Bistability versus bimodal distributions in gene regulatory processes from population balance

In recent times, stochastic treatments of gene regulatory processes have appeared in the literature in which a cell exposed to a signaling molecule in its environment triggers the synthesis of a specific protein through a network of intracellular reactions. The stochastic nature of this process leads to a distribution of protein levels in a population of cells as determined by a Fokker-Planck equation. Often instability occurs as a consequence of two (stable) steady state protein levels, one at the low end representing the "off" state, and the other at the high end representing the "on" state for a given concentration of the signaling molecule within a suitable range. A consequence of such bistability has been the appearance of bimodal distributions indicating two different populations, one in the "off" state and the other in the "on" state. The bimodal distribution can come about from stochastic analysis of a single cell. However, the concerted action of the population altering the extracellular concentration in the environment of individual cells and hence their behavior can only be accomplished by an appropriate population balance model which accounts for the reciprocal effects of interaction between the population and its environment. In this study, we show how to formulate a population balance model in which stochastic gene expression in individual cells is incorporated. Interestingly, the simulation of the model shows that bistability is neither sufficient nor necessary for bimodal distributions in a population. The original notion of linking bistability with bimodal distribution from single cell stochastic model is therefore only a special consequence of a population balance model.     

Transient dynamics of reduced-order models of genetic regulatory networks

In systems biology, a number of detailed genetic regulatory networks models have been proposed that are capable of modeling the fine-scale dynamics of gene expression. However, limitations on the type and sampling frequency of experimental data often prevent the parameter estimation of the detailed models. Furthermore, the high computational complexity involved in the simulation of a detailed model restricts its use. In such a scenario, reduced-order models capturing the coarse-scale behavior of the network are frequently applied. In this paper, we analyze the dynamics of a reduced-order Markov Chain model approximating a detailed Stochastic Master Equation model. Utilizing a reduction mapping that maintains the aggregated steady-state probability distribution of stochastic master equation models, we provide bounds on the deviation of the Markov Chain transient distribution from the transient aggregated distributions of the stochastic master equation model.     

Deciphering the Dynamical Origin of Mixed Population during Neural Stem Cell Development

Neural stem cells (NSCs) often give rise to a mixed population of cells during differentiation. However, the dynamical origin of these mixed states is poorly understood. In this article, our mathematical modeling study demonstrates that the bone morphogenetic protein 2 (BMP2) mediated disparate differentiation dynamics of NSCs in central and peripheral nervous systems essentially function through two distinct bistable switches that are mutually interconnected via a mushroom-like bifurcation. Stochastic simulations of the model reveal that the mixed population originates due to the existence of these bistable switching regulations and that the maintenance of such mixed states depends on the level of stochastic fluctuations of the system. It further demonstrates that due to extrinsic variability, cells in an NSC population can dynamically transit from mushroom to a unique isola kind of bifurcation state, which essentially extends the range of the BMP2-driven mixed population state during differentiation. Importantly, the model predicts that by individually altering the expression level of key regulatory proteins, the NSCs can be converted entirely to a preferred phenotype for BMP2 doses that previously resulted in a mixed population. Our findings show that efficient neuronal regeneration can be achieved by systematically maneuvering the differentiation dynamics.     

Laplacian Dynamics on General Graphs

In previous work, we have introduced a “linear framework” for time-scale separation in biochemical systems, which is based on a labelled, directed graph, G, and an associated linear differential equation, $dx/dt = \mathcal{L}(G)\cdot x$ , where $\mathcal{L}(G)$ is the Laplacian matrix of G. Biochemical nonlinearity is encoded in the graph labels. Many central results in molecular biology can be systematically derived within this framework, including those for enzyme kinetics, allosteric proteins, G-protein coupled receptors, ion channels, gene regulation at thermodynamic equilibrium, and protein post-translational modification. In the present paper, in response to new applications, which accommodate nonequilibrium mechanisms in eukaryotic gene regulation, we lay out the mathematical foundations of the framework. We show that, for any graph and any initial condition, the dynamics always reaches a steady state, which can be algorithmically calculated. If the graph is not strongly connected, which may occur in gene regulation, we show that the dynamics can exhibit flexible behavior that resembles multistability. We further reveal an unexpected equivalence between deterministic Laplacian dynamics and the master equations of continuous-time Markov processes, which allows rigorous treatment within the framework of stochastic, single-molecule mechanisms.     

Dynamical modeling of drug effect using hybrid systems

Drug discovery today is a complex, expensive, and time-consuming process with high attrition rate. A more systematic approach is needed to combine innovative approaches in order to lead to more effective and efficient drug development. This article provides systematic mathematical analysis and dynamical modeling of drug effect under gene regulatory network contexts. A hybrid systems model, which merges together discrete and continuous dynamics into a single dynamical model, is proposed to study dynamics of the underlying regulatory network under drug perturbations. The major goal is to understand how the system changes when perturbed by drugs and give suggestions for better therapeutic interventions. A realistic periodic drug intake scenario is considered, drug pharmacokinetics and pharmacodynamics information being taken into account in the proposed hybrid systems model. Simulations are performed using MATLAB/SIMULINK to corroborate the analytical results.     

The Potential Landscape of Genetic Circuits Imposes the Arrow of Time in Stem Cell Differentiation

Differentiation from a multipotent stem or progenitor state to a mature cell is an essentially irreversible process. The associated changes in gene expression patterns exhibit time-directionality. This “arrow of time” in the collective change of gene expression across multiple stable gene expression patterns (attractors) is not explained by the regulated activation, the suppression of individual genes which are bidirectional molecular processes, or by the standard dynamical models of the underlying gene circuit which only account for local stability of attractors. To capture the global dynamics of this nonequilibrium system and gain insight in the time-asymmetry of state transitions, we computed the quasipotential landscape of the stochastic dynamics of a canonical gene circuit that governs branching cell fate commitment. The potential landscape reveals the global dynamics and permits the calculation of potential barriers between cell phenotypes imposed by the circuit architecture. The generic asymmetry of barrier heights indicates that the transition from the uncommitted multipotent state to differentiated states is inherently unidirectional. The model agrees with observations and predicts the extreme conditions for reprogramming cells back to the undifferentiated state.     

A model of TLR4 signaling and tolerance using a qualitative, particle-event-based method: introduction of spatially configured stochastic reaction chambers (SCSRC)

Introduction

There have been great advances in the examination and characterization of intracellular signaling and synthetic pathways. However, these pathways are generally represented using static diagrams when in reality they exist with considerable dynamic complexity. In addition to the expansion of existing mathematical pathway representation tools (many utilizing systems biology markup language format), there is a growing recognition that spatially explicit modeling methods may be necessary to capture essential aspects of intracellular dynamics. This paper introduces spatially configured stochastic reaction chambers (SCSRC), an agent-based modeling (ABM) framework that incorporates an abstracted molecular ‘event’ rule system with a spatially explicit representation of the relationship between signaling and synthetic compounds. Presented herein is an example of the SCSRC as applied to Toll-like receptor (TLR) 4 signaling and the inflammatory response.

Methods

The underlying rationale for the architecture of the SCSRC is described. A SCSRC model of TLR-4 signaling was developed after a review of the literature regarding TLR-4 signaling and downstream synthetic events. The TLR-4 SCSRC was implemented in the free-ware software platform, Netlogo. A series of in silico experiments were performed to evaluate the response of the TLR-4 SCSRC with respect to response to simulated administration of lipopolysaccharide (LPS). The pro-inflammatory response was represented by simulated secretion of tumor necrosis factor (TNF). Subsequent in silico experiments examined the response to of the TLR-4 SCSRC in terms of a simulated preconditioning effect represented as tolerance of pro-inflammatory signaling to a second dose of LPS.

Results

The SCSRC produces simulated dynamics of TLR-4 signaling in response to LPS stimulation that are qualitatively similar to that reported in the literature. The expression of various components of the signaling cascade demonstrated stochastic noise, consistent with molecular expression data reported in the literature. There is a dose dependent pro-inflammatory response effect seen with increasing initial doses of LPS, and there was also a dose dependent response with respect to preconditioning effect and the establishment of tolerance. Both of these dynamics are consistent with published responses to LPS.

Conclusions

The particle-based, spatially oriented SCSRC model of TLR-4 signaling captures the essential dynamics of the TLR-4 signal transduction cascade, including stochastic signal behavior, dose dependent response, negative feedback control, and preconditioning effect. This is accomplished even given a high degree of molecular event abstraction. The component detail of the SCSRC may allow for sequential parsing of various preconditioning effects, something not possible without computational modeling and simulation, and may give insight into the expected consequences and responses resulting from manipulation of one or many of these modulating factors. The SCSRC is admittedly a work in evolution, and future work will sequentially incorporate additional regulatory mechanisms, both intracellular and paracrine/autocrine, and improved mapping between the spatial chamber configuration and molecular event rules, and experimentally define biochemical reaction rate constants. However, the SCSRC has promise as a highly modular and flexible modeling method that is suited to the dynamic knowledge representation of intracellular processes.     

The molecular and mathematical basis of Waddington's epigenetic landscape: A framework for post‐Darwinian biology?

The Neo-Darwinian concept of natural selection is plausible when one assumes a straightforward causation of phenotype by genotype. However, such simple 1:1 mapping must now give place to the modern concepts of gene regulatory networks and gene expression noise. Both can, in the absence of genetic mutations, jointly generate a diversity of inheritable randomly occupied phenotypic states that could also serve as a substrate for natural selection. This form of epigenetic dynamics challenges Neo-Darwinism. It needs to incorporate the non-linear, stochastic dynamics of gene networks. A first step is to consider the mathematical correspondence between gene regulatory networks and Waddington's metaphoric 'epigenetic landscape', which actually represents the quasi-potential function of global network dynamics. It explains the coexistence of multiple stable phenotypes within one genotype. The landscape's topography with its attractors is shaped by evolution through mutational re-wiring of regulatory interactions - offering a link between genetic mutation and sudden, broad evolutionary changes.     

Parameter estimation in stochastic biochemical reactions

Gene regulatory, signal transduction and metabolic networks are major areas of interest in the newly emerging field of systems biology. In living cells, stochastic dynamics play an important role; however, the kinetic parameters of biochemical reactions necessary for modelling these processes are often not accessible directly through experiments. The problem of estimating stochastic reaction constants from molecule count data measured, with error, at discrete time points is considered. For modelling the system, a hidden Markov process is used, where the hidden states are the true molecule counts, and the transitions between those states correspond to reaction events following collisions of molecules. Two different algorithms are proposed for estimating the unknown model parameters. The first is an approximate maximum likelihood method that gives good estimates of the reaction parameters in systems with few possible reactions in each sampling interval. The second algorithm, treating the data as exact measurements, approximates the number of reactions in each sampling interval by solving a simple linear equation. Maximising the likelihood based on these approximations can provide good results, even in complex reaction systems.