Question

2. Consider the following interconnected biochemical reactions: \[ X \underset{k_{-1}}{\stackrel{k_{1}}{\rightleftarrows}} A, \quad B \stackrel{k_{2}}{\rightarrow} Y, \quad 2 X+Y \stackrel{k_{3}}{\rightarrow} 3 X \] Here, the $ k $ 's denote the positive reaction rate constants for the first, second and third reactions. The concentrations of $ X $ and $ Y $ at time $ t $ are denoted as $ x(t) $ and $ y(t) $, respectively. The concentrations of $ A $ and $ B $ are treated as positive constants and denoted as $ a $ and $ b $, respectively. a) Using the law of mass action, write down the ordinary differential equations that describe how the concentrations of $ X $ and $ Y $ evolve over time. [6] b) By making the change of variables \[ u=\sqrt{\frac{k_{3}}{k_{1}}} x, \quad v=\sqrt{\frac{k_{3}}{k_{1}}} y, \quad \tau=k_{1} t \] show that the mass equations of part a) become \[ \begin{array}{l} \frac{d u}{d \tau}=c-u+u^{2} v \\ \frac{d v}{d \tau}=d-u^{2} v \end{array} \] where $ c $ and $ d $ are positive constants that you should define. [6] c) Show that the system described by (1) has exactly one positive equilibrium and that is unstable if and only if $ 2 d>(c+d)\left\{1+(c+d)^{2}\right\} $. [14] d) Assume that the inequality in part c) holds. Let the region $ D $ be bounded by four lines given by $ u=c, v=0, v=\frac{d}{c^{2}} $ and $ v=\frac{d}{c^{2}}+c+d-u $. Using a graphical proof and a suitable theorem, prove that the region $ D $ contains a limit cycle. [12]

          2. Consider the following interconnected biochemical reactions:
\[
X \underset{k_{-1}}{\stackrel{k_{1}}{\rightleftarrows}} A, \quad B \stackrel{k_{2}}{\rightarrow} Y, \quad 2 X+Y \stackrel{k_{3}}{\rightarrow} 3 X
\]
Here, the \( k \) 's denote the positive reaction rate constants for the first, second and third reactions. The concentrations of \( X \) and \( Y \) at time \( t \) are denoted as \( x(t) \) and \( y(t) \), respectively. The concentrations of \( A \) and \( B \) are treated as positive constants and denoted as \( a \) and \( b \), respectively.
a) Using the law of mass action, write down the ordinary differential equations that describe how the concentrations of \( X \) and \( Y \) evolve over time.
[6]
b) By making the change of variables
\[
u=\sqrt{\frac{k_{3}}{k_{1}}} x, \quad v=\sqrt{\frac{k_{3}}{k_{1}}} y, \quad \tau=k_{1} t
\]
show that the mass equations of part a) become
\[
\begin{array}{l}
\frac{d u}{d \tau}=c-u+u^{2} v \\
\frac{d v}{d \tau}=d-u^{2} v
\end{array}
\]
where \( c \) and \( d \) are positive constants that you should define.
[6]
c) Show that the system described by (1) has exactly one positive equilibrium and that is unstable if and only if \( 2 d>(c+d)\left\{1+(c+d)^{2}\right\} \).
[14]
d) Assume that the inequality in part c) holds. Let the region \( D \) be bounded by four lines given by \( u=c, v=0, v=\frac{d}{c^{2}} \) and \( v=\frac{d}{c^{2}}+c+d-u \). Using a graphical proof and a suitable theorem, prove that the region \( D \) contains a limit cycle.
[12]

$2. Consider the following interconnected biochemical reactions: X k-1k1⇄ A, B k2→ Y, 2 X+Y k3→ 3 X Here, the k 's denote the positive reaction rate constants for the first, second and third reactions. The concentrations of X and Y at time t are denoted as x(t) and y(t), respectively. The concentrations of A and B are treated as positive constants and denoted as a and b, respectively. a) Using the law of mass action, write down the ordinary differential equations that describe how the concentrations of X and Y evolve over time. [6] b) By making the change of variables u=√((k3)/(k1)) x, v=√((k3)/(k1)) y, τ=k1 t show that the mass equations of part a) become (d u)/(d τ)=c-u+u^2 v (d v)/(d τ)=d-u^2 v where c and d are positive constants that you should define. [6] c) Show that the system described by (1) has exactly one positive equilibrium and that is unstable if and only if 2 d>(c+d){1+(c+d)^2}. [14] d) Assume that the inequality in part c) holds. Let the region D be bounded by four lines given by u=c, v=0, v=(d)/(c^2) and v=(d)/(c^2)+c+d-u. Using a graphical proof and a suitable theorem, prove that the region D contains a limit cycle. [12]$

Added by Anand K.

Question

Please give Ace some feedback