Agent growth

To model the growth of an ActiveAgent in iDynoMiCS we loop through each reaction, finding the cellular mass of the particle catalysing the reaction and the specific reaction rate for that cell. We then loop through all particles in the cell, calculating the new particle mass in one of two different ways depending on whether the reaction is autocatalytic or not.

Non-autocatalytic reaction dynamics

If a reaction is not autocatalytic then we assume the mass of the catalytic particle is constant throughout the timestep; this is a simplification as it may be changed by other reactions, but it makes calculation a lot easier.

The rate of change of the mass of some particle P due to a reaction with specific rate μ, catalytic particle mass C and yield Y_P, is

dP/dt = Y_P C μ.

This is easily solved to

P(t) = Y_P C μ t

and so from time t to time (t + Δt) we add Y_P C μ Δt to the particle mass.

Autocatalytic reaction dynamics

By autocatalytic, we mean that the particle catalysing the reaction is itself changed by the reaction.

The rate of change of the mass of this catalytic particle is then

dC/dt = Y_C C(t) μ.

This has solution

C(t + Δt) = C(t) e^{Y_C μ Δt}

and so from time t to time (t + Δt) we add C(t) (e^{Y_C μ Δt} - 1) to the catalytic particle mass.

Now, the rate of change of the mass of any other particle due to this reaction is

dP/dt = Y_P C(t) μ

as before, except that C is no longer constant. Using the formula for C(t) found above,

dP/dt = Y_P μ C(t) e^{Y_C μ t}.

Integrating, we find that

P(t + Δt) = [Y_P μ C(t)]/[Y_C μ] e^{Y_C μ Δt} + k,

where k is the constant of integration. By rearranging and setting Δt = 0 we find that

k = P(t) - C(t) Y_P/Y_C,

and so,

P(t + Δt) = P(t) + C(t) Y_P/Y_C (e^{Y_C μ Δt} - 1).

From time t to time (t + Δt) we add C(t) Y_P/Y_C (e^{Y_C μ Δt} - 1) to the particle mass.

Now, as for the catalytic particle Y_P = Y_C, and so Y_P/Y_C = 1, we don't need to worry about calculating these differently.