Kinetics of aggregation with a finite number of particles and application to viral capsid assembly (Q888299)

A time-dependent process of aggregation is considered, which is generated by the arrival of aggregates of various sizes at a nucleation site, that arises in biology. It is assumed that a cluster is formed when it arrives at size \(N_0\) (= contains \(N_0\) particles) and if a cluster contains \(n\) particles, then it can accept aggregates of sizes \(\leq N_0-n\). In the deterministic model, the time dynamics of the size \(n(t)\) of a cluster at time \(t\geq 0\) are depicted by a first order differential equation \[ \dot{n}(t)=(1-n(t))\biggl(\sum_{k=1}^{N_0-n(t)} \lambda kn_k\biggr),\quad t\geq 0, \] where \(n_k\) is the number of aggregates of size \(k\) and \(\lambda>0\) is the Poisson rate of arrival of aggregates. The approximate solution of this equation for large and small \(t\) is discussed. The paper also suggests a stochastic version of the aggregation model, which is based on the assumption that the infinitesimal transitions are given by \[ x(t+ \Delta t)=\begin{cases} x(t), & \text{ w.p. }1-\mu(x(t))\Delta t \\ x(t)+\xi(x(t)), & \text{ w.p. }\mu(x(t))\Delta t,\end{cases} \] where \(x(t)\) is the size cluster at time \(t\) and \(\xi(x(t))\) is a random jump. The paper evaluates the expected value of \[ \tau(x)=\inf \{\tau>0: x(t)=N_0, x(0)=x\} \] and some related quantities.