Spatial Stochastic Simulation of the Hes1 gene regulatory network

Individual mouse embryonic stem cells have been found to exhibit highly variable differentiation responses under the same environmental conditions. Recent experimatal evidence suggest that the noisy cyclic expression of Hes1 and its downstream genes are  responsible for this, but the mechanism underlying this variability in expression is not well understood.

Together with Mark Chaplain’s group, we have recenly published a new paper in Journal of the Royal Society Interface, where we propose a spatial stochastic model of the Hes1 regulatory network. Simulations of this model with URDME suggest that the Hes1 oscillations will intrinsically give rise to broad period distributions, and hence very heterogenous cell popuations with respect to Hes1 expression. Our work suggests a simple mechanism to explain the observations that cells that were sorted according to high or low expression of Hes1 relaxed back to a heterogenous mixture of Hes1 expression. We also propose experiments to control the precise differentiation response using drug treatment.

 

URDME paper in BMC Systems Biology

Ever since the first version of URDME, a software framework based on our theoretical work on RDME simulations on unstructured meshes, was made public in 2008, we have wanted to write up a journal publication that describes the software. For variuos reasons we have not gotten around to it, until now.

The paper, published in BMC systems Biology, contains two modeling examples and one example that illustrates how one can use the framework as a tool in methods development. We also discuss the relationship between URDME and two other great RDME simulators, mesoRD and STEPS.

 

On the reaction-diffusion master equation in the microscopic limit

The RDME will break down in the limit of vanishing voxel sizes, in the sense that contributions from bimolecular reactions will be lost. The problem sets on earlier (for larger voxels), the more diffusion limited the reaction is. This is a problem that has attracted a lot of interest since it was pointed out by Samuel Isaacson  in this paper.

Recently, corrections to the bimolecular rates that are explicitly mesh-dependent has been proposed to deal with the problem. Erban and Chapman finds an expression in 3D that works down to a critical size of the mesh.

In this paper, we use a theorem from Montroll to show that there will always be such a critical mesh size for which no local correction to the RDME can make it agree with the Smoluchowski model in the sense that the mean binding time between two particles should be the same in both models. In the limit of perfect diffusion control, we find analytical values for the critical size in both 2D and 3D. Interestingly, the value we find in 3D agrees with the value found by Erban and Chapman. We also discuss the relationship between the local corrections of Erban and Chapman and ours to those derived by Fange et. al.

 

Multiscale and hybrid algorithms for stochastic chemical kinetics

Many biochemical network models display scale separation with respect to reaction rates and/or molecular copy numbers. Depending on the type of question under study, different models are best suited to simulate the system. For some parts of the system, a macroscopic model might be appropriate. For other parts, a mesoscopic model may provide additional insight into the models dynamics. In other cases, a microscopic model might be needed to capture the fine-grained features of the model. To efficiently simulate systems that have different requirements with respect to modeling levels, hybrid methods offer an attractive approach.