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Chapter 4: Problem 16

The Michaelis-Menten kinetics reaction is given by $$ E+S \frac{k_{3}}{k_{1}}+E S \underset{k_{2}}{ } E+P $$ The resulting system of equations for the chemical concentrations is $$ \begin{aligned} \frac{d[S]}{d t} &=-k_{1}[E][S]+k_{3}[E S] \\ \frac{d[E]}{d t} &=-k_{1}[E][S]+\left(k_{2}+k_{2}\right)[E S] \\ \frac{d[E S]}{d t} &=k_{1}[E][S]-\left(k_{2}+k_{2}\right)[E S] \\ \frac{d[P]}{d t} &=k_{3}[E S] \end{aligned} $$ In chemical kinetics, one seeks to determine the rate of product formation \(\left(v=d[P] / d t=k_{3}[E S]\right)\). Assuming that \([E S]\) is a constant, find \(v\) as a function of \([S]\) and the total enzyme concentration \(\left[E_{T}\right]=[E]+[E S] .\) As a nonlinear dynamical system, what are the equilibrium points?

Short Answer

Expert verified
The rate of product formation is \(v = \frac{k_1k_3E_{T}[S]}{k_2 + k_3 + k_1[S]}\). The equilibrium points will be the solution set of the system of equations \(\frac{d[S]}{dt} =0, \frac{d[E]}{dt} =0, \frac{d[ES]}{dt} =0,\) and \(\frac{d[P]}{dt} =0\).

Step by step solution

01

Determine v as a function of [S] and [E_T]

Assuming that [ES] is constant, we can replace [E] in the expression for \(v = k_{3}[ES]\) with \(E = E_{T} - [ES]\). If we recall that \(\frac{d[ES]}{dt} = k_1[E][S] - \left(k_2 + k_3\right)[ES]\), and solve for [ES] to get \([ES] = \frac{k_1[E][S]}{k_2 + k_3}\), we can substitute \([E] = E_{T} - [ES]\) into this equation and rearrange to give \([ES] = \frac{k_1E_{T}[S]}{k_2 + k_3 + k_1[S]}\). Substituting this back into the expression for v then gives \(v = \frac{k_1k_3E_{T}[S]}{k_2 + k_3 + k_1[S]}\).
02

Finding the Equilibrium Points

At equilibrium, the rates of change of [S], [E], [ES] and [P] are all zero. Setting the given differential equations to zero will provide a system of equations that can be solved for equilibrium concentrations for [S], [E], [ES] and [P]. These concentrations form the equilibrium points of the system.

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Most popular questions from this chapter

The Lorenz Model is a simple model for atmospheric convection developed by Edward Lorenz in \(1963 .\) The system is given by three equations: $$ \begin{aligned} &\frac{d x}{d t}=\sigma(y-x) \\ &\frac{d y}{d t}=x(\rho-z)-y \\ &\frac{d z}{d t}=x y-\beta z \end{aligned} $$ a. Find the equilibrium points of the system. b. Find the Jacobian matrix for the system and evaluate it at the equilibrium points. c. Determine any bifurcation points and describe what happens near bifurcation point(s). Consider \(\sigma=10, \beta=8 / 3\), and vary \(\rho\) d. This system is known to exhibit chaotic behavior. Lorenz found a so-called strange attractor for parameter values \(\sigma=10, \beta=8 / 3\), and \(\rho=28\). Using a computer, locate this strange attractor.

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