Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 6B / Solution

Applying the law of mass action to the sequential two-site enzyme reaction network givesAdding the appropriate equations gives the conservation laws $e+c_1+c_2=e_0$ and $s+c_1+2c_2+p=s_0$. In particular $e=e_0(1-v_1-v_2)$ after nondimensionalization.

Define the dimensionless reaction rates $a_1=k_{-1}/(k_1{s}_0)$, $a_3=k_{-3}/(k_1{s}_0)$, $b_2=k_2/(k_1{s}_0)$, $b_4=k_4/(k_1{s}_0)$ and $r_3=k_3/k_1$. Substitution, with primes denoting $d/d\tau$, yieldsThese right sides are $f,g_1,g_2$, respectively, with $u(0)=1$ and $v_1(0)=v_2(0)=0$. When $ϵ\ll 1$, the form of these ordinary differential equations displays the faster relaxation of the complexes, which motivates a subsequent quasi-steady-state approximation.