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 --> 00:00:12.730 align:middle line:90% is available at ocw.mit.edu. 00:00:12.730 --> 00:00:17.500 align:middle line:90% 00:00:17.500 --> 00:00:24.430 align:middle line:84% PROFESSOR: OK, welcome to RNA 2, which of course begins 00:00:24.430 --> 00:00:29.440 align:middle line:84% with a RNA 1 overview where we talked 00:00:29.440 --> 00:00:36.520 align:middle line:84% about the secondary and tertiary structure of RNA, 00:00:36.520 --> 00:00:40.120 align:middle line:84% and how one integrates dynamic programming 00:00:40.120 --> 00:00:44.020 align:middle line:90% in those algorithms. 00:00:44.020 --> 00:00:49.240 align:middle line:84% And then that is important in the way 00:00:49.240 --> 00:00:52.510 align:middle line:84% we go about measurements and it affects 00:00:52.510 --> 00:00:54.160 align:middle line:90% certain technical senses. 00:00:54.160 --> 00:00:57.220 align:middle line:84% And at an interpretation level, affects 00:00:57.220 --> 00:00:59.680 align:middle line:84% how we think about the quantitation of RNAm, which 00:00:59.680 --> 00:01:01.930 align:middle line:90% was the main topic last time. 00:01:01.930 --> 00:01:07.510 align:middle line:84% And then today, after we have the data analyzed 00:01:07.510 --> 00:01:12.850 align:middle line:84% so that we have RNA quantitation and 00:01:12.850 --> 00:01:17.440 align:middle line:84% the random systematic errors established some idea of what 00:01:17.440 --> 00:01:23.000 align:middle line:84% the interpretation consequences are and maybe time series data, 00:01:23.000 --> 00:01:25.410 align:middle line:84% the question is, what do we do next? 00:01:25.410 --> 00:01:28.960 align:middle line:84% What we do next is basically two things. 00:01:28.960 --> 00:01:30.790 align:middle line:90% At least for today's topic. 00:01:30.790 --> 00:01:34.630 align:middle line:84% We cluster, ask which gene expression 00:01:34.630 --> 00:01:36.460 align:middle line:84% products, whether they're RNA or protein, 00:01:36.460 --> 00:01:38.440 align:middle line:90% go up and down together. 00:01:38.440 --> 00:01:40.720 align:middle line:84% And if they go up and down together under a variety 00:01:40.720 --> 00:01:44.260 align:middle line:84% of conditions or time points in various conditions, 00:01:44.260 --> 00:01:46.330 align:middle line:90% then we want to know why. 00:01:46.330 --> 00:01:48.880 align:middle line:84% What is the mechanism by which they go up and down? 00:01:48.880 --> 00:01:53.490 align:middle line:84% And to what common goal are these gene products directed? 00:01:53.490 --> 00:01:56.200 align:middle line:84% In other words, two different whys. 00:01:56.200 --> 00:01:57.610 align:middle line:90% Why mechanistically. 00:01:57.610 --> 00:02:00.700 align:middle line:84% And why in terms of the way that they 00:02:00.700 --> 00:02:04.010 align:middle line:90% can help the entire system. 00:02:04.010 --> 00:02:06.610 align:middle line:84% So in order to deal with clustering, 00:02:06.610 --> 00:02:12.640 align:middle line:84% we'll go into quite some detail about the options 00:02:12.640 --> 00:02:14.470 align:middle line:84% that we have for doing clustering. 00:02:14.470 --> 00:02:17.200 align:middle line:84% And you'll see they're quite a number of combinations. 00:02:17.200 --> 00:02:21.040 align:middle line:84% We'll go through distance of similarity measures, 00:02:21.040 --> 00:02:27.520 align:middle line:84% hierarchical to non-hierarchical clustering and classification. 00:02:27.520 --> 00:02:28.960 align:middle line:84% Now this is kind of the roadmap-- 00:02:28.960 --> 00:02:32.230 align:middle line:84% the overview of all the different decisions 00:02:32.230 --> 00:02:37.600 align:middle line:84% that we need to make in order to establish gene expression 00:02:37.600 --> 00:02:39.310 align:middle line:90% clustering. 00:02:39.310 --> 00:02:44.110 align:middle line:84% Going from left to right, we've got data normalization choices, 00:02:44.110 --> 00:02:47.710 align:middle line:84% got distance metrics to choose from. 00:02:47.710 --> 00:02:54.100 align:middle line:84% Linkage methods, when we link two clusters together 00:02:54.100 --> 00:02:58.070 align:middle line:84% or two RNA types together, what methods do we use. 00:02:58.070 --> 00:03:02.700 align:middle line:84% And finally, the clustering method itself 00:03:02.700 --> 00:03:05.850 align:middle line:84% on the far right-hand side of slide three. 00:03:05.850 --> 00:03:10.530 align:middle line:84% And then the working backwards from the clustering method, 00:03:10.530 --> 00:03:14.455 align:middle line:84% you've got two basic goals, you can think. 00:03:14.455 --> 00:03:16.080 align:middle line:84% Typically, when we think of clustering, 00:03:16.080 --> 00:03:18.780 align:middle line:84% we're mainly talking about unsupervised methods. 00:03:18.780 --> 00:03:21.750 align:middle line:84% That is to say where we're really letting the data tell us 00:03:21.750 --> 00:03:27.390 align:middle line:84% what it has to say, what gene expression products go 00:03:27.390 --> 00:03:28.350 align:middle line:90% together. 00:03:28.350 --> 00:03:34.560 align:middle line:84% In a possibly an alternative or a [INAUDIBLE] 00:03:34.560 --> 00:03:40.020 align:middle line:84% to that would be to ask, can we use those discoveries 00:03:40.020 --> 00:03:43.680 align:middle line:84% in a sense to supervise classification? 00:03:43.680 --> 00:03:48.360 align:middle line:84% So rather than discovering what gene products go up and down 00:03:48.360 --> 00:03:51.810 align:middle line:84% together, ask those that go up and down together to use-- 00:03:51.810 --> 00:03:55.020 align:middle line:84% to allow us to classify the different conditions 00:03:55.020 --> 00:03:58.020 align:middle line:84% from which the gene expression has been ascertained. 00:03:58.020 --> 00:04:02.430 align:middle line:84% So classified as a pathological states, infectious states, 00:04:02.430 --> 00:04:05.050 align:middle line:90% cancer states and so on. 00:04:05.050 --> 00:04:06.570 align:middle line:84% So now we're going to kind of work 00:04:06.570 --> 00:04:11.190 align:middle line:84% backwards from the unsupervised clustering methods 00:04:11.190 --> 00:04:16.920 align:middle line:84% and then move into distance metrics and linkage. 00:04:16.920 --> 00:04:21.480 align:middle line:84% So we're basically working from right to left on this chart. 00:04:21.480 --> 00:04:26.250 align:middle line:84% First, with an overview of what the goals of such quantitation 00:04:26.250 --> 00:04:28.040 align:middle line:84% classification methods should be, 00:04:28.040 --> 00:04:30.310 align:middle line:84% this has been in a previous lecture. 00:04:30.310 --> 00:04:33.450 align:middle line:84% But basically, we can start with the RNA data 00:04:33.450 --> 00:04:39.780 align:middle line:84% which we reduced to a table in the previous lecture. 00:04:39.780 --> 00:04:43.620 align:middle line:84% You can think of it as a table of RNA expressions 00:04:43.620 --> 00:04:47.550 align:middle line:84% along the vertical axis and different conditions 00:04:47.550 --> 00:04:53.310 align:middle line:84% along the horizontal axis where we can have full change, say 00:04:53.310 --> 00:04:55.230 align:middle line:90% ratios or absolute levels. 00:04:55.230 --> 00:04:59.610 align:middle line:84% And we can do either clustering or classification. 00:04:59.610 --> 00:05:04.120 align:middle line:84% And when we do get to clustering and discovery, 00:05:04.120 --> 00:05:06.630 align:middle line:84% one of the things we can do is use motifs 00:05:06.630 --> 00:05:08.550 align:middle line:90% to get at direct causality. 00:05:08.550 --> 00:05:11.370 align:middle line:90% 00:05:11.370 --> 00:05:13.260 align:middle line:84% These are just some buzzwords that you 00:05:13.260 --> 00:05:17.430 align:middle line:84% will find coming up in this lecture and problem set 00:05:17.430 --> 00:05:18.810 align:middle line:90% and outside. 00:05:18.810 --> 00:05:22.800 align:middle line:84% Just examples of the two types of goals 00:05:22.800 --> 00:05:28.740 align:middle line:84% of analyzing gene expression or even more general collections 00:05:28.740 --> 00:05:31.290 align:middle line:90% of quantitative data. 00:05:31.290 --> 00:05:40.200 align:middle line:84% Four examples-- four examples of unsupervised clustering, 00:05:40.200 --> 00:05:43.290 align:middle line:84% k-means clustering, self-organizing maps, 00:05:43.290 --> 00:05:45.730 align:middle line:84% single value decomposition, cluster [INAUDIBLE] analysis. 00:05:45.730 --> 00:05:48.330 align:middle line:84% You may have heard these in various contexts. 00:05:48.330 --> 00:05:51.660 align:middle line:84% I'm lumping of all together here under this category. 00:05:51.660 --> 00:05:55.690 align:middle line:84% And we'll particularly delve into k-means as one example. 00:05:55.690 --> 00:05:59.400 align:middle line:84% We could delve into any of them, but we need to get some depth. 00:05:59.400 --> 00:06:01.920 align:middle line:84% And then just for your reference, here's 00:06:01.920 --> 00:06:04.560 align:middle line:84% some examples of the supervised learning 00:06:04.560 --> 00:06:07.140 align:middle line:84% if you were going to go into the classification. 00:06:07.140 --> 00:06:13.480 align:middle line:90% 00:06:13.480 --> 00:06:20.200 align:middle line:84% Here's some examples of early attempts at clustering. 00:06:20.200 --> 00:06:23.230 align:middle line:84% These are particularly interesting to look 00:06:23.230 --> 00:06:30.580 align:middle line:84% at because they were early and very little prior literature. 00:06:30.580 --> 00:06:33.730 align:middle line:84% They tended to take a fresh fresher look at it 00:06:33.730 --> 00:06:35.920 align:middle line:84% than you might get in the most recent papers. 00:06:35.920 --> 00:06:38.380 align:middle line:84% Fewer assumptions and therefore, more exposition 00:06:38.380 --> 00:06:43.190 align:middle line:84% about where they feel clustering comes in from other fields 00:06:43.190 --> 00:06:46.570 align:middle line:90% and is applicable to this field. 00:06:46.570 --> 00:06:49.000 align:middle line:84% The main dichotomy that I'm pointing out here 00:06:49.000 --> 00:06:51.220 align:middle line:90% is can cluster by gene. 00:06:51.220 --> 00:06:55.630 align:middle line:84% That is to say, by RNA or gene product or RNA protein. 00:06:55.630 --> 00:07:00.370 align:middle line:84% Or you can cluster by condition cell type or even time course. 00:07:00.370 --> 00:07:03.700 align:middle line:84% So you can think of that as-- by gene as this vertical axis, 00:07:03.700 --> 00:07:07.210 align:middle line:84% at least in the formats that most articles and this lecture 00:07:07.210 --> 00:07:08.050 align:middle line:90% will have it in. 00:07:08.050 --> 00:07:10.180 align:middle line:84% And then by condition will be your horizontal axis. 00:07:10.180 --> 00:07:13.030 align:middle line:84% Or you can do by clustering, which is clustering by both. 00:07:13.030 --> 00:07:15.970 align:middle line:84% And then down here is an example of one 00:07:15.970 --> 00:07:18.820 align:middle line:84% of many sources of free software that you 00:07:18.820 --> 00:07:21.940 align:middle line:84% can look at, both for microarray analysis and for clustering. 00:07:21.940 --> 00:07:24.680 align:middle line:90% 00:07:24.680 --> 00:07:27.140 align:middle line:84% The general purpose of this is to divide samples 00:07:27.140 --> 00:07:29.390 align:middle line:90% into fairly homogeneous groups. 00:07:29.390 --> 00:07:32.300 align:middle line:84% Clearly, because of biological variation 00:07:32.300 --> 00:07:34.670 align:middle line:84% that can be meaningful or random, 00:07:34.670 --> 00:07:37.970 align:middle line:84% these will not be perfectly homogeneous. 00:07:37.970 --> 00:07:41.570 align:middle line:84% When we find the coregulated genes but some of the methods 00:07:41.570 --> 00:07:45.230 align:middle line:84% we talked about in previous classes, 00:07:45.230 --> 00:07:47.600 align:middle line:84% we'll want to know what the protein complexes are 00:07:47.600 --> 00:07:50.750 align:middle line:84% that are mechanistically regulating 00:07:50.750 --> 00:07:55.430 align:middle line:84% and the downstream functions of these. 00:07:55.430 --> 00:07:59.360 align:middle line:84% Again, the major dichotomy among the unsupervised learning 00:07:59.360 --> 00:08:01.070 align:middle line:84% is whether you're doing hierarchical 00:08:01.070 --> 00:08:03.440 align:middle line:90% or non-hierarchical. 00:08:03.440 --> 00:08:04.880 align:middle line:90% We'll show an example of each. 00:08:04.880 --> 00:08:06.860 align:middle line:84% Typically, hierarchical is represented 00:08:06.860 --> 00:08:08.990 align:middle line:84% by a tree very similar to the trees 00:08:08.990 --> 00:08:12.020 align:middle line:84% that we have for sequence similarity 00:08:12.020 --> 00:08:16.880 align:middle line:84% and for pedigrees, phylogeny and so on. 00:08:16.880 --> 00:08:21.320 align:middle line:84% These are basically the terminal branches of the tree 00:08:21.320 --> 00:08:25.340 align:middle line:84% or the leaves of the tree are the individual RNA 00:08:25.340 --> 00:08:30.770 align:middle line:84% species representing a vector of different RNA quantitation. 00:08:30.770 --> 00:08:35.720 align:middle line:84% With the non-hierarchical, you'll tend to have 00:08:35.720 --> 00:08:36.950 align:middle line:90% represented-- 00:08:36.950 --> 00:08:39.500 align:middle line:84% these are visual representations as well as 00:08:39.500 --> 00:08:40.760 align:middle line:90% underlying algorithms. 00:08:40.760 --> 00:08:46.400 align:middle line:84% They'll be represented more as a multidimensional envelope, say, 00:08:46.400 --> 00:08:50.960 align:middle line:84% a sphere or ellipse that tries to encompass 00:08:50.960 --> 00:08:56.960 align:middle line:84% a set of related gene expression values. 00:08:56.960 --> 00:08:58.640 align:middle line:84% Now we'll use diagrams like this-- 00:08:58.640 --> 00:09:01.520 align:middle line:84% mainly the two on the far left hand side of slide nine-- 00:09:01.520 --> 00:09:06.950 align:middle line:84% where you'll have a fairly tight circular or spherical clusters 00:09:06.950 --> 00:09:09.530 align:middle line:84% where it's pretty evident how they're connected. 00:09:09.530 --> 00:09:14.840 align:middle line:84% Or you can have more elongated or more inner penetrating 00:09:14.840 --> 00:09:15.890 align:middle line:90% clusters. 00:09:15.890 --> 00:09:18.740 align:middle line:90% And how do we deal with these? 00:09:18.740 --> 00:09:21.890 align:middle line:84% The key terms that we'll try to define this 00:09:21.890 --> 00:09:24.980 align:middle line:84% is actually very similar to the ones we talked about before. 00:09:24.980 --> 00:09:27.110 align:middle line:84% We had either distance of similarity. 00:09:27.110 --> 00:09:30.785 align:middle line:84% These are in flip sides of the same coin. 00:09:30.785 --> 00:09:37.430 align:middle line:90% 00:09:37.430 --> 00:09:41.690 align:middle line:84% The greater the distance, than the less similarity. 00:09:41.690 --> 00:09:45.710 align:middle line:84% The dendrograms are the same kind of the trees 00:09:45.710 --> 00:09:47.165 align:middle line:90% that we've been seeing before. 00:09:47.165 --> 00:09:49.830 align:middle line:90% 00:09:49.830 --> 00:09:56.540 align:middle line:84% Now the most general way of discussing distance measures 00:09:56.540 --> 00:09:59.070 align:middle line:90% is the Minkowski metric. 00:09:59.070 --> 00:10:02.923 align:middle line:84% This is actually a set of metrics. 00:10:02.923 --> 00:10:04.340 align:middle line:84% And what we're going to be talking 00:10:04.340 --> 00:10:07.490 align:middle line:84% about here are two objects which are really two-- 00:10:07.490 --> 00:10:09.500 align:middle line:84% for the purpose of discussion, two RNAs. 00:10:09.500 --> 00:10:15.480 align:middle line:84% Call them RNAx and RNAy, geneX and geneY, have key features. 00:10:15.480 --> 00:10:19.290 align:middle line:84% Meaning you have P different conditions, P time points. 00:10:19.290 --> 00:10:21.660 align:middle line:84% You will call them dimensions sometimes. 00:10:21.660 --> 00:10:25.640 align:middle line:84% And so this means that gene expression 00:10:25.640 --> 00:10:29.420 align:middle line:84% of x under conditions one through P 00:10:29.420 --> 00:10:32.582 align:middle line:84% is it compared to the gene expression of y 00:10:32.582 --> 00:10:34.040 align:middle line:84% under those various conditions too. 00:10:34.040 --> 00:10:38.850 align:middle line:84% You can think of these as vectors with P entries in them. 00:10:38.850 --> 00:10:42.380 align:middle line:84% And so the distance is going to be some Rth 00:10:42.380 --> 00:10:47.570 align:middle line:90% root of a sum to the R power. 00:10:47.570 --> 00:10:50.330 align:middle line:84% And we're going to go through three different examples 00:10:50.330 --> 00:10:51.300 align:middle line:90% of this. 00:10:51.300 --> 00:10:52.370 align:middle line:90% And I think you'll-- 00:10:52.370 --> 00:10:55.740 align:middle line:84% hopefully, by the time we've gone through it, 00:10:55.740 --> 00:10:58.760 align:middle line:84% you'll see the advantages of this general 00:10:58.760 --> 00:11:00.470 align:middle line:90% and the specific forms. 00:11:00.470 --> 00:11:02.150 align:middle line:84% So the three examples will have R 00:11:02.150 --> 00:11:07.010 align:middle line:84% equals 2, 1, and infinity on slide 12. 00:11:07.010 --> 00:11:09.650 align:middle line:84% These are the most common metrics. 00:11:09.650 --> 00:11:14.090 align:middle line:84% And you should see them as fairly familiar. 00:11:14.090 --> 00:11:16.700 align:middle line:84% When R equals 2 in that formula, you now 00:11:16.700 --> 00:11:19.850 align:middle line:84% have the square root of the sum of squares. 00:11:19.850 --> 00:11:24.440 align:middle line:84% And this should remind you of your simple Cartesian plotting 00:11:24.440 --> 00:11:28.340 align:middle line:84% of the distance between two points on graph paper 00:11:28.340 --> 00:11:31.970 align:middle line:84% where you can take any diagonal, the shortest path. 00:11:31.970 --> 00:11:34.130 align:middle line:84% On the other hand, if you are navigating 00:11:34.130 --> 00:11:38.240 align:middle line:84% the streets of Manhattan, you will tend not to take diagonals 00:11:38.240 --> 00:11:39.630 align:middle line:90% through stone walls. 00:11:39.630 --> 00:11:42.710 align:middle line:84% You'll tend to obey the blocks and you 00:11:42.710 --> 00:11:45.320 align:middle line:84% may have to go three blocks this way and four blocks that way 00:11:45.320 --> 00:11:47.510 align:middle line:90% rather than square roots. 00:11:47.510 --> 00:11:53.270 align:middle line:84% And then finally, the last one is the maximum distance 00:11:53.270 --> 00:11:55.980 align:middle line:84% you might have to go in any particular direction. 00:11:55.980 --> 00:12:00.365 align:middle line:84% So you can think that if you take the Rth 00:12:00.365 --> 00:12:05.690 align:middle line:84% root of the sum of the differences in these two 00:12:05.690 --> 00:12:06.950 align:middle line:90% measures, x and y-- 00:12:06.950 --> 00:12:12.170 align:middle line:84% measures of the two RNAs at the same condition-- 00:12:12.170 --> 00:12:13.670 align:middle line:84% that as R goes to infinity, you're 00:12:13.670 --> 00:12:17.540 align:middle line:84% going to wait up the biggest distance difference along all 00:12:17.540 --> 00:12:18.780 align:middle line:90% the different axes. 00:12:18.780 --> 00:12:20.603 align:middle line:84% And then you'll take the R through to that. 00:12:20.603 --> 00:12:22.520 align:middle line:84% And then it'll basically be the absolute value 00:12:22.520 --> 00:12:25.790 align:middle line:90% of that difference. 00:12:25.790 --> 00:12:28.460 align:middle line:84% And so those are the three measures. 00:12:28.460 --> 00:12:32.210 align:middle line:84% But let's see some specific examples. 00:12:32.210 --> 00:12:33.408 align:middle line:90% Here we have two points. 00:12:33.408 --> 00:12:35.450 align:middle line:84% So you have-- this is the simplest possible case. 00:12:35.450 --> 00:12:40.080 align:middle line:84% Two RNAs under two different conditions. 00:12:40.080 --> 00:12:45.950 align:middle line:84% And let's say on this arbitrary scale, the distance between y 00:12:45.950 --> 00:12:49.770 align:middle line:84% and x along the horizontal dimensions, 00:12:49.770 --> 00:12:53.600 align:middle line:84% which is say condition horizontal, is four 00:12:53.600 --> 00:12:55.400 align:middle line:84% and the condition vertical is three. 00:12:55.400 --> 00:12:58.470 align:middle line:84% That's the difference between them. 00:12:58.470 --> 00:13:00.810 align:middle line:84% And where it is absolutely relative 00:13:00.810 --> 00:13:05.120 align:middle line:84% the origin doesn't matter in any of these three metrics. 00:13:05.120 --> 00:13:09.812 align:middle line:84% The diagonal-- the direct distance or Euclidean distance 00:13:09.812 --> 00:13:11.270 align:middle line:84% is going to be the square root of 4 00:13:11.270 --> 00:13:14.210 align:middle line:84% squared plus 3 squared, which is going to be 5. 00:13:14.210 --> 00:13:17.300 align:middle line:84% And the Manhattan distance, you can't take that-- 00:13:17.300 --> 00:13:18.830 align:middle line:90% you can't go as the Crow flies. 00:13:18.830 --> 00:13:24.800 align:middle line:84% You have to go four blocks to the left and three blocks up. 00:13:24.800 --> 00:13:26.150 align:middle line:90% And that's seven. 00:13:26.150 --> 00:13:28.910 align:middle line:84% And then the maximum of the two measures, 00:13:28.910 --> 00:13:35.660 align:middle line:84% if you think of these as many different measures, 00:13:35.660 --> 00:13:38.690 align:middle line:84% the biggest distance in any particular direction 00:13:38.690 --> 00:13:41.540 align:middle line:90% would be four. 00:13:41.540 --> 00:13:48.530 align:middle line:84% Now here's an example where the Manhattan distance is called 00:13:48.530 --> 00:13:50.930 align:middle line:84% the Hamming distance when all the features are binary. 00:13:50.930 --> 00:13:52.550 align:middle line:90% And why is this interesting? 00:13:52.550 --> 00:13:55.160 align:middle line:84% I mentioned, I think, in the first lecture 00:13:55.160 --> 00:13:57.470 align:middle line:84% that many biologists and scientists in general, 00:13:57.470 --> 00:13:59.660 align:middle line:84% when they have the opportunity, will classify things 00:13:59.660 --> 00:14:02.810 align:middle line:84% as on and off even when there is some underlying quantitative 00:14:02.810 --> 00:14:03.380 align:middle line:90% nature. 00:14:03.380 --> 00:14:06.230 align:middle line:84% Transistor can be on or off for all intents and purposes. 00:14:06.230 --> 00:14:09.050 align:middle line:84% And a gene circuit or a particular gene expression 00:14:09.050 --> 00:14:12.620 align:middle line:84% can be considered off or on, 0 or 1. 00:14:12.620 --> 00:14:16.220 align:middle line:84% And so now if you have say 17 different gene expression 00:14:16.220 --> 00:14:22.340 align:middle line:84% levels, this can be considered a 17-digit binary string. 00:14:22.340 --> 00:14:27.305 align:middle line:84% And the two genes, A and B, here can be compared. 00:14:27.305 --> 00:14:29.827 align:middle line:90% 00:14:29.827 --> 00:14:31.910 align:middle line:84% If you talk about distance rather than similarity, 00:14:31.910 --> 00:14:36.200 align:middle line:84% every time there's a conflict of 01 or 10, 00:14:36.200 --> 00:14:40.730 align:middle line:84% then you add that to the sum and you have a total of five 00:14:40.730 --> 00:14:43.350 align:middle line:84% of these cases where there's a difference. 00:14:43.350 --> 00:14:47.550 align:middle line:84% So the Hamming distance is five in this case. 00:14:47.550 --> 00:14:50.240 align:middle line:84% So you can see that this has some intuitive appeal 00:14:50.240 --> 00:14:56.180 align:middle line:84% if you're going to be doing this Boolean system biology. 00:14:56.180 --> 00:14:59.990 align:middle line:90% 00:14:59.990 --> 00:15:00.840 align:middle line:90% Here's another one. 00:15:00.840 --> 00:15:07.010 align:middle line:84% This is a fourth measure of similarity or distance here. 00:15:07.010 --> 00:15:09.080 align:middle line:90% And we've brought it up before. 00:15:09.080 --> 00:15:10.350 align:middle line:90% The correlation coefficient. 00:15:10.350 --> 00:15:15.440 align:middle line:84% This is a way of comparing this vector of RNA expression levels 00:15:15.440 --> 00:15:19.490 align:middle line:90% x sub i with y. 00:15:19.490 --> 00:15:22.412 align:middle line:84% So now instead of taking the difference between x and y, 00:15:22.412 --> 00:15:24.620 align:middle line:84% sub i, which is what we were doing with the Minkowski 00:15:24.620 --> 00:15:27.650 align:middle line:84% metrics, we're taking the product of those two. 00:15:27.650 --> 00:15:33.950 align:middle line:84% But if x and y are on some arbitrary scale, 00:15:33.950 --> 00:15:36.650 align:middle line:84% then we won't really have a way of comparing one experiment 00:15:36.650 --> 00:15:37.790 align:middle line:90% to another. 00:15:37.790 --> 00:15:39.980 align:middle line:84% This is an example of normalization. 00:15:39.980 --> 00:15:42.620 align:middle line:84% We're going to use normalization a couple of different ways 00:15:42.620 --> 00:15:44.990 align:middle line:90% in this class. 00:15:44.990 --> 00:15:46.910 align:middle line:84% But they're all related in that you 00:15:46.910 --> 00:15:49.370 align:middle line:84% want to put them on a scale that's universally 00:15:49.370 --> 00:15:50.360 align:middle line:90% recognizable. 00:15:50.360 --> 00:15:53.030 align:middle line:90% Typically 0 to 1 or -1 to 1. 00:15:53.030 --> 00:15:54.740 align:middle line:90% In this case, -1 to 1. 00:15:54.740 --> 00:16:01.370 align:middle line:84% And so what you do is in order to get it to the same center, 00:16:01.370 --> 00:16:04.040 align:middle line:84% you subtract the means from both the x and y. 00:16:04.040 --> 00:16:07.040 align:middle line:84% So now they're centroid is at 0 instead 00:16:07.040 --> 00:16:11.660 align:middle line:84% of at x bar, which is just the defined of the mean, as usual. 00:16:11.660 --> 00:16:13.880 align:middle line:84% And then to get the scale the same 00:16:13.880 --> 00:16:17.420 align:middle line:84% or on some commonly referenced scale, 00:16:17.420 --> 00:16:23.040 align:middle line:84% you divide by their product of the squares. 00:16:23.040 --> 00:16:26.143 align:middle line:84% So the result of this, as we previously 00:16:26.143 --> 00:16:27.560 align:middle line:84% discussed correlation coefficient, 00:16:27.560 --> 00:16:30.050 align:middle line:84% is that the correlation coefficient 00:16:30.050 --> 00:16:32.060 align:middle line:90% varies between minus 1 and 1. 00:16:32.060 --> 00:16:36.830 align:middle line:84% If it is 1 on slide 16, it means that they 00:16:36.830 --> 00:16:38.400 align:middle line:90% are perfectly correlated. 00:16:38.400 --> 00:16:41.560 align:middle line:84% Which is, of course rare, but bear with us. 00:16:41.560 --> 00:16:44.815 align:middle line:84% If the gene products go up and down 00:16:44.815 --> 00:16:46.940 align:middle line:84% perfectly under all the conditions and all the time 00:16:46.940 --> 00:16:48.482 align:middle line:84% points that you look at, then they're 00:16:48.482 --> 00:16:52.130 align:middle line:84% going to get a linear correlation coefficient of 1. 00:16:52.130 --> 00:16:55.520 align:middle line:84% If they're perfectly negatively correlated, 00:16:55.520 --> 00:16:59.450 align:middle line:84% then they will go up and down exactly out of phase 00:16:59.450 --> 00:17:01.220 align:middle line:84% or exactly when one is at this maximum, 00:17:01.220 --> 00:17:04.250 align:middle line:84% the other one will be at it's minimum. 00:17:04.250 --> 00:17:07.130 align:middle line:84% And if there's no linear correlation, 00:17:07.130 --> 00:17:11.430 align:middle line:84% then it will be a linear correlation coefficient 00:17:11.430 --> 00:17:12.410 align:middle line:90% of zero. 00:17:12.410 --> 00:17:16.130 align:middle line:84% Now there can be all kinds of complicated nonlinear 00:17:16.130 --> 00:17:17.119 align:middle line:90% relationships. 00:17:17.119 --> 00:17:20.839 align:middle line:84% I mean, they could be very, very codependent, 00:17:20.839 --> 00:17:24.599 align:middle line:84% say, quadratic and still have a zero 00:17:24.599 --> 00:17:27.670 align:middle line:84% for their linear correlation coefficient. 00:17:27.670 --> 00:17:29.730 align:middle line:90% So exercise for the reader. 00:17:29.730 --> 00:17:35.010 align:middle line:84% Which of these is 1 minus 1 and 0? 00:17:35.010 --> 00:17:37.770 align:middle line:84% We'll start with the upper left hand one. 00:17:37.770 --> 00:17:39.780 align:middle line:90% Is that 1? 00:17:39.780 --> 00:17:40.720 align:middle line:90% Minus 1. 00:17:40.720 --> 00:17:41.220 align:middle line:90% Good. 00:17:41.220 --> 00:17:43.740 align:middle line:90% And this one? 00:17:43.740 --> 00:17:45.180 align:middle line:90% 1 right. 00:17:45.180 --> 00:17:46.540 align:middle line:90% And zero. 00:17:46.540 --> 00:17:48.380 align:middle line:90% Great. 00:17:48.380 --> 00:17:51.870 align:middle line:84% And you will see that those have not 00:17:51.870 --> 00:17:54.930 align:middle line:84% been normalized because the correlation coefficient will 00:17:54.930 --> 00:17:56.610 align:middle line:90% do the normalization for us. 00:17:56.610 --> 00:17:58.870 align:middle line:90% In a moment, we will deal with-- 00:17:58.870 --> 00:18:00.420 align:middle line:84% we'll go back to Euclidean distances 00:18:00.420 --> 00:18:04.560 align:middle line:84% but we will do a normalization first. 00:18:04.560 --> 00:18:07.590 align:middle line:84% Now here's example of hierarchical clustering 00:18:07.590 --> 00:18:09.360 align:middle line:84% dendrograms-- just happened to be done 00:18:09.360 --> 00:18:11.680 align:middle line:90% for tumors and normal tissues. 00:18:11.680 --> 00:18:15.060 align:middle line:84% And you can see the tumors designated by T 00:18:15.060 --> 00:18:17.700 align:middle line:84% tend to cluster together, and the normal tissues 00:18:17.700 --> 00:18:21.265 align:middle line:84% on the right hand side of slide 18 tend to cluster together. 00:18:21.265 --> 00:18:22.140 align:middle line:90% But it's not perfect. 00:18:22.140 --> 00:18:23.390 align:middle line:90% There's some interpenetration. 00:18:23.390 --> 00:18:25.710 align:middle line:84% You can see this would be a challenging classification 00:18:25.710 --> 00:18:26.210 align:middle line:90% problem. 00:18:26.210 --> 00:18:31.830 align:middle line:90% 00:18:31.830 --> 00:18:36.510 align:middle line:84% The way that hierarchical tree was derived 00:18:36.510 --> 00:18:39.780 align:middle line:84% is you basically start by saying, each object-- 00:18:39.780 --> 00:18:41.850 align:middle line:84% gene-- and you're going to be measuring 00:18:41.850 --> 00:18:44.790 align:middle line:84% gene expression, which typically is RNA or protein. 00:18:44.790 --> 00:18:49.410 align:middle line:84% And you're going to call each individual RNA a cluster. 00:18:49.410 --> 00:18:50.760 align:middle line:90% It's a cluster of one. 00:18:50.760 --> 00:18:52.290 align:middle line:90% It's a trivial cluster. 00:18:52.290 --> 00:18:55.980 align:middle line:84% And then as you look through each step 00:18:55.980 --> 00:18:58.770 align:middle line:84% in the hierarchical clustering, very similar to some 00:18:58.770 --> 00:19:03.000 align:middle line:84% of the greedy algorithms we use for sequence alignment, 00:19:03.000 --> 00:19:04.710 align:middle line:84% you take the two closest clusters 00:19:04.710 --> 00:19:07.350 align:middle line:84% even if they're a cluster of one and you'll merge them. 00:19:07.350 --> 00:19:08.950 align:middle line:84% And now I call that the new cluster. 00:19:08.950 --> 00:19:11.340 align:middle line:84% Now it's a cluster of two and so on and so forth. 00:19:11.340 --> 00:19:15.400 align:middle line:84% Until finally, everything is in a cluster 00:19:15.400 --> 00:19:19.720 align:middle line:84% and you've kept tract of all the who's closest to who 00:19:19.720 --> 00:19:20.220 align:middle line:90% all the way. 00:19:20.220 --> 00:19:22.560 align:middle line:90% And that produces a tree. 00:19:22.560 --> 00:19:26.110 align:middle line:84% Now in order to generate that tree, 00:19:26.110 --> 00:19:28.740 align:middle line:84% you've got four other clustering methods. 00:19:28.740 --> 00:19:31.170 align:middle line:84% You've got the choice of the distance metric, 00:19:31.170 --> 00:19:36.490 align:middle line:84% the way of putting together the distances that you've measured. 00:19:36.490 --> 00:19:38.910 align:middle line:84% So the distance we measured can be some Minkowski 00:19:38.910 --> 00:19:40.830 align:middle line:90% or correlation coefficient. 00:19:40.830 --> 00:19:43.470 align:middle line:84% But you can put them together by either focusing 00:19:43.470 --> 00:19:46.680 align:middle line:84% on the nearest neighbor of the cluster 00:19:46.680 --> 00:19:47.820 align:middle line:90% or the furthest neighbor. 00:19:47.820 --> 00:19:49.820 align:middle line:84% That's the single link for this complete length. 00:19:49.820 --> 00:19:51.040 align:middle line:90% And we'll talk about that. 00:19:51.040 --> 00:19:53.165 align:middle line:84% And then the other methods that we won't talk about 00:19:53.165 --> 00:19:55.470 align:middle line:84% are centroid, if you can think of the center of mass 00:19:55.470 --> 00:19:57.840 align:middle line:90% for the cluster as it emerges. 00:19:57.840 --> 00:20:02.850 align:middle line:84% And the average, which is just to say the mean of all the 00:20:02.850 --> 00:20:05.580 align:middle line:90% cross cluster pairs. 00:20:05.580 --> 00:20:09.690 align:middle line:84% If you got two clusters and you do all pairwise. 00:20:09.690 --> 00:20:14.730 align:middle line:84% So let's do the single link versus the complete link. 00:20:14.730 --> 00:20:17.850 align:middle line:84% First, the single link in slide 21. 00:20:17.850 --> 00:20:21.510 align:middle line:84% And we're going to use exactly the same distance matrix 00:20:21.510 --> 00:20:23.980 align:middle line:90% for both of these examples. 00:20:23.980 --> 00:20:26.472 align:middle line:84% So you don't have to shift gears too much. 00:20:26.472 --> 00:20:28.680 align:middle line:84% The main thing is the only thing we're going to shift 00:20:28.680 --> 00:20:30.540 align:middle line:84% is between single and [INAUDIBLE].. 00:20:30.540 --> 00:20:34.540 align:middle line:84% And we're using Euclidean distance here, 00:20:34.540 --> 00:20:37.500 align:middle line:84% which is square root sum of square. 00:20:37.500 --> 00:20:41.655 align:middle line:84% And here you can see AB are the two closest and A 00:20:41.655 --> 00:20:43.570 align:middle line:90% and B are the furthest apart. 00:20:43.570 --> 00:20:46.500 align:middle line:84% And so the Euclidean distance for AB is 2 00:20:46.500 --> 00:20:50.020 align:middle line:90% and AB is 6, for example. 00:20:50.020 --> 00:20:54.510 align:middle line:84% And so in the single length method, 00:20:54.510 --> 00:20:59.460 align:middle line:84% this kicks in once you start collapsing the first link. 00:20:59.460 --> 00:21:01.470 align:middle line:84% So you make the link between A and B, 00:21:01.470 --> 00:21:03.720 align:middle line:84% that's obvious because it's the shortest distance. 00:21:03.720 --> 00:21:06.150 align:middle line:84% But how you collapse it depends on-- 00:21:06.150 --> 00:21:08.160 align:middle line:84% how you compare it to other points 00:21:08.160 --> 00:21:10.170 align:middle line:84% is what the single link method is about. 00:21:10.170 --> 00:21:16.440 align:middle line:84% So now AB is going to be treated as one unit-- one cluster. 00:21:16.440 --> 00:21:19.253 align:middle line:84% And you're going to ask, how far is AB from C? 00:21:19.253 --> 00:21:20.670 align:middle line:84% Well, since this is a single link, 00:21:20.670 --> 00:21:23.700 align:middle line:84% you're interested in the closest distance and that's BC. 00:21:23.700 --> 00:21:29.460 align:middle line:84% And BC, from the very first leftmost matrix was three. 00:21:29.460 --> 00:21:33.750 align:middle line:84% So you fill in for the AB to C at three. 00:21:33.750 --> 00:21:36.780 align:middle line:84% And similarly the D is the closest point. 00:21:36.780 --> 00:21:39.240 align:middle line:90% From AB to D is five. 00:21:39.240 --> 00:21:42.600 align:middle line:84% It's the diagonal from B to D and so on. 00:21:42.600 --> 00:21:45.960 align:middle line:84% And that's how you've lost the top row, 00:21:45.960 --> 00:21:47.310 align:middle line:90% and it's three and five. 00:21:47.310 --> 00:21:51.012 align:middle line:84% And now when you compare these, the next link 00:21:51.012 --> 00:21:52.470 align:middle line:84% you're going to make is going to be 00:21:52.470 --> 00:21:54.930 align:middle line:84% the smallest one in the whole table, which is three. 00:21:54.930 --> 00:21:59.188 align:middle line:84% And that happens to be the AB cluster is closest to C. 00:21:59.188 --> 00:22:01.230 align:middle line:84% And so that's going to be the next link you make. 00:22:01.230 --> 00:22:03.990 align:middle line:84% And then the rest of the game is over. 00:22:03.990 --> 00:22:07.170 align:middle line:84% It's just the ABC cluster is near D. 00:22:07.170 --> 00:22:09.843 align:middle line:84% So you can already imagine in your mind what that tree is 00:22:09.843 --> 00:22:10.635 align:middle line:90% going to look like. 00:22:10.635 --> 00:22:13.063 align:middle line:84% A and B are going closest, and then you bring in C. 00:22:13.063 --> 00:22:14.730 align:middle line:84% And then you finally bring in D. And you 00:22:14.730 --> 00:22:17.520 align:middle line:84% might think at this point, that's the only way to do this. 00:22:17.520 --> 00:22:20.375 align:middle line:84% But the complete length version of this 00:22:20.375 --> 00:22:21.690 align:middle line:90% is exactly the same matrix. 00:22:21.690 --> 00:22:23.480 align:middle line:90% You start up the same place. 00:22:23.480 --> 00:22:25.370 align:middle line:84% AB is still the closest one so that's 00:22:25.370 --> 00:22:27.230 align:middle line:84% the one you're going to link together first. 00:22:27.230 --> 00:22:30.650 align:middle line:84% But how you score it as you do this linking 00:22:30.650 --> 00:22:31.760 align:middle line:90% is a little different now. 00:22:31.760 --> 00:22:34.340 align:middle line:84% Because now you're concerned about completely 00:22:34.340 --> 00:22:39.740 align:middle line:84% all of the distances from the AB cluster to, say, C. 00:22:39.740 --> 00:22:42.410 align:middle line:84% Now B is close but A is far away. 00:22:42.410 --> 00:22:44.790 align:middle line:84% And we're interested in that greater distance as well. 00:22:44.790 --> 00:22:48.980 align:middle line:84% And so the whole cluster of AB gets the distance from A 00:22:48.980 --> 00:22:51.080 align:middle line:84% to C, the longest distance, five. 00:22:51.080 --> 00:22:53.180 align:middle line:84% And so five goes in that position. 00:22:53.180 --> 00:22:56.700 align:middle line:84% And six goes as long as this is from AB to D, which again, 00:22:56.700 --> 00:23:00.020 align:middle line:84% is A to D. And so now you have a completely different-- 00:23:00.020 --> 00:23:02.990 align:middle line:84% just toggle back and forth between slide 22 and, 21 00:23:02.990 --> 00:23:04.880 align:middle line:84% and you can see it went from three, five, 00:23:04.880 --> 00:23:07.250 align:middle line:90% four to five, six four. 00:23:07.250 --> 00:23:09.680 align:middle line:84% So now when you make the next link-- 00:23:09.680 --> 00:23:11.780 align:middle line:84% the first link is obvious in both case, AB. 00:23:11.780 --> 00:23:14.360 align:middle line:84% The next link is now CD, because the smallest 00:23:14.360 --> 00:23:20.750 align:middle line:84% one in that two-by-two matrix is four. 00:23:20.750 --> 00:23:22.190 align:middle line:84% And that happens between C and D. 00:23:22.190 --> 00:23:23.690 align:middle line:84% And now C and D are the next link. 00:23:23.690 --> 00:23:24.905 align:middle line:90% And then now the game's over. 00:23:24.905 --> 00:23:29.615 align:middle line:84% You connect CD and AB and the link is-- 00:23:29.615 --> 00:23:32.240 align:middle line:84% so now you can see you're going to get two very different trees 00:23:32.240 --> 00:23:37.490 align:middle line:84% from the single link method on the left hand side of slide 23 00:23:37.490 --> 00:23:43.070 align:middle line:84% is AB bringing in C and finally D. 00:23:43.070 --> 00:23:45.650 align:middle line:84% While the complete link, you have 00:23:45.650 --> 00:23:49.670 align:middle line:84% AB and CD as two separate pairs and then they come together. 00:23:49.670 --> 00:23:52.800 align:middle line:84% Now this is the simplest possible example 00:23:52.800 --> 00:23:54.360 align:middle line:90% I could have come up with. 00:23:54.360 --> 00:23:57.590 align:middle line:84% But I think it combined with the next couple of slides 00:23:57.590 --> 00:24:02.150 align:middle line:84% will drive home the importance of the clustering method 00:24:02.150 --> 00:24:03.980 align:middle line:90% that you're using. 00:24:03.980 --> 00:24:06.680 align:middle line:84% Here the linkage method, part of it. 00:24:06.680 --> 00:24:09.620 align:middle line:84% Again, focus in on the far left-hand side 00:24:09.620 --> 00:24:16.910 align:middle line:84% of where you have more compact spherical circular clusters 00:24:16.910 --> 00:24:19.430 align:middle line:90% or more elongated ones. 00:24:19.430 --> 00:24:23.090 align:middle line:84% We're going to take three examples here. 00:24:23.090 --> 00:24:25.230 align:middle line:84% Spherical, elongated, something in between. 00:24:25.230 --> 00:24:28.790 align:middle line:84% A single link in the middle of slide 25. 00:24:28.790 --> 00:24:31.520 align:middle line:84% And then complete link on the far right-hand side. 00:24:31.520 --> 00:24:33.442 align:middle line:84% In a single length, Now, you can kind of 00:24:33.442 --> 00:24:35.650 align:middle line:84% see why they're called single link and complete link. 00:24:35.650 --> 00:24:37.520 align:middle line:84% This is a different way of visualizing them. 00:24:37.520 --> 00:24:41.570 align:middle line:84% Here the single length does a great job 00:24:41.570 --> 00:24:44.450 align:middle line:84% for the top and bottom clusters-- 00:24:44.450 --> 00:24:47.522 align:middle line:84% the circular and the linear forms. 00:24:47.522 --> 00:24:49.730 align:middle line:84% But when you start getting something that's somewhere 00:24:49.730 --> 00:24:54.680 align:middle line:84% in between, you get this weird single link 00:24:54.680 --> 00:24:58.010 align:middle line:84% that, at least to my eye, connects up 00:24:58.010 --> 00:25:00.290 align:middle line:84% the two clusters along the bottom here 00:25:00.290 --> 00:25:04.700 align:middle line:84% and then leaves this little cluster as the second cluster. 00:25:04.700 --> 00:25:06.240 align:middle line:84% The complete link on the other hand, 00:25:06.240 --> 00:25:09.770 align:middle line:84% where you measure all the distances 00:25:09.770 --> 00:25:12.980 align:middle line:84% between previous clusters and the new clusters 00:25:12.980 --> 00:25:16.160 align:middle line:84% you're going to be adding goes well on the top one. 00:25:16.160 --> 00:25:18.500 align:middle line:84% And the middle one but does this weird thing 00:25:18.500 --> 00:25:22.310 align:middle line:84% with the elongated clusters where 00:25:22.310 --> 00:25:25.280 align:middle line:84% it takes a small cluster that seems, to my eye, 00:25:25.280 --> 00:25:29.130 align:middle line:84% to include things that are not that related. 00:25:29.130 --> 00:25:33.440 align:middle line:84% So the single link does well on the top and bottom 00:25:33.440 --> 00:25:36.330 align:middle line:84% and the complete link does well on the top middle. 00:25:36.330 --> 00:25:37.885 align:middle line:84% And so you can see that depending 00:25:37.885 --> 00:25:40.010 align:middle line:84% on what you think your data are going to look like, 00:25:40.010 --> 00:25:42.710 align:middle line:84% whether they're going to look closely spaced 00:25:42.710 --> 00:25:47.780 align:middle line:84% but compact clusters that might be single length 00:25:47.780 --> 00:25:51.410 align:middle line:84% and more elongated but separated by distance, then 00:25:51.410 --> 00:25:54.960 align:middle line:90% you might want a complete link. 00:25:54.960 --> 00:25:59.010 align:middle line:84% So now where are we in this overall road map in slide 26? 00:25:59.010 --> 00:26:00.660 align:middle line:84% We've been moving from the right where 00:26:00.660 --> 00:26:02.250 align:middle line:84% we've gone from clustering methods, 00:26:02.250 --> 00:26:04.920 align:middle line:84% supervised, unsupervised, hierarchical, non-hierarchical. 00:26:04.920 --> 00:26:08.040 align:middle line:84% We've gone through distance metrics and linkage metrics. 00:26:08.040 --> 00:26:10.380 align:middle line:84% Now let's see how it plays out with one 00:26:10.380 --> 00:26:13.680 align:middle line:84% particular non-hierarchical method. 00:26:13.680 --> 00:26:15.180 align:middle line:84% We've been focusing on hierarchical. 00:26:15.180 --> 00:26:17.820 align:middle line:84% Now we're going to go non-hierarchical k-means 00:26:17.820 --> 00:26:21.480 align:middle line:84% and bring in issues of data normalization. 00:26:21.480 --> 00:26:24.520 align:middle line:90% 00:26:24.520 --> 00:26:30.700 align:middle line:84% In this case, gene normalization where 00:26:30.700 --> 00:26:35.320 align:middle line:84% we're trying to put genes that are wildly 00:26:35.320 --> 00:26:38.620 align:middle line:84% different in their absolute value of expression 00:26:38.620 --> 00:26:42.610 align:middle line:90% on the same scale. 00:26:42.610 --> 00:26:45.340 align:middle line:84% So one might be a very small fluctuation 00:26:45.340 --> 00:26:47.770 align:middle line:90% at a kind of medium level. 00:26:47.770 --> 00:26:49.750 align:middle line:84% Another one could be very large fluctuation 00:26:49.750 --> 00:26:53.380 align:middle line:84% from baseline up to a very high level. 00:26:53.380 --> 00:26:55.960 align:middle line:84% And you want to account for this difference in baseline 00:26:55.960 --> 00:26:58.160 align:middle line:90% and this difference in scale. 00:26:58.160 --> 00:26:59.530 align:middle line:90% And so what you do-- 00:26:59.530 --> 00:27:02.500 align:middle line:84% and that's what all these three little normalized expression 00:27:02.500 --> 00:27:03.220 align:middle line:90% plots are. 00:27:03.220 --> 00:27:09.250 align:middle line:84% Is they represent this table, as I've mentioned, 00:27:09.250 --> 00:27:15.225 align:middle line:84% of genes along the vertical axis or gene expression levels-- 00:27:15.225 --> 00:27:16.600 align:middle line:84% genes that are going-- were we're 00:27:16.600 --> 00:27:19.017 align:middle line:84% going to measure expression levels along the vertical axis 00:27:19.017 --> 00:27:22.730 align:middle line:84% and the points or the conditions along the horizontal axis. 00:27:22.730 --> 00:27:24.700 align:middle line:84% And so we have two representations. 00:27:24.700 --> 00:27:30.520 align:middle line:84% One is this kind of dot cluster envelope representation 00:27:30.520 --> 00:27:35.230 align:middle line:84% in the middle where you have, in this case, three dimensions. 00:27:35.230 --> 00:27:38.230 align:middle line:84% But in a case that's a little harder to visualize, 00:27:38.230 --> 00:27:39.220 align:middle line:90% multi dimensions-- 00:27:39.220 --> 00:27:43.360 align:middle line:90% 17, 15 dimensions. 00:27:43.360 --> 00:27:47.740 align:middle line:84% That's one representation where the origin is essentially 00:27:47.740 --> 00:27:52.240 align:middle line:84% the mean where you normalize it, the mean becomes zero. 00:27:52.240 --> 00:27:55.090 align:middle line:84% And then the distance from that origin 00:27:55.090 --> 00:27:57.485 align:middle line:84% can be either positive or negative 00:27:57.485 --> 00:27:59.860 align:middle line:84% and it's the number of standard deviations from the mean. 00:27:59.860 --> 00:28:02.480 align:middle line:84% That's the way that we're going to normalize it. 00:28:02.480 --> 00:28:04.270 align:middle line:84% So each of these individual plots 00:28:04.270 --> 00:28:07.510 align:middle line:84% would be the average behavior in each of these clusters. 00:28:07.510 --> 00:28:12.112 align:middle line:84% And we'll take a look at that, the average and the deviation 00:28:12.112 --> 00:28:12.820 align:middle line:90% from the average. 00:28:12.820 --> 00:28:14.673 align:middle line:84% But the units here in the vertical axis 00:28:14.673 --> 00:28:16.840 align:middle line:84% of these little plots will be normalized expression. 00:28:16.840 --> 00:28:19.330 align:middle line:84% Number of standard deviations within the cluster 00:28:19.330 --> 00:28:20.725 align:middle line:90% from the mean of the cluster. 00:28:20.725 --> 00:28:25.170 align:middle line:90% 00:28:25.170 --> 00:28:27.950 align:middle line:84% Now when we're going to be measuring distances 00:28:27.950 --> 00:28:34.880 align:middle line:84% between clusters where we have the same normalized expression 00:28:34.880 --> 00:28:36.860 align:middle line:90% data table-- 00:28:36.860 --> 00:28:38.600 align:middle line:84% and this is the three dimension-- 00:28:38.600 --> 00:28:40.430 align:middle line:84% this is the three dimension, in this case, 00:28:40.430 --> 00:28:42.860 align:middle line:84% or multi-dimensional representation. 00:28:42.860 --> 00:28:45.710 align:middle line:84% Where the origin is zero or the mean for each 00:28:45.710 --> 00:28:51.770 align:middle line:84% of for each of the axes and the distance from that zero mean 00:28:51.770 --> 00:28:53.460 align:middle line:84% is the number of standard deviations. 00:28:53.460 --> 00:28:56.180 align:middle line:84% And when we'll measure it will measure this hidden distance 00:28:56.180 --> 00:28:58.640 align:middle line:84% the square root of sum of the squares over all 00:28:58.640 --> 00:29:00.410 align:middle line:90% the dimensions. 00:29:00.410 --> 00:29:06.530 align:middle line:84% And I want to emphasize that each of these clusters 00:29:06.530 --> 00:29:11.540 align:middle line:84% is not point-- if gene expression were regulated 00:29:11.540 --> 00:29:13.130 align:middle line:84% by transcription factors which bound 00:29:13.130 --> 00:29:18.050 align:middle line:84% to every site with exactly the same binding constant, 00:29:18.050 --> 00:29:19.700 align:middle line:84% then you might-- and if there were 00:29:19.700 --> 00:29:23.330 align:middle line:84% selection pressures forcing this to happen where 00:29:23.330 --> 00:29:27.110 align:middle line:84% the forcing everybody to be precisely-- 00:29:27.110 --> 00:29:29.540 align:middle line:84% everything in a cluster be precisely regulated, 00:29:29.540 --> 00:29:31.400 align:middle line:84% then these clusters would be really tight. 00:29:31.400 --> 00:29:33.560 align:middle line:84% There'd be almost a point and there'd 00:29:33.560 --> 00:29:35.100 align:middle line:90% be no overlap between them. 00:29:35.100 --> 00:29:38.210 align:middle line:84% But in reality, there is no such selective pressures. 00:29:38.210 --> 00:29:40.310 align:middle line:84% And the transcription factors, as a result, 00:29:40.310 --> 00:29:44.000 align:middle line:84% are possibly purposefully diverse. 00:29:44.000 --> 00:29:46.160 align:middle line:84% And you get these spread clusters. 00:29:46.160 --> 00:29:54.050 align:middle line:84% And so these little blue bars on each of the points on these 00:29:54.050 --> 00:29:56.480 align:middle line:84% time series plots of normalized expression-- 00:29:56.480 --> 00:29:58.760 align:middle line:90% the three times three plots-- 00:29:58.760 --> 00:30:00.890 align:middle line:84% those little blue bars don't necessarily 00:30:00.890 --> 00:30:02.330 align:middle line:90% represent experimental error. 00:30:02.330 --> 00:30:07.160 align:middle line:84% They represent the diversity of gene expression 00:30:07.160 --> 00:30:08.820 align:middle line:90% within a cluster. 00:30:08.820 --> 00:30:13.070 align:middle line:84% Now if you've accidentally made more-- 00:30:13.070 --> 00:30:16.160 align:middle line:84% assigned fewer clusters than is say the natural number 00:30:16.160 --> 00:30:19.940 align:middle line:84% of clusters, then you'll get more dispersion in that number 00:30:19.940 --> 00:30:21.920 align:middle line:90% than you might want. 00:30:21.920 --> 00:30:24.200 align:middle line:84% And that might be a tip off that you actually 00:30:24.200 --> 00:30:27.140 align:middle line:84% need more cluster-- you need to divide it up into more clusters 00:30:27.140 --> 00:30:28.680 align:middle line:90% and bring this down. 00:30:28.680 --> 00:30:34.040 align:middle line:84% Obviously, if you break it up in too many clusters that 00:30:34.040 --> 00:30:36.620 align:middle line:84% will have a different set of pathologies, 00:30:36.620 --> 00:30:41.480 align:middle line:84% you'll have the distance between clusters, some of the clusters 00:30:41.480 --> 00:30:43.580 align:middle line:90% will be abnormally close. 00:30:43.580 --> 00:30:46.170 align:middle line:84% They'll be almost as if they were right touching each other. 00:30:46.170 --> 00:30:49.160 align:middle line:84% And so that's the tip off that you have too many clusters. 00:30:49.160 --> 00:30:51.440 align:middle line:84% And the number of clusters is something 00:30:51.440 --> 00:30:54.890 align:middle line:84% that you can either determine in advance 00:30:54.890 --> 00:30:57.080 align:middle line:90% or you can discover as you go. 00:30:57.080 --> 00:31:00.680 align:middle line:84% But those are the examples of criteria you might use. 00:31:00.680 --> 00:31:02.780 align:middle line:84% Too much dispersion in those little blue air 00:31:02.780 --> 00:31:05.690 align:middle line:84% bars means that you've tried to lump too many things into one 00:31:05.690 --> 00:31:06.440 align:middle line:90% cluster. 00:31:06.440 --> 00:31:09.890 align:middle line:84% And too short a distance between adjacent clusters 00:31:09.890 --> 00:31:13.730 align:middle line:84% means that you probably divided them too finally. 00:31:13.730 --> 00:31:18.380 align:middle line:84% Now how do we begin to assess whether the clustering methods 00:31:18.380 --> 00:31:21.772 align:middle line:90% that we're using are optimal? 00:31:21.772 --> 00:31:23.480 align:middle line:84% We've talked about all the different kind 00:31:23.480 --> 00:31:26.570 align:middle line:84% of clustering methods that you can use. 00:31:26.570 --> 00:31:31.640 align:middle line:84% One of the ways to assess whether they're are optimal-- 00:31:31.640 --> 00:31:33.080 align:middle line:90% we'll talk about many. 00:31:33.080 --> 00:31:37.910 align:middle line:84% But one is to look way outside the box to some resource 00:31:37.910 --> 00:31:40.970 align:middle line:84% that maybe the biological community has 00:31:40.970 --> 00:31:43.520 align:middle line:90% curated functions. 00:31:43.520 --> 00:31:47.270 align:middle line:84% Now they may mean this in very vague and frustrating ways, 00:31:47.270 --> 00:31:51.230 align:middle line:84% but we believe that they have done a good job. 00:31:51.230 --> 00:31:54.590 align:middle line:84% And certainly an independent job of the experiment 00:31:54.590 --> 00:31:55.880 align:middle line:90% that's being done. 00:31:55.880 --> 00:32:01.160 align:middle line:84% The experiment that's being done is a fresh comprehensive gene 00:32:01.160 --> 00:32:02.550 align:middle line:90% expression analysis. 00:32:02.550 --> 00:32:05.090 align:middle line:84% And so if you find a cluster from a gene expression 00:32:05.090 --> 00:32:08.390 align:middle line:84% analysis that coincides with this completely independently 00:32:08.390 --> 00:32:12.177 align:middle line:84% curated database of functional categories, 00:32:12.177 --> 00:32:13.760 align:middle line:84% doesn't matter what [INAUDIBLE] means. 00:32:13.760 --> 00:32:17.120 align:middle line:84% This is some abbreviation for an Institute. 00:32:17.120 --> 00:32:19.880 align:middle line:84% Nor does it really matter what the gene names here mean. 00:32:19.880 --> 00:32:23.720 align:middle line:84% But what you will find is that a particular set of genes, 00:32:23.720 --> 00:32:25.760 align:middle line:84% once you look it up in the database, 00:32:25.760 --> 00:32:29.810 align:middle line:84% will set off a flag that says ribosome. 00:32:29.810 --> 00:32:31.730 align:middle line:84% And you know what ribosome means. 00:32:31.730 --> 00:32:34.220 align:middle line:90% And others will be unknowns. 00:32:34.220 --> 00:32:36.490 align:middle line:84% But the point is that these will be 00:32:36.490 --> 00:32:43.190 align:middle line:84% an orderly set, a set that's perhaps enriched-- 00:32:43.190 --> 00:32:44.490 align:middle line:90% unexpectedly enriched. 00:32:44.490 --> 00:32:48.590 align:middle line:84% And you want to have some way of quantitative your surprise 00:32:48.590 --> 00:32:52.010 align:middle line:84% at finding this many of one type of function 00:32:52.010 --> 00:32:53.540 align:middle line:90% in your RNA cluster. 00:32:53.540 --> 00:32:55.250 align:middle line:84% In a way, this is what you hope to find. 00:32:55.250 --> 00:32:56.330 align:middle line:90% It's a pleasant surprise. 00:32:56.330 --> 00:32:59.000 align:middle line:84% You want your clusters to have some coherence 00:32:59.000 --> 00:32:59.750 align:middle line:90% in their function. 00:32:59.750 --> 00:33:02.450 align:middle line:84% You also want to find some surprises, either unknowns 00:33:02.450 --> 00:33:07.180 align:middle line:84% or new combinations of functions that you didn't expect. 00:33:07.180 --> 00:33:11.045 align:middle line:84% Now this is an example of a clustering experiment. 00:33:11.045 --> 00:33:14.030 align:middle line:84% It's a popular way of representing it. 00:33:14.030 --> 00:33:17.540 align:middle line:84% Here is the trees we've been talking about here 00:33:17.540 --> 00:33:20.300 align:middle line:90% that the closest-- 00:33:20.300 --> 00:33:23.990 align:middle line:84% the tips, the leaves here are individual genes. 00:33:23.990 --> 00:33:27.320 align:middle line:84% You can barely see them at this scale. 00:33:27.320 --> 00:33:32.095 align:middle line:84% This is a small subset of the human genes. 00:33:32.095 --> 00:33:33.470 align:middle line:84% This is a RNA expression that has 00:33:33.470 --> 00:33:36.830 align:middle line:84% been measured over time course of serum stimulation. 00:33:36.830 --> 00:33:42.210 align:middle line:84% And considering the previous slide 00:33:42.210 --> 00:33:45.960 align:middle line:84% of different functional categorization, what 00:33:45.960 --> 00:33:48.930 align:middle line:84% you want as you hierarchically arrange these things, 00:33:48.930 --> 00:33:51.390 align:middle line:84% you've got time as the natural axis horizontally. 00:33:51.390 --> 00:33:53.190 align:middle line:84% And then you've tried to sort them 00:33:53.190 --> 00:33:57.165 align:middle line:84% so they're close together in the hierarchical tree. 00:33:57.165 --> 00:34:00.300 align:middle line:90% 00:34:00.300 --> 00:34:03.960 align:middle line:84% And you've and you've represented whether they're 00:34:03.960 --> 00:34:07.200 align:middle line:84% greatly induced or greatly suppressed 00:34:07.200 --> 00:34:10.739 align:middle line:90% during this serum stimulation. 00:34:10.739 --> 00:34:13.500 align:middle line:84% You take one of the time-- zero time point is the reference 00:34:13.500 --> 00:34:14.050 align:middle line:90% point. 00:34:14.050 --> 00:34:16.380 align:middle line:84% And then greatly increase or decrease 00:34:16.380 --> 00:34:19.929 align:middle line:84% represented by red and green respectively. 00:34:19.929 --> 00:34:21.600 align:middle line:84% And then within each of these clusters, 00:34:21.600 --> 00:34:25.620 align:middle line:84% you have little zones where they all 00:34:25.620 --> 00:34:29.230 align:middle line:84% have the same kind of pattern of black, gray, green, and red. 00:34:29.230 --> 00:34:34.080 align:middle line:84% And so for example, the E at the bottom in the red zone 00:34:34.080 --> 00:34:36.727 align:middle line:84% here is wound healing and tissue remodeling. 00:34:36.727 --> 00:34:38.310 align:middle line:84% And these are the genes that you might 00:34:38.310 --> 00:34:45.179 align:middle line:84% expect to be enriched in a growth stimulation 00:34:45.179 --> 00:34:47.250 align:middle line:84% paradigm, such as the one here where you're 00:34:47.250 --> 00:34:50.070 align:middle line:84% simulating with serum stimulation of fibroblasts. 00:34:50.070 --> 00:34:53.488 align:middle line:84% This is a particular example of how you might-- 00:34:53.488 --> 00:34:55.530 align:middle line:84% but you might want to quantitate this rather than 00:34:55.530 --> 00:34:56.940 align:middle line:90% just kind of showing it here. 00:34:56.940 --> 00:34:59.023 align:middle line:84% And we're going to walk through exactly how you do 00:34:59.023 --> 00:35:00.780 align:middle line:90% that quantitation in a moment. 00:35:00.780 --> 00:35:04.650 align:middle line:84% This is just a quick snapshot of how far this clustering goes. 00:35:04.650 --> 00:35:06.630 align:middle line:84% Actually, it goes well beyond biology. 00:35:06.630 --> 00:35:10.650 align:middle line:84% But here's something that's for slide 32 out 00:35:10.650 --> 00:35:13.800 align:middle line:90% of the range of RNA expression. 00:35:13.800 --> 00:35:17.820 align:middle line:84% Here we have compounds on the vertical axis and targets, 00:35:17.820 --> 00:35:20.958 align:middle line:84% meaning proteins on the horizontal axis. 00:35:20.958 --> 00:35:22.500 align:middle line:84% And you can see all these connections 00:35:22.500 --> 00:35:27.630 align:middle line:84% between different cancer therapeutics, different cancer 00:35:27.630 --> 00:35:33.150 align:middle line:84% cell lines and potential targets. 00:35:33.150 --> 00:35:36.480 align:middle line:90% But now back to the RNA. 00:35:36.480 --> 00:35:44.550 align:middle line:84% And we want to ask how do we assess the RNA array data 00:35:44.550 --> 00:35:48.150 align:middle line:84% collection, the clustering methods? 00:35:48.150 --> 00:35:49.950 align:middle line:90% And how do we go-- 00:35:49.950 --> 00:35:52.690 align:middle line:84% and how do we go beyond that in various directions, 00:35:52.690 --> 00:35:55.290 align:middle line:84% both as validation of the technical aspects 00:35:55.290 --> 00:35:59.310 align:middle line:84% but as showing that we're actually doing discovery 00:35:59.310 --> 00:36:03.360 align:middle line:90% and getting at mechanism. 00:36:03.360 --> 00:36:07.770 align:middle line:84% So one of the various methods we've used-- we already 00:36:07.770 --> 00:36:11.310 align:middle line:84% mentioned looking for functional categories, 00:36:11.310 --> 00:36:13.380 align:middle line:84% but another one is looking for motifs. 00:36:13.380 --> 00:36:15.480 align:middle line:84% If we find a consistent set of motifs, 00:36:15.480 --> 00:36:19.000 align:middle line:84% this is part of the validation process as well. 00:36:19.000 --> 00:36:26.520 align:middle line:84% And these are some of the examples of algorithms. 00:36:26.520 --> 00:36:30.600 align:middle line:84% The first one that leaps to mind when mathematicians 00:36:30.600 --> 00:36:33.270 align:middle line:84% and physicists enter the field and that one 00:36:33.270 --> 00:36:36.150 align:middle line:84% that we've used a great advantage in the sequence 00:36:36.150 --> 00:36:40.680 align:middle line:84% searching part of this course was oligonucleotide frequency. 00:36:40.680 --> 00:36:43.260 align:middle line:84% So you can use short oligonucleotides 00:36:43.260 --> 00:36:48.300 align:middle line:84% as convenient hashing keys or as ways of doing the lookup-- 00:36:48.300 --> 00:36:53.790 align:middle line:84% a very rapid lookup for sequences and in finding 00:36:53.790 --> 00:36:54.660 align:middle line:90% matches. 00:36:54.660 --> 00:36:58.620 align:middle line:84% And this is even more appropriate 00:36:58.620 --> 00:37:01.770 align:middle line:84% here for the motifs involved in transcriptional regulation 00:37:01.770 --> 00:37:06.030 align:middle line:84% because we from a variety of biological and chemical 00:37:06.030 --> 00:37:08.160 align:middle line:84% crystallographic studies that the motifs 00:37:08.160 --> 00:37:13.050 align:middle line:84% are in the range of 7 to 10 nucleotides often-- 00:37:13.050 --> 00:37:15.550 align:middle line:84% base pairs in double stranded DNA. 00:37:15.550 --> 00:37:18.520 align:middle line:84% And so you can use oligonucleotide frequencies. 00:37:18.520 --> 00:37:20.820 align:middle line:84% However, they're limited in that they're not 00:37:20.820 --> 00:37:24.240 align:middle line:84% as rich as the weight matrices that we got when we've 00:37:24.240 --> 00:37:25.765 align:middle line:90% got a multi-sequence alignment. 00:37:25.765 --> 00:37:28.140 align:middle line:84% And when we were talking about multi-sequence alignments, 00:37:28.140 --> 00:37:34.290 align:middle line:84% we pointed out that it was hard to get the algorithms to scale 00:37:34.290 --> 00:37:35.700 align:middle line:90% beyond the pairwise. 00:37:35.700 --> 00:37:42.450 align:middle line:84% Because pairwise was n squared where n is the sequence length. 00:37:42.450 --> 00:37:49.020 align:middle line:84% And then as you go to multi-sequence alignment, 00:37:49.020 --> 00:37:51.300 align:middle line:84% it goes up exponentially with the number of sequences. 00:37:51.300 --> 00:37:53.730 align:middle line:84% You want the number of sequences to be large though. 00:37:53.730 --> 00:37:55.770 align:middle line:84% Because the larger it is, the more 00:37:55.770 --> 00:37:58.080 align:middle line:84% you learn about the characteristics 00:37:58.080 --> 00:38:00.420 align:middle line:90% of that family of sequences. 00:38:00.420 --> 00:38:02.753 align:middle line:84% So anyway, Gibbs sampling was one of the methods 00:38:02.753 --> 00:38:04.920 align:middle line:84% that we said that we would put off to a later class. 00:38:04.920 --> 00:38:05.940 align:middle line:90% This is the later class. 00:38:05.940 --> 00:38:09.390 align:middle line:84% We'll talk about Gibbs sampling as a way of-- 00:38:09.390 --> 00:38:13.320 align:middle line:84% the idea of sampling this very large space 00:38:13.320 --> 00:38:18.810 align:middle line:84% where the large number of multi-sequences-- the multiple 00:38:18.810 --> 00:38:21.615 align:middle line:84% sequences you're comparing is that you 00:38:21.615 --> 00:38:23.490 align:middle line:84% don't want to get trapped in a local minimum. 00:38:23.490 --> 00:38:26.160 align:middle line:84% You can have these really greedy steepest descent algorithms, 00:38:26.160 --> 00:38:28.380 align:middle line:84% but you'll get to the bottom of that pit 00:38:28.380 --> 00:38:31.740 align:middle line:84% but you won't necessarily find the global. 00:38:31.740 --> 00:38:33.300 align:middle line:84% If the sampling space is too large, 00:38:33.300 --> 00:38:35.880 align:middle line:84% even sampling won't save you because you'll 00:38:35.880 --> 00:38:39.370 align:middle line:84% sample a lot of little local [INAUDIBLE].. 00:38:39.370 --> 00:38:41.290 align:middle line:84% But, anyway, Gibbs is an example where you 00:38:41.290 --> 00:38:44.740 align:middle line:90% use randomization to find it. 00:38:44.740 --> 00:38:50.620 align:middle line:84% Mean as example of maximization of expectations and [INAUDIBLE] 00:38:50.620 --> 00:38:52.970 align:middle line:84% and so forth are other ways of doing it. 00:38:52.970 --> 00:38:55.180 align:middle line:84% We're going to really focus in on one of these. 00:38:55.180 --> 00:38:56.350 align:middle line:90% Can't cover everything. 00:38:56.350 --> 00:38:59.050 align:middle line:84% We've talked about Gibbs sampling. 00:38:59.050 --> 00:39:01.690 align:middle line:84% And we want to put it in the context of-- 00:39:01.690 --> 00:39:03.980 align:middle line:84% and the thing that might be appealing. 00:39:03.980 --> 00:39:06.100 align:middle line:90% Why can't we just-- 00:39:06.100 --> 00:39:11.230 align:middle line:84% if the program for transcription factor regulation 00:39:11.230 --> 00:39:13.655 align:middle line:84% is inherent in the genome, then we 00:39:13.655 --> 00:39:15.280 align:middle line:84% should just look at the genome sequence 00:39:15.280 --> 00:39:20.350 align:middle line:84% and be able to see patterns of motifs in front of genes. 00:39:20.350 --> 00:39:23.380 align:middle line:90% 00:39:23.380 --> 00:39:27.400 align:middle line:84% And then find clusters of genes that are expressed and so on. 00:39:27.400 --> 00:39:29.200 align:middle line:90% The problem with that picture-- 00:39:29.200 --> 00:39:32.330 align:middle line:84% even for one of the best case scenarios, [INAUDIBLE],, 00:39:32.330 --> 00:39:35.830 align:middle line:90% which is about 12 mega bases-- 00:39:35.830 --> 00:39:37.750 align:middle line:84% as I said, these transcriptional control sites 00:39:37.750 --> 00:39:41.320 align:middle line:84% are about seven bases, let's say, of inflammation. 00:39:41.320 --> 00:39:44.110 align:middle line:84% Here's one that will be a star for a few 00:39:44.110 --> 00:39:49.900 align:middle line:84% slides here today after now and I have break. 00:39:49.900 --> 00:39:51.920 align:middle line:90% This is GCM4. 00:39:51.920 --> 00:39:54.580 align:middle line:84% You can see it has five really full scale, 00:39:54.580 --> 00:39:57.980 align:middle line:90% two-bit conserved bases. 00:39:57.980 --> 00:40:01.240 align:middle line:84% And then the rest of the bases in this motif-- 00:40:01.240 --> 00:40:03.520 align:middle line:84% the other five bases might add up 00:40:03.520 --> 00:40:06.700 align:middle line:84% to another two bases of information or 14 bits 00:40:06.700 --> 00:40:07.985 align:middle line:90% all together. 00:40:07.985 --> 00:40:09.610 align:middle line:84% Now 14 bits, you can think of that as 4 00:40:09.610 --> 00:40:16.330 align:middle line:84% to the seventh power about 1 match every 16,000 bases. 00:40:16.330 --> 00:40:19.210 align:middle line:84% Now if you have a 12-megabase genome, 00:40:19.210 --> 00:40:21.070 align:middle line:84% and since it's not symmetric, you 00:40:21.070 --> 00:40:22.420 align:middle line:90% have to look at both strands. 00:40:22.420 --> 00:40:26.500 align:middle line:84% You have to think of the transcription factor scanning 00:40:26.500 --> 00:40:28.060 align:middle line:90% the DNA in both directions. 00:40:28.060 --> 00:40:32.980 align:middle line:84% Then you have 24 megabytes mega bases of sites. 00:40:32.980 --> 00:40:35.530 align:middle line:90% 24 million sites. 00:40:35.530 --> 00:40:38.200 align:middle line:84% And at random, you expect 1 over 1,600. 00:40:38.200 --> 00:40:41.230 align:middle line:90% So you have a mean of 1,500. 00:40:41.230 --> 00:40:42.970 align:middle line:84% Now here we can bring in our old friend 00:40:42.970 --> 00:40:44.560 align:middle line:90% the Poisson distribution. 00:40:44.560 --> 00:40:46.600 align:middle line:84% And we will remember that the mean 00:40:46.600 --> 00:40:50.282 align:middle line:84% and the variance of a Poisson distribution are the same. 00:40:50.282 --> 00:40:51.865 align:middle line:84% And so the standard deviation is going 00:40:51.865 --> 00:40:53.282 align:middle line:84% to be the square root of variance, 00:40:53.282 --> 00:40:55.630 align:middle line:84% as it is for all variances and all standard deviations. 00:40:55.630 --> 00:40:59.440 align:middle line:84% And so the standard deviation is going to be about 40. 00:40:59.440 --> 00:41:03.190 align:middle line:84% So if you expect to convince yourself 00:41:03.190 --> 00:41:05.530 align:middle line:84% that you have something interesting 00:41:05.530 --> 00:41:08.170 align:middle line:84% then you want it to be about two or three standard deviations 00:41:08.170 --> 00:41:09.260 align:middle line:90% above the mean. 00:41:09.260 --> 00:41:11.230 align:middle line:84% So your noise that you're fighting 00:41:11.230 --> 00:41:14.620 align:middle line:84% is about-- you want to get 2 and 1/2 00:41:14.620 --> 00:41:17.500 align:middle line:90% times 40 or about 100 sites. 00:41:17.500 --> 00:41:21.460 align:middle line:84% Well, many biological phenomenon do not have 100 sites. 00:41:21.460 --> 00:41:25.300 align:middle line:84% They're not 100-- there may not be 100 GCM4 sites 00:41:25.300 --> 00:41:27.265 align:middle line:90% in the genome, for example. 00:41:27.265 --> 00:41:29.810 align:middle line:90% 00:41:29.810 --> 00:41:32.520 align:middle line:84% And so what you need is a way of winnowing down the genomes. 00:41:32.520 --> 00:41:34.270 align:middle line:84% We're not looking through the whole genome 00:41:34.270 --> 00:41:35.830 align:middle line:84% but we're enriching in various ways. 00:41:35.830 --> 00:41:38.470 align:middle line:84% What are the various ways that we can enrich? 00:41:38.470 --> 00:41:45.690 align:middle line:84% Well, the first three we'll lumped together as ways 00:41:45.690 --> 00:41:50.280 align:middle line:84% that we can biologically cluster. 00:41:50.280 --> 00:41:53.310 align:middle line:84% Basically, that was the theme of the first few minutes 00:41:53.310 --> 00:41:54.390 align:middle line:90% of this lecture. 00:41:54.390 --> 00:41:59.580 align:middle line:84% Ways that we can put together five genes that 00:41:59.580 --> 00:42:03.750 align:middle line:84% are-- where the gene expression products broken down together. 00:42:03.750 --> 00:42:07.320 align:middle line:84% And that would be, for example, whole genome [INAUDIBLE] data. 00:42:07.320 --> 00:42:11.370 align:middle line:90% That's the top line of slide 36. 00:42:11.370 --> 00:42:14.430 align:middle line:84% Or they could be-- and we had a little slide 00:42:14.430 --> 00:42:15.900 align:middle line:84% on this earlier of different ways 00:42:15.900 --> 00:42:18.972 align:middle line:84% that genes could show that they should go together. 00:42:18.972 --> 00:42:20.430 align:middle line:84% They could have a shared phenotype. 00:42:20.430 --> 00:42:24.240 align:middle line:84% You could do knockouts and they have 00:42:24.240 --> 00:42:28.680 align:middle line:84% similar biochemical or morphological characteristics. 00:42:28.680 --> 00:42:31.770 align:middle line:84% And so you put them in the same functional category. 00:42:31.770 --> 00:42:34.740 align:middle line:84% That might be the source of some of the functional categories 00:42:34.740 --> 00:42:36.960 align:middle line:90% we've been talking about today. 00:42:36.960 --> 00:42:39.120 align:middle line:84% They can be conserved among different species. 00:42:39.120 --> 00:42:41.040 align:middle line:90% Species will inherit them-- 00:42:41.040 --> 00:42:44.920 align:middle line:84% will tend to inherit them as a group and others. 00:42:44.920 --> 00:42:47.640 align:middle line:84% So this is the example why genes should go together. 00:42:47.640 --> 00:42:50.070 align:middle line:84% And then you'll reduce the sequence space 00:42:50.070 --> 00:42:52.350 align:middle line:84% to be the regulatory elements that 00:42:52.350 --> 00:42:55.380 align:middle line:84% go with those genes and not the rest of the genome. 00:42:55.380 --> 00:42:59.070 align:middle line:84% And those are the ways of selecting the genes. 00:42:59.070 --> 00:43:05.280 align:middle line:84% But then selecting the sequence itself near those genes 00:43:05.280 --> 00:43:06.665 align:middle line:90% or in those genes. 00:43:06.665 --> 00:43:08.790 align:middle line:84% You might want to eliminate protein coding regions, 00:43:08.790 --> 00:43:11.220 align:middle line:84% repetitive sequences or any other sequence 00:43:11.220 --> 00:43:13.710 align:middle line:90% is not likely to control sites. 00:43:13.710 --> 00:43:15.960 align:middle line:84% This helps you by reducing your sequence space. 00:43:15.960 --> 00:43:19.080 align:middle line:90% That's kind of a trivial help. 00:43:19.080 --> 00:43:20.580 align:middle line:90% Actually, an important help. 00:43:20.580 --> 00:43:23.820 align:middle line:84% But in addition to that, you want-- 00:43:23.820 --> 00:43:28.710 align:middle line:84% they help you by removing traps where you're 00:43:28.710 --> 00:43:30.660 align:middle line:84% going to find motifs but that they 00:43:30.660 --> 00:43:33.330 align:middle line:84% are unlikely a priori to be relevant to transcriptional 00:43:33.330 --> 00:43:33.940 align:middle line:90% control. 00:43:33.940 --> 00:43:35.357 align:middle line:84% Which is what you're really trying 00:43:35.357 --> 00:43:39.660 align:middle line:84% to get at here to validate and to extend the discoveries 00:43:39.660 --> 00:43:43.650 align:middle line:84% you find from the unsupervised clustering. 00:43:43.650 --> 00:43:44.850 align:middle line:90% And why do I say that? 00:43:44.850 --> 00:43:48.210 align:middle line:84% Why would protein coding regions and repetitive regions-- 00:43:48.210 --> 00:43:53.280 align:middle line:90% repetitive elements be a bias? 00:43:53.280 --> 00:43:55.650 align:middle line:84% Well, protein coding regions that 00:43:55.650 --> 00:43:59.100 align:middle line:84% for genes that cluster together, for some reason or other, 00:43:59.100 --> 00:44:03.090 align:middle line:84% likely a priori to have proteins that have similar functions. 00:44:03.090 --> 00:44:05.012 align:middle line:84% They're clustering together because they 00:44:05.012 --> 00:44:05.970 align:middle line:90% have similar functions. 00:44:05.970 --> 00:44:08.560 align:middle line:84% They might share protein domains in common. 00:44:08.560 --> 00:44:11.790 align:middle line:84% So you will find nucleic acid motifs 00:44:11.790 --> 00:44:15.210 align:middle line:84% that are similar to one another, not because they're 00:44:15.210 --> 00:44:18.840 align:middle line:84% involved in regulation but because the geniculate code 00:44:18.840 --> 00:44:21.400 align:middle line:84% turns into protein motifs that are similar to one another. 00:44:21.400 --> 00:44:23.290 align:middle line:84% So they can accomplish a similar function. 00:44:23.290 --> 00:44:25.710 align:middle line:90% And that's why they-- 00:44:25.710 --> 00:44:29.190 align:middle line:84% and repetitive regions are definitely 00:44:29.190 --> 00:44:32.880 align:middle line:84% destined to give motifs in common 00:44:32.880 --> 00:44:38.280 align:middle line:84% because of their selfish replication properties. 00:44:38.280 --> 00:44:41.340 align:middle line:84% The entire repetitive sequence from edge to edge 00:44:41.340 --> 00:44:43.230 align:middle line:90% will jump around the genome. 00:44:43.230 --> 00:44:46.020 align:middle line:90% 00:44:46.020 --> 00:44:49.440 align:middle line:84% And so there won't these little seven base pair motifs. 00:44:49.440 --> 00:44:53.010 align:middle line:90% They'll be a 10 kilo base motif. 00:44:53.010 --> 00:44:55.510 align:middle line:84% And that won't tell you much about transcription. 00:44:55.510 --> 00:44:59.790 align:middle line:84% Now having said that, we're in the business of sequence space 00:44:59.790 --> 00:45:00.330 align:middle line:90% reduction. 00:45:00.330 --> 00:45:04.140 align:middle line:84% Both the top three methods and this bottom method 00:45:04.140 --> 00:45:08.700 align:middle line:84% will exclude certain kinds of discoveries. 00:45:08.700 --> 00:45:12.645 align:middle line:84% But once you find the motif by severely restricting 00:45:12.645 --> 00:45:16.500 align:middle line:84% a sequence, you can then search for that motif 00:45:16.500 --> 00:45:18.780 align:middle line:84% and pick up the examples that you 00:45:18.780 --> 00:45:22.590 align:middle line:84% might have eliminated in the first pass in a much less noisy 00:45:22.590 --> 00:45:23.170 align:middle line:90% manner. 00:45:23.170 --> 00:45:25.290 align:middle line:84% You've got this bona fide motif, now 00:45:25.290 --> 00:45:26.970 align:middle line:84% you want to find all the other examples. 00:45:26.970 --> 00:45:29.250 align:middle line:84% In a way, you're testing the specificity of the motif. 00:45:29.250 --> 00:45:32.520 align:middle line:84% So for example, there could be RNA regulatory elements 00:45:32.520 --> 00:45:34.110 align:middle line:90% in protein coding regions. 00:45:34.110 --> 00:45:35.940 align:middle line:84% They could be some in repetitive regions. 00:45:35.940 --> 00:45:38.010 align:middle line:84% In the lecture that we gave on single nucleotide 00:45:38.010 --> 00:45:44.130 align:middle line:84% polymorphisms, I perversely chose a very interesting one 00:45:44.130 --> 00:45:47.310 align:middle line:84% that occurs in one of the most common dispersed repeats 00:45:47.310 --> 00:45:49.650 align:middle line:84% in the human genome, which is the ALU repeat. 00:45:49.650 --> 00:45:54.510 align:middle line:84% That one has regulatory significance, 00:45:54.510 --> 00:45:57.720 align:middle line:84% but we will exclude it from our search space 00:45:57.720 --> 00:46:01.320 align:middle line:84% initially so that we can get plenty of good examples 00:46:01.320 --> 00:46:04.720 align:middle line:90% in a small box. 00:46:04.720 --> 00:46:09.030 align:middle line:84% So these are the main ways of reducing search space. 00:46:09.030 --> 00:46:13.860 align:middle line:84% And we're going to illuminate this 00:46:13.860 --> 00:46:17.220 align:middle line:90% with a particular algorithm-- 00:46:17.220 --> 00:46:19.900 align:middle line:84% a modification that gives motif sampling, 00:46:19.900 --> 00:46:21.510 align:middle line:84% which is this one where you sample 00:46:21.510 --> 00:46:25.200 align:middle line:84% the multi-sequence alignment states randomly 00:46:25.200 --> 00:46:27.450 align:middle line:84% so you don't get past the local minimum. 00:46:27.450 --> 00:46:31.320 align:middle line:84% And this is called a [INAUDIBLE] nucleic acid 00:46:31.320 --> 00:46:33.060 align:middle line:90% conserved elements. 00:46:33.060 --> 00:46:34.545 align:middle line:90% The emphasis on nucleic acid. 00:46:34.545 --> 00:46:37.440 align:middle line:90% 00:46:37.440 --> 00:46:39.810 align:middle line:84% And what are the advantages to give 00:46:39.810 --> 00:46:42.050 align:middle line:90% a deep sampling [INAUDIBLE]? 00:46:42.050 --> 00:46:43.810 align:middle line:90% Why are we focusing in on it? 00:46:43.810 --> 00:46:45.560 align:middle line:84% Well, the [INAUDIBLE] sampling, as I said, 00:46:45.560 --> 00:46:47.380 align:middle line:90% keeps you out of local minimums. 00:46:47.380 --> 00:46:49.340 align:middle line:84% There are a number of sites per input sequence. 00:46:49.340 --> 00:46:52.400 align:middle line:84% It could be that in the genes that you've 00:46:52.400 --> 00:46:56.150 align:middle line:84% found in your cluster, some of them 00:46:56.150 --> 00:46:58.820 align:middle line:84% may have three of these motifs in front of it. 00:46:58.820 --> 00:47:00.680 align:middle line:84% Others will have one or even zero, 00:47:00.680 --> 00:47:04.310 align:middle line:84% because it could be that particular gene co-clusters is 00:47:04.310 --> 00:47:06.373 align:middle line:84% because of some other set of motifs that happen 00:47:06.373 --> 00:47:08.040 align:middle line:84% to have the same properties as the motif 00:47:08.040 --> 00:47:09.960 align:middle line:84% you're looking at it at any given moment. 00:47:09.960 --> 00:47:13.250 align:middle line:84% So you can have zero to a large number of motifs. 00:47:13.250 --> 00:47:14.420 align:middle line:90% And that's important. 00:47:14.420 --> 00:47:15.650 align:middle line:90% This algorithm handles it. 00:47:15.650 --> 00:47:19.010 align:middle line:84% Other algorithms assume there's exactly one site per sequence. 00:47:19.010 --> 00:47:22.730 align:middle line:90% And that introduces noise. 00:47:22.730 --> 00:47:25.220 align:middle line:84% You can distribute the information content 00:47:25.220 --> 00:47:26.420 align:middle line:90% in various ways. 00:47:26.420 --> 00:47:31.880 align:middle line:84% You'll see, we can fine tune the shape of a motif in a way. 00:47:31.880 --> 00:47:34.040 align:middle line:84% Some of these algorithms were based on proteins. 00:47:34.040 --> 00:47:35.660 align:middle line:90% Proteins have only one strand. 00:47:35.660 --> 00:47:38.870 align:middle line:84% They don't have a Watson and a Crick strand going in reverse 00:47:38.870 --> 00:47:40.290 align:middle line:90% complements one another. 00:47:40.290 --> 00:47:42.740 align:middle line:84% And so you need to make a conscious effort to adapt that 00:47:42.740 --> 00:47:44.280 align:middle line:90% algorithm so that it's-- 00:47:44.280 --> 00:47:47.420 align:middle line:84% that it, in a certain sense, recognizes 00:47:47.420 --> 00:47:52.070 align:middle line:84% the duality and the reverse complements of DNA strands. 00:47:52.070 --> 00:47:53.810 align:middle line:90% And you have to-- 00:47:53.810 --> 00:47:55.485 align:middle line:84% there's multiple distinct motifs that's 00:47:55.485 --> 00:47:57.860 align:middle line:84% different from the variable number of sites per sequence. 00:47:57.860 --> 00:48:01.430 align:middle line:84% Once you find motif number one, it 00:48:01.430 --> 00:48:03.860 align:middle line:84% may be the dominant motif that you find again and again 00:48:03.860 --> 00:48:06.320 align:middle line:90% in a multi-sequence alignment. 00:48:06.320 --> 00:48:08.000 align:middle line:84% You have to go back and find number two. 00:48:08.000 --> 00:48:12.230 align:middle line:84% Because it could be number one isn't the only 00:48:12.230 --> 00:48:15.320 align:middle line:84% or isn't the major biologically significant motif. 00:48:15.320 --> 00:48:17.930 align:middle line:84% It could be any two or three motifs acting in concert. 00:48:17.930 --> 00:48:20.030 align:middle line:84% So you can't just rest on your laurels 00:48:20.030 --> 00:48:22.160 align:middle line:90% when you find the first motif. 00:48:22.160 --> 00:48:24.920 align:middle line:84% And for each motif, there can be multiple examples of them 00:48:24.920 --> 00:48:26.030 align:middle line:90% per sequence. 00:48:26.030 --> 00:48:28.910 align:middle line:90% Anywhere from zero on up. 00:48:28.910 --> 00:48:30.860 align:middle line:84% So let's make this much more concrete 00:48:30.860 --> 00:48:34.230 align:middle line:84% and really drill down to a specific example. 00:48:34.230 --> 00:48:37.190 align:middle line:84% This example-- the real example-- 00:48:37.190 --> 00:48:45.260 align:middle line:84% it's taken from the amino acid biosynthetic genes in the yeast 00:48:45.260 --> 00:48:48.170 align:middle line:90% saccharomyces. 00:48:48.170 --> 00:48:51.200 align:middle line:84% So here we've applied the two major classes 00:48:51.200 --> 00:48:52.800 align:middle line:90% to sequence reduction. 00:48:52.800 --> 00:48:57.830 align:middle line:84% The first is by biological function here. 00:48:57.830 --> 00:49:00.320 align:middle line:84% These are all amino acid biosynthetic genes, 00:49:00.320 --> 00:49:05.270 align:middle line:84% histidine, aromatic amino acids, [INAUDIBLE].. 00:49:05.270 --> 00:49:08.120 align:middle line:84% These are all on the right hand side of slide 39. 00:49:08.120 --> 00:49:10.640 align:middle line:84% But in addition to the biological reduction 00:49:10.640 --> 00:49:15.230 align:middle line:84% of just maybe 116 genes that are involved in this process, 00:49:15.230 --> 00:49:19.970 align:middle line:84% we've also done the sequence space reduction near the gene 00:49:19.970 --> 00:49:22.250 align:middle line:84% to exclude the protein coding regions 00:49:22.250 --> 00:49:24.725 align:middle line:84% and only look at 300 to 600 bases upstream. 00:49:24.725 --> 00:49:27.410 align:middle line:90% 00:49:27.410 --> 00:49:30.230 align:middle line:90% Why 300 or 600? 00:49:30.230 --> 00:49:32.420 align:middle line:84% If the genes are really close together, 00:49:32.420 --> 00:49:35.133 align:middle line:84% you don't want to go much beyond 300 00:49:35.133 --> 00:49:36.800 align:middle line:84% because you can enter the protein coding 00:49:36.800 --> 00:49:38.438 align:middle line:90% region of an adjacent gene. 00:49:38.438 --> 00:49:40.730 align:middle line:84% If the genes are very far apart in this particular part 00:49:40.730 --> 00:49:43.670 align:middle line:84% of the genome, you don't want to get much more than 600 00:49:43.670 --> 00:49:45.410 align:middle line:84% or else you'll end up in the repetitive 00:49:45.410 --> 00:49:48.050 align:middle line:84% sequences or other things that are 00:49:48.050 --> 00:49:50.850 align:middle line:84% other regulatory elements unrelated to your particular 00:49:50.850 --> 00:49:51.350 align:middle line:90% protein. 00:49:51.350 --> 00:49:53.930 align:middle line:84% Or you might end up in an RNA encoding gene. 00:49:53.930 --> 00:49:57.920 align:middle line:84% So 300 to 600 is good for this particular organism. 00:49:57.920 --> 00:50:00.320 align:middle line:84% But you might need a different one for, say, human. 00:50:00.320 --> 00:50:03.470 align:middle line:84% You're going to have to look in introns and much 00:50:03.470 --> 00:50:06.230 align:middle line:84% further upstream, which makes it a much more difficult problem. 00:50:06.230 --> 00:50:08.580 align:middle line:84% Anyway this is the sequence reduction phase. 00:50:08.580 --> 00:50:13.190 align:middle line:84% And now let's say, well, do you see the motifs in here? 00:50:13.190 --> 00:50:16.550 align:middle line:84% I mean, those of you who are good computing 00:50:16.550 --> 00:50:20.720 align:middle line:84% should be able to do this algorithm in your head. 00:50:20.720 --> 00:50:22.628 align:middle line:90% But here's the answer. 00:50:22.628 --> 00:50:24.920 align:middle line:84% And then we're going to-- now we're going to go through 00:50:24.920 --> 00:50:26.920 align:middle line:84% and we're going to say how we got to that answer 00:50:26.920 --> 00:50:29.600 align:middle line:84% with the Gibbs sampling alignments algorithm. 00:50:29.600 --> 00:50:32.630 align:middle line:90% The answer here is GCN4. 00:50:32.630 --> 00:50:34.370 align:middle line:84% This is the one we used to illustrate 00:50:34.370 --> 00:50:36.530 align:middle line:84% we have about seven bits of information 00:50:36.530 --> 00:50:39.080 align:middle line:90% here in this Snyder logo format. 00:50:39.080 --> 00:50:43.280 align:middle line:84% And on the lower right, it has a map score 00:50:43.280 --> 00:50:46.160 align:middle line:90% that we'll define soon enough. 00:50:46.160 --> 00:50:48.420 align:middle line:84% Basically the higher the maps score, the better. 00:50:48.420 --> 00:50:50.990 align:middle line:84% It has to be greater than 0 to be non-random. 00:50:50.990 --> 00:50:53.300 align:middle line:84% And here's on the left hand side of slide 40 00:50:53.300 --> 00:50:56.300 align:middle line:84% is the multisequence alignment, just like the multi sequence 00:50:56.300 --> 00:50:59.870 align:middle line:84% alignments we talked about in the last lecture-- 00:50:59.870 --> 00:51:02.810 align:middle line:90% two lectures ago. 00:51:02.810 --> 00:51:06.110 align:middle line:84% And here in red are all these arrows. 00:51:06.110 --> 00:51:08.090 align:middle line:84% They point either left to right or right 00:51:08.090 --> 00:51:10.580 align:middle line:84% to left, depending on which strand they're on so they're 00:51:10.580 --> 00:51:12.590 align:middle line:90% not exact reverse complements. 00:51:12.590 --> 00:51:14.990 align:middle line:84% Although, this does have a little bit of symmetry in it. 00:51:14.990 --> 00:51:16.700 align:middle line:84% But you can see that you have anywhere 00:51:16.700 --> 00:51:22.040 align:middle line:84% from one to two of these in front of each of these genes. 00:51:22.040 --> 00:51:23.790 align:middle line:90% OK, so now how we get there? 00:51:23.790 --> 00:51:25.130 align:middle line:90% Let's go step by step. 00:51:25.130 --> 00:51:27.680 align:middle line:84% And some of you may find this algorithm counterintuitive 00:51:27.680 --> 00:51:32.270 align:middle line:84% at first so don't be surprised if it is. 00:51:32.270 --> 00:51:36.200 align:middle line:84% The first step is we randomly seed. 00:51:36.200 --> 00:51:41.260 align:middle line:84% We plop down, say, 10 more sequences 00:51:41.260 --> 00:51:44.680 align:middle line:84% 10 nucleotides long, arbitrarily picked that as our length 00:51:44.680 --> 00:51:49.310 align:middle line:84% and plop them down randomly on these sequences here. 00:51:49.310 --> 00:51:56.530 align:middle line:84% So we have represented seven of the 116 amino acid biosynthetic 00:51:56.530 --> 00:51:58.180 align:middle line:90% genes upstream regions here. 00:51:58.180 --> 00:52:01.390 align:middle line:84% And we've just highlighted red arbitrarily 00:52:01.390 --> 00:52:04.900 align:middle line:84% two red, 10 [INAUDIBLE] on the top one, 00:52:04.900 --> 00:52:07.700 align:middle line:84% and then none on the second one, and then one on the third one, 00:52:07.700 --> 00:52:09.460 align:middle line:90% and so on. 00:52:09.460 --> 00:52:14.200 align:middle line:84% And then since those are given and which is the first position 00:52:14.200 --> 00:52:16.810 align:middle line:84% is given, then it's a trivial matter to line them up. 00:52:16.810 --> 00:52:20.410 align:middle line:84% Just take all the first positions and you take a sum, 00:52:20.410 --> 00:52:23.880 align:middle line:90% and that's the weight matrix. 00:52:23.880 --> 00:52:26.380 align:middle line:84% Now you wouldn't expect since these were all randomly chosen 00:52:26.380 --> 00:52:28.030 align:middle line:84% for real sequences, you wouldn't expect 00:52:28.030 --> 00:52:34.508 align:middle line:84% this to be an astoundingly non-random weight matrix. 00:52:34.508 --> 00:52:35.050 align:middle line:90% And it's not. 00:52:35.050 --> 00:52:36.370 align:middle line:84% It has a maps score that's negative. 00:52:36.370 --> 00:52:37.995 align:middle line:84% And as I said, that's basically random. 00:52:37.995 --> 00:52:39.940 align:middle line:84% A few bases tend to stick their head 00:52:39.940 --> 00:52:44.243 align:middle line:84% up a little bit above the random noise of 0.25 00:52:44.243 --> 00:52:45.910 align:middle line:84% if this were a random genome or whatever 00:52:45.910 --> 00:52:47.260 align:middle line:90% the base composition is. 00:52:47.260 --> 00:52:49.710 align:middle line:84% And none of them are full scale it 2 bits. 00:52:49.710 --> 00:52:53.830 align:middle line:84% I'd say none of them are perfectly represented. 00:52:53.830 --> 00:52:55.038 align:middle line:90% So now what's the next step? 00:52:55.038 --> 00:52:56.830 align:middle line:84% That's the initial seeding and it gives you 00:52:56.830 --> 00:52:58.870 align:middle line:84% a flavor for what's going to happen next. 00:52:58.870 --> 00:53:00.970 align:middle line:84% But there's some interesting things 00:53:00.970 --> 00:53:03.640 align:middle line:84% that you can do to increase the chances of getting 00:53:03.640 --> 00:53:04.810 align:middle line:90% a good motif. 00:53:04.810 --> 00:53:09.880 align:middle line:84% So the next thing you do is either you add another site. 00:53:09.880 --> 00:53:11.260 align:middle line:90% You add another 10 [INAUDIBLE]. 00:53:11.260 --> 00:53:14.500 align:middle line:90% So the top row of side 42-- 00:53:14.500 --> 00:53:17.080 align:middle line:84% the top sequence already has two, but you add another one. 00:53:17.080 --> 00:53:18.280 align:middle line:90% You add a third one. 00:53:18.280 --> 00:53:22.180 align:middle line:84% Sequence number, arrow four still doesn't have any. 00:53:22.180 --> 00:53:24.318 align:middle line:84% But you added a third one randomly at the top 00:53:24.318 --> 00:53:26.110 align:middle line:84% and now you've got two sequence alignments. 00:53:26.110 --> 00:53:27.730 align:middle line:84% You really haven't been able to do anything up to this point. 00:53:27.730 --> 00:53:29.740 align:middle line:84% You've got now two multi-sequence alignments. 00:53:29.740 --> 00:53:31.460 align:middle line:84% And you ask, which one is better? 00:53:31.460 --> 00:53:34.750 align:middle line:84% Well, let's say the one on the right is a little bit better, 00:53:34.750 --> 00:53:37.270 align:middle line:84% the one you add of the sequence to is a little bit better. 00:53:37.270 --> 00:53:38.905 align:middle line:84% Now you don't just blind the program. 00:53:38.905 --> 00:53:43.090 align:middle line:84% It doesn't just blindly accept this as the better 00:53:43.090 --> 00:53:44.710 align:middle line:90% multi-sequence alignment. 00:53:44.710 --> 00:53:49.750 align:middle line:84% It has a probability that you will accept this. 00:53:49.750 --> 00:53:52.480 align:middle line:84% And that's again, to keep you from going through a completely 00:53:52.480 --> 00:53:54.160 align:middle line:90% greedy algorithm. 00:53:54.160 --> 00:53:59.980 align:middle line:84% Every improvement is going to be probabilistic. 00:53:59.980 --> 00:54:02.890 align:middle line:84% But you will definitely very greatly tend 00:54:02.890 --> 00:54:07.270 align:middle line:90% to accept each improvement. 00:54:07.270 --> 00:54:11.330 align:middle line:90% So this was adding a sequence. 00:54:11.330 --> 00:54:13.360 align:middle line:90% That's how you might improve it. 00:54:13.360 --> 00:54:14.710 align:middle line:90% Or you can remove one. 00:54:14.710 --> 00:54:18.430 align:middle line:90% 00:54:18.430 --> 00:54:23.920 align:middle line:84% You can add and remove another two from the top sequence here. 00:54:23.920 --> 00:54:25.030 align:middle line:90% Add one, remove one. 00:54:25.030 --> 00:54:27.400 align:middle line:84% And I asked you this if the multi sequence on the right 00:54:27.400 --> 00:54:28.358 align:middle line:90% is a little bit better. 00:54:28.358 --> 00:54:30.220 align:middle line:84% If it is, then you have a high probability 00:54:30.220 --> 00:54:32.560 align:middle line:90% of accepting those two. 00:54:32.560 --> 00:54:35.620 align:middle line:90% The add and the remove changes. 00:54:35.620 --> 00:54:38.560 align:middle line:84% These are adding or removing entire sequences. 00:54:38.560 --> 00:54:41.780 align:middle line:90% 00:54:41.780 --> 00:54:44.840 align:middle line:90% Keep going, adding and removing. 00:54:44.840 --> 00:54:50.000 align:middle line:84% Another thing you can do is can say, well, maybe 00:54:50.000 --> 00:54:52.340 align:middle line:84% the important bases aren't all smack in a row-- 00:54:52.340 --> 00:54:53.390 align:middle line:90% 10 in a row. 00:54:53.390 --> 00:54:55.872 align:middle line:84% Maybe you want to make it a little bit longer? 00:54:55.872 --> 00:54:57.830 align:middle line:84% Maybe the motifs should be a little bit longer? 00:54:57.830 --> 00:54:59.480 align:middle line:84% Maybe some of the ones in the middle 00:54:59.480 --> 00:55:02.960 align:middle line:84% aren't important, so we'll toggle one of them 00:55:02.960 --> 00:55:05.390 align:middle line:90% off and move the columns over. 00:55:05.390 --> 00:55:07.220 align:middle line:84% So now the motifs are a little bit wider, 00:55:07.220 --> 00:55:09.720 align:middle line:84% but it still has the same number of columns. 00:55:09.720 --> 00:55:13.010 align:middle line:84% And if that improves-- if that gives you a better map score, 00:55:13.010 --> 00:55:19.490 align:middle line:84% a greater surprise in a sense of probability that you've got-- 00:55:19.490 --> 00:55:21.560 align:middle line:84% that you would have this number of sites 00:55:21.560 --> 00:55:26.540 align:middle line:84% that are shared to this degree in this number of sequences, 00:55:26.540 --> 00:55:29.270 align:middle line:84% then you have a high probability of accepting that change. 00:55:29.270 --> 00:55:32.720 align:middle line:84% Now you're not just changing the collection of sequences 00:55:32.720 --> 00:55:34.550 align:middle line:84% you think belong to that motif family, 00:55:34.550 --> 00:55:41.840 align:middle line:84% but you're actually changing the structure of the elements 00:55:41.840 --> 00:55:43.730 align:middle line:84% that you're going to call the weight matrix. 00:55:43.730 --> 00:55:45.350 align:middle line:84% You're changing the column structure. 00:55:45.350 --> 00:55:48.410 align:middle line:90% And that's also probabilistic. 00:55:48.410 --> 00:55:53.090 align:middle line:84% And out of all this randomness, given many cycles, 00:55:53.090 --> 00:55:55.430 align:middle line:84% you eventually get the best motif. 00:55:55.430 --> 00:56:02.480 align:middle line:84% This might be the best motif for this particular learning set. 00:56:02.480 --> 00:56:04.820 align:middle line:84% But now you want to get the second best motif. 00:56:04.820 --> 00:56:08.010 align:middle line:84% Because this isn't necessarily the biologically best motif. 00:56:08.010 --> 00:56:09.680 align:middle line:90% And this one may not act alone. 00:56:09.680 --> 00:56:11.750 align:middle line:84% It may have another one that's also enriched 00:56:11.750 --> 00:56:14.480 align:middle line:84% and it could be that their co-occurrence is 00:56:14.480 --> 00:56:17.010 align:middle line:84% even more significant than either one of them occurring 00:56:17.010 --> 00:56:17.510 align:middle line:90% singly. 00:56:17.510 --> 00:56:21.920 align:middle line:90% 00:56:21.920 --> 00:56:27.053 align:middle line:90% So what do we do? 00:56:27.053 --> 00:56:28.470 align:middle line:84% And I think what we're going to do 00:56:28.470 --> 00:56:30.053 align:middle line:84% is we're going to take a little break. 00:56:30.053 --> 00:56:34.860 align:middle line:84% And then when we come back, your incredible curiosity 00:56:34.860 --> 00:56:39.390 align:middle line:84% will be satisfied as to how we get the second motif. 00:56:39.390 --> 00:56:41.750 align:middle line:90% So take a little break. 00:56:41.750 --> 00:56:42.328 align:middle line:90%