WEBVTT 00:00:00.000 --> 00:00:01.960 align:middle line:84% NARRATOR: The following content is 00:00:01.960 --> 00:00:05.290 align:middle line:84% provided by MIT OpenCourseWare under a Creative Commons 00:00:05.290 --> 00:00:06.340 align:middle line:90% license. 00:00:06.340 --> 00:00:10.120 align:middle line:84% Additional information about our license and MIT OpenCourseWare, 00:00:10.120 --> 00:00:13.090 align:middle line:84% in general, is available at ocw.mit.edu. 00:00:13.090 --> 00:00:16.255 align:middle line:90% 00:00:16.255 --> 00:00:18.720 align:middle line:84% DR. GEORGE CHURCH: As promised, we're 00:00:18.720 --> 00:00:22.110 align:middle line:84% going into flux balance analysis. 00:00:22.110 --> 00:00:25.080 align:middle line:84% The simplifications here are even greater than the ones 00:00:25.080 --> 00:00:28.020 align:middle line:90% that we have been making so far. 00:00:28.020 --> 00:00:30.330 align:middle line:84% But hopefully, you will see some benefit. 00:00:30.330 --> 00:00:35.600 align:middle line:90% 00:00:35.600 --> 00:00:37.550 align:middle line:90% We're here. 00:00:37.550 --> 00:00:39.710 align:middle line:84% Assuming the time constant metabolic reactions 00:00:39.710 --> 00:00:42.480 align:middle line:84% are very fast compared to cell growth. 00:00:42.480 --> 00:00:45.230 align:middle line:84% Cell growth, we've seen the time constants are 00:00:45.230 --> 00:00:48.860 align:middle line:84% in the order of seconds or sometimes less, while growth 00:00:48.860 --> 00:00:50.450 align:middle line:90% might be on the order of hours. 00:00:50.450 --> 00:00:54.890 align:middle line:90% 00:00:54.890 --> 00:00:56.990 align:middle line:84% Even though we've seen interesting dynamics that 00:00:56.990 --> 00:00:59.030 align:middle line:84% can occur, those phase diagrams that we 00:00:59.030 --> 00:01:03.810 align:middle line:84% had in the ordinary differentiable equations, 00:01:03.810 --> 00:01:05.880 align:middle line:90% there are also many-- 00:01:05.880 --> 00:01:09.142 align:middle line:84% it's very typical for a cell to be in fairly stable conditions. 00:01:09.142 --> 00:01:11.100 align:middle line:84% There'll be a transition time, and then they'll 00:01:11.100 --> 00:01:11.850 align:middle line:90% be stable again. 00:01:11.850 --> 00:01:13.725 align:middle line:84% And that will be the steady-state assumption. 00:01:13.725 --> 00:01:19.060 align:middle line:84% So we'll say there's no net accumulation of metabolites. 00:01:19.060 --> 00:01:21.090 align:middle line:84% So even though there may be fluxes 00:01:21.090 --> 00:01:23.460 align:middle line:84% into and out of every metabolite, 00:01:23.460 --> 00:01:25.170 align:middle line:90% the net change is zero. 00:01:25.170 --> 00:01:28.320 align:middle line:84% And so that means this equation we've seen a few times, 00:01:28.320 --> 00:01:32.220 align:middle line:84% the change in X will reflect the time is zero. 00:01:32.220 --> 00:01:34.470 align:middle line:84% And that means that stoichiometric matrix 00:01:34.470 --> 00:01:40.110 align:middle line:84% times the flux rate for that for all relevant reactions, 00:01:40.110 --> 00:01:41.970 align:middle line:84% you can think of as a matrix representation. 00:01:41.970 --> 00:01:45.150 align:middle line:84% Sum of all the inputs and outputs of that particular X 00:01:45.150 --> 00:01:47.830 align:middle line:84% position, minus transport vector, 00:01:47.830 --> 00:01:51.702 align:middle line:84% which is kept separately, is all zero. 00:01:51.702 --> 00:01:52.660 align:middle line:90% So what does that mean? 00:01:52.660 --> 00:01:54.160 align:middle line:84% That means the stoichiometric matrix 00:01:54.160 --> 00:01:56.400 align:middle line:84% times the internal fluxes is equal to transport 00:01:56.400 --> 00:01:57.870 align:middle line:90% across the membrane. 00:01:57.870 --> 00:02:00.210 align:middle line:90% And this is something-- 00:02:00.210 --> 00:02:03.070 align:middle line:84% I'll show you-- we're going to do this in two steps. 00:02:03.070 --> 00:02:05.070 align:middle line:84% First, we're going to solve for that as if there 00:02:05.070 --> 00:02:08.669 align:middle line:84% was an exact set of equations, just 00:02:08.669 --> 00:02:10.232 align:middle line:84% the right number of equations, just 00:02:10.232 --> 00:02:11.440 align:middle line:90% the right number of unknowns. 00:02:11.440 --> 00:02:14.910 align:middle line:84% So you can actually get a solve for the unknown fluxes. 00:02:14.910 --> 00:02:16.410 align:middle line:84% And then we'll work through it where 00:02:16.410 --> 00:02:18.450 align:middle line:84% it's actually underdetermined, but there 00:02:18.450 --> 00:02:21.300 align:middle line:90% are inequality constraints. 00:02:21.300 --> 00:02:25.560 align:middle line:84% First, let's take the exact case because the stoichiometry 00:02:25.560 --> 00:02:29.640 align:middle line:84% is known that's S. These are the zeros and ones, minus ones, 00:02:29.640 --> 00:02:31.950 align:middle line:90% twos, and so on. 00:02:31.950 --> 00:02:34.620 align:middle line:84% The uptake rates can be known in the sense 00:02:34.620 --> 00:02:42.730 align:middle line:84% that you can regulate the amount of input and output 00:02:42.730 --> 00:02:46.960 align:middle line:84% by the rate at which you remove substrates and add them. 00:02:46.960 --> 00:02:49.320 align:middle line:84% And the metabolic flux's are what you actually want. 00:02:49.320 --> 00:02:51.150 align:middle line:84% This is in contrast to the red blood 00:02:51.150 --> 00:02:54.158 align:middle line:84% cell or other ordinary difference equation cases 00:02:54.158 --> 00:02:55.200 align:middle line:90% we've been talking about. 00:02:55.200 --> 00:02:57.720 align:middle line:84% Where the rate equations were known, 00:02:57.720 --> 00:02:59.692 align:middle line:84% but what we were looking for concentrations. 00:02:59.692 --> 00:03:01.650 align:middle line:84% Here we're less concerned about concentrations, 00:03:01.650 --> 00:03:04.890 align:middle line:84% and we want to learn what it might be the fluxes 00:03:04.890 --> 00:03:07.620 align:middle line:84% and later what might be the optimal fluxes. 00:03:07.620 --> 00:03:13.380 align:middle line:90% So Slide 42, let's say that we-- 00:03:13.380 --> 00:03:17.503 align:middle line:84% as, in a way, nomenclature, we focus on positive fluxes. 00:03:17.503 --> 00:03:19.170 align:middle line:84% If we want to go the opposite direction, 00:03:19.170 --> 00:03:23.730 align:middle line:84% we'll make that another reverse flux, separate flux. 00:03:23.730 --> 00:03:27.120 align:middle line:84% So focusing on positive fluxes, we 00:03:27.120 --> 00:03:29.190 align:middle line:84% might know the flux level in certain reactions 00:03:29.190 --> 00:03:33.930 align:middle line:84% that will help us have the equations appropriately 00:03:33.930 --> 00:03:36.080 align:middle line:90% constrained. 00:03:36.080 --> 00:03:37.990 align:middle line:90% We can control update rates. 00:03:37.990 --> 00:03:43.720 align:middle line:84% We can have maximum values for uptake in internal fluxes. 00:03:43.720 --> 00:03:44.663 align:middle line:90% So here's an example. 00:03:44.663 --> 00:03:45.580 align:middle line:90% Let's walk through it. 00:03:45.580 --> 00:03:47.980 align:middle line:90% It's nice to have a couple of-- 00:03:47.980 --> 00:03:53.100 align:middle line:84% one or two examples where we walk through in a given class. 00:03:53.100 --> 00:03:58.970 align:middle line:84% So here, we're going to have an input molecule A is transported 00:03:58.970 --> 00:04:01.850 align:middle line:84% across the membrane at a rate R sub A. 00:04:01.850 --> 00:04:04.790 align:middle line:84% It then has essentially a decision fork here. 00:04:04.790 --> 00:04:07.130 align:middle line:84% Some fraction of the A is going to go 00:04:07.130 --> 00:04:10.160 align:middle line:84% through the X1 reaction, and some fraction through the X2. 00:04:10.160 --> 00:04:14.664 align:middle line:84% The amount that goes through X1 and X2 is what we want to know. 00:04:14.664 --> 00:04:18.040 align:middle line:90% We are going to be given-- 00:04:18.040 --> 00:04:21.490 align:middle line:84% we're going to essentially clamp control the rate it 00:04:21.490 --> 00:04:22.730 align:middle line:90% goes across membrane RA. 00:04:22.730 --> 00:04:23.980 align:middle line:90% So that's going to be a given. 00:04:23.980 --> 00:04:25.313 align:middle line:90% You can see the constraint here. 00:04:25.313 --> 00:04:26.530 align:middle line:90% RA is going to be 3. 00:04:26.530 --> 00:04:30.400 align:middle line:84% And we're going to also know the rate at which B is removed, 00:04:30.400 --> 00:04:32.530 align:middle line:84% and that's going to be kept constant at one. 00:04:32.530 --> 00:04:34.990 align:middle line:84% So we're going to solve for the two internal fluxes X, 00:04:34.990 --> 00:04:40.362 align:middle line:84% and maybe this external transport flux are C. Now, 00:04:40.362 --> 00:04:42.070 align:middle line:84% there's going to be conservation of mass, 00:04:42.070 --> 00:04:44.260 align:middle line:84% so we can start setting up the flux balances 00:04:44.260 --> 00:04:48.970 align:middle line:84% in the upper right-hand corner here of Slide 43. 00:04:48.970 --> 00:04:51.250 align:middle line:90% Take a look at A. A is created-- 00:04:51.250 --> 00:04:54.250 align:middle line:84% essentially, the intracellular concentration of A 00:04:54.250 --> 00:04:59.350 align:middle line:84% is dependent on R sub A, and then it splits into X1 and X2. 00:04:59.350 --> 00:05:03.160 align:middle line:84% So you know that R sub A minus X1, minus X2 00:05:03.160 --> 00:05:04.457 align:middle line:90% is going to be zero. 00:05:04.457 --> 00:05:06.040 align:middle line:84% They're going to cancel out because A, 00:05:06.040 --> 00:05:09.280 align:middle line:84% a steady-state assumption, is going to be constant. 00:05:09.280 --> 00:05:13.420 align:middle line:84% Even though the fluxes are not zero, there sum is. 00:05:13.420 --> 00:05:14.815 align:middle line:84% And for B, a similar argument can 00:05:14.815 --> 00:05:21.520 align:middle line:84% be made X1, which creates B has to sum will also be zero. 00:05:21.520 --> 00:05:25.393 align:middle line:84% And same thing for C, all the inputs have to be the outputs. 00:05:25.393 --> 00:05:27.310 align:middle line:84% We have these two constraints where we've just 00:05:27.310 --> 00:05:30.490 align:middle line:84% said we're going to clamp experimentally 00:05:30.490 --> 00:05:33.490 align:middle line:84% the amount of A going in and the amount of B coming out. 00:05:33.490 --> 00:05:36.070 align:middle line:84% And we want to solve for the other three kinetic parameters. 00:05:36.070 --> 00:05:39.340 align:middle line:84% So we have three equations and three unknowns. 00:05:39.340 --> 00:05:42.640 align:middle line:84% The unknowns are the two internal fluxes in R sub B. 00:05:42.640 --> 00:05:46.180 align:middle line:84% And so, another way of stating this equation X1 plus X2 is 00:05:46.180 --> 00:05:49.030 align:middle line:84% equal to 3 because we know three going in 00:05:49.030 --> 00:05:51.730 align:middle line:84% has to equal X1, X2 by flux balance. 00:05:51.730 --> 00:05:54.400 align:middle line:84% This is, essentially, the flux balance for A 00:05:54.400 --> 00:05:56.770 align:middle line:84% can be summarized as X1 plus X2 equals 3. 00:05:56.770 --> 00:06:02.440 align:middle line:84% Or in matrix formation, it's the top row here 1, 1, 0. 00:06:02.440 --> 00:06:12.660 align:middle line:84% All right, and the 3 is in the transport column vector 00:06:12.660 --> 00:06:14.502 align:middle line:90% at the far right hand side. 00:06:14.502 --> 00:06:15.960 align:middle line:84% And similarly, you fill up the rest 00:06:15.960 --> 00:06:18.180 align:middle line:84% of the symmetric matrix with zeros and ones 00:06:18.180 --> 00:06:22.170 align:middle line:84% and twos and minus ones and the transport, 00:06:22.170 --> 00:06:28.040 align:middle line:84% the flux vectors are 3, 1, 0, for the constraints 00:06:28.040 --> 00:06:29.310 align:middle line:90% that we have. 00:06:29.310 --> 00:06:30.480 align:middle line:90% And you solve for this. 00:06:30.480 --> 00:06:32.620 align:middle line:90% Three equations, three unknowns. 00:06:32.620 --> 00:06:34.140 align:middle line:90% You can solve it-- 00:06:34.140 --> 00:06:41.620 align:middle line:84% my standard linear algebra tricks S times B equals. 00:06:41.620 --> 00:06:44.550 align:middle line:84% So V is equal to inverse matrix S 00:06:44.550 --> 00:06:48.270 align:middle line:84% times B. The matrix we had in the previous slide, which was-- 00:06:48.270 --> 00:06:50.670 align:middle line:90% first row was 1, 1, 0. 00:06:50.670 --> 00:06:53.070 align:middle line:84% Now, when you do the inverse matrix, 00:06:53.070 --> 00:06:57.270 align:middle line:84% you get this upper right-hand portion, 00:06:57.270 --> 00:06:59.520 align:middle line:84% and you multiply it out, and you get the column vector 00:06:59.520 --> 00:07:04.420 align:middle line:84% solution which is 1, 2, 4, for X1, X2, and R, C, respectively. 00:07:04.420 --> 00:07:07.410 align:middle line:84% Now, they're plugged into this diagram, 00:07:07.410 --> 00:07:11.860 align:middle line:84% and you can see that the whole masses balance out. 00:07:11.860 --> 00:07:19.200 align:middle line:84% 3 is equal to 1 plus 2, 2 times C, 2 times 2 is equal to 4. 00:07:19.200 --> 00:07:21.810 align:middle line:90% OK. 00:07:21.810 --> 00:07:24.960 align:middle line:84% That's an example where it's heavily constrained, 00:07:24.960 --> 00:07:26.880 align:middle line:90% and so we can solve it exactly. 00:07:26.880 --> 00:07:29.640 align:middle line:84% However, very often, there are not 00:07:29.640 --> 00:07:34.680 align:middle line:84% [AUDIO OUT] of the measurements that have been made. 00:07:34.680 --> 00:07:37.310 align:middle line:84% We can't clamp all these fluxes, and there 00:07:37.310 --> 00:07:39.840 align:middle line:84% are many more internal fluxes than external. 00:07:39.840 --> 00:07:42.900 align:middle line:84% And so we have an under-determined system. 00:07:42.900 --> 00:07:47.080 align:middle line:84% What is a good system biologist to do at this point? 00:07:47.080 --> 00:07:49.080 align:middle line:84% Well, the formal solution is it's 00:07:49.080 --> 00:07:51.960 align:middle line:84% no longer a point, a single point, a nice column 00:07:51.960 --> 00:07:54.300 align:middle line:90% vector of fluxes. 00:07:54.300 --> 00:07:57.030 align:middle line:84% It's now an entire feasible space. 00:07:57.030 --> 00:07:59.100 align:middle line:84% This is a kind of a cop-out answer, 00:07:59.100 --> 00:08:02.700 align:middle line:84% is that, OK, we have fewer equations 00:08:02.700 --> 00:08:03.870 align:middle line:90% than we have unknowns. 00:08:03.870 --> 00:08:05.550 align:middle line:84% That means the unknowns can occupy 00:08:05.550 --> 00:08:08.410 align:middle line:84% this entire multi-dimensional space, where however many 00:08:08.410 --> 00:08:10.410 align:middle line:84% fluxes you have, if you have hundreds of fluxes, 00:08:10.410 --> 00:08:11.868 align:middle line:84% you'll have hundreds of dimensions. 00:08:11.868 --> 00:08:13.590 align:middle line:84% You have three fluxes, A, B, and C. 00:08:13.590 --> 00:08:18.780 align:middle line:84% Then you'll get this kind of multi-dimensional polyhedron 00:08:18.780 --> 00:08:21.300 align:middle line:84% region, and anything inside that polyhedron 00:08:21.300 --> 00:08:25.380 align:middle line:84% is the acceptable solution to the under-determined set. 00:08:25.380 --> 00:08:28.320 align:middle line:84% Well, this is still progress in that you now 00:08:28.320 --> 00:08:34.460 align:middle line:84% know that your solution is in there somewhere. 00:08:34.460 --> 00:08:36.429 align:middle line:90% But what if we wanted to find-- 00:08:36.429 --> 00:08:38.740 align:middle line:84% add some more constraints, which are not now 00:08:38.740 --> 00:08:46.210 align:middle line:84% exact constraints anymore, but they're more an optimization 00:08:46.210 --> 00:08:46.930 align:middle line:90% process. 00:08:46.930 --> 00:08:49.960 align:middle line:84% You might have this multi-dimensional polyhedron 00:08:49.960 --> 00:08:52.150 align:middle line:90% will be bounded by inequalities. 00:08:52.150 --> 00:08:54.220 align:middle line:84% That it's less than-- is greater than zero. 00:08:54.220 --> 00:08:58.090 align:middle line:84% Right, that it's positive, or it's less of some maximum flux. 00:08:58.090 --> 00:09:01.270 align:middle line:84% But then you want to find some optimal solution, 00:09:01.270 --> 00:09:05.860 align:middle line:84% and the optimization is the thing 00:09:05.860 --> 00:09:12.250 align:middle line:84% that math has been harnessed for in some other fields outside 00:09:12.250 --> 00:09:14.290 align:middle line:90% of ours, such as economics. 00:09:14.290 --> 00:09:19.450 align:middle line:84% When you're doing planning of transport or planning 00:09:19.450 --> 00:09:21.670 align:middle line:84% of economic investments, and so forth, you 00:09:21.670 --> 00:09:23.500 align:middle line:84% want to know how much of your resources 00:09:23.500 --> 00:09:25.418 align:middle line:84% do you want to send to the X1 route? 00:09:25.418 --> 00:09:27.460 align:middle line:84% And how much do you want to send to the X2 route? 00:09:27.460 --> 00:09:32.450 align:middle line:84% Very analogous to the situation we had in the previous slide. 00:09:32.450 --> 00:09:37.070 align:middle line:84% And that can be solved by linear programming. 00:09:37.070 --> 00:09:41.800 align:middle line:84% Linear programming often finding economic applications. 00:09:41.800 --> 00:09:44.653 align:middle line:84% But we need to ask what is it we want to optimize? 00:09:44.653 --> 00:09:46.570 align:middle line:84% In economics, you want to optimize, typically, 00:09:46.570 --> 00:09:49.150 align:middle line:90% the bottom line. 00:09:49.150 --> 00:09:54.550 align:middle line:84% You want to lower costs, and increase profits. 00:09:54.550 --> 00:10:02.300 align:middle line:84% What we have here is we have this very commonly 00:10:02.300 --> 00:10:03.950 align:middle line:84% encountered in many metabolic systems 00:10:03.950 --> 00:10:05.900 align:middle line:84% where you have this convex polyhedron cone. 00:10:05.900 --> 00:10:11.900 align:middle line:84% Convex just means that there are no little gaps in it. 00:10:11.900 --> 00:10:13.550 align:middle line:90% No little carved-out regions. 00:10:13.550 --> 00:10:19.190 align:middle line:90% It's all contained. 00:10:19.190 --> 00:10:23.510 align:middle line:84% That convexity allows you to now take a linear objective 00:10:23.510 --> 00:10:25.790 align:middle line:84% function, say a multi-dimensional plane 00:10:25.790 --> 00:10:30.000 align:middle line:84% or a line that you then move through this convex space. 00:10:30.000 --> 00:10:31.430 align:middle line:84% And since there are no gaps in it, 00:10:31.430 --> 00:10:33.627 align:middle line:84% you will eventually move off as you 00:10:33.627 --> 00:10:35.960 align:middle line:84% move this objective function to-- the objective function 00:10:35.960 --> 00:10:38.640 align:middle line:84% gets better and better as it goes off through this feasible 00:10:38.640 --> 00:10:39.140 align:middle line:90% space. 00:10:39.140 --> 00:10:41.480 align:middle line:84% It eventually gets to a point where 00:10:41.480 --> 00:10:44.480 align:middle line:84% it leaves the feasible space, and that is the maximum value. 00:10:44.480 --> 00:10:49.210 align:middle line:84% The optimal value for that objective function. 00:10:49.210 --> 00:10:51.520 align:middle line:84% So we can use this feasible space 00:10:51.520 --> 00:10:54.980 align:middle line:84% combined with this objective function to find some optimal. 00:10:54.980 --> 00:10:58.060 align:middle line:84% Now, what is the objective function that we want to use? 00:10:58.060 --> 00:11:01.180 align:middle line:84% And if we were doing red blood cell, 00:11:01.180 --> 00:11:03.670 align:middle line:84% the objective function might be ATP or redox 00:11:03.670 --> 00:11:06.400 align:middle line:90% or both or delivery of oxygen. 00:11:06.400 --> 00:11:08.440 align:middle line:84% And for a variety of other systems 00:11:08.440 --> 00:11:13.960 align:middle line:84% of importance to understanding health or biotechnology 00:11:13.960 --> 00:11:19.960 align:middle line:84% production, or other medical and engineering goals, what 00:11:19.960 --> 00:11:22.600 align:middle line:90% we want to optimize is biomass. 00:11:22.600 --> 00:11:28.420 align:middle line:84% The ability of the cells to grow and produce other cells 00:11:28.420 --> 00:11:32.020 align:middle line:84% or produce a particular subset of molecules in the cell, 00:11:32.020 --> 00:11:36.100 align:middle line:84% but let's deal mainly with the case of cells producing cells. 00:11:36.100 --> 00:11:39.580 align:middle line:84% And what this is is a sum over all the monomers, 00:11:39.580 --> 00:11:41.830 align:middle line:84% all the components of the cell, which 00:11:41.830 --> 00:11:44.360 align:middle line:90% represent the body of the cell. 00:11:44.360 --> 00:11:45.977 align:middle line:90% So as you move-- 00:11:45.977 --> 00:11:48.310 align:middle line:84% as you transport small molecules in from the environment 00:11:48.310 --> 00:11:49.698 align:middle line:84% and incorporate it into the cell, 00:11:49.698 --> 00:11:51.490 align:middle line:84% as certainly you can think of it as a sync, 00:11:51.490 --> 00:11:53.500 align:middle line:84% it's removing those molecules from circulation 00:11:53.500 --> 00:11:55.480 align:middle line:90% from some solution. 00:11:55.480 --> 00:12:02.800 align:middle line:84% And this sum over all the components, the ratios, 00:12:02.800 --> 00:12:04.570 align:middle line:84% the monomers, you can think of there 00:12:04.570 --> 00:12:08.230 align:middle line:84% is a fixed ratio of alanine to glycine 00:12:08.230 --> 00:12:11.590 align:middle line:84% to leucine to all the other components of the cell. 00:12:11.590 --> 00:12:13.270 align:middle line:90% And this is known. 00:12:13.270 --> 00:12:14.920 align:middle line:84% You can know this without knowing 00:12:14.920 --> 00:12:17.170 align:middle line:84% lots of other parts of the system we want to know just 00:12:17.170 --> 00:12:19.870 align:middle line:84% by taking the cell and doing a chemical composition on it. 00:12:19.870 --> 00:12:21.070 align:middle line:90% Very simple experiment. 00:12:21.070 --> 00:12:24.070 align:middle line:84% There are tables of this known provide variety of cells. 00:12:24.070 --> 00:12:28.780 align:middle line:84% And it hardly changes depending much on how a cell is growing. 00:12:28.780 --> 00:12:31.630 align:middle line:84% What sources of carbon and nitrogen 00:12:31.630 --> 00:12:33.995 align:middle line:84% do not greatly affect the ratio of alanine, 00:12:33.995 --> 00:12:36.370 align:middle line:84% leucine, and glycine because those are determined by what 00:12:36.370 --> 00:12:38.690 align:middle line:90% it takes to run the cell. 00:12:38.690 --> 00:12:40.660 align:middle line:84% So this is important, flux you can 00:12:40.660 --> 00:12:43.000 align:middle line:84% think of this as a kind of a lumped flux, 00:12:43.000 --> 00:12:45.410 align:middle line:84% and this will be our objective function as well. 00:12:45.410 --> 00:12:47.870 align:middle line:84% So the objective function Z sometimes called, 00:12:47.870 --> 00:12:53.500 align:middle line:84% so Z is going to be equal to the flux of growth. 00:12:53.500 --> 00:12:55.750 align:middle line:84% So you can see that you have, again, the same equation 00:12:55.750 --> 00:12:58.030 align:middle line:84% against the chemistry matrix times the internal flux 00:12:58.030 --> 00:13:00.730 align:middle line:90% is equal to the uptake fluxes. 00:13:00.730 --> 00:13:05.890 align:middle line:84% And we've got this now-- this constraint, this optimization 00:13:05.890 --> 00:13:06.590 align:middle line:90% function. 00:13:06.590 --> 00:13:13.420 align:middle line:84% So let's take-- just like we had a very simple exact solution 00:13:13.420 --> 00:13:16.930 align:middle line:84% before, now let's take this very simple linear programming 00:13:16.930 --> 00:13:20.470 align:middle line:84% or LPM solution where now it's underdetermined so we 00:13:20.470 --> 00:13:21.860 align:middle line:90% can't get the exact solution. 00:13:21.860 --> 00:13:23.560 align:middle line:84% But we can ask what's the maximum 00:13:23.560 --> 00:13:25.323 align:middle line:84% for a particular objective function. 00:13:25.323 --> 00:13:26.740 align:middle line:84% The objective function here is not 00:13:26.740 --> 00:13:29.170 align:middle line:84% going to be the biomass production of the entire cell. 00:13:29.170 --> 00:13:32.260 align:middle line:84% It's going to be either maximizing the production of D 00:13:32.260 --> 00:13:37.090 align:middle line:84% or C. Now, this is a slightly different diagram 00:13:37.090 --> 00:13:38.090 align:middle line:90% than we had before. 00:13:38.090 --> 00:13:40.330 align:middle line:84% We still have the rate of uptake of A 00:13:40.330 --> 00:13:43.540 align:middle line:84% on the left side of this kind of circular pseudo cell. 00:13:43.540 --> 00:13:44.500 align:middle line:90% A goes in. 00:13:44.500 --> 00:13:48.130 align:middle line:84% It makes the same binary decision-- 00:13:48.130 --> 00:13:52.563 align:middle line:84% two-way decision of X1 versus X2. 00:13:52.563 --> 00:13:53.230 align:middle line:90% It's not binary. 00:13:53.230 --> 00:13:58.360 align:middle line:84% It's these quantitative real numbers determine how much X 00:13:58.360 --> 00:14:01.100 align:middle line:90% and how much X2. 00:14:01.100 --> 00:14:04.010 align:middle line:84% And then, if it takes the upper X1 route, 00:14:04.010 --> 00:14:07.220 align:middle line:84% then it splits into B and C. If it takes the X2 route, 00:14:07.220 --> 00:14:10.820 align:middle line:84% it turns into B and D. Both cases it produces a B molecule. 00:14:10.820 --> 00:14:13.320 align:middle line:84% You can already see a constraint coming up here, 00:14:13.320 --> 00:14:16.100 align:middle line:84% which is the RA is going to be for RB. 00:14:16.100 --> 00:14:20.090 align:middle line:84% However you go from A to the outside again, 00:14:20.090 --> 00:14:22.160 align:middle line:84% it's going to produce one molecule of B 00:14:22.160 --> 00:14:26.000 align:middle line:84% for every molecule A. This is a perfect conservation reaction. 00:14:26.000 --> 00:14:29.420 align:middle line:84% And we've said that we're only interested in positive fluxes, 00:14:29.420 --> 00:14:33.590 align:middle line:84% so you get this little triangle here of X1 and X2 00:14:33.590 --> 00:14:34.910 align:middle line:90% are greater than zero. 00:14:34.910 --> 00:14:37.280 align:middle line:84% And they're constrained, and we've 00:14:37.280 --> 00:14:40.040 align:middle line:84% said that RA we're going to clamp so that it 00:14:40.040 --> 00:14:45.450 align:middle line:84% can't have more than one arbitrary molds per liter, 00:14:45.450 --> 00:14:45.950 align:middle line:90% per minute. 00:14:45.950 --> 00:14:48.710 align:middle line:90% 00:14:48.710 --> 00:14:52.220 align:middle line:84% And so X1 plus X2 are constrained to be less than RA, 00:14:52.220 --> 00:14:53.850 align:middle line:90% so they are less than 1 as well. 00:14:53.850 --> 00:14:56.120 align:middle line:84% So you get this feasible space, which 00:14:56.120 --> 00:15:00.930 align:middle line:84% is this diagonally cross-hatched region, 00:15:00.930 --> 00:15:03.150 align:middle line:90% and that's the exact solution. 00:15:03.150 --> 00:15:05.730 align:middle line:84% That's the set of all exact solutions. 00:15:05.730 --> 00:15:09.240 align:middle line:84% But now, if we want to maximize a particular objective function 00:15:09.240 --> 00:15:12.780 align:middle line:84% Z, in this case, let's maximize the production of D, 00:15:12.780 --> 00:15:16.650 align:middle line:84% and we're not too concerned about anything else. 00:15:16.650 --> 00:15:18.510 align:middle line:90% Then you get this line. 00:15:18.510 --> 00:15:21.300 align:middle line:84% You can think this is a hyper-plane going-- 00:15:21.300 --> 00:15:24.807 align:middle line:84% a line basically going up through feasible space 00:15:24.807 --> 00:15:26.640 align:middle line:84% starting at the bottom of the slide going up 00:15:26.640 --> 00:15:28.800 align:middle line:84% and up and up until it just barely leaves the feasible 00:15:28.800 --> 00:15:29.300 align:middle line:90% space. 00:15:29.300 --> 00:15:32.160 align:middle line:84% And when it does, the last point it gets to is the maximum, 00:15:32.160 --> 00:15:35.850 align:middle line:84% and the maximum happens here to be X1 equals 0, 00:15:35.850 --> 00:15:38.280 align:middle line:90% X2 equal to the max RD. 00:15:38.280 --> 00:15:41.130 align:middle line:84% Maximum rate of production of the molecule 00:15:41.130 --> 00:15:42.090 align:middle line:90% we're interested in. 00:15:42.090 --> 00:15:45.810 align:middle line:84% If we had an objective function that went off the other axis, 00:15:45.810 --> 00:15:49.870 align:middle line:84% it would be X2 zero, and X1 equal to maximum RC. 00:15:49.870 --> 00:15:51.120 align:middle line:90% So you can see how this works. 00:15:51.120 --> 00:15:55.050 align:middle line:84% We create the feasible space with this design and objective 00:15:55.050 --> 00:16:00.570 align:middle line:84% function Z, and we run the Z off the edge of this convex space. 00:16:00.570 --> 00:16:03.120 align:middle line:84% It's important that it be convex. 00:16:03.120 --> 00:16:07.060 align:middle line:84% OK, so how applicable is this linear programming 00:16:07.060 --> 00:16:09.750 align:middle line:84% and so-called flux balance analysis? 00:16:09.750 --> 00:16:12.510 align:middle line:84% It works when the stoichiometry is well known. 00:16:12.510 --> 00:16:14.340 align:middle line:90% For E coli, it's well known. 00:16:14.340 --> 00:16:18.660 align:middle line:84% For newly sequenced genomes, we can make connections 00:16:18.660 --> 00:16:22.120 align:middle line:90% to previous enzymatic reactions. 00:16:22.120 --> 00:16:26.322 align:middle line:84% And we can guess at what the stoichiometric matrix would be. 00:16:26.322 --> 00:16:27.780 align:middle line:84% But in that case, the stoichiometry 00:16:27.780 --> 00:16:29.412 align:middle line:84% is less well known, and in which case, 00:16:29.412 --> 00:16:31.620 align:middle line:84% you're going to have to really embrace your outliers. 00:16:31.620 --> 00:16:34.470 align:middle line:84% When you get to the N, you're going to see all the errors 00:16:34.470 --> 00:16:36.068 align:middle line:84% and go back and figure out what was 00:16:36.068 --> 00:16:38.610 align:middle line:84% wrong with our stoichiometric matrix derived from our genome. 00:16:38.610 --> 00:16:41.550 align:middle line:90% 00:16:41.550 --> 00:16:47.510 align:middle line:84% You don't need much experimental information to run this. 00:16:47.510 --> 00:16:51.950 align:middle line:84% But you need some, and we'll explore two of them or at least 00:16:51.950 --> 00:16:56.000 align:middle line:84% to test how it's going to do data. 00:16:56.000 --> 00:17:00.600 align:middle line:84% So what are the precursors to cell growth 00:17:00.600 --> 00:17:05.220 align:middle line:84% that we are monitoring as a Z function? 00:17:05.220 --> 00:17:09.119 align:middle line:84% We want to come up with-- define this growth function in terms 00:17:09.119 --> 00:17:10.170 align:middle line:90% of the biomass. 00:17:10.170 --> 00:17:14.069 align:middle line:84% And there's like I said there's tables of composition we'll 00:17:14.069 --> 00:17:16.540 align:middle line:90% show on in a couple of slides. 00:17:16.540 --> 00:17:20.998 align:middle line:84% But you can use this as a part of 00:17:20.998 --> 00:17:22.290 align:middle line:90% the complete metabolic network. 00:17:22.290 --> 00:17:25.780 align:middle line:84% You can also use this as observation function. 00:17:25.780 --> 00:17:27.720 align:middle line:84% It can be described as some small number 00:17:27.720 --> 00:17:33.220 align:middle line:84% of biosynthetic precursors, plus the energy and redox cofactors. 00:17:33.220 --> 00:17:35.440 align:middle line:84% Now there are many ways of doing in silico cells. 00:17:35.440 --> 00:17:38.380 align:middle line:84% We show the red blood cells ordinary equation. 00:17:38.380 --> 00:17:41.380 align:middle line:84% The kind of in silico cells here where the optimization 00:17:41.380 --> 00:17:44.170 align:middle line:84% is fairly limited, the stoichiometric matrices 00:17:44.170 --> 00:17:47.470 align:middle line:84% have only been worked out for three, maybe five 00:17:47.470 --> 00:17:50.950 align:middle line:90% cells, three published. 00:17:50.950 --> 00:17:54.480 align:middle line:90% Yeast is on its way. 00:17:54.480 --> 00:17:56.700 align:middle line:84% And these are mostly hand curated. 00:17:56.700 --> 00:17:59.970 align:middle line:84% There is definitely a need for getting a more automated input 00:17:59.970 --> 00:18:05.480 align:middle line:84% from genomic models to these flux analyzes. 00:18:05.480 --> 00:18:06.720 align:middle line:90% Here's some references here. 00:18:06.720 --> 00:18:08.830 align:middle line:90% We'll be talking in a moment. 00:18:08.830 --> 00:18:10.580 align:middle line:84% First, we'll talk about the wild-type case 00:18:10.580 --> 00:18:13.038 align:middle line:84% for each of these cells under a variety of different growth 00:18:13.038 --> 00:18:13.710 align:middle line:90% conditions. 00:18:13.710 --> 00:18:16.320 align:middle line:84% Then we'll move on to mutants and ask whether mutants 00:18:16.320 --> 00:18:19.690 align:middle line:90% we expect to be optimal or not. 00:18:19.690 --> 00:18:21.660 align:middle line:84% And that will be called minimization 00:18:21.660 --> 00:18:24.440 align:middle line:90% of metabolic adjustment. 00:18:24.440 --> 00:18:27.500 align:middle line:84% Now, where do these stoichiometric matrices 00:18:27.500 --> 00:18:29.480 align:middle line:90% come from? 00:18:29.480 --> 00:18:31.430 align:middle line:84% Parenthetically the kinetic parameters 00:18:31.430 --> 00:18:33.590 align:middle line:84% that we are talking about are known 00:18:33.590 --> 00:18:36.140 align:middle line:84% for some of these complicated biochemical systems, 00:18:36.140 --> 00:18:38.030 align:middle line:90% not just for red blood cell. 00:18:38.030 --> 00:18:39.920 align:middle line:84% And where both the stoichiometric matrices 00:18:39.920 --> 00:18:41.920 align:middle line:84% and the kinetic parameters come from, of course, 00:18:41.920 --> 00:18:44.300 align:middle line:84% as a vast literature is an unwieldy literature 00:18:44.300 --> 00:18:45.770 align:middle line:84% in the sense it was done at a time 00:18:45.770 --> 00:18:47.353 align:middle line:84% before anybody thought that we were 00:18:47.353 --> 00:18:48.770 align:middle line:84% going to be-- that they were going 00:18:48.770 --> 00:18:53.240 align:middle line:84% to be responsible for getting this into computer passable 00:18:53.240 --> 00:18:54.660 align:middle line:90% databases. 00:18:54.660 --> 00:19:00.620 align:middle line:84% So it's mostly been re-entered by technicians into databases, 00:19:00.620 --> 00:19:02.640 align:middle line:84% and you'll end up with these diagrams, 00:19:02.640 --> 00:19:06.800 align:middle line:84% such as the central one where each of these boxes contains 00:19:06.800 --> 00:19:09.680 align:middle line:84% is a node that contains a substrate. 00:19:09.680 --> 00:19:12.930 align:middle line:84% And the lines have inside information numbers 00:19:12.930 --> 00:19:14.420 align:middle line:90% this dotted-- 00:19:14.420 --> 00:19:18.950 align:middle line:84% this multiple decimal point showing 00:19:18.950 --> 00:19:21.740 align:middle line:84% a hierarchical classification going 00:19:21.740 --> 00:19:24.860 align:middle line:84% from left to right getting more and more detailed 00:19:24.860 --> 00:19:26.600 align:middle line:90% in the enzymatic reaction. 00:19:26.600 --> 00:19:30.150 align:middle line:84% And this is built up in a database. 00:19:30.150 --> 00:19:35.840 align:middle line:84% And from this, you can access such things 00:19:35.840 --> 00:19:42.730 align:middle line:84% as effects of kinetic constants, effects of pH, 00:19:42.730 --> 00:19:45.730 align:middle line:90% and other details. 00:19:45.730 --> 00:19:49.120 align:middle line:84% And also, from this, you can get the stoichiometric matrices 00:19:49.120 --> 00:19:52.030 align:middle line:84% in principle because here, you can see that you've got, 00:19:52.030 --> 00:20:00.460 align:middle line:84% say, an input of ADP plus glucose 6 phosphate going 00:20:00.460 --> 00:20:02.680 align:middle line:84% into this reaction or coming out depending on which 00:20:02.680 --> 00:20:04.940 align:middle line:90% direction you're going. 00:20:04.940 --> 00:20:09.220 align:middle line:84% And so those convert to zeros and ones in the matrix. 00:20:09.220 --> 00:20:11.930 align:middle line:84% That's the source of the stoichiometric matrix, 00:20:11.930 --> 00:20:14.560 align:middle line:84% which tells you what reactions are allowed. 00:20:14.560 --> 00:20:16.990 align:middle line:84% What two things can come together, 00:20:16.990 --> 00:20:20.530 align:middle line:84% or one thing can be converted by the various enzymes that 00:20:20.530 --> 00:20:21.820 align:middle line:90% actually exist in cells. 00:20:21.820 --> 00:20:25.060 align:middle line:84% And you can toggle those on and off, either by regulation 00:20:25.060 --> 00:20:28.690 align:middle line:84% or by evolutionary change or by genetic manipulation 00:20:28.690 --> 00:20:29.860 align:middle line:90% or mutagenesis. 00:20:29.860 --> 00:20:32.020 align:middle line:84% You can basically have a universal matrix, 00:20:32.020 --> 00:20:34.510 align:middle line:84% and you toggle on only the ones that you 00:20:34.510 --> 00:20:37.705 align:middle line:90% think happen in your organism. 00:20:37.705 --> 00:20:39.580 align:middle line:84% That's how you get the stoichiometric matrix. 00:20:39.580 --> 00:20:41.350 align:middle line:84% How you get the optimization function, 00:20:41.350 --> 00:20:44.860 align:middle line:84% the Z function, which is how you want 00:20:44.860 --> 00:20:47.740 align:middle line:84% to make those decisions of how much X1 and how much X2. 00:20:47.740 --> 00:20:50.890 align:middle line:84% This is dependent on the biomass composition. 00:20:50.890 --> 00:20:57.040 align:middle line:84% And the biomass composition, say we have on the vertical axis 00:20:57.040 --> 00:21:04.260 align:middle line:84% here, is the coefficient in the growth reaction, which 00:21:04.260 --> 00:21:07.140 align:middle line:84% ranges here over almost eight logs 00:21:07.140 --> 00:21:12.510 align:middle line:84% from the lowest compositional fraction, which 00:21:12.510 --> 00:21:14.370 align:middle line:90% are some of these coenzymes. 00:21:14.370 --> 00:21:18.930 align:middle line:84% So just like enzymes, they're only needed in small amounts 00:21:18.930 --> 00:21:24.660 align:middle line:84% to these that are needed both for major participants 00:21:24.660 --> 00:21:27.060 align:middle line:84% in hundreds of reactions, like ATP. 00:21:27.060 --> 00:21:29.910 align:middle line:84% Some of which contribute heavily to biomass. 00:21:29.910 --> 00:21:36.390 align:middle line:84% So ATP also contributes to large biomass like ribosomal RNA. 00:21:36.390 --> 00:21:37.870 align:middle line:84% And then the various amino acids, 00:21:37.870 --> 00:21:40.560 align:middle line:84% which tend to be many of these stars 00:21:40.560 --> 00:21:42.480 align:middle line:84% at the higher levels, like lysine, leucine, 00:21:42.480 --> 00:21:45.880 align:middle line:90% and the other 18 amino acids. 00:21:45.880 --> 00:21:49.260 align:middle line:84% OK, so these are examples of the numbers 00:21:49.260 --> 00:21:52.230 align:middle line:84% that would go into the optimization curve here. 00:21:52.230 --> 00:21:57.690 align:middle line:84% These red hyper planes would be that linear some in the ratios 00:21:57.690 --> 00:21:59.710 align:middle line:90% in the previous slide. 00:21:59.710 --> 00:22:03.653 align:middle line:84% Now, we've already seen, we've already 00:22:03.653 --> 00:22:05.070 align:middle line:84% worked through an example where we 00:22:05.070 --> 00:22:07.410 align:middle line:84% slid one of these hyperplanes off the edge 00:22:07.410 --> 00:22:10.710 align:middle line:84% of in a two-dimensional model and got the optimum production 00:22:10.710 --> 00:22:14.190 align:middle line:90% of a particular substance D. 00:22:14.190 --> 00:22:15.720 align:middle line:84% But now, we're trying to optimize 00:22:15.720 --> 00:22:19.170 align:middle line:84% this linear sum of all the metabolites 00:22:19.170 --> 00:22:21.640 align:middle line:84% that go into the body of the organism. 00:22:21.640 --> 00:22:23.860 align:middle line:84% And just focus on the green feasible space. 00:22:23.860 --> 00:22:26.110 align:middle line:84% It's actually-- some of it's hidden behind the yellow, 00:22:26.110 --> 00:22:31.180 align:middle line:84% but imagine this whole convex polyhedron of green which 00:22:31.180 --> 00:22:35.290 align:middle line:84% is feasible space, FS for wild type, WT, 00:22:35.290 --> 00:22:36.670 align:middle line:90% and you can get an optimum. 00:22:36.670 --> 00:22:38.590 align:middle line:84% You assume that this optimum, whatever 00:22:38.590 --> 00:22:42.010 align:middle line:84% the conditions that you're looking at 00:22:42.010 --> 00:22:47.420 align:middle line:84% is likely to be achieved because for millions of years, 00:22:47.420 --> 00:22:49.750 align:middle line:84% this organism has been living through all 00:22:49.750 --> 00:22:51.310 align:middle line:90% the different growth scenarios. 00:22:51.310 --> 00:22:55.340 align:middle line:84% Glucose on glucose, galactose, various nitrogen sources, 00:22:55.340 --> 00:22:58.810 align:middle line:84% and so forth, so no matter what reasonable set of conditions 00:22:58.810 --> 00:23:00.310 align:middle line:84% you throw at it, it's seen something 00:23:00.310 --> 00:23:02.560 align:middle line:84% like that before or some other combinations, 00:23:02.560 --> 00:23:03.830 align:middle line:90% and so it's optimal for it. 00:23:03.830 --> 00:23:07.900 align:middle line:84% And that's what that top right red 00:23:07.900 --> 00:23:10.180 align:middle line:84% dot means is that's the optimum for the wild type. 00:23:10.180 --> 00:23:12.407 align:middle line:84% Moving this hyperplane, which is the sum 00:23:12.407 --> 00:23:13.990 align:middle line:84% of all the different growth components 00:23:13.990 --> 00:23:15.470 align:middle line:90% in the correct ratio. 00:23:15.470 --> 00:23:19.030 align:middle line:84% So if you're going to optimize all the decisions, X1s and X2s 00:23:19.030 --> 00:23:21.320 align:middle line:84% so that you get the right ratios, 00:23:21.320 --> 00:23:24.430 align:middle line:84% so you don't get way too much, say, of some rare amino acid, 00:23:24.430 --> 00:23:27.400 align:middle line:84% like tryptophan and not enough of the glycine, 00:23:27.400 --> 00:23:30.430 align:middle line:90% say as alanine is most common. 00:23:30.430 --> 00:23:32.408 align:middle line:84% That's great for a wild type under a variety 00:23:32.408 --> 00:23:33.700 align:middle line:90% of different growth conditions. 00:23:33.700 --> 00:23:36.580 align:middle line:84% But what if you throw a real curveball 00:23:36.580 --> 00:23:42.700 align:middle line:84% and say not give it a condition, but give it a perturbation 00:23:42.700 --> 00:23:45.410 align:middle line:84% it really doesn't see very often, if at all, 00:23:45.410 --> 00:23:49.030 align:middle line:84% which would be to knockout of gene completely by deletion 00:23:49.030 --> 00:23:51.100 align:middle line:90% or conceivably knock in a gene. 00:23:51.100 --> 00:23:54.820 align:middle line:84% Add a gene that it hasn't seen before very often. 00:23:54.820 --> 00:23:56.920 align:middle line:84% Well, now you could say we're going 00:23:56.920 --> 00:23:58.120 align:middle line:90% to do the same optimization. 00:23:58.120 --> 00:24:01.570 align:middle line:84% We'll run the same red hyperplane through the new-- 00:24:01.570 --> 00:24:03.820 align:middle line:84% so the new feasible space is reduced 00:24:03.820 --> 00:24:06.550 align:middle line:84% if it's a knockout, could be increased if it's a knock in, 00:24:06.550 --> 00:24:07.660 align:middle line:90% where we're adding a gene. 00:24:07.660 --> 00:24:11.170 align:middle line:84% But in this reduced space, the original optima 00:24:11.170 --> 00:24:12.520 align:middle line:90% is no longer accessible. 00:24:12.520 --> 00:24:15.490 align:middle line:84% And if you rerun the optimization running this plane 00:24:15.490 --> 00:24:17.230 align:middle line:84% off the edge, you'll find a new red dot 00:24:17.230 --> 00:24:21.080 align:middle line:84% in the yellow space, which is a knockout optimum. 00:24:21.080 --> 00:24:22.610 align:middle line:90% But that could take-- 00:24:22.610 --> 00:24:25.430 align:middle line:84% after you do the knockout, it could take evolutionary time 00:24:25.430 --> 00:24:30.890 align:middle line:84% or at least long lb times to allow all the other genes 00:24:30.890 --> 00:24:34.040 align:middle line:84% to mutate and be selected so that they accommodate 00:24:34.040 --> 00:24:35.930 align:middle line:90% this new knockout. 00:24:35.930 --> 00:24:37.820 align:middle line:84% That the wild type had plenty of time 00:24:37.820 --> 00:24:40.700 align:middle line:84% to do that had millions of years. 00:24:40.700 --> 00:24:42.793 align:middle line:84% The yellow feasible space for the knockout 00:24:42.793 --> 00:24:44.210 align:middle line:84% may not have had that, so you want 00:24:44.210 --> 00:24:46.040 align:middle line:84% to know what's this immediate response. 00:24:46.040 --> 00:24:47.690 align:middle line:84% And for its immediate response, you 00:24:47.690 --> 00:24:51.770 align:middle line:84% might imagine is showing here as an orthogonal. 00:24:51.770 --> 00:24:54.860 align:middle line:84% A closest distance think of this is a Euclidean distance 00:24:54.860 --> 00:24:57.890 align:middle line:84% in this multi-dimensional space between the wild type 00:24:57.890 --> 00:25:00.710 align:middle line:84% optimum and the projection onto the feasible space 00:25:00.710 --> 00:25:01.730 align:middle line:90% of the mutant. 00:25:01.730 --> 00:25:05.030 align:middle line:84% Now that distance, you should already 00:25:05.030 --> 00:25:07.810 align:middle line:84% be thinking that this is no longer a linear programming. 00:25:07.810 --> 00:25:09.560 align:middle line:84% This may be quadratic because the distance 00:25:09.560 --> 00:25:10.580 align:middle line:90% is a quadratic function. 00:25:10.580 --> 00:25:14.010 align:middle line:90% 00:25:14.010 --> 00:25:16.350 align:middle line:84% And we'll see in particular since there 00:25:16.350 --> 00:25:18.630 align:middle line:84% are certain pathologies where you really 00:25:18.630 --> 00:25:23.190 align:middle line:84% are forced to take-- you can't just take a projection. 00:25:23.190 --> 00:25:27.450 align:middle line:84% Because sometimes, the projections 00:25:27.450 --> 00:25:31.590 align:middle line:84% can end up in a part of space, which is not feasible. 00:25:31.590 --> 00:25:34.590 align:middle line:84% This projection does not land on the yellow feasible space, 00:25:34.590 --> 00:25:36.870 align:middle line:90% FS of the knockout. 00:25:36.870 --> 00:25:42.030 align:middle line:84% So what you need to do is nudge this purple symbol 00:25:42.030 --> 00:25:46.200 align:middle line:84% up just to the nearest point of the feasible space, which 00:25:46.200 --> 00:25:50.140 align:middle line:84% minimizes the distance to the wild type optimum, 00:25:50.140 --> 00:25:52.000 align:middle line:84% and still falls in feasible space. 00:25:52.000 --> 00:25:56.760 align:middle line:84% Now, this is a quadratic programming algorithm. 00:25:56.760 --> 00:25:59.920 align:middle line:84% It's really just-- the linear is very simple 00:25:59.920 --> 00:26:03.490 align:middle line:84% to think of as just moving this plane off the edge 00:26:03.490 --> 00:26:04.672 align:middle line:90% of the convex space. 00:26:04.672 --> 00:26:06.130 align:middle line:84% We'll go up-- quadratic programming 00:26:06.130 --> 00:26:07.870 align:middle line:84% is a bit more complicated, but I think 00:26:07.870 --> 00:26:11.340 align:middle line:84% you get the idea you're minimizing that distance. 00:26:11.340 --> 00:26:16.050 align:middle line:84% OK, now, with any good modeling exercise, 00:26:16.050 --> 00:26:19.710 align:middle line:84% there should be some data lurking in the wings. 00:26:19.710 --> 00:26:21.810 align:middle line:84% Many of the network models that we will do 00:26:21.810 --> 00:26:24.960 align:middle line:84% are getting more ambitious than even the massive amounts 00:26:24.960 --> 00:26:27.510 align:middle line:90% of data that are coming in. 00:26:27.510 --> 00:26:31.080 align:middle line:84% There are two types of data that might leap to mind 00:26:31.080 --> 00:26:33.240 align:middle line:90% as being appropriate for this. 00:26:33.240 --> 00:26:36.660 align:middle line:84% Remember, we said that what the organism has done 00:26:36.660 --> 00:26:41.810 align:middle line:84% is optimize the use of these networks 00:26:41.810 --> 00:26:43.560 align:middle line:90% so that you can maximize growth. 00:26:43.560 --> 00:26:46.910 align:middle line:84% So the two sources of data that you'd want to test this with 00:26:46.910 --> 00:26:50.090 align:middle line:84% would be the flux data itself because you're saying-- 00:26:50.090 --> 00:26:52.790 align:middle line:84% you're predicting how much is going to be 00:26:52.790 --> 00:26:55.220 align:middle line:90% going in each flux direction. 00:26:55.220 --> 00:26:59.150 align:middle line:84% And the growth data, so you're making predictions 00:26:59.150 --> 00:27:02.390 align:middle line:84% that you'll optimally use the fluxes in the network 00:27:02.390 --> 00:27:05.450 align:middle line:84% in order to maximize growth for various mutants 00:27:05.450 --> 00:27:08.120 align:middle line:84% and various different conditions of growth, different carbon 00:27:08.120 --> 00:27:09.390 align:middle line:90% and nitrogen sources. 00:27:09.390 --> 00:27:15.130 align:middle line:84% So here you have a group in Zurich, 00:27:15.130 --> 00:27:17.670 align:middle line:84% which is among the very few groups that can actually 00:27:17.670 --> 00:27:25.320 align:middle line:84% measure the internal fluxes for metabolic pathways 00:27:25.320 --> 00:27:27.720 align:middle line:84% that begin with, say, isotopically 00:27:27.720 --> 00:27:31.110 align:middle line:84% labeled glucose and end in the various amino acids. 00:27:31.110 --> 00:27:33.720 align:middle line:90% You can measure these-- 00:27:33.720 --> 00:27:36.750 align:middle line:84% we prepared ourselves for this a little bit in the last class 00:27:36.750 --> 00:27:38.640 align:middle line:84% where we talked about the isotopomers 00:27:38.640 --> 00:27:44.070 align:middle line:84% where you can have different chemical compounds whose only 00:27:44.070 --> 00:27:48.660 align:middle line:84% difference-- they have basically identical chemically. 00:27:48.660 --> 00:27:50.490 align:middle line:84% But in different carbon positions, 00:27:50.490 --> 00:27:54.610 align:middle line:84% you have a carbon-13 in one place, carbon-12 in another. 00:27:54.610 --> 00:27:57.540 align:middle line:84% And the exact arrangement of carbon-12 and carbon-13s 00:27:57.540 --> 00:27:59.800 align:middle line:84% in this will determine the isotopomer. 00:27:59.800 --> 00:28:05.100 align:middle line:84% And if you have mixtures of carbon-12 and carbon-13 00:28:05.100 --> 00:28:06.030 align:middle line:90% glucose-- 00:28:06.030 --> 00:28:10.410 align:middle line:84% in the upper left-hand corner of this diagram 00:28:10.410 --> 00:28:15.480 align:middle line:84% includes glycolysis and pentose phosphate and TCA cycle. 00:28:15.480 --> 00:28:18.240 align:middle line:84% This isotopically labeled glucose goes through here. 00:28:18.240 --> 00:28:20.220 align:middle line:84% And then you need a little bit of modeling 00:28:20.220 --> 00:28:23.740 align:middle line:90% that we won't go into. 00:28:23.740 --> 00:28:26.102 align:middle line:84% But that does require a stoichiometry matrix, 00:28:26.102 --> 00:28:27.810 align:middle line:84% just like our optimization modeling does. 00:28:27.810 --> 00:28:30.210 align:middle line:84% But now, very independently of that, 00:28:30.210 --> 00:28:33.540 align:middle line:84% you need to ask how the isotopes of glucose 00:28:33.540 --> 00:28:35.525 align:middle line:84% would make their way into the amino acids. 00:28:35.525 --> 00:28:36.900 align:middle line:84% And then once you have that, then 00:28:36.900 --> 00:28:40.427 align:middle line:84% given the amino acid ratios by mass spect in NMR, 00:28:40.427 --> 00:28:42.510 align:middle line:84% you can go back and calculate what the fluxes must 00:28:42.510 --> 00:28:43.530 align:middle line:90% have been in here. 00:28:43.530 --> 00:28:45.030 align:middle line:84% Once you have those fluxes, then you 00:28:45.030 --> 00:28:47.220 align:middle line:84% can ask how close to optimal are they 00:28:47.220 --> 00:28:50.820 align:middle line:84% for the wild type under one condition. 00:28:50.820 --> 00:28:53.010 align:middle line:84% For the mutants, under the same condition. 00:28:53.010 --> 00:28:57.180 align:middle line:84% For the wild type and mutant under different conditions. 00:28:57.180 --> 00:28:59.700 align:middle line:84% OK, so this is the first class of data that we'll use, 00:28:59.700 --> 00:29:05.890 align:middle line:84% and it's the internal fluxes for wild-type and mutants. 00:29:05.890 --> 00:29:08.680 align:middle line:84% Look in the upper right-hand corner where-- 00:29:08.680 --> 00:29:10.690 align:middle line:84% so you've got in the upper left-hand corner 00:29:10.690 --> 00:29:13.850 align:middle line:84% is the icon color-coded from the previous slide. 00:29:13.850 --> 00:29:16.360 align:middle line:84% The upper right-hand corner is the predictions 00:29:16.360 --> 00:29:20.930 align:middle line:84% for wild type using the linear programming or FBA model. 00:29:20.930 --> 00:29:23.770 align:middle line:84% Get predicted fluxes on the vertical axis 00:29:23.770 --> 00:29:27.250 align:middle line:84% and experimental fluxes on the horizontal axis. 00:29:27.250 --> 00:29:32.410 align:middle line:84% You see, here is a good correlation coefficient 00:29:32.410 --> 00:29:36.060 align:middle line:90% of about 90 plus percent. 00:29:36.060 --> 00:29:40.150 align:middle line:84% And a probability that this would not be random. 00:29:40.150 --> 00:29:44.980 align:middle line:84% That this is a positive linear correlation is better than 10 00:29:44.980 --> 00:29:46.630 align:middle line:90% to the minus 7. 00:29:46.630 --> 00:29:49.570 align:middle line:84% Very unlikely to happen at random. 00:29:49.570 --> 00:29:53.320 align:middle line:84% Then you can focus in on the outliers like number 18 00:29:53.320 --> 00:29:55.660 align:middle line:90% here, but overall, it's a good-- 00:29:55.660 --> 00:29:58.150 align:middle line:84% I mean, this can allow you to ask what 00:29:58.150 --> 00:30:01.120 align:middle line:84% experimental or model problems you might have. 00:30:01.120 --> 00:30:04.900 align:middle line:84% But let's ask-- that's the wild type under common growth 00:30:04.900 --> 00:30:06.740 align:middle line:90% conditionns. 00:30:06.740 --> 00:30:09.070 align:middle line:84% What about mutant under the same growth conditions? 00:30:09.070 --> 00:30:14.920 align:middle line:84% Same stoichiometric modeling, same measurements, 00:30:14.920 --> 00:30:16.750 align:middle line:84% now the experimental fluxes versus 00:30:16.750 --> 00:30:21.160 align:middle line:84% predicted fluxes in the lower left-hand quadrant 00:30:21.160 --> 00:30:22.780 align:middle line:90% is almost completely random. 00:30:22.780 --> 00:30:25.090 align:middle line:84% There's no positive or negative correlation 00:30:25.090 --> 00:30:29.680 align:middle line:84% that's significant, as we might have expected for the mutant. 00:30:29.680 --> 00:30:32.780 align:middle line:90% Remember, it isn't optimal. 00:30:32.780 --> 00:30:35.800 align:middle line:84% So you don't expect linear programming method 00:30:35.800 --> 00:30:38.890 align:middle line:84% which produces the optimum to be appropriate. 00:30:38.890 --> 00:30:42.130 align:middle line:84% Unless we had allowed this to evolve 00:30:42.130 --> 00:30:45.670 align:middle line:84% in the laboratory for hundreds of generations 00:30:45.670 --> 00:30:47.410 align:middle line:84% or a sufficient number, however many 00:30:47.410 --> 00:30:50.080 align:middle line:84% that is, to get all the other genes 00:30:50.080 --> 00:30:53.260 align:middle line:84% or enough of the other genes to adapt so 00:30:53.260 --> 00:30:54.610 align:middle line:90% that you get near an optimum. 00:30:54.610 --> 00:30:56.500 align:middle line:90% In any case, this one is random. 00:30:56.500 --> 00:30:59.260 align:middle line:84% If you now use the quadratic programming approach, 00:30:59.260 --> 00:31:02.800 align:middle line:84% the MOMA or minimization of metabolic adjustment. 00:31:02.800 --> 00:31:05.890 align:middle line:84% Now get the very exciting result that it now 00:31:05.890 --> 00:31:09.760 align:middle line:84% becomes statistically significant again. 00:31:09.760 --> 00:31:14.927 align:middle line:84% Still a few outliers, 17 and 16 here in magenta, 00:31:14.927 --> 00:31:16.510 align:middle line:84% and these are things where you can now 00:31:16.510 --> 00:31:18.670 align:middle line:90% make very specific tests. 00:31:18.670 --> 00:31:20.980 align:middle line:84% Because now you know which ones are 00:31:20.980 --> 00:31:24.520 align:middle line:84% most discrepant between the experimental 00:31:24.520 --> 00:31:25.450 align:middle line:90% and predicted fluxes. 00:31:25.450 --> 00:31:26.908 align:middle line:84% And you can go in and ask what part 00:31:26.908 --> 00:31:29.320 align:middle line:84% of that model or what part of that data collection 00:31:29.320 --> 00:31:34.540 align:middle line:84% might be accounting for the poor fit. 00:31:34.540 --> 00:31:36.700 align:middle line:84% But overall, this starts to give one 00:31:36.700 --> 00:31:41.930 align:middle line:84% the impression that maybe the mutants will not be eligible 00:31:41.930 --> 00:31:44.630 align:middle line:84% and that they can be somewhat better served 00:31:44.630 --> 00:31:50.280 align:middle line:84% by this quasi-non-optimal solution. 00:31:50.280 --> 00:31:56.240 align:middle line:84% And if you walk through various different conditions 00:31:56.240 --> 00:31:57.920 align:middle line:84% with different knockouts, you can 00:31:57.920 --> 00:32:00.450 align:middle line:90% see multiple examples of this. 00:32:00.450 --> 00:32:04.730 align:middle line:84% So here in the upper left-hand quadrant of Slide 59, 00:32:04.730 --> 00:32:06.530 align:middle line:84% where we have the carbon limitation 00:32:06.530 --> 00:32:10.190 align:middle line:84% condition on the far left and then 00:32:10.190 --> 00:32:14.780 align:middle line:84% the comparison of the method for flux balance on wild type. 00:32:14.780 --> 00:32:19.280 align:middle line:84% Here's that 0.9 correlation coefficient and good P-value, 00:32:19.280 --> 00:32:21.560 align:middle line:90% and then comparing MOMA and FBA. 00:32:21.560 --> 00:32:25.870 align:middle line:84% And again, the significance of switching from FBA to MoMA 00:32:25.870 --> 00:32:29.770 align:middle line:84% is 3 times 10 to the minus 3, indicating 00:32:29.770 --> 00:32:32.380 align:middle line:84% that the quadratic programming is the more appropriate 00:32:32.380 --> 00:32:33.550 align:middle line:90% application here. 00:32:33.550 --> 00:32:35.050 align:middle line:84% And as you walk through this, you'll 00:32:35.050 --> 00:32:38.020 align:middle line:84% see many good correlation coefficients 00:32:38.020 --> 00:32:43.180 align:middle line:84% and many improvements when you go to the quadratic or MOMA, 00:32:43.180 --> 00:32:48.040 align:middle line:84% not every case, but certainly many of them. 00:32:48.040 --> 00:32:52.540 align:middle line:84% Now that's one class of test, which is the internal fluxes. 00:32:52.540 --> 00:32:59.100 align:middle line:84% The idea is that you adjust the internal fluxes until they're 00:32:59.100 --> 00:33:00.840 align:middle line:90% optimal to produce growth. 00:33:00.840 --> 00:33:05.190 align:middle line:84% Well, how about measuring growth itself on a set of mutants. 00:33:05.190 --> 00:33:11.670 align:middle line:84% Again, these mutants you expect will follow the MOMA prediction 00:33:11.670 --> 00:33:13.840 align:middle line:90% slightly better. 00:33:13.840 --> 00:33:17.820 align:middle line:84% And so Slide 60 is a particular way 00:33:17.820 --> 00:33:22.140 align:middle line:84% of measuring a large genome-wide set of mutants. 00:33:22.140 --> 00:33:24.000 align:middle line:84% This is not a trivial undertaking. 00:33:24.000 --> 00:33:27.900 align:middle line:84% We know how to measure a transcriptome set of RNAs. 00:33:27.900 --> 00:33:29.520 align:middle line:84% Where on microarrays, maybe there's 00:33:29.520 --> 00:33:33.810 align:middle line:84% an analogous way to measure genomes worth of mutants. 00:33:33.810 --> 00:33:37.530 align:middle line:84% Ideally, you would have a mutant, not just one per gene. 00:33:37.530 --> 00:33:41.220 align:middle line:84% One knockout per gene, which is maybe 00:33:41.220 --> 00:33:45.330 align:middle line:84% a very expensive handcrafted deletion, 00:33:45.330 --> 00:33:48.090 align:middle line:84% but you'd have a little targeted mutation 00:33:48.090 --> 00:33:51.780 align:middle line:84% in every domain that contributes to that gene function. 00:33:51.780 --> 00:33:56.910 align:middle line:84% Mutations in the DNA regulatory elements, various protein 00:33:56.910 --> 00:33:59.820 align:middle line:84% domains, RNA stability domain, and so 00:33:59.820 --> 00:34:04.200 align:middle line:84% on, not quite at that level of ideal, but getting close to it. 00:34:04.200 --> 00:34:05.970 align:middle line:84% It's transposon mutagenesis, where 00:34:05.970 --> 00:34:11.159 align:middle line:84% you insert small bits of DNA randomly throughout the genome. 00:34:11.159 --> 00:34:15.659 align:middle line:84% The modern set of commonly used transposons 00:34:15.659 --> 00:34:18.989 align:middle line:84% are pretty random in their insertion site choice. 00:34:18.989 --> 00:34:20.610 align:middle line:84% And then, you need a way of turning 00:34:20.610 --> 00:34:24.630 align:middle line:84% this collection of transposons into a readout of growth rates. 00:34:24.630 --> 00:34:28.989 align:middle line:84% Well, one way to do this is to have a population of cells, 00:34:28.989 --> 00:34:33.600 align:middle line:84% each having their own distinctive transposon 00:34:33.600 --> 00:34:36.010 align:middle line:90% and growing them as a mixture. 00:34:36.010 --> 00:34:41.460 align:middle line:84% And as they grow as a mixture, the ones that grow better 00:34:41.460 --> 00:34:45.320 align:middle line:90% will dominate the population. 00:34:45.320 --> 00:34:47.650 align:middle line:84% And so their transposon hit the junction 00:34:47.650 --> 00:34:49.900 align:middle line:84% between the transposon-- the transposons is universal. 00:34:49.900 --> 00:34:52.389 align:middle line:90% It's present in every cell. 00:34:52.389 --> 00:34:58.150 align:middle line:84% But where it sits is unique to that cell, or nearly so. 00:34:58.150 --> 00:35:00.840 align:middle line:90% So it sits in-- 00:35:00.840 --> 00:35:03.090 align:middle line:84% the junction between transposon and genome is unique, 00:35:03.090 --> 00:35:04.590 align:middle line:84% and you want a way of assaying that. 00:35:04.590 --> 00:35:08.760 align:middle line:84% The way you assay that is you take the complete DNA 00:35:08.760 --> 00:35:10.950 align:middle line:84% from the entire cell mixture, and you 00:35:10.950 --> 00:35:13.590 align:middle line:84% want to say how much of each transposon exists? 00:35:13.590 --> 00:35:15.502 align:middle line:84% So you cut the entire DNA mixture 00:35:15.502 --> 00:35:17.460 align:middle line:84% with a restriction enzyme that cuts frequently. 00:35:17.460 --> 00:35:21.420 align:middle line:84% So called 4 cutter, cuts with about every 236 base pairs 00:35:21.420 --> 00:35:22.382 align:middle line:90% on average. 00:35:22.382 --> 00:35:24.090 align:middle line:84% And you like it on this very special kind 00:35:24.090 --> 00:35:26.460 align:middle line:84% of linker, which is a linker which will not 00:35:26.460 --> 00:35:30.980 align:middle line:84% amplify with the corresponding primer 00:35:30.980 --> 00:35:34.520 align:middle line:84% until some other DNA synthesis has gone through because you 00:35:34.520 --> 00:35:38.460 align:middle line:84% see this little Y linker is not perfectly logically based 00:35:38.460 --> 00:35:38.960 align:middle line:90% paired. 00:35:38.960 --> 00:35:41.588 align:middle line:90% 00:35:41.588 --> 00:35:46.110 align:middle line:84% It requires DNA polymerase to make a complement. 00:35:46.110 --> 00:35:48.210 align:middle line:84% Complement then will bind the primer. 00:35:48.210 --> 00:35:53.540 align:middle line:84% It's a long way of saying that only in cases where you have 00:35:53.540 --> 00:35:57.410 align:middle line:84% a primer in the transposon near one of these Y linkers will 00:35:57.410 --> 00:35:58.730 align:middle line:90% you get amplification. 00:35:58.730 --> 00:36:00.470 align:middle line:84% Transposon alone, you won't get it. 00:36:00.470 --> 00:36:02.000 align:middle line:84% Y linker alone, you won't get it. 00:36:02.000 --> 00:36:04.437 align:middle line:84% That's one step of enrichment, and the reason 00:36:04.437 --> 00:36:06.770 align:middle line:84% you have to enrich is if you just threw this whole thing 00:36:06.770 --> 00:36:08.990 align:middle line:84% on the microarray, you'd get everything 00:36:08.990 --> 00:36:10.580 align:middle line:84% lighting up because every genome has 00:36:10.580 --> 00:36:15.240 align:middle line:84% every piece of the genome in it, and the transposons 00:36:15.240 --> 00:36:17.410 align:middle line:90% are present in trace amounts. 00:36:17.410 --> 00:36:19.770 align:middle line:84% So you need to amplify the junction fragment 00:36:19.770 --> 00:36:23.290 align:middle line:84% by first this trick, followed by a T7 promoter. 00:36:23.290 --> 00:36:25.230 align:middle line:84% This is a phage promoter a very commonly used. 00:36:25.230 --> 00:36:28.770 align:middle line:84% Very clean RNA polymerase background 00:36:28.770 --> 00:36:31.200 align:middle line:84% that's been incorporated into any transpose 00:36:31.200 --> 00:36:32.730 align:middle line:90% on your favorite transposon. 00:36:32.730 --> 00:36:34.120 align:middle line:90% So now you have two steps. 00:36:34.120 --> 00:36:39.120 align:middle line:84% First, this ligation-mediated PCR, and second the T7 in vitro 00:36:39.120 --> 00:36:41.183 align:middle line:90% transcription to make RNA. 00:36:41.183 --> 00:36:42.600 align:middle line:84% Now, you've basically reduced this 00:36:42.600 --> 00:36:46.830 align:middle line:84% to a problem similar to transcriptome microarrays 00:36:46.830 --> 00:36:48.070 align:middle line:90% that we've done before. 00:36:48.070 --> 00:36:52.560 align:middle line:84% Now the RNAs are surrogates for the amount of each strain. 00:36:52.560 --> 00:36:53.670 align:middle line:90% We're not measuring RNA. 00:36:53.670 --> 00:36:56.040 align:middle line:84% We've made an RNA that represents 00:36:56.040 --> 00:36:58.260 align:middle line:84% the transposon junction fragment and hence 00:36:58.260 --> 00:37:00.540 align:middle line:84% allows you to quantitate the amount 00:37:00.540 --> 00:37:04.920 align:middle line:84% of that particular mutant in a population. 00:37:04.920 --> 00:37:08.500 align:middle line:84% As that mutant goes up or down in the population, 00:37:08.500 --> 00:37:12.040 align:middle line:90% so does the RNA from this assay. 00:37:12.040 --> 00:37:17.040 align:middle line:84% So you might ask at this point, well, this is great, 00:37:17.040 --> 00:37:17.940 align:middle line:90% but I don't trust it. 00:37:17.940 --> 00:37:21.630 align:middle line:84% Just like some people might not have trusted mass spectrometry 00:37:21.630 --> 00:37:24.550 align:middle line:84% as being reproducible enough to be quantitative. 00:37:24.550 --> 00:37:26.760 align:middle line:84% So the way that you determine whether something 00:37:26.760 --> 00:37:28.890 align:middle line:84% is sufficiently reproducible to be quantitative, 00:37:28.890 --> 00:37:31.980 align:middle line:84% one way to do it is to do two independent selection 00:37:31.980 --> 00:37:34.620 align:middle line:90% experiments. 00:37:34.620 --> 00:37:38.460 align:middle line:84% Evolutionists might say, oh, if we reran evolution 00:37:38.460 --> 00:37:41.460 align:middle line:84% all over again, we would not get the same set of organisms 00:37:41.460 --> 00:37:42.470 align:middle line:90% we have on Earth today. 00:37:42.470 --> 00:37:44.220 align:middle line:84% That may be true, but that's because there 00:37:44.220 --> 00:37:47.850 align:middle line:84% are all sorts of bottlenecks in historical events. 00:37:47.850 --> 00:37:57.070 align:middle line:84% In this case, we specifically tried 00:37:57.070 --> 00:38:00.070 align:middle line:84% to make it reproducible by avoiding bottlenecks 00:38:00.070 --> 00:38:04.690 align:middle line:84% by having every mutation, every transposon type represented 00:38:04.690 --> 00:38:09.170 align:middle line:84% by 1,000 different independent events. 00:38:09.170 --> 00:38:13.340 align:middle line:84% As well as 1,000 copies of every transposon hit, 00:38:13.340 --> 00:38:15.710 align:middle line:84% so you keep the population size large, 00:38:15.710 --> 00:38:17.330 align:middle line:84% and so it becomes more reproducible. 00:38:17.330 --> 00:38:19.820 align:middle line:84% And the way you measure the reproducibility is you 00:38:19.820 --> 00:38:21.830 align:middle line:84% do the two selection experiments. 00:38:21.830 --> 00:38:26.870 align:middle line:84% The two ideally independent sets of transposons 00:38:26.870 --> 00:38:30.230 align:middle line:84% subjected to exactly the same selection procedure. 00:38:30.230 --> 00:38:34.610 align:middle line:84% And you see a very gratifying curve 00:38:34.610 --> 00:38:40.280 align:middle line:84% here with an R-squared regression 00:38:40.280 --> 00:38:43.460 align:middle line:90% measure in excess of 98%. 00:38:43.460 --> 00:38:45.320 align:middle line:84% And the kind of scatter that you would 00:38:45.320 --> 00:38:48.720 align:middle line:84% be pleased to get in an RNA microarray experiment. 00:38:48.720 --> 00:38:52.940 align:middle line:84% So this is reproducible and, therefore, quite capable 00:38:52.940 --> 00:38:54.470 align:middle line:90% of producing quantitative data. 00:38:54.470 --> 00:38:56.480 align:middle line:84% Well, now let's go back and compare it 00:38:56.480 --> 00:39:02.000 align:middle line:84% to the two different methods of modeling the flux optimization. 00:39:02.000 --> 00:39:04.140 align:middle line:84% There's the FBA or linear programming, 00:39:04.140 --> 00:39:06.950 align:middle line:84% which assumes that the mutant is immediately optimal. 00:39:06.950 --> 00:39:09.650 align:middle line:84% And then there's the MOMA or minimization 00:39:09.650 --> 00:39:13.403 align:middle line:84% of metabolic adjustment, which doesn't assume 00:39:13.403 --> 00:39:15.320 align:middle line:84% immediate ophthalmology, but assumes that it's 00:39:15.320 --> 00:39:18.830 align:middle line:90% close to wild-type optimal. 00:39:18.830 --> 00:39:23.420 align:middle line:84% So starting at the top with the linear flux balance, 00:39:23.420 --> 00:39:30.800 align:middle line:84% you can classify the predictions of the model for each gene. 00:39:30.800 --> 00:39:35.870 align:middle line:90% And here we have almost 400-- 00:39:35.870 --> 00:39:39.170 align:middle line:84% almost 500 different genes mutated, 00:39:39.170 --> 00:39:41.330 align:middle line:84% and first you run through in silico predictions 00:39:41.330 --> 00:39:44.762 align:middle line:84% you can classify them as being essential for a particular 00:39:44.762 --> 00:39:45.470 align:middle line:90% growth condition. 00:39:45.470 --> 00:39:47.420 align:middle line:84% They aren't necessarily essential for every growth 00:39:47.420 --> 00:39:47.920 align:middle line:90% condition. 00:39:47.920 --> 00:39:51.520 align:middle line:84% But the growth conditions here the essential or nonessential 00:39:51.520 --> 00:39:52.520 align:middle line:90% or something in between. 00:39:52.520 --> 00:39:55.145 align:middle line:84% We'll just focus on the extremes of essential and nonessential. 00:39:55.145 --> 00:39:57.980 align:middle line:84% And then, the experiment can be classified as 00:39:57.980 --> 00:40:01.970 align:middle line:84% to whether there's significant negative selection 00:40:01.970 --> 00:40:04.550 align:middle line:84% or no noticeable selection in the particular growth 00:40:04.550 --> 00:40:05.120 align:middle line:90% conditions. 00:40:05.120 --> 00:40:07.550 align:middle line:84% And so you expect the ones to be essential in these growth 00:40:07.550 --> 00:40:10.100 align:middle line:84% conditions to be heavily negative selectively, 00:40:10.100 --> 00:40:14.310 align:middle line:84% and you have 80 examples of that out of 142 predicted. 00:40:14.310 --> 00:40:17.720 align:middle line:84% However, you wouldn't expect any to be missing in selection 00:40:17.720 --> 00:40:18.720 align:middle line:90% since they're essential. 00:40:18.720 --> 00:40:21.560 align:middle line:84% So the 62 should be a zero for the 62. 00:40:21.560 --> 00:40:25.640 align:middle line:84% This is an example of how we're going to try to explain or use 00:40:25.640 --> 00:40:28.100 align:middle line:84% this as a way of generating interesting hypotheses 00:40:28.100 --> 00:40:31.130 align:middle line:84% about the exceptions or outliers. 00:40:31.130 --> 00:40:34.370 align:middle line:84% Similarly, the nonessential genes should have no selection. 00:40:34.370 --> 00:40:39.350 align:middle line:84% This 180 in the lower right-hand part of the upper FBA, 00:40:39.350 --> 00:40:42.140 align:middle line:90% but the 119 should be zero. 00:40:42.140 --> 00:40:45.260 align:middle line:84% So the first pass explanation is OK. 00:40:45.260 --> 00:40:49.260 align:middle line:84% We knew that these weren't going to be optimal right away. 00:40:49.260 --> 00:40:54.860 align:middle line:84% So let's use the other model, the sub-optimal, nearly optimal 00:40:54.860 --> 00:40:57.920 align:middle line:84% model MOMA and see how well it does. 00:40:57.920 --> 00:41:01.040 align:middle line:84% Well, on the far right-hand side of Slide 62, 00:41:01.040 --> 00:41:05.630 align:middle line:84% you see the chi-square P-value is how well the predictions 00:41:05.630 --> 00:41:08.250 align:middle line:90% agree with the observations. 00:41:08.250 --> 00:41:12.620 align:middle line:84% And it is significant for the linear programming. 00:41:12.620 --> 00:41:17.630 align:middle line:84% It's 4 times 10, then minus 3, fairly significant. 00:41:17.630 --> 00:41:20.750 align:middle line:84% And the MOMA is much more significant than the minus 5. 00:41:20.750 --> 00:41:25.400 align:middle line:84% It has improved some of those upper right 00:41:25.400 --> 00:41:28.070 align:middle line:90% 66 and lower left 108. 00:41:28.070 --> 00:41:30.980 align:middle line:84% Those should still be zero, and they're not. 00:41:30.980 --> 00:41:35.840 align:middle line:84% And the extent that they deviate from the ideal expectation 00:41:35.840 --> 00:41:38.070 align:middle line:90% means that either the data-- 00:41:38.070 --> 00:41:40.070 align:middle line:84% or somehow the way we're collecting the data not 00:41:40.070 --> 00:41:41.180 align:middle line:90% the way we expected it. 00:41:41.180 --> 00:41:49.140 align:middle line:84% Or that the model, more likely the model 00:41:49.140 --> 00:41:50.250 align:middle line:90% has some problems in it. 00:41:50.250 --> 00:41:54.360 align:middle line:84% And examples of problems for those two classes, 00:41:54.360 --> 00:41:57.030 align:middle line:84% the predicted essential genes which 00:41:57.030 --> 00:42:00.810 align:middle line:84% show those-- which show no selection, 00:42:00.810 --> 00:42:02.880 align:middle line:84% might be examples of redundancies 00:42:02.880 --> 00:42:04.770 align:middle line:84% that we failed to model when we put together 00:42:04.770 --> 00:42:06.450 align:middle line:90% the stoichiometric matrix. 00:42:06.450 --> 00:42:08.700 align:middle line:84% The stoichiometric matrix we take every known 00:42:08.700 --> 00:42:10.470 align:middle line:90% biochemical reaction. 00:42:10.470 --> 00:42:15.480 align:middle line:84% We might take convincing homologs sequenced 00:42:15.480 --> 00:42:17.430 align:middle line:84% homology level, and say those might 00:42:17.430 --> 00:42:18.690 align:middle line:90% be examples of redundancy. 00:42:18.690 --> 00:42:20.550 align:middle line:84% We might take analogous or parallel 00:42:20.550 --> 00:42:23.190 align:middle line:84% roots that are known documented biochemically, where 00:42:23.190 --> 00:42:26.040 align:middle line:84% you can get to the same point by a series of possibly 00:42:26.040 --> 00:42:29.340 align:middle line:84% non-sequence homologous enzymatic steps. 00:42:29.340 --> 00:42:30.780 align:middle line:90% So those are novel redundancies. 00:42:30.780 --> 00:42:33.690 align:middle line:84% This is an example of potential discovery, 00:42:33.690 --> 00:42:37.380 align:middle line:84% and can be pursued in a very directed way 00:42:37.380 --> 00:42:40.950 align:middle line:84% because these are 66 very specific predictions. 00:42:40.950 --> 00:42:45.335 align:middle line:84% 108 essential gene knockouts, which nevertheless 00:42:45.335 --> 00:42:46.710 align:middle line:84% show negative selection, could be 00:42:46.710 --> 00:42:48.300 align:middle line:90% examples of position effects. 00:42:48.300 --> 00:42:53.010 align:middle line:84% Where a mutation in one gene affects other genes that 00:42:53.010 --> 00:42:56.020 align:middle line:84% are close along the DNA position, 00:42:56.020 --> 00:42:59.700 align:middle line:84% meaning along the DNA, an example of this 00:42:59.700 --> 00:43:05.610 align:middle line:84% is that a variety of mutations in a gene which 00:43:05.610 --> 00:43:08.160 align:middle line:84% is upstream from another gene in an operon, 00:43:08.160 --> 00:43:11.700 align:middle line:84% have poorer effects on downstream genes 00:43:11.700 --> 00:43:14.790 align:middle line:84% due to the coupling of transcription and translation. 00:43:14.790 --> 00:43:18.610 align:middle line:84% You can have other position effects as well. 00:43:18.610 --> 00:43:23.940 align:middle line:84% These are examples of possibly many explanations 00:43:23.940 --> 00:43:25.920 align:middle line:84% and discoveries that can contribute 00:43:25.920 --> 00:43:28.770 align:middle line:90% to the wonderful nature of-- 00:43:28.770 --> 00:43:29.820 align:middle line:90% it's a win-win situation. 00:43:29.820 --> 00:43:32.010 align:middle line:84% Either you get great correlation, 00:43:32.010 --> 00:43:36.620 align:middle line:84% or you get great exceptions and discoveries, or both. 00:43:36.620 --> 00:43:42.400 align:middle line:84% OK, now, when we talked about redundancy 00:43:42.400 --> 00:43:43.870 align:middle line:84% as one of the possible explanations 00:43:43.870 --> 00:43:45.412 align:middle line:84% in the previous slide, that brings up 00:43:45.412 --> 00:43:48.640 align:middle line:84% really important conceptual component 00:43:48.640 --> 00:43:52.310 align:middle line:84% of the post-genomic functional genomics world. 00:43:52.310 --> 00:43:57.805 align:middle line:84% Which is what do we do about multiple homologous domains? 00:43:57.805 --> 00:43:59.680 align:middle line:84% When you go through and you sequence genomes, 00:43:59.680 --> 00:44:01.130 align:middle line:90% you find parallels. 00:44:01.130 --> 00:44:04.540 align:middle line:84% You find either whole genes or pieces 00:44:04.540 --> 00:44:08.200 align:middle line:84% of genes which have high sequence homology, which 00:44:08.200 --> 00:44:10.600 align:middle line:84% we talked about at the beginning of the course. 00:44:10.600 --> 00:44:14.560 align:middle line:84% And here's examples of three protein-coding regions. 00:44:14.560 --> 00:44:17.410 align:middle line:84% We encode enzymes involved in the biosynthesis 00:44:17.410 --> 00:44:19.910 align:middle line:90% of amino acids. 00:44:19.910 --> 00:44:24.490 align:middle line:84% So you can imagine that when you grow the cells on minimal media 00:44:24.490 --> 00:44:26.860 align:middle line:84% that you are not providing the amino acids, 00:44:26.860 --> 00:44:28.930 align:middle line:84% so the cell has to make the amino acids. 00:44:28.930 --> 00:44:31.600 align:middle line:84% If it's got a mutation, a transposon 00:44:31.600 --> 00:44:35.080 align:middle line:84% hit in one of these genes that's key 00:44:35.080 --> 00:44:38.755 align:middle line:84% in the biosynthesis synthesis of lysine 3 and methionine. 00:44:38.755 --> 00:44:40.630 align:middle line:84% And you might expect that it won't grow well. 00:44:40.630 --> 00:44:42.838 align:middle line:84% It'll be at a selective disadvantage in minimal media 00:44:42.838 --> 00:44:45.940 align:middle line:84% unless one of the other genes covers for it. 00:44:45.940 --> 00:44:50.140 align:middle line:84% As possibly as green-- so these have two or three domains here 00:44:50.140 --> 00:44:50.770 align:middle line:90% color coded. 00:44:50.770 --> 00:44:54.970 align:middle line:84% So the green domain is shared by these three 00:44:54.970 --> 00:44:57.100 align:middle line:84% biosynthetic proteins that are otherwise 00:44:57.100 --> 00:45:01.970 align:middle line:84% very different in their metabolic contributions. 00:45:01.970 --> 00:45:05.000 align:middle line:84% And you might imagine that some of these green domains 00:45:05.000 --> 00:45:08.210 align:middle line:84% might cover for others when one of them is mutated. 00:45:08.210 --> 00:45:10.520 align:middle line:84% One of them, however, the one in lysine 00:45:10.520 --> 00:45:13.950 align:middle line:84% when it's got a transposon hit in that domain, 00:45:13.950 --> 00:45:17.090 align:middle line:84% the replication rate selective disadvantage in minimal media 00:45:17.090 --> 00:45:17.900 align:middle line:90% is severe. 00:45:17.900 --> 00:45:18.510 align:middle line:90% It's a fact. 00:45:18.510 --> 00:45:20.420 align:middle line:84% It's an order of magnitude, while the others 00:45:20.420 --> 00:45:21.380 align:middle line:90% are more subtle. 00:45:21.380 --> 00:45:24.350 align:middle line:84% It could be that the lysine covers for the other two 00:45:24.350 --> 00:45:28.730 align:middle line:84% because it's made in large amounts, is very active. 00:45:28.730 --> 00:45:31.310 align:middle line:84% But the other two can't cover for the lysine, 00:45:31.310 --> 00:45:33.770 align:middle line:84% or there's a variety of other possible explanations 00:45:33.770 --> 00:45:35.960 align:middle line:90% for this observation. 00:45:35.960 --> 00:45:39.380 align:middle line:84% The point is that this generates hypotheses 00:45:39.380 --> 00:45:44.870 align:middle line:84% that allow us to follow up on the flux balance or MOMA 00:45:44.870 --> 00:45:46.500 align:middle line:90% type of modeling. 00:45:46.500 --> 00:45:49.495 align:middle line:84% And I think the reason I spent so much time on it 00:45:49.495 --> 00:45:51.680 align:middle line:84% is I think the concept of optimization 00:45:51.680 --> 00:45:54.200 align:middle line:84% is something that's important both in the sense that we're 00:45:54.200 --> 00:45:56.750 align:middle line:84% in engineering, as engineers, we're 00:45:56.750 --> 00:45:59.600 align:middle line:90% optimizing living systems. 00:45:59.600 --> 00:46:03.140 align:middle line:84% And also, as students of evolutionary systems 00:46:03.140 --> 00:46:09.950 align:middle line:84% where to understand what those systems are optimized to do, 00:46:09.950 --> 00:46:14.600 align:middle line:84% we need to look at from this perspective. 00:46:14.600 --> 00:46:17.290 align:middle line:84% So this is just what we've covered today. 00:46:17.290 --> 00:46:21.880 align:middle line:84% We've covered both continuous and discrete ways 00:46:21.880 --> 00:46:24.490 align:middle line:90% of modeling molecular systems. 00:46:24.490 --> 00:46:27.970 align:middle line:84% And, in particular, the red blood cell and the copy number 00:46:27.970 --> 00:46:32.860 align:middle line:84% control, as a way of dealing with metabolism and biopolymers 00:46:32.860 --> 00:46:35.860 align:middle line:84% separately, and the flux balance has brought these together, 00:46:35.860 --> 00:46:38.370 align:middle line:84% and brought in the notion of optimization. 00:46:38.370 --> 00:46:42.000 align:middle line:90% OK, so until next time.