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Chapter 3: Problem 5

Zebra stripes and butterfly wing patterns are two of the most spectacular examples of biological pattern formation. Explaining the development of these patterns is one of the outstanding problems of biology; see Murray (1989) for an excellent review of our current knowledge. As one ingredient in a model of pattern formation, Lewis et al. (1977) considered a simple example of a biochemical switch, in which a gene \(G\) is activated by a biochemical signal substance \(S\). For example, the gene may normally be inactive but can be "switched on" to produce a pigment or other gene product when the concentration of \(S\) exceeds a certain threshold. Let \(g(t)\) denote the concentration of the gene product, and assume that the concentration \(s_{0}\) of \(S\) is fixed. The model is $$ \dot{g}=k_{1} s_{0}-k_{2} g+\frac{k_{3} g^{2}}{k_{4}{ }^{2}+g^{2}} $$ where the \(k\) 's are positive constants. The production of \(g\) is stimulated by \(s_{0}\) at a rate \(k_{1}\), and by an autocatalytic or positive feedback process (the nonlinear term). There is also a linear degradation of \(g\) at a rate \(k_{2}\). a) Show that the system can be put in the dimensionless form $$ \frac{d x}{d \tau}=s-r x+\frac{x^{2}}{1+x^{2}} $$ where \(r>0\) and \(s \geq 0\) are dimensionless groups. b) Show that if \(s=0\), there are two positive fixed points \(x\) * if \(r

Short Answer

Expert verified
In summary, we analyzed a biochemical switch model that describes the activation of a gene by a signal substance \(S\). We converted the model into a dimensionless form, found fixed points for \(s=0\), and discussed the gene system's behavior under changes in signal substance concentration. We then derived parametric equations for bifurcation curves and classified the bifurcations. Finally, we mentioned how to use a computer to plot the stability diagram in the \((r, s)\) space, shedding light on the gene system's behavior under various conditions.

Step by step solution

01

Convert to dimensionless form

We start by redefining time and concentration. Let \(\tau = k_2t\) and \(x(\tau) = \frac{g}{k_4}\). Then, the differential equation becomes: $$\frac{d x}{d \tau} = \frac{k_1 s_0}{k_2k_4}-\frac{k_2}{k_2} x+\frac{(xk_4)^{2}}{(k_4)^{2}+(xk_4)^{2}}$$ Now, define \(s = \frac{k_1 s_0}{k_2k_4}\) and \(r=\frac{1}{k_4}\). It simplifies to: $$\frac{d x}{d \tau}=s-r x+\frac{x^{2}}{1+x^{2}}$$
02

Find fixed points for \(s=0\)

Let's find the fixed points of the system when \(s=0\). Set \(\frac{d x}{d \tau} = 0\), giving us: $$0=-r x+\frac{x^{2}}{1+x^{2}}$$ Rearrange the equation to: $$x(r+x) = x^2$$ The fixed points are given by \(x=0\) and \(x=r\). We know that \(r>0\), so we have two positive fixed points if \(r
03

Analyze the behavior

If \(s\) is slowly increased from zero, it indirectly affects the fixed points. The gene system \(g(t)\) will respond accordingly, which we need to determine by analyzing the fixed points. If \(s\) then goes back to zero, we need to know whether the gene system turns off again. This is a matter of bifurcation in the system and we need to analyze the stability of the fixed points under these changes.
04

Find parametric equations for bifurcation curves

To find the parametric equations for the bifurcation curves and classify the bifurcations that occur, we need to analyze the stability of the fixed points as functions of the parameters \(r\) and \(s\). We will use the parametric representation and analyze the eigenvalues of the Jacobian matrix, which determine the stability of fixed points. For the dimensionless form of the equation, let's differentiate it with respect to \(x\): $$\frac{d^2 x}{d \tau^2}=-r+\frac{2x}{(1+x^2)^2}-\frac{2x^3}{(1+x^2)^2}$$ The eigenvalues are given by the sign of the second derivative at the fixed points: $$ \begin{cases} \text{At } x=0: &\lambda_1 = -r \\ \text{At } x=r: &\lambda_2 = -r+\frac{2r}{(1+r^2)^2} - \frac{2r^3}{(1+r^2)^2} \end{cases} $$ Now, we can derive the parametric equations for the bifurcation curves by setting \(\lambda_1=0\) and \(\lambda_2=0\), and analyzing the stability of the fixed points.
05

Plot the stability diagram

To give a quantitatively accurate plot of the stability diagram in \((r, s)\) space, we need to use a computer. Implementing the calculations in a mathematical or numerical software package like Mathematica, MATLAB, or Python (with NumPy and Matplotlib libraries) will give you the results. Plots of the stability diagram will show the regions of stability and instability, which can help explain when and how the gene system switches on and off as \(s\) varies.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Biochemical Switch
In biological systems, a biochemical switch is a crucial regulatory mechanism that can change the state of a cell or organism in response to specific signals. Imagine flipping a light switch; similarly, certain genes can be 'turned on' or 'off' in response to specific biochemical signals.

In our example, the gene product is activated when the concentration of a signal substance, denoted as S, exceeds a certain threshold. This model incorporates production, degradation, and autocatalytic processes to govern the concentration of the gene product. The autocatalytic term, where the gene product promotes its own production, is an example of a positive feedback loop 鈥� a common feature in biological switches.

Understanding these switches helps explain how complex patterns, like zebra stripes or butterfly wings, develop as the activation of different genes leads to the expression of specific characteristics in precise locations. It is the interplay of these switches that allows for the intricate designs and functions we find in nature.
Fixed Points Analysis
In mathematical models, such as our gene activation model, fixed points analysis is critical for understanding the system's behavior over time. A fixed point is where the system remains unchanged if it starts in that state 鈥� analogous to a ball resting at the bottom of a bowl, it has no reason to move.

To find the fixed points, we set the rate of change to zero, which, in our equation, means setting \(\frac{dx}{d\tau} = 0\). For the given gene model, we determine the fixed points as \(x=0\) and \(x=r\) under a certain condition on \(r\). The presence of multiple fixed points could mean that the system has different 'resting states' depending on the initial conditions or external signals. Analyzing these points allows us to predict how the gene product concentration will behave over time and under different circumstances.
Bifurcation Curves
A bifurcation curve represents the critical points at which a small change in the parameters of a system can lead to a sudden qualitative change in its behavior 鈥� it's like plotting a road map for the system states. As parameters cross certain thresholds, the system may flip from one state to another, which is particularly important for a biochemical switch.

In our exercise, we focus on finding these curves for different values of \(r\) and \(s\), using parametric equations. By analyzing these bifurcation curves, we can predict under what conditions the gene is likely to be turned on and when it might switch off. This is akin to charting the conditions necessary for a light to go from off to on and back again.
Stability Diagram
The stability diagram complements bifurcation analysis by illustrating stable and unstable regions for the system states. Think of it like a weather map, showing areas of calm versus turbulent weather patterns. By inputting the parameters \(r\) and \(s\) into a phase plot, we create a visual tool that helps predict the system's behavior.

For the gene activation model, plotting the stability diagram helps us understand the conditions that lead to the gene being 'on' or 'off' and provides insights into the dynamics that control pattern formation in biological systems. With a stability diagram, we can visually grasp how robust the system is against perturbations and how it might transition between states as conditions change.

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Most popular questions from this chapter

The next exercises are designed to test your ability to distinguish among the various types of bifurcations - it's easy to confuse them! In each case, find the values of \(r\) at which bifurcations occur, and classify those as saddle- node, transcritical, supercritical pitchfork, or subcritical pitchfork. Finally, sketch the bifureation diagram of fixed points \(x^{*}\) vs. \(r\). $$ \dot{x}=r x+\frac{x^{3}}{1+x^{2}} $$

In parts (a)-(c), let \(V(x)\) be the potential, in the sense that \(\dot{x}=-d V / d x .\) Sketch the potential as a function of \(r .\) Be sure to show all the qualitatively different cases, including bifurcation values of \(r\). a) (Saddle-node) \(\dot{x}=r-x^{2}\) b) (Transcritical) \(\hat{x}=r x-x^{2}\) c) (Subcritical pitchfork) \(\dot{x}=r x+x^{3}-x^{5}\)

The Maxwell-Bloch equations provide an even more sophisticated model for a laser. These equations describe the dynamics of the electric field \(E\), the mean polarization \(P\) of the atoms, and the population inversion \(D\) : $$ \begin{aligned} &\dot{E}=\kappa(P-E) \\ &\dot{P}=\gamma_{1}(E D-P) \\ &\dot{D}=\gamma_{2}(\lambda+1-D-\lambda E P) \end{aligned} $$ where \(\kappa\) is the decay rate in the laser cavity due to beam transmission, \(\gamma_{1}\) and \(\gamma_{2}\) are decay rates of the atomic polarization and population inversion, respectively, and \(\lambda\) is a pumping energy parameter. The parameter \(\lambda\) may be positive, negative, or zero; all the other parameters are positive. These equations are similar to the Lorenz equations and can exhibit chaotic behavior (Haken 1983, Weiss and Vilaseca 1991). However, many practical lasers do not operate in the chaotic regime. In the simplest case \(\gamma_{1}, \gamma_{2} \gg \kappa ;\) then \(P\) and \(D\) relax rapidly to steady values, and hence may be adiabatically eliminated, as follows. a) Assuming \(\dot{P}=0, \dot{D} \approx 0\), express \(P\) and \(D\) in terms of \(E\), and thereby derive a first-order equation for the evolution of \(E .\) b) Find all the fixed points of the equation for \(E\). c) Draw the bifurcation diagram of \(E^{*}\) vs. \(\lambda .\) (Be sure to distinguish between stable and unstable branches.)

The next exercises are designed to test your ability to distinguish among the various types of bifurcations - it's easy to confuse them! In each case, find the values of \(r\) at which bifurcations occur, and classify those as saddle- node, transcritical, supercritical pitchfork, or subcritical pitchfork. Finally, sketch the bifureation diagram of fixed points \(x^{*}\) vs. \(r\). $$ \dot{x}=x+\tanh (r x) $$

In pioneering work in epidemiology, Kermack and McKendrick ( 1927 ) proposed the following simple model for the evolution of an epidemic. Suppose that the population can be divided into three classes: \(x(t)=\) number of healthy people; \(y(t)=\) number of sick people; \(z(t)=\) number of dead people. Assume that the total population remains constant in size, except for deaths due to the epidemic. (That is, the epidemic evolves so rapidly that we can ignore the slower changes in the populations due to births, emigration, or deaths by other causes.) Then the model is $$ \begin{aligned} &\dot{x}=-k x y \\ &\dot{y}=k x y-\ell y \\ &\dot{z}=\ell y \end{aligned} $$ where \(k\) and \(\ell\) are positive constants. The equations are based on two assumptions: (i) Healthy people get sick at a rate proportional to the product of \(x\) and \(y\). This would be true if healthy and sick people encounter each other at a rate proportional to their numbers, and if there were a constant probability that each such encounter would lead to transmission of the disease. (ii) Sick people die at a constant rate \(\ell\). The goal of this exercise is to reduce the model, which is a third-order system, to a first-order system that can analyzed by our methods. (In Chapter 6 we will see a simpler analysis.) a) Show that \(x+y+z=N\), where \(N\) is constant. b) Use the \(\dot{x}\) and \(\dot{z}\) equation to show that \(x(t)=x_{0} \exp (-k z(t) / \ell)\), where \(x_{0}=x(0)\) c) Show that \(z\) satisfies the first-order equation \(\dot{z}=\ell\left[N-z- x_{0} \exp (-k z / \ell)\right]\). d) Show that this equation can be nondimensionalized to $$ \frac{d u}{d \tau}=a-b u-e^{-4} $$ by an appropriate rescaling. e) Show that \(a \geq 1\) and \(b>0\). f) Determine the number of fixed points \(u^{*}\) and classify their stability. g) Show that the maximum of \(\dot{u}(t)\) occurs at the same time as the maximum of both \(\tilde{z}(t)\) and \(y(t)\), (This time is called the peak of the epidemic, denoted \(t_{\text {peak }}\). At this time, there are more sick people and a higher daily death rate than at any other time.) h) Show that if \(b<1\), then \(\dot{u}(t)\) is increasing at \(t=0\) and reaches its maximum at some time \(t_{\text {peak }}>0\). Thus things get worse before they get better. (The term epidemic is reserved for this case.) Show that \(\dot{u}(t)\) eventually decreases to 0 . i) On the other hand, show that \(t_{\text {peak }}=0\) if \(b>1\). (Hence no epidemic occurs if \(b>1 .\) j) The condition \(b=1\) is the threshold condition for an epidemic to occur. Can you give a biological interpretation of this condition? k) Kermack and McKendrick showed that their model gave a good fit to data from the Bombay plague of 1906 . How would you improve the model to make it more appropriate for AIDS? Which assumptions need revising? For an introduction to models of epidemics, see Murray (1989), Chapter 19, or Edelstein-Keshet (1988). Models of AIDS are discussed by Murray (1989) and May and Anderson (1987). An excellent review and commentary on the Kermack- McKendrick papers is given by Anderson (1991).
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