Single molecule simulations in complex geometries with embedded dynamic one-dimensional structures

Abstract

Stochastic models of reaction-diffusion systems are important for the study of biochemical reaction networks where species are present in low copy numbers or if reactions are highly diffusion limited. In living cells many such systems include reactions and transport on one-dimensional structures, such as DNA and microtubules. The cytoskeleton is a dynamic structure where individual fibers move, grow, and shrink. In this paper we present a simulation algorithm that combines single molecule simulations in three-dimensional space with single molecule simulations on one-dimensional structures of arbitrary shape. Molecules diffuse and react with each other in space, they associate with and dissociate from one-dimensional structures as well as diffuse and react with each other on the one-dimensional structure. A general curve embedded in space can be approximated by a piecewise linear curve to arbitrary accuracy. The resulting algorithm is hence very flexible. Molecules bound to a curve can move by pure diffusion or via active transport, and the curve can move in space as well as grow and shrink. The flexibility and accuracy of the algorithm is demonstrated in five numerical examples.

Stochastic Reaction–Diffusion Processes with Embedded Lower-Dimensional Structures

“…To compare results obtained with the mesoscopic simulations to the corresponding microscopic model we have also simulated the system according to the version of the GFRD algorithm described in Hellander (2013), Hellander and Lötstedt (2011). The new position of a molecule or a reaction is sampled from a probability distribution with a PDF satisfying a Smoluchowski equation.…”

“…The accuracy of the method is improved by introducing protective domains around the separate molecules in Donev et al (2010), Takahashi et al (2010). In recent work (Mauro et al 2013; Hellander 2013), the single molecule simulation technique has been extended to 1D polymers embedded in 3D. Hybrid methods where some species are treated at the microscale and other species are modeled at the mesoscale are found in Flegg et al (2012), Hellander et al (2012a), Klann et al (2012).…”

“…The GFRD method in Hellander and Lötstedt (2011) is extended to curved 1D structures embedded in 3D and compared to mesoscopic simulations for a circular polymer. This method is further developed in Hellander (2013). The DNA and its interaction with transcription factors and repressors at a specific binding site are modeled in a bacterial cell using our method in the two final examples.…”

The effect of cell geometry on polarization in budding yeast

“…Because the actin cables have a physical extent, they can be constrained by the geometry of the cell, providing a possible mechanism to overcome geometry effects. While there are methods to model such systems rigorously (see [ 39 ]), it is not common.…”

Hierarchical Reaction-Diffusion Master Equation

“…In this paper we determine the accuracy of our simulations by comparing to corresponding simulations on the microscopic scale. To ensure the highest possible accuracy on the microscopic scale, we have compared to results obtained with the eGFRD algorithm, as well as another efficient implementation of a similar algorithm 32,33 .…”