Reaction-diffusion systems in and out of equilibrium - methods for simulation and inference

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

Reversible Interacting-Particle Reaction Dynamics

Interacting-Particle Reaction Dynamics (iPRD) simulates the spatiotemporal evolution of particles that experience interaction forces and can react with one another. The combination of interaction forces and reactions enable a wide range of complex reactive systems in biology and chemistry, but give rise to new questions such as how to evolve the dynamical equations in a computationally efficient and statistically correct manner. Here we consider reversible reactions such as A + B C with interacting particles and derive expressions for the microscopic iPRD simulation parameters such that desired values for the equilibrium constant and the dissociation rate are obtained in the dilute limit. We then introduce a Monte-Carlo algorithm that ensures detailed balance in the iPRD time-evolution (iPRD-DB). iPRD-DB guarantees the correct thermodynamics at all concentrations and maintains the desired kinetics in the dilute limit, where chemical rates are well-defined and kinetic measurement experiments usually operate. We show that in dense particle systems, the incorporation of detailed balance is essential to obtain physically realistic solutions. iPRD-DB is implemented in ReaDDy 2 (https://readdy.github.io)

Reactive SINDy: Discovering governing reactions from concentration data

The inner workings of a biological cell or a chemical reaction can be rationalized by the network of reactions, whose structure reveals the most important functional mechanisms. For complex systems, these reaction networks are not known a priori and cannot be efficiently computed with ab initio methods, therefore an important approach goal is to estimate effective reaction networks from observations, such as time series of the main species. Reaction networks estimated with standard machine learning techniques such as least-squares regression may fit the observations, but will typically contain spurious reactions. Here we extend the sparse identification of nonlinear dynamics (SINDy) method to vector-valued ansatz functions, each describing a particular reaction process. The resulting sparse tensor regression method “reactive SINDy” is able to estimate a parsimonious reaction network. We illustrate that a gene regulation network can be correctly estimated from observed time series

Contribution to understanding the mathematical structure of quantum mechanics

Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, commutation and uncertainty relations, probability density current, momentum operator, rules for including the scalar and vector potentials and antiparticles can be obtained from the probabilistic description of results of measurement of the space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrodinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the space-time Fisher information. The limit case of the delta-like probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many particle systems and the postulates of quantum mechanics are also discussed.Comment: 21 page