Derivation of continuum models from discrete models of mechanical forces in cell populations.

In certain discrete models of populations of biological cells, the mechanical forces between the cells are center based or vertex based on the microscopic level where each cell is individually represented. The cells are circular or spherical in a center based model and polygonal or polyhedral in a vertex based model. On a higher, macroscopic level, the time evolution of the density of the cells is described by partial differential equations (PDEs). We derive relations between the modelling on the micro and macro levels in one, two, and three dimensions by regarding the micro model as a discretization of a PDE for conservation of mass on the macro level. The forces in the micro model correspond on the macro level to a gradient of the pressure scaled by quantities depending on the cell geometry. The two levels of modelling are compared in numerical experiments in one and two dimensions.

The Linear Noise Approximation for Spatially Dependent Biochemical Networks.

An algorithm for computing the linear noise approximation (LNA) of the reaction-diffusion master equation (RDME) is developed and tested. The RDME is often used as a model for biochemical reaction networks. The LNA is derived for a general discretization of the spatial domain of the problem. If M is the number of chemical species in the network and N is the number of nodes in the discretization in space, then the computational work to determine approximations of the mean and the covariances of the probability distributions is proportional to [Formula: see text] in a straightforward implementation. In our LNA algorithm, the work is proportional to [Formula: see text]. Since N usually is larger than M, this is a significant reduction. The accuracy of the approximation in the algorithm is estimated analytically and evaluated in numerical experiments.

Stochastic Simulation Service: Bridging the Gap between the Computational Expert and the Biologist.

We present StochSS: Stochastic Simulation as a Service, an integrated development environment for modeling and simulation of both deterministic and discrete stochastic biochemical systems in up to three dimensions. An easy to use graphical user interface enables researchers to quickly develop and simulate a biological model on a desktop or laptop, which can then be expanded to incorporate increasing levels of complexity. StochSS features state-of-the-art simulation engines. As the demand for computational power increases, StochSS can seamlessly scale computing resources in the cloud. In addition, StochSS can be deployed as a multi-user software environment where collaborators share computational resources and exchange models via a public model repository. We demonstrate the capabilities and ease of use of StochSS with an example of model development and simulation at increasing levels of complexity.

ANALYSIS AND DESIGN OF JUMP COEFFICIENTS IN DISCRETE STOCHASTIC DIFFUSION MODELS.

In computational systems biology, the mesoscopic model of reaction-diffusion kinetics is described by a continuous time, discrete space Markov process. To simulate diffusion stochastically, the jump coefficients are obtained by a discretization of the diffusion equation. Using unstructured meshes to represent complicated geometries may lead to negative coefficients when using piecewise linear finite elements. Several methods have been proposed to modify the coefficients to enforce the nonnegativity needed in the stochastic setting. In this paper, we present a method to quantify the error introduced by that change. We interpret the modified discretization matrix as the exact finite element discretization of a perturbed equation. The forward error, the error between the analytical solutions to the original and the perturbed equations, is bounded by the backward error, the error between the diffusion of the two equations. We present a backward analysis algorithm to compute the diffusion coefficient from a given discretization matrix. The analysis suggests a new way of deriving nonnegative jump coefficients that minimizes the backward error. The theory is tested in numerical experiments indicating that the new method is superior and also minimizes the forward error.

MESOSCOPIC MODELING OF STOCHASTIC REACTION-DIFFUSION KINETICS IN THE SUBDIFFUSIVE REGIME.

Subdiffusion has been proposed as an explanation of various kinetic phenomena inside living cells. In order to fascilitate large-scale computational studies of subdiffusive chemical processes, we extend a recently suggested mesoscopic model of subdiffusion into an accurate and consistent reaction-subdiffusion computational framework. Two different possible models of chemical reaction are revealed and some basic dynamic properties are derived. In certain cases those mesoscopic models have a direct interpretation at the macroscopic level as fractional partial differential equations in a bounded time interval. Through analysis and numerical experiments we estimate the macroscopic effects of reactions under subdiffusive mixing. The models display properties observed also in experiments: for a short time interval the behavior of the diffusion and the reaction is ordinary, in an intermediate interval the behavior is anomalous, and at long times the behavior is ordinary again.

Stochastic diffusion processes on Cartesian meshes.

Diffusion of molecules is simulated stochastically by letting them jump between voxels in a Cartesian mesh. The jump coefficients are first derived using finite difference, finite element, and finite volume approximations of the Laplacian on the mesh. An alternative is to let the first exit time for a molecule in random walk in a voxel define the jump coefficient. Such coefficients have the advantage of always being non-negative. These four different ways of obtaining the diffusion propensities are compared theoretically and in numerical experiments. A finite difference and a finite volume approximation generate the most accurate coefficients.

Simulation of stochastic diffusion via first exit times.

In molecular biology it is of interest to simulate diffusion stochastically. In the mesoscopic model we partition a biological cell into unstructured subvolumes. In each subvolume the number of molecules is recorded at each time step and molecules can jump between neighboring subvolumes to model diffusion. The jump rates can be computed by discretizing the diffusion equation on that unstructured mesh. If the mesh is of poor quality, due to a complicated cell geometry, standard discretization methods can generate negative jump coefficients, which no longer allows the interpretation as the probability to jump between the subvolumes. We propose a method based on the mean first exit time of a molecule from a subvolume, which guarantees positive jump coefficients. Two approaches to exit times, a global and a local one, are presented and tested in simulations on meshes of different quality in two and three dimensions.

Stochastic reaction-diffusion processes with embedded lower-dimensional structures.

Small copy numbers of many molecular species in biological cells require stochastic models of the chemical reactions between the molecules and their motion. Important reactions often take place on one-dimensional structures embedded in three dimensions with molecules migrating between the dimensions. Examples of polymer structures in cells are DNA, microtubules, and actin filaments. An algorithm for simulation of such systems is developed at a mesoscopic level of approximation. An arbitrarily shaped polymer is coupled to a background Cartesian mesh in three dimensions. The realization of the system is made with a stochastic simulation algorithm in the spirit of Gillespie. The method is applied to model problems for verification and two more detailed models of transcription factor interaction with the DNA.

Transcription factor binding kinetics constrain noise suppression via negative feedback.

Negative autoregulation, where a transcription factor regulates its own expression by preventing transcription, is commonly used to suppress fluctuations in gene expression. Recent single molecule in vivo imaging has shown that it takes significant time for a transcription factor molecule to bind its chromosomal binding site. Given the slow association kinetics, transcription factor mediated feedback cannot at the same time be fast and strong. Here we show that with a limited association rate follows an optimal transcription factor binding strength where noise is maximally suppressed. At the optimal binding strength the binding site is free a fixed fraction of the time independent of the transcription factor concentration. One consequence is that high-copy number transcription factors should bind weakly to their operators, which is observed for transcription factors in Escherichia coli. The results demonstrate that a binding site's strength may be uncorrelated to its functional importance.

Delay-induced anomalous fluctuations in intracellular regulation.

Direct negative feedback decreases fluctuations and is a ubiquitous mechanism for homoeostatic control. However, intracellular regulation frequently operates indirectly, resulting in delayed responses. Here we derive an analytical expression that quantifies the consequences from delayed negative feedback resulting from typical multistep synthesis pathways, for example, transcription or translation. We find that indirect feedback leads to more fluctuations than without feedback for intermediate delays, but surprisingly not for long delays. The anomalous fluctuations at intermediate delays emerge from positive correlations between the delayed regulatory events, and are shown to be equivalent to an increased stoichiometry in the synthesis of new molecules. The results primarily give us insight about the design principles of delayed stochastic control systems and why a fixed feedback delay gives more fluctuations than a broadly distributed feedback delay. It is also shown that the feedback delay of auto-repressed regulators can result in more sensitive regulation of downstream processes through stochastic focusing.

Costs and constraints from time-delayed feedback in small gene regulatory motifs.

The multistep character of transcription, translation, and protein modification inevitably leads to time delays between sensing gene regulatory signals and responding with changed concentrations of functional proteins. However, the interplay between the time-delayed and the stochastic nature of gene regulation has been poorly investigated. Here we present an extension of the linear noise approximation which makes it possible to estimate second moments--variances and covariances--of fluctuations around stationary states in time-delayed systems. The usefulness of the method is exemplified by analyzing two ubiquitous regulatory motifs. In the first system, we show that there is an optimal combination of transcriptional repression and direct product inhibition in determining the activity of an enzyme system. In particular, we demonstrate that direct product inhibition is necessary to avoid deleterious fluctuations in a system when the gene regulatory response is delayed. The second system is an anabolic motif where the substrate fluxes are balanced by time-delayed regulation responding to the substrate concentrations. The extended linear noise approximation makes it possible to show analytically that increased association rate between the substrates leads to a lower product flux because of increasing unbalance in substrate pools.