Math 19. Lecture 31SwitchesT. JudsonFall 20051 Pocket Gophers and Mite ParasitesConsider the co-evolution of pocket gophers and their mice parasites. Sup-pose that g(t) represents the fraction of gophers of a certain blood type andm(t) the fra ction of lice that like to feed on gophers of that blood type.• This interaction might be governed by the following system.dgdt= F (g, m) (1)dmdt= H(g, m). (2)• Since the parasites r eproduce at a much faster rate than the gophers,(1) must be the slow moving system and (2) must be the fast mov-ing system. This dictates that |F (g, m)| should be much smaller than|H(g, m)| except near those values o f (g, m) where is H = 0.• The requirement that |F | ≪ |H| says that for most values of (g, m) thefunction g(t) is changing at a much slower rate that m(t).• For any given initial value for m, a good first approximation to ( 1) and(2) can be obtained by regarding g as a constant in (2) and using it asa parameter for m. Using this approximation,dmdt= H(g, m)1predicts that m(t) will be found equal to one of the solutions to theequationH(g, m) = 0, (3)where∂H∂m(g, ·)|m< 0. (4)The condition in (3) says that m is an equilibrium point todmdt= H(g, m).The condition in (4) says that this equilibrium point is stable.• In general, g will not be truly constant as its motion is controlled bydgdt= F (g, m).That is, g will change slowly a nd thus the conditions in (3) and (4) thatdepend on g will change slowly, and thus m will change slowly even a sit stays close to obeying (3) for each value of t.• An exception occurs when g evolves inH(g, m) = 0so as to make one of the stable equilibria of the m equation disappear.In the figure below, we get a sudden switching from 0 .1 to 0.7. Suchswitches are sometimes called catastrophes.2 Thresholds in DevelopmentA fundamental a pplication of these fast-slow ideas can be seen in the arti-cle Th resholds i n Development.1The po int of the article is to present a ndgive evidence for a model that explains how nearest neighbor cells in a nembryo might naturally develop in drastically different ways. Lewis, Slack,and Wolpert sought a mechanism that was compatible with the notion thatdevelopment is determined by relative concentrations of ambient chemicals(e.g. morphogens).1Lewis, Slack, and Wolpert. “Thresholds in Development,” Journal of TheoreticalBiology 65 (1977) 579–590. See Reading 26.1 (pp. 421–428).22.1 The Historical Context of the ArticleThe proposed explanations f or such catastrophic difference in offspring fatehad two fundamental flaws.• They required drastic and unrealistic changes in the size of the mor-phogen concentration over very small distances.• They couldn’t explain how cells “remember” morphogen signals afterthe morphogen dissipates.2.2 The Proposed ModelLewis, Slack, and Wolpert considered the activation of a gene G by a signalingsubstance S.• The amount of G’s product at time t is denoted by g(t).• The amount of S at time t is given by S(t).• Lewis, Slack, and Wolpert proposed that the rate of change of g de-pends linearly on the amount of S, there are feedbacks so that relativelysmall concentrations of g promote g’s growth, while large concentra-tions inhibit it.• They considered t he following equation for g:dgdt= k1S +k2g2k3+ g2− k4g, (5)where the ki’s are constants.2.3 The Analysis of the ModelLewis, Slack, and Wolpert considered the behavior of g for different valuesof S. The plot of the right-hand side of (5) has different numbers of stableequilibrium points.When S < Sc, there are two stable equilibria, one near g = 0 and onewith g much larger than zero. There is also one unstable equilibrium pointbetween the two stable ones. Thus, as S → Sc, the small g stable equilibriumpoint cancels against the unstable one so that when S > Sc, there is onlyone stable equilibrium point, and this one is where g is relatively large.32.4 An ExplanationThis model explains how two adjoining cells can have different values of geven though they are close together. All we need is that S is near Scatthese two cells but with S slightly larger than Scin one cell and slightlysmaller that Scin the other cell. the result is that the former cell has g nearzero while t he latter cell has relatively large g. Moreover, if S subsequentlydecreases to zero (because the signaling cells are no longer active), then thedrastic difference in g output by these two neighboring cells still remains.Readings and References• C. Taubes. Mod e l i ng Differential Equations in Bio l ogy. Prentice Hall,Upper Saddle River, NJ, 2001. Chapter 26.• “Thresholds in Development,” pp. 421–
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