Solution to the one-dimensional telegrapher's equation subject to a backreaction boundary condition

We discuss solutions of the one-dimensional telegrapher's equation in the presence of boundary conditions. We revisit the case of a radiation boundary condition and obtain an alternative expression for the already known Green's function. Furthermore, we formulate a backreaction boundary condition, which has been widely used in the context of diffusion-controlled reversible reactions, for a one-dimensional telegrapher's equation and derive the corresponding Green's function.Comment: 13 pages, 1 figur

Exact Green's function of the reversible ABCD reaction in two space dimensions

We derive an exact expression for the Green's functions in the time domain of the reversible diffusion-influenced ABCD reaction $A+B\leftrightarrow C+D$ in two space dimensions. Furthermore, we calculate the corresponding survival and reaction probabilities. The obtained expressions should prove useful for the study of reversible membrane-bound reactions in cell biology and can serve as a useful ingredient of enhanced stochastic particle-based simulation algorithms.Comment: 9 pages, 1 figur

Survival probabilities and rates derived from an exact Green's function of the reversible diffusion-influenced reaction for an isolated pair in 2D

Recently, an exact Green's function of the diffusion equation for a pair of spherical interacting particles in two dimensions subject to a backreaction boundary condition was derived. Here, we use the obtained Green's function to calculate exact expressions for the survival probability, the time-dependent reaction rate coefficient for the initially unbound pair and the survival probability of the bound state in the time domain. Moreover, we derive an exact expression for the off-rate

Note on an integral expression for the average lifetime of the bound state in 2D

Recently, an exact Green's function of the diffusion equation for a pair of spherical interacting particles in two dimensions subject to a backreaction boundary condition was used to derive an exact expression for the average lifetime of the bound state. Here, we show that the corresponding divergent integral may be considered as the formal limit of a Stieltjes transform. Upon analytically calculating the Stieltjes transform one can obtain an exact expression for the finite part of the divergent integral and hence for the average lifetime

General theory of area reactivity models: rate coefficients, binding probabilities and all that

We further develop the general theory of the area reactivity model that provides an alternative description of the diffusion-influenced reaction of an isolated receptor-ligand pair in terms of a generalized Feynman-Kac equation. We analyze both the irreversible and reversible reaction and derive the equation of motion for the survival and separation probability. Furthermore, we discuss the notion of a time-dependent rate coefficient within the alternative model and obtain a number of relations between the rate coefficient, the survival and separation probabilities and the reaction rate. Finally, we calculate asymptotic and approximate expressions for the (irreversible) rate coefficient, the binding probability, the average lifetime of the bound state and discuss on- and off-rates in this context. Throughout our treatment, we will point out similarities and differences between the area and the classical contact reactivity model. The presented analysis and obtained results provide a theoretical framework that will facilitate the comparison of experiment and model predictions

Exact solution of the area reactivity model of an isolated pair

We investigate the reversible diffusion-influenced reaction of an isolated pair in two space dimensions in the context of the area reactivity model. We compute the exact Green's function in the Laplace domain for the initially unbound molecule. Furthermore, we calculate the exact expression for the Green's function in the time domain by inverting the Laplace transform via the Bromwich contour integral. The obtained results should be useful for comparing the behavior of the area reactivity model with more conventional models based on contact reactivity.Comment: 10 pages, 1 figur

Quantum mechanical inspired factorization of the molecule pair propagator in theories of diffusion-influenced reactions

Building on mathematical similarities between quantum mechanics and theories of diffusion-influenced reactions, we discuss how the propagator of a reacting molecule pair can be represented as a product of three factors in the Laplace domain. This representation offers several advantages. First, the full propagator can be calculated without ever having to solve the corresponding partial differential equation or path integral. Second, the representation is quite general and capable of capturing not only the classical Smoluchowski-Collins-Kimball model, but also alternative theories, as is here exemplified by the case of a delta- and step-function potential in one and two dimensions, respectively. Third, the three factors correspond to physical quantities that feature prominently in stochastic spatially-resolved simulation algorithms and hence the interpretation of current and the design of future algorithms may benefit. Finally, the representation may serve as a suitable starting point for numerical approximations that could be employed to enhance the efficiency of stochastic simulations.Comment: 11 page

Non-Markovian reversible diffusion-influenced reactions in two dimensions

We investigate the reversible diffusion-influenced reaction of an isolated pair in the presence of a non-Markovian generalization of the backreaction boundary condition in two space dimensions. Following earlier work by Agmon and Weiss, we consider residence time probability densities that decay slower than an exponential and that are characterized by a parameter $0<\sigma\leq 1$. We calculate an exact expression for the probability $S(t|\ast)$ that the initially bound particle is unbound, which is valid for arbitrary $\sigma$ and for all times. Furthermore, we derive an approximate solution for long times. We show that the ultimate fate of the bound state is complete dissociation, as in the 2D Markovian case. However, the limiting value is approached quite differently: Instead of a $\sim t^{-1}$ decay, we obtain $1-S(t|\ast)\sim t^{-\sigma}\ln t$.Comment: 9 pages, 1 figur

Coarse-Grained Stochastic Particle-based Reaction-Diffusion Simulation Algorithm

In recent years, several particle-based stochastic simulation algorithms (PSSA) have been developed to study the spatially resolved dynamics of biochemical networks at a molecular scale. A challenge all these approaches have to address is to allow for simulations at cell-biologically relevant timescales without neither neglecting important spatial and biochemical properties of the simulated system nor introducing ad-hoc assumptions not based on physical principles. Here we describe a PSSA that permits large time steps while still retaining a high degree of accuracy. The approach addresses the typical disadvantage of Brownian dynamics, namely the need to use small time steps to resolve bimolecular encounters accurately, by estimating the number of otherwise unnoticed encounters with the help of the Green's functions of the diffusion equation incorporating molecular interactions. This method has previously been proposed for purely absorbing boundary conditions and irreversible bimolecular reactions. Building on those ideas, we developed a general-purpose PSSA that is applicable to a broad class of reaction-diffusion problems by incorporating reflective and radiation boundary conditions and reversible reactions. We furthermore discuss how reaction-diffusion systems on 2D membranes can be described and derive small time expansions of the Green's functions that substantially speed up key calculations, particularly in the problematic case of molecules in close proximity. Finally, we point out the formal relationship between our and exact algorithms. The proposed algorithm may serve as an easily implementable and flexible, computationally efficient, coarse-grained description of reaction-diffusion systems in 2D and 3D that nevertheless provides a stochastic, detailed representation at the level of individual particle trajectories in space and time

Stochastic single-particle based simulations of cellular signaling embedded into computational models of cellular morphology

Cells exhibit a wide variety of different shapes. This diversity poses a challenge for computational approaches that attempt to shed light on the role cell geometry plays in regulating cell physiology and behavior. The simulation platform Simmune is capable of embedding the computational representation of signaling pathways into realistic models of cellular morphology. However, Simmune's current approach to account for the cell geometry is limited to deterministic models of reaction-diffusion processes, thus providing a coarse-grained description that ignores stochastic local fluctuations. Here we present an extension of Simmune that removes these limitations by employing an alternative computational representation of cellular geometry that is smooth and grid-free. These features make it possible to incorporate a fully stochastic, spatially resolved description of the cellular biochemistry. The alternative computational representation is compatible with Simmune's current approach for specifying molecular interactions. This means that a modeler using the approach needs to create a model of cellular biochemistry and morphology only once to be able to use it for both, deterministic and stochastic simulations.Comment: 26 pages, 7 figure