Catastrophe, Chaos and Complexity

Transcription

1 Catastrophe, Chaos and Complexity A workshop by Professor Richard C. Lewontin (Harvard University) Conducted in the Old Geology Lecture Theatre, University of Sydney November 28th, 1997 Professor Richard Lewontin is Professor of Zoology at Harvard University, and is on the Science Board of the Santa Fe Institute of Complexity Studies. Following his presentation of the 1997 Templeton Lecture, he conducted a three-hour workshop exploring the theories of catastrophe, chaos and complexity in biology. While Professor Lewontin spoke for most of the time, the audience and a panel of three invited experts, provided many comments, questions and responses throughout the workshop. The invited guests were: David Green, Professor of Information Technology, Charles Sturt University in Albury. Keith Williams, Professor of Biology in the Department of Molecular Biology and Development, Macquarie University, Sydney. Dr. Arran Gare, Department of Philosophy and Cultural Inquiry, Swinburne University in Melbourne. The workshop was chaired by CHAST committee member, Peter Farleigh. What follows is the transcription from the workshop, with apologies for any errors in the text due to the less-than-perfect recording of the event. Part 1: Catastrophe Richard Lewontin: I gave the title "Catastrophe, Chaos & Complexity" just as a way of organising a rather disorganised subject. And also because all three of those words at one time or another in biology have been very trendy. And I want to ask where does this trendiness come from, how do those notions get into biology. In order to motivate the whole affair I want to go back to what I said in my formal talk last night and in my answers to questions. I want to emphasize a couple of points, which will lead into the issues of catastrophe, chaos and complexity. I want to assert that organisms are a particular part of the physical world that have certain properties which stem from the place they occupy in the space of physical phenomena. I hope you will understand that everything I say assumes that living organisms are in some sense nothing but physical systems. This is a totally materialist, realist point of view. But they occupy a region of the space of physical systems, which makes for particular difficulties. In particular organisms are intermediate in size. They are not as big as planets, they are not as small as molecules or nuclei and they are internally heterogeneous. And the fact that they are internally heterogeneous and intermediate in size, introduces another property, characteristic of such systems, and that is that the interactions or dynamical systems that are occurring within the organism are not of greater magnitude in general then the coupling between the systems within the organism and the systems outside the organism. That is to say - it is an error or an historical relic of the development of biology that one talks about the organism on the one hand, and the

2 environment on the other and in some way illuminates them and sees the dynamical systems outside as somehow independent of the dynamical systems inside. What I want to talk about is the coupling of dynamical systems inside and those outside. The coupling of multiple pathways of causation within the system and the consequence of its intermediate size. Physicists are not very good with dealing with such systems. They work with wind tunnels and computers but outside of that they wouldn t know what to do. If I throw this key on the floor one would not want to try to write equations for what just happened. There is an elastic collision, but noise is generated so there is some dissipation of energy. It came down through the air in a funny way. Whereas the laws of Newton are clearly applicable in some sense, they don t do all that good a job. And of course every time you turn on the tap the water comes out in a way that involves complex hydrodynamics. The problem with objects that are discussed in the second volume of Newtown's Principia, about objects which move through viscous fluids experiencing friction and are intermediate in size and bump and bash into one another and deform rather than springing apart - those are all problems we have and that makes the study of biological systems rather more difficult. But more important than that is the internal heterogeneity of the system. That is to say, and we have a slogan here: "Living organisms are the nexus between a very large number of weakly determining causal pathways." That is the part of the space of physical systems that they occupy. And that has very serious consequences for the biologist as scientist, that the biologist would like to try to avoid if possible. That is because in what we consider the normal organism there are small perturbations in very large numbers of interactive causal pathways with the result that there are small outcomes of those perturbations, and the system has buffered properties such that perturbations are not propagated to all aspects of the system; some parts of the system may absorb the change so they become static systems. And the result of that is that the biologists cannot understand biology by looking at the objects of interest unlike the astronomer, who doesn t have much choice but to look. The astronomer has of course the advantage of some physical knowledge and theory and learns about the system by watching it. The biologist can learn virtually nothing by watching. And the reason is that the causal pathways are many and weakly determining and no single one of them is normally perturbed to such a degree to make a difference to the system, until the organism is sick. An organism is sick when it is dominated by one of the causal pathways. (My liver is destroying my life. Or my kidneys, or I have an idée fixe and am obsessed.) So sickness occurs when what is normally determined by a large number of causal pathways is dominated by only one. Now I feel very strongly as an evolutionary biologist and a population geneticist and ecologist, that we cannot really reconstruct or estimate the parameters of the dynamical systems which are operating in the evolution of an organism just by looking at it. The static observational data are not rich enough. So what does the biologist do? Then he becomes an experimentalist. But what is an experiment? An experiment is where you take a system and you destroy it, so to speak, by taking one causal pathway, emphasising it, making it extreme in its effect, while holding all the others constant in the hope that the response of the system to this unusual perturbation ceteris paribus will tell you how that pathway is operating in the undisturbed system. But the problem with organisms is that we have a scaling problem. That is to say we do not, we sometimes can, but can't always, tell how the organism actually operates by doing this trick. 2

3 I tried to illustrate that last night using developmental genetics as an example. There is a classic method in genetics that I inherited as a student that you try to understand biochemical pathways by studying mutations. So you had a mutation which prevented the colour (prevented the pigment development) in the eye of Drosophila, called the white eye mutant and you looked to try to find out the enzyme that was blocked in that mutant and then you would say : "Now I know something about the formation of eye pigment". You do indeed discover something about pigment formation but there are many, many different points of intervention that affect eye pigment. And it turns out that what you don't know is how to account for small variations in the amount of pigment in Drosophila eye, from one individual to another, or between species. Because the perturbation you have performed is not related to the variation that is actually the cause, or one of the contributing causes, of natural variation. It is a pathological condition. An awful lot of mutation studies in the history of developmental genetics have been studies on what could be called pathological changes. So the current problematic of developmental genetics, for example, is to work out the signalling pathways among the genes that are concerned with determining the difference between the front and the back of an organism or between the ventral and dorsal surface. But that problematic does not include the question of why our noses are different shapes from one another or, indeed, why a nose is nose-shaped. It's not even in the questions. Why is a nose a particular shape? Shape is not in the problematic. What I am saying is that when you go down an experimental path you necessarily make big perturbations and when you make those perturbations you push the system outside of its normal set of homeostatic buffering channels. If I plot the ph of the system against an amount of added acid it looks like Figure 1 and it s buffered in the region A. outside the buffering range the line looks like this, and this flat region (A) is what we call a buffered region where acid or base does not change the ph, and there may be multiple buffer regions. Now organisms have morphogenic buffer regions in which all kinds of perturbations in either the environment or the genotype have no effect whatsoever on the phenotype. [Jimmy] Rendell who used to be in the CSIRO here did very famous experiments to show that genetic variation had no effect whatsoever on certain phenotypic aspects of the organism nor did environmental variation until you put enough genetic variation into it to push the system into the response region. Once you made the system respond you discovered that this genetic variation was relevant. It wasn't that it was irrelevant it was just buffered out. As you move the system outside the zone of buffering you could observe this variation having its effect. So that is our experimental problem. Within the zones of buffering which have built up within the organism through evolution, an awful lot of the variation of causal pathways has little or no effect on the phenotype. Nevertheless the causal pathways are there inside the organism. So we are in an awful dilemma, that if we perform experiments which succeed in pushing the system outside the buffer region, it is then in a state which does not normally occur and which does not adequately describe what is normally going on. And if we don't do that, we say there is nothing varying and that we have nothing to study. This is not a problem, I gather, which occurs in other regions of physical study. We assume that the scaling problems are not so severe in other areas, although I think they must occur to some extent. 3

4 Bob Hunter: What you are talking about here is the macroscopic behaviour of the organism. But you could feasibly see, in that buffered region, a variation in, say, the concentration of some particular metabolite. Lewontin: When we say the system is buffered, we mean some aspect of the system is buffered, which means that that part of the system does not change; other parts may change but you have to know where to look. And the problem is that we come back to the internal heterogeneity of the organism. There are so many causal pathways you don't know where to look. Keith Williams: Can I suggest that is changing? I think we are heading for a paradigm shift here, because the reason that you are stuck with that problem is that we did not know about the totality of the system and so we were forced to focus on particular issues and, as you say, do dramatic things to them in order to get some insights into what is going on. I think one of the things that is happening in the world at the moment is that we are beginning to get a complete read-out of what the ingredients are. We have never had that before, so we are starting to get to the stage with the DNA sequencing programs where we understand all the informational components to the organism. So we understand basically what the raw material to be worked on is. The problem at the DNA level is, I think, that it is information but nothing more. And if you translate that to the protein level, to actually what gets made, then you can start working in that middle region and see what happens as you move along that buffered zone. And the sort of thing we are seeing because we are looking at mass protein display of the codes is that you have got these whole networks of changes that occur. The lovely thing about that is that you do not need to make decisions about what the important componentry is. You can now start to look at holistic analysis and actually work in that domain. Lewontin: I agree that that is the program of developmental biology. If I could caricature developmental biology, it is to make a completely connected graph of the signalling pathways among all the genes. That is the program and the real issue is whether we can cash it out. People say, as you just have, that we are beginning to do that. I am less sanguine than you that you will be able to sort all those pathways out in a reasonable time with the resources available. David Green: Can I take the point further? Some physical systems do have that property. Think about water, ice and steam where you get water remaining as a liquid over a large range of temperatures but you get to those particular temperatures where something dramatic happens. Lewontin: So we have phase transitions. Sure, but on this question of whether developmental genetics can complete its program, I say it can t complete its program if it doesn't ask all the questions. Let me give an example: a great deal is known about the genes and certain signalling pathways for wing formation, (for example the UBX system) but the question is does that tell us anything about variation in wing size between individuals and between species? Are the genetic variations in wing size connectable with variations in the UBX system and its proteins more than in other genes? Now we have some information about that. Greg Gibson recently has found indeed that flies with different size wings have somewhat different DNA sequences for UBX, so UBX is somehow involved, but it is not the only thing that is involved. There is quite a lot of genetic variation out there that is not in UBX and we don t know how to look for it. So I agree with you that the program, as a program, is clear. The question is how much of genome can we explore and perturb in ways which will tell us how much of it is connected. That's the issue. Keith Williams: The problem that we are facing at the moment is that the informational people, the DNA people, are trying to talk about function from an informational base and it might be that does not work. What I think is really exciting is that, and if you think of it like building a city like Sydney, if we want to work out how people build Sydney, you do need all the specs, you need the 4

5 specifications how to make the chairs in this lecture theatre, and how to make a telephone booth and all of those things, and without all of that information you are not going to be able to build a city. But having all that information is pretty useless. What you need to be able to do is to start looking at what actually happens. And this is where I think we are in a revolution because that is what looking at protein in a mass sense is. Because now we can start asking questions like `that fly's got a bigger wing than that one', what has happened in terms of gene turn on, etc. to produce proteins that have brought about that change. I think that is a way forward. But you have to understand that protein chemistry is changing and you don't have to work on proteins as individuals anymore. We are beginning to develop technologies to globally display how an organism is composed. You then are still in the same sort of dilemma and if you think then about having demonstrated what the organism is composed of, the analogy there might be it is a bit like having a chess set where you ve described all the pieces and you have some idea where they are on the board, but you don't have the dynamic picture, you don't know how to play the game. I have no idea how we get into learning how to play the game. But it is clear to me that there are tremendous changes happening because we are at least learning what all the componentry of the game is, and I think that is a huge change. Lewontin: Well you and I agree that we have got to know the circuitry, and we are also in agreement that there is a lot more to it than that. But you are more hopeful perhaps than I am. Charles Birch: But what has all this got to do with catastrophe? Lewontin: Well I am trying to get there. What I was trying to do is to describe the situation that exists in biology and the problems of biology and our puzzle about it. Now let me get to chaos and catastrophe. Biologists in my view have not been happy with this situation. They do not really like the fact that the objects of their study are so heterogeneous and have such nasty interactive pathways and what is on the inside is interacting with what is on the outside and in addition to which I claim that stochastic elements within cells are real phenomena which have importance in many of the things we see in biology. Biologists who are not happy about that have been very receptive to theories or world views or principles, I don't know what you want to call them, which would seem to tame that complexity, and tame that stochasticity. I have lived through three of them and they were others before that. The first one that I can recall was catastrophe theory. Now let me tell you what catastrophe theory is. We have the three Cs. I am intrigues as to why it is catastrophe, chaos and complexity. I have been waiting for another C to come along. If you take a dynamical physical system and describe it by a set of differential equations, it turns out that the topology of that system can undergo radical shifts for certain values of the variables, not the parameters but the variables. As x changes through time, the topology of the object remains the same, it is compact. And then as x moves a bit more in time things start to break up. Sydneysiders know this better than anybody in the world because the famous example is the breaking wave. This is what surfers depend on, they depend on the catastrophic nature, the structural instability of the equations which would describe the movement of water off the Pacific as it comes into the land. You get the thing building up, building up, building up and then at a critical moment it turns over and crashes and changes its entire shape. And René Thom s catastrophe theory wasn t a theory about the world. It was about the development of certain notions of mathematical structure and stability and the way in which the topology of a system would suddenly go into a new state from compact to breaking to pieces, as one or two variables were changing continuously. Another example would be if I dropped this glass from a low height it would come out as a glass, and I could drop it from different heights and there would be a point at 5

6 which when I reached that height, if I dropped it its topology would change rapidly into bits and pieces. And that is a mathematical catastrophe, although the value of the variable here is simply the kinetic energy, and as the energy increases suddenly it shatters. Now what René Thom suggested to Conrad Waddington, who was always on the look out for some way to tame biological complexity, was that a lot of what goes on in organisms, could be described by catastrophe theory. For example, you begin with a cell and it becomes two cells and then it begins to change shape, and the problem of developmental biology is a problem of shape. And that is the problem which no developmental biologists are working on. That is to say, this thing is a shape and it goes like that, it didn't always go like that, things start to bud out and things come apart and organs separate, and what Thom suggested to Waddington was that what you are observing is the unfoldings of topological catastrophes. The argument was that these systems were very simple dynamically, they are not complicated, they are described by simple mathematical sets of equations. And the terrible complexity you see in the development of the organism is nothing but the unfolding like the breaking wave or the breaking glass or the breaking stick. And Waddington and other developmental biologists at the time thought well maybe this is the way to describe things. What they liked about it was that it tamed the messiness and the apparent inability to predict the randomness, the stochastic elements. What appeared to be messy and radically different was nothing but the continuous unfolding of very simple systems and if you only knew the equations you could then unfold the whole thing. That is to say what you are looking for is the magical formula. It is the ultimate Faustian problem, you know. If you could really know the secret formula of the world you could unfold the whole course. And that was the catastrophe theory, and nobody that I know of currently works on catastrophe theory. David Green: Isn't Brian Goodwin still doing it? Lewontin: Oh, is Brian still doing it? I thought Brian had become a complexicologist. But Brian was at the same meeting in Bologia that I was when Waddington introduced René Thom to the biological world and Brian took it up. Arran Gare: Wasn't it the other way about? That René Thom was inspired by Waddington's ideas, and he formalized it? Lewontin: But surely notions of structural instability, and the basic elements of topology, must have been known before. I don t know. I m just guessing. Green: What he did was to prove a series of mathematical theorems which showed that these structures fall into several different categories which you can actually draw, so that all these biological forms fall into these different categories. Lewontin: Right - but I am interested in that suggestion that Waddington pushed him into this direction in his topological studies. Is that so? I didn't know that. Green: Yes. He says that in the latest edition of theoretical biology, the one edited by Goodwin. In the first paper he says that mathematics has got more from biology then biology has got from mathematics. Lewontin: Well, it couldn't have got less. I say that as a mathematical biologist. So what we are trying to fill you in on is how catastrophe theory came together with biology. Intervention: I think that comment is also made in the preface to René Thom's book. Lewontin: So speaking purely from the biological standpoint, I would say that, however Thom got 6

7 his ideas, that was the original attraction; it would solve the complexity problem. Hugh Murdoch: Catastrophe Theory as I understand it has no predictive power, it is only descriptive. Lewontin: Well, if you knew to which regime the system belonged you ought to be able to unfold that using Catastrophe Theory. In that sense it would be predictive, but you would have to know to which of the set of regimes it belonged. David Green: But there is another problem as well - basically any model of the world has to be valid. That is to say the assumptions have to satisfy what you see out in the real world, and as I see it the big problem with Catastrophe Theory is that it is just not valid. Most of the examples that are touted about this, for example one is the retreat or attack by a dog, it is fear versus rage - if you plot these on different axis you can chart as the animal gets more and more angry, or more and more fearful, the situation evolves until it will either attack or retreat. And it flips from one behaviour to the other. The problem is how do you map fear and rage? How do you measure those precisely to map them onto that surface? The second problem is how do you map things onto that geometric surface? It involves, for example, probability notions - and so far as I know, no one has ever come up with satisfactory formulae for plotting these on the model. Lewontin: You raise a problem that we will come back to. These are essentially metaphors, and the problem is to catch the metaphor out in the real system. Keith Williams: The other thing about that is, I think Hans Meinhart is one of the people who has really had a go at being predictive with some of the Drosophila modelling and he has actually done quite a good job. It is one of the rare cases in biology where he has actually told some Drosophila geneticists to go and do this and this is what will happen. The problem for a biologist is where you have this general theory that has some predictive power but the models that Hans is setting up, when I talk to him about trying now to understand what is going on, in terms of how you form a discontinuity. There are a couple of ways that people have argued about doing that. Working with Dictyostelium it is a very simple system. Basically you just take one type of cell and you end up with two types of cells. There is a decision between a bunch of cells whether you will be A or B and there are two theories about how this happens. One is that the cells get together and make decisions as to whether they will be A's or B's they do it in a neighbourly fashion. A quarter of the cells are A, so when a cell says it is A three cells around it say they will be B - and having decided what they will be they then sort out to form a pattern. So that is one of the models. The other model is we will have a hierarchical situation where we will take all of the cells. Someone will make a decision about where the line will be drawn and all the ones to the left of the line will be A and to the right will be B. So there are two quite different models or views about how you make a pattern. One is positional, if you like: you decide where the line is going to be drawn, and do it - whereas the other is local and it is followed by sorting out. The problem I have had with the theoretical guys is that their models did not distinguish between the two. They would say it is easy to switch between those two types, and I would come back and say well your model is not much use to me if you can't distinguish between them. If it is so easy to fix your models so that either can be correct, you are not giving me any guidance on how to go. I think that might be why biologists have walked away from Catastrophe Theory. It hasn't helped. Part 2: Chaos 7

8 Lewontin: It is at the wrong level of abstraction. You can't cash it out. So let us go on briefly to the next one of the three Cs, which is quite trendy now, that is Chaos theory. There is a lot of misunderstanding about chaos. Some very, very simple - extremely simple differential equations in a single variable describing the rate of change of the system with time, for example some variable x and you describe its rate of change with time, dx/dt is some simple function of x, let's say the ratio of two cubics: It turns out that for some choices of these constants, if you look at this not as a differential equation but as a discrete time function you can calculate the state at time (t+1) as some function of the state at time t. For lots of choices for these parameters (a,b,...a,b...), the state at time (t+1) if plotted over time, if you look at it as a statistician it would appear to be undergoing completely random unpredictable behaviour. No matter how much information you were given, you would require infinite information, to predict what will happen. Because it never revisits the same pattern, it does not go through cycles. If you look at it forever and you look at the fluctuations, that particular set of states is never repeated. It is totally incoherent. Now I could write down hundreds of such simple equations that have that quality that undergoes this so-called chaotic behaviour. And also what you can show is that for some set of parameters the system behaves in a perfectly nice way; it starts here and approaches some steady state and as I change the parameters continuously I reach a critical point where if a is a little bigger than some critical value the thing just goes nuts. It did not quite go nuts immediately. It went say smoothly from two to four and then it went nuts. And what I started out by saying is that biological systems show very nasty apparent stochastic variation. We will take an ecological problem. The abundances of some particular organism over time, for example: the number of aphids on roses. We can count the number of aphids on rosebushes. They have some correlation with some environmental variables like highest summer temperature, but it is only a correlation. You cannot predict the numbers from time to time because it is a stochastic relation and the problem is the stochastic variations are big; you have just got to throw up your hands and say: "I don't know whether it is going to be an outbreak year next year or not because there is so much stochastic variation." There are too many variables entering the system, and it's a mess. It's chaotic! But what this system of equations does is it generates a chaotic sequence of states of the system based on an extremely simple formula. It says that what appears to you to be chaotic is nothing but the time dependent behaviour of a system that has a very simple underlying law: 3x 3 + 9x 2 +..etc. and therefore, if I could discover that law, then if you tell me what the state is I will be able to tell you exactly what the subsequent states are. So this not a stochastic system. If I know the state at any time, t, and I know the law, I can then write down forever the complete set of subsequent states. And I have turned it from a hopeless jumble of stochastic variations with no apparent law into a very simple law-like behaviour. If I were to do that, biologists would be out of their wits, because they would once again have accomplished what is supposed to be probably science, that is the taming of this bewildering stochastic complexity and bringing it under a very simple law of behaviour, which gives you complete predictability. Intervention: But isn't that predictability only possible if the initial state is known exactly? 8

9 Lewontin: Indeed if I don't know exactly where I am at time t then I can't make predictions about the subsequent behaviour of the system. To make predictions I have to know exactly what the initial conditions are and if this initial situation is slightly different then the subsequent behaviour pattern is entirely different - and that is a very important point. Let me put it this way - what you have said is a very good criticism of the usefulness of chaos theory in biology. Even if it were right, then precisely for the reason you say, we come down to an argument among biologists about whether these really are stochastic events and whether we know the initial state, and that is a very good criticism. But my criticism is that this is another attempt to tame unpredictability and complex interactions by knowing the secret formula which can't work. First of all there are real stochastic events going on. I don't want to get into the question all the way down to the basement about whether stochasticity is ontological or not because for a biologist it does not matter. For every science there is a level which is the basement for which stochasticity is good enough. I don't have to answer the ultimate philosophical problem. So long as it is Brownian motion which satisfies Einstein's equation it is good enough for me. So that is where chaos is. And everybody is talking about chaos. Why? Because they somehow feel better about it. They feel that all this bewildering activity has been explained. Now chaos does indeed exist, you can buy a little device in a toy store which will show you chaotic behaviour a little pendulum. It is not that you cannot make simple physical systems that show chaos. The question is: is that what genes in biology do? Keith Williams: One of the first systems which was shown to be chaotic was looked at by Bob May who looked at logarithmic growth, which is growth of populations where you have two parameters, the growth rate of the population and the carrying capacity for the environment. What happens is that you get this famous S- shaped curve, where the population grows rapidly at first, and then as it gets closer and closer to the carrying capacity the rate drops off and then smoothly and asymptotically approaches the carrying capacity. Lewontin: Here is a plot of the numbers of individuals at time t as function of t, increasing at first exponentially but then falling off to reach an equilibrium value and the equation to describe that is d n/dt = a log Keith Williams: The reason I mention this is that it shows up a) what this sort of theory can demonstrate and b) what the practical limitations are: the issues you have to take into account. But again it comes back to the validity of the model. The first thing to point out is that if you assume that growth is continuous in a system like that you will always get that curve. You only get the chaotic property which we can talk about if you have discrete growth. For example if you have a population which is reproducing annually or seasonally, say crops that are planted every year. Then what happens is that the pattern can vary and you get the onset of chaos. When you increase the growth rate it first starts to do funny things, you get little blips in the curve, and you will get oscillations. Those oscillations will then become faster and more pronounced as you increase the 9

10 growth rate. The populations with higher growth rate are obviously more pronounced until you get the onset of chaos. Now a lot of the models you see talking about this are looking at an abstract system where the equations have been converted so that they always lie in the range of 0 to 1 so you are dealing with continuous numbers. Populations, of course, are not continuous numbers. You either have an organism or you don't. And you find that when you get to this chaotic region what almost inevitability happens is that very quickly the population crashes to zero, and so that is where the whole process stops. Lewontin: I would like to add to this. Bob May's demonstration that population growth is chaotic depends upon a particular form of the population growth equation. Now no one in the world knows whether there is any such equation. That is to say we write down the population growth as a very simple second order equation in numbers with a collision model in mind. For organisms to reproduce they need another organism of their own kind and if you reduce their reproduction you reduce the number of interactions. We don't have a universally agreed upon set of dynamical laws which control populations. We have a model we use all the time but I don't know if it is true. It turns out to be the simplest linear equation I can write, the second order equation to describe the population growth made out of... law. The chaotic behaviour depends entirely on whether it is that form or some other form. So there again is the question of validity. Why are we even talking about the chaotic properties of that equation, when we don't even know if it is the equation for population growth? Populations grow, that's all, and then they shrink. They grow and they shrink, and they may not follow any law, but it is so much more appealing to say yes we have a law of population growth and if I increase the value of R to the critical level suddenly you get a multiplication... John Bennett: How do we know that these big variations are not due to some error in the computer? Lewontin: You don't need a computer to show this behaviour. The behaviour can be shown analytically. This is not a computational problem. It has to do with the mathematics of bifurcation, so it is not rounding error. Keith Williams: In fact that's how Lorenz discovered the butterfly effect by detecting round-off errors in his computer. When he fed the results back in they had these round-off errors. Lewontin: But the round-off errors led to the discovery that the chaotic regime existed independent of the round off. It is a mathematical phenomenon. Hugh Murdoch: This is the point that even a very simple equation can lead to chaotic behaviour. So anything in real life is likely to be much more complex. Lewontin: Yes, things probably are a lot more complex, but then what do we accomplish by showing that a simple equation if we turn the parameters will become chaotic? That's the bill of goods I am trying to present to you. That all the talk about chaos is precisely the attempt to say that the underlying laws are simple, very simple, and the parameters have just driven us into chaos. Intervention: A lot of these things just come back to validity, don't they? That any model is an abstraction of the real world, but to achieve that you are making a lot of assumptions. To write something down as a mathematical formula you are assuming that it can be expressed in those forms as numbers, as quantities and that any other constraints or factors are just not entering into the system. Lewontin: If I avoid Complexity Theory for now and leave that for later and instead go on to a further discussion of stochasticity would you agree with that? I want to talk about stochasticity and 10

11 the questions raised about it. This is not a question of whether chance in an ontological property of the universe at all levels. I have nothing to say about that question, so much has already been said about that it is hard to say anything useful. The fact is that cells have complicated machinery for the production of bits and pieces and interactions. That machinery involves large molecules and some small molecules. These molecules are manufactured by the cell, the large molecules are interacting with one another, and in the process they fold up during protein synthesis for example. A protein is not specified by its DNA sequence. What specifies the protein, if anything, is the amino-acid sequence, but that sequence of amino-acids is not a protein. A protein is at the very minimum a folded sequence of amino-acids in three dimensional space. And the folding of that polypeptide is not predictable from its amino-acid sequence. There exist multiple local free energy minima on the folding surface, ie. there are different ways of folding the chain, each one of which represents a local minimum in free energy. The consequence of this is that we do not have any computer program which will predict the folding of a polypeptide and to get people to work on this is a problem. We just don't know how to do it. You can't even do it after a little local unfolding. With these programs, if you join partially unfolded proteins you cannot put them back. The conditions in the cell in which the folding occurs are critical and this has had a practical engineering application in the use of bacteria as factories for making proteins like insulin. It turns out you can take the gene for insulin and put it in a bacterium and the bacteria can be put in a vat and they will grind out insulin polypeptide, but they will not grind out insulin protein, because it will form the wrong folded intermediary and you have to search around for the right system. It has been found, so it is now possible to make human insulin because we know how to arrange the external conditions in the vat to get the right folding. This folding is of molecules which are of very low concentration and it is under the influence of other molecules that are in very low concentration. Most of the molecules that really matter to a cell are not present in Avogadro's number in the cell. They are present as one of this and two of that and nine of that and fifteen of that and some of them are over here and some over there, and some are inside a cell organelle and some are outside. To do their business they have to be brought together in exactly the right place. There are cell organelles that process them. So the metaphor of a factory is a good one in a sense, although you must be careful of all metaphors. [Norbert Wiener said that the price of metaphors is eternal vigilance. We must be vigilant, but I will use the metaphor of the factory.] There is a temporal passage through the system of molecules one at a time, and the consequence of that is that cells do not do identical jobs even though they are right next door to one another and have the same genes, and are surrounded by the same external medium. Because they do not have exactly the same number of molecules in each case. Indeed the reductio is the DNA molecule which is present in only [depending on how you say it] one or two copies, and the cell has to have a very special complicated device for guaranteeing that those two molecules of DNA are reproduced as exactly the same two molecules when the cell divides.... the concentration of DNA in the cell is quite interesting. You have this DNA molecule and there are two of them, if they are diploids, and when that cell divides into two, if everything is working all right, each of those cells will have exactly two DNA molecules, not more and not less, and there is very complicated machinery to guarantee that. You cannot rely on the laws of mass action, because the laws of mass action are not working here. There are lots of other molecules in a cell that are present in low concentration, for example certain vitamins like Biotin which are important for fungal growth, are present at an average of one molecule per cell. 11

12 Given that many molecules in cells are spatially located and they are in small numbers and it takes real time to make them and to move them around, you get stochastic effects. I do not care whether the thermal model which has them jiggling around is based on a real ultimate stochasticity. It is irrelevant. The fact of the matter is that when I am dealing with systems of interacting molecules at low concentrations I get stochastic processes because they are in different thermal states, they are wriggling around, they have varying bond energies. It comes down to quantum chemistry, it really does. It is where quantum chemistry and the physical chemistry of molecules, like their vibrational states and rotational states really enter into the picture. The consequence of that can be large or small depending on what you are interested in. Here is an example which I think perhaps I gave in the discussion last night of a bacterial colony in which it is not the case that all cells divide in phase with each other, and that is because they are parcelling out small numbers of molecules and each daughter cell does not get exactly equal numbers. If there is a critical number involved that you have to have before you can divide then you have to wait different amounts of time. We do not really have a very good idea of how much macro developmental events depend on these stochastic events, but I want to give you one example. I mentioned it last night but did not give the details. I want to get a little technical here. This has to do with the formation of sensory bristles or hairs on the bodies of fruit flies. The hair on a fruit fly is a sensory apparatus and it consists of the result of three cells. There is the cell which gives rise to the hair itself, the so-called trichogen cell. There is a collar out of which the bristle protrudes and that is the result of a secretion of another cell called a tormogen cell and that cell has a nerve cell that grows out of it that connects it with the central nervous system of the fly because it is a sensory apparatus. So there are three different cells which give rise to this little sensory device. Now those three cells arise from two divisions of a single stem cell. It divides into three cells, one will become the trichogen, the second becomes the tormogen and the third is the nerve cell. The cell division which gives rise to these three cells is occurring deep down in the sub-surface layers of the developing Drosophila. And up here is the integument, that is the hard coat or outer skin of the fly. It is hardening while the cell division is going on. And when these cells finish dividing they migrate as a cluster to the surface and if they make it to the surface before that surface is hardened they can succeed in making a bristle. And if they do not make it to the surface before it hardens, they get excluded and they will not make a bristle. Now you have two processes that are going on simultaneously. The process of hardening and the process of cell division and migration, and they have obviously some constants and variability because it takes different amounts of time for them to divide and to get there. And the consequence of all that is a lot of variation from place to place on the fly of whether a bristle gets there or not. We have a famous mutation in Drosophila which is different from almost any other mutation known. Most mutations in fruit flies either reduce the size of something or increase its size - you get too much or too little - but we have this one mutation which is called the...? which actually causes variation on both sides of the organisms. Some flies have too many bristles and some flies have too few. All have the same mutation, and the reason turns out to be that this mutation causes an additional division of the original stem cell so that it forms two or even three stem cells and each one of those undergoes bristle formation. Now each division takes time and if it takes too much time none of these cells gets included in the surface. And if it does not take too much time, all of them may get included in the surface. And that is how stochasticity becomes involved. We have to cope with that kind of stochasticity in actual systems. That kind of process applies all the way up the scale. It is not interesting to me 12

13 whether being struck by lightning, before producing any offspring is regarded as stochastic in some ontological sense. "The race is not to the swift, nor the battle to the strong, nor riches to the wise man, but time and chance happen to all." And that is true at every level of organisation in biology. Of two female Drosophila individuals of identical genotype, one may be completely fertile and the other completely sterile, and I do not know why that is; something bad happened on the way. And it does not matter whether that is true. I want to emphasise that there are effectively stochastic events in the development and life history patterns of cells, organisms and entire species. Some people think that is very important. The attempt of biologists to tame that stochasticity and remove it from biological explanation is I think doomed to failure but I will regard it when we get to complexity. Keith Williams: I think there are interesting things that come out of that because in the case of the hairs it does not really matter to the organism. The stochasticity can happen there because the organism still survives. We can give chromosomal examples where it is absolutely crucial, because if you have too much stochastics going on, you do not have an organism. I think that, particularly with complex organisms, much of what happens when you make proteins, for example, in processes like folding them up, you may have anti-stochastic processes or processes that minimize the stochasticity. It may be you can use this to probe crucial functions versus ones that you can allow to be more sloppy. Lewontin: There I want to disagree with you very strongly. Speaking now as a population geneticist, I think that we should not talk about differences in organisms which have no fitness consequences because I cannot find them. That is to say it is certainly true that having two chromosomes (instead of one may be lethal) but having the wrong number of sense sensillae, although allowing the organism to survive, I would guess does make a difference to the population. For a living what I do is sequence DNA. And for years we have been saying that, because of the nature of the DNA code, there are redundant positions where the code is degenerate and it does not make any difference what the third position is like. If there is one thing I now know at the end of my career, it is that there is no such thing as a non-selective nucleotide position. Every DNA position even if it is not translated, not transcribed, all show patterns of constraint, which cannot be explained except by assuming that there is some kind of selective difference. And there is our problem: it is impossible to measure physiologically those differences because they make too little difference to the outcome. You cannot measure it by counting eggs. The difference is too small. The only thing you can do is go back to the static data and say there is a pattern in DNA sequence data which is clearly not random, so there must be something going on. I do not know what it is and I cannot tell a physiological story because the difference is small. Let me point out to you that if it makes a difference of one 100th of 1% in the probability of survival on introduction into the organism, whether it has four bristles or three, then that is quite enough to drive that system evolutionarily within 100,000 generations, and that is not all that long. Bennett: Are you suggesting that you can have a viable theory of evolution based on this. Some of these things survive and may be they would do so...? (Inaudible) Lewontin: I do not understand the question. Bennett: Theories of why there are changes of species came to invoke cosmic rays and all kinds of things but you have described a possibility that you might get change due to the random effects you are talking about. Lewontin: You are confusing it with the mutation issue. And certainly chemical stochasticity has 13

14 to do with mutations. So-called spontaneous mutations have to do with mis-copying of DNA and that probability is a consequence purely of free energy considerations for crystal structure. You cannot have a perfect chance of reproduction. It would not correspond to a minimum free energy. So of course you have mutations. They must happen stochastically. Intervention: I would like to ask you a question about what are considered non-random patterns of what are naively considered neutral mutations. Could that possibly be due to non-randomicity in the evolutionary processes of that organism. Lewontin: No, when I say non-random I am not involving a stochastic process here with an ergodic state. This is at a much simpler level than that. If I look at the DNA sequences of a lot of different genes in Drosophila, and I look at the variation among individuals of a gene species... What I find that for almost every protein gene I have looked at, there is a region of 100 to 150 base pairs just off the end of translation but still with a transcriber which is totally conserved to a species... I do not need any notion of ergodicity to deal with that. Then why the hell doesn't that... and that is true of gene after gene, not just one particular gene. There is some mechanical something happening involving that 150 base pairs where maintenance of constancy not only between individuals but across species boundaries has been extreme, so I can smell something going on here but I do not know what it is. But I can say what it could be for instance and we do not have any data about this: "How much of the constancy, how much of the lack of variation of nucleotides can have nothing whatsoever to do with what amino-acids are produced even when those are coding for these, is entirely a consequence of the necessity to keep a certain two or three dimensional structure for the RNA molecule. That is how much amino acid composition constancy is not due to the necessity to have certain amino acids but because the nucleotide sequence has to be maintained. We do not know anything about that. Intervention: Is there any such thing as genetic apoptosis wherein if that 150 amino acid sequence is not there, the molecule self-destructs and that perhaps is why it has to be there. If that 150 amino acid sequence is quite resilient in all the genes that you are looking at... Lewontin: (Interjects) No, no - you misunderstood what I said. I referred to 150 nucleotide positions not translated in different genes and there is no similarity between what is maintained in say the dpp gene and what is maintained in the esterase gene. There is no motif - it is 150 base pairs or two hundred base pairs at the end of the gene and different genes have a different set. Hunter: Are you saying that those base pairs have to be there so that the RNA molecule sits against the gene in such a way as to read it properly. Lewontin: Yes. I am saying it has something to do with the translation and transcription mechanism. But there is no common motif - that is what really drives me nuts! I did these completely conserved regions and I searched the entire genetic database and I cannot find it anywhere else. Intervention: Is that at the level of DNA or is that motif at the level of DNA? Lewontin: It is a motif at the level of AGCTCCAT, not at the level of secondary structure. Of course you said exactly the right thing that what you need to have is good secondary structure programs that will predict whether there is a loop or a hairpin. Intervention: There is another thing. What implications does this have to do in the way the mutation variation occurs at the molecular level. Is it by point mutation, or do you think that exon shuffling is involved? 14