Reading: Most of the material in today s lecture comes from pages 1 6 of Chapter 1in

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1 Lecture 1 First Steps Reading: Most of the material in today s lecture comes from pages 1 6 of Chapter 1in James D. Murray (2002), Mathematical Biology I: An Introduction, 3rd edition. If you are at the university, either physically or via the VP, you can read an online version at The lack of real contact between mathematics and biology is either a tragedy, a scandal, or a challenge, it is hard to decide which. Gian-Carlo Rota Mathematical modelling in biology This course is about the application of mathematics to the Life Sciences. We ll begin with some very simple population models, then move on to more interesting and sophisticated models, eventually applying the mathematical tools we ll develop to the study of chemical reactions, metabolism and gene regulation. One of the highlights of the course will be a famous paper by Alan Turing s about the appearance of patterns during the development of an embryo: here is a brief outline of the course. (1) The first 5 6 weeks of the term will be modelled closely on Jim Murray s famous book Mathematical Biology: an introduction. Our main aims will be to: learn to read, interpret, critique and develop models based on di erential equations, including ODEs, PDEs and delay di erential equations (DDEs); study some standard examples including logistic growth, Lotka-Volterra (predatorprey) and mass-action chemical kinetics; 1 In M. ac, G.-C. Rota and J. T. Schwartz (1986), Discrete Thoughts, Birkhäuser, Boston. 1.1

2 emphasize the extraction of qualitative information about the behaviour of solutions. (2) The latter half of course will be devoted to reading famous or recent papers, including: Alan Turing s celebrated paper On the Chemical Basis of Morphogenesis ; papers related to Christine Reder s influential work on Metabolic Flux Analysis; papers on the Gillespie algorithm for modelling chemical reactions among small numbers of molecules; Yuri Alon s work on regulatory motifs. But before we plunge into these interesting matters, it s worth pausing briefly to think about what it is that people are doing when they use mathematics to model biological systems. To this end, I ve borrowed a figure which appears here as Fig. 1.1 from a recent paper 2 by (former Manchester lecturer) Prof. Helen Byrne. She reviews the contributions of mathematical modelling to the study of cancer and an important feature of the approach illustrated in her diagrams is that the process of model development is cyclical, starting from data and the hypotheses they inspire, passing through the formulation of explicit mathematical models and their analysis and returning, via biological interpretation of the analytic results and predictions, to new experiments and data. Also note that the skills one typically develops during a mathematical education analysing mathematical objects and solving equations are only part of the process: the formulation of explicit models from verbal descriptions and the reinterpretation of mathematical results in the language of the original problem are at least as important. Although I do not expect you to memorise this diagram, nor to shoehorn every model we develop into this scheme, similar concerns will motivate a lot of the discussion this term. But rather than speak abstractly, the remainder of this first lecture will be devoted to developing simple models for the population of a single species that we will refine, through a few cycles of mathematical analysis and biological critique. 1.2 Models for a single species The examples we ll discuss in the remainder of the lecture are very straightforward and you may have seen them before, but I want to use them as vehicles to introduce several themes that will occupy us for the rest of the term. Although both of the models we ll discuss have exact solutions in terms of well-known functions, I want to study them for a number of reasons. Firstly, one can get a good qualitative understanding of the population s behaviour without reference to the exact solutions 2 Helen M. Byrne (2010), Dissecting cancer through mathematics: from the cell to the animal model, ature Reviews Cancer, 10(3): DOI /nrc

3 a b Experimental data Model predictions Biological hypothesis Biological hypotheses Model solution and validation State model assumptions and identify key physic al variables Model formulation and refinement From biological hypothesis to testable prediction by mathematical modelling a The modelling cycle, highlighting its iterative and multidisciplinary nature. Analysis of experimental data is used to generate biological hypotheses that are then formulated as a mathematical model. Preliminary model validation involves establishing consistency between model solutions and the experimental data. If this is not obtained, then the original hypotheses and mathematical model must be revised until there is qualitative agreement. Parameter values can be estimated by fitting the model to the data and the model used to generate new, experimentally testable predictions. b The stages involved in the development and validation of a mathematical model. Develop word equations for all variables and identify the physical processes that regulate their dynamics Convert word equations into mathematical equations, specifying all functional forms Solve mathematical model using appropriate numerical and analytical techniques Check consistency of r esults with original biological hypothesis Figure 1.1: The figure above comes from Helen M. Byrne (2010), ature Reviews Cancer, 10(3):

4 and this sort of qualitative analysis will prove useful for models that aren t exactly soluble. Secondly, these models are very well-studied and many more sophisticated models use them as a foundation and a source of intuition. Both models evolve in continuous time (as opposed to advancing time in, say, one-year jumps, as might be appropriate for animals that only mate once a year) and both represent the population-as-a-function-of-time with a continuously-varying function. They re of the general form =births deaths + migration. (1.1) where (t) isthenumberof beings (bacteria,say,oryeastorplanktonorpu on some isolated arctic island...). ns Malthusian growth Perhaps the simplest model of the form (1.1) is one proposed by Thomas Malthus 3 that has: no migration; abirthrate(measuredino spring-per-unit-time)proportionaltothepopulation; adeathratethat salsoproportionaltothepopulation. The model is thus = b {z} births {z} deaths = (b d) = r (1.2) where b and d are positive real numbers and I have defined r = b has solution (t) = 0 e rt d. This ODE where 0 = (t = 0) is the initial population. It also has dull qualitative dynamics: if r<0(equivalenttob<d)thenthepopulationdiesawayexponentially; if r =0(sob = d) thepopulationholdssteadyatitsinitialvalue; if r>0(sob>d)thepopulationgrowsexponentially. Thus, in the absence of an exact balance between birth and death rates, there can be no steady population. ow, clearly, this model leaves out a lot of complicating, and perhaps relevant, phenomena. The following factors help explain why one wouldn t expect the model in Eqn. (1.2) to give a good account of, say, the growth of the human population: 3 T.R. Malthus (1798), An Essay on the Principle of Population. 1.4

5 The population (t) variescontinuouslyintime,butthetruepopulationis a whole number. When the population is large enough this doesn t matter much, but when the population falls to just a few individuals 4 one needs a di erent sort of model, involving discrete, random jumps in the population rather than di erential equations. We ll learn how to do this later in the term. There is no account taken of sexual reproduction: o spring just appear at a rate proportional to the total population. There is no account of gestation, the time it takes for a child to mature in the womb. We ll learn one interesting approach to this, and related issues, in the next lecture. There is no account of the age structure of the population. Modern humans bear their children during a few of decades in the middle of a much longer life, but the Malthusian growth law does not reflect this. The model has no spatial component and makes no distinction between various subpopulations, but human fertility (measured by, say, the average number of children born to a woman over her life) varies strongly between industrialised urban societies and more rural ones. One sometimes deals with such issues by making the population vary in both space and time, introducing a PDE for (x, t) wherex is a space variable. We will touch on this approach briefly a few weeks from now. The Malthusian growth law (1.2) permits unbounded growth, but something disease, war or famine, for example would eventually limit the growth of a human populations. We will see one very simple way to model such limits in the next section Logistic growth A slightly more recent 5 model due to Verhulst, who was a contemporary of Darwin, suggests the growth law = r 1, (1.3) which is similar in form to the Malthusian law, but with the role of the net growth rate (b d) playedinsteadbytheformr(1 (/)): Eqn. (1.3) is usually called 4 Modern cheetahs have unusually low genetic diversity they re more inbred than many strains of laboratory mice suggesting that the species nearly went extinct in the recent (on the evolutionary timescale) past. Some scientists estimate that at the end of the last ice age (around 10,000 years ago) the cheetah population may have fallen to no more than about a dozen animals. See, e.g., M. Menotti-Raymond and S. J. O Brien (1993), Dating the genetic bottleneck of the African cheetah, PAS, 90: DOI: /pnas Murray cites Pierre-François Verhulst (1838), otice sur la loi que la population poursuit dans son accroissement, Correspondance mathématique et physique, 10: and Pierre-François Verhulst (1845), Recherches mathématiques sur la loi d accroissement de la population [Mathematical Researches into the Law of Population Growth Increase], ouveaux Mémoires de l Académie Royale des Sciences et Belles-Lettres de Bruxelles, 18:

6 the logistic growth law. The parameter is called the carrying capacity and it represents some sort of constraint on the population food supply or number of nesting sites or any other critical resource whose availability limits the growth of the populaton. To see why, note that when the logistic law becomes r and the population grows approximately exponentially. But as increases the number of o spring per member of the population starts to decline and, for >, actually becomes negative (in terms of Eqn. (1.1), the per-being death rate exceeds the birth rate). It is possible to solve this equation exactly to obtain (t) = 0 e rt ( + 0 (e rt 1)), but it will prove more fruitful to see what we can learn about the solution without reference to this result. The logistic growth law (1.3) has very di erent qualitative behaviour than the Malthusian one (1.2). In particular, it has two equilibria (values of the population for which / =0): =0 ) 0=r 1 ) =0or =. This is especially easy to see if one makes a sketch of / as a function of : see Figure 1.2. The equilibrium at =0issaiobeunstable because any small, nonzero population increases, while the equilibrium at = is stable in the sense that small deviations from (t) = tend to die away. The sketch in Figure 1.2 illustrates this, but one can make the same point with a calculation that will serve as a prototype for an analysis we ll do repeatedly throughout the term. Suppose we introduce a new variable n(t) =(t) and suppose further that n(t) or, equivalently, that n/ 1. Then dn = d ( ) = ( + n) = r( + n) 1 = r( + n) 1 1+ n = r 1+ n n rn where, in passing to the final line, I have used the assumption (n/) 1toneglect the term of order (n/) 2.Thismeansthatifn(t) > 0, dn/ < 0anhepopulation declines, approximately exponentially, toward n(t) =0. Similarly,n(t) < 0, then dn/ > 0anhepopulationgrows,againapproaching(t) = exponentially. This behaviour is illustrated in Figure

7 Logistic ODE / 0 r/4 -r/4 0 /4 /2 3/4 Population Figure 1.2: The logistic growth law: note that it doesn t make any sense to consider <0 as is supposed to be a population. The orange points at =0and = are the equilibria (places where / =0) and the orange arrows are meant to indicate that / > 0 when 0 <<, but / < 0 when >. Thus for any initial condition 0 satisfying 0 < 0 <the solution (t) is monotone increasing, while if < 0 then (t) is monotone decreasing. Logistic Growth Population (t) Time 8 Figure 1.3: Solutions to the logistic growth law (1.3) with r = 1 and four di erent initial conditions, 0 / 2{0.1, 0.3, 0.75, 1.25}. All four solutions tend asymptotically to (t) =. 1.7

8 1.3 Linearisation in general The techniques we ve applied here can be generalised: the graphical technique illustrated in Figure 1.2 clearly works for any ODE of the form = f() (1.4) where f() isacontinuousfunction. Further,thecalculationsoftheprevious section can be made to apply near any equilibrium?.thatis,suppose? is such that f(? ) = 0. ow consider solutions of the ODE (1.4) whose initial conditions lie near to?.definen(t) sothat(t) =? + n(t) or,equivalently, Then we have dn n(t) =(t)?. = d ((t)?) = = f() = f(? + n) and if we expand the rightmost expression in a Taylor series around? we obtain dn f(?)+nf 0 (? )+... (1.5) If f 0 (? ) 6= 0,thisis tofirstnonvanishingorder alinearodewhosesolutions are n(t) =n 0 e f 0 (?)t. If f 0 (? ) < 0thisisexponentialdecaywithacharacteristictimescale 1/f 0 (? ). 1.4 A word about dimensionless variables It s a very common practice to simplify models by introducing new variables and new units of time and the logistic growth law provides a good example of how to do this and why it can be useful. Table 1.1 provides a summary of the variables appearing in the original model and the units in which they are measured. In light of this table, there are two natural dimensionless variables: x = and = rt. The first amounts to measuring the population in units of the carrying capacity, while the second involves measuring time in units of 1/r, which is, in linear approximation, the time it takes for a single animal to produce a single o spring under optimal conditions. We d then like to find an ODE for dx/d. First, by thinking of t as a function of and invoking the chain rule, we get dx d = dx d 1.8 = 1 r dx, (1.6)

9 Variable or Parameter r t Units the population: units of beings the carrying capacity: units of beings maximal per-animal growth rate: units of 1/time units of time Table 1.1: The variables and parameters of the logistic growth law (1.3) along with their units. ote that the units of r are determined by the form of the ODE: the lefthand side has units of beings-per-unit-time, so the the righthand side must have the same units. The quantity in parentheses is dimensionless and has units of beings, so r must have units of 1/time. where, to get the second equality, I have used t = /r,so/d =1/r. We then continue the calculation as follows: dx = d = 1 = 1 apple r 1. (1.7) Finally, by combining Eqns. (1.6) and (1.7), we obtain dx d = 1 dx r apple 1 = 1 r = 1 r 1 = x(1 x). (1.8) otice that the final equation above has no parameters at all. If one looks at the logistic law in its natural units measuring the population in terms of the carrying capacity and time in units of 1/r then there is really only a single underlying system and thus only a single underlying pattern of dynamics: findings such as this are among the main motivations for converting models into dimensionless form. In general one can eliminate one parameter for each dimensionless quantity defined and so reduce the apparent complexity of the model. 1.9