81 research outputs found
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 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 . We
calculate an exact expression for the probability that the
initially bound particle is unbound, which is valid for arbitrary 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 decay, we obtain .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
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