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 --> 00:00:06.340 align:middle line:84% by MIT OpenCourseWare under a Creative Commons license. 00:00:06.340 --> 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 --> 00:00:14.050 align:middle line:90% is available at ocw.mit.edu. 00:00:14.050 --> 00:00:38.790 align:middle line:90% 00:00:38.790 --> 00:00:41.010 align:middle line:84% GEORGE CHURCH: Is there a section? 00:00:41.010 --> 00:00:43.860 align:middle line:84% No, there are no sections for a few days. 00:00:43.860 --> 00:00:46.770 align:middle line:84% The sections, both the weekly sections 00:00:46.770 --> 00:00:49.170 align:middle line:84% and the extra sections, will be on the website. 00:00:49.170 --> 00:00:50.820 align:middle line:90% That's your best place to look. 00:00:50.820 --> 00:00:55.695 align:middle line:84% If you have questions after you've looked at the website, 00:00:55.695 --> 00:00:57.570 align:middle line:84% then you should contact your teaching fellow. 00:00:57.570 --> 00:00:59.028 align:middle line:84% But definitely look at the website. 00:00:59.028 --> 00:01:02.880 align:middle line:84% It will be updated daily, and your sections 00:01:02.880 --> 00:01:06.816 align:middle line:84% will be assigned probably by Thursday or Friday, OK. 00:01:06.816 --> 00:01:13.350 align:middle line:90% 00:01:13.350 --> 00:01:16.230 align:middle line:90% OK, welcome back from the break. 00:01:16.230 --> 00:01:18.360 align:middle line:84% As promised, we're going to discuss 00:01:18.360 --> 00:01:22.350 align:middle line:84% the innards of computers that are going to be helping us 00:01:22.350 --> 00:01:25.050 align:middle line:90% with computational biology. 00:01:25.050 --> 00:01:31.110 align:middle line:84% We have a schematic diagram here, illustrating in part-- 00:01:31.110 --> 00:01:34.710 align:middle line:84% we'll have similar schematic diagrams for biological systems 00:01:34.710 --> 00:01:36.240 align:middle line:90% or biochemical systems. 00:01:36.240 --> 00:01:37.680 align:middle line:90% This is one for transistors. 00:01:37.680 --> 00:01:43.170 align:middle line:84% We have these nonlinear transistor elements here 00:01:43.170 --> 00:01:45.870 align:middle line:84% with an input voltage and output voltage in a controlled voltage 00:01:45.870 --> 00:01:50.790 align:middle line:84% VDD, and a ground here, this little triangle in the lower 00:01:50.790 --> 00:01:52.710 align:middle line:90% left. 00:01:52.710 --> 00:01:58.290 align:middle line:84% These transistors are in this circuit here that is certainly 00:01:58.290 --> 00:01:59.520 align:middle line:90% a higher-level description. 00:01:59.520 --> 00:02:05.940 align:middle line:84% It allows you to put down the more detailed description 00:02:05.940 --> 00:02:12.160 align:middle line:84% of this voltage curve as a function of time. 00:02:12.160 --> 00:02:14.220 align:middle line:84% So time is the horizontal axis here, 00:02:14.220 --> 00:02:16.980 align:middle line:84% ranging out to 200 nanoseconds, and this is all 00:02:16.980 --> 00:02:22.110 align:middle line:84% done in a program, a simulator called Spice. 00:02:22.110 --> 00:02:24.570 align:middle line:84% of this complementary metal oxide inverter. 00:02:24.570 --> 00:02:26.910 align:middle line:84% And this sort of simulator is one of the things 00:02:26.910 --> 00:02:30.257 align:middle line:84% that we will be talking about in biological systems. 00:02:30.257 --> 00:02:32.340 align:middle line:84% And it's very useful for designing these circuits, 00:02:32.340 --> 00:02:36.120 align:middle line:84% and you can see as the voltage, VDD, goes up 00:02:36.120 --> 00:02:40.890 align:middle line:84% in this straight blue line from 0 volts, which is off, 00:02:40.890 --> 00:02:42.600 align:middle line:90% to 5 volts, which is on. 00:02:42.600 --> 00:02:46.620 align:middle line:84% You get this almost all or none, but certainly 00:02:46.620 --> 00:02:49.080 align:middle line:84% nonlinear response, where the output 00:02:49.080 --> 00:02:52.320 align:middle line:84% voltage on the vertical axis goes from 5 volts, 00:02:52.320 --> 00:02:54.090 align:middle line:90% on, to 0 volts, off. 00:02:54.090 --> 00:02:55.500 align:middle line:90% It's basically the opposite. 00:02:55.500 --> 00:02:56.640 align:middle line:90% When it's 0, it's 5. 00:02:56.640 --> 00:02:58.410 align:middle line:90% When 5, it's zero. 00:02:58.410 --> 00:03:00.570 align:middle line:84% But in between, you can see there's this gray zone. 00:03:00.570 --> 00:03:03.060 align:middle line:84% You want to stay away from this in digital circuitry. 00:03:03.060 --> 00:03:08.820 align:middle line:84% You want to stay fully saturated or basically 0 volts. 00:03:08.820 --> 00:03:15.630 align:middle line:84% And then these inverters can be wired together 00:03:15.630 --> 00:03:17.430 align:middle line:84% in even higher-level diagram into what 00:03:17.430 --> 00:03:20.340 align:middle line:84% are called registers, which allow you to store 00:03:20.340 --> 00:03:23.190 align:middle line:90% multi-digit binary numbers. 00:03:23.190 --> 00:03:25.540 align:middle line:84% These can then be coupled together. 00:03:25.540 --> 00:03:30.450 align:middle line:84% So you can take two registers and add, digit by digit, 00:03:30.450 --> 00:03:31.950 align:middle line:90% the contents of those registers. 00:03:31.950 --> 00:03:33.200 align:middle line:90% That's called an adder. 00:03:33.200 --> 00:03:37.320 align:middle line:84% Adders and a variety of other higher-level electronic 00:03:37.320 --> 00:03:40.740 align:middle line:84% components can be put together and then addressed 00:03:40.740 --> 00:03:43.290 align:middle line:90% by software called a compiler. 00:03:43.290 --> 00:03:45.258 align:middle line:84% Well, how did you get this software in there 00:03:45.258 --> 00:03:46.050 align:middle line:90% in the first place? 00:03:46.050 --> 00:03:49.290 align:middle line:84% We have to toggle the hardware until it 00:03:49.290 --> 00:03:51.600 align:middle line:84% gets into a state where you sort of manually 00:03:51.600 --> 00:03:54.840 align:middle line:84% get the transistors to be in the right set of on and off 00:03:54.840 --> 00:03:56.850 align:middle line:90% voltages, 5 and 0 volts. 00:03:56.850 --> 00:03:59.460 align:middle line:84% Once you get the first compiler, you can now 00:03:59.460 --> 00:04:00.990 align:middle line:90% work at a much higher level. 00:04:00.990 --> 00:04:02.500 align:middle line:84% A compiler is basically something-- 00:04:02.500 --> 00:04:04.500 align:middle line:84% all this code that I've been showing you so far, 00:04:04.500 --> 00:04:07.230 align:middle line:84% the Perl, the Fortran 77, the Mathematica, so forth, 00:04:07.230 --> 00:04:13.110 align:middle line:84% those are all things like, print x above 1. 00:04:13.110 --> 00:04:16.500 align:middle line:90% That is like a compiler. 00:04:16.500 --> 00:04:18.510 align:middle line:84% It's a code which you can type in. 00:04:18.510 --> 00:04:20.579 align:middle line:90% It's almost English. 00:04:20.579 --> 00:04:24.510 align:middle line:84% Some of you may say it's not nearly close enough to English. 00:04:24.510 --> 00:04:26.070 align:middle line:84% But it's much closer than dealing 00:04:26.070 --> 00:04:28.110 align:middle line:90% with these little voltages, OK. 00:04:28.110 --> 00:04:31.170 align:middle line:84% And once you have one compiler, you can make another one. 00:04:31.170 --> 00:04:33.930 align:middle line:84% You can use that pseudo English to write 00:04:33.930 --> 00:04:35.580 align:middle line:84% a more complicated compiler that can 00:04:35.580 --> 00:04:38.347 align:middle line:84% deal with a even higher-level language, and then reduce it. 00:04:38.347 --> 00:04:39.930 align:middle line:84% Then it takes care of the bookkeeping, 00:04:39.930 --> 00:04:42.750 align:middle line:84% or reduces it down to telling the computer what 00:04:42.750 --> 00:04:45.930 align:middle line:90% voltages to put where and when. 00:04:45.930 --> 00:04:47.460 align:middle line:84% As you go up the still higher ones, 00:04:47.460 --> 00:04:49.920 align:middle line:84% we have these high level application programs, 00:04:49.920 --> 00:04:53.760 align:middle line:84% which might have intense graphics or something 00:04:53.760 --> 00:04:55.650 align:middle line:84% like that, OK, so that you really 00:04:55.650 --> 00:05:00.690 align:middle line:84% get a world view that is much more in resonance 00:05:00.690 --> 00:05:06.120 align:middle line:84% with our primate visual and auditory senses. 00:05:06.120 --> 00:05:10.770 align:middle line:84% Now, this idea of self-compiling and self-assembling 00:05:10.770 --> 00:05:12.840 align:middle line:84% is very interesting, very self-referential, 00:05:12.840 --> 00:05:16.110 align:middle line:84% as many of the things in this course will be. 00:05:16.110 --> 00:05:21.540 align:middle line:84% We have biological components which have these very 00:05:21.540 --> 00:05:24.210 align:middle line:84% interesting complementary surfaces that I talked about 00:05:24.210 --> 00:05:29.203 align:middle line:84% a couple of slides ago, where the two strands of DNA which 00:05:29.203 --> 00:05:31.620 align:middle line:84% are not covalently linked-- covalently means that you have 00:05:31.620 --> 00:05:35.520 align:middle line:84% these strong bonds along this one ribbon-- 00:05:35.520 --> 00:05:37.770 align:middle line:84% and connected by a series of stacking 00:05:37.770 --> 00:05:42.090 align:middle line:84% interactions, where plates, the sort of planar bases stack up. 00:05:42.090 --> 00:05:46.230 align:middle line:84% And they form weak bonds from one strand to the other 00:05:46.230 --> 00:05:48.220 align:middle line:84% where the rules I mentioned before-- 00:05:48.220 --> 00:05:51.720 align:middle line:84% now slightly more realistic symbols here for the GC and AT 00:05:51.720 --> 00:05:55.050 align:middle line:84% base pairs rather than just the alphabetics. 00:05:55.050 --> 00:05:56.310 align:middle line:90% But this is still symbols. 00:05:56.310 --> 00:05:57.810 align:middle line:84% This is not really electron density, 00:05:57.810 --> 00:05:59.893 align:middle line:84% where you're using letters instead of the electron 00:05:59.893 --> 00:06:03.210 align:middle line:84% density of nitrogens, carbons, oxygens, hydrogens. 00:06:03.210 --> 00:06:06.630 align:middle line:84% But these hydrogens make this hydrogen bond a weak bond, 00:06:06.630 --> 00:06:09.370 align:middle line:84% and the surfaces are complementary, 00:06:09.370 --> 00:06:11.610 align:middle line:84% so that if you try to pair a C with a T, 00:06:11.610 --> 00:06:13.200 align:middle line:90% you have the wrong spacing. 00:06:13.200 --> 00:06:18.750 align:middle line:84% You have steric clashes and so on. 00:06:18.750 --> 00:06:24.390 align:middle line:84% This process by which one sequence will not 00:06:24.390 --> 00:06:26.970 align:middle line:84% make the same sequence right away-- replication 00:06:26.970 --> 00:06:28.690 align:middle line:84% does not make an identical sequence. 00:06:28.690 --> 00:06:31.380 align:middle line:84% It actually makes a complementary surface. 00:06:31.380 --> 00:06:34.370 align:middle line:84% And then one more cycle, and you get back to the original, 00:06:34.370 --> 00:06:36.360 align:middle line:84% just like that example, I gave before 00:06:36.360 --> 00:06:40.350 align:middle line:84% of the trinucleotide going to a hexanucleotide. 00:06:40.350 --> 00:06:42.360 align:middle line:84% And then that helps the second round. 00:06:42.360 --> 00:06:45.840 align:middle line:84% This is very analogous, except now these molecules, 00:06:45.840 --> 00:06:47.760 align:middle line:84% instead of being six nucleotides long, 00:06:47.760 --> 00:06:52.200 align:middle line:84% they can be hundreds of millions of nucleotides long. 00:06:52.200 --> 00:06:59.400 align:middle line:84% So this introduction to minimal life, 00:06:59.400 --> 00:07:03.600 align:middle line:84% all these life have in common self-assembly, catalysis, 00:07:03.600 --> 00:07:06.750 align:middle line:84% replication, mutation and selection. 00:07:06.750 --> 00:07:09.780 align:middle line:84% The monomers, meaning the simple molecules 00:07:09.780 --> 00:07:13.230 align:middle line:84% taken from the environment, the environment being defined 00:07:13.230 --> 00:07:16.560 align:middle line:84% by some boundary-- this could be a fairly flexible boundary. 00:07:16.560 --> 00:07:19.110 align:middle line:84% It could be a membranous boundary. 00:07:19.110 --> 00:07:21.390 align:middle line:84% It could just be some kind of aggregate. 00:07:21.390 --> 00:07:26.400 align:middle line:84% The small, simple monomers, these 00:07:26.400 --> 00:07:29.610 align:middle line:84% combine to make complicated polymers. 00:07:29.610 --> 00:07:32.670 align:middle line:84% And then this replication process continues. 00:07:32.670 --> 00:07:35.430 align:middle line:84% When we go to more complicated systems, 00:07:35.430 --> 00:07:37.080 align:middle line:90% now this is a central dogma. 00:07:37.080 --> 00:07:41.100 align:middle line:84% The DNA, the long-term stable storage, 00:07:41.100 --> 00:07:45.300 align:middle line:84% is accessed by making ephemeral RNA, which 00:07:45.300 --> 00:07:47.370 align:middle line:90% then encodes proteins. 00:07:47.370 --> 00:07:50.370 align:middle line:84% And in some systems, RNA can replicate. 00:07:50.370 --> 00:07:52.250 align:middle line:84% RNA can be reverse-transcribed to DNA. 00:07:52.250 --> 00:07:54.750 align:middle line:84% Unfortunately, the AIDS virus is one of them that does this. 00:07:54.750 --> 00:07:58.260 align:middle line:84% DNA can replicate itself, and certain pathological cases, 00:07:58.260 --> 00:08:00.810 align:middle line:84% or actually certain cases, I should say, of proteins, 00:08:00.810 --> 00:08:06.030 align:middle line:84% can in an autocatalytic cycle recruit other proteins 00:08:06.030 --> 00:08:07.410 align:middle line:90% to their particular state. 00:08:07.410 --> 00:08:09.720 align:middle line:84% These are called prions, and they're 00:08:09.720 --> 00:08:12.010 align:middle line:84% based in Mad Cow disease and things like that. 00:08:12.010 --> 00:08:16.950 align:middle line:84% But again, you've got a boundary for the replication, 00:08:16.950 --> 00:08:20.850 align:middle line:84% and simple structures come in and complicated structures 00:08:20.850 --> 00:08:21.570 align:middle line:90% are generated. 00:08:21.570 --> 00:08:24.570 align:middle line:84% When we talk about the polymers, we 00:08:24.570 --> 00:08:28.230 align:middle line:84% want to quantitate their amounts in their interactions. 00:08:28.230 --> 00:08:30.540 align:middle line:84% They initiate, elongate, terminate. 00:08:30.540 --> 00:08:31.230 align:middle line:90% They fold. 00:08:31.230 --> 00:08:33.549 align:middle line:84% They get modified in various ways. 00:08:33.549 --> 00:08:36.929 align:middle line:84% So it's not just simple linear polymers. 00:08:36.929 --> 00:08:38.909 align:middle line:84% Their position in space is important, 00:08:38.909 --> 00:08:41.070 align:middle line:84% and they are either degraded or diluted 00:08:41.070 --> 00:08:43.289 align:middle line:90% during the replication process. 00:08:43.289 --> 00:08:44.970 align:middle line:84% This gets even a little more complicated 00:08:44.970 --> 00:08:47.220 align:middle line:84% when we talk about functional genomics. 00:08:47.220 --> 00:08:51.690 align:middle line:84% Here, we measure the growth rate. 00:08:51.690 --> 00:08:54.570 align:middle line:84% We measure the concentration of RNA and proteins. 00:08:54.570 --> 00:08:56.440 align:middle line:84% Sometimes, their localization, it's 00:08:56.440 --> 00:08:57.690 align:middle line:90% important to measure that too. 00:08:57.690 --> 00:08:59.550 align:middle line:90% That's called gene expression. 00:08:59.550 --> 00:09:01.860 align:middle line:84% And the interactions are important to be part 00:09:01.860 --> 00:09:03.270 align:middle line:90% of this measurement process. 00:09:03.270 --> 00:09:05.580 align:middle line:84% These measures and models that I'm talking about 00:09:05.580 --> 00:09:10.990 align:middle line:84% are how we get to defining enough about living systems 00:09:10.990 --> 00:09:12.010 align:middle line:90% that we can model. 00:09:12.010 --> 00:09:16.190 align:middle line:90% So here is another model. 00:09:16.190 --> 00:09:18.560 align:middle line:84% This is a Rorschach test as you might take. 00:09:18.560 --> 00:09:22.360 align:middle line:84% You go into a psychologist's office maybe long ago, 00:09:22.360 --> 00:09:24.910 align:middle line:84% and they'd say, look at this inkblot. 00:09:24.910 --> 00:09:26.905 align:middle line:90% What does it remind you of? 00:09:26.905 --> 00:09:30.730 align:middle line:84% You know, bad things your father did, so forth. 00:09:30.730 --> 00:09:33.260 align:middle line:84% Here, the Rorschach test is, what does this curve remind you 00:09:33.260 --> 00:09:33.760 align:middle line:90% of. 00:09:33.760 --> 00:09:37.700 align:middle line:90% 00:09:37.700 --> 00:09:39.358 align:middle line:90% Give me some hints. 00:09:39.358 --> 00:09:41.143 align:middle line:90% STUDENT: Exponential curve. 00:09:41.143 --> 00:09:42.810 align:middle line:84% GEORGE CHURCH: Exponential curve, great. 00:09:42.810 --> 00:09:45.870 align:middle line:84% And how do you get-- and this is like the stock crisis 00:09:45.870 --> 00:09:49.350 align:middle line:90% before a dotcom crash. 00:09:49.350 --> 00:09:51.328 align:middle line:90% How do you get this? 00:09:51.328 --> 00:09:53.370 align:middle line:84% How do you find it in biology for the biologists, 00:09:53.370 --> 00:09:55.453 align:middle line:84% or how do you find it mathematically for the rest? 00:09:55.453 --> 00:09:58.040 align:middle line:90% 00:09:58.040 --> 00:09:58.946 align:middle line:90% Yeah. 00:09:58.946 --> 00:10:00.890 align:middle line:84% STUDENT: It kind of reminds me of human population growth. 00:10:00.890 --> 00:10:02.515 align:middle line:84% GEORGE CHURCH: Human population growth. 00:10:02.515 --> 00:10:05.360 align:middle line:84% Yeah, that's a good biological example. 00:10:05.360 --> 00:10:07.770 align:middle line:84% Maybe, and Malthus and others have said, 00:10:07.770 --> 00:10:08.850 align:middle line:90% this can't go on forever. 00:10:08.850 --> 00:10:12.470 align:middle line:90% But this is not a fact. 00:10:12.470 --> 00:10:15.890 align:middle line:90% It's a pretty solid speculation. 00:10:15.890 --> 00:10:18.860 align:middle line:90% So how do you get this? 00:10:18.860 --> 00:10:20.240 align:middle line:84% Well, for those of you who prefer 00:10:20.240 --> 00:10:24.050 align:middle line:84% to see or expect to see stocks go down as well as up, 00:10:24.050 --> 00:10:27.050 align:middle line:84% we've got an exponential decay curve in magenta. 00:10:27.050 --> 00:10:30.290 align:middle line:84% It sort of is a reflection of the exponential growth curve. 00:10:30.290 --> 00:10:34.370 align:middle line:84% These are just e to the kt, or e to the minus-- 00:10:34.370 --> 00:10:36.590 align:middle line:90% or where k is positive. 00:10:36.590 --> 00:10:40.010 align:middle line:84% And this is the world's simplest differentiable equation. 00:10:40.010 --> 00:10:44.720 align:middle line:84% y, the y-axis, is a function of time, the horizontal axis. 00:10:44.720 --> 00:10:52.370 align:middle line:84% And the small changes in y with small changes in t, time, t, 00:10:52.370 --> 00:10:53.600 align:middle line:90% dydt-- 00:10:53.600 --> 00:10:57.530 align:middle line:84% that's sort of the slope of the y as it goes up, 00:10:57.530 --> 00:10:59.090 align:middle line:90% the blue exponential curve-- 00:10:59.090 --> 00:11:01.340 align:middle line:90% is related to how much y. 00:11:01.340 --> 00:11:08.120 align:middle line:84% The more humans are, the faster the human race replicates, OK. 00:11:08.120 --> 00:11:09.890 align:middle line:84% And it just keeps getting more and more. 00:11:09.890 --> 00:11:12.110 align:middle line:84% And that's why it has this exponential curve. 00:11:12.110 --> 00:11:14.860 align:middle line:84% This is much steeper than quadratic. 00:11:14.860 --> 00:11:17.820 align:middle line:84% And its origins are way back here. 00:11:17.820 --> 00:11:19.370 align:middle line:84% It's similar to exponential decay, 00:11:19.370 --> 00:11:21.245 align:middle line:84% as you might get with radioactive substances. 00:11:21.245 --> 00:11:25.370 align:middle line:90% It follows a reverse process. 00:11:25.370 --> 00:11:31.202 align:middle line:84% And if you integrate this, you get a very simple integral. 00:11:31.202 --> 00:11:32.660 align:middle line:84% It's what we've been talking about. 00:11:32.660 --> 00:11:34.460 align:middle line:90% It's e to the kt. 00:11:34.460 --> 00:11:38.330 align:middle line:84% So it's an exponential function of time, y, say 00:11:38.330 --> 00:11:40.490 align:middle line:84% the human population or your stocks, 00:11:40.490 --> 00:11:44.955 align:middle line:84% where e is this number that we highlighted before, about 2.7. 00:11:44.955 --> 00:11:47.330 align:middle line:84% If you're interested in half life, which sometimes people 00:11:47.330 --> 00:11:50.480 align:middle line:84% are, like radioactive decay or half-life of replication 00:11:50.480 --> 00:11:54.140 align:middle line:84% of bacteria in a solution, it's a very simple formula 00:11:54.140 --> 00:11:56.330 align:middle line:84% that gets you from the rate constant k. 00:11:56.330 --> 00:12:02.600 align:middle line:84% This is like a biochemical rate constant to a half-life. 00:12:02.600 --> 00:12:04.550 align:middle line:90% This is growth and decay. 00:12:04.550 --> 00:12:05.530 align:middle line:90% So what limits this? 00:12:05.530 --> 00:12:09.430 align:middle line:84% Why doesn't it just keep going up? 00:12:09.430 --> 00:12:13.060 align:middle line:84% What we've been looking at is the lower left-hand corner 00:12:13.060 --> 00:12:16.930 align:middle line:84% of this graph in slide 29, where it goes exponentially 00:12:16.930 --> 00:12:19.960 align:middle line:90% up from close to 0, not quite. 00:12:19.960 --> 00:12:23.440 align:middle line:84% And eventually, it will plateau, or worse yet, it 00:12:23.440 --> 00:12:24.700 align:middle line:90% might come down. 00:12:24.700 --> 00:12:27.382 align:middle line:84% And what causes this plateau is exhaustion of resources. 00:12:27.382 --> 00:12:29.590 align:middle line:84% And if you get enough accumulation of waste products, 00:12:29.590 --> 00:12:31.060 align:middle line:84% or enough exhaustion of resources. 00:12:31.060 --> 00:12:32.860 align:middle line:90% You can plummet. 00:12:32.860 --> 00:12:35.140 align:middle line:84% If you just zoom in on this little part here, 00:12:35.140 --> 00:12:40.360 align:middle line:84% one way of analyzing it, very hopefully known to some of you, 00:12:40.360 --> 00:12:45.010 align:middle line:84% is that you take the logarithm of y and plot it versus t. 00:12:45.010 --> 00:12:49.960 align:middle line:84% So t is linear axis, and the vertical axis is logarithmic. 00:12:49.960 --> 00:12:52.600 align:middle line:84% Now you get a straight line, at least for the beginning here. 00:12:52.600 --> 00:12:56.155 align:middle line:84% And eventually it will plateau the same as this one does. 00:12:56.155 --> 00:12:58.780 align:middle line:84% And that's a way of telling that you have a simple exponential. 00:12:58.780 --> 00:13:02.110 align:middle line:84% If you have e to the t power, or 2 to the t power, 00:13:02.110 --> 00:13:04.563 align:middle line:84% or anything to the t power, simple. 00:13:04.563 --> 00:13:05.980 align:middle line:84% Those are all simple exponentials, 00:13:05.980 --> 00:13:09.590 align:middle line:84% and they'll give a line when you take the logarithm. 00:13:09.590 --> 00:13:13.750 align:middle line:84% Now, what does Mathematica do to help us here? 00:13:13.750 --> 00:13:15.820 align:middle line:90% You set up this equation. 00:13:15.820 --> 00:13:18.443 align:middle line:84% Instead of saying dydt, you could say, y prime of t. 00:13:18.443 --> 00:13:19.360 align:middle line:90% That's just shorthand. 00:13:19.360 --> 00:13:22.240 align:middle line:84% It's very commonly used in calculus. 00:13:22.240 --> 00:13:25.240 align:middle line:84% y prime of t the first derivative of t, 00:13:25.240 --> 00:13:30.700 align:middle line:84% of y with respect to t, is directly proportional to y. 00:13:30.700 --> 00:13:34.840 align:middle line:84% That is to say, your slope of the human race expanding 00:13:34.840 --> 00:13:38.330 align:middle line:84% is directly proportional to y, the number of humans, OK. 00:13:38.330 --> 00:13:40.570 align:middle line:84% And then you're going to start at time equals 0. 00:13:40.570 --> 00:13:42.555 align:middle line:90% We've got one human. 00:13:42.555 --> 00:13:43.930 align:middle line:84% Well, that's probably not enough. 00:13:43.930 --> 00:13:44.980 align:middle line:90% Well, OK. 00:13:44.980 --> 00:13:45.970 align:middle line:90% Maybe a bacteria. 00:13:45.970 --> 00:13:48.520 align:middle line:90% OK. 00:13:48.520 --> 00:13:51.130 align:middle line:90% You have initial conditions, OK. 00:13:51.130 --> 00:13:55.600 align:middle line:84% And then so you just say solve it. 00:13:55.600 --> 00:13:57.430 align:middle line:84% You tell the computer, so everything 00:13:57.430 --> 00:14:00.040 align:middle line:84% to the right of this equation is something 00:14:00.040 --> 00:14:02.470 align:middle line:90% you can type in to Mathematica. 00:14:02.470 --> 00:14:05.740 align:middle line:84% Do a differentiable equation solve of this string 00:14:05.740 --> 00:14:07.480 align:middle line:84% that I typed in here, which tells you 00:14:07.480 --> 00:14:09.670 align:middle line:84% the initial conditions and the formula. 00:14:09.670 --> 00:14:11.860 align:middle line:90% And boom, out comes the out. 00:14:11.860 --> 00:14:12.880 align:middle line:90% You didn't type this in. 00:14:12.880 --> 00:14:15.580 align:middle line:84% Mathematica came up with this, e to the t power. 00:14:15.580 --> 00:14:16.570 align:middle line:90% That's pretty cool. 00:14:16.570 --> 00:14:18.070 align:middle line:84% And even though it's the world's simplest differentiable 00:14:18.070 --> 00:14:19.630 align:middle line:90% equation, it solved it. 00:14:19.630 --> 00:14:22.030 align:middle line:84% Try to do that in your other favorite programming 00:14:22.030 --> 00:14:28.720 align:middle line:84% languages, Excel, or Fortran, or Perl, or Python or whatever, C. 00:14:28.720 --> 00:14:30.730 align:middle line:90% This is really powerful. 00:14:30.730 --> 00:14:34.060 align:middle line:84% Now, this is analytic or symbolic or formal. 00:14:34.060 --> 00:14:36.970 align:middle line:84% These are various terms you would say for this trick. 00:14:36.970 --> 00:14:39.160 align:middle line:84% And as the equations get more and more complicated, 00:14:39.160 --> 00:14:42.185 align:middle line:84% this becomes more and more amazing, almost intelligent. 00:14:42.185 --> 00:14:43.810 align:middle line:84% Eventually, they get complicated enough 00:14:43.810 --> 00:14:47.270 align:middle line:84% that neither humans nor Mathematica can solve them. 00:14:47.270 --> 00:14:50.650 align:middle line:84% And so what you do then is use a numerical approximation 00:14:50.650 --> 00:14:56.590 align:middle line:84% where you take little steps and you solve it 00:14:56.590 --> 00:14:58.960 align:middle line:90% by numeric approximation. 00:14:58.960 --> 00:15:00.580 align:middle line:90% But you set it up the same way. 00:15:00.580 --> 00:15:03.580 align:middle line:84% You tell it that the derivative y with respect to t 00:15:03.580 --> 00:15:05.140 align:middle line:90% is proportional to y. 00:15:05.140 --> 00:15:07.960 align:middle line:84% Or in this case, proportionality counts as one. 00:15:07.960 --> 00:15:11.920 align:middle line:84% Same initial conditions, one starting bacterium. 00:15:11.920 --> 00:15:14.350 align:middle line:84% But now you tell it what interval you want to do this. 00:15:14.350 --> 00:15:16.850 align:middle line:84% You don't want it to have to do these little steps all over, 00:15:16.850 --> 00:15:19.690 align:middle line:84% all negative deposit of infinity for time. 00:15:19.690 --> 00:15:22.090 align:middle line:84% You just want to say, I'm just interested in time from 0 00:15:22.090 --> 00:15:26.290 align:middle line:84% to 3 minutes or hours or whatever years, OK. 00:15:26.290 --> 00:15:28.240 align:middle line:84% And then you evaluate it, and you 00:15:28.240 --> 00:15:32.620 align:middle line:84% can plot this, which appeals to the primate visual system, 00:15:32.620 --> 00:15:33.580 align:middle line:90% these plots. 00:15:33.580 --> 00:15:35.440 align:middle line:84% And you'll see lots of plots in this course. 00:15:35.440 --> 00:15:39.680 align:middle line:84% But now here, y, a function of t, is this exponential curve. 00:15:39.680 --> 00:15:41.620 align:middle line:84% And if we plotted log of y as a function of t 00:15:41.620 --> 00:15:46.363 align:middle line:84% with this numerical solver, it would be a straight line. 00:15:46.363 --> 00:15:48.530 align:middle line:84% Now, I give you some where it isn't a straight line. 00:15:48.530 --> 00:15:51.220 align:middle line:84% These are all logarithmic on the y-axis, 00:15:51.220 --> 00:15:55.105 align:middle line:84% and they're all linear on the horizontal axis. 00:15:55.105 --> 00:15:58.780 align:middle line:84% And they're all time on the horizontal axis, linear time. 00:15:58.780 --> 00:16:01.900 align:middle line:84% And they're more than simple exponentials. 00:16:01.900 --> 00:16:03.190 align:middle line:90% They're going up faster. 00:16:03.190 --> 00:16:04.870 align:middle line:84% Rather than going out slower, which 00:16:04.870 --> 00:16:07.520 align:middle line:84% we think human population, bacterial population 00:16:07.520 --> 00:16:11.260 align:middle line:84% and so forth, they'll go up linear on a log plot 00:16:11.260 --> 00:16:13.090 align:middle line:84% and then flatten out, these things 00:16:13.090 --> 00:16:14.690 align:middle line:84% are just going faster and faster. 00:16:14.690 --> 00:16:15.950 align:middle line:90% What are these things? 00:16:15.950 --> 00:16:17.920 align:middle line:84% Well, even though your dotcoms didn't work out, 00:16:17.920 --> 00:16:21.730 align:middle line:84% if you had had a stock portfolio in Western European commerce 00:16:21.730 --> 00:16:25.060 align:middle line:84% in the year 1000, you'd be in really good shape 00:16:25.060 --> 00:16:26.380 align:middle line:90% now in the year 2000. 00:16:26.380 --> 00:16:28.060 align:middle line:84% This is not only going exponential, 00:16:28.060 --> 00:16:30.100 align:middle line:84% it's going steeper than exponential. 00:16:30.100 --> 00:16:32.575 align:middle line:84% And we all hope that this will keep going forever, 00:16:32.575 --> 00:16:35.290 align:middle line:84% that the gross domestic product of people 00:16:35.290 --> 00:16:37.850 align:middle line:84% in Western Europe and the world will just keep going up. 00:16:37.850 --> 00:16:39.100 align:middle line:90% And this is due to technology. 00:16:39.100 --> 00:16:41.720 align:middle line:84% And technology keeps reinventing itself. 00:16:41.720 --> 00:16:44.060 align:middle line:84% And hopefully, it can keep doing that. 00:16:44.060 --> 00:16:45.230 align:middle line:90% Here's another example . 00:16:45.230 --> 00:16:47.620 align:middle line:84% This is more drilling down to specific technologies 00:16:47.620 --> 00:16:50.225 align:middle line:84% that have been on a superexponential 00:16:50.225 --> 00:16:51.100 align:middle line:90% or hyperexponential-- 00:16:51.100 --> 00:16:53.230 align:middle line:84% I don't know quite what the right term here is-- 00:16:53.230 --> 00:16:54.200 align:middle line:90% for a long time. 00:16:54.200 --> 00:16:57.580 align:middle line:84% These are greater than linear, steeper than linear. 00:16:57.580 --> 00:17:01.000 align:middle line:90% These are close to quadratic. 00:17:01.000 --> 00:17:04.980 align:middle line:84% And so it's an exponent of a quadratic. 00:17:04.980 --> 00:17:07.190 align:middle line:84% And these are for transmission rate in pink, 00:17:07.190 --> 00:17:12.680 align:middle line:84% of data from the Morse code in the 1830s to optical fibers 00:17:12.680 --> 00:17:14.030 align:middle line:90% here in the present. 00:17:14.030 --> 00:17:18.829 align:middle line:84% And then the blue are digital processing 00:17:18.829 --> 00:17:24.859 align:middle line:84% from the first census in the 1890s to modern computers. 00:17:24.859 --> 00:17:30.230 align:middle line:84% And this is in instructions per second per $1,000. 00:17:30.230 --> 00:17:34.160 align:middle line:84% Now, this unit, the little piece of this, these integrated 00:17:34.160 --> 00:17:36.320 align:middle line:84% circuits that Moore's law refers to from just 00:17:36.320 --> 00:17:38.700 align:middle line:84% the tiny end of this from 1965 onward, 00:17:38.700 --> 00:17:39.950 align:middle line:90% refers to integrated circuits. 00:17:39.950 --> 00:17:41.390 align:middle line:84% And these will run out of gas pretty soon, 00:17:41.390 --> 00:17:42.380 align:middle line:90% everybody tells us. 00:17:42.380 --> 00:17:45.470 align:middle line:84% But this curve may not, because it goes beyond it. 00:17:45.470 --> 00:17:49.700 align:middle line:84% It predates integrated circuits and it will post date them. 00:17:49.700 --> 00:17:52.452 align:middle line:90% And who knows where this leads. 00:17:52.452 --> 00:17:53.307 align:middle line:90% STUDENT: Question. 00:17:53.307 --> 00:17:54.140 align:middle line:90% GEORGE CHURCH: Yeah. 00:17:54.140 --> 00:17:55.528 align:middle line:90% STUDENT: What's the r squared? 00:17:55.528 --> 00:17:56.570 align:middle line:90% GEORGE CHURCH: Oh, sorry. 00:17:56.570 --> 00:17:58.790 align:middle line:84% We'll get to that at the end of this lecture, 00:17:58.790 --> 00:18:00.900 align:middle line:84% but it's a correlation coefficient, 00:18:00.900 --> 00:18:06.170 align:middle line:84% which is, it's to what extent is there a fit between one 00:18:06.170 --> 00:18:06.920 align:middle line:90% curve and another. 00:18:06.920 --> 00:18:08.480 align:middle line:84% How well does the calculated curve 00:18:08.480 --> 00:18:12.090 align:middle line:90% fit the observed data collected? 00:18:12.090 --> 00:18:16.400 align:middle line:84% And so these are around 99, which is very good correlation. 00:18:16.400 --> 00:18:18.290 align:middle line:84% And it's better than the linear plot, 00:18:18.290 --> 00:18:22.270 align:middle line:84% but of course, you have more adjustable parameters. 00:18:22.270 --> 00:18:22.770 align:middle line:90% OK. 00:18:22.770 --> 00:18:27.480 align:middle line:84% Another sign of hopefulness is data are coming in faster. 00:18:27.480 --> 00:18:30.770 align:middle line:90% So our life is getting better. 00:18:30.770 --> 00:18:32.720 align:middle line:84% Our computers are getting faster, 00:18:32.720 --> 00:18:36.565 align:middle line:84% and data is coming in from the Genome Project. 00:18:36.565 --> 00:18:37.940 align:middle line:84% And this little inflection point, 00:18:37.940 --> 00:18:40.110 align:middle line:84% where it was log linear for a while 00:18:40.110 --> 00:18:43.160 align:middle line:84% and then a new log linear-- so overall, it's superexponential. 00:18:43.160 --> 00:18:45.290 align:middle line:84% And this is for the number of base pairs 00:18:45.290 --> 00:18:47.540 align:middle line:84% we can get per dollar, starting with transfer 00:18:47.540 --> 00:18:51.590 align:middle line:84% RNAs in the late '60s and ending with who 00:18:51.590 --> 00:18:54.860 align:middle line:84% knows how many human genomes that we will have by the year 00:18:54.860 --> 00:18:58.010 align:middle line:90% 2010. 00:18:58.010 --> 00:19:01.393 align:middle line:84% Now, where does this all this exponential growth go? 00:19:01.393 --> 00:19:03.560 align:middle line:84% Some people think we will be creating computers that 00:19:03.560 --> 00:19:05.270 align:middle line:90% are smarter than we are soon. 00:19:05.270 --> 00:19:06.710 align:middle line:90% What would this require? 00:19:06.710 --> 00:19:10.520 align:middle line:84% Here's a nice back of the envelope example of systems 00:19:10.520 --> 00:19:13.490 align:middle line:84% analysis where biology meets computers. 00:19:13.490 --> 00:19:16.310 align:middle line:90% Let's analyze our retina. 00:19:16.310 --> 00:19:18.890 align:middle line:84% All of our retinas are processing right now, 00:19:18.890 --> 00:19:20.870 align:middle line:90% hopefully. 00:19:20.870 --> 00:19:30.310 align:middle line:84% And Hans Moravec simulated a retina for video imaging 00:19:30.310 --> 00:19:32.590 align:middle line:84% where he did edge and motion detection, 00:19:32.590 --> 00:19:39.370 align:middle line:84% and it required about a billion instructions per second 00:19:39.370 --> 00:19:41.860 align:middle line:84% to match the 10 times per second which 00:19:41.860 --> 00:19:43.600 align:middle line:90% you're updating the retina. 00:19:43.600 --> 00:19:47.660 align:middle line:84% The brain is about 100,000 times larger than the retina, 00:19:47.660 --> 00:19:50.540 align:middle line:84% and if this scales linearly, which is speculative, 00:19:50.540 --> 00:19:57.460 align:middle line:84% then you need a computer that has about 100 million MIPs 00:19:57.460 --> 00:20:04.760 align:middle line:84% or about 10 to the 14 instructions per second 00:20:04.760 --> 00:20:08.330 align:middle line:84% of compute power, and a similar number of bytes. 00:20:08.330 --> 00:20:10.950 align:middle line:84% Now, back in 1998, that was still quite a ways away. 00:20:10.950 --> 00:20:14.480 align:middle line:84% But here in 2002, the best supercomputer-- and this site 00:20:14.480 --> 00:20:17.570 align:middle line:84% keeps tract of the top 500 supercomputers. 00:20:17.570 --> 00:20:20.970 align:middle line:84% And trust me, your computer is not on that list. 00:20:20.970 --> 00:20:23.510 align:middle line:84% But anyway, the top one is within a factor 00:20:23.510 --> 00:20:25.520 align:middle line:90% of 10 of this compute power. 00:20:25.520 --> 00:20:29.690 align:middle line:84% Now probably, Earth's scientists that own this thing 00:20:29.690 --> 00:20:35.240 align:middle line:84% will not bother to try to see if it can do ordinary human things 00:20:35.240 --> 00:20:36.470 align:middle line:90% like watching soap operas. 00:20:36.470 --> 00:20:39.890 align:middle line:90% But we're in that range. 00:20:39.890 --> 00:20:43.350 align:middle line:84% We need to be cognizant of the possibility. 00:20:43.350 --> 00:20:44.510 align:middle line:90% Here's another model. 00:20:44.510 --> 00:20:47.177 align:middle line:84% I've tried to put it in the same units we've been talking about, 00:20:47.177 --> 00:20:48.710 align:middle line:90% this exponential growth. 00:20:48.710 --> 00:20:50.900 align:middle line:84% Again, we have the rate constant k. 00:20:50.900 --> 00:20:54.260 align:middle line:84% We have the y, the human or bacterial population growing 00:20:54.260 --> 00:20:56.340 align:middle line:90% exponentially. 00:20:56.340 --> 00:21:02.350 align:middle line:84% And here now, we try to model the case where yes, 00:21:02.350 --> 00:21:04.840 align:middle line:84% as you have a great population size, 00:21:04.840 --> 00:21:07.780 align:middle line:84% it means greater growth until it gets 00:21:07.780 --> 00:21:12.880 align:middle line:84% close to the maximum carrying capacity, the 100%, the 1, 00:21:12.880 --> 00:21:14.170 align:middle line:90% the maximum it can go. 00:21:14.170 --> 00:21:15.730 align:middle line:90% And then it will plateau. 00:21:15.730 --> 00:21:22.660 align:middle line:90% So you have a plateauing near 1. 00:21:22.660 --> 00:21:25.030 align:middle line:84% And this is called the logistic map. 00:21:25.030 --> 00:21:27.160 align:middle line:84% It is the basis of that complexity calculation 00:21:27.160 --> 00:21:29.260 align:middle line:90% we talked about earlier. 00:21:29.260 --> 00:21:32.470 align:middle line:84% And here, the population size is a function of rate constant, 00:21:32.470 --> 00:21:37.900 align:middle line:84% and it has both grows when y is small. 00:21:37.900 --> 00:21:40.227 align:middle line:84% As y scales up, it goes up exponentially. 00:21:40.227 --> 00:21:42.310 align:middle line:84% And then finally, as it approaches a maximum of 1, 00:21:42.310 --> 00:21:43.510 align:middle line:90% it plateaus. 00:21:43.510 --> 00:21:46.660 align:middle line:84% However, if you get greedy, and you increase your growth rate 00:21:46.660 --> 00:21:48.520 align:middle line:90% beyond, say-- 00:21:48.520 --> 00:21:52.600 align:middle line:84% here, it's very, very small, very ungreedy, just 1.01. 00:21:52.600 --> 00:21:55.690 align:middle line:84% That's like a 1% interest in your bank account, OK. 00:21:55.690 --> 00:21:59.320 align:middle line:84% But still, you'd grow exponentially given 00:21:59.320 --> 00:22:01.383 align:middle line:90% enough time. 00:22:01.383 --> 00:22:02.800 align:middle line:84% However, you get greedy and say, I 00:22:02.800 --> 00:22:05.060 align:middle line:84% want a 300% return on my investment. 00:22:05.060 --> 00:22:07.060 align:middle line:84% Well, then you start getting these little cycles 00:22:07.060 --> 00:22:10.953 align:middle line:84% of like stock market going up and down, OK. 00:22:10.953 --> 00:22:12.370 align:middle line:84% And if you get really greedy where 00:22:12.370 --> 00:22:17.320 align:middle line:84% you need a 400% improvement each cycle, you get chaos. 00:22:17.320 --> 00:22:20.770 align:middle line:84% And then you can eventually drop down very close to 0 and crash, 00:22:20.770 --> 00:22:23.770 align:middle line:84% and the population can go extinct 00:22:23.770 --> 00:22:29.140 align:middle line:84% because it used up as resources or made non-optimal use 00:22:29.140 --> 00:22:32.260 align:middle line:90% and maybe toxic side products. 00:22:32.260 --> 00:22:34.730 align:middle line:90% OK, graphs. 00:22:34.730 --> 00:22:38.780 align:middle line:90% We have directed acyclic graphs. 00:22:38.780 --> 00:22:42.650 align:middle line:84% Just as an example, graphs are made up of nodes. 00:22:42.650 --> 00:22:45.050 align:middle line:84% You can think of these here as, the nodes 00:22:45.050 --> 00:22:47.570 align:middle line:90% are people or organisms. 00:22:47.570 --> 00:22:51.740 align:middle line:84% And you start with one bacterium here on the far left-hand side 00:22:51.740 --> 00:22:52.910 align:middle line:90% of slide 35. 00:22:52.910 --> 00:22:55.550 align:middle line:90% And you have a direction. 00:22:55.550 --> 00:22:57.830 align:middle line:90% You can only go forward in time. 00:22:57.830 --> 00:23:03.350 align:middle line:84% So the node is the bacterium individual, 00:23:03.350 --> 00:23:06.860 align:middle line:84% and the lines connecting them are 00:23:06.860 --> 00:23:09.770 align:middle line:90% edges in the graph terminology. 00:23:09.770 --> 00:23:10.910 align:middle line:90% And they're directed. 00:23:10.910 --> 00:23:12.860 align:middle line:84% And they can't go in cycles because you 00:23:12.860 --> 00:23:17.870 align:middle line:84% can't have a daughter giving rise to a mother, OK. 00:23:17.870 --> 00:23:19.385 align:middle line:84% So this all makes intuitive sense. 00:23:19.385 --> 00:23:22.430 align:middle line:90% 00:23:22.430 --> 00:23:25.310 align:middle line:84% But you can use these kind of graphs for a whole variety 00:23:25.310 --> 00:23:26.420 align:middle line:90% of interesting things. 00:23:26.420 --> 00:23:31.400 align:middle line:84% You have not just the pedigree we talked about, 00:23:31.400 --> 00:23:34.400 align:middle line:84% but phylogeny in general, ancient connections 00:23:34.400 --> 00:23:35.990 align:middle line:90% between organisms. 00:23:35.990 --> 00:23:39.440 align:middle line:84% The biopolymer backbone, you have a simple linear backbone 00:23:39.440 --> 00:23:40.970 align:middle line:90% or a branch backbone. 00:23:40.970 --> 00:23:42.830 align:middle line:90% This can be represented. 00:23:42.830 --> 00:23:47.450 align:middle line:84% It doesn't covalently cross back on itself. 00:23:47.450 --> 00:23:49.970 align:middle line:84% If you want to know what's near one another 00:23:49.970 --> 00:23:53.615 align:middle line:84% as this polymer folds up, like that transfer I showed you 00:23:53.615 --> 00:23:58.910 align:middle line:84% in the first slides, those contacts are indicated. 00:23:58.910 --> 00:24:02.460 align:middle line:84% Now you start getting cycles, because A can contact B, B, C, 00:24:02.460 --> 00:24:04.350 align:middle line:90% D, back to A again. 00:24:04.350 --> 00:24:07.760 align:middle line:84% You get cycles in a three-dimensional structure. 00:24:07.760 --> 00:24:09.710 align:middle line:84% You get cycles in a regulatory network. 00:24:09.710 --> 00:24:12.650 align:middle line:84% You can have, in order to maintain homeostasis 00:24:12.650 --> 00:24:17.210 align:middle line:84% in your body, A can regulate B. B can regulate C, and back 00:24:17.210 --> 00:24:18.930 align:middle line:90% to A again. 00:24:18.930 --> 00:24:23.000 align:middle line:90% But that's all directed. 00:24:23.000 --> 00:24:27.380 align:middle line:84% There are system models that we and others will study. 00:24:27.380 --> 00:24:29.000 align:middle line:90% They have in common-- 00:24:29.000 --> 00:24:31.430 align:middle line:84% this is slide 37 on the left-hand side-- 00:24:31.430 --> 00:24:33.920 align:middle line:90% the system models. 00:24:33.920 --> 00:24:36.755 align:middle line:84% And they've been chosen mainly because in the pre-genomic era, 00:24:36.755 --> 00:24:38.210 align:middle line:84% it was very hard to get data sets. 00:24:38.210 --> 00:24:40.400 align:middle line:84% Certain systems were just technically easier 00:24:40.400 --> 00:24:44.870 align:middle line:84% to get large data sets, genetic or biochemical. 00:24:44.870 --> 00:24:48.560 align:middle line:84% These include E. Coli, going toward food and away 00:24:48.560 --> 00:24:50.000 align:middle line:90% from toxins. 00:24:50.000 --> 00:24:53.150 align:middle line:84% Red blood cell is a nice metabolic system 00:24:53.150 --> 00:24:55.520 align:middle line:84% because it doesn't have any polymer synthesis. 00:24:55.520 --> 00:24:56.480 align:middle line:90% Makes it simpler. 00:24:56.480 --> 00:24:59.390 align:middle line:84% Cell division cycle is really key for understanding pathogen 00:24:59.390 --> 00:25:01.400 align:middle line:90% replication, cancer. 00:25:01.400 --> 00:25:09.065 align:middle line:84% Circadian rhythm, a huge number, many if not 00:25:09.065 --> 00:25:13.190 align:middle line:84% possibly all organisms, have some circadian rhythms 00:25:13.190 --> 00:25:18.740 align:middle line:84% that keep their biochemistry optimal, and keep us hopefully 00:25:18.740 --> 00:25:23.940 align:middle line:84% awake, right now, anyway, until it's time. 00:25:23.940 --> 00:25:24.440 align:middle line:90% OK. 00:25:24.440 --> 00:25:26.810 align:middle line:84% Plasmid DNA replication is an example 00:25:26.810 --> 00:25:28.020 align:middle line:90% of single-molecule precision. 00:25:28.020 --> 00:25:30.950 align:middle line:84% And we'll talk about the DNA single molecules 00:25:30.950 --> 00:25:31.790 align:middle line:90% in just a moment. 00:25:31.790 --> 00:25:35.510 align:middle line:90% 00:25:35.510 --> 00:25:38.170 align:middle line:84% So that's where we're aiming right now, 00:25:38.170 --> 00:25:40.960 align:middle line:84% is from graphs and pedigrees down to the single molecules 00:25:40.960 --> 00:25:44.260 align:middle line:90% that allow replication to work. 00:25:44.260 --> 00:25:47.950 align:middle line:84% This replication is achieved by interconnected machines 00:25:47.950 --> 00:25:49.700 align:middle line:90% they are somewhat modular. 00:25:49.700 --> 00:25:52.120 align:middle line:84% "Modular" is also a computer term 00:25:52.120 --> 00:25:54.130 align:middle line:84% where you try to put code that works together 00:25:54.130 --> 00:25:57.760 align:middle line:84% into something that's defined spatially and functionally. 00:25:57.760 --> 00:26:03.220 align:middle line:84% So you have these little modules that replicate the DNA, 00:26:03.220 --> 00:26:04.630 align:middle line:90% make RNA from it. 00:26:04.630 --> 00:26:06.670 align:middle line:84% A different module does the protein synthesis. 00:26:06.670 --> 00:26:08.543 align:middle line:84% There is some interconnection between these. 00:26:08.543 --> 00:26:10.960 align:middle line:84% This will be discussed. these kind of complicated machines 00:26:10.960 --> 00:26:14.110 align:middle line:84% that biologists love to simplify in diagrams 00:26:14.110 --> 00:26:19.010 align:middle line:84% will be described in more detail next week. 00:26:19.010 --> 00:26:21.850 align:middle line:84% But the idea is this idea of modules 00:26:21.850 --> 00:26:24.460 align:middle line:84% versus extensively coupled networks. 00:26:24.460 --> 00:26:28.560 align:middle line:84% This is how we get the replication. 00:26:28.560 --> 00:26:32.400 align:middle line:84% The way we analyze the replication is somewhere here 00:26:32.400 --> 00:26:36.540 align:middle line:84% in the middle, where I've had a scale here 00:26:36.540 --> 00:26:43.770 align:middle line:84% that goes from high resolution, very accurate descriptions 00:26:43.770 --> 00:26:46.770 align:middle line:84% of physical processes sort of in the nanometer 00:26:46.770 --> 00:26:51.090 align:middle line:84% femtosecond range on the far right-hand side of slide 40, 00:26:51.090 --> 00:26:54.030 align:middle line:84% to things that are very long timescale, 00:26:54.030 --> 00:26:56.460 align:middle line:84% very large scale sort of kilometers, 00:26:56.460 --> 00:26:59.160 align:middle line:84% years that happen in population dynamics, 00:26:59.160 --> 00:27:02.513 align:middle line:84% sometimes global population dynamics. 00:27:02.513 --> 00:27:04.680 align:middle line:84% You should understand that all of these models we'll 00:27:04.680 --> 00:27:06.900 align:middle line:84% be talking about are approximations. 00:27:06.900 --> 00:27:09.840 align:middle line:84% As we go down the scale, it gets more 00:27:09.840 --> 00:27:17.520 align:middle line:84% and more computable to compute more and more complicated 00:27:17.520 --> 00:27:19.830 align:middle line:84% things, but at the cost of greater 00:27:19.830 --> 00:27:22.350 align:middle line:90% and greater approximations. 00:27:22.350 --> 00:27:26.700 align:middle line:84% Even the molecular mechanics that we use in conjunction 00:27:26.700 --> 00:27:29.700 align:middle line:84% with crystallographic diffraction data 00:27:29.700 --> 00:27:33.733 align:middle line:84% is amazing computational chemistry, 00:27:33.733 --> 00:27:36.150 align:middle line:84% but it's a great approximation of quantum mechanics, which 00:27:36.150 --> 00:27:38.692 align:middle line:84% in turn is an approximation of quantum electrodynamics, which 00:27:38.692 --> 00:27:39.900 align:middle line:90% itself is an approximation. 00:27:39.900 --> 00:27:43.140 align:middle line:84% And all of these things are very hard to compute 00:27:43.140 --> 00:27:48.870 align:middle line:84% on any even reasonable atomic system multi-body problem. 00:27:48.870 --> 00:27:50.790 align:middle line:84% The big approximation for molecular mechanics 00:27:50.790 --> 00:27:52.930 align:middle line:84% is that you have spherical atoms, 00:27:52.930 --> 00:27:55.800 align:middle line:84% so you don't have the distortion of the dipole that 00:27:55.800 --> 00:27:58.110 align:middle line:84% occurs in very useful bonds, such as the hydrogen 00:27:58.110 --> 00:28:01.530 align:middle line:84% bonds and almost all non-bond interactions. 00:28:01.530 --> 00:28:03.810 align:middle line:84% That's poorly approximated, but it's 00:28:03.810 --> 00:28:06.840 align:middle line:84% the best we have that can be computed right now with most 00:28:06.840 --> 00:28:10.860 align:middle line:84% computers and even modestly large molecules. 00:28:10.860 --> 00:28:14.040 align:middle line:84% Then now, as we go down, we can think of it 00:28:14.040 --> 00:28:16.298 align:middle line:84% as higher and higher-level abstraction. 00:28:16.298 --> 00:28:18.090 align:middle line:84% Just like high-level programming languages, 00:28:18.090 --> 00:28:20.310 align:middle line:84% we're now programming chemical systems 00:28:20.310 --> 00:28:21.840 align:middle line:90% and thinking about them. 00:28:21.840 --> 00:28:23.940 align:middle line:84% Now instead of dealing with single atoms 00:28:23.940 --> 00:28:26.760 align:middle line:84% in molecular mechanics that produced the tRNA structure 00:28:26.760 --> 00:28:29.580 align:middle line:84% that I showed you, we now deal with that whole tRNA 00:28:29.580 --> 00:28:31.000 align:middle line:90% as a single molecule. 00:28:31.000 --> 00:28:34.200 align:middle line:84% But it's still a great depth of precision, 00:28:34.200 --> 00:28:35.845 align:middle line:84% because each molecule has its own life, 00:28:35.845 --> 00:28:37.470 align:middle line:84% and you track each one on the computer. 00:28:37.470 --> 00:28:40.410 align:middle line:84% And that's stochastic simulation. 00:28:40.410 --> 00:28:43.110 align:middle line:84% The next higher level beyond that is now, 00:28:43.110 --> 00:28:44.610 align:middle line:84% we don't deal with single molecules. 00:28:44.610 --> 00:28:46.590 align:middle line:84% We deal with populations of molecules, 00:28:46.590 --> 00:28:49.387 align:middle line:84% or we deal with a concentration as a function of time. 00:28:49.387 --> 00:28:51.720 align:middle line:84% The ordinary differentiable equations that we've already 00:28:51.720 --> 00:28:55.230 align:middle line:84% been talking about, like that exponential growth curve, 00:28:55.230 --> 00:28:59.430 align:middle line:84% is one way of dealing with concentration of bacteria 00:28:59.430 --> 00:29:01.390 align:middle line:90% as a function of time. 00:29:01.390 --> 00:29:02.940 align:middle line:90% That's appropriate. 00:29:02.940 --> 00:29:05.440 align:middle line:84% There are other cases where we want to do this optimization. 00:29:05.440 --> 00:29:08.880 align:middle line:84% We want to study how close to optimal a system is. 00:29:08.880 --> 00:29:14.250 align:middle line:84% One way to study that is with these economic functions, 00:29:14.250 --> 00:29:18.210 align:middle line:84% these linear programming, to look at the fluxes. 00:29:18.210 --> 00:29:20.730 align:middle line:84% Now, we're no longer talking about concentration and time, 00:29:20.730 --> 00:29:29.100 align:middle line:84% because we're interested in the rates through which chemicals 00:29:29.100 --> 00:29:33.300 align:middle line:84% are flowing through a biological system, where 00:29:33.300 --> 00:29:37.800 align:middle line:84% any particular chemical concentration is 00:29:37.800 --> 00:29:39.090 align:middle line:90% at a steady-state level. 00:29:39.090 --> 00:29:42.027 align:middle line:84% And that means you have things going in and things going out, 00:29:42.027 --> 00:29:43.860 align:middle line:84% but stuff in the middle is staying the same. 00:29:43.860 --> 00:29:47.910 align:middle line:84% That's a very useful approximation. 00:29:47.910 --> 00:29:48.960 align:middle line:90% It's used time and again. 00:29:48.960 --> 00:29:51.630 align:middle line:84% Even though these are dynamic systems, you can find them. 00:29:51.630 --> 00:29:55.110 align:middle line:84% In a pseudo-steady state, you can apply these very powerful 00:29:55.110 --> 00:29:56.100 align:middle line:90% computing tools. 00:29:56.100 --> 00:29:57.850 align:middle line:84% And then you can do computations that 00:29:57.850 --> 00:30:00.540 align:middle line:84% would be very hard to do in these more 00:30:00.540 --> 00:30:03.150 align:middle line:90% precise and complete methods. 00:30:03.150 --> 00:30:05.070 align:middle line:84% And we'll go through these in much more detail 00:30:05.070 --> 00:30:07.560 align:middle line:84% later on you'll find very interesting connections 00:30:07.560 --> 00:30:09.930 align:middle line:84% between the larger-scale things we're talking about, 00:30:09.930 --> 00:30:13.320 align:middle line:84% where we talk about stochastics of whole organisms 00:30:13.320 --> 00:30:18.174 align:middle line:84% in big populations, and the stochastic of single molecules. 00:30:18.174 --> 00:30:18.990 align:middle line:90% OK. 00:30:18.990 --> 00:30:20.640 align:middle line:84% We're talking about single molecules. 00:30:20.640 --> 00:30:22.560 align:middle line:90% This is our last topic today. 00:30:22.560 --> 00:30:25.860 align:middle line:84% And each of you do single-molecule manipulations 00:30:25.860 --> 00:30:30.270 align:middle line:84% on a regular basis, and your ancestors 00:30:30.270 --> 00:30:33.240 align:middle line:84% have been doing it for $10,000 years without a license, 00:30:33.240 --> 00:30:34.860 align:middle line:90% without a computer. 00:30:34.860 --> 00:30:37.590 align:middle line:84% And they've been doing a pretty good job of it. 00:30:37.590 --> 00:30:40.110 align:middle line:84% They've taken this little weedlike thing, teosinte, 00:30:40.110 --> 00:30:43.080 align:middle line:84% and turned it into this corn that would make the 4th of July 00:30:43.080 --> 00:30:44.130 align:middle line:90% quite proud. 00:30:44.130 --> 00:30:47.950 align:middle line:84% And dogs, who knows what their ancestors looked like. 00:30:47.950 --> 00:30:54.900 align:middle line:84% But right now, they span about three logarithms of mass. 00:30:54.900 --> 00:30:59.550 align:middle line:84% And this was all done with the awesome power of single DNA 00:30:59.550 --> 00:31:04.380 align:middle line:84% molecule technology, crosses basically. 00:31:04.380 --> 00:31:07.090 align:middle line:84% And what happens in each of these cells in your body 00:31:07.090 --> 00:31:09.450 align:middle line:84% if all goes well is you start out 00:31:09.450 --> 00:31:13.800 align:middle line:90% with one chromosome of interest. 00:31:13.800 --> 00:31:17.010 align:middle line:84% And it divides, and then the cell divides. 00:31:17.010 --> 00:31:19.260 align:middle line:84% And that chromosome-- we'll forget about all the other 00:31:19.260 --> 00:31:21.060 align:middle line:84% chromosomes in there for a moment-- 00:31:21.060 --> 00:31:24.960 align:middle line:84% that chromosome has a choice of when it divides, 00:31:24.960 --> 00:31:28.790 align:middle line:84% it can go one each into each of the daughter cells. 00:31:28.790 --> 00:31:31.310 align:middle line:84% Or both chromosomes can hang out together, 00:31:31.310 --> 00:31:33.060 align:middle line:84% since they're all tangled or something, 00:31:33.060 --> 00:31:34.477 align:middle line:84% and then one of the daughter cells 00:31:34.477 --> 00:31:36.380 align:middle line:84% doesn't get any copy of that chromosome. 00:31:36.380 --> 00:31:38.150 align:middle line:84% That obviously is not a good thing. 00:31:38.150 --> 00:31:40.010 align:middle line:84% Even if two copies of the chromosome 00:31:40.010 --> 00:31:43.220 align:middle line:84% is OK, if you can tolerate that extra dosage, 00:31:43.220 --> 00:31:47.410 align:middle line:84% you certainly can't tolerate zero chromosomes. 00:31:47.410 --> 00:31:49.180 align:middle line:84% Well, what's the chances this will happen? 00:31:49.180 --> 00:31:51.000 align:middle line:84% Well this is really elementary probability. 00:31:51.000 --> 00:31:52.250 align:middle line:90% I'm going to ease you into it. 00:31:52.250 --> 00:31:55.697 align:middle line:84% It's that it's about 50/50 chance. 00:31:55.697 --> 00:31:56.280 align:middle line:90% These are all. 00:31:56.280 --> 00:31:58.590 align:middle line:84% This is the exhaustive list of the possibilities. 00:31:58.590 --> 00:32:06.630 align:middle line:84% And about 50% of them have the wrong dosage. 00:32:06.630 --> 00:32:10.020 align:middle line:84% Well, what if we have a more realistic situation, 00:32:10.020 --> 00:32:12.440 align:middle line:90% human cells with 46 chromosomes? 00:32:12.440 --> 00:32:15.180 align:middle line:84% What are the odds that they'll all 00:32:15.180 --> 00:32:19.260 align:middle line:84% be right, that we'll get exactly one of each? 00:32:19.260 --> 00:32:22.492 align:middle line:84% We have 23 chromosomes from mom and 23 from pop. 00:32:22.492 --> 00:32:24.450 align:middle line:84% What's the odds that this is going to work out? 00:32:24.450 --> 00:32:26.340 align:middle line:84% Well, we're going to take a couple of slides 00:32:26.340 --> 00:32:27.450 align:middle line:90% to get to that answer. 00:32:27.450 --> 00:32:30.940 align:middle line:90% 00:32:30.940 --> 00:32:32.730 align:middle line:84% But first, to motivate you, this is 00:32:32.730 --> 00:32:35.430 align:middle line:84% extremely important in a health care sense. 00:32:35.430 --> 00:32:39.420 align:middle line:84% It's the most common form of mutation, happens all the time. 00:32:39.420 --> 00:32:42.870 align:middle line:84% Unfortunately, at every chromosome, 00:32:42.870 --> 00:32:50.580 align:middle line:84% duplication or losses is a big change in the human state. 00:32:50.580 --> 00:32:54.198 align:middle line:84% And the mildest of all of the additions or subtractions 00:32:54.198 --> 00:32:55.740 align:middle line:84% of a chromosome-- here, you just have 00:32:55.740 --> 00:32:58.770 align:middle line:84% three copies of chromosome 21, everything else normal. 00:32:58.770 --> 00:33:03.030 align:middle line:84% So 1.5 dosage of one of the smallest human chromosomes 00:33:03.030 --> 00:33:04.590 align:middle line:90% has an enormous impact. 00:33:04.590 --> 00:33:07.770 align:middle line:84% Most of you have seen someone with Down syndrome, which 00:33:07.770 --> 00:33:12.510 align:middle line:84% is severe mental retardation, and heart defects 00:33:12.510 --> 00:33:13.765 align:middle line:90% in various other organs. 00:33:13.765 --> 00:33:14.640 align:middle line:90% STUDENT: [INAUDIBLE]. 00:33:14.640 --> 00:33:16.350 align:middle line:90% GEORGE CHURCH: Yeah, question. 00:33:16.350 --> 00:33:20.220 align:middle line:84% STUDENT: This problem that you just described, in reality, 00:33:20.220 --> 00:33:23.700 align:middle line:84% though, it's not random, because there are mechanisms 00:33:23.700 --> 00:33:28.977 align:middle line:84% in the cell that would bias the tool in order 00:33:28.977 --> 00:33:30.060 align:middle line:90% not to segregate one cell. 00:33:30.060 --> 00:33:30.935 align:middle line:90% GEORGE CHURCH: Right. 00:33:30.935 --> 00:33:32.880 align:middle line:90% This is a good point. 00:33:32.880 --> 00:33:37.860 align:middle line:84% This is what I'm setting us up for, exactly that conclusion. 00:33:37.860 --> 00:33:39.150 align:middle line:90% It cannot be random. 00:33:39.150 --> 00:33:44.040 align:middle line:84% Single molecules are subject to stochastics. 00:33:44.040 --> 00:33:47.220 align:middle line:84% And so to overcome that stochastic process that 00:33:47.220 --> 00:33:49.650 align:middle line:84% should be random, you have to have machines 00:33:49.650 --> 00:33:53.333 align:middle line:84% that involve multiple molecules, because only 00:33:53.333 --> 00:33:54.750 align:middle line:84% through multiple molecules can you 00:33:54.750 --> 00:33:57.400 align:middle line:84% get the statistics to overcome the single molecule. 00:33:57.400 --> 00:33:59.460 align:middle line:90% And that's quite a trick. 00:33:59.460 --> 00:34:01.260 align:middle line:84% You can't just casually say, oh, there's 00:34:01.260 --> 00:34:05.100 align:middle line:84% some machine in there that takes care of this, OK. 00:34:05.100 --> 00:34:06.850 align:middle line:84% Just to expand this a little bit more, 00:34:06.850 --> 00:34:08.520 align:middle line:84% we know that certainly, the DNA is 00:34:08.520 --> 00:34:11.409 align:middle line:84% the case where the single molecule is always a problem. 00:34:11.409 --> 00:34:15.370 align:middle line:84% It has to be aided by molecular machines where you use energy. 00:34:15.370 --> 00:34:17.940 align:middle line:84% You expend energy in order to make sure the DNA molecules 00:34:17.940 --> 00:34:18.500 align:middle line:90% work. 00:34:18.500 --> 00:34:20.250 align:middle line:84% We'll get back to that calculation of what 00:34:20.250 --> 00:34:22.590 align:middle line:84% the odds are at random, but I should say also, 00:34:22.590 --> 00:34:26.940 align:middle line:84% RNAs in many systems appear to be, on average-- remember, 00:34:26.940 --> 00:34:28.440 align:middle line:84% it's the population average to be 00:34:28.440 --> 00:34:30.090 align:middle line:90% close to 1 molecule per cell. 00:34:30.090 --> 00:34:31.949 align:middle line:84% They're produce in bursts, following 00:34:31.949 --> 00:34:34.199 align:middle line:84% stochastic bursts of RNA where you get 00:34:34.199 --> 00:34:36.239 align:middle line:90% transcription factors binding. 00:34:36.239 --> 00:34:39.090 align:middle line:84% And then that burst of RNA produces even bigger bursts 00:34:39.090 --> 00:34:40.260 align:middle line:90% of proteins. 00:34:40.260 --> 00:34:42.900 align:middle line:84% But on average, it comes out to be a very small number, 00:34:42.900 --> 00:34:44.639 align:middle line:90% because the proteins preserve. 00:34:44.639 --> 00:34:47.250 align:middle line:84% They persist through many cell divisions 00:34:47.250 --> 00:34:50.909 align:middle line:84% while the RNAs maybe can turn over more rapidly. 00:34:50.909 --> 00:34:54.389 align:middle line:84% To get back to that question of how much variation 00:34:54.389 --> 00:34:57.300 align:middle line:84% is tolerable in biological systems, 00:34:57.300 --> 00:35:01.320 align:middle line:84% here's the very beginnings of your statistics. 00:35:01.320 --> 00:35:03.450 align:middle line:84% Some of you may have this hopefully already. 00:35:03.450 --> 00:35:05.910 align:middle line:84% There will be sections where you can cover this. 00:35:05.910 --> 00:35:10.260 align:middle line:84% But here are some of the really, really useful, easy statistics. 00:35:10.260 --> 00:35:12.580 align:middle line:84% What do we want to know about a distribution? 00:35:12.580 --> 00:35:15.990 align:middle line:84% Making the fewest assumptions for now, 00:35:15.990 --> 00:35:20.160 align:middle line:84% we want to know its mean, its arithmetic average. 00:35:20.160 --> 00:35:23.190 align:middle line:84% What is the average number of chromosomes in a cell? 00:35:23.190 --> 00:35:26.760 align:middle line:84% If they're supposed to be 1, how close to the average are they? 00:35:26.760 --> 00:35:27.675 align:middle line:90% That's the variance. 00:35:27.675 --> 00:35:30.660 align:middle line:90% 00:35:30.660 --> 00:35:33.000 align:middle line:84% To get the arithmetic mean, you basically 00:35:33.000 --> 00:35:34.980 align:middle line:84% add up all the numbers and divide 00:35:34.980 --> 00:35:37.530 align:middle line:90% by the number that you counted. 00:35:37.530 --> 00:35:42.460 align:middle line:84% Add up the values which are the axes. 00:35:42.460 --> 00:35:44.580 align:middle line:84% And so here you take a weighted sum. 00:35:44.580 --> 00:35:48.210 align:middle line:84% The sigma means the sum of the x values weighted 00:35:48.210 --> 00:35:50.850 align:middle line:84% by the frequency, f of x, the frequency 00:35:50.850 --> 00:35:53.580 align:middle line:84% that they occur in the sample that you take. 00:35:53.580 --> 00:35:55.830 align:middle line:90% Here, r for the mean is just 1. 00:35:55.830 --> 00:36:02.250 align:middle line:84% It's taking the first moment to get the mean. 00:36:02.250 --> 00:36:07.830 align:middle line:84% The analogous thing is, now you correct all the values 00:36:07.830 --> 00:36:10.230 align:middle line:84% that you're measuring, these variables, 00:36:10.230 --> 00:36:11.790 align:middle line:90% the number of chromosomes, say. 00:36:11.790 --> 00:36:13.360 align:middle line:90% And you subtract the mean. 00:36:13.360 --> 00:36:18.180 align:middle line:84% So now the mean for this x minus mu is now effectively 0. 00:36:18.180 --> 00:36:21.000 align:middle line:84% The mean is 0, and you want to ask, how far from 0 00:36:21.000 --> 00:36:23.040 align:middle line:90% do you deviate. 00:36:23.040 --> 00:36:25.770 align:middle line:84% For chromosomes, you want that deviation to be very small. 00:36:25.770 --> 00:36:28.410 align:middle line:84% You want the variance to be very small. 00:36:28.410 --> 00:36:30.270 align:middle line:84% And it's just the sum of the squares. 00:36:30.270 --> 00:36:34.410 align:middle line:84% We want to take the squares because if it deviates either 00:36:34.410 --> 00:36:37.920 align:middle line:84% more or less, it's still a tragedy, 00:36:37.920 --> 00:36:39.800 align:middle line:84% and you want to keep tract of that. 00:36:39.800 --> 00:36:40.800 align:middle line:90% So these are two things. 00:36:40.800 --> 00:36:42.600 align:middle line:84% They don't make any assumption about what kind 00:36:42.600 --> 00:36:43.380 align:middle line:90% of distribution. 00:36:43.380 --> 00:36:45.300 align:middle line:84% The distribution can be anything. 00:36:45.300 --> 00:36:46.920 align:middle line:84% You can calculate the arithmetic mean 00:36:46.920 --> 00:36:51.180 align:middle line:90% and the standard deviation. 00:36:51.180 --> 00:36:53.670 align:middle line:84% Another useful concept that starts 00:36:53.670 --> 00:36:55.560 align:middle line:84% to make more assumptions in interpretation 00:36:55.560 --> 00:36:59.880 align:middle line:84% is, now you have two variables, say x and y. 00:36:59.880 --> 00:37:01.757 align:middle line:84% And you want to ask, do they covariate. 00:37:01.757 --> 00:37:03.090 align:middle line:90% Are they related to one another? 00:37:03.090 --> 00:37:07.530 align:middle line:84% When you do two different experiments, 00:37:07.530 --> 00:37:09.955 align:middle line:84% you want to ask whether they're giving similar results. 00:37:09.955 --> 00:37:12.330 align:middle line:84% If they're two completely different kinds of experiments, 00:37:12.330 --> 00:37:14.913 align:middle line:84% you might want to know whether they're reinforcing each other. 00:37:14.913 --> 00:37:16.770 align:middle line:84% You want to know whether they're redundant. 00:37:16.770 --> 00:37:18.930 align:middle line:84% If you're observing two biological facts, 00:37:18.930 --> 00:37:21.370 align:middle line:84% you don't know whether they are related to one another. 00:37:21.370 --> 00:37:23.100 align:middle line:90% It's a discovery if they covary. 00:37:23.100 --> 00:37:24.090 align:middle line:90% That's what this means. 00:37:24.090 --> 00:37:27.060 align:middle line:84% Covariance is, again, using this concept of expectation. 00:37:27.060 --> 00:37:32.370 align:middle line:84% The sum of the x's corrected, so that you subtract 00:37:32.370 --> 00:37:34.980 align:middle line:84% the means, their averages, and divide 00:37:34.980 --> 00:37:38.380 align:middle line:84% by their standard deviations, square root of the variance. 00:37:38.380 --> 00:37:42.210 align:middle line:84% So you basically end up with a mean of 0 and a variance of 1. 00:37:42.210 --> 00:37:47.340 align:middle line:84% And then whenever these normalize, when x goes up 00:37:47.340 --> 00:37:52.060 align:middle line:84% and y goes up, the product will be reflected in this sum. 00:37:52.060 --> 00:37:56.450 align:middle line:84% So now this has an interesting property 00:37:56.450 --> 00:37:59.150 align:middle line:84% that when x and y are independent, unrelated, 00:37:59.150 --> 00:38:04.060 align:middle line:84% then the C or the Pearson correlation coefficient is 0. 00:38:04.060 --> 00:38:07.130 align:middle line:84% However, the reverse is not true. 00:38:07.130 --> 00:38:10.370 align:middle line:84% If C is 0, it does not imply that the x and y 00:38:10.370 --> 00:38:11.780 align:middle line:90% are independent. 00:38:11.780 --> 00:38:15.260 align:middle line:84% An example, a simple example, is the curve, the quadratic curve, 00:38:15.260 --> 00:38:16.550 align:middle line:90% y equals x squared. 00:38:16.550 --> 00:38:20.420 align:middle line:84% Here, they are completely related to one another, 00:38:20.420 --> 00:38:22.670 align:middle line:84% but they give a correlation coefficient of 0. 00:38:22.670 --> 00:38:25.190 align:middle line:84% That's because this is a linear correlation coefficient. 00:38:25.190 --> 00:38:27.890 align:middle line:84% The model that you're testing is that they are linear related. 00:38:27.890 --> 00:38:29.960 align:middle line:84% They are either positively correlated, 00:38:29.960 --> 00:38:34.130 align:middle line:84% which means the extreme case of C will be equal to 1, 00:38:34.130 --> 00:38:38.120 align:middle line:84% or negatively correlated C is equal to minus 1. 00:38:38.120 --> 00:38:40.130 align:middle line:84% You can plug this little formula here, 00:38:40.130 --> 00:38:42.260 align:middle line:84% handy-dandy practical form you can plug in 00:38:42.260 --> 00:38:43.790 align:middle line:90% to calculate probabilities. 00:38:43.790 --> 00:38:45.770 align:middle line:90% And this is an Excel formula. 00:38:45.770 --> 00:38:50.210 align:middle line:84% The probability that a correlation is far from 0, 00:38:50.210 --> 00:38:54.380 align:middle line:84% that's dependent upon the sample size where you sampled 00:38:54.380 --> 00:38:55.760 align:middle line:90% different x's and y's. 00:38:55.760 --> 00:39:01.010 align:middle line:84% You know, it could be head size and weight, or length 00:39:01.010 --> 00:39:02.660 align:middle line:90% and weight, and so forth. 00:39:02.660 --> 00:39:05.330 align:middle line:84% If they're correlated, then this probability 00:39:05.330 --> 00:39:09.210 align:middle line:84% will be significant if your sample size is large enough. 00:39:09.210 --> 00:39:11.330 align:middle line:84% There are some very practical things. 00:39:11.330 --> 00:39:15.950 align:middle line:84% And now let's put these in the context of a particular class 00:39:15.950 --> 00:39:16.650 align:middle line:90% of distribution. 00:39:16.650 --> 00:39:18.322 align:middle line:84% Now, most of those did not require 00:39:18.322 --> 00:39:20.030 align:middle line:84% that we state what kind of distributions, 00:39:20.030 --> 00:39:21.980 align:middle line:84% but there's a big interesting set which 00:39:21.980 --> 00:39:25.310 align:middle line:90% are roughly bell-shaped curves. 00:39:25.310 --> 00:39:29.300 align:middle line:84% And I've rigged this so that these three wildly different 00:39:29.300 --> 00:39:34.280 align:middle line:84% types of distribution happen to give similar curves. 00:39:34.280 --> 00:39:36.885 align:middle line:84% And you see in the next couple slides how I rigged it, 00:39:36.885 --> 00:39:38.760 align:middle line:84% but basically, this is a normal distribution, 00:39:38.760 --> 00:39:42.080 align:middle line:84% the Poisson distribution and the binomial. 00:39:42.080 --> 00:39:44.630 align:middle line:84% The binomial has a limited range. 00:39:44.630 --> 00:39:47.730 align:middle line:84% n goes from 0 to 40 in this case. 00:39:47.730 --> 00:39:50.030 align:middle line:90% It has a maximum n of 40. 00:39:50.030 --> 00:39:56.810 align:middle line:90% 00:39:56.810 --> 00:40:02.360 align:middle line:84% The Poisson has a mean which in this case is 20. 00:40:02.360 --> 00:40:04.640 align:middle line:84% The normal distribution has a mean that's similar. 00:40:04.640 --> 00:40:09.770 align:middle line:84% The normal distribution can have any range, 00:40:09.770 --> 00:40:10.940 align:middle line:90% the standard deviation. 00:40:10.940 --> 00:40:17.630 align:middle line:84% And this is set here to be, the standard deviation to be, 00:40:17.630 --> 00:40:20.360 align:middle line:84% the square root of the mean of the Poisson distribution. 00:40:20.360 --> 00:40:24.140 align:middle line:84% That's how you can rig these to be similar to one another. 00:40:24.140 --> 00:40:27.290 align:middle line:90% 00:40:27.290 --> 00:40:28.790 align:middle line:84% I think time is not going to permit 00:40:28.790 --> 00:40:32.990 align:middle line:84% me to go through all this in detail, 00:40:32.990 --> 00:40:35.900 align:middle line:84% but you will cover these in your statistics sections 00:40:35.900 --> 00:40:37.310 align:middle line:90% if you don't already know it. 00:40:37.310 --> 00:40:40.160 align:middle line:84% But suffice it to say that the binomial distribution is 00:40:40.160 --> 00:40:40.955 align:middle line:90% limit x. 00:40:40.955 --> 00:40:43.910 align:middle line:84% It Has to be an integer, and the integer 00:40:43.910 --> 00:40:46.160 align:middle line:90% is limited to going from 0 to n. 00:40:46.160 --> 00:40:49.040 align:middle line:84% This distinguishes it from a Poisson distribution, where 00:40:49.040 --> 00:40:53.570 align:middle line:84% it goes from 0 to infinity, and the normal curve, 00:40:53.570 --> 00:40:56.420 align:middle line:84% where it can go from negative infinity to positive infinity. 00:40:56.420 --> 00:40:59.300 align:middle line:84% Binomial and Poisson are discrete. 00:40:59.300 --> 00:41:03.890 align:middle line:84% They happen at integers while the Gaussians are continuous. 00:41:03.890 --> 00:41:06.920 align:middle line:90% 00:41:06.920 --> 00:41:12.320 align:middle line:84% The way you calculate this is, the probability 00:41:12.320 --> 00:41:17.990 align:middle line:84% of an individual n happening has probability p, 00:41:17.990 --> 00:41:21.650 align:middle line:84% so say 0.01 in the previous slide. 00:41:21.650 --> 00:41:26.660 align:middle line:84% And then getting exactly x of those 00:41:26.660 --> 00:41:30.260 align:middle line:84% is p to the x power as a first approximation. 00:41:30.260 --> 00:41:34.520 align:middle line:84% But there are actually two cases here, hence the name binomial. 00:41:34.520 --> 00:41:36.830 align:middle line:84% If it were multiple cases, they'd be multinomial. 00:41:36.830 --> 00:41:39.950 align:middle line:84% But the two cases are basically p and 1 minus p. 00:41:39.950 --> 00:41:41.000 align:middle line:90% They have to sum to 1. 00:41:41.000 --> 00:41:42.950 align:middle line:84% All these probabilities have to sum to 1. 00:41:42.950 --> 00:41:45.980 align:middle line:84% So now you have the probability that you have exactly x, 00:41:45.980 --> 00:41:50.120 align:middle line:84% and the leftovers go into 1 minus p. 00:41:50.120 --> 00:41:51.620 align:middle line:84% But then you also have to correct 00:41:51.620 --> 00:41:53.030 align:middle line:84% for the number of different ways that you 00:41:53.030 --> 00:41:54.950 align:middle line:84% could get this, the number of combinations, 00:41:54.950 --> 00:42:00.380 align:middle line:84% which is the total number of possibilities choosing 00:42:00.380 --> 00:42:01.610 align:middle line:90% x at a time. 00:42:01.610 --> 00:42:05.120 align:middle line:84% And then again, the leftover is n minus x. 00:42:05.120 --> 00:42:07.977 align:middle line:84% And this is n factorial over x factorial times 00:42:07.977 --> 00:42:08.810 align:middle line:90% n minus x factorial. 00:42:08.810 --> 00:42:10.310 align:middle line:84% This is the number of combinations. 00:42:10.310 --> 00:42:12.590 align:middle line:84% And so now the binomial distribution is this, 00:42:12.590 --> 00:42:16.370 align:middle line:84% and the sum of all the terms here has to add to 1. 00:42:16.370 --> 00:42:20.810 align:middle line:84% That's one of the properties of probability distribution, 00:42:20.810 --> 00:42:23.570 align:middle line:84% is if you think of all the possibilities, 00:42:23.570 --> 00:42:25.700 align:middle line:90% they add up to 1. 00:42:25.700 --> 00:42:28.730 align:middle line:84% So the sum of the binomial distribution of all the x's is 00:42:28.730 --> 00:42:30.050 align:middle line:90% 1. 00:42:30.050 --> 00:42:34.340 align:middle line:84% Now, just to remind you that computers are fallible, 00:42:34.340 --> 00:42:36.590 align:middle line:84% here's what, when you take a fairly-- you know, 00:42:36.590 --> 00:42:41.270 align:middle line:84% x is equal to exactly 300 taken from a population of 700. 00:42:41.270 --> 00:42:45.455 align:middle line:84% A probability of 1 for a unit event. 00:42:45.455 --> 00:42:47.660 align:middle line:84% The probability of getting exactly 350 00:42:47.660 --> 00:42:52.100 align:middle line:84% from that kind of bell-shaped curve is very small, but not 0. 00:42:52.100 --> 00:42:58.030 align:middle line:84% And Mathematica gets it right, and Excels gueses it at 0. 00:42:58.030 --> 00:42:59.638 align:middle line:90% Good guess, but wrong. 00:42:59.638 --> 00:43:02.820 align:middle line:90% 00:43:02.820 --> 00:43:09.130 align:middle line:84% Poisson, you now can and must go out to positive infinity. 00:43:09.130 --> 00:43:14.380 align:middle line:84% And there, you often will make the approximation 00:43:14.380 --> 00:43:17.510 align:middle line:84% that for large n and small probability in the binomial 00:43:17.510 --> 00:43:20.410 align:middle line:84% we've been talking about-- n is the total number of objects 00:43:20.410 --> 00:43:22.300 align:middle line:84% you're looking at, you're choosing x from. 00:43:22.300 --> 00:43:25.270 align:middle line:84% And t is the unit probability of each of those. 00:43:25.270 --> 00:43:29.350 align:middle line:84% The mean is now approximately n times p, 00:43:29.350 --> 00:43:32.560 align:middle line:84% and the binomial and Poisson are very similar. 00:43:32.560 --> 00:43:34.585 align:middle line:84% That's why they look similar in that plot. 00:43:34.585 --> 00:43:38.050 align:middle line:90% 00:43:38.050 --> 00:43:41.080 align:middle line:84% And here's some practical magic you can do with the Poisson. 00:43:41.080 --> 00:43:43.300 align:middle line:84% If you have a library of [INAUDIBLE] 00:43:43.300 --> 00:43:46.120 align:middle line:84% or combinatorial chemistry, or genomic clones and so forth, 00:43:46.120 --> 00:43:49.540 align:middle line:84% and x is a number of hits, and you 00:43:49.540 --> 00:43:55.690 align:middle line:84% want that to be greater than 0 so your thesis can proceed, 00:43:55.690 --> 00:44:01.510 align:middle line:84% you want the 0 hit term to be very small, e to the minus mu. 00:44:01.510 --> 00:44:04.900 align:middle line:84% So you want the mean to be greater 00:44:04.900 --> 00:44:07.030 align:middle line:90% than 1 or 2 or maybe even-- 00:44:07.030 --> 00:44:08.852 align:middle line:84% if the mean is 10, the probability 00:44:08.852 --> 00:44:10.810 align:middle line:84% that for a given experiment you'll have a 0 hit 00:44:10.810 --> 00:44:12.760 align:middle line:90% is very small. 00:44:12.760 --> 00:44:17.180 align:middle line:84% And you can estimate this from the number and 1 2 and 3 hits 00:44:17.180 --> 00:44:17.680 align:middle line:90% you get. 00:44:17.680 --> 00:44:19.510 align:middle line:84% You can estimate the 0 hit, and you 00:44:19.510 --> 00:44:22.510 align:middle line:84% can estimate whether it actually fits a Poisson or not. 00:44:22.510 --> 00:44:25.128 align:middle line:84% The final of this trio is the normal. 00:44:25.128 --> 00:44:27.420 align:middle line:84% Now you go from negative infinity to positive infinity. 00:44:27.420 --> 00:44:30.880 align:middle line:84% It's not just 0 to n or 0 to infinity. 00:44:30.880 --> 00:44:32.200 align:middle line:90% And it's continuous. 00:44:32.200 --> 00:44:34.220 align:middle line:90% That means everything takes a p. 00:44:34.220 --> 00:44:37.870 align:middle line:84% So now instead of summing up to 1, you integrate up to 1, 00:44:37.870 --> 00:44:42.880 align:middle line:84% because now the little delta x's are infinitely small. 00:44:42.880 --> 00:44:46.818 align:middle line:84% And so now, here's an exponential of a quadratic, 00:44:46.818 --> 00:44:48.860 align:middle line:84% just like the ones we were talking about earlier. 00:44:48.860 --> 00:44:50.985 align:middle line:90% So this is a negative quadratic. 00:44:50.985 --> 00:44:53.110 align:middle line:84% And that gives you a nice little bell-shaped curve. 00:44:53.110 --> 00:44:58.900 align:middle line:84% And the 2 pi sigma square root is a normalization, 00:44:58.900 --> 00:45:02.260 align:middle line:84% so it does actually integrate sum to 1. 00:45:02.260 --> 00:45:04.540 align:middle line:84% Here, another approximation sometimes applied 00:45:04.540 --> 00:45:07.070 align:middle line:84% is when n times p times q is large, 00:45:07.070 --> 00:45:09.130 align:middle line:84% the normal is very similar to the binomial. 00:45:09.130 --> 00:45:14.075 align:middle line:84% People will abuse this and use one of these three 00:45:14.075 --> 00:45:15.700 align:middle line:84% distributions in the place of the other 00:45:15.700 --> 00:45:16.783 align:middle line:90% when it isn't appropriate. 00:45:16.783 --> 00:45:18.680 align:middle line:84% And we'll give some examples as we go. 00:45:18.680 --> 00:45:20.500 align:middle line:90% So back to this calculation. 00:45:20.500 --> 00:45:22.990 align:middle line:84% If we apply the binomial that we had in those previous 00:45:22.990 --> 00:45:25.210 align:middle line:84% slides-- and I urge you to do this as an exercise. 00:45:25.210 --> 00:45:29.590 align:middle line:84% It's not on the problem set, but just do it. 00:45:29.590 --> 00:45:32.770 align:middle line:84% Just getting some for any 46 chromosomes, 00:45:32.770 --> 00:45:36.220 align:middle line:84% you get the right number of chromosomes, is about 8%. 00:45:36.220 --> 00:45:37.090 align:middle line:90% That's not too bad. 00:45:37.090 --> 00:45:41.770 align:middle line:84% But it's still fairly lethal, because you really want exactly 00:45:41.770 --> 00:45:45.340 align:middle line:84% the correct 46, which is 0.5 to the 46th power, which 00:45:45.340 --> 00:45:47.650 align:middle line:84% is infinitesimally unlikely to happen at random, which 00:45:47.650 --> 00:45:49.930 align:middle line:84% gets back to the point made by the audience 00:45:49.930 --> 00:45:52.690 align:middle line:90% here that this is not random. 00:45:52.690 --> 00:45:55.330 align:middle line:84% But you can't fight the fact that single molecules 00:45:55.330 --> 00:45:56.530 align:middle line:90% are based in stochastics. 00:45:56.530 --> 00:45:59.860 align:middle line:84% So you have to have a lot of events adding up, 00:45:59.860 --> 00:46:03.040 align:middle line:84% energy being input to overcome that. 00:46:03.040 --> 00:46:04.810 align:middle line:84% We have selection that's optimizing 00:46:04.810 --> 00:46:07.120 align:middle line:90% this over long periods of time. 00:46:07.120 --> 00:46:09.970 align:middle line:84% We can use random numbers that underlie this 00:46:09.970 --> 00:46:12.160 align:middle line:84% for the simulations of these stochastic events, 00:46:12.160 --> 00:46:14.320 align:middle line:84% and also for permutation statistics, 00:46:14.320 --> 00:46:17.200 align:middle line:84% that when you have some data and you want to know whether it's 00:46:17.200 --> 00:46:21.790 align:middle line:84% significant or not, you can kind of do a Monte Carlo 00:46:21.790 --> 00:46:23.230 align:middle line:90% simulation of it. 00:46:23.230 --> 00:46:24.700 align:middle line:84% Here's how you code it in a couple 00:46:24.700 --> 00:46:29.565 align:middle line:84% of different languages, Perl, Excel, Fortran 77, Mathematica. 00:46:29.565 --> 00:46:30.940 align:middle line:84% Even though you can't evaluate it 00:46:30.940 --> 00:46:33.880 align:middle line:84% by looking at these numbers on the screen, trust me. 00:46:33.880 --> 00:46:35.563 align:middle line:84% There are bad random number generators. 00:46:35.563 --> 00:46:36.980 align:middle line:84% There are random number generators 00:46:36.980 --> 00:46:38.290 align:middle line:90% which are not very random, OK. 00:46:38.290 --> 00:46:41.740 align:middle line:90% 00:46:41.740 --> 00:46:45.580 align:middle line:84% Where they come from is this remainder operation, operated 00:46:45.580 --> 00:46:46.990 align:middle line:90% on very special numbers. 00:46:46.990 --> 00:46:48.910 align:middle line:84% You'll have to look these up in this reference 00:46:48.910 --> 00:46:50.800 align:middle line:90% to get a full feeling for it. 00:46:50.800 --> 00:46:54.012 align:middle line:84% But these are deterministic formulas 00:46:54.012 --> 00:46:55.220 align:middle line:90% that give you random numbers. 00:46:55.220 --> 00:46:57.800 align:middle line:84% It's not really the same as flipping a coin. 00:46:57.800 --> 00:47:00.000 align:middle line:84% The computer actually will give you-- 00:47:00.000 --> 00:47:01.750 align:middle line:84% It's the same random numbers over and over 00:47:01.750 --> 00:47:04.210 align:middle line:84% again unless you do something very special. 00:47:04.210 --> 00:47:06.970 align:middle line:84% And typically, these give a uniform distribution 00:47:06.970 --> 00:47:11.115 align:middle line:84% between 0 and 1, or over some integer range. 00:47:11.115 --> 00:47:13.240 align:middle line:84% And then you can turn it into a normal distribution 00:47:13.240 --> 00:47:15.070 align:middle line:84% with this kind of trick here where 00:47:15.070 --> 00:47:16.540 align:middle line:90% you make a transformation. 00:47:16.540 --> 00:47:19.030 align:middle line:84% There's a difference between a uniform random distribution 00:47:19.030 --> 00:47:20.710 align:middle line:90% and a bell-shaped one. 00:47:20.710 --> 00:47:23.560 align:middle line:84% And you can generate both of them just with this slide 00:47:23.560 --> 00:47:26.170 align:middle line:90% alone. 00:47:26.170 --> 00:47:29.800 align:middle line:84% So we full cycle back to these three different bell curves. 00:47:29.800 --> 00:47:33.220 align:middle line:84% They have very different properties that can be applied. 00:47:33.220 --> 00:47:36.640 align:middle line:84% Binomial only when you have a limited range, Poisson 00:47:36.640 --> 00:47:38.020 align:middle line:90% when you're positive infinity. 00:47:38.020 --> 00:47:42.430 align:middle line:84% Normal, negative, positive and continuous. 00:47:42.430 --> 00:47:45.670 align:middle line:84% Thank you for participating in this. 00:47:45.670 --> 00:47:48.175 align:middle line:84% These are the topics that we covered. 00:47:48.175 --> 00:47:50.860 align:middle line:90% 00:47:50.860 --> 00:47:52.640 align:middle line:90% See you back here in a week. 00:47:52.640 --> 00:47:54.960 align:middle line:84% Please hand in your questionnaires, 00:47:54.960 --> 00:47:59.270 align:middle line:84% and the sections should be assigned by Thursday or Friday. 00:47:59.270 --> 00:48:00.840 align:middle line:90% STUDENT: Thank you. 00:48:00.840 --> 00:48:40.000 align:middle line:90%