chemical reaction:
E + S <=> E~S <=> E-S
<=> indicates a bidirectional reaction
with different forward and backward rate constants. we can call these k1, k2, k3 and k4 in the above equation.
Here, the enzyme binds to the substrate; the initial complex E~S then undergoes a conformational change to E–S.
How do we write differential equations for the changes of concentrations of E~S and E-S with time? and solve them?
To write the differential equations for the changes in concentrations of E~S and E-S with time, we need to consider the rates of the forward and backward reactions. Let's denote the concentrations of E~S and E-S as [ES] and [ES'], respectively, at any given time (t).
To write the differential equations, we apply the law of mass action. The rate of change of [ES] with time is determined by the forward and backward reactions:
d[ES]/dt = k1[E][S] - k2[ES] - k3[ES] + k4[E-S]
Here, the first term on the right side represents the formation of E~S through the forward reaction with rate constant k1, where [E] and [S] are the concentrations of enzyme and substrate, respectively. The second term represents the dissociation of E~S to E and S with rate constant k2. The third term accounts for the forward isomerization of E~S to E-S with rate constant k3. Lastly, the fourth term represents the backward isomerization of E-S to E~S with rate constant k4.
Similarly, the rate of change of [ES'] with time can be written as:
d[ES']/dt = k3[ES] - k4[E-S]
Here, the first term on the right side represents the formation of E-S through the forward isomerization of E~S with rate constant k3, and the second term represents the backward isomerization of E-S to E~S with rate constant k4.
To solve these differential equations, you will need to specify initial conditions, such as the initial concentrations of enzyme ([E]), substrate ([S]), E~S ([ES]), and E-S ([ES']). Additionally, you'll need values for the rate constants (k1, k2, k3, and k4).
There are numerical methods available to solve these differential equations, such as the Euler method or the Runge-Kutta method. These methods calculate the concentrations of E~S and E-S at different time points based on specified time intervals and the differential equations. The resulting concentration-time profiles can provide insights into the dynamics of the chemical reaction.