Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation

We present computer-assisted methods for analyzing stochastic models of gene regulatory networks. The main idea that underlies this equation-free analysis is the design and execution of appropriately-initialized short bursts of stochastic simulations; the results of these are processed to estimate coarse-grained quantities of interest, such as mesoscopic transport coefficients. In particular, using a simple model of a genetic toggle switch, we illustrate the computation of an effective free energy and of a state-dependent effective diffusion coefficient that characterize an unavailable effective Fokker-Planck equation. Additionally we illustrate the linking of equation-free techniques with continuation methods for performing a form of stochastic "bifurcation analysis"; estimation of mean switching times in the case of a bistable switch is also implemented in this equation-free context. The accuracy of our methods is tested by direct comparison with long-time stochastic simulations. This type of equation-free analysis appears to be a promising approach to computing features of the long-time, coarse-grained behavior of certain classes of complex stochastic models of gene regulatory networks, circumventing the need for long Monte Carlo simulations.Comment: 33 pages, submitted to The Journal of Chemical Physic

Biochemical Network Stochastic Simulator (BioNetS): software for stochastic modeling of biochemical networks

BACKGROUND: Intrinsic fluctuations due to the stochastic nature of biochemical reactions can have large effects on the response of biochemical networks. This is particularly true for pathways that involve transcriptional regulation, where generally there are two copies of each gene and the number of messenger RNA (mRNA) molecules can be small. Therefore, there is a need for computational tools for developing and investigating stochastic models of biochemical networks. RESULTS: We have developed the software package Biochemical Network Stochastic Simulator (BioNetS) for efficiently and accurately simulating stochastic models of biochemical networks. BioNetS has a graphical user interface that allows models to be entered in a straightforward manner, and allows the user to specify the type of random variable (discrete or continuous) for each chemical species in the network. The discrete variables are simulated using an efficient implementation of the Gillespie algorithm. For the continuous random variables, BioNetS constructs and numerically solves the appropriate chemical Langevin equations. The software package has been developed to scale efficiently with network size, thereby allowing large systems to be studied. BioNetS runs as a BioSpice agent and can be downloaded from . BioNetS also can be run as a stand alone package. All the required files are accessible from . CONCLUSIONS: We have developed BioNetS to be a reliable tool for studying the stochastic dynamics of large biochemical networks. Important features of BioNetS are its ability to handle hybrid models that consist of both continuous and discrete random variables and its ability to model cell growth and division. We have verified the accuracy and efficiency of the numerical methods by considering several test systems

Mathematical model reveals role of nucleotide signaling in airway surface liquid homeostasis and its dysregulation in cystic fibrosis

The intrapulmonary airways conduct air to the alveoli and are defended from inhaled pathogens by a highly regulated protective system of mucus, cilia, and liquid. In healthy lungs, a well-hydrated mucus layer is cleared by cilia from airway surfaces. In cystic fibrosis (CF), airway surfaces are dehydrated, leading to a failure of cilia-mediated mucus clearance and accumulation of pathogen-infected mucus. In this study, we created a mathematical model of airway surface liquid regulation in normal and CF cells and used the model to investigate a potential therapy to rehydrate CF airways and restore proper mucus clearance

Time-Domain Methods for Diffusive Transport in Soft Matter

Passive microrheology [12] utilizes measurements of noisy, entropic fluctuations (i.e., diffusive properties) of micron-scale spheres in soft matter to infer bulk frequency-dependent loss and storage moduli. Here, we are concerned exclusively with diffusion of Brownian particles in viscoelastic media, for which the Mason-Weitz theoretical-experimental protocol is ideal, and the more challenging inference of bulk viscoelastic moduli is decoupled. The diffusive theory begins with a generalized Langevin equation (GLE) with a memory drag law specified by a kernel [7, 16, 22, 23]. We start with a discrete formulation of the GLE as an autoregressive stochastic process governing microbead paths measured by particle tracking. For the inverse problem (recovery of the memory kernel from experimental data) we apply time series analysis (maximum likelihood estimators via the Kalman filter) directly to bead position data, an alternative to formulas based on mean-squared displacement statistics in frequency space. For direct modeling, we present statistically exact GLE algorithms for individual particle paths as well as statistical correlations for displacement and velocity. Our time-domain methods rest upon a generalization of well-known results for a single-mode exponential kernel [1, 7, 22, 23] to an arbitrary M-mode exponential series, for which the GLE is transformed to a vector Ornstein-Uhlenbeck process

Variable-free exploration of stochastic models: a gene regulatory network example

Finding coarse-grained, low-dimensional descriptions is an important task in the analysis of complex, stochastic models of gene regulatory networks. This task involves (a) identifying observables that best describe the state of these complex systems and (b) characterizing the dynamics of the observables. In a previous paper [13], we assumed that good observables were known a priori, and presented an equation-free approach to approximate coarse-grained quantities (i.e, effective drift and diffusion coefficients) that characterize the long-time behavior of the observables. Here we use diffusion maps [9] to extract appropriate observables ("reduction coordinates") in an automated fashion; these involve the leading eigenvectors of a weighted Laplacian on a graph constructed from network simulation data. We present lifting and restriction procedures for translating between physical variables and these data-based observables. These procedures allow us to perform equation-free coarse-grained, computations characterizing the long-term dynamics through the design and processing of short bursts of stochastic simulation initialized at appropriate values of the data-based observables.Comment: 26 pages, 9 figure

In Silico Generation of Alternative Hypotheses Using Causal Mapping (CMAP)

Previously, we introduced causal mapping (CMAP) as an easy to use systems biology tool for studying the behavior of biological processes that occur at the cellular and molecular level. CMAP is a coarse-grained graphical modeling approach in which the system of interest is modeled as an interaction map between functional elements of the system, in a manner similar to portrayals of signaling pathways commonly used by molecular cell biologists. CMAP describes details of the interactions while maintaining the simplicity of other qualitative methods (e.g., Boolean networks)

A cut-cell method for simulating spatial models of biochemical reaction networks in arbitrary geometries

Cells use signaling networks consisting of multiple interacting proteins to respond to changes in their environment. In many situations, such as chemotaxis, spatial and temporal information must be transmitted through the network. Recent computational studies have emphasized the importance of cellular geometry in signal transduction, but have been limited in their ability to accurately represent complex cell morphologies. We present a finite volume method that addresses this problem. Our method uses Cartesian cut cells and is second order in space and time. We use our method to simulate several models of signaling systems in realistic cell morphologies obtained from live cell images and examine the effects of geometry on signal transduction

From physics to pharmacology?

Over the last fifty years there has been an explosion of biological data, leading to the realization that to fully explain biological mechanisms it is necessary to interpret them as complex dynamical systems. The first stage of this interpretation is to determine which components (proteins, genes or metabolites) of the system interact. This is usually represented by a graph, or network. The behavior of this network can then be investigated using mathematical modeling. In vivo these biological networks show several remarkable (and seemingly paradoxical) properties including robustness, plasticity and sensitivity. Erroneous behavior of these networks is often associated with disease. Hence understanding the system-level properties can have important implications for the treatment of disease. Systems biology is an organized approach to quantitatively describe and elucidate the behavior of these complex networks. This review focuses on the progress and future challenges of a systems approach to biology

Biophysics at the coffee shop: lessons learned working with George Oster

Over the past 50 years, the use of mathematical models, derived from physical reasoning, to describe molecular and cellular systems has evolved from an art of the few to a cornerstone of biological inquiry. George Oster stood out as a pioneer of this paradigm shift from descriptive to quantitative biology not only through his numerous research accomplishments, but also through the many students and postdocs he mentored over his long career. Those of us fortunate enough to have worked with George agree that his sharp intellect, physical intuition and passion for scientific inquiry not only inspired us as scientists but also greatly influenced the way we conduct research. We would like to share a few important lessons we learned from George in honor of his memory and with the hope that they may inspire future generations of scientists.Comment: 22 pages, 3 figures, accepted in Molecular Biology of the Cel

Compression and dilation of the membrane-cortex layer generates rapid changes in cell shape

A cyclic process of membrane-cortex compression and dilation generates a traveling wave of cortical actin density that in turn generates oscillations in cell morphology.Rapid changes in cellular morphology require a cell body that is highly flexible yet retains sufficient strength to maintain structural integrity. We present a mechanism that meets both of these requirements. We demonstrate that compression (folding) and subsequent dilation (unfolding) of the coupled plasma membrane–cortex layer generates rapid shape transformations in rounded cells. Two- and three-dimensional live-cell images showed that the cyclic process of membrane-cortex compression and dilation resulted in a traveling wave of cortical actin density. We also demonstrate that the membrane-cortex traveling wave led to amoeboid-like cell migration. The compression–dilation hypothesis offers a mechanism for large-scale cell shape transformations that is complementary to blebbing, where the plasma membrane detaches from the actin cortex and is initially unsupported when the bleb extends as a result of cytosolic pressure. Our findings provide insight into the mechanisms that drive the rapid morphological changes that occur in many physiological contexts, such as amoeboid migration and cytokinesis