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Chapter 7: Problem 17

The following chemical reaction mechanism was studied by Lotka in 1920 and later in 1956: $$ \begin{gathered} \mathrm{A}+\mathrm{X} \stackrel{k_{1}}{\longrightarrow} 2 \mathrm{X} \\ \mathrm{X}+\mathrm{Y} \stackrel{k_{2}}{\longrightarrow} 2 \mathrm{Y} \\ \mathrm{Y} \stackrel{k_{3}}{\longrightarrow} \mathrm{B} \end{gathered} $$ Assume that \(\mathbf{A}\) and \(\mathbf{B}\) are kept at a constant concentration. (a) Write a set of equations for the concentrations of \(\mathrm{X}\) and \(\mathrm{Y}\) using the law of mass action. Suggest a dimensionless form of the equations. (b) Show that there are two steady states, and use the methods of Chapter 5 to demonstrate that Lotka's system has oscillatory solutions. Compare with the Lotka-Volterra predator-prey system.

Short Answer

Expert verified
The system has two steady states: (0,0) and \(\left(\frac{1}{a}, \frac{a-c}{b}\right)\), with oscillatory solutions demonstrated via the existence of a Hopf bifurcation.

Step by step solution

01

Write the law of mass action equations

Using the law of mass action, we can write the rate of change of concentrations for species \(\text{X}\) and \(\text{Y}\): \(\frac{d[\text{X}]}{dt} = k_1[A][X] - k_2[X][Y]\) and \(\frac{d[\text{Y}]}{dt} = k_2[X][Y] - k_3[Y]\).
02

Introduce dimensionless variables

Introduce dimensionless variables \(x = \frac{[X]}{[A]}\) and \(y = \frac{[Y]}{[A]}\), and let \( \tau = k_1[A]\). Rewrite the previous equations in terms of these variables to obtain \( \frac{dx}{d\tau} = x - ax - bxy \) and \( \frac{dy}{d\tau} = bxy - cy \), where \(a = \frac{k_2[A]}{k_1[A]}\), \(b = \frac{k_2}{k_1}\) and \(c = \frac{k_3}{k_1[A]}\).
03

Find the steady states

To find the steady states, set the time derivatives to zero: \( \frac{dx}{d\tau} = 0\) and \( \frac{dy}{d\tau} = 0\). This results in two equations \(0 = x - ax - bxy \) and \( 0 = bxy - cy \). Solving these, we get \(y=0\) or \(y=\frac{a-c}{b}\). For \(y=0\), \(x=0\) or \(x=\frac{1}{a}\). Thus, the two steady states are \((0, 0) \) and \(\frac{1}{a}, \frac{a-c}{b} \).
04

Determine the stability of the steady states

To determine stability, linearize the system around the steady states by finding the Jacobian matrix. Evaluate the Jacobian at \( (0,0) \) and \(\frac{1}{a}, \frac{a-c}{b} \). The eigenvalues of the Jacobian will determine stability: If all eigenvalues have negative real parts, then the steady state is stable; otherwise, it is unstable.
05

Show the existence of oscillatory solutions

Analyzing the Jacobian at the non-trivial steady state \(\frac{1}{a}, \frac{a-c}{b} \), find conditions under which the real part of the complex eigenvalues is zero, indicative of a Hopf bifurcation. This implies the existence of a periodic orbit or oscillations. Comparing the dynamics with the Lotka-Volterra model reveals similarities in predator-prey interactions leading to oscillations.
06

Compare with the Lotka-Volterra predator-prey system

The Lotka system is similar to the Lotka-Volterra predator-prey model, which also exhibits oscillatory behavior due to the inherent interactions between predator and prey populations. In both models, the interactions between species create feedback loops, leading to sustained cycles in population dynamics.

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Key Concepts

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

Law of Mass Action
The Law of Mass Action is fundamental in chemical kinetics. It states that the rate of a reaction is proportional to the product of the concentrations of the reactants. For Lotka's reactions, we apply this law to express the system's dynamics. For instance, consider the reaction \(\text{A} + \text{X} \rightarrow 2 \text{X}\). The rate of this reaction is proportional to the concentration of \(\text{A}\) and \(\text{X}\). Thus, the rate can be written as \(k_1[\text{A}][\text{X}]\). Similarly, for the reaction \(\text{X} + \text{Y} \rightarrow 2 \text{Y}\), the rate is \(k_2[\text{X}][\text{Y}]\). Utilizing these rates, we set up the differential equations to describe changes in concentrations of \(\text{X}\) and \(\text{Y}\) over time.
Steady States
Steady states occur when the concentrations of reactants and products in a system remain constant over time. To find these states in Lotka's model, we set the time derivatives \(\frac{dx}{d\tau}\) and \(\frac{dy}{d\tau}\) to zero. This allows us to solve for \(x\) and \(y\) when the system is at equilibrium. For example, setting \(\frac{dx}{d\tau} = x - ax - bxy = 0\) gives potential solutions for \(x\). Similarly, \(\frac{dy}{d\tau} = bxy - cy = 0\) must be satisfied. Solving these simultaneously shows that the steady states in the system are \((0,0)\) and \(\frac{1}{a}, \frac{a-c}{b}\). These points represent equilibrium points where concentrations do not change over time.
Oscillatory Solutions
Oscillatory solutions are periodic changes in concentrations, indicating that the system does not settle into a steady state but instead continually cycles through different states. In Lotka's model, these can be found by analyzing the Jacobian matrix of the system at the identified steady states. If the eigenvalues of the Jacobian have non-zero imaginary parts, the steady state becomes unstable and the system exhibits oscillations. This type of behavior is common in many biological and chemical systems, reflecting natural cycles like predator-prey interactions or cell cycles.
Predator-Prey Dynamics
Lotka's chemical reaction model shares similarities with the Lotka-Volterra predator-prey model in ecology. In predator-prey systems, the population of predators and prey influences each other in a cyclic manner. For example, an increase in prey population (\text{X}) leads to more food for predators (\text{Y}), which then increase until they overconsume prey, causing their own population to decrease. In Lotka's system, the chemicals \(\text{X}\) and \(\text{Y}\) play roles analogous to prey and predators. The reaction rates regulate the population changes, creating feedback loops that lead to sustained oscillations.
Dimensionless Variables
Converting to dimensionless variables simplifies the analysis of the system by reducing the number of parameters and making the equations more manageable. In Lotka's model, we introduce dimensionless variables \(\frac{[X]}{[A]}\) and \(\frac{[Y]}{[A]}\), allowing us to describe the system's behavior without explicitly tracking individual concentrations. By scaling time with \(k_1[A]\), we redefine the system's dynamics in a more generalized form: \(\frac{dx}{d\tau} = x - ax - bxy\) and \(\frac{dy}{d\tau} = bxy - cy\). Here, \(a\), \(b\), and \(c\) are new dimensionless parameters capturing the intrinsic properties of the reactions, facilitating mathematical and computational analysis.

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Most popular questions from this chapter

The glycolytic oscillator. A biochemical reaction that is ubiquitous in metabolic systems contains the following sequence of steps: $$ \text { glucose } \rightarrow \mathrm{GGP} \rightarrow \mathrm{FGP} \rightarrow \frac{\mathrm{PFK}}{\text { ATP ADP }} \text { FDP } \rightarrow \text { products, } $$ where GGP \(=\) glucose-6-phosphate, FGP \(=\) fructose-6-phosphate, FDP \(=\) fructose-1, 6 -diphosphate, ATP \(=\) adenosine triphosphate, ADP \(=\) adenosine diphosphate, PFK \(=\) phosphofructokinase. An assumption generally made is that the enzyme phosphofructokinase has two states, one of which has a higher activity. ADP stimulates this allosteric regulatory enzyme and produces the more active form. Thus a product of the reaction step mediated by PFK enhances the rate of reaction. A schematic version of the kinetics is Equations for this system, where \(x\) stands for \(F G P\) and \(y\) for \(A D P\), are as follows: $$ \begin{aligned} &\frac{d x}{d t}=\delta-k x-x y^{2} \\ &\frac{d y}{d t}=k x+x y^{2}-y \end{aligned} $$ These equations, derived in many sources (see references), are known to have stable oscillations as well as other interesting properties. (a) Show that the steady state of these equations is \((\bar{x}, \bar{y})=\left(\frac{\delta}{\left(k+\delta^{2}\right)}, \delta\right)\) (b) Find the Jacobian of the glycolytic oscillator equations at the above steady state. (c) Find conditions on the parameters so that the system is a positive- feedback system as described in Section 7.8.

The following model was proposed by Othmer and Aldridge (1978): A cell can produce two chemical species \(x\) and \(y\) from a substrate according to the reaction substrate \(\rightarrow x \rightarrow y \rightarrow\) products. Species \(x\) can diffuse across the cell membrane at a rate that depends linearly on its concentration gradient. The ratio of the volume of cells to the volume of extemal medium is given by a parameter \(\epsilon ; x\) and \(y\) are intracellular concentrations of \(X\) and \(Y\) and \(x^{0}\) is the extracellular concentration of \(X .\) The equations they studied were $$ \begin{aligned} &\frac{d x}{d t}=\delta-F(x, y)+P\left(x^{0}-x\right) m \\ &\frac{d y}{d t}=\alpha[F(x, y)-G(y)] \\ &\frac{d x^{0}}{d t}=\epsilon P\left(x-x^{0}\right) \end{aligned} $$ (a) Explain the equations. Determine the values of \(\bar{x}, F(\bar{x}, \bar{y})\) and \(G(\bar{y})\) at the steady state \(\left(\bar{x}, \bar{y}, \bar{x}^{0}\right)\). (b) The matrix of linearization of these equations about this steady state is $$ \mathbf{J}=\left(\begin{array}{ccc} k_{11}-P & k_{12} & P \\ k_{21} & k_{22} & 0 \\ \epsilon P & 0 & -\epsilon P \end{array}\right) $$ What are the constants \(k_{j j} ?\) (c) For the characteristic equation $$ \lambda^{3}+a_{1} \lambda^{2}+a_{2} \lambda+a_{3}=0 $$ find \(a_{1}, a_{2}\), and \(a_{3}\) in terms of \(k_{i j}\) and in terms of partial derivatives of \(F\) and \(G\).

Consider the following hypothetical chemical system: $$ \begin{array}{ll} M_{0} \text { (a catalyst) }+\mathrm{X} \underset{k_{1}}{\stackrel{k_{1}}{\mathrm{M}}} \mathrm{M}_{\mathrm{t}} & \text { (active complex), } \\ \mathrm{M}_{1}+\mathrm{X} \stackrel{k_{0}}{k_{-2}} \mathrm{M}_{2} & \text { (inactive complex), } \\ \mathrm{M}_{1}+\mathrm{Y} \stackrel{k_{2}}{\longrightarrow} \mathrm{P}+\mathrm{Q}+\mathrm{M}_{0} & \text { (products plus catalyst). } \end{array} $$ This system is called a substrate-inhibited reaction since the chemical \(\mathrm{X}\) can deactivate the complex \(M_{1}\) which is required in forming the products. (a) Write equations for the chemical components. (b) Assume \(M_{0}+M_{1}+M_{2}=C\) (where \(m_{0}, m_{1}\), and \(m_{2}\) are concentrations of \(M_{0}, M_{1}\), and \(M_{2}\), and \(C\) is a constant). Make a quasi-steady-state assumption for \(M_{0}, M_{1}\), and \(M_{2}\) and show that $$ m_{0}=m_{1} \frac{k_{-1}+k_{3} y}{k_{1} x}, \quad m_{2}=k_{2} x \frac{m_{1}}{k-2} . $$ (c) Use these results to show that $$ m_{1}=\frac{C k_{1} x}{k_{-1}+k_{3} y+k_{1} x\left[1+\left(k_{2} / k-2\right) x\right]} $$ (d) Now show that \(x\) will satisfy an equation whose dimensionless form is $$ \frac{d x}{d t}=\gamma-x-\frac{\beta x y}{1+x+y+(\alpha / \delta) x^{2}} $$ Identify the various combinations of parameters. When can the term \(y\) in the denominator be neglected?

An activator-inhibitor system (Gierer-Meinhardt). A system also studied in connection with spatial pattems (see Chapter 10) consists of two substances. The activator enhances its own synthesis as well as that of the inhibitor. The inhibitor causes the formation of both substances to decline. Several versions have been studied, among them is the following system: $$ \begin{aligned} &\frac{d x}{d t}=\rho+\frac{x^{2}}{y}-x \\ &\frac{d y}{d t}=x^{2}-\gamma y \end{aligned} $$ (a) Find the steady state for this system. (b) Show that if the input of activator is sufficiently small compared to thedecay rate of inhibitor, the system is an activator-inhibitor system as described in Section \(7.8\). (c) In a modified version of these equations the term \(x^{2} / y\) is replaced by \(x^{2} / y\left(1+k x^{2}\right)\). Suggest what sort of chemical interactions may be occurring in this and in the original system. (d) Why is it not possible to solve for the steady state of the modified GiererMeinhardt equations? (e) Find the Jacobian of the modified system. When does the Jacobian have the sign pattern of an activator-inhibitor system?

The Brusselator. This hypothetical system was first proposed by a group working in Brussels [see Prigogine and Lefever \((1968)]\) in connection with spatially nonuniform chemical patterns. Because certain steps involve trimolecular reactions, it is not a model of any real chemical system but rather a prototype that has been studied extensively. The reaction steps are $$ \begin{aligned} & A & \rightarrow X \\ B+& X & \rightarrow Y+D \\ 2 X+Y & \rightarrow 3 X \\ X & \rightarrow E \end{aligned} $$ It is assumed that concentrations of \(A, B, D\), and \(E\) are kept artificially constant so that only \(\mathrm{X}\) and \(\mathrm{Y}\) vary with time. (a) Show that if all rate constants are chosen appropriately, the equations describing a Brusselator are: $$ \begin{aligned} &\frac{d x}{d t}=A-(B+1) x+x^{2} y \\ &\frac{d y}{d t}=B x-x^{2} y \end{aligned} $$ (b) Find the steady state. (c) Calculate the Jacobian and show that if \(\mathrm{B}>1\), the Brusselator is a positive-feedback system as described in Section \(7.8\).
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