.900 --> 00:12:56.800 This is if you have a test tube, and this 00:12:56.800 --> 00:13:03.040 is the amount of this sugar that you add in there. 00:13:03.040 --> 00:13:09.170 So S total, then, is the sum of that and S. 00:13:09.170 --> 00:13:12.450 Do you need more time? 00:13:12.450 --> 00:13:12.950 Ready. 00:13:12.950 --> 00:13:16.640 Three, two, one. 00:13:16.640 --> 00:13:21.760 So we have maybe an island of people that disagree. 00:13:25.370 --> 00:13:28.070 At least a majority are saying, in this case, 00:13:28.070 --> 00:13:31.120 it should have to go to 1. 00:13:31.120 --> 00:13:35.454 Of course, this is a limit and a limit. 00:13:35.454 --> 00:13:37.870 It's always going to be between 0 and 1, but in the limit, 00:13:37.870 --> 00:13:38.890 it does go to 1. 00:13:38.890 --> 00:13:41.560 So if you just add so much of the substrate 00:13:41.560 --> 00:13:44.240 then you should be able to saturate that binding 00:13:44.240 --> 00:13:47.210 and drive all of that enzyme into the bound state. 00:13:50.900 --> 00:13:54.460 So this one is actually, maybe, a little bit more subtle. 00:13:54.460 --> 00:14:02.627 So what if we take this limit? 00:14:02.627 --> 00:14:04.960 I'm going to give you 20 seconds to think about it, just 00:14:04.960 --> 00:14:08.446 because it's roughly three times as hard as the last one. 00:14:34.454 --> 00:14:35.370 Do you need more time? 00:14:35.370 --> 00:14:37.660 Or do you think that you have something you believe in, 00:14:37.660 --> 00:14:39.493 that you're willing to turn to your neighbor 00:14:39.493 --> 00:14:41.990 and-- Let's see where we are. 00:14:41.990 --> 00:14:42.840 Ready. 00:14:42.840 --> 00:14:44.445 Three, two, one. 00:14:47.426 --> 00:14:48.550 All right, so this is good. 00:14:48.550 --> 00:14:54.170 So we have a fair distribution, and this is one 00:14:54.170 --> 00:14:57.970 that reasonable people might be able to argue about. 00:14:57.970 --> 00:15:00.890 It's worth having the argument. 00:15:00.890 --> 00:15:03.040 Give yourself 30 seconds, turn your neighbor, 00:15:03.040 --> 00:15:05.280 preferably a neighbor that disagrees with you, 00:15:05.280 --> 00:15:07.780 and tell them why you said what you said. 00:16:08.890 --> 00:16:10.640 If in a pair, you've already convinced 00:16:10.640 --> 00:16:12.890 each other of something, then go ahead and look around 00:16:12.890 --> 00:16:15.000 to see if there's another pair that has maybe 00:16:15.000 --> 00:16:16.380 settled on a different answer. 00:16:43.990 --> 00:16:47.380 I think that you guys are still kind of passionately arguing, 00:16:47.380 --> 00:16:50.800 but maybe we'll go ahead and convene, 00:16:50.800 --> 00:16:55.379 and try to get a sense of-- I think, from the sound of it 00:16:55.379 --> 00:16:57.420 at least, there's some disagreement about the way 00:16:57.420 --> 00:17:00.390 to think about this. 00:17:00.390 --> 00:17:01.890 In general, if you want to volunteer 00:17:01.890 --> 00:17:04.780 an opinion or an explanation, what I like to do is, 00:17:04.780 --> 00:17:08.440 I like to tell the group what your neighbor thought. 00:17:08.440 --> 00:17:13.276 So go ahead, anybody, it could be a neighbor in quotes. 00:17:13.276 --> 00:17:15.359 Anybody want to volunteer one possible explanation 00:17:15.359 --> 00:17:18.109 of how to think about this? 00:17:18.109 --> 00:17:21.109 AUDIENCE: Well, what my neighbors thought 00:17:21.109 --> 00:17:27.440 was if E total is defined as E plus ES, 00:17:27.440 --> 00:17:33.450 then as E total goes to 0, then this goes to 0 over 0 00:17:33.450 --> 00:17:37.930 at some vague, unclear-- 00:17:37.930 --> 00:17:39.700 PROFESSOR: Yeah, although, right. 00:17:39.700 --> 00:17:41.750 So you're saying, maybe it's all going to 0, 00:17:41.750 --> 00:17:43.580 and then this is just philosophy. 00:17:45.984 --> 00:17:48.150 Not that I'm putting words in your neighbor's mouth. 00:17:52.040 --> 00:17:55.885 So there's a sense in which this is true, but mathematically, 00:17:55.885 --> 00:17:59.630 and actually also physically, there are well-defined ways 00:17:59.630 --> 00:18:01.430 of taking such a limit, right? 00:18:01.430 --> 00:18:05.350 So if you get 0 over 0, then you can use L'Hopital's Rule, 00:18:05.350 --> 00:18:07.840 which we'll have the opportunity to pull out sometime 00:18:07.840 --> 00:18:08.710 within the class. 00:18:08.710 --> 00:18:10.395 Of course, the most difficult part of L'Hopital's Rule 00:18:10.395 --> 00:18:11.760 is knowing how to spell it. 00:18:21.214 --> 00:18:23.130 So that's one answer, the mathematical answer, 00:18:23.130 --> 00:18:25.860 that you should be able to just take this limit and so forth. 00:18:25.860 --> 00:18:28.110 But I think there is another physical answer, which is 00:18:28.110 --> 00:18:30.870 that this could happen, right? 00:18:30.870 --> 00:18:36.206 And something is going to occur, right? 00:18:36.206 --> 00:18:37.580 You could, in principle, measure. 00:18:37.580 --> 00:18:39.980 And even if you just had a single enzyme there, 00:18:39.980 --> 00:18:45.170 this Fb would be the fraction of time that that enzyme is bound. 00:18:45.170 --> 00:18:49.820 So this is a well-defined experimental question 00:18:49.820 --> 00:18:52.430 and the answer should arise from these interactions. 00:18:56.150 --> 00:19:00.290 But then how do we edit it-- what does it all mean? 00:19:00.290 --> 00:19:02.850 How do we figure out the answer? 00:19:02.850 --> 00:19:06.930 What's another possible view on this? 00:19:06.930 --> 00:19:08.242 Maybe in the back. 00:19:08.242 --> 00:19:09.700 AUDIENCE: My neighbor thought that, 00:19:09.700 --> 00:19:12.694 if there's a non-zero concentration of S, 00:19:12.694 --> 00:19:15.480 and the concentration of E goes to 0, 00:19:15.480 --> 00:19:18.047 then all E will be bound [? at some point ?]. 00:19:18.047 --> 00:19:19.630 PROFESSOR: This is interesting, right? 00:19:19.630 --> 00:19:22.800 So, if there's a finite concentration of S, 00:19:22.800 --> 00:19:24.870 if E goes to 0, then you say well there's 00:19:24.870 --> 00:19:28.934 plenty of S to go around, so I should always get bound. 00:19:28.934 --> 00:19:35.326 Is that what the neighbor-- that's another option. 00:19:35.326 --> 00:19:36.700 So, so far, we've had an argument 00:19:36.700 --> 00:19:40.630 for can't be determined, it's philosophy. 00:19:40.630 --> 00:19:44.800 We've had an argument for 1. 00:19:44.800 --> 00:19:45.870 Other possibilities? 00:19:45.870 --> 00:19:49.694 This is an interesting question, because depending 00:19:49.694 --> 00:19:51.860 on how you think about it, you can convince yourself 00:19:51.860 --> 00:19:54.670 that it's anything, right? 00:19:54.670 --> 00:19:55.726 Other possible answers? 00:19:59.270 --> 00:20:00.695 No. 00:20:00.695 --> 00:20:02.595 AUDIENCE: So, I don't hear the good answer. 00:20:05.868 --> 00:20:10.206 I said E because, when there isn't very much E, 00:20:10.206 --> 00:20:19.035 then it's true that a lot of people go into yes, but yeah. 00:20:19.035 --> 00:20:20.520 I don't know what to say. 00:20:24.490 --> 00:20:26.870 PROFESSOR: Now, that's OK. 00:20:26.870 --> 00:20:29.312 Another take on that answer, or a different one? 00:20:29.312 --> 00:20:30.270 AUDIENCE: I don't know. 00:20:30.270 --> 00:20:31.644 If I had to complete that answer, 00:20:31.644 --> 00:20:34.210 maybe something like, but there still might not 00:20:34.210 --> 00:20:36.480 be a total large concentration of either, 00:20:36.480 --> 00:20:38.736 so there might still be a decent chance for E 00:20:38.736 --> 00:20:41.434 to be around for a while without encountering some of S. 00:20:41.434 --> 00:20:43.100 AUDIENCE 2: So the forward reaction rate 00:20:43.100 --> 00:20:45.400 will also go to 0. 00:20:45.400 --> 00:20:47.650 Even though S is very large, the forward reaction rate 00:20:47.650 --> 00:20:50.290 will also go to 0, as the concentration [INAUDIBLE]. 00:20:50.290 --> 00:20:51.940 PROFESSOR: By the forward rate, it's 00:20:51.940 --> 00:20:53.481 not necessarily a variable. it's just 00:20:53.481 --> 00:20:56.197 that it depends on the substrate concentration in this case. 00:21:00.830 --> 00:21:01.765 Other takes on it? 00:21:07.060 --> 00:21:08.810 This is interesting. 00:21:08.810 --> 00:21:12.750 So I'm going to argue that the most reasonable way to view 00:21:12.750 --> 00:21:14.970 this would give you-- as long as there 00:21:14.970 --> 00:21:17.620 is some finite concentration of that substrate around, 00:21:17.620 --> 00:21:20.840 and I think that there's a very well-defined sense in which 00:21:20.840 --> 00:21:24.170 it's going to go to some finite fraction. 00:21:24.170 --> 00:21:26.990 And I think that when you're thinking about this 00:21:26.990 --> 00:21:32.250 in the context of molecular kinetics, the chemistry of you, 00:21:32.250 --> 00:21:34.660 it still is all well-defined, but the way 00:21:34.660 --> 00:21:37.130 that I think that I get the most clear intuition is just 00:21:37.130 --> 00:21:41.480 to imagine myself as being that one and only enzyme in the test 00:21:41.480 --> 00:21:42.870 tube. 00:21:42.870 --> 00:21:44.600 Now there's going to be some rate that I 00:21:44.600 --> 00:21:48.000 bind to the substrates, right? 00:21:48.000 --> 00:21:49.910 And what's going to determine that rate? 00:21:53.776 --> 00:21:55.740 AUDIENCE: The concentration of the substrate? 00:21:55.740 --> 00:21:57.781 PROFESSOR: The concentration of substrate, right. 00:21:57.781 --> 00:21:59.690 So if I double the substrate concentration, 00:21:59.690 --> 00:22:02.690 what should that do to the rate of me binding? 00:22:02.690 --> 00:22:03.875 It should double it, yeah. 00:22:03.875 --> 00:22:05.250 And then of course there's always 00:22:05.250 --> 00:22:07.700 this Kf somewhere in there, and some units, right? 00:22:07.700 --> 00:22:10.070 But there's going to be some rate that I bind. 00:22:10.070 --> 00:22:11.550 And then when I bind again? 00:22:11.550 --> 00:22:14.180 Now I'm just an enzyme substrate. 00:22:14.180 --> 00:22:16.739 Now, instead thinking about this in the context of chemical 00:22:16.739 --> 00:22:18.280 kinetics, I can just think about this 00:22:18.280 --> 00:22:20.920 from the standpoint of an individual molecule, where 00:22:20.920 --> 00:22:24.080 I'm a complex, S-bound, and there's some rate 00:22:24.080 --> 00:22:25.680 that I fall apart. 00:22:25.680 --> 00:22:27.640 And it's just the balance of those two rates 00:22:27.640 --> 00:22:30.050 of finding a substrate and falling part 00:22:30.050 --> 00:22:32.430 that's going to lead to this fraction bound. 00:22:35.150 --> 00:22:36.858 AUDIENCE: Taking what you just explained, 00:22:36.858 --> 00:22:41.425 couldn't you think about it as E always either 0 or 1, 00:22:41.425 --> 00:22:42.300 because when you're-- 00:22:42.300 --> 00:22:42.625 PROFESSOR: Yeah 00:22:42.625 --> 00:22:45.041 AUDIENCE: --in the case of having one [? and the same-- ?] 00:22:45.041 --> 00:22:46.020 PROFESSOR: Yes. 00:22:46.020 --> 00:22:51.290 So, if you'd like, we could put an average sign here, 00:22:51.290 --> 00:22:52.821 to say an average over time. 00:22:52.821 --> 00:22:55.320 Because in many, many cases, what we're really interested in 00:22:55.320 --> 00:22:58.872 is the fraction of time that, say, the promoter is bound, 00:22:58.872 --> 00:23:00.330 or the enzyme is bound, or whatnot. 00:23:00.330 --> 00:23:04.550 And if you have many, many, many molecules, then 00:23:04.550 --> 00:23:07.911 at any moment in time, the average is the time average. 00:23:07.911 --> 00:23:09.660 But once you're down to a single molecule, 00:23:09.660 --> 00:23:11.493 then you really want to take a time average. 00:23:17.445 --> 00:23:20.967 We'll revisit this in just a moment using the math, 00:23:20.967 --> 00:23:23.300 but what's interesting is the math also can mislead you. 00:23:26.680 --> 00:23:29.890 Other questions? 00:23:29.890 --> 00:23:32.582 If somebody wants to argue forcefully 00:23:32.582 --> 00:23:34.040 what their neighbor said was right, 00:23:34.040 --> 00:23:42.420 then I'm happy to-- we will come back to this in a little bit. 00:23:42.420 --> 00:23:44.910 If I want to do the fourth possibility, which 00:23:44.910 --> 00:23:47.697 is the total concentration of the enzyme, 00:23:47.697 --> 00:23:48.905 it's going to go to infinity. 00:23:52.100 --> 00:23:55.270 I'll give you, again, eight seconds to think and then 00:23:55.270 --> 00:23:56.860 get your card ready. 00:24:10.450 --> 00:24:12.380 Are you ready? 00:24:12.380 --> 00:24:13.870 Three, two, one. 00:24:16.461 --> 00:24:16.960 OK. 00:24:16.960 --> 00:24:21.100 So we actually have a fair amount of disagreement, 00:24:21.100 --> 00:24:25.260 between As and Bs it seems. 00:24:25.260 --> 00:24:27.890 Go ahead and, again, turn to your neighbor, 00:24:27.890 --> 00:24:31.760 but maybe find somebody that disagrees with you. 00:24:31.760 --> 00:24:33.565 Sometimes there are pockets of people that 00:24:33.565 --> 00:24:34.690 agree one way or the other. 00:24:34.690 --> 00:24:38.140 So try to find each other. 00:25:10.560 --> 00:25:11.570 Yeah, I know. 00:25:11.570 --> 00:25:12.162 I understand. 00:25:14.740 --> 00:25:18.280 You can try to figure out the expression for the fraction 00:25:18.280 --> 00:25:23.409 bound, and how that behaves. 00:25:23.409 --> 00:25:25.534 And that's actually kind of weird as well, frankly. 00:25:32.450 --> 00:25:35.707 Why don't we go ahead and reconvene, just so I can see. 00:25:35.707 --> 00:25:37.290 And maybe, let's go ahead and re-vote, 00:25:37.290 --> 00:25:38.870 so I can see if anybody convinced 00:25:38.870 --> 00:25:41.050 anybody of anything else. 00:25:41.050 --> 00:25:41.640 Ready. 00:25:41.640 --> 00:25:44.570 Three, two, one. 00:25:44.570 --> 00:25:47.035 So it seems like now there's pretty good agreement. 00:25:47.035 --> 00:25:48.410 The answer to this is going to be 00:25:48.410 --> 00:25:52.880 A. There were a fair number of people that said B before. 00:25:52.880 --> 00:25:54.724 So here comes our curvy lines. 00:25:57.830 --> 00:26:00.890 And so the idea here is that in the limit 00:26:00.890 --> 00:26:03.410 of the enzyme concentration going to infinity, 00:26:03.410 --> 00:26:05.740 in that limit the fraction bound has to go to 0, 00:26:05.740 --> 00:26:09.170 because you just don't have any substrate to bind. 00:26:09.170 --> 00:26:12.016 And even if all the substrate's bound-- and indeed 00:26:12.016 --> 00:26:13.890 in this limit, what fraction of the substrate 00:26:13.890 --> 00:26:16.051 ends up being bound? 00:26:16.051 --> 00:26:17.110 AUDIENCE: All of it. 00:26:17.110 --> 00:26:19.090 PROFESSOR: All of it, right. 00:26:19.090 --> 00:26:20.760 So then, indeed, the fraction bound 00:26:20.760 --> 00:26:21.780 is just going to be the concentration 00:26:21.780 --> 00:26:24.279 of the substrate divided by the concentration of the enzyme. 00:26:27.895 --> 00:26:29.270 In the limit of the concentration 00:26:29.270 --> 00:26:31.353 of the enzyme going to infinity, then the fraction 00:26:31.353 --> 00:26:33.470 of the enzyme bound is going to go to 0. 00:26:33.470 --> 00:26:34.550 You're unhappy. 00:26:34.550 --> 00:26:38.090 AUDIENCE: I think having made an extremely eloquent argument, 00:26:38.090 --> 00:26:42.560 number 3 for B, I actually think that that answer is wrong. 00:26:42.560 --> 00:26:44.020 PROFESSOR: OK. 00:26:44.020 --> 00:26:45.664 So this one, right? 00:26:45.664 --> 00:26:47.080 Yes, I mean, you had convinced me. 00:26:51.450 --> 00:26:56.160 AUDIENCE: So if you take E total to 0 at fixed S-- 00:26:56.160 --> 00:26:57.160 PROFESSOR: Fixed S, yes. 00:26:57.160 --> 00:27:01.228 AUDIENCE: --then the reaction rate is-- 00:27:04.306 --> 00:27:06.722 PROFESSOR: You have to take the limits carefully, I think. 00:27:06.722 --> 00:27:10.658 AUDIENCE: --is a ratio of the forward propensity 00:27:10.658 --> 00:27:12.134 and the backwards propensity. 00:27:14.894 --> 00:27:15.560 PROFESSOR: Yeah. 00:27:18.242 --> 00:27:20.200 I think that the clearest way to think about it 00:27:20.200 --> 00:27:22.460 is as just that single enzyme. 00:27:22.460 --> 00:27:25.632 Because, it's certainly going to have some rate of binding, 00:27:25.632 --> 00:27:27.090 and then once it's bound it's going 00:27:27.090 --> 00:27:28.548 to have some rate of falling apart. 00:27:30.565 --> 00:27:32.690 And that ratio is not a function of whether there's 00:27:32.690 --> 00:27:35.230 one enzyme-- I mean, that ratio is well-defined. 00:27:38.350 --> 00:27:40.890 The time average of the probability 00:27:40.890 --> 00:27:42.710 of that enzyme being bound is, indeed, 00:27:42.710 --> 00:27:43.710 a well-defined quantity. 00:27:46.320 --> 00:27:50.080 AUDIENCE: So can Ethan say D and it depends on the Kd? 00:27:50.080 --> 00:27:54.270 PROFESSOR: Well, people always like to argue Ds. 00:27:58.170 --> 00:28:00.590 We can write down what the expression is, actually, now. 00:28:00.590 --> 00:28:05.560 And then you can decide whether you think D is justified. 00:28:05.560 --> 00:28:07.310 We can also just try to figure out, 00:28:07.310 --> 00:28:09.250 what's the equilibrium of this thing? 00:28:09.250 --> 00:28:14.972 So the change in the complex concentration 00:28:14.972 --> 00:28:16.430 as a function of time is just going 00:28:16.430 --> 00:28:19.020 to be the rate of creation minus the rate of destruction. 00:28:19.020 --> 00:28:21.640 So there's just going to be this K forward, 00:28:21.640 --> 00:28:35.470 ES-- oh sorry, these brackets are awful-- Kr, ES. 00:28:35.470 --> 00:28:37.320 And we just want to set this equal to 0, 00:28:37.320 --> 00:28:40.950 if we want to figure out what the equilibrium there is. 00:28:40.950 --> 00:28:46.840 And in one line, you can find that the fraction bound can 00:28:46.840 --> 00:28:50.860 be written as a concentration of the substrate here, 00:28:50.860 --> 00:28:56.996 divided by Kd, plus S. Is this correct from standpoint 00:28:56.996 --> 00:28:57.495 of units? 00:29:04.440 --> 00:29:09.300 So there's something that you might find troubling 00:29:09.300 --> 00:29:11.114 about this expression, though. 00:29:15.650 --> 00:29:17.090 Is anybody troubled by it? 00:29:19.850 --> 00:29:22.317 AUDIENCE: But it equals 0. 00:29:22.317 --> 00:29:24.275 PROFESSOR: You could say, what if E total is 0? 00:29:27.820 --> 00:29:31.760 I'd say that if E total is actually zero, now 00:29:31.760 --> 00:29:33.430 I think that's an ill-defined quantity. 00:29:33.430 --> 00:29:36.620 So, at some point, I'll side with the philosophers here. 00:29:36.620 --> 00:29:41.160 But if there is an enzyme to talk about fraction bound, 00:29:41.160 --> 00:29:44.840 then-- So it's related to this, but it's-- 00:29:47.756 --> 00:29:52.032 AUDIENCE: But whenever you have [INAUDIBLE] E [INAUDIBLE]. 00:29:52.032 --> 00:29:52.740 PROFESSOR: Right. 00:29:52.740 --> 00:29:55.730 So we've already discussed from our intuition 00:29:55.730 --> 00:29:59.390 that as E total goes to infinity, 00:29:59.390 --> 00:30:03.260 then the fraction bound is supposed to go what? 00:30:03.260 --> 00:30:05.360 To 0, we decided, right? 00:30:05.360 --> 00:30:08.436 And does this expression do that? 00:30:08.436 --> 00:30:10.266 AUDIENCE: Well this is S, not S total. 00:30:10.266 --> 00:30:10.890 PROFESSOR: Yes! 00:30:10.890 --> 00:30:12.360 So this is S, not S total. 00:30:12.360 --> 00:30:16.940 And once again, really easy to screw this up. 00:30:16.940 --> 00:30:22.790 Because, in many contexts, S and S total are the same thing. 00:30:22.790 --> 00:30:24.850 Especially, in context of enzyme kinetics, 00:30:24.850 --> 00:30:27.880 it's often the case that the concentration enzyme is really 00:30:27.880 --> 00:30:30.680 rather low, and then the substrate concentration 00:30:30.680 --> 00:30:33.630 is huge, so then S and S total we can really 00:30:33.630 --> 00:30:37.090 treat as being interchangeable. 00:30:37.090 --> 00:30:41.100 But here, in the general context, we can't. 00:30:41.100 --> 00:30:44.210 And this concentration of S, this thing 00:30:44.210 --> 00:30:49.470 is a function of the concentration of the enzyme. 00:30:49.470 --> 00:30:52.590 So this guy here, it's a function 00:30:52.590 --> 00:30:56.100 of the total amount of substrate you have 00:30:56.100 --> 00:31:00.695 and also E total, and for that matter, Kd. 00:31:06.960 --> 00:31:11.180 So really this thing is a true statement, 00:31:11.180 --> 00:31:14.337 but it's very misleading if you're not keeping track 00:31:14.337 --> 00:31:15.420 of what these things mean. 00:31:15.420 --> 00:31:19.594 Because, this thing is very simple, except for that 00:31:19.594 --> 00:31:22.010 it's really actually complicated because S section depends 00:31:22.010 --> 00:31:24.720 on everything right? 00:31:24.720 --> 00:31:26.220 Now there's one context in which you 00:31:26.220 --> 00:31:29.470 can be safe in assuming that S and S total are the same, 00:31:29.470 --> 00:31:32.540 and that's in the limit of, for example, E total going to 0. 00:31:32.540 --> 00:31:38.080 So if you just have 1 enzyme, then this expression 00:31:38.080 --> 00:31:41.400 is, essentially, always valid. 00:31:41.400 --> 00:31:43.344 Every now and then, you might be using up 00:31:43.344 --> 00:31:45.260 one of your substrate molecules, occasionally. 00:31:45.260 --> 00:31:46.240 Right? 00:31:46.240 --> 00:31:49.050 But it's pretty safe to say that in the limit of E total 00:31:49.050 --> 00:31:51.600 going 0, when it's just in the limit of one enzyme, 00:31:51.600 --> 00:31:54.130 then it's really just described by a curve that 00:31:54.130 --> 00:31:56.110 looks like this. 00:31:56.110 --> 00:31:59.220 And this curve is something that you see over, and over, 00:31:59.220 --> 00:32:02.080 and over again. 00:32:02.080 --> 00:32:03.720 And this is the fundamental reason 00:32:03.720 --> 00:32:08.020 that Michaelis Menten kinetics looks the way it does. 00:32:08.020 --> 00:32:10.390 But this is going to be very useful for us 00:32:10.390 --> 00:32:13.770 because, in many contexts that we're interested in, 00:32:13.770 --> 00:32:15.810 we want to think about, for example, 00:32:15.810 --> 00:32:19.610 the rate expression of some gene. 00:32:19.610 --> 00:32:22.180 And what we want to know about is the fraction of time 00:32:22.180 --> 00:32:25.890 that it's going to be bound by, say, a transcription factor. 00:32:25.890 --> 00:32:27.530 So then, in the simplest case, we 00:32:27.530 --> 00:32:31.010 get an input-output relationship that is just given by this. 00:32:31.010 --> 00:32:34.120 Because, there's just one, or few copies, of that DNA, 00:32:34.120 --> 00:32:36.740 so it doesn't really sequester the transcription factor 00:32:36.740 --> 00:32:37.990 that's going to be binding it. 00:32:43.810 --> 00:32:48.640 Do you guys understand why this is weird? 00:32:48.640 --> 00:32:50.240 A thing you have to be careful of? 00:32:54.040 --> 00:32:56.770 And more generally, I strongly recommend 00:32:56.770 --> 00:32:59.390 that, in all of these sorts of problems, 00:32:59.390 --> 00:33:02.884 it's good to just plot some things. 00:33:02.884 --> 00:33:04.300 For example, the fraction bound is 00:33:04.300 --> 00:33:08.850 a function of if you vary the substrate concentration. 00:33:08.850 --> 00:33:12.420 Because, often you think that you know what's going on, 00:33:12.420 --> 00:33:15.460 and then when you just go and sit down to draw some curve, 00:33:15.460 --> 00:33:17.070 just get your intuition, you realize 00:33:17.070 --> 00:33:18.860 you don't know where it starts, you don't where it ends, 00:33:18.860 --> 00:33:20.630 you don't know what it does in between. 00:33:20.630 --> 00:33:22.010 It's embarrassing, but it's only when 00:33:22.010 --> 00:33:24.301 you sit down and try to do something like that that you 00:33:24.301 --> 00:33:26.450 realize that it's not obvious. 00:33:26.450 --> 00:33:29.700 So just, for example, it's useful to imagine a situation, 00:33:29.700 --> 00:33:32.280 just between a similar E and S, where 00:33:32.280 --> 00:33:36.390 we, for the sake of argument, say 00:33:36.390 --> 00:33:39.690 that S total is around this Kd. 00:33:43.380 --> 00:33:47.460 Now what we want to do is ask, what's the fraction 00:33:47.460 --> 00:34:00.000 bound as a function of E total? 00:34:00.000 --> 00:34:02.530 So we're going to fix total substrate, 00:34:02.530 --> 00:34:05.555 vary the enzyme concentration. 00:34:09.270 --> 00:34:11.770 So we already know what the limit here should be. 00:34:11.770 --> 00:34:13.560 We go to infinity, what should this go to? 00:34:17.429 --> 00:34:18.651 0. 00:34:18.651 --> 00:34:20.400 We know that eventually it should go to 0, 00:34:20.400 --> 00:34:24.360 and we already figured that out for any finite substrate 00:34:24.360 --> 00:34:25.330 concentration. 00:34:25.330 --> 00:34:28.739 Incidentally, on many of the exams, 00:34:28.739 --> 00:34:33.210 I will ask for plots of curves like this. 00:34:33.210 --> 00:34:36.940 So basically you want to indicate 00:34:36.940 --> 00:34:38.941 where it goes on one end, where it starts-- 00:34:38.941 --> 00:34:41.440 and where is it going to start in the limit of E total going 00:34:41.440 --> 00:34:44.239 to 0? 00:34:44.239 --> 00:34:46.460 Half. 00:34:46.460 --> 00:34:50.489 Then it's actually accurately described by that. 00:34:50.489 --> 00:34:51.250 So we start here. 00:34:51.250 --> 00:34:52.210 This is 1. 00:34:52.210 --> 00:34:55.199 So we start at 1/2. 00:34:55.199 --> 00:34:59.590 And it's going to have to go in between those two, right? 00:34:59.590 --> 00:35:02.480 So what's the characteristic concentration here 00:35:02.480 --> 00:35:05.360 where something-- where it's changing a lot? 00:35:14.162 --> 00:35:15.107 AUDIENCE: Kd? 00:35:15.107 --> 00:35:15.940 PROFESSOR: Yeah, Kd. 00:35:15.940 --> 00:35:18.470 Kd's actually the only concentration in the problem. 00:35:18.470 --> 00:35:20.260 So that means that's what sets scale. 00:35:20.260 --> 00:35:22.340 And I don't know, Kd, exactly where it should 00:35:22.340 --> 00:35:24.680 be, but something in there. 00:35:24.680 --> 00:35:26.980 So in these sorts of situations, you 00:35:26.980 --> 00:35:30.910 want to get the limits and what is it that sets the scale? 00:35:30.910 --> 00:35:32.960 If there's a peak, is where is it? 00:35:37.390 --> 00:35:39.639 I encourage for a few toy examples, 00:35:39.639 --> 00:35:40.930 just draw some of these things. 00:35:40.930 --> 00:35:45.460 It's a fun way to spend a Saturday afternoon. 00:35:45.460 --> 00:35:49.215 Are there any questions about what we've said so far? 00:35:49.215 --> 00:35:49.715 Yes. 00:35:49.715 --> 00:35:53.352 AUDIENCE: What's the Fb when E total equals Kd? 00:35:53.352 --> 00:35:54.810 PROFESSOR: So the question is, what 00:35:54.810 --> 00:35:57.790 is the fraction of the enzyme that's bound 00:35:57.790 --> 00:36:03.270 when E total is equal to Kd? 00:36:03.270 --> 00:36:06.790 I think we could figure it out, but it might actually 00:36:06.790 --> 00:36:09.770 be a little bit of math. 00:36:09.770 --> 00:36:10.270 Let me see. 00:36:14.980 --> 00:36:17.070 I would have to think about it, but it's 00:36:17.070 --> 00:36:21.070 going to be around a third or a fifth, somewhere in there. 00:36:24.117 --> 00:36:25.950 If somebody gets bored with what I'm saying, 00:36:25.950 --> 00:36:27.590 they can do the calculation and report to us 00:36:27.590 --> 00:36:28.423 at the end of class. 00:36:34.830 --> 00:36:41.120 So what we've just done, it feels like a lot of time 00:36:41.120 --> 00:36:44.000 to spend on two molecules binding to each other, 00:36:44.000 --> 00:36:46.020 but I think that it's good to just make sure 00:36:46.020 --> 00:36:48.629 that you're comfortable with the simplest kind of process 00:36:48.629 --> 00:36:50.420 before you start thinking about things that 00:36:50.420 --> 00:36:53.650 are super complicated, for example, Michaelis Menten 00:36:53.650 --> 00:36:55.030 kinetics. 00:36:55.030 --> 00:36:57.380 So it's not super complicated. 00:36:57.380 --> 00:37:01.260 E plus S. So now it's the same thing here, 00:37:01.260 --> 00:37:05.760 where we have K forward, K reverse, to this complex. 00:37:05.760 --> 00:37:13.290 But here, at some rate, Kcat, enzyme does something, 00:37:13.290 --> 00:37:15.710 turns it into a product. 00:37:15.710 --> 00:37:19.410 Now this is a model of how an enzyme works. 00:37:19.410 --> 00:37:26.170 It is not a perfect description of reality in any given case, 00:37:26.170 --> 00:37:27.740 or in general. 00:37:27.740 --> 00:37:30.560 What's the most obvious possible point of concern? 00:37:34.402 --> 00:37:35.860 AUDIENCE: There's no way to go back 00:37:35.860 --> 00:37:37.160 PROFESSOR: No way to go back, well that's OK. 00:37:37.160 --> 00:37:38.368 What's that matter with that? 00:37:38.368 --> 00:37:39.870 AUDIENCE: Well, sometimes there is. 00:37:39.870 --> 00:37:40.980 PROFESSOR: Well sometimes there is. 00:37:40.980 --> 00:37:41.479 OK. 00:37:44.089 --> 00:37:45.630 I'd say the problem is, in some ways, 00:37:45.630 --> 00:37:47.691 more fundamental than just, sometimes there is. 00:37:47.691 --> 00:37:48.190 Right? 00:37:53.239 --> 00:37:55.530 It's true that sometimes-- but sometimes lots of things 00:37:55.530 --> 00:37:56.029 happen. 00:37:56.029 --> 00:38:01.430 Sometimes the enzyme binds 2 substrates. 00:38:01.430 --> 00:38:03.660 On any specific case, there are many ways 00:38:03.660 --> 00:38:08.050 that this thing can fail, but there's 00:38:08.050 --> 00:38:10.285 a more fundamental sense which is a problem. 00:38:10.285 --> 00:38:15.430 AUDIENCE: The rate at which it produces the-- P doesn't depend 00:38:15.430 --> 00:38:17.020 on any other small molecules. 00:38:17.020 --> 00:38:18.520 It depends on other concentrations-- 00:38:18.520 --> 00:38:19.990 PROFESSOR: OK, right. 00:38:19.990 --> 00:38:22.124 So what Sam is saying is, well this Kcat 00:38:22.124 --> 00:38:23.540 is not a function of other things. 00:38:23.540 --> 00:38:25.359 AUDIENCE: Yeah. 00:38:25.359 --> 00:38:27.400 PROFESSOR: It's true, and in many cases in might, 00:38:27.400 --> 00:38:37.200 but there's a real sense in which this thing is failing 00:38:37.200 --> 00:38:39.140 fundamentally for any enzyme. 00:38:39.140 --> 00:38:41.848 And I just want to make sure that we're all-- 00:38:41.848 --> 00:38:43.560 AUDIENCE: Dissociation of P. 00:38:43.560 --> 00:38:45.370 PROFESSOR: Dissociation of P, and what-- 00:38:45.370 --> 00:38:47.320 AUDIENCE: From the enzyme. 00:38:47.320 --> 00:38:50.500 PROFESSOR: So, you don't like the dissociation? 00:38:50.500 --> 00:38:52.374 AUDIENCE: We don't have an association there. 00:38:52.374 --> 00:38:54.248 You're assuming that the Kf of disocciation-- 00:38:54.248 --> 00:38:55.940 PROFESSOR: Oh, right. 00:38:55.940 --> 00:39:00.690 Although, I could argue, Kcat is some kind of bulk parameter 00:39:00.690 --> 00:39:03.340 that tells you about the rate of breaking some bond 00:39:03.340 --> 00:39:04.570 and dissociating. 00:39:04.570 --> 00:39:08.150 And it's just a simple model. 00:39:08.150 --> 00:39:10.595 We don't want to ask too much of it. 00:39:14.554 --> 00:39:16.842 AUDIENCE: K reverse is huge. 00:39:16.842 --> 00:39:18.050 PROFESSOR: K reverse is huge? 00:39:18.050 --> 00:39:19.841 Well, I haven't told you what K reverse is. 00:39:19.841 --> 00:39:23.100 So, it's not huge. 00:39:23.100 --> 00:39:24.020 I mean, it could be. 00:39:24.020 --> 00:39:26.061 So far we haven't said anything about what it is. 00:39:29.230 --> 00:39:32.540 What are the fundamental properties of an enzyme? 00:39:32.540 --> 00:39:34.220 Or a catalyst, for that matter? 00:39:38.570 --> 00:39:40.460 When you go home for Thanksgiving, 00:39:40.460 --> 00:39:44.030 your grandmother asks you, honey, tell me, 00:39:44.030 --> 00:39:46.094 what's a catalyst? 00:39:46.094 --> 00:39:48.260 AUDIENCE: It's doesn't get used during the reaction. 00:39:48.260 --> 00:39:48.710 PROFESSOR: What's that? 00:39:48.710 --> 00:39:50.310 AUDIENCE: It doesn't get used up during the reaction. 00:39:50.310 --> 00:39:52.393 PROFESSOR: It doesn't get used up in the reaction. 00:39:52.393 --> 00:39:55.110 OK, perfect, not used up. 00:39:55.110 --> 00:39:58.084 And does this model violate that? 00:39:58.084 --> 00:39:58.890 AUDIENCE: No. 00:39:58.890 --> 00:40:00.630 PROFESSOR: So all right. 00:40:00.630 --> 00:40:01.500 Grandma's happy. 00:40:06.750 --> 00:40:10.145 AUDIENCE: You can deactivate or activate these catalysts? 00:40:10.145 --> 00:40:11.610 I don't know. 00:40:11.610 --> 00:40:13.350 PROFESSOR: Right, so it's true, there's 00:40:13.350 --> 00:40:15.230 some enzymes you can activate, deactivate. 00:40:15.230 --> 00:40:20.702 How you deactivate a protein enzyme, if you wanted to? 00:40:20.702 --> 00:40:21.670 AUDIENCE: Denature. 00:40:21.670 --> 00:40:24.600 PROFESSOR: You could denature it from heat or salt. 00:40:24.600 --> 00:40:31.620 But that's maybe not one of the most fundamental. 00:40:31.620 --> 00:40:34.766 AUDIENCE: You should have a rate from S to P without the enzyme. 00:40:34.766 --> 00:40:37.140 PROFESSOR: Right so maybe there should be a rate, S to P, 00:40:37.140 --> 00:40:39.230 without the enzyme. 00:40:39.230 --> 00:40:40.970 Although, I'm just trying to tell you 00:40:40.970 --> 00:40:43.200 about the rate of what the enzyme is doing, 00:40:43.200 --> 00:40:47.620 so you could write a difference equation. 00:40:47.620 --> 00:40:51.132 If I just let this model go to infinity, then what happens? 00:40:51.132 --> 00:40:52.980 AUDIENCE: You get P. 00:40:52.980 --> 00:40:55.800 PROFESSOR: You get P. And how much P is it? 00:40:55.800 --> 00:40:56.340 A lot of P? 00:40:56.340 --> 00:40:57.090 A little bit of P? 00:40:57.090 --> 00:40:58.805 AUDIENCE: As much as it can make. 00:40:58.805 --> 00:41:00.370 PROFESSOR: It's all pee. 00:41:00.370 --> 00:41:01.170 Right? 00:41:01.170 --> 00:41:03.390 And how much substrate? 00:41:03.390 --> 00:41:05.250 None, right? 00:41:05.250 --> 00:41:08.600 So if I just let this go, you have 0 substrate, all product. 00:41:11.210 --> 00:41:11.990 Is that OK? 00:41:11.990 --> 00:41:18.160 I mean is that, in general-- I like product. 00:41:23.490 --> 00:41:25.330 AUDIENCE: Time. 00:41:25.330 --> 00:41:26.661 PROFESSOR: Time? 00:41:26.661 --> 00:41:29.455 AUDIENCE: How much time it would make too. 00:41:29.455 --> 00:41:30.830 PROFESSOR: Well, we can calculate 00:41:30.830 --> 00:41:32.690 what the V is in this model, and then we 00:41:32.690 --> 00:41:33.981 could figure out what the time. 00:41:36.650 --> 00:41:40.310 But there's something wrong with that. 00:41:40.310 --> 00:41:42.880 And normally I wouldn't want to belabor the point, 00:41:42.880 --> 00:41:45.105 but it's worth belaboring maybe. 00:41:45.105 --> 00:41:45.730 AUDIENCE: Yeah. 00:41:45.730 --> 00:41:48.705 If we had a back reaction, then that wouldn't happen. 00:41:48.705 --> 00:41:50.330 PROFESSOR: Right, OK, so this gets back 00:41:50.330 --> 00:41:52.490 to your back reaction, right? 00:41:52.490 --> 00:41:54.260 And I like the back rate. 00:41:54.260 --> 00:41:56.790 It's just there was something a little more 00:41:56.790 --> 00:42:00.540 fundamental than the way you phrased it, was my concern. 00:42:00.540 --> 00:42:04.250 Because what you said is, there might be some back rate. 00:42:04.250 --> 00:42:05.020 Right? 00:42:05.020 --> 00:42:09.960 And I guess what I would say is that there's kind of always 00:42:09.960 --> 00:42:13.850 some back rate, or that the equilibrium-- this is 00:42:13.850 --> 00:42:17.160 fundamental-- the equilibrium ratio between S and P, 00:42:17.160 --> 00:42:18.775 how does the enzyme change it? 00:42:18.775 --> 00:42:19.650 AUDIENCE: Not at all. 00:42:19.650 --> 00:42:21.510 PROFESSOR: It doesn't change it, right? 00:42:21.510 --> 00:42:24.950 So if you take the enzyme, invertase, 00:42:24.950 --> 00:42:28.050 and you put it in a test tube with sucrose, 00:42:28.050 --> 00:42:32.040 it's going to break down almost all that sucrose, really fast. 00:42:32.040 --> 00:42:36.530 It's going to speed things up by a factor of 10 to the 5, 00:42:36.530 --> 00:42:38.690 or I don't know, by a lot. 00:42:38.690 --> 00:42:42.869 But if you leave the test tube for a year, 00:42:42.869 --> 00:42:44.660 it comes to an equilibrium with the enzyme. 00:42:44.660 --> 00:42:47.490 If you left it in the test you without the enzyme 00:42:47.490 --> 00:42:52.010 for a million years, you would get to the same outcome. 00:42:52.010 --> 00:42:55.605 You come to some equilibrium between the substrate 00:42:55.605 --> 00:42:56.630 and the product. 00:42:56.630 --> 00:43:00.200 And that's a function of the kinetics. 00:43:00.200 --> 00:43:03.215 There's a delta G and so forth, but the important point 00:43:03.215 --> 00:43:06.210 is that the enzyme does not change that equilibrium. 00:43:08.739 --> 00:43:10.114 AUDIENCE: I just have a question. 00:43:10.114 --> 00:43:13.042 This molecule is effectively [INAUDIBLE]. 00:43:13.042 --> 00:43:15.482 If you leave it in a test tube for a million years, 00:43:15.482 --> 00:43:16.950 then the ATP will all be consumed. 00:43:16.950 --> 00:43:18.300 PROFESSOR: Yes. 00:43:18.300 --> 00:43:20.268 AUDIENCE: But if you keep providing 00:43:20.268 --> 00:43:22.649 ATP, then you should-- 00:43:22.649 --> 00:43:24.940 PROFESSOR: Well, first of all, not all enzymes actually 00:43:24.940 --> 00:43:26.005 are coupled to ATP. 00:43:26.005 --> 00:43:29.560 So ATP is a way of putting out a big delta G, right? 00:43:29.560 --> 00:43:33.180 So that you can really push things far. 00:43:33.180 --> 00:43:36.340 And ATP could, in principle, be included as a co-factor, 00:43:36.340 --> 00:43:38.670 and then you take the overall delta G of that, 00:43:38.670 --> 00:43:40.380 and then calculate it. 00:43:40.380 --> 00:43:41.985 But if you want to keep on adding, 00:43:41.985 --> 00:43:43.110 then it complicates things. 00:43:43.110 --> 00:43:44.545 But I think, for many enzymes, it's more straightforward 00:43:44.545 --> 00:43:46.900 just to think about enzymes that don't require 00:43:46.900 --> 00:43:48.530 any extra input of energy. 00:43:48.530 --> 00:43:50.390 So they're just lowering the energy barrier 00:43:50.390 --> 00:43:52.740 and they're just speeding up the rate of reaction. 00:43:52.740 --> 00:43:53.900 But the important point there is that they're 00:43:53.900 --> 00:43:55.020 speeding up both rates. 00:43:57.960 --> 00:44:01.670 So the equilibrium between those two is not going to change. 00:44:01.670 --> 00:44:05.550 And that's why, this thing, it's a great model, 00:44:05.550 --> 00:44:08.290 but like all models you have to make 00:44:08.290 --> 00:44:11.680 sure you keep track of what the assumptions are going into it. 00:44:11.680 --> 00:44:14.659 Because this is going to violate the laws of physics 00:44:14.659 --> 00:44:16.200 if you take this model too seriously. 00:44:18.971 --> 00:44:21.429 AUDIENCE: I don't understand what the fundamental principle 00:44:21.429 --> 00:44:22.674 that's being violated is. 00:44:22.674 --> 00:44:26.160 Because why is it not that if you have it stable, 00:44:26.160 --> 00:44:29.148 everything is product. 00:44:29.148 --> 00:44:31.140 You never see as in nature. 00:44:31.140 --> 00:44:34.973 I mean, how is that not a physical situation? 00:44:34.973 --> 00:44:37.014 AUDIENCE 2: So if you can get a really small test 00:44:37.014 --> 00:44:41.365 tube with one G and one S, isn't it just like-- 00:44:41.365 --> 00:44:42.740 PROFESSOR: But the same statement 00:44:42.740 --> 00:44:44.260 that we talked about for single, then 00:44:44.260 --> 00:44:46.490 you would want-- if you had just a single substrate going 00:44:46.490 --> 00:44:48.281 to product-- then you want to look probably 00:44:48.281 --> 00:44:49.290 at the time average. 00:44:49.290 --> 00:44:53.100 Because, the thing is that the equilibrium is determined 00:44:53.100 --> 00:44:57.280 by the delta G of the reaction. 00:44:57.280 --> 00:45:00.670 And that's going to determine the equilibrium, whether you 00:45:00.670 --> 00:45:05.380 have the enzyme there or not. 00:45:05.380 --> 00:45:08.960 So if the delta G is such that it's 00:45:08.960 --> 00:45:12.110 at equilibrium-- sort of 90% product, 00:45:12.110 --> 00:45:14.812 10% substrate-- then what you can do is go, 00:45:14.812 --> 00:45:16.520 well if you start out with all substrate, 00:45:16.520 --> 00:45:18.240 this model may work wonderfully. 00:45:18.240 --> 00:45:20.490 But then as you're getting closer to that equilibrium, 00:45:20.490 --> 00:45:22.281 then this model's going to be breaking down 00:45:22.281 --> 00:45:23.930 because this model is not accounting 00:45:23.930 --> 00:45:26.780 for the back reaction, as you were saying. 00:45:26.780 --> 00:45:29.990 But I just want to stress that it's not just a detailed model, 00:45:29.990 --> 00:45:32.124 or it's not just a failure for some enzymes, 00:45:32.124 --> 00:45:33.540 this is the way that enzymes work. 00:45:37.980 --> 00:45:39.536 Are there other questions about this? 00:45:39.536 --> 00:45:41.160 Or different ways of thinking about it? 00:45:44.375 --> 00:45:45.250 So, it's not used up. 00:45:45.250 --> 00:45:51.990 It speeds up reaction in both directions. 00:46:03.655 --> 00:46:05.738 AUDIENCE: I mean, but that's not necessarily true. 00:46:05.738 --> 00:46:07.702 You can have an enzyme that is only 00:46:07.702 --> 00:46:11.160 really capable of going in one direction. 00:46:11.160 --> 00:46:12.140 PROFESSOR: Really? 00:46:12.140 --> 00:46:15.510 We should meet after class and you can give me your-- 00:46:15.510 --> 00:46:18.456 AUDIENCE: It basically binds in a particular direction. 00:46:18.456 --> 00:46:19.830 PROFESSOR: It's just not allowed. 00:46:26.280 --> 00:46:28.744 So it's true that enzymes can be-- 00:46:28.744 --> 00:46:31.410 and this is getting to the other fundamental point of an enzyme, 00:46:31.410 --> 00:46:35.090 which is that they, especially enzymes in biology, 00:46:35.090 --> 00:46:39.690 can be exquisitely specific. 00:46:39.690 --> 00:46:43.420 What you're saying is that it's really 00:46:43.420 --> 00:46:46.010 only catalyzing this one, weird reaction, 00:46:46.010 --> 00:46:50.380 going from some funny substrate to some funny product, right? 00:46:50.380 --> 00:46:55.360 But that enzyme also speeds up that back reaction, 00:46:55.360 --> 00:46:59.590 going from the funny product to the funny substrate. 00:46:59.590 --> 00:47:04.560 And that's just like the nature of the beast. 00:47:04.560 --> 00:47:09.140 I'm try to think of what I can-- 00:47:09.140 --> 00:47:11.575 AUDIENCE: That's where you have one enzyme going one way, 00:47:11.575 --> 00:47:15.900 and another going the other way in biology. 00:47:15.900 --> 00:47:17.510 PROFESSOR: So it does happen there, 00:47:17.510 --> 00:47:20.400 but then what they are often doing 00:47:20.400 --> 00:47:24.670 is they're coupling things to ATP hydrolysis or something, 00:47:24.670 --> 00:47:31.240 in order to actually make that reaction go in the single way. 00:47:31.240 --> 00:47:34.440 Just as kind of like a general statement-- because the way 00:47:34.440 --> 00:47:40.010 these things work is that there's some over here 00:47:40.010 --> 00:47:43.080 and it's over here somehow, and these enzymes, 00:47:43.080 --> 00:47:46.236 they just lower this energy barrier. 00:47:46.236 --> 00:47:47.736 AUDIENCE: So the thing that confused 00:47:47.736 --> 00:47:49.950 me at first is that I was just thinking of rates, 00:47:49.950 --> 00:47:52.019 and I think the thing that's important 00:47:52.019 --> 00:47:55.931 is to just realize again that the enzyme doesn't change 00:47:55.931 --> 00:47:59.090 the thermodynamics, it only changes that variable 00:47:59.090 --> 00:48:00.180 to change where they are. 00:48:00.180 --> 00:48:02.600 So the key thing is that it doesn't 00:48:02.600 --> 00:48:06.071 change the ratio of the product through the substrate, 00:48:06.071 --> 00:48:08.020 the rates are realatively-- 00:48:08.020 --> 00:48:10.580 PROFESSOR: Right, because from a thermodynamic standpoint, 00:48:10.580 --> 00:48:13.350 it's not used up, which means there's an enzyme here 00:48:13.350 --> 00:48:15.230 and an enzyme here. 00:48:15.230 --> 00:48:17.840 So these final states, you can think about only 00:48:17.840 --> 00:48:19.620 in terms of the substrate and the product, 00:48:19.620 --> 00:48:23.582 because the enzyme was there in both beginning and ending. 00:48:23.582 --> 00:48:25.040 So from a thermodynamic standpoint, 00:48:25.040 --> 00:48:26.980 it's just you're not allowed to change 00:48:26.980 --> 00:48:28.413 one rate without the other. 00:48:32.200 --> 00:48:42.970 Now in the reading, you saw the Michaelis Menten kinetics, 00:48:42.970 --> 00:48:48.980 where you found that once you reach 00:48:48.980 --> 00:48:52.510 this equilibrium between the enzyme substrate complex, 00:48:52.510 --> 00:48:54.330 the velocity can be described by something 00:48:54.330 --> 00:48:57.140 that is rather simple. 00:48:57.140 --> 00:49:03.820 There's some Km plus S, and then there's some Vmax. 00:49:06.500 --> 00:49:08.940 And if the substrate concentration, 00:49:08.940 --> 00:49:10.540 the total concentration is very large, 00:49:10.540 --> 00:49:14.810 then you can just think about this is the S total. 00:49:14.810 --> 00:49:18.520 Now in this case, this, once again, 00:49:18.520 --> 00:49:21.450 can be thought of in this limit of if the enzyme concentration 00:49:21.450 --> 00:49:23.950 is really small, then this is really just the fraction 00:49:23.950 --> 00:49:25.350 of the enzyme that's bound. 00:49:29.825 --> 00:49:31.700 So we've already spent a lot of time thinking 00:49:31.700 --> 00:49:36.830 about how to get at the fraction bound, 00:49:36.830 --> 00:49:42.670 and the question is, what should this Km be here? 00:49:42.670 --> 00:49:45.000 Now that I've told you that it's the fraction bound, 00:49:45.000 --> 00:49:49.480 is it just going to be the same thing that we had before? 00:49:49.480 --> 00:49:51.020 Is the Km the same thing is the Kd? 00:49:55.490 --> 00:50:00.075 So remember, before, we found that Kd was just Kr over Kf. 00:50:03.710 --> 00:50:07.770 But you should, in principle, be able ti just look at that 00:50:07.770 --> 00:50:10.420 and say what fraction bound should be. 00:50:13.294 --> 00:50:16.647 AUDIENCE: Is it Kr over Kf plus-- other way around, 00:50:16.647 --> 00:50:19.050 Kr plus Kcat over Kf 00:50:19.050 --> 00:50:21.475 PROFESSOR: Yes, because now, from the standpoint 00:50:21.475 --> 00:50:27.330 of the enzyme, there's some rate at which you form the complex. 00:50:27.330 --> 00:50:29.650 And now the lifetime of that complex 00:50:29.650 --> 00:50:31.496 has been reduced, because now there 00:50:31.496 --> 00:50:33.790 are two ways for the complex to fall apart, right? 00:50:33.790 --> 00:50:36.680 One, is could just go back where it came from, 00:50:36.680 --> 00:50:40.200 but the other is that you can catalyze the reaction. 00:50:40.200 --> 00:50:43.350 So, from the standpoint of the enzyme and the fraction bound, 00:50:43.350 --> 00:50:45.330 then we can just-- the entire discussion 00:50:45.330 --> 00:50:47.130 that we had before-- we can just replace Kd 00:50:47.130 --> 00:50:51.680 with this new Michaelis constant, Km. 00:50:51.680 --> 00:50:56.130 Where now, we say now it's the Kr up in the numerator still, 00:50:56.130 --> 00:50:58.810 but now, instead of just being Kf at the bottom, 00:50:58.810 --> 00:51:00.760 we have to add Kcat, because there are just 00:51:00.760 --> 00:51:04.510 two ways that that enzyme substrate 00:51:04.510 --> 00:51:06.430 complex can fall apart. 00:51:13.160 --> 00:51:15.354 Oh I'm sorry, I've already messed up. 00:51:18.258 --> 00:51:26.815 Kf over-- So Kcat has to be with Kr. 00:51:36.490 --> 00:51:39.265 So it just kind of speeds up the effective rate of dissociation. 00:51:42.730 --> 00:51:46.300 And of course, depending whether Kcat is large or small 00:51:46.300 --> 00:51:50.860 as compared to Kr, this can be either a large or small effect. 00:51:50.860 --> 00:51:55.100 But these rates, they just add. 00:51:55.100 --> 00:51:58.630 And we'll spend a lot of time thinking about how rates add 00:51:58.630 --> 00:52:02.259 and so forth in a few weeks. 00:52:02.259 --> 00:52:04.008 AUDIENCE: For this expression to be valid, 00:52:04.008 --> 00:52:06.664 don't you need Kcat to be much longer than the other rates? 00:52:06.664 --> 00:52:07.580 PROFESSOR: Right, yes. 00:52:07.580 --> 00:52:10.485 So you want Kcat to be-- So there's 00:52:10.485 --> 00:52:13.110 various kinds of limits in which you can talk about this thing. 00:52:13.110 --> 00:52:15.600 So in general, what you want is Kcat to be small, 00:52:15.600 --> 00:52:19.652 and you also want the initial transient to have gone away. 00:52:19.652 --> 00:52:21.360 Because when you first add the substrate, 00:52:21.360 --> 00:52:24.290 you don't yet have any enzyme substrate complex. 00:52:24.290 --> 00:52:26.460 So you have to wait until you've gotten 00:52:26.460 --> 00:52:30.454 to this so-called steady state, where the Michaelis Menten 00:52:30.454 --> 00:52:31.120 formula applies. 00:52:31.120 --> 00:52:33.150 And then you also can't have let it 00:52:33.150 --> 00:52:34.890 go too far, because then of course you're 00:52:34.890 --> 00:52:36.556 going to start running out of substrate. 00:52:49.326 --> 00:52:51.200 In the homework, you're going to get a chance 00:52:51.200 --> 00:52:53.366 to play with Michaelis Menten kinetics a little bit, 00:52:53.366 --> 00:52:56.420 and think about the dynamics when you have 00:52:56.420 --> 00:52:58.270 different kinds of inhibitors. 00:52:58.270 --> 00:53:02.320 So you can imagine having inhibitors that inhibit 00:53:02.320 --> 00:53:03.390 multiple different ways. 00:53:03.390 --> 00:53:08.270 You could have an inhibitor the binds the enzyme, 00:53:08.270 --> 00:53:12.230 and prevents the enzyme from providing the substrate. 00:53:12.230 --> 00:53:13.875 Now should this effect the Vmax? 00:53:21.802 --> 00:53:23.260 We'll think about it for 10 seconds 00:53:23.260 --> 00:53:28.050 and we'll vote because it's so much fun. 00:53:28.050 --> 00:53:29.760 We have these cards. 00:53:29.760 --> 00:53:35.930 Vmax change-- and this is with an inhibitor 00:53:35.930 --> 00:53:41.650 that binds here-- and forming an EI complex, reversibly. 00:53:41.650 --> 00:53:43.655 The question is, does Vmax change? 00:53:55.020 --> 00:53:58.560 A is yes and B is no. 00:54:04.320 --> 00:54:06.630 I'll give you 10 seconds to think about this. 00:54:06.630 --> 00:54:12.570 So Vmax is, again, defined as this rate of product formation 00:54:12.570 --> 00:54:15.370 at saturation, when you have a lot of the substrate. 00:54:34.460 --> 00:54:36.290 Do you need time? 00:54:36.290 --> 00:54:39.489 Or will time help? 00:54:39.489 --> 00:54:40.530 Well who wants more time? 00:54:40.530 --> 00:54:41.821 Just nod if you want more time. 00:54:45.260 --> 00:54:46.930 OK, well let's see how we feel. 00:54:46.930 --> 00:54:48.820 Let's go ahead and vote. 00:54:48.820 --> 00:54:53.510 If I add this competitive inhibitor, 00:54:53.510 --> 00:54:55.640 the question is, will be Vmax change? 00:54:55.640 --> 00:54:56.300 Ready. 00:54:56.300 --> 00:54:59.930 Three, two, one. 00:54:59.930 --> 00:55:04.830 So we have a majority of Bs, but some As. 00:55:04.830 --> 00:55:07.570 Can somebody give the intuition for why the Vmax should not 00:55:07.570 --> 00:55:08.070 change? 00:55:11.120 --> 00:55:12.156 Yes. 00:55:12.156 --> 00:55:18.108 AUDIENCE: Vmax is when substrate is 00:55:18.108 --> 00:55:21.580 far excess to the enzyme and, at that time, 00:55:21.580 --> 00:55:25.260 all of the enzymes bond to the substrate not to the inhibitor. 00:55:25.260 --> 00:55:26.260 PROFESSOR: Right, right. 00:55:26.260 --> 00:55:30.415 So Vmax occurs when you have lots and lots of substrate. 00:55:30.415 --> 00:55:31.790 And, of course, the condition you 00:55:31.790 --> 00:55:33.170 have to be a little bit careful, because it's not just 00:55:33.170 --> 00:55:35.120 having more substrate than the enzyme, 00:55:35.120 --> 00:55:38.490 but it's when the substrate is saturating. 00:55:38.490 --> 00:55:40.470 So if you have lots and lots of substrate, 00:55:40.470 --> 00:55:42.210 then the important point there is 00:55:42.210 --> 00:55:45.436 that it's when you've pushed this reaction all 00:55:45.436 --> 00:55:49.414 the way over here, all the enzyme is bound, 00:55:49.414 --> 00:55:51.580 and that's when you get this maximal rate of product 00:55:51.580 --> 00:55:52.680 formation. 00:55:52.680 --> 00:55:55.328 And that's true, you might need more substrate in order 00:55:55.328 --> 00:55:57.036 to get all that enzyme bound, because you 00:55:57.036 --> 00:56:00.870 have to pull the enzyme away from this side reaction. 00:56:00.870 --> 00:56:03.330 And, indeed, this kind of inhibitor 00:56:03.330 --> 00:56:07.820 alters the Km, the effect of Km of the reaction. 00:56:07.820 --> 00:56:13.380 But it does not affect this Vmax, 00:56:13.380 --> 00:56:18.300 whereas other inhibitors can bind this complex 00:56:18.300 --> 00:56:22.300 and prevent it from catalyzing the reaction. 00:56:22.300 --> 00:56:26.800 And that will instead affect Vmax, but won't affect Km. 00:56:26.800 --> 00:56:30.340 So this was a powerful way that enzymologists 00:56:30.340 --> 00:56:34.020 have used to try to get at mechanism of inhibitors. 00:56:34.020 --> 00:56:35.550 So if you have some small molecule 00:56:35.550 --> 00:56:38.309 you know somehow inhibits some enzymatic reaction 00:56:38.309 --> 00:56:40.100 and you want to know, how is it doing that? 00:56:40.100 --> 00:56:43.630 One thing you can do is you can titrate in that inhibitor 00:56:43.630 --> 00:56:47.130 and then measure the Michaelis Menten 00:56:47.130 --> 00:56:50.520 curve to get out the Vmax and Km to try 00:56:50.520 --> 00:56:52.820 to get a sense mechanism. 00:56:52.820 --> 00:56:55.340 And I always say we should be drawing these things. 00:56:55.340 --> 00:56:58.240 So V is a function of-- and this is 00:56:58.240 --> 00:56:59.950 in the [? lit ?] for a lot of substrate 00:56:59.950 --> 00:57:02.780 relative to the enzyme-- then we can indeed 00:57:02.780 --> 00:57:10.620 say it's going to plateau in Vmax at concentration Km. 00:57:10.620 --> 00:57:11.150 It's at 1/2. 00:57:15.580 --> 00:57:17.040 And then it plateaus. 00:57:17.040 --> 00:57:20.390 This is a very, very, very common curve. 00:57:20.390 --> 00:57:26.090 Lots of things in biology and life start at 0 and plateau, 00:57:26.090 --> 00:57:28.455 and there are almost only two ways you can do that. 00:57:28.455 --> 00:57:29.830 OK, there are more than two ways, 00:57:29.830 --> 00:57:32.880 but there are a very small number of ways you can do that. 00:57:32.880 --> 00:57:33.919 This is one of them. 00:57:38.050 --> 00:57:42.480 Any questions on these Michaelis and Menten kinetics inhibitors? 00:57:42.480 --> 00:57:46.990 You're going to spend a couple hours over the next few days 00:57:46.990 --> 00:57:48.270 thinking about this. 00:57:54.720 --> 00:57:57.190 So what I want to do for the last 20 minutes 00:57:57.190 --> 00:57:59.250 is switch gears a little bit and to think 00:57:59.250 --> 00:58:03.757 about the simple dynamics of gene expression. 00:58:03.757 --> 00:58:05.590 The ideas that we've just been talking about 00:58:05.590 --> 00:58:10.915 end up being just very relevant for the simple models here. 00:58:10.915 --> 00:58:13.040 So what we want to think about is a situation where 00:58:13.040 --> 00:58:18.740 we have some transcription factor, X, that is activating 00:58:18.740 --> 00:58:26.930 expression of gene Y. So we have X activating Y. Now, 00:58:26.930 --> 00:58:28.810 the way we can think about this, for example, 00:58:28.810 --> 00:58:34.350 is that we may have X, which together with some 00:58:34.350 --> 00:58:42.530 signal S of X, turns into some X star 00:58:42.530 --> 00:58:48.150 It's X star that can bind to the promoter 00:58:48.150 --> 00:58:59.360 and lead to expression of Y. 00:58:59.360 --> 00:59:02.040 Now in Uri's book, he talks about this idea 00:59:02.040 --> 00:59:04.160 of a separation of time scales that 00:59:04.160 --> 00:59:08.360 is often useful to invoke when thinking about gene expression. 00:59:08.360 --> 00:59:11.860 In this context, what was the fast event? 00:59:17.527 --> 00:59:18.610 AUDIENCE: Activation of X? 00:59:18.610 --> 00:59:21.360 PROFESSOR: Activation of X. So in many cases, 00:59:21.360 --> 00:59:26.730 if this is a sugar or a small molecule that is going to be, 00:59:26.730 --> 00:59:31.150 in this case, activating X, that can occur really quite quickly. 00:59:31.150 --> 00:59:34.180 Often maybe less than a second. 00:59:34.180 --> 00:59:36.900 The rate-limiting step would then, in many cases, 00:59:36.900 --> 00:59:38.980 be getting the signal into the cell, so 00:59:38.980 --> 00:59:42.450 depending on how that works. 00:59:42.450 --> 00:59:45.680 So this occurs very rapidly. 00:59:45.680 --> 00:59:49.160 What that means is if we look at a signal Sx, 00:59:49.160 --> 00:59:53.790 as a function of time, where it starts out being absent 00:59:53.790 --> 00:59:58.430 and then, all of a sudden, sugar appears in the environment, 00:59:58.430 --> 01:00:02.100 we can think about the concentration of X and X star 01:00:02.100 --> 01:00:09.400 So X starts out high and then quickly goes down, right? 01:00:09.400 --> 01:00:16.270 Whereas X star will do the reverse here, quickly comes up. 01:00:16.270 --> 01:00:17.345 And this should be flat. 01:00:20.090 --> 01:00:23.180 Now, what is it that Y will do as a function of time? 01:00:27.050 --> 01:00:31.360 So if X is an activator that means that before X star became 01:00:31.360 --> 01:00:36.500 available, there was no expression of Y. 01:00:36.500 --> 01:00:37.490 So it should be low. 01:00:42.130 --> 01:00:44.550 So X is quickly activated, turns into X star. 01:00:44.550 --> 01:00:52.171 So we start expressing Y. So, roughly, 01:00:52.171 --> 01:00:53.462 what does this curve look like? 01:01:03.100 --> 01:01:05.666 Somebody please help me. 01:01:05.666 --> 01:01:08.155 AUDIENCE: It's S-shaped. 01:01:08.155 --> 01:01:09.780 PROFESSOR: OK, so it could be S-shaped. 01:01:15.590 --> 01:01:18.420 The thing that's very fast is activation of X, 01:01:18.420 --> 01:01:21.570 and then what's really still rather fast 01:01:21.570 --> 01:01:24.450 is equilibration of X star on this promoter. 01:01:24.450 --> 01:01:27.270 So that might still be very rapid 01:01:27.270 --> 01:01:31.610 because these things were nearly instantaneous, But. 01:01:31.610 --> 01:01:34.590 Coming to equilibrium here still might happen over 01:01:34.590 --> 01:01:37.580 time scales of seconds. 01:01:37.580 --> 01:01:39.730 So that means that you actually, sort of quickly, 01:01:39.730 --> 01:01:43.710 start getting expression, at least on time scales 01:01:43.710 --> 01:01:46.180 are relevant in terms of hours kind of time scales. 01:01:46.180 --> 01:01:48.750 Of course, it still does take time to express. 01:01:48.750 --> 01:01:54.950 So it takes minutes for the RNA polymerase to transcribe, 01:01:54.950 --> 01:01:56.720 and then of course the ribosome's 01:01:56.720 --> 01:01:57.970 going to have to do something. 01:02:04.024 --> 01:02:04.940 What do I want to ask? 01:02:09.530 --> 01:02:11.500 Let's write down the equation that Uri 01:02:11.500 --> 01:02:15.569 invokes because there's a very real sense in which it 01:02:15.569 --> 01:02:17.360 does, maybe, look a little bit more S-like. 01:02:17.360 --> 01:02:21.010 But at least in terms of Uri's kind of formalism, 01:02:21.010 --> 01:02:23.510 he often would say, the change in the concentration 01:02:23.510 --> 01:02:28.260 of this protein, it's going to be some function of, 01:02:28.260 --> 01:02:29.510 in this case X star. 01:02:32.590 --> 01:02:35.740 And then there's another term here, which was the minus alpha 01:02:35.740 --> 01:02:39.714 Y. What was the minus alpha Y due to? 01:02:39.714 --> 01:02:40.630 AUDIENCE: Degradation. 01:02:40.630 --> 01:02:43.210 PROFESSOR: Right, so there are two terms. 01:02:43.210 --> 01:02:47.150 So there's alpha, and it's going to be the sum of two things. 01:02:47.150 --> 01:02:51.060 There's alpha due to degradation. 01:02:51.060 --> 01:02:55.230 So if the protein is degraded actively in some way. 01:02:55.230 --> 01:02:56.960 If the protein is not degraded, then 01:02:56.960 --> 01:02:59.940 does that mean that alpha is equal to 0? 01:02:59.940 --> 01:03:00.750 No. 01:03:00.750 --> 01:03:04.504 So what is this other term? 01:03:04.504 --> 01:03:05.420 AUDIENCE: Cell growth. 01:03:05.420 --> 01:03:07.544 PROFESSOR: Right, so it's alpha due to some growth. 01:03:07.544 --> 01:03:13.800 And cell growth leads to some dilution effect. 01:03:13.800 --> 01:03:16.260 So if you have the same number of proteins in the cell 01:03:16.260 --> 01:03:17.760 that the cell is growing, that means 01:03:17.760 --> 01:03:19.240 the concentration is shrinking. 01:03:19.240 --> 01:03:19.780 Right? 01:03:19.780 --> 01:03:22.210 Now the reality of this process is 01:03:22.210 --> 01:03:26.380 that it's complicated, because cell growth is not uniform. 01:03:26.380 --> 01:03:29.610 But if you kind of average over things, 01:03:29.610 --> 01:03:32.230 then a reasonable description is just to say, 01:03:32.230 --> 01:03:35.540 just a first order effective dilution rate. 01:03:35.540 --> 01:03:41.960 If you want to, you can write down a more detailed formula, 01:03:41.960 --> 01:03:44.900 or a model where, you say if cell growth does this, 01:03:44.900 --> 01:03:47.164 then-- It's going to kind of wiggle a little bit 01:03:47.164 --> 01:03:48.580 over the course of the cell cycle, 01:03:48.580 --> 01:03:50.860 but this is a reasonable description. 01:03:50.860 --> 01:03:55.700 Now what this is saying is that even if there 01:03:55.700 --> 01:03:57.550 is no active degradation, then there still 01:03:57.550 --> 01:04:00.760 is an effective term due to this dilution. 01:04:00.760 --> 01:04:05.610 And this means that if we immediately activate, 01:04:05.610 --> 01:04:09.070 and if F of X star-- at time T equal to 0 here-- if it just 01:04:09.070 --> 01:04:12.880 goes to some beta, then what is the long time 01:04:12.880 --> 01:04:13.990 solution of this equation? 01:04:17.750 --> 01:04:18.700 AUDIENCE: Beta/alpha. 01:04:18.700 --> 01:04:19.970 PROFESSOR: Beta/alpha, right? 01:04:19.970 --> 01:04:23.140 So we know it should eventually come to beta/alpha. 01:04:27.538 --> 01:04:29.912 What's the characteristic time scale for it to get there? 01:04:36.716 --> 01:04:40.582 AUDIENCE: Cell alpha's rate, it's 1/alpha. 01:04:40.582 --> 01:04:41.290 PROFESSOR: Right. 01:04:41.290 --> 01:04:44.141 So characteristic time is 1/alpha. 01:04:44.141 --> 01:04:45.890 The solution to this differential equation 01:04:45.890 --> 01:04:53.850 is just an exponential where, if extend this line here, 01:04:53.850 --> 01:04:56.580 this is 1/alpha. 01:04:56.580 --> 01:04:58.120 So that's time. 01:04:58.120 --> 01:05:00.060 And then, of course, the T 1/2, the time 01:05:00.060 --> 01:05:03.340 it takes to get to 1/2, is indeed different by log 2, 01:05:03.340 --> 01:05:04.877 and that's the cell division time. 01:05:10.330 --> 01:05:17.220 This point here-- this is at T 1/2-- is cell division. 01:05:17.220 --> 01:05:21.155 This is for a stable protein, assuming that alpha degradation 01:05:21.155 --> 01:05:21.738 is equal to 0. 01:05:25.950 --> 01:05:29.150 So the thing to remember is that this basic differential 01:05:29.150 --> 01:05:34.450 equation of Y dot is equal to a minus alpha Y, is 01:05:34.450 --> 01:05:37.264 an exponential by going to 0. 01:05:37.264 --> 01:05:39.180 Whereas if you have a constant term here, then 01:05:39.180 --> 01:05:41.180 it's an exponential going to some nonzero value. 01:05:43.750 --> 01:05:49.990 So, indeed, if the signal here goes away, 01:05:49.990 --> 01:05:52.250 then we quickly come back here. 01:05:52.250 --> 01:05:53.755 So this comes here. 01:05:53.755 --> 01:05:59.620 This comes here, and then this-- does it go back down to 0? 01:05:59.620 --> 01:06:02.840 Is it more or less rapid returning to 0 01:06:02.840 --> 01:06:04.810 than it took to come up? 01:06:09.160 --> 01:06:09.820 All right. 01:06:09.820 --> 01:06:10.320 OK. 01:06:15.860 --> 01:06:17.585 So, how can I phrase this? 01:06:20.570 --> 01:06:27.330 OK, faster decay, question mark. 01:06:27.330 --> 01:06:32.840 A is yes, and B is no, and you can always 01:06:32.840 --> 01:06:35.500 do C or something if you don't know what I'm asking. 01:06:35.500 --> 01:06:40.460 The question is, we've turned off the signal, 01:06:40.460 --> 01:06:43.580 is it going to go away faster, or slower, or the same? 01:06:47.830 --> 01:06:48.455 This is faster. 01:06:51.440 --> 01:06:54.530 B can even be slower maybe. 01:06:54.530 --> 01:06:55.175 C is same. 01:06:59.640 --> 01:07:01.900 Do you understand the options now? 01:07:01.900 --> 01:07:06.734 So we stopped expressing Y, so concentration of X 01:07:06.734 --> 01:07:07.900 is going to decrease, right? 01:07:07.900 --> 01:07:12.110 Question is, it is going to go away faster, slower, 01:07:12.110 --> 01:07:15.310 or the same as the rate that it came up? 01:07:20.330 --> 01:07:23.150 Do you need more time? 01:07:23.150 --> 01:07:23.980 Ready. 01:07:23.980 --> 01:07:26.060 Three, two, one. 01:07:30.100 --> 01:07:35.000 OK so we have a fair agreement that, this thing, 01:07:35.000 --> 01:07:36.140 it's going to be the same. 01:07:36.140 --> 01:07:38.550 So there's a characteristic time for it to come 01:07:38.550 --> 01:07:45.350 and it's the same characteristic time for it to degrade away. 01:07:45.350 --> 01:07:51.570 So this, I would say, is not a priori obvious, 01:07:51.570 --> 01:07:55.450 but it's really just the nature of when you 01:07:55.450 --> 01:07:58.122 have these sorts of situations. 01:07:58.122 --> 01:07:59.580 This sets the time scale for if you 01:07:59.580 --> 01:08:01.650 want to change the concentration-- doesn't matter 01:08:01.650 --> 01:08:03.441 whether you're going to 0, a finite number, 01:08:03.441 --> 01:08:05.630 or if you go from high to low, but not 0. 01:08:05.630 --> 01:08:07.796 Again, it's going to be exponential in the same time 01:08:07.796 --> 01:08:09.930 scale. 01:08:09.930 --> 01:08:12.722 So if you want that to be faster, 01:08:12.722 --> 01:08:14.680 if you want to be able to respond more rapidly, 01:08:14.680 --> 01:08:21.420 then one solution would be to actively degrade the protein. 01:08:21.420 --> 01:08:21.970 Right? 01:08:21.970 --> 01:08:25.500 Now it's obvious that degrading the protein actively 01:08:25.500 --> 01:08:28.470 will allow it to go way more rapidly. 01:08:28.470 --> 01:08:30.700 What's perhaps less obvious is that there's 01:08:30.700 --> 01:08:34.229 a real sense in which degrading the protein allows 01:08:34.229 --> 01:08:37.930 this response to be more rapid as well. 01:08:37.930 --> 01:08:42.440 But of course, did we keep everything constant? 01:08:42.440 --> 01:08:44.502 If I say, oh I want the curve to look like this, 01:08:44.502 --> 01:08:46.210 can I just increase the degradation rate? 01:08:49.472 --> 01:08:52.279 AUDIENCE: [INAUDIBLE]. 01:08:52.279 --> 01:08:55.500 PROFESSOR: Well let's say that X star is already maximal 01:08:55.500 --> 01:08:56.380 saturating. 01:08:56.380 --> 01:08:59.193 So we're already-- well, OK. 01:08:59.193 --> 01:08:59.859 So I understand. 01:08:59.859 --> 01:09:02.630 OK, now I understand what you're saying. 01:09:02.630 --> 01:09:04.630 You need to increase beta, and that could either 01:09:04.630 --> 01:09:06.609 be by increasing X star or it could 01:09:06.609 --> 01:09:11.060 be just by increasing the strength of that promoter. 01:09:11.060 --> 01:09:15.039 So the idea is that if you want a more rapid on or off, 01:09:15.039 --> 01:09:16.830 you can also increase the degradation rate. 01:09:16.830 --> 01:09:18.640 But there's a cost to that, which 01:09:18.640 --> 01:09:22.109 is that you have to make more protein. 01:09:22.109 --> 01:09:27.640 And, indeed, many transcription factors are actively degraded. 01:09:27.640 --> 01:09:31.520 And that may be because if you are using these transcription 01:09:31.520 --> 01:09:33.069 factors to turn things on and off, 01:09:33.069 --> 01:09:37.350 then you want to get rapid responses. 01:09:37.350 --> 01:09:39.560 And, also, transcription factors are often not 01:09:39.560 --> 01:09:42.314 expressed at the same high levels as structural proteins. 01:09:42.314 --> 01:09:44.189 So that means that the cost of degrading them 01:09:44.189 --> 01:09:46.450 is not going to be as severe. 01:09:46.450 --> 01:09:52.024 If you're actively degrading cell wall type of things, 01:09:52.024 --> 01:09:53.399 that's going to be really costly. 01:09:56.175 --> 01:09:57.550 Are there any questions of what I 01:09:57.550 --> 01:09:59.790 mean by this discussion of active degradation, 01:09:59.790 --> 01:10:03.370 why it might help, costs? 01:10:03.370 --> 01:10:05.540 Because, over the next week or two, 01:10:05.540 --> 01:10:08.560 we're going to see multiple possible solutions 01:10:08.560 --> 01:10:09.320 to this problem. 01:10:09.320 --> 01:10:12.260 If you want to increase the rate that you respond 01:10:12.260 --> 01:10:14.490 to some environmental change, one way you can do it 01:10:14.490 --> 01:10:17.824 is by actively degrading some of the signaling proteins. 01:10:17.824 --> 01:10:20.240 But there are other solutions we're going to come up with, 01:10:20.240 --> 01:10:21.240 such as auto regulation. 01:10:29.167 --> 01:10:30.750 In the last few minutes here, I wanted 01:10:30.750 --> 01:10:36.240 to say something about this question of ultrasensitivity. 01:10:36.240 --> 01:10:37.660 So there are many cases where you 01:10:37.660 --> 01:10:43.350 would like to get very sensitive responses, i.e. 01:10:43.350 --> 01:10:45.570 you'd like to be able to make a modest change 01:10:45.570 --> 01:10:48.080 in the concentration of some, for example, transcription 01:10:48.080 --> 01:10:51.965 factor, and get a significant change in-- I don't know why 01:10:51.965 --> 01:10:55.272 I erased that but-- and you want to be 01:10:55.272 --> 01:10:58.370 able to get a significant change in the output. 01:10:58.370 --> 01:11:00.050 And one way that you can do this is 01:11:00.050 --> 01:11:02.490 by having some sort of cooperative binding. 01:11:02.490 --> 01:11:05.070 If you have dimerization of a transcription factor 01:11:05.070 --> 01:11:06.930 before binding then you, for example, 01:11:06.930 --> 01:11:09.470 can get a more sensitive response. 01:11:09.470 --> 01:11:14.760 So one way to think that this is if you have an X activating Y 01:11:14.760 --> 01:11:21.280 right in this simple case, then the rate 01:11:21.280 --> 01:11:27.440 of Y expression as a function of-- 01:11:27.440 --> 01:11:30.680 and here we're going to write X for now 01:11:30.680 --> 01:11:35.500 and we'll just assume that all Xs are indeed active. 01:11:35.500 --> 01:11:37.380 OK? 01:11:37.380 --> 01:11:39.910 Now if it just is a single X binding Y, 01:11:39.910 --> 01:11:43.580 then this should behave just like this Michaelis Menten 01:11:43.580 --> 01:11:46.280 formula, where there's some maximal rate of expression 01:11:46.280 --> 01:11:47.730 here. 01:11:47.730 --> 01:11:54.100 There's going to be some Kd, which is bound 1/2 the time, 01:11:54.100 --> 01:11:56.454 and then some curve that looks like this. 01:11:56.454 --> 01:11:58.120 So this would be an example of something 01:11:58.120 --> 01:12:01.310 that is not ultrasensitive and that you 01:12:01.310 --> 01:12:06.350 don't get a significant change in the rate of Y expression-- 01:12:06.350 --> 01:12:12.500 or the equilibrium Y value, if you'd like-- 01:12:12.500 --> 01:12:15.110 as a function of changing X. 01:12:15.110 --> 01:12:17.212 As you start having more and more X, 01:12:17.212 --> 01:12:18.670 you would need get more and more Y, 01:12:18.670 --> 01:12:23.080 but that ratio, if you double X, you always 01:12:23.080 --> 01:12:29.260 get less than a doubling of Y. So the question is, 01:12:29.260 --> 01:12:33.230 what can you do to make things somehow more sensitive? 01:12:33.230 --> 01:12:37.580 You'd like something that looks a little bit more-- 01:12:37.580 --> 01:12:44.160 well the ultimate would be 0 and then beta. 01:12:44.160 --> 01:12:49.170 And indeed this would be this logic kind of limit. 01:12:49.170 --> 01:12:53.380 So this is Y expression as a function of X. 01:12:53.380 --> 01:12:56.760 If you didn't get any until some Kd and then all of a sudden you 01:12:56.760 --> 01:12:59.880 had beta, that would be as sensitive as you could 01:12:59.880 --> 01:13:02.550 possibly-- this is ultra-, ultra- sensitive. 01:13:06.450 --> 01:13:09.164 So there's one solution that was talked about in the book 01:13:09.164 --> 01:13:11.330 to get something that's a little bit more like this. 01:13:14.460 --> 01:13:15.710 AUDIENCE: Cooperative binding. 01:13:15.710 --> 01:13:17.251 PROFESSOR: Yeah, cooperative binding. 01:13:17.251 --> 01:13:23.700 So we often describe these functions-- 01:13:23.700 --> 01:13:26.080 this is the rate of expression as a function of X-- 01:13:26.080 --> 01:13:28.420 as via some hill equation. 01:13:28.420 --> 01:13:30.580 So I'm just going to write Xs here. 01:13:30.580 --> 01:13:37.720 So it could be there's X, K plus X here. 01:13:37.720 --> 01:13:41.180 Now if you have cooperative binding 01:13:41.180 --> 01:13:43.560 either at the side of the promoter or dimerization, 01:13:43.560 --> 01:13:45.950 trimerzation, something before binding, 01:13:45.950 --> 01:13:49.810 you can get some effective hill coverage in here, 01:13:49.810 --> 01:13:52.516 where this is going to be up X to the n, X to the n, K 01:13:52.516 --> 01:13:53.240 to then. 01:13:53.240 --> 01:13:55.965 We put K to n here just so that all the units are still 01:13:55.965 --> 01:13:58.260 reasonable. 01:13:58.260 --> 01:14:01.920 And as n increases, this thing becomes more and more sensitive 01:14:01.920 --> 01:14:03.220 or ultrasensitive. 01:14:03.220 --> 01:14:05.950 So this is with n just equal to 1, 01:14:05.950 --> 01:14:09.430 just a monomer binding in a simple way. 01:14:09.430 --> 01:14:11.150 Kd is still the 1/2 mark. 01:14:11.150 --> 01:14:15.590 So things always cross here, but if it's 2 01:14:15.590 --> 01:14:20.610 then it might look like this, now three, four. 01:14:20.610 --> 01:14:24.121 So it gets steeper and steeper as that hill coefficient 01:14:24.121 --> 01:14:24.620 increases. 01:14:34.090 --> 01:14:38.065 As Uri mentions, for many input in many genes, if you go in 01:14:38.065 --> 01:14:39.690 and you measure these things, you often 01:14:39.690 --> 01:14:42.265 get something that's reasonably well-defined by the S 01:14:42.265 --> 01:14:45.060 with a n somewhere between 1 and 4. 01:14:45.060 --> 01:14:47.251 So things are often moderately cooperative. 01:14:57.020 --> 01:15:00.120 Maybe I'll just tell you about this other mechanism 01:15:00.120 --> 01:15:04.620 for ultrasensitivity, this idea of molecular titration. 01:15:04.620 --> 01:15:08.010 I'm going to leave you with the basic model, or the basic idea, 01:15:08.010 --> 01:15:10.820 and then at the beginning of class on Thursday, 01:15:10.820 --> 01:15:12.390 we will try to figure out what are 01:15:12.390 --> 01:15:15.227 the requirements for the various binding affinities 01:15:15.227 --> 01:15:16.560 in order for that model to work. 01:15:22.920 --> 01:15:24.460 So what's neat about this is it's 01:15:24.460 --> 01:15:26.440 a situation where you can get something 01:15:26.440 --> 01:15:29.705 that is ultrasensitive without any cooperativity. 01:15:33.950 --> 01:15:40.530 The idea is that you have some X that is indeed 01:15:40.530 --> 01:15:46.250 binding to the promoter to activate expression of Y, 01:15:46.250 --> 01:15:51.720 but, in addition, you have some other protein, say W, 01:15:51.720 --> 01:15:58.535 that can bind to X and turn it into this complex XW. 01:16:01.710 --> 01:16:06.580 So we can always describe things as there's some Kw here, 01:16:06.580 --> 01:16:11.900 some Kd for X to bind to the promoter, 01:16:11.900 --> 01:16:20.960 and for some relations of Kw, Kd, and W 01:16:20.960 --> 01:16:24.450 total, you can get ultrasensitivity. 01:16:24.450 --> 01:16:28.810 What happens is that if you look at this rate of expression-- 01:16:28.810 --> 01:16:34.250 so this is Y expression-- as a function of X, 01:16:34.250 --> 01:16:39.260 if you don't have any W here, then indeed it just looks 01:16:39.260 --> 01:16:40.540 like our standard thing here. 01:16:43.290 --> 01:16:47.960 Whereas in the this is when you add W, 01:16:47.960 --> 01:16:51.790 when you add this molecular titration phenomenon, 01:16:51.790 --> 01:16:54.710 you can make this curve slide over so you don't 01:16:54.710 --> 01:16:59.310 get significant expression until X-- 01:16:59.310 --> 01:17:00.790 or this is X total if you'd like-- 01:17:00.790 --> 01:17:03.974 is larger than W total and this whole curve just slides over. 01:17:09.170 --> 01:17:12.367 You can see this is ultrasensitive. 01:17:12.367 --> 01:17:14.200 So nothing happens until all of a sudden you 01:17:14.200 --> 01:17:16.280 start getting expression. 01:17:16.280 --> 01:17:18.530 So what we're going to do is on the beginning of class 01:17:18.530 --> 01:17:21.330 on Thursday, we'll try to figure out 01:17:21.330 --> 01:17:26.390 what is the relationship between these different binding 01:17:26.390 --> 01:17:30.240 parameters in order to get something that looks like this. 01:17:34.280 --> 01:17:36.910 Are there any questions about anything 01:17:36.910 --> 01:17:38.527 that we've said so far today? 01:17:42.350 --> 01:17:45.180 With that, why don't we go ahead and quit.