33 research outputs found
On the Reaction Diffusion Master Equation in the Microscopic Limit
Stochastic modeling of reaction-diffusion kinetics has emerged as a powerful
theoretical tool in the study of biochemical reaction networks. Two frequently
employed models are the particle-tracking Smoluchowski framework and the
on-lattice Reaction-Diffusion Master Equation (RDME) framework. As the mesh
size goes from coarse to fine, the RDME initially becomes more accurate.
However, recent developments have shown that it will become increasingly
inaccurate compared to the Smoluchowski model as the lattice spacing becomes
very fine. In this paper we give a new, general and simple argument for why the
RDME breaks down. Our analysis reveals a hard limit on the voxel size for which
no local RDME can agree with the Smoluchowski model
A Comparison of Bimolecular Reaction Models for Stochastic Reaction Diffusion Systems
Stochastic reaction-diffusion models have become an important tool in
studying how both noise in the chemical reaction process and the spatial
movement of molecules influences the behavior of biological systems. There are
two primary spatially-continuous models that have been used in recent studies:
the diffusion limited reaction model of Smoluchowski, and a second approach
popularized by Doi. Both models treat molecules as points undergoing Brownian
motion. The former represents chemical reactions between two reactants through
the use of reactive boundary conditions, with two molecules reacting instantly
upon reaching a fixed separation (called the reaction-radius). The Doi model
uses reaction potentials, whereby two molecules react with a fixed probability
per unit time, , when separated by less than the reaction radius. In
this work we study the rigorous relationship between the two models. For the
special case of a protein diffusing to a fixed DNA binding site, we prove that
the solution to the Doi model converges to the solution of the Smoluchowski
model as , with a rigorous
error bound (for any fixed ). We investigate by numerical
simulation, for biologically relevant parameter values, the difference between
the solutions and associated reaction time statistics of the two models. As the
reaction-radius is decreased, for sufficiently large but fixed values of
, these differences are found to increase like the inverse of the
binding radius.Comment: 21 pages, 3 Figures, Fixed typo in titl
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
A Reactive Signaling Approach to Ensure Coexistence Between Molecular Communication and External Biochemical Systems
International audienceIn molecular communication systems operating in a crowded biochemical environment, there is the potential for unintended chemical or physical interactions with external biochemical systems. In order to avoid these interactions, or ensure coexistence, it is necessary to tailor the signaling scheme. In this paper, we propose a signaling strategy exploiting chemical reactions between different transmitted chemical species. While intuitively appealing, the non-linear nature of the governing partial differential equations (PDE) means that selecting the signaling strategy to minimize the probability of error is com-putationally challenging. To reduce this computational burden, we introduce a new proxy metric called the modified signal-to-interference difference (mSID). We show that optimizing the mSID yields low complexity and near-optimal solutions, requiring only deterministic nonlinear programming rather than standard brute force Monte Carlo methods
Simulating Stochastic Reaction-Diffusion Systems on and within Moving Boundaries
Chemical reactions inside cells are generally considered to happen within
fixed-size compartments. Needless to say, cells and their compartments are
highly dynamic. Thus, such stringent assumptions may not reflect biochemical
reality, and can highly bias conclusions from simulation studies. In this work,
we present an intuitive algorithm for particle-based diffusion in and on moving
boundaries, for both point particles and spherical particles. We first
benchmark in appropriate scenarios our proposed stochastic method against
solutions of partial differential equations, and further demonstrate that
moving boundaries can give rise to super diffusive motion as well as
time-inhomogeneous reaction rates. Finally, we conduct a numerical experiment
representing photobleaching of diffusing fluorescent proteins in dividing
Saccharomyces cerevisiae cells to demonstrate that moving boundaries might
cause important effects neglected in previously published studies.Comment: 22 pages, 7 figure
Stochastic modelling of reaction-diffusion processes:\ud algorithms for bimolecular reactions
Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this paper, two commonly used SSAs are studied. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. In both cases, it is shown that the commonly used implementation of bimolecular reactions (i.e. the reactions of the form A+B → C, or A+A → C) might lead to incorrect results. Improvements of both SSAs are suggested which overcome the difficulties highlighted. In particular, a formula is presented for the smallest possible compartment size (lattice spacing) which can be correctly implemented in the first model. This implementation uses a new formula for the rate of bimolecular reactions per compartment (lattice site)
Stochastic modelling of reaction-diffusion processes: algorithms for bimolecular reactions
Several stochastic simulation algorithms (SSAs) have been recently proposed
for modelling reaction-diffusion processes in cellular and molecular biology.
In this paper, two commonly used SSAs are studied. The first SSA is an
on-lattice model described by the reaction-diffusion master equation. The
second SSA is an off-lattice model based on the simulation of Brownian motion
of individual molecules and their reactive collisions. In both cases, it is
shown that the commonly used implementation of bimolecular reactions (i.e. the
reactions of the form A + B -> C, or A + A -> C) might lead to incorrect
results. Improvements of both SSAs are suggested which overcome the
difficulties highlighted. In particular, a formula is presented for the
smallest possible compartment size (lattice spacing) which can be correctly
implemented in the first model. This implementation uses a new formula for the
rate of bimolecular reactions per compartment (lattice site).Comment: 33 pages, submitted to Physical Biolog