And here lies an important difference between the physical and life scientists at
the ... living systems we want to find a different perspective based on Physics and
.
The Physics of Living Matter. A discussion-course Introduction: a biologist’s perspective Alfonso Martinez Arias Department of Genetics, University of Cambridge, Cambridge UK ama11@hermes,cam.ac.uk These lectures want to provide a flavour for the physical underpinning of Biology. They are addressed to an audience with mixed interests and backgrounds. For this reason the need to make them accessible to everybody will run the risk of , at times, being too superficial to some of the participants. Physicists and engineers have a penchant for Principles and Reasoning based on careful quantitative measurements and experiments. Some of these have been part of certain biological traditions, like hard core cell Biology and Biochemistry but, for the most part modern Biology has drifted into qualitative arguments derived from a a long (and necessary) treck into a gene hunt. And here lies an important difference between the physical and life scientists at the moment. While biologists are most comfortable in the midst of complexity using oversimplified (but complex) diagrams to describe their processs through qualitative and descriptive languages, physicists and engineers want to see a mechanistic basis to the solution of a problem with a more quantitative underpinning than Biology can offer at the moment. This is changing and a blend is emerging that will benefit both areas of enquiry. From the point of view of the Life Sciences the prospect is very exciting. Having seen how far a semi-quantitative approach has got us. It is going to be interesting to see where the era that is dawning will lead. Introduction Biology exerts a fascination on physical scientists which has often turned into a fatal attraction as the complexity of biological systems (and often the incompleteness of their description) gives way to frustration. Biology is fascinating because it is both, teasingly complex and beautiful to the eye. It can be frustrating because, at least from the perspective of our current understanding explanations of the kind that Physics and Engineering are used to, remain elusive for most of biological facts. For the moment we do not know how to reduce it to simple principles that encapsulate this behaviour. No Schrodinger or Newton or Maxwell equations behind the activity of genes At least for the moment. In these lectures we want to attempt a fresh look at Biology, and rather than, once again, describe the well known facts that underlie the organization and function of living systems we want to find a different perspective based on Physics and Engineering. We also want to emphasize what we do not know and the ways that are either available or needed to tackle that ignorance. In this first lecture I would like to tell you how a biologist sees the interface of Physics and Biology. What is there in Biology that should grab the interest of Physics and what can biologists gain from a physics/engineering approach to biological questions. The main topic of the lecture is what is Living Matter and I have decided t focus the answer, at least for today in one message: living matter is non equilibrium chemistry, matter organized as a system that contains a code to generate dissipative
structures which are maintained for variable amounts of time in a steady state which we call Life and that manifests itself in what we call organisms. In the first part of the lecture I shall describe the main features of Living Matter, as I see them, and in doing so I shall outline some of the important issues that challenge Biology today. Then I shall show how modelling of specific biological issues can provide an insight into them. I hope to convince biologists that in order to progress we have to adopt some of the attitudes and ways of physicists and that there is a beauty and an insight and much to be gained in this undertaking. To the physicists I would like to suggest that even though there are no great principles that we can gauge (at the moment) in Biology of the form that they are familiar with, simple equations mixed with some biological insight can give interesting views of what is living matter. Finally I want to stress that these notes (which accompany the slides of the lecture) are just that ‘notes’ and do not represent, nor aim to represent, a polished, finished text. They are simply elements for you to think and argue with. 1. What is Living Matter? Biology is the study of Living Matter, a configuration of organic matter in a dynamic non equilibrium state that exhibits complex patterns of spatial and temporal organization. These patterns, which we call cells, tissues or organs, depending on the level and degree of organization of the components, are robust to perturbations and can evolve i.e. change over time in a manner that makes them (adaptively) optimal for the environment. From the point of view of Physics, Living Matter lies within the mesoscopic realm, with the quantum world at one end and the astronomical/cosmological at the other. From the point of view of humans, Living Matter has length scales that can be observed directly and one can forget about the whims of the quantum and relativistic realms. Biological systems have behaviour that is predictable, and reproducible. The study of Living Matter is not only about its organization but also about how this organization yields activities. Physics has made some very significant contributions to the study of Living Matter. In some ways the development of microscopy is a bridge between the two disciplines and the development of structural biology, another in the direction of building a useful bridge. Proteins and nucleic acids are more complex than the inorganic crystals that were being analyzed with X rays at the turn of the XX century and for this reason their structural analysis required a dialogue between the two fields which informed, challenged and developed practicioners of either of the, In the end, the structure of these most fundamental elements of Living Matter, does provide a sound basis from which to understand the organization of a cell (Judson, 1979 for a history with a lot of Science; Nickell et al 2006 for the most recent developments at the interface of structural and cell biology). A development that went hand in hand with the growth of structural biology was Molecular Biology which owes a great deal not to the techniques per se that could be borrowed from the Physical Sciences, but to the ways of thinking and approaching problems with a reductionistic approach at the center (Judson, 1979 again for a thorough examination of the history and the Science; de Chadaverian 2002 for an in depth and focused analysis of the development). However, Living Matter has many aspects that are not covered by the structural
biology programme and some of them, particularly the thermodynamic constraints under which biological systems operate, have been in the mind of many physicists for a long time. In his seminal (but also some times harshly criticized) book “What is life”, Schrodinger (1944) raises this issue and musing over it, ponders whether or not there might be new laws of Nature to stem from a physical understanding of biological systems. He might well have been right but it has taken a long time to be in a position to have a glimpse of the answer to this question. In many ways, this course will explire this statement. Some of these new ‘laws’ or ‘principles’ (not the same thing, of course) Schrödinger was talking about might lurk in two areas. One is the thermodynamics of living systems, the other what we can call the ‘dimensionality’ of life. The first area has been explored at length by I. Prigogine and his school and I shall leave you to decide whether his claims (see Prigogine 1996 for a lesser technical discussion of his views on the matter; also Coveney and Highfield, 1991 as an interesting history of the evolution of thermodynamic ideas with some discussion of the biological issues) are as he describes them or feel into a more basic plot that we sort of knew already. The second one we are just beginning to explore and we need to know a bit more: we need to know more about the molecular dynamics of living systems in order to actually figure out how it works. Hopefully this need and what it means practically will become clear in this course. Living matter is easier to describe than to understand and perhaps for this reason most of the contributions of Physics to Biology up to now, have been of ways of looking: microscopes of different kinds and length scales. But there is also an input which in the gene hunt of the last few years we, biologists, have forgotten about: measurement. We have forgotten how to measure and, of course, this is a very good realm for the physicists. In this regard we shall see repeatedly in the course that it is the ability to obtain accurate measurements of biological phenomena that is changing the face of Biology. The appreciation that Living Matter is about rates and proportions, about scales and efficiencies is going to transform our understanding of Biology. Finally we also have to model and not only in the qualitative diagrammatic manner that we overuse today but in a more engineer like manner. We have to model because as we shall see today, it is the only way to understand. If as R. Feynman allegedly said, we only understand something if we can build it, we are far from understanding biological systems. And the reason for this is not that we do not know the component elements of the system, we do know them but rather (and forgive me for insisting) because we do not know how they are arranged in the cell, We do not know their stoichiometries and dynamics. Wherever we look in some detail we become amazed at the efficiency of the activities that the elements can perform (see Bustamante et al. 2005 and Phillips and Quake, 2006 for a very accessible review of some of these issues) and this in the background of a huge energy expenditure. To understand Living Matter will require a more quantitative Biology, it demands the development of new vistas of biological processes to get a better balance between structure and function. It could be argued that some of this is being catered for by the current impetus on Systems Biology (see Hartwell et al. 1999 for a very lucid discussion of these issues) however, while it is true that Systems Biology is a very broad church, it is also true that for the most part Systems Biology is devoted to the generation and analysis of high throughput data. There are small pockets of activity, amongst which I would include Synthetic Biology, that deal with other kinds
of problems but, for the most part (in my opinion) Systems Biology is a place of encounter for advanced Molecular Biology, Genetics, Computer Science and Bioinformatics. The harvest from these interactions has enriched and enlightened the fabric of Biology and will continue to do so. The analysis of Living Matter, what some of us have called the Physics of Living Matter (PLM) is a different and complementary perspective on the challenges of modern Biology. The aim is to be able to obtain accurate measurements of biological processes, to gain insight into dynamic relationships of structure and function, to understand emergent properties of systems from the constituent elements in their native length and time scales rather than from the phenomenological properties generated by high throughput data. PLM straddles the boundary between Biochemistry, Cell Biology and Physics with a tinge of Genetics as a way to perturb the systems. PLM looks to integrate primarily physicists and biologists with a defined course of action: Advanced Microscopy Development, Cell Biology and Biochemistry are at the core of the initiative. While in a very wide sense PLM could be construed as part of Systems Biology, such a broadening of its definition runs the risk of leading to a loss of focus and research co-ordination that would in our opinion reduce the chance for ground breaking discoveries. Having set the stage we can get into the two main topics of todays lecture, with an interlude about language. 2. Principles of the structure and function of Living Matter I would like to begin by listing some of the properties of Living Matter thus putting limits to what is Living Matter and also trying to gauge something about its foundations and understanding. Living Matter is dazzling to the senses. The large number organisms with their adaptations, their ever changing organization and their evolution grab our imaginations with ease. Think of butterflies, leopards, whales, orchids and of course dinosaurs! And upon casting our eyes over this questions rush into our minds. What is the engine that drives this panoply of shapes and colours? What is the substance that allows them to change over time and sometimes to convert one into another? The work of the last one hundreds years has laid down a body of knowledge that allows us to begin to ask questions not about the make up of living systems, but about the rules of their organization and their function. Maybe there are some principles that rule the functioning and operation of Living Matter. Here is a list of some possibilities (you can look at the PP for some illustrations of the points): (i)
Living matter is a dynamic mixture of organic polymers, nucleic acids and polypeptides, with significant contributions from carbohydrate and lipids. Inorganic ions and other organic compounds e.g vitamins, play significant additional roles in the mixture. It is a mixture far from thermodynamic equilibrium that generates dissipative structures. It is a mixture in a nonequilibrium steady state.
(ii)
Biology is about information which, at the highest and most universal level, flows according to the “central dogma” : the information to make,
replicate and mold a cell is stored in genes, DNA, and is decoded by making transient copies in the form of mRNAs from which it is translated into proteins, that encode the functions of the cells (enzymes, structural role, processing of information). Note: some RNAs act as regulators in the form of RNA. Corollary: genes are software, proteins configure hardware. (iii)
Reproduction, self assembly, self organization and evolvability are trademarks of Living Matter. The last one is a particularly important element of Living Matter that makes it different from other structures that can self organized or assemble (tornados, sand dunes). Reproduction but above all evolvability which is built into the information flow that runs the system, are completely specific to Living Matter.
(iv)
Proteins are the workhorses of Living Matter and represent thermodynamically very efficient nanomachines that operate far from equilibrium: many are enzymes and some are catalytic, others process information and most of them play structural roles (pure structure, scaffold or adaptors). Corollary: An inconvenience: a protein might have more than one activity and although proteins are modular, often the functional domains are overlapping.
(v)
The generation of an organism requires the generation of different kinds of cells. The essence of this is cell type specific gene expression. Different cells are different because the contain different RNA populations. A cell state is a multidimensional genetic space (maybe some numbers for flavour).
(vi)
Living Matter has a hierarchical organization with emergent properties as a significant underlying theme.
(vii)
The flow of information from the DNA feeds two boxes: an information box, dedicated to matters of cell diversification, patterning, coordination of events in time and space at different length scales, etc. And an energy box, dedicated to providing the fuel that allows a cell to run and operate its informational devices. Living matter organizes its activity around the interplay of these two boxes.
(viii) The cell is the fundamental unit of biological systems. Cells are the essential link between molecules and organisms. Cells integrate the energy and information boxes. With these statements (principles?) in mind, perhaps we can look at Biology in a different light and try to see if the different specific cases that we all study are manifestations of these. A pervasive view of Biology is that there are no or very few rules from which one can work out how things work, that all is manifestations of contingency and that this is why it is hard for physicists to understand. At the other
extreme are the few (and I count myself amongst those on occasions) that believe that there are principles, that we should find rules that will allow us to understand and predict. But maybe in the end one has to come to the compromise of the engineer that uses rules and principles that exist to build structures that, sometimes, are contingent and cover specific needs. Maybe Biology is more like Engineering but even engineers have to use the principles from Physics. So, to summarize, the main attractions that Living Matter has for a physicist are that: 1. it begets complexity in a non chaotic manner as it is reproducible, robust and evolvable. Yes, underlying this complexity is basic chemistry in numbers and organization we know little abour. 2. it deals with an interesting thermodynamic realm and might highlight new principles if not new laws. 3. Because it is an important part of the physical world However, from the perspective of a physicist the study of Living Matter is a tall challenge because: 1. Deals with complex mixtures of very difficult resolution (see Hartwell etyt all 1999 and Phillips and Quake 2006 for interesting discussions of these issues) 2. It is very far from thermodynamic equilibrium and mathematics are complex. In fact there is very little by way of serious theoretical work about obtaining emergent properties from components. From a biologist’s point of view, a physico-chemical approach to Biological problems has the promise of revealing new dimensions and perspectives which in the long run will provide real understanding. Through this it might beget a new kind of Medicine. 3. Hierarchies and organization It seems to me that the essence of Living Matter is the way the different hierarchical levels are woven: genes encode proteins that organize themselves into networks which feedback onto the genes to create gene regulatory networks (GRNs). This is very, very important and we have spent the last many years describing and understanding this interaction between the gene and the protein level. By necessity we have extrapolated to other levels but our understanding is limited to these too levels. Now, GRNs have no large spatial element and one can, in principle, describe them in time (and we shall do this below). Proteins, on the other hand, can assemble into higher order spatial structures and the most significant ones are cells which they construct together with lipids and carbohydrates. Cells are remarkable inventions. The acticity of a protein or a protein network in a cell is different than in isolation and the structural organization of a cell will feedback on the activity of a PRN (Protein Regulatory Network) and through this onto that of the GRNs. Then cells organize into tissues and these into organs and thus into organisms. In this structure, the elements that give rise
to one level, when assembled, acquire new properties which impinge upon their function at the lower levels. Much of the study of Living Matter goes into describing and understanding the organization and functioning of each of these levels and their relationships. 4. Issues of language The Life and Physical sciences share much of the approach and method to the solution of problems but do emphasize different languages. For this reason, it might be useful to stop and make some remarks on these differences so that each camp can take stock of how the other speaks (Lazebnik, 2002 for a somewhat comic, but in certain aspects realistic, appraisal of the ways of the differences between Biology and Engineering). 4.1 An interlude for biologists, equations Equations, in particular differential equations, are a way to represent the world. Physicists have known this since Newton discovered (invented?) calculus and, at the same time, applied it to draw general laws for the motion of bodies. Equations allow us to describe relationships between variables and more significantly to draw behaviours and to describe the changes in those variables. Physics and engineering are built around equations and much of everyday life, from the behaviour of your money in the Bank, the way the water flows down your shower or the signals exchanged between mobile phones have underlying laws, trends and principles which can be captured by differential equations. Like everything else the appearance of computers has changed the way we perceive and do Mathematics and this is being reflected in the way differential equations are used. If you are a physicist you will have seen this already, if you are a biologist, you also have a feel for the computational aspects of your work and you will appreciate what they are doing to mathematics. So, in a way, a differential equation is a tool we can use to describe a process. It is a powerful tool in that application of a number of rules and operations allow us to solve the equation and find some general relationships between the elements of the process. And modern computers are coming to help in solving these equations. Of course biologists are used to some equations as most elementary Biochemistry covers Michaelis Menten kinetics and cooperativity and if you have taken any population genetics or evolutionary biology you have had your fair share of muc inserting an equationh of what I am saying here. Thus equations allow us to draw relationships between the elements of a system, the variables, and this is often represented in a Phase Space, an ideal space where a point results from the relationship between two or more variables. You are familiar with two and three dimensional spaces but there are others. In an abstract manner you will remember your thermpodynamics and will remember the relationships between Pressure, Volume and Temperature, which define a three dimensional space for every point
PV = nRT
(1)
Also, for mechanical systems, the phase space usually consists of all possible values of position and momentum variables. However, spaces need not be three dimensional, ! we are all familiar at some level with the concept of space-time, which leads us to
think about 4 dimensions. If dimensions are the number of variables needed to define a system i.e its coordinates, certain biological realities will need multidimensional spaces. For example, if a gene is a variable, a cell state will be define by n dimensions, where n is the number of genes that contribute to that state. One cannot visualize n dimensional spaces for n>3 but there are ways to sort of ‘see’ them. Trajectories in phase spaces represent laws that link the different variables and these are often extracted from Differential Equations. In a differential equation one simply established a relationship between two variables and draws a function which describes this relationship. Thus, Newton’s second law can be written
F = ma
(2)
where F is a force, m is the mass and a is the acceleration. Since the term a is the acceleration, the equation cak also be written as ! dv (3) F=m dt dx and since the velocity is v = , the equation can also be written dt
! F=m
!
d 2x dt 2
(4)
For a constant m, the nature of the force will determine the trajectories and the dynamics. The world of mechanics (earthly and celestial) as well as engineering has ! large number of variations on this theme. Now, every solution of a differential equation can be viewed as a trajectory in a phase space and this, which has become very easy to visualized with numerical methods and computer graphics. Since in most instances a model is based on two or three variables, this is a great aid to understanding the dynamics of a processes and the relationships between the different variables. As all Biology deals with dynamics and processes involving more than one variable, it is very important to know HOW TO DESCRIBE A PROCESS WITH A DIFFERENTIAL EQUATION as well as to have some ideas of how to solve a differential equation or a system of differential equations. In fact there are very famous and well established differential equations in Biology, for example the Hodgkin and Huxley equation which describe the initiation and propagation of an action potential in a nerve cell:
IM = Cm
dV + = IK + INa + IL dt
(5)
Where I is the current with M for the total membrane, K and Na for the two ions and L for the leakage. Cm is the capacitance of the lipid bilayer and V is the membrane voltage. The!solution to the equation is, of course, the action potential. There are two interesting features of differential equations. The first one is that most of them, and certainly most of the interesting ones, do not have ‘easy’ (analytical) solutions, sometimes the solutions do not exist. One has to use computers to visualize them and it is here that the notion of phase space and its relationship to the solutions
of a differential equation is important. The second feature is that innocent looking equations can have, under certain conditions very complex and sometimes wicked behaviours and the phenomenon of chaos is based on this premise (there are many website dedicated to Chaos and you may find some that will suit you but this is one which, I think, will help biologist appreciate the issue –which has some roots in Ecology- and also see the thin boundary between Science, Fun and Art: http://pass.maths.org.uk/issue9/features/lyapunov/index.html) Both of the features mentioned above arise very easily in equations (models, remember that an equation is a model) of biological systems which usually imply the use of more than one element. A classic example of the power of equations and some of these features is the Lotka-Volterra model which describes the prey (x) predator (y) relationship and is widely used in Ecology but has much wider uses. In its simplest/ideal formulation, the relationship is described by a system of two coupled, time dependent differential equations: dx = "x # $xy dt dy = %xy # &y dt
(6)
The system says that the dynamics of the prey (x) will depend on its birth rate (α) minus its death determined by encounters with the predator (y) which will occur at a frequency (ß). In the!simplest model the prey population has an unlimited birth rate. On the other hand, the population of predator (y) will depend on its renewal rate plus its encounters with predators, represented by the parameter δ, and its death rate, represented by γ. The equations have periodic solutions (see figure below) which when plotted in a phase diagram produce a very visual representation of the oscillations of the populations in this ideal system (see the Fig below the oscillayions which shows that the two species alternate; the different diamters represent different initial conditions of the system.. More complicated models of the situation (competition for resources, multiples interactions) produce more complicated solutions that help ecologists understand the situation. Thus, differential equations provide a very powerful weapon to understand Nature through a description of the phenomena. They are better than simple words because they provide an insight into mechanism. NB for biologists. It is not easy to find a proper reference which at one stroke could inform a biologist about the mathematics and physics that could be of use to him.
These days the web is full of useful hints and aids which, with appropriate guidance, can be helpful in particular situations. Nothing, really, like taking on a problem and getting into the mathematics and the physics through the curiosity of solving that problem. What nonetheless should be an important element in the conceptual kit. Advice to biologists: talk to physicists and engineers, ask questions and try to formulate your problems in terms of variables and dynamics. The web contains a large number of aids for quantitatively challenged individuals; surf it!. 4.2 An interlude for physicists, the genetic approach In many respect, Genetics has been the great triumph of Biology. It has become a discipline on its own right and a truly ‘scale invariant’ tool that has allowed us to solve very hard problems. We are all familiar with Mendel’s laws and sort of know what a gene is (do not get bogged down in philosophies about the concept of a gene, biologists are very good at getting bogged down with words), what is important is to know that some of the major advances in our understanding of embryonic development, the activity of the nervous system or metabolism have come from the application of Genetics to the solution of these problems. The reason for this is that Genetics is a language that plays a role in Biology similar to what mathematics does for Physics. Genetics is a body of tools, tricks and loose principles that allows you to gather information from function. Implicitly it acknowledges that Biology is about information and ploughs along with this as an implicit premise. If you want to know how something works, you take it apart and the way you do this with genetics is by searching for ways to disrupt its functioning. Genetic analysis has three sequential stages which are summarized in the left hand Figure which is easy to follow. It begins by assuming (as proven) that all that happens in an organism is mediated through a protein (which is true) and that affecting the gene that codes for the protein will affect the process. Thus, given a specific question (how is a wing or a leg is made, what are the elements that configure the synthesis of a particular molecules, etc …)..one looks for mutants (perturbations in genes) that affect that process, through a screen. Then one organizes the mutants into genes through an exercise called ‘complementation” (collecting the mutants that affect the same gene) and then one tries to order the genes, through an analytical process, with rules called ‘epistasis’ (you can look this up. An attempt to summarize how to do it in the context of modern Biology can be found in Chapter 5 of Martinez Arias and Stewart, 2002; see also page on epistasis in the wormbook http://wormbook.org/chapters/www_epistasis.2/epistasis.html, there is more here than some of you may like, but it is good). This analysis allows you to organize the genes into linear functional series which will provide some insight into the process and
allow some understanding. Screens need not be functional, they can be molecular, and thus they always have to get back to the functional genetics. Genetic analysis not only gives you the elements of a process but also provides you with some hints of their functional organization. It has problems in that it cannot deal with more than two genes at a time, that it does not tell you much about mechanism and that it cannot deal very well with multicompnent machines. NB for physicists and engineers. There are some good books of Genetics and, in some ways, it is not too difficult to pick up, though like with maths, practice is all and there are some conceptual issues that are not in the textbook. Talk to a biologist! 5. Principles and constraints of self organizing systems Having established some definitions about Living Matter it might be good to begin to discuss how it operates. As we have pointed out above, Living Matter is nonequilibrium chemistry and for this reason in this section I shall outline briefly what this means from the practical point of view. I shall avoid thermodynamic arguments here and will go straight to explore the conditions under which chemical reactions can generate temporal and spatial patterns of organization and how these relate to Biology. 5.1 Chemical dynamical systems Living Matter is about chemistry and as we have discussed above, a very special kind of chemistry: non equilibrium chemistry. We are familiar with the description of a chemical process by a differential equation and thus, the velocity of the production of a product P from the interaction of an enzyme E with its substrate S
E+S
k1 k"1
k2 ES ## $E + P
(7)
can be described by the equation (after some algebra which you can find in any Biochemistry textbook…..or do it yourself as an exercise!): ! dP S S = k2 ES = k 2 E o = Vmax dt Km + S Km + S
(8)
This has the familiar hyperbolic shape and there is very little else that one can extract from here. The enzyme might exhibit cooperativity, which leads to a slightly more ! complicate version of the equation, but when the substrate is over, the reaction is over and the system moves on. This set of equations has been extremely useful in understanding a large number of phenomena and enzyme kinetics has profited from this. This is all typical of an equilibrium system. Whereas an equilibrium system will stay and die, a nonequilibrium system can generate patterns and this has consequences. This was first realize by A. Lotka in his analysis of the following chemical reaction:
A + X " 2X X + Y " 2Y Y " B
(9)
There are several things that make this equation different from the enzymatic one. The net change is A B, which is very similar to the one above from S to P, but here all reactions are irreversible, there are two components and, most importantly, ! there is an autocatalytic term in that the interaction of one molecule of X with one molecule of A produces two molecules of X. One can write the kinetic equations for these reactions which bear a resemblance to the Lotka-Volterra predator-prey equations (as a matter of fact it was Lotka the one that wrote the system first to describe the chemical equation and then Volterra worked over it on the ecological case). Thus the equations:
dA = "k1 AX dt dX = k1 AX " k 2 XY dt (10) dY = k2 XY " k 3Y dt dB = k3Y dt One can plot the solutions of the equations as solutions and in phase space and one can see that they suggest oscillations. Such behaviour was not known in the 1920s, which is when Lotka was thinking about this, but it was intriguing that under certain ! conditions chemical reactions COULD, MIGHT exhibit spatial oscillations. A very important observation between the classical enzymatic kinetics, which has been extremely helpful in understanding metabolism, and the kind of processes revealed by Lotka’s gedanken reaction, is that the first is purely an equlibrium reaction which over time reaches a stable steady state, whereas the second one, which begins to hint at non equilibrium, has tempotal patterns built in its structure (see figure on the right for a comparison). Around the early 1950s a Russian biochemist was trying to reproduce the Krebs cycle (one of the energetic powerhouses of the cell) in a test tube when he observed that a solution of citric acid, acidified bromate (BrO3- ) and ceric salt exhibited periodic oscillations between yellow and clear. This was extraordinary as chemistry was all about equilibrium. Another Russian chemist A. Zhabotinski refined the reaction by changing citric acid for malonica acid not only saw the oscillations but when the reaction was performed on a thin layer, spatial patterns emerge: the reaction generates pattern (the web is full of images of the reaction, two websites that discuss it: hhttp://www.uniregensburg.de/Fakultaeten/nat_Fak_IV/Organische_Chemie/Didaktik/Keusch/Doscill-e.htm
And one by Zhabotinsky himself http://www.scholarpedia.org/article/Belousov Zhabotinsky_reaction). The reaction, is the Belusov Zhabotinski.reaction. and can be modelled as the Lotka reaction but the equations are more complicated, still they explain why there are oscillations. The model that explains this is called the ‘oregonator; as it was a group at the University of Oregon that discovered the principle. Thus chemical systems, with more than one component, under non equilibrium conditions and with feedbacks undergo temporal and spatial (! More about this below) patterns. Once you are here you know chemistry and mathematics have something else to tell Biology than Michaelis-Menten kinetics. 5.2 The structure and activity of gene regulatory networks The chemistry of Living Matter is somewhat more complicated. Indeed, oscillations of the type observe in the Belusov Zhabotinski reaction have been observed, for example during Glycolysis and understanding these plays an in the manipulation of certain industrial processes. However, while metabolism is interesting intriguing bits emerge when thinking about pattern formation or neural networks. At the base of these events there are genes which, as we have discussed above, code for proteins which assemble them into networks which, in a software kind of way, run the information processing properties of the organism. In the last few years, particularly after the deluge of information generated by the genetic molecular analysis of biological systems, the realization is dawning on us that rather than looking at the power of individual genes, we should be looking at gene networks and that much f the activity of these networks forms the fabric of an organism. Whether the networks are dedicated to the generation of energy, the homeostasis of the cell or the generation of pattern intra or inter cells, gene regulatory networks (GRNs) lie at the bottom of it all. The modelling and analysis of GRNs occupies a large amount of the time of molecular biologists. Some times this is useful as it shows something about the behaviour of very complex systems. The premise of these studies is the analysis of chemical reactions that we have outlined above, with a little bit of enzyme kinetics. In general the models treat genes as chemicals with the proviso, which sometimes becomes important, that one has to make sure one accounts for the fact that from a gene to a protein there are two steps: transcription (which always involves an amplification from 1 (genes) to many) and the translation, where there are spatial as well as quantitative controls which we are still learning about (see Alon, 2006 for an excellent exposition of the theory and practice of the current modelling of GRNs). Thus, if one says that protein A regulates the expression of protein B (and this is a lot of what GRNs are about), one has to write down that gene A is transcribed and translated to make protein A which then finds gene B and transcribes it, then the mRNA from gene B is translated into protein B. Often one can approximate things, particularly if all one wants to deal with is how protein A lead to the expression of gene B, but it is important to know what one has to do. Note on terminology: often, in Biology, genes are written in italics and proteins in roman.
GRNs can have interesting properties which can come handy to explain biological observations. A very nice demonstration of this was performed by assemblying a synthetic GRN, called the repressilator (Elowitz and Leibler, 2000; I think this is a classic) in which three repressors were functionally linked to each other in E. coli in the way indicated on the right. A reporter of the activity of the system, in the form of GFP, was linked to one of the elements and the system was both run and modelled. The kinetics of the system is described by six coupled first ordered differential equations of the form:
dmi # = "mi + + #0 dt (1+ p nj ) dpi = "$ ( pi " mi ) dt
(11)
with i and j being lacI, tetR or cI. The repressor!protein concentration is pi and mi is its mRNA concentration. αo represents the leakiness of the promoter and α + αo is the amount of protein I in the absence of the repressor.ß denotes the ratio of protein decay to the rate of mRNA decay rate and n is the Hill coefficent. This system of equations has periodic solutions which reveal oscillations (see figure on the right which indicates solutions of the equastions: left deterministic). A stochastic model (right hand side in figure), probably more accurate of reality, also shows the oscillations of the different elements. As you can see the situation is similar to those described above but here we are looking at the activity of a gene network, albeit a synthetic GRN. There are many situations in nature which assemble periodic phenomena. Oscillations are really common and probably represent one of the stable solutions to the problems posed by the structure (in the sense of organization)-function relationships in Living Matter. Two classic ones are circadian rhythms and the generation of somites in vertebrate embryos. No time to go into this here, perhaps we can look at it later. Suffice to say the oscillatory behaviour of chemicals or populations is fairly well understood. 5.3 Diffusion and spatial patterns The fact that chemical reactions can generate temporal patterns is intriguing. Let us not forget the conditions of this feat: non linearity, feedback and coupling. Also, for this to be sustained in the manner that life is, the system has to be thermodynamically open and far from equilibrium. We have seen that a modern version of these notions
applied to GRNs can give us important insights into certain processes. However, Biology is as much about the conquest of space as it is of time and this, physicists have known for a long time. It was A. Turing who in 1951 (have a look at the paper which might look faded to the physicists but has much to think about from the perspective of the biologists….and it is easier to read than you think when you first look at it) decided to explore the possibility that in a stable system of chemical reactions, random fluctuations in local concentrations can trigger spatial patterns. He does this by using a system of two equations representing a chemical system much in the spirit of Lotka and Volterra, but now he assumes that these reactions occur in a line or a circle of cells and that the concentrations of the compounds can be exchanged between cells i.e the chemicals diffuse. Furthermore, he allows for random fluctuations in the local concentrations and studies what happens with these fluctuations over time under the influence of the diffusion process . He studied a case of two substances and observed that the fluctuations in concentration can be amplified and, furthermore, that they can give rise to spatial patterns i.e. Turing discovers that coupling of diffusion to a reaction system can give rise to spatial patterns. The Turing equation in a general form for two reactants, a continuum and one dimension is "A "2A = f (A,B) + DA 2 "t "x : (12) "B " 2B = g(A,B) + DB 2 "t "x He could only deal with linearized equations (see Figire below) and was very limited by the computing power of the time ! him!). But not (which owed much to bad for a start (see Figure on the right which shows one of Turing’s original equations, its solution and a pattern in the skin of a jaguar). It is worth pointing out that the influence of diffusion in biological pattern formation was much in the air and that it was felt that it did play an important role in the process. The insight of Turing is not only to couple it to chemical kinetics, but to realize the importance of fluctuations. This is a seminal piece of work which with time provides the seeds for the explanation of the 2D Belousov Zhabotinski reaction and many other spatial patterns of chemical reaction. It can be generalized to multiple chemicals and three dimensions with an equation of the type:
"X i + F(X i ) + Di# 2 X i "t
(13)
where Xi represents different chemicals each with particular diffusion constants. The model has seen many variations through the years. Of particular interest are those of !
Nicolis and Prigogine with a sound basis in thermodynamics and emphasizing the need for the system to be open and non equilibrium. Also,.A. Gierer and H. Meinhardt (see the long look back at their work in Meinhardt and Gierer, 2000) provided an important variation of Turing’s equation which includes some constraints for such systems to generate patterns with the stability that we observe in Nature: short range activator and long range inhibitor, coupled (for some of the solutions look to the left and if you wished you can visit the websites of Gierer and Meinhardt which contain some interesting stuff; (Gierer: http://www.eb.tuebingen.mpg.de/departments/former-departments/a-gierer and Meinhardt, http://www.eb.tuebingen.mpg.de/departments/former-departments/hmeinhardt/hans_meinhardt.html). This work clearly makes the point that chemistry can generate spatial patterns of the kind that we observe in biological systems but the work, largely developed by non biologists missed some important points and in doing so has led to a lack of appreciation by the biologists. The point is that biological patterns are not generated de novo from a small set of reactions but rather arise progressively, have different levels of organization and are robust. The Turing insight and its variations shows us what is possible but we still have some way to go to get the full worth that is there. We can invent chemical dynamical systems whose solutions mimic biological patterns but this is not the way those patterns are generated. The most famous case is the early patterning of insects which was a ripe field for the Turing fans for a long time until it was found that there was no single global mechanism that generated all stripes but rather they emerge one by one (Akam,, 1989 an old comment, a comment from the time but it does make a point). Where Turing/Prigogine type models might have something to say is in telling is that the elements of biological systems have pattern forming capabilities. This has been nicely demonstrated in the work of E Karsenti and colleagues on the ability of microtubules and associated proteins to generate patterns depending on the parameters (Surrey et al, 2001 for a beautiful demonstration of how to do IT; Karsenti et al. 2006 for a review of the midset of that school) 6. A comment on the strengths, limitations and scope of these approaches. A few general points to conclude. When talking about physical sciences one means Physics but also Chemistry and Engineering. The engineers are an interesting group in the context of this lectures and offer interesting approaches to biological systems which in some ways differ from those of the physicists and a debate of what is a most profitable line of pursuit when dealing with Biology is not sterile. Some problems (issues of structure, organization and dynamics) might require more of a physicist point of view while others (particularly when dealing with the activity of networka or the properties of tissues) might require much more of an engineer.
Thus, while understanding the properties of Living Matter does require a great deal of Physics and Chemistry, understanding the operation of the assemblies of Living Matter might require a great deal of Engineering approaches (without forgetting the Physics). We shall see much of this in the series. There is much in Biology which is intuitive, much that is based in the experience that one accumulates working with a system. But this is the beginning. I am sure that the quantitation and the modelling will bring out much that is hidden behind the surface. Perhaps those laws that Schrödinger was talking about. Finally, as I hope to have hinted at (particularly in the lecture) the fact that one can describe an observation or a phenomenon with an equation does not mean that one understand more or anything at all. Equations are workds and are only useful in so far as they provide insight and make quantitative predictions and statements. The qualitative analysis of the behaviour of variables is interesting and maybe useful for some experimental situations but only as a preamble, Turing showed that by connection a chemical reaction with diffusion one could generate patterns and it is indeed very striking that some of the patterns developed from the theoretical models mimic those encountered in Nature. However, what is important is the basis of those patterns and Turing knew this (read the paper!). Although it is possible to write equations that generate patters that look like those of zebras and leopards, the basis for those patterns lie in the migration and interactions of neural crest cells which are neither represented nor predicted in the equations. If one could write a set of equations that generate the patterns from the activity of those cells, that would be impressive. In the meantime we have to look at different realms as does the inspiring and interesting work of the Karsenti and Nedelec groups (http://www.cytosim.org/). In summary the fact that one can write an equation to describe a system does not mean that one understands the system and this is particularly true of biological systems. If the equation has real data (numbers) and produces the properties of the system from the activity of the components, then it is useful. References Akam M. (1989) Drosophila development: making stripes inelegantly. Nature 341, 282-283. Alon, U. (2007) An introduction to systems biology. Design and principles of biological circuits. Chapman and Hall/CRC. Bustamante, C, , Liphardt, J and Ritort, F. The Nonequilibrium Thermodynamics of Small Systems Physics Today July 2005 page 43. Cohen JE. (2004) Mathematics is biology's next microscope, only better; biology is mathematics' next physics, only better. PLoS Biol. 2004 Dec;2(12):e439. Coveney, P. and Highfield, R. (1991) The arrow of time. Flamingo Elowitz MB, Leibler S. (2000)A synthetic oscillatory network of transcriptional regulators. Nature 403, 335-338.
Hartwell LH, Hopfield JJ, Leibler S, Murray AW. (1999) From molecular to modular cell biology. Nature. 402, Suppl: C47-52. Judson, H. (1979) The eighth day of creation. Karsenti E, Nédélec F, Surrey T. (2006) Modelling microtubule patterns. Nat Cell Biol. 8, 1204-1211. Lazebnik Y. (2002) Can a biologist fix a radio?--Or, what I learned while studying apoptosis. Cancer Cell. 2, 179-182. Lewis J. (2003)Autoinhibition with transcriptional delay: a simple mechanism for the zebrafish somitogenesis oscillator. Curr Biol. 13, 1398-1408. Martinez Arias, A and Stewart, A. (2003) Molecular Principles of Development. Oxford University Press Meinhardt H, Gierer A. (2000) Pattern formation by local self-activation and lateral inhibition. Bioessays. 22, 753-760. Nickell S, Kofler C, Leis AP, Baumeister W. (2006) A visual approach to proteomics. Nat Rev Mol Cell Biol. 7, 225-230. Phillips R and Quake, SR (2006) The Biological Frontier of Physics. Physics Today May 2006 page 38. Prigogine, I. (1996) The end od certainty: time, chaos and the new laws of Nature. The free Press. Schrodinger, E. (1944)‘What is life’ Cambridge Univ. Press. A classic which you should read with perspective. There are two comments on the book that are worth looking at. SJ Gould (1995) ” “what is life? As a problem in history in ‘What is Life? The next fifty years. Cambridge University Press. And a very negative view: Perutz, M. (1987) Physics and the riddle of life. Nature 326, 555-558. Surrey T, Nedelec F, Leibler S, Karsenti E. (2001) Physical properties determining self-organization of motors and microtubules. Science 292, 1167-1171. Turing AM. (1952) The chemical basis of morphogenesis. Phil. Trans. Roy. Soc Lond. Ser B. 237, 37-72.
