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 \(G\).

Step by step solution

01

Understand the Differential Equations

The system of differential equations describes the dynamics of concentrations of chemical species inside and outside a cell: \( \frac{d x}{d t}=\delta -F(x, y)+P(x^0-x) \) represents the rate of change of the intracellular concentration of species \(x\). The term \(\delta\) is a constant source term, \(F(x,y)\) is the rate of conversion of \(x\) to \(y\), and \(P(x^0-x)\) describes diffusion across the cell membrane, with \(P\) being the proportionality constant.\(\frac{d y}{d t}=\alpha[F(x, y)-G(y)] \) represents the rate of change of the intracellular concentration of species \(y\). \(\alpha\) is a constant, and \(G(y)\) is the rate of conversion of \(y\) to products.\(\frac{d x^0}{d t}=\epsilon P(x-x^0) \) represents the rate of change of the extracellular concentration of species \(x\). \(\epsilon\) is a parameter representing the ratio of the volume of cells to the volume of external medium.

02

Determine Steady-State Values

At steady state, the derivatives with respect to time are zero. Thus, set each differential equation to zero and solve for \( \bar{x} \), \( \bar{y} \), and \( \bar{x}^0 \):\[ \delta - F(\bar{x}, \bar{y}) + P(\bar{x}^0 - \bar{x}) = 0 \]\[ \alpha [ F(\bar{x}, \bar{y}) - G(\bar{y}) ] = 0 \]\[ \epsilon P(\bar{x} - \bar{x}^0) = 0 \]From the third equation, \( \bar{x} = \bar{x}^0 \). Substitute this into the first equation to get: \[ \delta = F(\bar{x}, \bar{y}) \]

03

Apply Steady-State Conditions to Second Equation

From \( \alpha[ F(\bar{x}, \bar{y}) - G(\bar{y}) ] = 0 \), since \( \alpha \) is non-zero, we get: \[ F(\bar{x}, \bar{y}) = G(\bar{y}) \] This means that at steady state, the rate at which \(x\) is converted to \(y\) is equal to the rate at which \(y\) is converted to products.

04

Understand the Linearization Matrix J

The matrix of linearization \( \mathbf{J} \) is given as:\( \mathbf{J}=\left( \begin{array}{ccc} k_{11}-P & k_{12} & P \ k_{21} & k_{22} & 0 \ \epsilon P & 0 & -\epsilon P \end{array} \right) \)Each element \( k_{ij} \) represents the partial derivatives evaluated at steady state.

05

Identify the Constants \(k_{jj}\)

The constants \( k_{jj} \) are defined as:\[ k_{11} = \left. \frac{\partial (\delta - F(x, y) + P(x^0 - x))}{\partial x} \right|_{(\bar{x}, \bar{y}, \bar{x}^0)} \]\[ k_{12} = \left. \frac{\partial (\delta - F(x, y) + P(x^0 - x))}{\partial y} \right|_{(\bar{x}, \bar{y}, \bar{x}^0)} \]\[ k_{21} = \left. \frac{\partial \left( \alpha [F(x, y) - G(y)] \right)}{\partial x} \right|_{(\bar{x}, \bar{y})} \]\[ k_{22} = \left. \frac{\partial \left( \alpha [F(x, y) - G(y)] \right)}{\partial y} \right|_{(\bar{x}, \bar{y})} \]Using the steady-state conditions, these will become:\[ k_{11} = -P - \frac{\partial F}{\partial x} \quad , \quad k_{12} = -\frac{\partial F}{\partial y} \]\[ k_{21} = \alpha \frac{\partial F}{\partial x} \quad , \quad k_{22} = \alpha (\frac{\partial F}{\partial y} - \frac{\partial G}{\partial y}) \]

06

Derive Characteristic Equation

To find the characteristic equation, consider the determinant of \( \mathbf{J} - \lambda \mathbf{I} \):\( \lambda^3 + a_1 \lambda^2 + a_2 \lambda + a_3 = 0 \)The coefficients \(a_1, a_2,\) and \(a_3\) are given by:\[ a_1 = -(k_{11} - P + k_{22} - \epsilon P) \]\[ a_2 = (k_{11} - P) k_{22} + \epsilon P (k_{11} - P) - P k_{21} \]\[ a_3 = -\epsilon P k_{12} k_{21} \]

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Key Concepts

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

Differential Equations

Differential equations are mathematical equations that describe how a quantity changes over time. In the context of biological systems modeling, they help us understand the dynamics of biological processes. For the given model, the differential equations represent the concentration changes of species inside and outside a cell.
The first equation \[ \frac{dx}{dt} = \delta - F(x, y) + P(x^0 - x) \] describes the rate of change of the intracellular concentration of species \(x\).
The second equation \[ \frac{dy}{dt} = \alpha [F(x, y) - G(y)] \] describes the rate of change of the intracellular concentration of species \(y\).
The third equation \[ \frac{dx^0}{dt} = \epsilon P(x - x^0) \] represents the rate of change of the extracellular concentration of species \(x\).
These equations incorporate various factors such as conversion rates and diffusion across the cell membrane.

Steady State

A steady state in a biological system is a condition where the concentrations of the reacting species no longer change over time, meaning that the rates of change become zero. When we set the derivatives in our differential equations to zero, we enter a steady state scenario.
For our system, setting \[ \frac{dx}{dt} = 0 \] \[ \frac{dy}{dt} = 0 \] \[ \frac{dx^0}{dt} = 0 \] allows us to solve for the steady-state concentrations. From the given equations, we derive that \(\bar{x} = \bar{x^0}\) and \[ F(\bar{x}, \bar{y}) = \delta \] and \[ F(\bar{x}, \bar{y}) = G(\bar{y}) \].
This means, at steady state, the production and consumption rates balance each other out.

Linearization

Linearization is a method used to analyze systems near their steady states. It involves approximating nonlinear functions by linear ones based on their partial derivatives.
For the provided system, the linearization matrix \(\mathbf{J}\) is given as: \[ \mathbf{J}=\left(\begin{array}{ccc} k_{11}-P & k_{12} & P \ k_{21} & k_{22} & 0 \ \epsilon P & 0 & -\epsilon P \end{array}\right) \].
The elements of this matrix \( k_{ij} \) represent the partial derivatives of the functions with respect to their variables, evaluated at the steady state.

Characteristic Equation

The characteristic equation is derived from the linearization matrix and is used to determine the stability of the system. It is found by calculating the determinant of \(\mathbf{J} - \lambda \mathbf{I}\), where \(\mathbf{I}\) is the identity matrix and \(\lambda\) are the eigenvalues.
The characteristic equation for our system is: \[ \lambda^{3}+a_{1} \lambda^{2}+a_{2} \lambda+a_{3}=0 \]
Where: \[ a_{1} = -(k_{11} - P + k_{22} - \epsilon P) \]
\[ a_{2} = (k_{11} - P) k_{22} + \epsilon P (k_{11} - P) - P k_{21} \]
\[ a_{3} = -\epsilon P k_{12} k_{21} \].
These coefficients must be determined to understand the system’s behavior near the steady state.

Reaction Kinetics

Reaction kinetics involves studying the rates of chemical reactions and how they are affected by various factors. In our model, the terms \(F(x, y)\) and \(G(y)\) represent the rates at which species \(x\) and \(y\) are converted.
Kinetics can be affected by:

• Concentration of reactants.
• Temperature.
• Catalysts or inhibitors.

The specific forms of \(F(x, y)\) and \(G(y)\) are essential to describe how \(x\) converts to \(y\) and how \(y\) proceeds to the final product. Understanding these rates is crucial for accurately modeling biological systems.