Binding equilibria

We suppose two agents called A and B, each with one site called s. Consider the rule

A(s), B(s) <-> A(s!0), B(s!0) @ kon, koff

that, read from left-to-right, performs the binding of A to B and, read from right-to-left, undoes that binding. The rate constant for unbinding is k_off with units s^-1, i.e. number of unbinding events per second. The rate constant for binding is k_on with units molecule^-1 s^-1. We use 'molecules' rather than 'moles' because we are typically interested in reactions taking place inside cells - which are small and so do not contain large numbers of molecules. Many binding rate constants reported in the literature have units M^-1 s^-1, i.e. 'per molar per second', which is subtly different: molar means 'moles per liter' and so is a density, i.e. a relative rather than an absolute number of molecules. We translate from 'molar' to 'molecules per liter' by multiplying by Avagadro's number A ~ 6 x 10²³. We further multiply by the volume to obtain an absolute number of molecules. Typical values of binding rate constants are 10⁶ - 10⁹ M^-1 s^-1. A typical eukaryote cell's volume is V ~ 1.5 x 10^-12 litres. So AV ~10¹² meaning that typical binding rate constants for rules are between 10⁶ / AV and 10⁹ / AV, i.e. 10^-6 - 10^-3 molecule^-1 s^-1.

We use four typical values: 10^-6 for 'slow', 10^-5 for 'medium', 10^-4 for 'fast' and 10^-3 for 'very fast'. We have none of this complication for unbinding rules. Typical values are between 0.01 - 1.0 s^-1. Let us use 0.01 as 'slow', 0.1 as 'medium' and 1.0 as 'fast'. Note that 'fast' means 'weaker' for unbinding: slow binding and fast unbinding give the weakest bond strength; (very) fast binding and slow unbinding give the strongest bond strength.

Try running the system:

A(s), B(s) <-> A(s!0), B(s!0) @ 0.00001, 0.1

%init: 1000 * A(s)

%init: 1000 * B(s)

%obs: A(s!1), B(s!1)

We have used the 'medium' values for both binding and unbinding. What is the equilibrium value of the observable A(s!1), B(s!1) ? How long does the system take to reach that equilibrium? What happens if you use 'fast' values for both rate constants? What if you use 'fast' only for binding? Play around with the rate constants and also the initial numbers of As and Bs. What happens if we add a new rule

A(s), C(s) <-> A(s!0), C(s!0)

to the system? Play around with the rate constants and initial numbers of As, Bs and Cs...