We show that polymers displaying dynamic instability (DI) have at least two experimentally distinguishable critical concentrations (CCs), typical DI occurs between these two CCs, and the separation between the CCs depends on the NTP hydrolysis rate. We demonstrate how these CCs relate to various existing experimental and theoretical definitions of CC.

Stochastic reaction-diffusion models are now a popular tool for studying physical systems in which both the explicit diffusion of molecules and noise in the chemical reaction process play important roles. The Smoluchowski diffusion-limited reaction model (SDLR) is one of several that have been used to study biological systems. Exact realizations of the underlying stochastic processes described by the SDLR model can be generated by the recently proposed First-Passage Kinetic Monte Carlo (FPKMC) method. This exactness relies on sampling analytical solutions to one and two-body diffusion equations in simplified protective domains.In this work we extend the FPKMC to allow for drift arising from fixed, background potentials. As the corresponding Fokker-Planck equations that describe the motion of each molecule can no longer be solved analytically, we develop a hybrid method that discretizes the protective domains. The discretization is chosen so that the driftdiffusion of each molecule within its protective domain is approximated by a continuous-time random walk on a lattice. New lattices are defined dynamically as the protective domains are updated, hence we will refer to our method as Dynamic Lattice FPKMC or DL-FPKMC. We focus primarily on the one-dimensional case in this manuscript, and demonstrate the numerical convergence and accuracy of our method in this case for both smooth and discontinuous potentials. We also present applications of our method, which illustrate the impact of drift on reaction kinetics.

The diffusion of a reactant to a binding target plays a key role in many biological processes. The reaction-radius at which the reactant and target may interact is often a small parameter relative to the diameter of the domain in which the reactant diffuses. We develop uniform in time asymptotic expansions in the reaction-radius of the full solution to the corresponding diffusion equations for two separate reactant-target interaction mechanisms: the Doi or volume reactivity model, and the Smoluchowski-Collins-Kimball partial absorption surface reactivity model. In the former, the reactant and target react with a fixed probability per unit time when within a specified separation. In the latter, upon reaching a fixed separation, they probabilistically react or the reactant reflects away from the target. Expansions of the solution to each model are constructed by projecting out the contribution of the first eigenvalue and eigenfunction to the solution of the diffusion equation, and then developing matched asymptotic expansions in Laplace-transform space. Our approach offers an equivalent, but alternative, method to the pseudo-potential approach we previously employed in [1] for the simpler Smoluchowski pure absorption reaction mechanism. We find that the resulting asymptotic expansions of the diffusion equation solutions are identical with the exception of one parameter: the diffusion limited reaction rates of the Doi and partial absorption models. This demonstrates that for biological systems in which the reaction-radius is a small parameter, properly calibrated Doi and partial absorption models may be functionally equivalent.

The concept of critical concentration (CC) is central to understanding behaviors of microtubules and other cytoskeletal polymers. Traditionally, these polymers are understood to have one CC, measured multiple ways and assumed to be the subunit concentration necessary for polymer assembly. However, this framework does not incorporate dynamic instability (DI), and there is work indicating that microtubules have two CCs. We use our previously established simulations to confirm that microtubules have (at least) two experimentally relevant CCs and to clarify the behaviors of individuals and populations relative to the CCs. At free subunit concentrations above the lower CC (CC IndGrow ), growth phases of individual filaments can occur transiently; above the higher CC (CC PopGrow ), the population's polymer mass will increase persistently. Our results demonstrate that most experimental CC measurements correspond to CC PopGrow , meaning "typical" DI occurs below the concentration traditionally considered necessary for polymer assembly. We report that [free tubulin] at steady state does not equal CC PopGrow , but instead approaches CC PopGrow asymptotically as [total tubulin] increases and depends on the number of stable microtubule seeds. We show that the degree of separation between CC IndGrow and CC PopGrow depends on the rate of nucleotide hydrolysis. This clarified framework helps explain and unify many experimental observations.