Reaction-diffusion methods allow treatment of mesoscopic dynamic phenomena of soft condensed matter especially in the context of cellular biology. Macromolecules such as proteins consist of thousands of atoms, in reaction-diffusion models their interaction is described by effective dynamics with much fewer degrees of freedom. Reaction-diffusion methods can be categorized by the spatial and temporal length-scales involved and the amount of molecules, e.g. classical reaction kinetics are macroscopic equations for fast diffusion and many molecules described by average concentrations. The focus of this work however is interacting-particle reaction-dynamics (iPRD), which operates on length scales of few nanometers and time scales of nanoseconds, where proteins can be represented by coarse-grained beads, that interact via effective potentials and undergo reactions upon encounter. In practice these systems are often studied using time-stepping computer simulations. Reactions in such iPRD simulations are discrete events which rapidly interchange beads, e.g. in the scheme A + B <-> C the two interacting particles A and B will be replaced by a C complex and vice-versa. Such reactions in combination with the interaction potentials pose two practical problems: 1. To achieve a well defined state of equilibrium, it is of vital importance that the reaction transitions obey microscopic reversiblity (detailed balance). 2. The mean rate of a bimolecular association reaction changes when the particles interact via a pair-potential. In this work the first question is answered both theoretically and algorithmically. Theoretically by formulating the state of equilibrium for a closed iPRD system and the requirements for detailed balance. Algorithmically by implementing the detailed balance reaction scheme in a publicly available simulator ReaDDy~2 for iPRD systems. The second question is answered by deriving concrete formulae for the macroscopic reaction rate as a function of the intrinsic parameters for the Doi reaction model subject to pair interactions. Especially this work addresses two important scenarios: Reversible reactions in a closed container and irreversible bimolecular reactions in the diffusion-influenced regime.
A characteristic of reactions occurring in cellular environments is that the number of species involved in a physiological response is very large. Unveiling the network of necessary reactions is a task that can be addressed by a data-driven approach. In particular, analyzing observation data of such processes can be used to learn the important governing dynamics. This work gives an overview of the inference of dynamical reactive systems for the different reaction-diffusion models. For the case of reaction kinetics a method called Reactive Sparse Identification of Nonlinear Dynamics (Reactive SINDy) is developed that allows to obtain a sparse reaction network out of candidate reactions from time-series observations of molecule concentrations.
