WEBVTT 00:00:00.000 --> 00:00:00.610 align:middle line:90% 00:00:00.610 --> 00:00:02.440 align:middle line:84% The following content is provided 00:00:02.440 --> 00:00:06.370 align:middle line:84% by MIT OpenCourseWare under a Creative Commons license. 00:00:06.370 --> 00:00:08.470 align:middle line:84% Additional information about our license 00:00:08.470 --> 00:00:10.870 align:middle line:84% and MIT OpenCourseWare in general 00:00:10.870 --> 00:00:12.730 align:middle line:90% is available at ocw.mit.edu. 00:00:12.730 --> 00:00:18.640 align:middle line:90% 00:00:18.640 --> 00:00:20.530 align:middle line:90% GEORGE CHURCH: OK. 00:00:20.530 --> 00:00:23.710 align:middle line:84% Well, even though we've been hinting at networks pretty 00:00:23.710 --> 00:00:25.400 align:middle line:84% heavily all the way through the course, 00:00:25.400 --> 00:00:27.910 align:middle line:84% these are the three lectures where we actually take it on. 00:00:27.910 --> 00:00:34.090 align:middle line:84% We really started this at the end of last week's lecture, 00:00:34.090 --> 00:00:35.440 align:middle line:90% so-called protein 2. 00:00:35.440 --> 00:00:38.830 align:middle line:84% Where in the process of talking about protein modifications 00:00:38.830 --> 00:00:41.865 align:middle line:84% and quantitating metabolites and their interactions 00:00:41.865 --> 00:00:43.240 align:middle line:84% with proteins, we started talking 00:00:43.240 --> 00:00:47.200 align:middle line:84% about the sorts of sources of data that you would have that 00:00:47.200 --> 00:00:54.430 align:middle line:84% would allow you to get at a quantitative analysis 00:00:54.430 --> 00:00:58.310 align:middle line:84% of protein networks, such as red blood cell. 00:00:58.310 --> 00:01:01.540 align:middle line:84% So we're going to pick up on that theme 00:01:01.540 --> 00:01:06.070 align:middle line:84% by talking about macroscopic continuous concentration 00:01:06.070 --> 00:01:07.780 align:middle line:84% gradients, and then contrast that 00:01:07.780 --> 00:01:11.020 align:middle line:84% with mesoscopic or discrete molecular numbers. 00:01:11.020 --> 00:01:14.410 align:middle line:84% We're just going to very briefly touch upon the issues 00:01:14.410 --> 00:01:17.230 align:middle line:84% in discriminating between the stochastic modeling 00:01:17.230 --> 00:01:19.960 align:middle line:90% and the continuous modeling. 00:01:19.960 --> 00:01:23.915 align:middle line:84% And it's a very interesting connection 00:01:23.915 --> 00:01:25.540 align:middle line:84% between the red blood cell model, where 00:01:25.540 --> 00:01:28.630 align:middle line:84% you have a few examples of cooperativity 00:01:28.630 --> 00:01:30.670 align:middle line:84% with modest Hill's coefficients-- 00:01:30.670 --> 00:01:33.550 align:middle line:84% to take that to an extreme case where you have actually 00:01:33.550 --> 00:01:36.760 align:middle line:84% bistability, where you have two stable modes which 00:01:36.760 --> 00:01:40.750 align:middle line:84% are separated by a very cooperative interaction. 00:01:40.750 --> 00:01:44.050 align:middle line:84% And then we'll talk about copy number controls, 00:01:44.050 --> 00:01:46.810 align:middle line:84% another opportunity for doing modeling 00:01:46.810 --> 00:01:49.930 align:middle line:84% either by macroscopic continuous modeling 00:01:49.930 --> 00:01:53.950 align:middle line:90% or this stochastic modeling. 00:01:53.950 --> 00:01:56.770 align:middle line:84% And then, after the break, we'll talk about flux balance 00:01:56.770 --> 00:02:01.930 align:middle line:84% optimization, which I think is a really exciting and clever way 00:02:01.930 --> 00:02:04.210 align:middle line:84% of leveraging the little bits of information 00:02:04.210 --> 00:02:07.690 align:middle line:84% that you have about very complicated 00:02:07.690 --> 00:02:11.120 align:middle line:84% regulatory networks, biochemical networks. 00:02:11.120 --> 00:02:14.980 align:middle line:84% So I think we've kind of just barely touched 00:02:14.980 --> 00:02:16.650 align:middle line:90% upon this before. 00:02:16.650 --> 00:02:21.880 align:middle line:84% The particular networks that are being studied, how did they 00:02:21.880 --> 00:02:22.810 align:middle line:90% come to be studied? 00:02:22.810 --> 00:02:25.450 align:middle line:90% Why and how? 00:02:25.450 --> 00:02:27.670 align:middle line:84% And typically, what they have in common 00:02:27.670 --> 00:02:32.143 align:middle line:84% is that they have large genetic and biochemical kinetic data 00:02:32.143 --> 00:02:33.310 align:middle line:90% sets to go with it-- and/or. 00:02:33.310 --> 00:02:36.010 align:middle line:90% 00:02:36.010 --> 00:02:39.100 align:middle line:84% Right now there is no model that describes 00:02:39.100 --> 00:02:43.360 align:middle line:84% all of the interesting aspects of an entire cell 00:02:43.360 --> 00:02:45.215 align:middle line:90% or an entire organism. 00:02:45.215 --> 00:02:46.840 align:middle line:84% Usually, there are little pieces of it. 00:02:46.840 --> 00:02:49.180 align:middle line:84% The closest to a whole cell is the red blood cell. 00:02:49.180 --> 00:02:53.320 align:middle line:84% And this is because there's no biopolymer components 00:02:53.320 --> 00:02:56.860 align:middle line:90% to it, no biopolymer synthesis. 00:02:56.860 --> 00:02:58.450 align:middle line:84% Today, we'll talk a little bit more 00:02:58.450 --> 00:03:00.190 align:middle line:84% about these two related topics, which 00:03:00.190 --> 00:03:05.530 align:middle line:84% is cell division cycle and the segregation of chromosomes 00:03:05.530 --> 00:03:07.570 align:middle line:90% during the cell division. 00:03:07.570 --> 00:03:15.040 align:middle line:84% Here, the key point is the critical nature 00:03:15.040 --> 00:03:20.530 align:middle line:84% of single molecules in these that come into play 00:03:20.530 --> 00:03:22.360 align:middle line:90% in the dividing cells. 00:03:22.360 --> 00:03:25.540 align:middle line:84% And in this, they'll be talking about bistability, how there's 00:03:25.540 --> 00:03:29.200 align:middle line:84% a decision of either to take the next step in dividing 00:03:29.200 --> 00:03:30.520 align:middle line:90% the cell or not. 00:03:30.520 --> 00:03:34.200 align:middle line:84% And how that bistability can be achieved either stochastically, 00:03:34.200 --> 00:03:35.950 align:middle line:84% where you're dealing with the fluctuations 00:03:35.950 --> 00:03:38.470 align:middle line:84% of single molecules greatly affecting a switch, 00:03:38.470 --> 00:03:41.337 align:middle line:84% such as a phase lambda switch, or you 00:03:41.337 --> 00:03:43.420 align:middle line:84% can have it involving a large number of molecules, 00:03:43.420 --> 00:03:47.440 align:middle line:84% where stochastic seems to play less of a role. 00:03:47.440 --> 00:03:50.350 align:middle line:84% And then, at the end, we'll talk about how 00:03:50.350 --> 00:03:52.660 align:middle line:84% we can do comparative metabolism, where we literally 00:03:52.660 --> 00:03:58.660 align:middle line:84% integrate genomics with a network model of biochemistry 00:03:58.660 --> 00:04:00.400 align:middle line:90% at many different levels. 00:04:00.400 --> 00:04:02.900 align:middle line:84% Of course, there's the genome encoding 00:04:02.900 --> 00:04:04.580 align:middle line:90% the components of that network. 00:04:04.580 --> 00:04:09.280 align:middle line:84% But there's also the systematic knockouts of genes 00:04:09.280 --> 00:04:12.140 align:middle line:84% and their effects on the network. 00:04:12.140 --> 00:04:14.892 align:middle line:90% Now, this slide is also review. 00:04:14.892 --> 00:04:16.600 align:middle line:84% But it puts in context where we'll mainly 00:04:16.600 --> 00:04:18.375 align:middle line:90% begin talking today. 00:04:18.375 --> 00:04:20.500 align:middle line:84% We'll talk a little bit about ordinary differential 00:04:20.500 --> 00:04:22.540 align:middle line:84% equations, both for-- at the beginning, 00:04:22.540 --> 00:04:27.850 align:middle line:84% under red blood cell, and in the bistability discussion, 00:04:27.850 --> 00:04:31.480 align:middle line:84% how you can get, even with just concentration and time, 00:04:31.480 --> 00:04:33.700 align:middle line:84% you can get these very interesting behaviors, 00:04:33.700 --> 00:04:35.290 align:middle line:90% highly cooperative. 00:04:35.290 --> 00:04:38.020 align:middle line:84% And then we'll drop concentration and time 00:04:38.020 --> 00:04:39.560 align:middle line:84% by a steady-state approximation when 00:04:39.560 --> 00:04:42.260 align:middle line:84% we talk about flux balance in the second half. 00:04:42.260 --> 00:04:45.220 align:middle line:84% And of course, before, we were talking 00:04:45.220 --> 00:04:49.810 align:middle line:84% about molecular mechanics in the context of protein structure. 00:04:49.810 --> 00:04:51.790 align:middle line:84% And master's equations are the way 00:04:51.790 --> 00:04:53.890 align:middle line:84% of looking at stochastic single molecules, which 00:04:53.890 --> 00:04:57.490 align:middle line:90% we'll mention in passing. 00:04:57.490 --> 00:04:59.920 align:middle line:84% Eventually, in the network, discussions 00:04:59.920 --> 00:05:02.860 align:middle line:84% will get to spatially inhomogeneous models, where 00:05:02.860 --> 00:05:07.090 align:middle line:84% you actually care or realize the importance of where 00:05:07.090 --> 00:05:11.920 align:middle line:84% particular molecules lie in terms of their function. 00:05:11.920 --> 00:05:14.650 align:middle line:84% Now, what are the limits and problems 00:05:14.650 --> 00:05:20.300 align:middle line:84% in connecting the in vitro parameters that 00:05:20.300 --> 00:05:25.080 align:middle line:84% were so key last time and this time in developing a system 00:05:25.080 --> 00:05:25.580 align:middle line:90% model? 00:05:25.580 --> 00:05:27.980 align:middle line:84% By isolating particular molecules, 00:05:27.980 --> 00:05:32.600 align:middle line:84% you can cut off, make the system simpler. 00:05:32.600 --> 00:05:34.830 align:middle line:84% But then there's the problem of reintegration. 00:05:34.830 --> 00:05:38.180 align:middle line:84% And here, this is more historical artifact 00:05:38.180 --> 00:05:42.350 align:middle line:84% than really critical to this discussion. 00:05:42.350 --> 00:05:46.610 align:middle line:84% And originally, enzyme kinetics would ignore the products 00:05:46.610 --> 00:05:49.530 align:middle line:84% for technical mathematical reasons. 00:05:49.530 --> 00:05:51.027 align:middle line:84% But as you can see from last time, 00:05:51.027 --> 00:05:53.360 align:middle line:84% we showed that you could represent those equations quite 00:05:53.360 --> 00:05:56.380 align:middle line:84% well and even make the measurements. 00:05:56.380 --> 00:05:58.130 align:middle line:84% In the presence of products, it would just 00:05:58.130 --> 00:06:03.170 align:middle line:84% be another few terms in the rate equations. 00:06:03.170 --> 00:06:08.875 align:middle line:84% More critical, however, is that including the product along 00:06:08.875 --> 00:06:11.000 align:middle line:84% with substrate in the measurements and the modeling 00:06:11.000 --> 00:06:13.880 align:middle line:84% is just the first step, because how many different products 00:06:13.880 --> 00:06:17.480 align:middle line:84% and substrates and other regulatory molecules 00:06:17.480 --> 00:06:20.900 align:middle line:84% might be involved that you don't know about initially? 00:06:20.900 --> 00:06:24.020 align:middle line:84% In addition, the conditions for doing 00:06:24.020 --> 00:06:28.370 align:middle line:84% the in vitro measurements, it's hard to do them 00:06:28.370 --> 00:06:30.620 align:middle line:90% at the concentrations in cells. 00:06:30.620 --> 00:06:33.440 align:middle line:84% And in cells, they're nearly crystalline densities, 00:06:33.440 --> 00:06:42.980 align:middle line:84% as I say, up to 30% or so, which is basically 00:06:42.980 --> 00:06:47.600 align:middle line:84% the very high concentration of solute that occurs-- 00:06:47.600 --> 00:06:50.090 align:middle line:84% proteins and other macromolecules. 00:06:50.090 --> 00:06:54.980 align:middle line:84% And the substrates, the small molecules, 00:06:54.980 --> 00:06:58.280 align:middle line:84% are typically in vast excess in in vitro reactions. 00:06:58.280 --> 00:07:00.810 align:middle line:84% But they're very close to equal molar in vivo 00:07:00.810 --> 00:07:02.810 align:middle line:84% because a lot of them are bound up with enzymes. 00:07:02.810 --> 00:07:04.352 align:middle line:84% And you get interesting observations, 00:07:04.352 --> 00:07:06.680 align:middle line:84% such as this one mentioned at the bottom where 00:07:06.680 --> 00:07:09.620 align:middle line:84% a chemical reaction, which is spontaneous in solution, which 00:07:09.620 --> 00:07:14.630 align:middle line:84% is the epimerization of galactose, 00:07:14.630 --> 00:07:20.930 align:middle line:84% does not occur in normal cells of E. coli 00:07:20.930 --> 00:07:24.020 align:middle line:84% unless they have an enzyme that catalyzes 00:07:24.020 --> 00:07:25.920 align:middle line:84% this normally spontaneous reaction. 00:07:25.920 --> 00:07:28.730 align:middle line:84% So this is curious that it happens in solution. 00:07:28.730 --> 00:07:31.955 align:middle line:84% Doesn't happen in the cell unless you have the enzyme. 00:07:31.955 --> 00:07:35.900 align:middle line:90% 00:07:35.900 --> 00:07:36.470 align:middle line:90% OK. 00:07:36.470 --> 00:07:38.840 align:middle line:84% So with all those caveats in mind, 00:07:38.840 --> 00:07:42.765 align:middle line:84% and recognizing that even though this model has more 00:07:42.765 --> 00:07:45.140 align:middle line:84% measured parameters than almost any other cellular model, 00:07:45.140 --> 00:07:47.723 align:middle line:84% they were all measured in vitro, with the caveats that we just 00:07:47.723 --> 00:07:48.530 align:middle line:90% mentioned. 00:07:48.530 --> 00:07:51.140 align:middle line:90% And the cell is aimed-- 00:07:51.140 --> 00:07:53.390 align:middle line:84% the function of all these network 00:07:53.390 --> 00:07:55.730 align:middle line:84% is mainly to provide the redox that 00:07:55.730 --> 00:07:59.510 align:middle line:84% keeps the hemoglobin reduced and the ATP that 00:07:59.510 --> 00:08:02.255 align:middle line:84% keeps the osmotic pressure under control. 00:08:02.255 --> 00:08:06.350 align:middle line:90% 00:08:06.350 --> 00:08:08.360 align:middle line:84% And also, the cell is-- even though we've 00:08:08.360 --> 00:08:10.540 align:middle line:84% shown a little structure in this network, 00:08:10.540 --> 00:08:12.860 align:middle line:84% the structure is assumed to be fairly 00:08:12.860 --> 00:08:15.140 align:middle line:84% continuous within the cell, which 00:08:15.140 --> 00:08:18.660 align:middle line:84% is a fairly good approximation in this case. 00:08:18.660 --> 00:08:21.140 align:middle line:84% So this is not merely the stapling 00:08:21.140 --> 00:08:25.340 align:middle line:84% together of some kinetic in vitro parameters. 00:08:25.340 --> 00:08:33.610 align:middle line:84% We have other considerations in a real cell. 00:08:33.610 --> 00:08:38.679 align:middle line:84% We have the sort of physical parameters 00:08:38.679 --> 00:08:44.290 align:middle line:84% of the mass balance, energy balance, and redox balance. 00:08:44.290 --> 00:08:46.510 align:middle line:84% We've already mentioned energy and redox. 00:08:46.510 --> 00:08:49.720 align:middle line:84% But you have to have conservation of mass, 00:08:49.720 --> 00:08:52.810 align:middle line:84% as we'll see develop quite a bit more in the second half 00:08:52.810 --> 00:08:54.910 align:middle line:90% of this talk. 00:08:54.910 --> 00:08:58.780 align:middle line:84% In addition, there are physics, such as the osmotic pressure 00:08:58.780 --> 00:08:59.950 align:middle line:90% and electron neutrality. 00:08:59.950 --> 00:09:04.390 align:middle line:84% There are cells which do have transience 00:09:04.390 --> 00:09:06.205 align:middle line:90% of non-electrode neutrality. 00:09:06.205 --> 00:09:09.520 align:middle line:84% But for the red blood cells, certainly, one of the goals 00:09:09.520 --> 00:09:12.130 align:middle line:84% is to maintain very close to electron neutrality 00:09:12.130 --> 00:09:14.860 align:middle line:90% and osmotically stable. 00:09:14.860 --> 00:09:17.110 align:middle line:84% You want to have as many non-adjustable constraints as 00:09:17.110 --> 00:09:18.700 align:middle line:90% possible. 00:09:18.700 --> 00:09:23.620 align:middle line:84% As in other modeling systems, if these are measurements, rather 00:09:23.620 --> 00:09:26.440 align:middle line:84% than adjustable model parameters, 00:09:26.440 --> 00:09:32.950 align:middle line:84% then it allows you to test the few hypotheses you have, 00:09:32.950 --> 00:09:35.290 align:middle line:84% and have them overdetermined, and look 00:09:35.290 --> 00:09:39.610 align:middle line:90% for contradictions, outliers. 00:09:39.610 --> 00:09:42.820 align:middle line:84% And then, eventually, we'll see advantages 00:09:42.820 --> 00:09:45.010 align:middle line:84% to knowing what the maximum fluxes-- 00:09:45.010 --> 00:09:49.360 align:middle line:84% the maximum rates that you can have in these complex networks. 00:09:49.360 --> 00:09:51.700 align:middle line:84% And we could incorporate gene regulation, 00:09:51.700 --> 00:09:56.110 align:middle line:84% as we've seen lots of wholesale, increasingly 00:09:56.110 --> 00:09:59.320 align:middle line:84% accurate data on gene regulation is becoming available. 00:09:59.320 --> 00:10:01.930 align:middle line:84% It would be nice to integrate these, 00:10:01.930 --> 00:10:04.662 align:middle line:84% because the expression of the proteins 00:10:04.662 --> 00:10:05.620 align:middle line:90% affects their activity. 00:10:05.620 --> 00:10:07.990 align:middle line:84% The activity affects these fluxes. 00:10:07.990 --> 00:10:11.650 align:middle line:84% So these fluxes are represented here in slide number 8 00:10:11.650 --> 00:10:15.880 align:middle line:84% as dx dt, where each of the x sub i's represents 00:10:15.880 --> 00:10:18.910 align:middle line:84% one of these blue dots, a node in the network, where 00:10:18.910 --> 00:10:22.900 align:middle line:84% you have four basic processes that affect it-- up to four. 00:10:22.900 --> 00:10:25.420 align:middle line:84% You can have synthesis steps, degradation steps. 00:10:25.420 --> 00:10:29.740 align:middle line:84% So synthesis produces transport, can bring it into the cell. 00:10:29.740 --> 00:10:31.000 align:middle line:90% Degradation removes it. 00:10:31.000 --> 00:10:32.740 align:middle line:84% And it can be utilized, incorporated 00:10:32.740 --> 00:10:34.120 align:middle line:90% into the body of the cell. 00:10:34.120 --> 00:10:37.120 align:middle line:84% And you can think of this as a sink removing it 00:10:37.120 --> 00:10:41.320 align:middle line:90% from the free population of-- 00:10:41.320 --> 00:10:42.850 align:middle line:90% for each x molecule. 00:10:42.850 --> 00:10:47.020 align:middle line:84% And this can be restated as a stoichiometric matrix [? so ?] 00:10:47.020 --> 00:10:52.330 align:middle line:84% ij, where that's mainly 1's and minus 1's. 00:10:52.330 --> 00:10:57.580 align:middle line:84% As you can see, in front of each of these fluxes, 00:10:57.580 --> 00:11:00.280 align:middle line:84% synthesis degradation and so forth will be 1's and minus 00:11:00.280 --> 00:11:03.100 align:middle line:84% 1's that refer to the stoichiometry, 00:11:03.100 --> 00:11:05.980 align:middle line:90% and sometimes 2's and 0's. 00:11:05.980 --> 00:11:09.175 align:middle line:84% And then the transport is its own vector. 00:11:09.175 --> 00:11:11.870 align:middle line:90% 00:11:11.870 --> 00:11:14.560 align:middle line:84% And you'll see the utility of the stoichiometric matrix 00:11:14.560 --> 00:11:17.020 align:middle line:84% where i is the metabolite number, 00:11:17.020 --> 00:11:20.260 align:middle line:84% and j is the reaction number or the enzyme number 00:11:20.260 --> 00:11:25.790 align:middle line:84% for all the possible reactions that can occur in a cell. 00:11:25.790 --> 00:11:27.290 align:middle line:84% These can be all possible reactions. 00:11:27.290 --> 00:11:30.490 align:middle line:84% Then you can toggle them on and off with mutations or changing 00:11:30.490 --> 00:11:32.530 align:middle line:90% different cell types. 00:11:32.530 --> 00:11:38.260 align:middle line:84% Now, this particularly rich system, the red blood cell, 00:11:38.260 --> 00:11:42.280 align:middle line:84% has been modeled many times and continues to be modeled since 00:11:42.280 --> 00:11:46.210 align:middle line:84% the mid '70s and now into the 2000s, 00:11:46.210 --> 00:11:49.390 align:middle line:84% starting originally just with glycolysis, 00:11:49.390 --> 00:11:53.890 align:middle line:84% later adding pentose phosphate, nucleotide metabolism, 00:11:53.890 --> 00:11:59.260 align:middle line:84% various [? pumps, ?] [? osmotic ?] consideration. 00:11:59.260 --> 00:12:02.860 align:middle line:84% Hemoglobin ligands have been treated from time to time, 00:12:02.860 --> 00:12:07.450 align:middle line:84% and less so issues having to do with [INAUDIBLE] and shape. 00:12:07.450 --> 00:12:10.930 align:middle line:84% No model includes all of these in one model, 00:12:10.930 --> 00:12:14.470 align:middle line:90% although it's really very close. 00:12:14.470 --> 00:12:18.340 align:middle line:84% But there are models that include all of the metabolism 00:12:18.340 --> 00:12:22.240 align:middle line:84% that we know and the transport osmotic properties. 00:12:22.240 --> 00:12:25.330 align:middle line:90% 00:12:25.330 --> 00:12:30.550 align:middle line:84% Relatively few of those were made available broadly. 00:12:30.550 --> 00:12:34.360 align:middle line:84% But now they're increasingly being made available freely 00:12:34.360 --> 00:12:39.550 align:middle line:90% on the web, as models should be. 00:12:39.550 --> 00:12:42.430 align:middle line:84% The assumptions behind this is that-- 00:12:42.430 --> 00:12:45.970 align:middle line:90% 00:12:45.970 --> 00:12:47.890 align:middle line:84% like I said, not everything is modeled. 00:12:47.890 --> 00:12:51.370 align:middle line:84% Some of it, for mathematical convenience, 00:12:51.370 --> 00:12:55.120 align:middle line:84% you will typically, in order to do differential equation-- 00:12:55.120 --> 00:12:57.490 align:middle line:84% since there's vast ranges of time constants, 00:12:57.490 --> 00:13:02.050 align:middle line:84% from things that happen extremely quickly to things 00:13:02.050 --> 00:13:04.870 align:middle line:84% that happen extremely slowly, typically, what you do 00:13:04.870 --> 00:13:09.550 align:middle line:84% is you'll model with a window in the middle, where you'll say, 00:13:09.550 --> 00:13:11.110 align:middle line:84% things that happened very quickly 00:13:11.110 --> 00:13:13.030 align:middle line:84% can be treated as a pseudo-equilibrium, 00:13:13.030 --> 00:13:16.510 align:middle line:84% as we've listed in this middle line here on slide 10. 00:13:16.510 --> 00:13:18.310 align:middle line:84% Things that happen extremely slowly 00:13:18.310 --> 00:13:19.630 align:middle line:90% can be treated as a constant. 00:13:19.630 --> 00:13:22.870 align:middle line:84% If they happen over the period of years, 00:13:22.870 --> 00:13:26.020 align:middle line:84% then in the course of an experiment that might be hours, 00:13:26.020 --> 00:13:29.020 align:middle line:84% they can be treated as a constant or something 00:13:29.020 --> 00:13:30.625 align:middle line:90% that you systematically explore. 00:13:30.625 --> 00:13:34.450 align:middle line:90% 00:13:34.450 --> 00:13:38.500 align:middle line:84% Although we'll typically be ignoring little pieces 00:13:38.500 --> 00:13:40.420 align:middle line:84% of the metabolism, like [INAUDIBLE] metabolism 00:13:40.420 --> 00:13:40.920 align:middle line:90% calcium-- 00:13:40.920 --> 00:13:44.290 align:middle line:90% 00:13:44.290 --> 00:13:47.080 align:middle line:84% either as data comes in from other systems 00:13:47.080 --> 00:13:51.490 align:middle line:84% and try to treat it by homology or analogy. 00:13:51.490 --> 00:13:55.180 align:middle line:84% In addition, when we talk about a typical cell, 00:13:55.180 --> 00:13:59.650 align:middle line:84% this can mean that we assume homogeneous distribution 00:13:59.650 --> 00:14:03.040 align:middle line:84% of molecules within the cell and homogeneity from cell 00:14:03.040 --> 00:14:05.170 align:middle line:84% to cell within a particular organism. 00:14:05.170 --> 00:14:07.810 align:middle line:84% It also means there's a tendency to model 00:14:07.810 --> 00:14:11.800 align:middle line:84% a wild type without respect to polymorphism. 00:14:11.800 --> 00:14:15.160 align:middle line:84% Although in human population, with red blood cell function 00:14:15.160 --> 00:14:18.160 align:middle line:84% in particular, there's quite a literature 00:14:18.160 --> 00:14:24.115 align:middle line:84% on mutations that affect the functioning of the proteins 00:14:24.115 --> 00:14:25.240 align:middle line:90% within the red blood cells. 00:14:25.240 --> 00:14:32.120 align:middle line:84% Probably one of the best studied human genetic systems. 00:14:32.120 --> 00:14:33.890 align:middle line:90% OK. 00:14:33.890 --> 00:14:36.260 align:middle line:84% Surface area is not absolutely constant, 00:14:36.260 --> 00:14:40.280 align:middle line:84% although for the time being, it's modeled as such. 00:14:40.280 --> 00:14:42.590 align:middle line:84% This is some examples of a subset of it. 00:14:42.590 --> 00:14:47.150 align:middle line:84% This is a subset that refers to glycolysis. 00:14:47.150 --> 00:14:49.250 align:middle line:84% You can see they all have the same form where 00:14:49.250 --> 00:14:52.100 align:middle line:84% you have a change in a concentration in moles 00:14:52.100 --> 00:14:55.970 align:middle line:84% per liter with respect to time of some small molecule-- here 00:14:55.970 --> 00:14:57.380 align:middle line:90% glucose-6-phosphate. 00:14:57.380 --> 00:15:03.590 align:middle line:84% It's some synthesis rate with the subscript being hexokinase. 00:15:03.590 --> 00:15:07.280 align:middle line:84% This is the upper left-hand corner. 00:15:07.280 --> 00:15:11.480 align:middle line:84% And then that's the synthesis minus the sinks, 00:15:11.480 --> 00:15:14.315 align:middle line:84% the degradation rates, which go through two other enzymes. 00:15:14.315 --> 00:15:20.540 align:middle line:90% 00:15:20.540 --> 00:15:22.170 align:middle line:84% Again, a reminder-- at the bottom, 00:15:22.170 --> 00:15:25.920 align:middle line:84% you'll see this come up a few times, just a reminder. 00:15:25.920 --> 00:15:27.380 align:middle line:84% Each of these is a form of change 00:15:27.380 --> 00:15:31.200 align:middle line:84% in the concentration of some small molecule with respect 00:15:31.200 --> 00:15:31.820 align:middle line:90% to time-- 00:15:31.820 --> 00:15:38.120 align:middle line:84% is the sum of the synthesis, subtracting the degradation 00:15:38.120 --> 00:15:40.400 align:middle line:90% transport utilization. 00:15:40.400 --> 00:15:45.740 align:middle line:90% 00:15:45.740 --> 00:15:47.360 align:middle line:90% OK. 00:15:47.360 --> 00:15:49.640 align:middle line:84% Now, remember we focused in on one little piece 00:15:49.640 --> 00:15:51.410 align:middle line:90% of this a couple of times now. 00:15:51.410 --> 00:15:55.490 align:middle line:84% This is a step that happens to be allosteric. 00:15:55.490 --> 00:15:58.223 align:middle line:84% That means that, depending on the concentration of the-- 00:15:58.223 --> 00:16:08.653 align:middle line:90% 00:16:08.653 --> 00:16:10.070 align:middle line:84% the top part of this formula tends 00:16:10.070 --> 00:16:13.230 align:middle line:84% to be composed of substrates and products having an effect. 00:16:13.230 --> 00:16:17.090 align:middle line:84% And then this term in the denominator 00:16:17.090 --> 00:16:21.440 align:middle line:84% has a fourth power dependence on a variety of other site 00:16:21.440 --> 00:16:23.300 align:middle line:90% effectors, including AMP. 00:16:23.300 --> 00:16:28.250 align:middle line:84% And you can see that the velocity is either hyperbolic, 00:16:28.250 --> 00:16:32.840 align:middle line:84% which is the upper curve-- which kind goes smoothly up 00:16:32.840 --> 00:16:33.890 align:middle line:90% and plateaus. 00:16:33.890 --> 00:16:37.400 align:middle line:84% While a sigmoid curve implies greater cooperatively, 00:16:37.400 --> 00:16:40.430 align:middle line:84% which can be affected by some of these other site effectors, 00:16:40.430 --> 00:16:42.380 align:middle line:90% which is more sigmoid in shape. 00:16:42.380 --> 00:16:46.100 align:middle line:84% And we'll use this as a stepping stone 00:16:46.100 --> 00:16:49.010 align:middle line:84% to talking about what are the various ways of getting 00:16:49.010 --> 00:16:50.325 align:middle line:90% that sigmoid shape? 00:16:50.325 --> 00:16:51.950 align:middle line:84% We've already talked about how proteins 00:16:51.950 --> 00:16:55.190 align:middle line:84% can be multimers, dimers, tetramers, and so forth. 00:16:55.190 --> 00:16:56.780 align:middle line:84% That could be one way of achieving 00:16:56.780 --> 00:17:00.080 align:middle line:84% the sigmoid shape via a conformation change 00:17:00.080 --> 00:17:02.510 align:middle line:90% which senses the second site. 00:17:02.510 --> 00:17:06.349 align:middle line:84% But we'll talk about another way in just a couple of slides. 00:17:06.349 --> 00:17:11.090 align:middle line:84% When we have the kinetic expressions here, 00:17:11.090 --> 00:17:12.750 align:middle line:84% they have the form of the previous one. 00:17:12.750 --> 00:17:17.510 align:middle line:84% Most of them are simpler than the one in slide 12. 00:17:17.510 --> 00:17:20.420 align:middle line:84% The model has a total of 44 rate expressions. 00:17:20.420 --> 00:17:24.060 align:middle line:84% They have about five constants, on average, 00:17:24.060 --> 00:17:26.690 align:middle line:90% so about 200 parameters. 00:17:26.690 --> 00:17:28.700 align:middle line:84% These are not truly adjustable in the sense 00:17:28.700 --> 00:17:35.420 align:middle line:84% that they're determined from the in vitro reactions. 00:17:35.420 --> 00:17:38.240 align:middle line:90% What kind of assumptions? 00:17:38.240 --> 00:17:40.820 align:middle line:84% We've already mentioned the difference between in vitro 00:17:40.820 --> 00:17:43.220 align:middle line:90% and in vivo. 00:17:43.220 --> 00:17:47.840 align:middle line:84% We have the lingering question of how many effectors 00:17:47.840 --> 00:17:50.060 align:middle line:84% might there be that we don't know about? 00:17:50.060 --> 00:17:51.620 align:middle line:84% Typically, these in vitro experiments 00:17:51.620 --> 00:17:54.470 align:middle line:84% were done with a small number of substrates and products 00:17:54.470 --> 00:17:56.060 align:middle line:90% that you know about. 00:17:56.060 --> 00:17:59.330 align:middle line:84% But as a worst case scenario, Mike Savageau you know, 00:17:59.330 --> 00:18:02.150 align:middle line:84% likes to trot out the glutamine synthetase, which fortunately 00:18:02.150 --> 00:18:04.160 align:middle line:90% is not in this particular model. 00:18:04.160 --> 00:18:06.350 align:middle line:84% But it could be that there's an enzyme that's 00:18:06.350 --> 00:18:08.720 align:middle line:84% just as complicated but hasn't been studied as much. 00:18:08.720 --> 00:18:10.220 align:middle line:84% In the case of glutamine synthetase, 00:18:10.220 --> 00:18:11.510 align:middle line:90% there are three substrates. 00:18:11.510 --> 00:18:13.430 align:middle line:84% Remember, the previous example, there 00:18:13.430 --> 00:18:15.848 align:middle line:84% were only two substrates and two products. 00:18:15.848 --> 00:18:18.140 align:middle line:84% Glutamine synthetase, three substrates, three products, 00:18:18.140 --> 00:18:20.600 align:middle line:84% and has nine allosteric effectors, rather than 00:18:20.600 --> 00:18:25.050 align:middle line:84% the three or so in the previous example. 00:18:25.050 --> 00:18:26.840 align:middle line:84% So this gives a grand total of 15 00:18:26.840 --> 00:18:28.820 align:middle line:84% different molecules you need to track. 00:18:28.820 --> 00:18:33.710 align:middle line:84% So the number of different measurements 00:18:33.710 --> 00:18:36.295 align:middle line:84% you might have to make, hypothetically-- 00:18:36.295 --> 00:18:38.420 align:middle line:84% no one's actually made this number of measurements. 00:18:38.420 --> 00:18:41.930 align:middle line:84% But Mike Savageau likes to point out 00:18:41.930 --> 00:18:44.330 align:middle line:84% that even if you only did four concentration 00:18:44.330 --> 00:18:46.940 align:middle line:84% points in this multi-dimensional space, 00:18:46.940 --> 00:18:48.830 align:middle line:84% you have 4 to the 15th measurements, 00:18:48.830 --> 00:18:51.260 align:middle line:90% or a billion measurements. 00:18:51.260 --> 00:18:53.420 align:middle line:84% A billion isn't nearly as intimidating 00:18:53.420 --> 00:18:56.030 align:middle line:84% as it was when he made this statement in '76, 00:18:56.030 --> 00:18:59.330 align:middle line:84% but it still is not something that's routinely done. 00:18:59.330 --> 00:19:03.217 align:middle line:90% 00:19:03.217 --> 00:19:04.175 align:middle line:90% What other constraints? 00:19:04.175 --> 00:19:06.110 align:middle line:84% There are these physical chemical constraints 00:19:06.110 --> 00:19:09.260 align:middle line:84% of osmotic pressure and electron neutrality, 00:19:09.260 --> 00:19:11.240 align:middle line:84% here stated a little more explicitly. 00:19:11.240 --> 00:19:14.690 align:middle line:90% You have pi i equal to pi e. 00:19:14.690 --> 00:19:16.790 align:middle line:84% That means the pressure on the inside 00:19:16.790 --> 00:19:18.575 align:middle line:84% is equal to the pressure on the outside. 00:19:18.575 --> 00:19:20.450 align:middle line:84% That sounds like a good way to balance things 00:19:20.450 --> 00:19:22.280 align:middle line:84% out so that the cell doesn't explode. 00:19:22.280 --> 00:19:27.050 align:middle line:84% And explicitly, what that means is these gas constant r 00:19:27.050 --> 00:19:31.820 align:middle line:84% times the absolute temperature, degrees Kelvin, 00:19:31.820 --> 00:19:40.070 align:middle line:84% times the sum of the pressure components for the j molecule, 00:19:40.070 --> 00:19:45.350 align:middle line:84% going up to m chemical species, for i, standing for Interior, 00:19:45.350 --> 00:19:48.110 align:middle line:84% is equal to the same sum, equivalent sum 00:19:48.110 --> 00:19:51.650 align:middle line:84% for the subscript e, for Exterior. 00:19:51.650 --> 00:19:54.260 align:middle line:84% Electroneutrality has the same set of concentrations 00:19:54.260 --> 00:19:59.240 align:middle line:84% for the i, interior, and j molecules, where now z 00:19:59.240 --> 00:20:02.180 align:middle line:84% is the charge, where charge is the same z that we 00:20:02.180 --> 00:20:05.690 align:middle line:84% had in m over z for the mass spectrometry. 00:20:05.690 --> 00:20:06.920 align:middle line:90% So, OK. 00:20:06.920 --> 00:20:11.690 align:middle line:84% Now, some of the models I give you today, 00:20:11.690 --> 00:20:15.710 align:middle line:84% we will compare it to the calculated and the observed, 00:20:15.710 --> 00:20:17.510 align:middle line:90% as we have done before. 00:20:17.510 --> 00:20:19.730 align:middle line:84% And here it's shown a little bit differently 00:20:19.730 --> 00:20:22.490 align:middle line:84% than how I've done it before and how we'll do it later on. 00:20:22.490 --> 00:20:25.388 align:middle line:84% Typically, we would have observed on one axis 00:20:25.388 --> 00:20:26.555 align:middle line:90% and calculated on the other. 00:20:26.555 --> 00:20:30.375 align:middle line:90% 00:20:30.375 --> 00:20:32.000 align:middle line:84% In general, we're looking for outliers. 00:20:32.000 --> 00:20:34.580 align:middle line:84% And here we're sorting by the degree to which they 00:20:34.580 --> 00:20:37.800 align:middle line:90% deviate from observation. 00:20:37.800 --> 00:20:41.180 align:middle line:84% And so the deviation is going to be observation 00:20:41.180 --> 00:20:43.580 align:middle line:90% minus the calculated. 00:20:43.580 --> 00:20:46.520 align:middle line:84% And the degree of deviation can either 00:20:46.520 --> 00:20:49.070 align:middle line:84% be normalized to the standard deviation-- which is basically 00:20:49.070 --> 00:20:51.350 align:middle line:84% normalizing it to how confident we are 00:20:51.350 --> 00:20:53.420 align:middle line:90% in the experimental measure. 00:20:53.420 --> 00:20:56.310 align:middle line:84% Or it can be normalized to the averaged value, 00:20:56.310 --> 00:20:59.780 align:middle line:84% which then becomes less dependent upon the accuracy 00:20:59.780 --> 00:21:01.730 align:middle line:84% of the experiment and more dependent 00:21:01.730 --> 00:21:04.010 align:middle line:84% upon-- with a fraction of the-- and so we've 00:21:04.010 --> 00:21:05.780 align:middle line:90% sorted it on the ladder. 00:21:05.780 --> 00:21:07.910 align:middle line:84% And you can see that most of them 00:21:07.910 --> 00:21:09.510 align:middle line:90% are less than two standard-- 00:21:09.510 --> 00:21:10.760 align:middle line:90% sorry. 00:21:10.760 --> 00:21:15.530 align:middle line:84% Deviation is less than twofold the average value 00:21:15.530 --> 00:21:18.350 align:middle line:84% and less than seven standard deviations 00:21:18.350 --> 00:21:21.080 align:middle line:90% in terms of the measurements. 00:21:21.080 --> 00:21:23.840 align:middle line:84% But the ones that are furthest to the right 00:21:23.840 --> 00:21:26.570 align:middle line:84% are clearly the ones that require the most attention, 00:21:26.570 --> 00:21:28.520 align:middle line:84% either in the experimental measurements 00:21:28.520 --> 00:21:29.917 align:middle line:90% or in the modeling. 00:21:29.917 --> 00:21:31.250 align:middle line:90% These are steady-state measures. 00:21:31.250 --> 00:21:34.790 align:middle line:84% This is kind of an abuse of a beautiful kinetic model. 00:21:34.790 --> 00:21:39.800 align:middle line:84% But it reflects the limited data that exists. 00:21:39.800 --> 00:21:42.830 align:middle line:84% And it's much easier to collect steady-state data where, 00:21:42.830 --> 00:21:44.420 align:middle line:90% basically, the red blood cell-- 00:21:44.420 --> 00:21:46.400 align:middle line:84% every particular molecule, even though there 00:21:46.400 --> 00:21:49.627 align:middle line:84% are fluxes in and out of every molecule, the molecule 00:21:49.627 --> 00:21:51.335 align:middle line:84% concentration itself is staying constant. 00:21:51.335 --> 00:21:53.970 align:middle line:90% 00:21:53.970 --> 00:21:55.730 align:middle line:90% So that's how you-- 00:21:55.730 --> 00:21:58.550 align:middle line:84% if you're assuming each molecular concentration 00:21:58.550 --> 00:22:01.070 align:middle line:84% is staying constant with respect to time, 00:22:01.070 --> 00:22:04.680 align:middle line:84% you're just measuring steady-state levels. 00:22:04.680 --> 00:22:06.920 align:middle line:84% But if you're more interested in looking 00:22:06.920 --> 00:22:11.090 align:middle line:84% at the dynamics, the movie of how molecules will change 00:22:11.090 --> 00:22:16.400 align:middle line:84% if you perturb the system, you can 00:22:16.400 --> 00:22:19.070 align:middle line:84% think of a wide variety of different curves 00:22:19.070 --> 00:22:21.440 align:middle line:90% that the timecourses can take. 00:22:21.440 --> 00:22:23.810 align:middle line:84% And then the challenge is how do you represent them? 00:22:23.810 --> 00:22:26.810 align:middle line:84% You have 40-some different small molecules. 00:22:26.810 --> 00:22:28.520 align:middle line:84% You're tracking the concentrations. 00:22:28.520 --> 00:22:31.680 align:middle line:84% And then the timecourse can range over hours. 00:22:31.680 --> 00:22:34.340 align:middle line:84% But one way of doing it is doing pairs 00:22:34.340 --> 00:22:38.990 align:middle line:84% of substrates at a time, substrate a and b, 00:22:38.990 --> 00:22:41.120 align:middle line:84% and then monitoring the timecourse as a vector. 00:22:41.120 --> 00:22:44.880 align:middle line:84% Think of these as a series of little points along here. 00:22:44.880 --> 00:22:51.650 align:middle line:84% And let's look in slide 17, look in the upper left, number 1. 00:22:51.650 --> 00:22:55.610 align:middle line:84% If, let's say, a is converted to b, 00:22:55.610 --> 00:22:58.380 align:middle line:84% so that a plus b is equal to some constant, 00:22:58.380 --> 00:23:03.260 align:middle line:84% you can see this slope of negative 1. 00:23:03.260 --> 00:23:07.040 align:middle line:84% This is exactly negative 1, because for every molecule 00:23:07.040 --> 00:23:09.750 align:middle line:84% of a that's consumed, a molecule of b is produced. 00:23:09.750 --> 00:23:12.620 align:middle line:84% And so when you see this perfect slope of negative 1, 00:23:12.620 --> 00:23:14.120 align:middle line:84% then that's the kind of relationship 00:23:14.120 --> 00:23:18.770 align:middle line:84% you expect between those two, even if you're randomly 00:23:18.770 --> 00:23:20.600 align:middle line:90% sampling a dynamic system. 00:23:20.600 --> 00:23:23.750 align:middle line:90% Here you're taking a timecourse. 00:23:23.750 --> 00:23:26.020 align:middle line:84% Number two is a pair of concentrations [? in the ?] 00:23:26.020 --> 00:23:26.660 align:middle line:90% equilibrium. 00:23:26.660 --> 00:23:28.130 align:middle line:84% You'll get an equilibrium constant, 00:23:28.130 --> 00:23:31.640 align:middle line:84% and the ratios here will not be negative 1. 00:23:31.640 --> 00:23:37.460 align:middle line:84% They will be some constant which determines that equilibrium. 00:23:37.460 --> 00:23:42.800 align:middle line:84% You have two dynamically independent metabolites, 00:23:42.800 --> 00:23:45.530 align:middle line:90% as in quadrant III here. 00:23:45.530 --> 00:23:51.740 align:middle line:84% Basically, as you march through increasing b, 00:23:51.740 --> 00:23:54.270 align:middle line:84% a can stay constant because it doesn't really care. 00:23:54.270 --> 00:23:57.950 align:middle line:84% It doesn't respond to changes in b or changes 00:23:57.950 --> 00:23:59.660 align:middle line:90% that result in b changing. 00:23:59.660 --> 00:24:02.990 align:middle line:84% And then maybe some other set of dynamics 00:24:02.990 --> 00:24:06.260 align:middle line:84% will cause a to change, and b stays constant. 00:24:06.260 --> 00:24:08.130 align:middle line:84% And if you sample enough timecourse, 00:24:08.130 --> 00:24:12.320 align:middle line:84% you may find that this fills up the entire space 00:24:12.320 --> 00:24:17.750 align:middle line:84% concentrations available to a and b showing no correlation. 00:24:17.750 --> 00:24:19.910 align:middle line:84% Another interesting type of phenomena you can see 00:24:19.910 --> 00:24:24.230 align:middle line:84% is that not every possible concentration of a-- 00:24:24.230 --> 00:24:27.260 align:middle line:84% and it's not completely independent of b. 00:24:27.260 --> 00:24:30.170 align:middle line:84% As you might start at a particular point in a time 00:24:30.170 --> 00:24:32.690 align:middle line:84% series at this gap here in the lower right-- 00:24:32.690 --> 00:24:36.320 align:middle line:84% and you start decreasing b and increasing a. 00:24:36.320 --> 00:24:38.840 align:middle line:84% And then, at some point, the dynamics of the entire network, 00:24:38.840 --> 00:24:43.160 align:middle line:84% not just a and b, contribute to now a taking a dive down 00:24:43.160 --> 00:24:44.300 align:middle line:90% and b increasing. 00:24:44.300 --> 00:24:46.820 align:middle line:84% And eventually, you return to that steady-state point, 00:24:46.820 --> 00:24:48.590 align:middle line:84% and you've described all the conditions 00:24:48.590 --> 00:24:54.698 align:middle line:84% that you might be able to achieve in this closed loop. 00:24:54.698 --> 00:24:56.990 align:middle line:84% So we're going to look at these kind of phase diagrams. 00:24:56.990 --> 00:25:02.120 align:middle line:84% The concentration of a versus concentration of b-- 00:25:02.120 --> 00:25:05.240 align:middle line:84% where a series of time points that would be color coded. 00:25:05.240 --> 00:25:08.300 align:middle line:84% These can be either lumped, as they are in this diagram, 00:25:08.300 --> 00:25:12.140 align:middle line:84% or in the next slide, we'll see them separated out 00:25:12.140 --> 00:25:13.192 align:middle line:90% one metabolite at a time. 00:25:13.192 --> 00:25:14.900 align:middle line:84% But it's the same concept, whether you've 00:25:14.900 --> 00:25:17.630 align:middle line:84% got a group of metabolites involved in glycolysis 00:25:17.630 --> 00:25:22.010 align:middle line:84% lumped together, compared to, say, a group that are redox. 00:25:22.010 --> 00:25:26.562 align:middle line:84% You have in the lower left of each of these quadrants 00:25:26.562 --> 00:25:28.520 align:middle line:84% is this time series that we've been looking at. 00:25:28.520 --> 00:25:32.270 align:middle line:84% And the upper part is the correlation coefficients, 00:25:32.270 --> 00:25:35.510 align:middle line:84% color coded so that blue is a negative correlation, 00:25:35.510 --> 00:25:38.240 align:middle line:84% and gray is very significant positive correlation, 00:25:38.240 --> 00:25:40.770 align:middle line:84% and everything else is something in between. 00:25:40.770 --> 00:25:43.250 align:middle line:84% So what you see from these is you see, for example, here's 00:25:43.250 --> 00:25:46.130 align:middle line:84% this up curve that's very close to the negative 1 curve, 00:25:46.130 --> 00:25:48.860 align:middle line:84% as if there's a conservation reaction here 00:25:48.860 --> 00:25:52.130 align:middle line:84% between glycolysis and the adjacent steps. 00:25:52.130 --> 00:25:55.500 align:middle line:84% You see little loops, for example, 00:25:55.500 --> 00:26:04.040 align:middle line:84% in the lower left, adenine biosynthesis, in that row, 00:26:04.040 --> 00:26:04.730 align:middle line:90% and so on. 00:26:04.730 --> 00:26:07.040 align:middle line:84% You see examples of each of these kind of behaviors 00:26:07.040 --> 00:26:10.040 align:middle line:84% here, where you're going from the red point-- 00:26:10.040 --> 00:26:11.540 align:middle line:90% these little dots in red-- 00:26:11.540 --> 00:26:16.820 align:middle line:84% to green to blue to yellow to the end gray in increasing time 00:26:16.820 --> 00:26:20.540 align:middle line:84% coarseness, starting with 0.1-hour resolution and ending 00:26:20.540 --> 00:26:24.950 align:middle line:90% with 300-hour resolution. 00:26:24.950 --> 00:26:26.360 align:middle line:84% This is lumping, where we're kind 00:26:26.360 --> 00:26:28.550 align:middle line:84% of looking at things like ATP and redox loads. 00:26:28.550 --> 00:26:31.130 align:middle line:84% Now, if we look at it one molecule at a time, 00:26:31.130 --> 00:26:32.750 align:middle line:84% you obviously get a more complex-- 00:26:32.750 --> 00:26:35.990 align:middle line:84% you get every possible pairwise combination of molecules. 00:26:35.990 --> 00:26:38.120 align:middle line:84% And you can see these full dynamics. 00:26:38.120 --> 00:26:39.410 align:middle line:90% Now, these are not data. 00:26:39.410 --> 00:26:40.520 align:middle line:90% These are all simulations. 00:26:40.520 --> 00:26:44.360 align:middle line:84% And unfortunately, we don't have that kind of dynamic resolution 00:26:44.360 --> 00:26:46.602 align:middle line:90% in experimental data just yet. 00:26:46.602 --> 00:26:47.810 align:middle line:90% But this gives you some idea. 00:26:47.810 --> 00:26:50.780 align:middle line:84% If you see some particularly interesting phenomenon here, 00:26:50.780 --> 00:26:53.870 align:middle line:84% then it might be a motivation for going in and looking 00:26:53.870 --> 00:26:56.195 align:middle line:90% at the data in more detail. 00:26:56.195 --> 00:26:58.730 align:middle line:90% 00:26:58.730 --> 00:27:01.210 align:middle line:90% OK. 00:27:01.210 --> 00:27:11.290 align:middle line:84% Now, we mentioned this difference between the ordinary 00:27:11.290 --> 00:27:16.810 align:middle line:84% hyperbolic curve-- in the lower right of this whole slide, 00:27:16.810 --> 00:27:18.790 align:middle line:90% the upper left of the insert-- 00:27:18.790 --> 00:27:20.650 align:middle line:84% and the sigmoid curve that you get 00:27:20.650 --> 00:27:26.810 align:middle line:90% from an allosteric interaction. 00:27:26.810 --> 00:27:30.190 align:middle line:84% And within-- this whole cell is set up 00:27:30.190 --> 00:27:33.250 align:middle line:84% for transporting oxygen. And as oxygen concentration 00:27:33.250 --> 00:27:36.550 align:middle line:84% increases for hemoglobin, you'll either 00:27:36.550 --> 00:27:40.360 align:middle line:84% get this hyperbolic or sigmoid curve 00:27:40.360 --> 00:27:45.280 align:middle line:84% as it gets increasingly sigmoid, increasingly cooperative, 00:27:45.280 --> 00:27:48.040 align:middle line:84% with increasing amounts of one of 00:27:48.040 --> 00:27:52.780 align:middle line:84% the intermediate metabolites, 2, 3 diphosphoglyserate. 00:27:52.780 --> 00:27:55.090 align:middle line:84% And you can see, this is now the connection 00:27:55.090 --> 00:27:57.760 align:middle line:90% between the glycolytic pathway. 00:27:57.760 --> 00:28:01.450 align:middle line:84% The regulation is sensing the state of the glycolytic pathway 00:28:01.450 --> 00:28:05.013 align:middle line:90% and the hemoglobin. 00:28:05.013 --> 00:28:06.430 align:middle line:84% In addition, there are connections 00:28:06.430 --> 00:28:08.890 align:middle line:84% with the pH, which is also regulated 00:28:08.890 --> 00:28:11.080 align:middle line:90% in this, and the redox. 00:28:11.080 --> 00:28:13.300 align:middle line:84% You can see just above it here, the hemoglobin 00:28:13.300 --> 00:28:16.700 align:middle line:84% going to the unproductive methemoglobin state. 00:28:16.700 --> 00:28:18.670 align:middle line:84% So you can see there are connections 00:28:18.670 --> 00:28:20.905 align:middle line:84% in the network between this ultimate function, which 00:28:20.905 --> 00:28:25.000 align:middle line:84% is transporting oxygen, and all these intermediate metabolism 00:28:25.000 --> 00:28:25.810 align:middle line:90% components. 00:28:25.810 --> 00:28:28.210 align:middle line:84% And it also brings us to this topic, again, 00:28:28.210 --> 00:28:31.900 align:middle line:84% of this cooperativity and how does it arise here? 00:28:31.900 --> 00:28:34.600 align:middle line:84% And the hemoglobin has a tetrameric conformation state 00:28:34.600 --> 00:28:37.900 align:middle line:84% change, which is there's a second site that 00:28:37.900 --> 00:28:44.750 align:middle line:84% binds this organic diphosphoglyserate. 00:28:44.750 --> 00:28:46.790 align:middle line:84% But another direction we can take-- 00:28:46.790 --> 00:28:50.780 align:middle line:84% so in the lower left-hand corner of slide 21 00:28:50.780 --> 00:28:53.600 align:middle line:84% is the same icon again of how we get increasingly 00:28:53.600 --> 00:28:57.890 align:middle line:84% cooperative from hyperbolic to sigmoidal, to the point where 00:28:57.890 --> 00:29:01.530 align:middle line:84% this becomes almost vertical and displaced from the origin 00:29:01.530 --> 00:29:06.050 align:middle line:84% so that the cell, at a good point in response 00:29:06.050 --> 00:29:09.410 align:middle line:84% to a stimulus, will make a decision 00:29:09.410 --> 00:29:13.580 align:middle line:84% to commit to the next phase in cell division. 00:29:13.580 --> 00:29:15.110 align:middle line:84% This is similar to the cell division 00:29:15.110 --> 00:29:17.300 align:middle line:84% that we talked about earlier in the context 00:29:17.300 --> 00:29:20.060 align:middle line:84% of the microarray analysis of yeast cell division. 00:29:20.060 --> 00:29:22.730 align:middle line:84% Here we're talking about a Xenopus amphibian 00:29:22.730 --> 00:29:27.050 align:middle line:84% oocyte, which has nice large cells to do this kind of study. 00:29:27.050 --> 00:29:30.050 align:middle line:84% Where you need to decide to come out of G1 00:29:30.050 --> 00:29:32.450 align:middle line:90% and commit to synthesizing DNA-- 00:29:32.450 --> 00:29:35.510 align:middle line:84% in the S phase, in the lower portion of the circle, 00:29:35.510 --> 00:29:37.100 align:middle line:84% where you get, now, two DNA molecules. 00:29:37.100 --> 00:29:38.750 align:middle line:84% And once you're convinced that you've 00:29:38.750 --> 00:29:41.120 align:middle line:84% finished replicating all your DNA, 00:29:41.120 --> 00:29:44.030 align:middle line:84% only at that point, then can you commit to mitosis, 00:29:44.030 --> 00:29:47.240 align:middle line:90% another major decision. 00:29:47.240 --> 00:29:49.130 align:middle line:84% And then you get two cells, and you go back, 00:29:49.130 --> 00:29:50.930 align:middle line:90% and you complete the cell cycle. 00:29:50.930 --> 00:29:52.940 align:middle line:84% The little timecourse at the bottom 00:29:52.940 --> 00:29:55.340 align:middle line:84% here should be reminiscent for you 00:29:55.340 --> 00:29:57.530 align:middle line:84% of the timecourse we had of RNA synthesis 00:29:57.530 --> 00:29:59.810 align:middle line:90% of various clusters of RNA. 00:29:59.810 --> 00:30:01.620 align:middle line:90% Here it's the DNA synthesis. 00:30:01.620 --> 00:30:05.030 align:middle line:84% You can see it ramps up in the red S phase and then ramps 00:30:05.030 --> 00:30:08.670 align:middle line:84% down in the metaphase due to the creation of two new cells. 00:30:08.670 --> 00:30:13.730 align:middle line:84% But we want to talk about how do we make this as responsive 00:30:13.730 --> 00:30:16.970 align:middle line:84% as, say, progesterone or something that is signaling 00:30:16.970 --> 00:30:21.710 align:middle line:84% cells that might be waiting for long periods of time 00:30:21.710 --> 00:30:24.080 align:middle line:84% to complete the next step in a division 00:30:24.080 --> 00:30:26.750 align:middle line:84% cycle for an external hormone stimulus 00:30:26.750 --> 00:30:31.955 align:middle line:84% that it's time to start the next step in cell cycle? 00:30:31.955 --> 00:30:33.830 align:middle line:84% We want this to be displaced from the origin. 00:30:33.830 --> 00:30:36.230 align:middle line:84% You don't want it to be just flipping on and off 00:30:36.230 --> 00:30:37.890 align:middle line:90% irrespective of a stimulus. 00:30:37.890 --> 00:30:40.830 align:middle line:84% But when it does flip, you want it to go very quickly. 00:30:40.830 --> 00:30:43.100 align:middle line:90% So how would we model this? 00:30:43.100 --> 00:30:49.130 align:middle line:84% So look at the upper right-hand diagram of slide 22. 00:30:49.130 --> 00:30:53.630 align:middle line:84% And you have a set of these oocytes kind 00:30:53.630 --> 00:30:57.230 align:middle line:84% of diagrammatically indicating their state. 00:30:57.230 --> 00:30:59.000 align:middle line:84% Their state determined by to what 00:30:59.000 --> 00:31:07.190 align:middle line:84% extent are they ready to commit to the next division state? 00:31:07.190 --> 00:31:09.800 align:middle line:84% Here we can think of it as the biomarker 00:31:09.800 --> 00:31:14.270 align:middle line:84% is the state of phosphorylation of a protein MAP kinase. 00:31:14.270 --> 00:31:17.240 align:middle line:84% If it's phosphorylated, then it's 00:31:17.240 --> 00:31:18.410 align:middle line:90% committed to this division. 00:31:18.410 --> 00:31:21.530 align:middle line:84% And we can think of this as the black side of this gradient, 00:31:21.530 --> 00:31:22.910 align:middle line:90% going from white to black. 00:31:22.910 --> 00:31:27.050 align:middle line:84% But if you grind up a whole population of these oocytes 00:31:27.050 --> 00:31:31.700 align:middle line:84% and measure the total MAP kinase phosphorylation as a function 00:31:31.700 --> 00:31:34.600 align:middle line:90% of increasing stimulus, S-- 00:31:34.600 --> 00:31:37.790 align:middle line:84% in this case progesterone-- the response-- 00:31:37.790 --> 00:31:39.560 align:middle line:84% that is to say, the phosphorylated state 00:31:39.560 --> 00:31:42.440 align:middle line:90% and the commitment to mitosis-- 00:31:42.440 --> 00:31:44.510 align:middle line:84% will gradually increase, as indicated 00:31:44.510 --> 00:31:47.060 align:middle line:90% in the gradient model. 00:31:47.060 --> 00:31:49.010 align:middle line:84% But that's if you ground up all the cells. 00:31:49.010 --> 00:31:52.220 align:middle line:84% But the other way of obtaining that result 00:31:52.220 --> 00:31:55.850 align:middle line:84% would be if each cell is making an all or none decision. 00:31:55.850 --> 00:31:58.820 align:middle line:84% And what happens is the probability of a cell being 00:31:58.820 --> 00:32:01.700 align:middle line:84% in that all or none state changes with increasing 00:32:01.700 --> 00:32:03.750 align:middle line:90% stimulus to progesterone. 00:32:03.750 --> 00:32:06.480 align:middle line:84% And that is the lower model and is, 00:32:06.480 --> 00:32:10.400 align:middle line:84% in fact, closer to reality, as indicated by the experiment 00:32:10.400 --> 00:32:16.610 align:middle line:84% at the bottom part of the slide here, where you have-- 00:32:16.610 --> 00:32:18.727 align:middle line:84% this is a part of a concentration curve 00:32:18.727 --> 00:32:20.810 align:middle line:84% where you're increasing the stimulus progesterone. 00:32:20.810 --> 00:32:22.670 align:middle line:84% But at this particular stimulus, you 00:32:22.670 --> 00:32:24.380 align:middle line:90% can sample individual cells. 00:32:24.380 --> 00:32:27.650 align:middle line:84% There's enough of the cell that you can actually do proteomics 00:32:27.650 --> 00:32:29.090 align:middle line:90% on individual cells. 00:32:29.090 --> 00:32:31.670 align:middle line:84% And the proteomics here is a western blot. 00:32:31.670 --> 00:32:36.050 align:middle line:84% We mentioned this a couple of lectures back. 00:32:36.050 --> 00:32:40.850 align:middle line:84% And here you can see the two states of the MAP kinase. 00:32:40.850 --> 00:32:42.950 align:middle line:84% The phosphorylated state is the slower. 00:32:42.950 --> 00:32:46.280 align:middle line:84% Electrophoretically, it's the upper band in this diagram. 00:32:46.280 --> 00:32:49.550 align:middle line:84% And the lower band is the unphosphorylated state. 00:32:49.550 --> 00:32:53.570 align:middle line:84% And you can see there's no example here of a cell which 00:32:53.570 --> 00:32:55.130 align:middle line:84% is in an intermediate state, where 00:32:55.130 --> 00:33:00.260 align:middle line:84% it has half and half or 40/60 of the two different protein 00:33:00.260 --> 00:33:01.440 align:middle line:90% forms. 00:33:01.440 --> 00:33:04.190 align:middle line:84% However, if you, as a thought experiment, 00:33:04.190 --> 00:33:07.370 align:middle line:84% took all those cells and mushed them up, and took this 00:33:07.370 --> 00:33:10.610 align:middle line:84% and ran it all in one lane, you would see a mixture. 00:33:10.610 --> 00:33:12.560 align:middle line:84% You would see all the intermediate states, 00:33:12.560 --> 00:33:15.210 align:middle line:84% and that would be a function of progesterone. 00:33:15.210 --> 00:33:18.980 align:middle line:84% So this is a warning, similar to ones that I've said before, 00:33:18.980 --> 00:33:23.780 align:middle line:84% that when you grind everything up and mix populations of cells 00:33:23.780 --> 00:33:26.780 align:middle line:84% or molecules, you need to be careful 00:33:26.780 --> 00:33:28.950 align:middle line:84% because different cells may be in different states, 00:33:28.950 --> 00:33:30.908 align:middle line:84% different molecules may be in different states. 00:33:30.908 --> 00:33:34.580 align:middle line:84% And the average behavior is not the same as the individual. 00:33:34.580 --> 00:33:37.850 align:middle line:84% But now, that's only part of the lesson here. 00:33:37.850 --> 00:33:42.000 align:middle line:84% The cells are going through this all or none process. 00:33:42.000 --> 00:33:44.900 align:middle line:84% We can monitor it by single-cell proteomics. 00:33:44.900 --> 00:33:46.700 align:middle line:90% But how do we model it? 00:33:46.700 --> 00:33:49.970 align:middle line:84% Well, in very abstract terms, the response 00:33:49.970 --> 00:33:53.090 align:middle line:84% here can be modeled as in this Hill coefficient, where 00:33:53.090 --> 00:33:57.950 align:middle line:84% you have a stimulus, s, some kinetic constant, k-- kind 00:33:57.950 --> 00:34:01.640 align:middle line:84% of like a Michaelis constant-- where it's basically, 00:34:01.640 --> 00:34:09.409 align:middle line:84% as gets s closer to k, it has a larger response here, 00:34:09.409 --> 00:34:10.699 align:middle line:90% effect on the response. 00:34:10.699 --> 00:34:17.420 align:middle line:84% And that's nonlinear because you have the exponent h. 00:34:17.420 --> 00:34:20.929 align:middle line:84% And the larger h is, the more nonlinear it is. 00:34:20.929 --> 00:34:25.050 align:middle line:84% So let's say that h is 1 in this little schematic to the right 00:34:25.050 --> 00:34:25.550 align:middle line:90% here. 00:34:25.550 --> 00:34:28.639 align:middle line:90% It's hyperbolic. 00:34:28.639 --> 00:34:30.500 align:middle line:90% No sigmoid character at all. 00:34:30.500 --> 00:34:33.418 align:middle line:84% In the case of hemoglobin and phosphofructokinase 00:34:33.418 --> 00:34:35.210 align:middle line:84% that we talked about in the red blood cell, 00:34:35.210 --> 00:34:42.800 align:middle line:84% it's more sigmoidal, like an h of 2.8, almost cube law there. 00:34:42.800 --> 00:34:45.830 align:middle line:84% And in fact, even within this system, one of the steps 00:34:45.830 --> 00:34:48.005 align:middle line:84% that we'll talk about in the next slide-- 00:34:48.005 --> 00:34:51.359 align:middle line:90% 00:34:51.359 --> 00:34:54.530 align:middle line:84% where you have a stimulus of a Mos protein 00:34:54.530 --> 00:34:58.700 align:middle line:84% and a response of this same MAP kinase phosphorylation 00:34:58.700 --> 00:35:03.830 align:middle line:84% response, it has a modest sigmoid Hill coefficient of 3, 00:35:03.830 --> 00:35:05.390 align:middle line:90% just like hemoglobin. 00:35:05.390 --> 00:35:10.160 align:middle line:84% But the overall response of MAP kinase to progesterone 00:35:10.160 --> 00:35:14.510 align:middle line:90% has an enormous exponent of 42. 00:35:14.510 --> 00:35:17.090 align:middle line:84% That means it's almost vertical, and it's 00:35:17.090 --> 00:35:19.230 align:middle line:90% displaced from the origin. 00:35:19.230 --> 00:35:20.880 align:middle line:90% How do we get that? 00:35:20.880 --> 00:35:21.380 align:middle line:90% OK. 00:35:21.380 --> 00:35:22.340 align:middle line:90% So-- whoops. 00:35:22.340 --> 00:35:25.090 align:middle line:90% 00:35:25.090 --> 00:35:27.700 align:middle line:90% So here is a proposed model. 00:35:27.700 --> 00:35:31.090 align:middle line:84% And it's an interesting snapshot in the inevitable evolution 00:35:31.090 --> 00:35:33.580 align:middle line:84% of a model from something very primitive, which 00:35:33.580 --> 00:35:38.950 align:middle line:84% might have just been the Mos effects on MAP kinase, 00:35:38.950 --> 00:35:42.470 align:middle line:84% kind of as a direct effect here, which as we said, 00:35:42.470 --> 00:35:45.240 align:middle line:84% has a Hill coefficient of about threefold-- 00:35:45.240 --> 00:35:46.580 align:middle line:90% sorry. 00:35:46.580 --> 00:35:49.150 align:middle line:84% Hill coefficient of 3, no threefold, 00:35:49.150 --> 00:35:51.010 align:middle line:90% because that's an exponential. 00:35:51.010 --> 00:35:54.130 align:middle line:84% But overall, combination of two other factors 00:35:54.130 --> 00:35:56.680 align:middle line:84% that you have a chain of modifiers, each 00:35:56.680 --> 00:35:59.830 align:middle line:84% close to saturation, meaning that each one has 00:35:59.830 --> 00:36:03.640 align:middle line:90% a slight sigmoid behavior. 00:36:03.640 --> 00:36:07.900 align:middle line:90% And you've got-- sorry. 00:36:07.900 --> 00:36:09.370 align:middle line:84% It could be a hyperbolic function, 00:36:09.370 --> 00:36:12.640 align:middle line:84% such as, let's just say, this dotted line that's 00:36:12.640 --> 00:36:16.030 align:middle line:84% going up smoothly from 0, where it says "neither." 00:36:16.030 --> 00:36:18.100 align:middle line:84% That would be the effect of, if you just 00:36:18.100 --> 00:36:22.330 align:middle line:84% have a normal enzymatic reaction with no allostery, 00:36:22.330 --> 00:36:25.908 align:middle line:84% no feedback, no ultrasensitivity. 00:36:25.908 --> 00:36:27.700 align:middle line:84% If each of those has a component and you're 00:36:27.700 --> 00:36:30.980 align:middle line:84% close to saturation-- meaning your substrate is very high. 00:36:30.980 --> 00:36:33.380 align:middle line:84% And so you're going as fast as the enzyme will go. 00:36:33.380 --> 00:36:34.660 align:middle line:90% You have a chain of those. 00:36:34.660 --> 00:36:37.450 align:middle line:84% You can show through kinetic modeling 00:36:37.450 --> 00:36:41.080 align:middle line:84% that that will create a high sensitivity to the reaction. 00:36:41.080 --> 00:36:46.840 align:middle line:84% And that's what the furthest dotted curve close to the axis 00:36:46.840 --> 00:36:49.660 align:middle line:90% is, ultrasensitivity alone. 00:36:49.660 --> 00:36:52.660 align:middle line:84% And then, here, you have progesterone 00:36:52.660 --> 00:36:59.740 align:middle line:84% going in, in the upper left, affecting a complicated rate. 00:36:59.740 --> 00:37:02.470 align:middle line:84% Now, these rate constants don't necessarily 00:37:02.470 --> 00:37:07.188 align:middle line:84% mean a unitary, simple kinetic step. 00:37:07.188 --> 00:37:08.980 align:middle line:84% They can represent something as complicated 00:37:08.980 --> 00:37:13.840 align:middle line:84% as going from Amino Acids, AA, to a particular protein, Mos. 00:37:13.840 --> 00:37:17.080 align:middle line:84% And then the reverse reaction, k sub minus 1, 00:37:17.080 --> 00:37:19.420 align:middle line:84% is the degradation of Mos back to amino acids, which 00:37:19.420 --> 00:37:24.790 align:middle line:84% is not a reverse of the same enzyme set by any means. 00:37:24.790 --> 00:37:27.260 align:middle line:90% The next step, k2, is simpler. 00:37:27.260 --> 00:37:30.010 align:middle line:84% It's just Mos being phosphorylated by, actually, 00:37:30.010 --> 00:37:34.960 align:middle line:84% our friend MAP kinase in its [? phosphorous ?] state, 00:37:34.960 --> 00:37:37.960 align:middle line:90% producing the-- 00:37:37.960 --> 00:37:41.950 align:middle line:84% for catalyzing the Mos phosphorylation, which 00:37:41.950 --> 00:37:43.630 align:middle line:84% catalyzes another phosphorylation 00:37:43.630 --> 00:37:46.990 align:middle line:84% of another protein, which then positively stimulates 00:37:46.990 --> 00:37:48.830 align:middle line:90% MAP kinase itself. 00:37:48.830 --> 00:37:51.460 align:middle line:84% So this whole thing is a set of positive reactions. 00:37:51.460 --> 00:37:54.760 align:middle line:84% Each of the phosphoproteins increases 00:37:54.760 --> 00:38:00.850 align:middle line:84% the enzymatic tendency for each of the other phosphoproteins 00:38:00.850 --> 00:38:01.810 align:middle line:90% to be produced. 00:38:01.810 --> 00:38:04.060 align:middle line:84% So you can see that this is kind of on a hair trigger. 00:38:04.060 --> 00:38:06.143 align:middle line:84% If any one of these phosphoproteins gets produced, 00:38:06.143 --> 00:38:08.230 align:middle line:84% then it'll increase all the other ones, 00:38:08.230 --> 00:38:11.680 align:middle line:84% and it'll be a very cooperative procedure. 00:38:11.680 --> 00:38:15.100 align:middle line:84% And that's what this, furthest towards the axis in the lower 00:38:15.100 --> 00:38:16.150 align:middle line:90% left-hand corner-- 00:38:16.150 --> 00:38:18.700 align:middle line:84% where you have this positive feedback 00:38:18.700 --> 00:38:23.410 align:middle line:84% alone causes this very great tendency 00:38:23.410 --> 00:38:29.470 align:middle line:84% to just jump up from 0 to very high response 00:38:29.470 --> 00:38:31.640 align:middle line:90% with very little stimulus. 00:38:31.640 --> 00:38:32.860 align:middle line:90% Well, that's dangerous. 00:38:32.860 --> 00:38:34.960 align:middle line:84% You want it to be nearly vertical, 00:38:34.960 --> 00:38:37.300 align:middle line:84% but you don't want to be nearly vertical at 0 stimulus, 00:38:37.300 --> 00:38:38.740 align:middle line:90% because that's unstable. 00:38:38.740 --> 00:38:40.022 align:middle line:90% So you want to move it over. 00:38:40.022 --> 00:38:41.980 align:middle line:84% And that's where the ultrasensitivity comes in, 00:38:41.980 --> 00:38:44.860 align:middle line:84% when you have this chain of modifiers putting both together 00:38:44.860 --> 00:38:49.810 align:middle line:84% as a solid black line, where it's shifted over so you have-- 00:38:49.810 --> 00:38:54.340 align:middle line:84% this assay was done with Mos, rather than progesterone, 00:38:54.340 --> 00:38:54.940 align:middle line:90% as the input. 00:38:54.940 --> 00:38:57.460 align:middle line:90% 00:38:57.460 --> 00:39:03.370 align:middle line:84% But you can see the overall increased cooperativity 00:39:03.370 --> 00:39:07.460 align:middle line:90% and shifting to the right. 00:39:07.460 --> 00:39:10.210 align:middle line:84% So that's an example of how you can get this very high Hill 00:39:10.210 --> 00:39:13.630 align:middle line:84% coefficient and where you can get bistability 00:39:13.630 --> 00:39:15.670 align:middle line:90% without stochastic. 00:39:15.670 --> 00:39:18.280 align:middle line:84% You can imagine that you can have stochastic bistability. 00:39:18.280 --> 00:39:20.950 align:middle line:84% If you have one molecule in the cell, and either it's there 00:39:20.950 --> 00:39:23.115 align:middle line:84% or it's not, then you have bistability. 00:39:23.115 --> 00:39:24.490 align:middle line:84% You have two states for the cell. 00:39:24.490 --> 00:39:27.160 align:middle line:84% Either it's the cell with the molecule or without. 00:39:27.160 --> 00:39:30.160 align:middle line:84% But here you can see that even with a very large cell-- 00:39:30.160 --> 00:39:32.410 align:middle line:84% Xenopus oocytes being one of the largest cells-- 00:39:32.410 --> 00:39:34.480 align:middle line:84% and very large amounts of proteins-- 00:39:34.480 --> 00:39:37.720 align:middle line:84% enough that you can easily see them in proteomics-- 00:39:37.720 --> 00:39:39.520 align:middle line:84% you can still achieve bistability 00:39:39.520 --> 00:39:40.930 align:middle line:90% with the right kinetic model. 00:39:40.930 --> 00:39:43.990 align:middle line:84% Not every random model would achieve that high Hill 00:39:43.990 --> 00:39:45.760 align:middle line:90% coefficient. 00:39:45.760 --> 00:39:46.270 align:middle line:90% OK. 00:39:46.270 --> 00:39:47.950 align:middle line:84% So we're just going to briefly mention 00:39:47.950 --> 00:39:49.790 align:middle line:84% the other way of getting bistability, 00:39:49.790 --> 00:39:54.050 align:middle line:84% which is via stochastic, so small molecules. 00:39:54.050 --> 00:39:59.510 align:middle line:90% And here, an example-- 00:39:59.510 --> 00:40:01.540 align:middle line:84% so instead of dealing with very large cells 00:40:01.540 --> 00:40:03.850 align:middle line:84% with very large numbers of molecules involved, 00:40:03.850 --> 00:40:07.090 align:middle line:84% here in, say, bacteria in particular, 00:40:07.090 --> 00:40:11.200 align:middle line:84% a phage-infected bacteria, you generally 00:40:11.200 --> 00:40:14.290 align:middle line:84% have the case of very small bursts of activity. 00:40:14.290 --> 00:40:18.550 align:middle line:84% A transcription factor will bind to a promoter. 00:40:18.550 --> 00:40:20.830 align:middle line:84% Before the transcription factor comes off, 00:40:20.830 --> 00:40:24.390 align:middle line:84% it may cause a small burst of a couple of RNA transcripts 00:40:24.390 --> 00:40:27.720 align:middle line:84% being made by a couple of RNA polymerases 00:40:27.720 --> 00:40:29.460 align:middle line:84% seeing those transcription factors. 00:40:29.460 --> 00:40:32.430 align:middle line:84% Then each of those RNAs caused its own bursts 00:40:32.430 --> 00:40:34.680 align:middle line:84% of protein synthesis where a whole series of ribosomes 00:40:34.680 --> 00:40:36.870 align:middle line:90% will bind in a polysome. 00:40:36.870 --> 00:40:41.220 align:middle line:84% And you'll get this double burst of RNA and protein. 00:40:41.220 --> 00:40:44.850 align:middle line:84% And the stochastic binding of that transcription factor 00:40:44.850 --> 00:40:50.850 align:middle line:84% that starts that burst can be modeled by reasonably 00:40:50.850 --> 00:40:53.370 align:middle line:84% measured parameters for each of these steps. 00:40:53.370 --> 00:40:55.620 align:middle line:84% And you can see that cells one, two, and three, 00:40:55.620 --> 00:40:58.380 align:middle line:90% in the lower left of slide 25-- 00:40:58.380 --> 00:41:00.510 align:middle line:84% where time is the horizontal axis up to, say, 00:41:00.510 --> 00:41:04.440 align:middle line:84% 45 minutes or a cell division or two. 00:41:04.440 --> 00:41:08.070 align:middle line:84% While the number of product proteins here, 00:41:08.070 --> 00:41:11.370 align:middle line:84% measured in dimers of protein, fluctuates, 00:41:11.370 --> 00:41:14.700 align:middle line:84% where cell one gets an early start, early burst, and cell 00:41:14.700 --> 00:41:17.880 align:middle line:84% three hasn't quite hit its burst yet. 00:41:17.880 --> 00:41:19.630 align:middle line:84% So you can see there's a lot of variation. 00:41:19.630 --> 00:41:22.560 align:middle line:84% And this is one way of achieving a bistable switch. 00:41:22.560 --> 00:41:24.240 align:middle line:84% But as you've seen-- not the only way. 00:41:24.240 --> 00:41:27.090 align:middle line:84% You can also do it where all the proteins are 00:41:27.090 --> 00:41:28.920 align:middle line:90% present in all large amounts. 00:41:28.920 --> 00:41:31.085 align:middle line:84% If you do choose to go the stochastic route-- 00:41:31.085 --> 00:41:33.460 align:middle line:84% and this might be an interesting project for some of you. 00:41:33.460 --> 00:41:36.210 align:middle line:84% It's by no means shown to be mission critical 00:41:36.210 --> 00:41:41.640 align:middle line:84% for the community of systems modelers. 00:41:41.640 --> 00:41:47.160 align:middle line:84% But many people believe it is a way to go. 00:41:47.160 --> 00:41:49.950 align:middle line:84% There has been great progress since 1977 00:41:49.950 --> 00:41:57.960 align:middle line:84% when Gillespie proposed the algorithm named after him 00:41:57.960 --> 00:42:01.260 align:middle line:84% for stochastic simulation, a couple of chemical 00:42:01.260 --> 00:42:04.080 align:middle line:84% reactions in general, not just biological, biochemical 00:42:04.080 --> 00:42:05.340 align:middle line:90% reactions. 00:42:05.340 --> 00:42:11.370 align:middle line:84% Since then, Gibson and Bruck, within the last couple 00:42:11.370 --> 00:42:14.460 align:middle line:84% of years, have come up with an algorithm, which is now 00:42:14.460 --> 00:42:16.320 align:middle line:84% time proportional to the logarithm of number 00:42:16.320 --> 00:42:18.403 align:middle line:84% of reactions, rather than the number of reactions. 00:42:18.403 --> 00:42:22.810 align:middle line:84% Any time you go from n to log n, this is big progress. 00:42:22.810 --> 00:42:27.900 align:middle line:84% And this is done by better tracking of calculations 00:42:27.900 --> 00:42:30.360 align:middle line:90% that you can reuse. 00:42:30.360 --> 00:42:32.550 align:middle line:84% And so I encourage you to take a look 00:42:32.550 --> 00:42:36.060 align:middle line:90% at this aspect of stochastics. 00:42:36.060 --> 00:42:39.900 align:middle line:84% Another aspect is people often think of the stochastics 00:42:39.900 --> 00:42:40.920 align:middle line:90% as kind of a nuisance. 00:42:40.920 --> 00:42:42.570 align:middle line:84% They increase the computer time that it 00:42:42.570 --> 00:42:44.160 align:middle line:90% takes to do simulations. 00:42:44.160 --> 00:42:47.130 align:middle line:84% They increase your uncertainty about the simulations 00:42:47.130 --> 00:42:48.990 align:middle line:90% that you then produce. 00:42:48.990 --> 00:42:50.940 align:middle line:84% But there is an aspect of it which 00:42:50.940 --> 00:42:52.410 align:middle line:84% is just beginning to be harnessed 00:42:52.410 --> 00:42:54.330 align:middle line:84% in various fields of engineering, 00:42:54.330 --> 00:42:57.130 align:middle line:84% and biological engineering is no exception. 00:42:57.130 --> 00:43:00.390 align:middle line:84% And I give you two examples here to just whet your appetite. 00:43:00.390 --> 00:43:02.070 align:middle line:84% Again, I'm not going to go through them. 00:43:02.070 --> 00:43:04.830 align:middle line:84% But you can see that you can actually 00:43:04.830 --> 00:43:08.460 align:middle line:84% make switches and amplifiers for gene expression-- gene 00:43:08.460 --> 00:43:10.270 align:middle line:84% expression being one of our favorite topics 00:43:10.270 --> 00:43:12.450 align:middle line:84% in this course-- which are based on noise 00:43:12.450 --> 00:43:16.170 align:middle line:84% and where you can get bistability 00:43:16.170 --> 00:43:19.770 align:middle line:90% using these fluctuations. 00:43:19.770 --> 00:43:21.718 align:middle line:84% And that's not too unexpected based 00:43:21.718 --> 00:43:23.010 align:middle line:90% on what we've just been saying. 00:43:23.010 --> 00:43:24.990 align:middle line:84% But in addition, you can get stochastic 00:43:24.990 --> 00:43:30.090 align:middle line:84% focusing where the fluctuation allows enhanced sensitivity. 00:43:30.090 --> 00:43:31.380 align:middle line:90% OK. 00:43:31.380 --> 00:43:33.480 align:middle line:84% So I encourage you to look at that. 00:43:33.480 --> 00:43:34.950 align:middle line:84% Now, a particular place where you 00:43:34.950 --> 00:43:37.980 align:middle line:84% might worry that stochastics is coming into play quite a bit 00:43:37.980 --> 00:43:39.780 align:middle line:84% is in chromosome copy number, whether this 00:43:39.780 --> 00:43:42.420 align:middle line:84% is eukaryotic chromosomes or in the case we'll illustrate, 00:43:42.420 --> 00:43:45.602 align:middle line:84% a very simple case of plasmid chromosomes. 00:43:45.602 --> 00:43:47.310 align:middle line:84% Now, the interesting thing about plasmids 00:43:47.310 --> 00:43:52.440 align:middle line:84% is they can either be in lockstep with cell division, 00:43:52.440 --> 00:43:53.940 align:middle line:84% the way that eukaryotic chromosomes 00:43:53.940 --> 00:43:56.398 align:middle line:84% are-- in the cases of Xenopus oocytes we just talked about, 00:43:56.398 --> 00:43:58.530 align:middle line:90% where it makes a big decision. 00:43:58.530 --> 00:44:00.420 align:middle line:84% And that's the case of the R1 plasmids. 00:44:00.420 --> 00:44:04.080 align:middle line:84% Or it can be more of kind of a cloud of copy number, 00:44:04.080 --> 00:44:08.010 align:middle line:84% where they're trying to be close to a target number 00:44:08.010 --> 00:44:11.590 align:middle line:84% where you'll have more copies than one per cell. 00:44:11.590 --> 00:44:15.330 align:middle line:84% And so as the cell divides, it kind of randomly 00:44:15.330 --> 00:44:21.180 align:middle line:84% takes a partition of that number of plasmids. 00:44:21.180 --> 00:44:24.130 align:middle line:90% And ColE1 is an example of that. 00:44:24.130 --> 00:44:26.730 align:middle line:84% And you model it in order to determine 00:44:26.730 --> 00:44:29.970 align:middle line:90% the factors that govern it. 00:44:29.970 --> 00:44:31.860 align:middle line:90% This has implication. 00:44:31.860 --> 00:44:36.450 align:middle line:84% The copy number will affect the expression levels. 00:44:36.450 --> 00:44:40.530 align:middle line:84% And the expression levels are of importance to biotechnology. 00:44:40.530 --> 00:44:43.840 align:middle line:84% And plasmids are, of course, also important in pathogenesis. 00:44:43.840 --> 00:44:47.010 align:middle line:84% The copy number affects pathogenesis 00:44:47.010 --> 00:44:49.590 align:middle line:84% since plasmids are a major way that drug resistance 00:44:49.590 --> 00:44:51.130 align:middle line:90% elements are passed around. 00:44:51.130 --> 00:44:55.170 align:middle line:84% So let's take a look at one, hopefully, 00:44:55.170 --> 00:44:58.170 align:middle line:84% highly oversimplified version of this. 00:44:58.170 --> 00:45:01.530 align:middle line:84% Here you have two RNAs, imaginatively 00:45:01.530 --> 00:45:03.870 align:middle line:90% called RNA 1 and RNA 2. 00:45:03.870 --> 00:45:08.220 align:middle line:84% We'll start-- RNA 2 is transcribed here, 00:45:08.220 --> 00:45:10.515 align:middle line:84% on the bottom strand, from right to left. 00:45:10.515 --> 00:45:13.560 align:middle line:90% 00:45:13.560 --> 00:45:16.230 align:middle line:84% So actually, look at the very bottom of slide 30. 00:45:16.230 --> 00:45:21.060 align:middle line:84% The magenta RNA polymerase is making RNA 2. 00:45:21.060 --> 00:45:27.910 align:middle line:84% And if nothing binds to RNA 2, if RNA 1 does not bind to it, 00:45:27.910 --> 00:45:29.290 align:middle line:90% RNase H will cleave it. 00:45:29.290 --> 00:45:31.420 align:middle line:84% It will then bind to blue DNA polymerase 00:45:31.420 --> 00:45:34.840 align:middle line:84% and will start replicating the plasmid. 00:45:34.840 --> 00:45:37.750 align:middle line:84% On the other hand, if RNA 1, which is made on the opposite 00:45:37.750 --> 00:45:40.510 align:middle line:84% strand of RNA 2-- it's this antisense story. 00:45:40.510 --> 00:45:44.110 align:middle line:84% It will then come and bind to RNA 2, sort of in trans. 00:45:44.110 --> 00:45:46.150 align:middle line:84% It acts as a transacting inhibitor. 00:45:46.150 --> 00:45:49.300 align:middle line:84% It's aided and abetted by the Rom protein. 00:45:49.300 --> 00:45:51.970 align:middle line:84% And now you don't get cleavage of RNase H. 00:45:51.970 --> 00:45:54.670 align:middle line:84% And so DNA polymerase doesn't have a primer 00:45:54.670 --> 00:45:57.190 align:middle line:84% upon which to act, and you don't get replication. 00:45:57.190 --> 00:46:00.260 align:middle line:84% And this is, of course, not just a yes or no thing. 00:46:00.260 --> 00:46:02.380 align:middle line:84% This is something that's regulated and allows 00:46:02.380 --> 00:46:04.870 align:middle line:84% it to feed back to get the right copy number. 00:46:04.870 --> 00:46:07.210 align:middle line:84% You don't want it to get an infinite copy number, 00:46:07.210 --> 00:46:10.330 align:middle line:84% or else it'll choke the cell that's harboring it. 00:46:10.330 --> 00:46:12.410 align:middle line:84% But you don't want it to drop down low enough 00:46:12.410 --> 00:46:14.080 align:middle line:84% so that then many cells will segregate 00:46:14.080 --> 00:46:19.420 align:middle line:90% with no copies of the plasmid. 00:46:19.420 --> 00:46:20.920 align:middle line:84% So you want to have a mass balance. 00:46:20.920 --> 00:46:24.340 align:middle line:84% You have to have conservation of mass. 00:46:24.340 --> 00:46:30.460 align:middle line:84% You want to be able to model both the initiation 00:46:30.460 --> 00:46:32.270 align:middle line:90% and degradation and inhibition. 00:46:32.270 --> 00:46:34.750 align:middle line:84% So you do this by making some simplifications 00:46:34.750 --> 00:46:36.730 align:middle line:90% that the RNase H rate is fast. 00:46:36.730 --> 00:46:38.800 align:middle line:84% Remember we had slow and fast reactions 00:46:38.800 --> 00:46:46.090 align:middle line:84% that we would eliminate, so too with the DNA polymerization. 00:46:46.090 --> 00:46:49.930 align:middle line:84% By subsuming the RNA 2 concentration in an RNA 00:46:49.930 --> 00:46:51.860 align:middle line:84% 1-based model, you can simplify it 00:46:51.860 --> 00:46:54.310 align:middle line:84% so that you're really only considering two species, RNA 00:46:54.310 --> 00:46:56.590 align:middle line:90% 1 and the plasmid DNA itself. 00:46:56.590 --> 00:47:02.420 align:middle line:84% We'll call these r and n for the two different molecules. 00:47:02.420 --> 00:47:04.900 align:middle line:84% So this is just a way of introducing 00:47:04.900 --> 00:47:06.100 align:middle line:90% a two-species model. 00:47:06.100 --> 00:47:11.800 align:middle line:84% We're going to come up with a rate equation for change 00:47:11.800 --> 00:47:13.670 align:middle line:90% of RNA 1 with respect to time. 00:47:13.670 --> 00:47:16.010 align:middle line:84% And then the next slide will show the change 00:47:16.010 --> 00:47:17.800 align:middle line:84% in the concentration of the plasmid. 00:47:17.800 --> 00:47:20.590 align:middle line:90% Concentration of the RNA is r. 00:47:20.590 --> 00:47:22.540 align:middle line:84% Concentration of the plasmid is n. 00:47:22.540 --> 00:47:25.130 align:middle line:90% We have dr dt and dn dt. 00:47:25.130 --> 00:47:26.240 align:middle line:90% And this is very simple. 00:47:26.240 --> 00:47:27.698 align:middle line:84% It's just like what we were talking 00:47:27.698 --> 00:47:30.250 align:middle line:90% about with the metabolites. 00:47:30.250 --> 00:47:32.620 align:middle line:90% You synthesize the RNA. 00:47:32.620 --> 00:47:33.610 align:middle line:90% That's a positive term. 00:47:33.610 --> 00:47:35.860 align:middle line:84% You degrade it, or you dilute it out. 00:47:35.860 --> 00:47:37.840 align:middle line:84% The dilution is based on the growth rate. 00:47:37.840 --> 00:47:39.160 align:middle line:90% Mu is a typical-- 00:47:39.160 --> 00:47:41.753 align:middle line:84% it's used in growth rates in population genetics, 00:47:41.753 --> 00:47:43.420 align:middle line:84% and here it's used in chemical kinetics. 00:47:43.420 --> 00:47:45.130 align:middle line:84% In fact, this is, in a certain sense, 00:47:45.130 --> 00:47:47.290 align:middle line:84% an example of a very exciting field where you're 00:47:47.290 --> 00:47:53.470 align:middle line:84% bringing together population genetics and chemical 00:47:53.470 --> 00:47:54.715 align:middle line:90% kinetics into one place. 00:47:54.715 --> 00:47:57.370 align:middle line:90% 00:47:57.370 --> 00:48:00.160 align:middle line:84% And population genetics and chemical kinetics, 00:48:00.160 --> 00:48:04.180 align:middle line:84% when they come together, unite some of the most 00:48:04.180 --> 00:48:06.650 align:middle line:90% disparate parts of this course. 00:48:06.650 --> 00:48:07.150 align:middle line:90% OK. 00:48:07.150 --> 00:48:08.890 align:middle line:84% So now we've got an equation here 00:48:08.890 --> 00:48:13.180 align:middle line:84% where the positive term is k1, the rate of initiation 00:48:13.180 --> 00:48:15.220 align:middle line:90% of RNA synthesis. 00:48:15.220 --> 00:48:18.638 align:middle line:84% And it's, of course, the more molecules, n, 00:48:18.638 --> 00:48:20.680 align:middle line:84% the higher the concentration of the plasmid, then 00:48:20.680 --> 00:48:21.850 align:middle line:84% the more RNA you're going to make. 00:48:21.850 --> 00:48:22.990 align:middle line:84% So that makes sense that you have 00:48:22.990 --> 00:48:25.690 align:middle line:84% the product, the rate constant times the number of plasmids. 00:48:25.690 --> 00:48:27.670 align:middle line:84% Similarly, the loss of it is going 00:48:27.670 --> 00:48:29.170 align:middle line:84% to be related to the number of RNAs. 00:48:29.170 --> 00:48:31.630 align:middle line:84% More RNAs you have, the more that you're 00:48:31.630 --> 00:48:34.000 align:middle line:84% going to lose, more that you have to lose. 00:48:34.000 --> 00:48:36.580 align:middle line:90% 00:48:36.580 --> 00:48:40.420 align:middle line:84% So that's the RNA, and this is for the DNA. 00:48:40.420 --> 00:48:47.470 align:middle line:84% Here you have dependence on the RNA. 00:48:47.470 --> 00:48:49.900 align:middle line:84% RNA 1, remember, is the thing that we 00:48:49.900 --> 00:48:51.430 align:middle line:90% modeled in the previous slide-- 00:48:51.430 --> 00:48:53.200 align:middle line:90% is an inhibitor. 00:48:53.200 --> 00:48:58.240 align:middle line:84% And so it's going to, when it binds to RNA 2-- 00:48:58.240 --> 00:49:00.380 align:middle line:84% which is implicitly modeled here-- 00:49:00.380 --> 00:49:03.400 align:middle line:84% it's going to have this inhibitory constant 00:49:03.400 --> 00:49:05.600 align:middle line:90% in the denominator. 00:49:05.600 --> 00:49:08.380 align:middle line:84% And so as the inhibitor RNA goes up, 00:49:08.380 --> 00:49:14.020 align:middle line:84% this inhibitory term goes up, the forward rate goes down. 00:49:14.020 --> 00:49:17.530 align:middle line:84% And the rate of replication is going 00:49:17.530 --> 00:49:19.570 align:middle line:84% to be also dependent on plasmid copy number, 00:49:19.570 --> 00:49:21.550 align:middle line:90% so it goes up within. 00:49:21.550 --> 00:49:25.070 align:middle line:84% The dilution rate is, of course, dependent on n as well. 00:49:25.070 --> 00:49:27.340 align:middle line:84% So the idea of this, in the next slide, 00:49:27.340 --> 00:49:30.280 align:middle line:84% is going to be to solve for the plasmid number. 00:49:30.280 --> 00:49:34.750 align:middle line:84% So slide 34, we have how you would implement those two 00:49:34.750 --> 00:49:38.770 align:middle line:84% equations that we had in the last two slides-- 00:49:38.770 --> 00:49:42.430 align:middle line:84% are shown on the very top left part of the slide. 00:49:42.430 --> 00:49:45.100 align:middle line:90% 00:49:45.100 --> 00:49:50.320 align:middle line:84% dr dt is abbreviated dr here in mathematical format. 00:49:50.320 --> 00:49:52.990 align:middle line:84% It's that same k1 constant times n 00:49:52.990 --> 00:49:55.800 align:middle line:84% is the concentration of plasmid molecules. 00:49:55.800 --> 00:50:00.370 align:middle line:84% Then the negative is the degradation rate. 00:50:00.370 --> 00:50:04.600 align:middle line:84% And the dilution by cell division, mu. 00:50:04.600 --> 00:50:07.780 align:middle line:84% And that's times r, which is the concentration of the inhibitory 00:50:07.780 --> 00:50:09.640 align:middle line:90% RNA 1. 00:50:09.640 --> 00:50:12.080 align:middle line:84% And in an [? analysis ?] equation, 00:50:12.080 --> 00:50:15.673 align:middle line:84% which we've already seen before for the plasmid molecules, n, 00:50:15.673 --> 00:50:17.965 align:middle line:84% the change in concentration of n is a function of time, 00:50:17.965 --> 00:50:23.130 align:middle line:84% dn dt, here abbreviated dn, is there. 00:50:23.130 --> 00:50:28.630 align:middle line:84% And then we're going to solve it. 00:50:28.630 --> 00:50:31.740 align:middle line:84% So these first three things are setting it up and asking 00:50:31.740 --> 00:50:34.320 align:middle line:90% the program to solve it. 00:50:34.320 --> 00:50:35.910 align:middle line:84% And we'll do it under the constraints 00:50:35.910 --> 00:50:38.880 align:middle line:84% of dr dt is equal to 0, dn dt is equal to 0. 00:50:38.880 --> 00:50:41.370 align:middle line:84% You will recognize this as the steady-state assumption. 00:50:41.370 --> 00:50:45.300 align:middle line:84% Even though there are fluxes in and out that are non-zero, 00:50:45.300 --> 00:50:47.970 align:middle line:90% the net effect is zero. 00:50:47.970 --> 00:50:50.260 align:middle line:84% And so that's the formula for steady state. 00:50:50.260 --> 00:50:53.850 align:middle line:84% We're going to start at a dilution rate of 1, 00:50:53.850 --> 00:50:55.770 align:middle line:84% where you'll have some steady-state level. 00:50:55.770 --> 00:50:57.770 align:middle line:84% And then we're going to watch the dynamics as it 00:50:57.770 --> 00:51:01.267 align:middle line:84% goes to a dilution rate that's slower, that is to say, 00:51:01.267 --> 00:51:02.350 align:middle line:90% the growth rate is slower. 00:51:02.350 --> 00:51:04.500 align:middle line:84% And as the growth rate is slower, 00:51:04.500 --> 00:51:06.060 align:middle line:84% you'll expect, maybe, to accumulate 00:51:06.060 --> 00:51:07.140 align:middle line:90% more plasmid molecules. 00:51:07.140 --> 00:51:10.230 align:middle line:84% Because they're not in lockstep with the cell division rate, 00:51:10.230 --> 00:51:12.360 align:middle line:90% the cell grows more slowly. 00:51:12.360 --> 00:51:13.860 align:middle line:84% And there's not some other feedback, 00:51:13.860 --> 00:51:16.027 align:middle line:84% and we haven't put any other feedback in this model, 00:51:16.027 --> 00:51:18.380 align:middle line:84% then it should go to about twice the level. 00:51:18.380 --> 00:51:22.140 align:middle line:90% 00:51:22.140 --> 00:51:29.580 align:middle line:84% So you do the symbolic solver in the top here. 00:51:29.580 --> 00:51:33.280 align:middle line:84% And you get this symbolic solution here. 00:51:33.280 --> 00:51:34.740 align:middle line:84% And if you do the numeric solver, 00:51:34.740 --> 00:51:41.233 align:middle line:84% NDSolve, then you get a very similar solution. 00:51:41.233 --> 00:51:42.150 align:middle line:90% And you could plot it. 00:51:42.150 --> 00:51:44.520 align:middle line:84% Here you're plotting y, which is a plasmid copy 00:51:44.520 --> 00:51:49.890 align:middle line:84% number from over a time range of 0 to 3. 00:51:49.890 --> 00:51:52.050 align:middle line:84% And you can see it goes up from slightly over 1 00:51:52.050 --> 00:51:57.000 align:middle line:84% to slightly over 2 in terms of a concentration of plasmid, 00:51:57.000 --> 00:52:04.140 align:middle line:84% as you might expect from lowering the dilution curve. 00:52:04.140 --> 00:52:08.100 align:middle line:84% Just as we had stochastic models for the bistability 00:52:08.100 --> 00:52:09.840 align:middle line:90% that we talked about earlier-- 00:52:09.840 --> 00:52:12.840 align:middle line:84% the Xenopus bistability could be continuous. 00:52:12.840 --> 00:52:17.040 align:middle line:84% And the lambda model could be stochastic. 00:52:17.040 --> 00:52:20.755 align:middle line:84% Here there are stochastic models for Copy Number Control, CNC, 00:52:20.755 --> 00:52:21.880 align:middle line:90% which are very interesting. 00:52:21.880 --> 00:52:23.580 align:middle line:90% I urge you to look at them-- 00:52:23.580 --> 00:52:28.680 align:middle line:84% where you can have, basically, using stochastic modeling to 00:52:28.680 --> 00:52:31.410 align:middle line:84% do molecular clocks, where you can reduce 00:52:31.410 --> 00:52:32.880 align:middle line:90% the rate at plasmid loss. 00:52:32.880 --> 00:52:35.850 align:middle line:84% You can see, in that last one, if you had a very small number 00:52:35.850 --> 00:52:41.340 align:middle line:84% of molecules, you would have a loss in some of the cells 00:52:41.340 --> 00:52:46.080 align:middle line:84% that would be more accurately modeled in a stochastic model. 00:52:46.080 --> 00:52:52.110 align:middle line:84% Now, we want to go from these models of red blood cell where 00:52:52.110 --> 00:52:54.630 align:middle line:84% you have metabolism without polymer synthesis, 00:52:54.630 --> 00:52:57.990 align:middle line:84% and the CNC model where you have polymer synthesis of RNA 00:52:57.990 --> 00:53:03.360 align:middle line:84% and DNA, but without metabolism, to a more integrated cell 00:53:03.360 --> 00:53:05.460 align:middle line:90% where you have both going on. 00:53:05.460 --> 00:53:09.000 align:middle line:84% And you want to represent the full optimization that 00:53:09.000 --> 00:53:13.740 align:middle line:84% must occur, getting metabolism optimally suited so that you 00:53:13.740 --> 00:53:18.000 align:middle line:84% get the right kinds of macromolecules made 00:53:18.000 --> 00:53:20.020 align:middle line:84% in a complicated cell like E. coli, 00:53:20.020 --> 00:53:23.010 align:middle line:84% which can adjust to a wide variety of different growth 00:53:23.010 --> 00:53:23.980 align:middle line:90% conditions. 00:53:23.980 --> 00:53:25.650 align:middle line:90% So what are the problems here? 00:53:25.650 --> 00:53:28.740 align:middle line:84% The number of parameters that we needed 00:53:28.740 --> 00:53:31.930 align:middle line:84% and had for the red blood cell was enormous. 00:53:31.930 --> 00:53:32.610 align:middle line:90% It was 200. 00:53:32.610 --> 00:53:34.590 align:middle line:84% It was a tour de force to get those. 00:53:34.590 --> 00:53:38.820 align:middle line:84% For E. coli, it's orders of magnitude more that are needed. 00:53:38.820 --> 00:53:41.520 align:middle line:84% Because instead of 40 enzymes, we 00:53:41.520 --> 00:53:48.150 align:middle line:84% have somewhere between 400 and 4,000. 00:53:48.150 --> 00:53:49.920 align:middle line:84% So measuring parameters is a problem. 00:53:49.920 --> 00:53:53.040 align:middle line:84% And we have the same problem about in vitro versus in vivo. 00:53:53.040 --> 00:53:56.790 align:middle line:84% We have the same set of constraints. 00:53:56.790 --> 00:53:59.280 align:middle line:84% And we want to focus more of our attention 00:53:59.280 --> 00:54:05.080 align:middle line:90% now on the flux constraints. 00:54:05.080 --> 00:54:07.950 align:middle line:84% So after a short break, we're going 00:54:07.950 --> 00:54:10.830 align:middle line:84% to come back and talk about the flux balance as a solution 00:54:10.830 --> 00:54:11.730 align:middle line:90% to this. 00:54:11.730 --> 00:54:16.050 align:middle line:84% And just as we had with the red blood cell, 00:54:16.050 --> 00:54:18.000 align:middle line:84% we're going to be focusing in on ways 00:54:18.000 --> 00:54:23.010 align:middle line:84% to relook at the way we think about the synthesis 00:54:23.010 --> 00:54:26.580 align:middle line:84% and degradation of the molecules in this network 00:54:26.580 --> 00:54:28.320 align:middle line:84% to see if we can rephrase in a way 00:54:28.320 --> 00:54:30.240 align:middle line:84% where we can ask interesting questions 00:54:30.240 --> 00:54:34.060 align:middle line:84% about the optimization of these systems. 00:54:34.060 --> 00:54:35.850 align:middle line:90% So take a quick break. 00:54:35.850 --> 00:54:40.610 align:middle line:84% And the second half, we'll talk about the flux balance. 00:54:40.610 --> 00:54:43.000 align:middle line:90%