6,596 research outputs found
Efficient Reactive Brownian Dynamics
We develop a Split Reactive Brownian Dynamics (SRBD) algorithm for particle
simulations of reaction-diffusion systems based on the Doi or volume reactivity
model, in which pairs of particles react with a specified Poisson rate if they
are closer than a chosen reactive distance. In our Doi model, we ensure that
the microscopic reaction rules for various association and disassociation
reactions are consistent with detailed balance (time reversibility) at
thermodynamic equilibrium. The SRBD algorithm uses Strang splitting in time to
separate reaction and diffusion, and solves both the diffusion-only and
reaction-only subproblems exactly, even at high packing densities. To
efficiently process reactions without uncontrolled approximations, SRBD employs
an event-driven algorithm that processes reactions in a time-ordered sequence
over the duration of the time step. A grid of cells with size larger than all
of the reactive distances is used to schedule and process the reactions, but
unlike traditional grid-based methods such as Reaction-Diffusion Master
Equation (RDME) algorithms, the results of SRBD are statistically independent
of the size of the grid used to accelerate the processing of reactions. We use
the SRBD algorithm to compute the effective macroscopic reaction rate for both
reaction- and diffusion-limited irreversible association in three dimensions.
We also study long-time tails in the time correlation functions for reversible
association at thermodynamic equilibrium. Finally, we compare different
particle and continuum methods on a model exhibiting a Turing-like instability
and pattern formation. We find that for models in which particles diffuse off
lattice, such as the Doi model, reactions lead to a spurious enhancement of the
effective diffusion coefficients.Comment: To appear in J. Chem. Phy
Varying the resolution of the Rouse model on temporal and spatial scales: application to multiscale modelling of DNA dynamics
A multi-resolution bead-spring model for polymer dynamics is developed as a
generalization of the Rouse model. A polymer chain is described using beads of
variable sizes connected by springs with variable spring constants. A numerical
scheme which can use different timesteps to advance the positions of different
beads is presented and analyzed. The position of a particular bead is only
updated at integer multiples of the timesteps associated with its connecting
springs. This approach extends the Rouse model to a multiscale model on both
spatial and temporal scales, allowing simulations of localized regions of a
polymer chain with high spatial and temporal resolution, while using a coarser
modelling approach to describe the rest of the polymer chain. A method for
changing the model resolution on-the-fly is developed using the
Metropolis-Hastings algorithm. It is shown that this approach maintains key
statistics of the end-to-end distance and diffusion of the polymer filament and
makes computational savings when applied to a model for the binding of a
protein to the DNA filament.Comment: Submitted to Multiscale Modeling and Simulatio
Dynamics of protein-protein encounter: a Langevin equation approach with reaction patches
We study the formation of protein-protein encounter complexes with a Langevin
equation approach that considers direct, steric and thermal forces. As three
model systems with distinctly different properties we consider the pairs
barnase:barstar, cytochrome c:cytochrome c peroxidase and p53:MDM2. In each
case, proteins are modeled either as spherical particles, as dipolar spheres or
as collection of several small beads with one dipole. Spherical reaction
patches are placed on the model proteins according to the known experimental
structures of the protein complexes. In the computer simulations, concentration
is varied by changing box size. Encounter is defined as overlap of the reaction
patches and the corresponding first passage times are recorded together with
the number of unsuccessful contacts before encounter. We find that encounter
frequency scales linearly with protein concentration, thus proving that our
microscopic model results in a well-defined macroscopic encounter rate. The
number of unsuccessful contacts before encounter decreases with increasing
encounter rate and ranges from 20-9000. For all three models, encounter rates
are obtained within one order of magnitude of the experimentally measured
association rates. Electrostatic steering enhances association up to 50-fold.
If diffusional encounter is dominant (p53:MDM2) or similarly important as
electrostatic steering (barnase:barstar), then encounter rate decreases with
decreasing patch radius. More detailed modeling of protein shapes decreases
encounter rates by 5-95 percent. Our study shows how generic principles of
protein-protein association are modulated by molecular features of the systems
under consideration. Moreover it allows us to assess different coarse-graining
strategies for the future modelling of the dynamics of large protein complexes
Reaction-diffusion kinetics on lattice at the microscopic scale
Lattice-based stochastic simulators are commonly used to study biological
reaction-diffusion processes. Some of these schemes that are based on the
reaction-diffusion master equation (RDME), can simulate for extended spatial
and temporal scales but cannot directly account for the microscopic effects in
the cell such as volume exclusion and diffusion-influenced reactions.
Nonetheless, schemes based on the high-resolution microscopic lattice method
(MLM) can directly simulate these effects by representing each finite-sized
molecule explicitly as a random walker on fine lattice voxels. The theory and
consistency of MLM in simulating diffusion-influenced reactions have not been
clarified in detail. Here, we examine MLM in solving diffusion-influenced
reactions in 3D space by employing the Spatiocyte simulation scheme. Applying
the random walk theory, we construct the general theoretical framework
underlying the method and obtain analytical expressions for the total rebinding
probability and the effective reaction rate. By matching Collins-Kimball and
lattice-based rate constants, we obtained the exact expressions to determine
the reaction acceptance probability and voxel size. We found that the size of
voxel should be about 2% larger than the molecule. MLM is validated by
numerical simulations, showing good agreement with the off-lattice
particle-based method, eGFRD. MLM run time is more than an order of magnitude
faster than eGFRD when diffusing macromolecules with typical concentrations in
the cell. MLM also showed good agreements with eGFRD and mean-field models in
case studies of two basic motifs of intracellular signaling, the protein
production-degradation process and the dual phosphorylation cycle. Moreover,
when a reaction compartment is populated with volume-excluding obstacles, MLM
captures the non-classical reaction kinetics caused by anomalous diffusion of
reacting molecules
Reactive Boundary Conditions as Limits of Interaction Potentials for Brownian and Langevin Dynamics
A popular approach to modeling bimolecular reactions between diffusing
molecules is through the use of reactive boundary conditions. One common model
is the Smoluchowski partial absorption condition, which uses a Robin boundary
condition in the separation coordinate between two possible reactants. This
boundary condition can be interpreted as an idealization of a reactive
interaction potential model, in which a potential barrier must be surmounted
before reactions can occur. In this work we show how the reactive boundary
condition arises as the limit of an interaction potential encoding a steep
barrier within a shrinking region in the particle separation, where molecules
react instantly upon reaching the peak of the barrier. The limiting boundary
condition is derived by the method of matched asymptotic expansions, and shown
to depend critically on the relative rate of increase of the barrier height as
the width of the potential is decreased. Limiting boundary conditions for the
same interaction potential in both the overdamped Fokker-Planck equation
(Brownian Dynamics), and the Kramers equation (Langevin Dynamics) are
investigated. It is shown that different scalings are required in the two
models to recover reactive boundary conditions that are consistent in the high
friction limit where the Kramers equation solution converges to the solution of
the Fokker-Planck equation.Comment: 23 pages, 2 figure
Reaction Brownian Dynamics and the effect of spatial fluctuations on the gain of a push-pull network
Brownian Dynamics algorithms are widely used for simulating soft-matter and
biochemical systems. In recent times, their application has been extended to
the simulation of coarse-grained models of cellular networks in simple
organisms. In these models, components move by diffusion, and can react with
one another upon contact. However, when reactions are incorporated into a
Brownian Dynamics algorithm, attention must be paid to avoid violations of the
detailed-balance rule, and therefore introducing systematic errors in the
simulation. We present a Brownian Dynamics algorithm for reaction-diffusion
systems that rigorously obeys detailed balance for equilibrium reactions. By
comparing the simulation results to exact analytical results for a bimolecular
reaction, we show that the algorithm correctly reproduces both equilibrium and
dynamical quantities. We apply our scheme to a ``push-pull'' network in which
two antagonistic enzymes covalently modify a substrate. Our results highlight
that the diffusive behaviour of the reacting species can reduce the gain of the
response curve of this network.Comment: 25 pages, 7 figures, submitted to Journal of Chemical Physic
Jump-Diffusion Approximation of Stochastic Reaction Dynamics: Error bounds and Algorithms
Biochemical reactions can happen on different time scales and also the
abundance of species in these reactions can be very different from each other.
Classical approaches, such as deterministic or stochastic approach, fail to
account for or to exploit this multi-scale nature, respectively. In this paper,
we propose a jump-diffusion approximation for multi-scale Markov jump processes
that couples the two modeling approaches. An error bound of the proposed
approximation is derived and used to partition the reactions into fast and slow
sets, where the fast set is simulated by a stochastic differential equation and
the slow set is modeled by a discrete chain. The error bound leads to a very
efficient dynamic partitioning algorithm which has been implemented for several
multi-scale reaction systems. The gain in computational efficiency is
illustrated by a realistically sized model of a signal transduction cascade
coupled to a gene expression dynamics.Comment: 32 pages, 7 figure
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