Exact Probability Landscapes of Stochastic Phenotype Switching in Feed-Forward Loops: Phase Diagrams of Multimodality.

Feed-forward loops (FFLs) are among the most ubiquitously found motifs of reaction networks in nature. However, little is known about their stochastic behavior and the variety of network phenotypes they can exhibit. In this study, we provide full characterizations of the properties of stochastic multimodality of FFLs, and how switching between different network phenotypes are controlled. We have computed the exact steady-state probability landscapes of all eight types of coherent and incoherent FFLs using the finite-butter Accurate Chemical Master Equation (ACME) algorithm, and quantified the exact topological features of their high-dimensional probability landscapes using persistent homology. Through analysis of the degree of multimodality for each of a set of 10,812 probability landscapes, where each landscape resides over 105-106 microstates, we have constructed comprehensive phase diagrams of all relevant behavior of FFL multimodality over broad ranges of input and regulation intensities, as well as different regimes of promoter binding dynamics. In addition, we have quantified the topological sensitivity of the multimodality of the landscapes to regulation intensities. Our results show that with slow binding and unbinding dynamics of transcription factor to promoter, FFLs exhibit strong stochastic behavior that is very different from what would be inferred from deterministic models. In addition, input intensity play major roles in the phenotypes of FFLs: At weak input intensity, FFL exhibit monomodality, but strong input intensity may result in up to 6 stable phenotypes. Furthermore, we found that gene duplication can enlarge stable regions of specific multimodalities and enrich the phenotypic diversity of FFL networks, providing means for cells toward better adaptation to changing environment. Our results are directly applicable to analysis of behavior of FFLs in biological processes such as stem cell differentiation and for design of synthetic networks when certain phenotypic behavior is desired.

Discrete and continuous models of probability flux of switching dynamics: Uncovering stochastic oscillations in a toggle-switch system.

The probability flux and velocity in stochastic reaction networks can help in characterizing dynamic changes in probability landscapes of these networks. Here, we study the behavior of three different models of probability flux, namely, the discrete flux model, the Fokker-Planck model, and a new continuum model of the Liouville flux. We compare these fluxes that are formulated based on, respectively, the chemical master equation, the stochastic differential equation, and the ordinary differential equation. We examine similarities and differences among these models at the nonequilibrium steady state for the toggle switch network under different binding and unbinding conditions. Our results show that at a strong stochastic condition of weak promoter binding, continuum models of Fokker-Planck and Liouville fluxes deviate significantly from the discrete flux model. Furthermore, we report the discovery of stochastic oscillation in the toggle-switch system occurring at weak binding conditions, a phenomenon captured only by the discrete flux model.

Sensitivities of Regulation Intensities in Feed-Forward Loops with Multistability.

Gene regulatory networks depict the interactions among genes, proteins, and other components of the cell. These interactions are stochastic when large differences in reaction rates and small copy number of molecules are involved. Discrete Chemical Master Equation (dCME) provides a general framework for understanding the stochastic nature of these networks. Here we used the Accurate Chemical Master Equation method to directly compute the exact steady state probability landscape of the feed-forward loop motif (FFL). FFL is one of the most abundant gene regulatory networks motifs where the regulation is carried out from the top nodes to the bottom ones. We examine the behavior of stochastic FFLs under different conditions of various regulation intensities. Under the conditions with slow promoter binding, we show how FFL can exhibit different multistabilities in their landscapes. We also study the sensitivities of regulations of FFLs and introduce a new definition of stochastic sensitivity to characterize how FFLs respond in their probability distributions at the steady state to perturbations of system parameters. We show how change in gene expression under FFL regulations are sensitive to system parameters, including the state of multistability in FFLs.

Discrete flux and velocity fields of probability and their global maps in reaction systems.

Stochasticity plays important roles in reaction systems. Vector fields of probability flux and velocity characterize time-varying and steady-state properties of these systems, including high probability paths, barriers, checkpoints among different stable regions, as well as mechanisms of dynamic switching among them. However, conventional fluxes on continuous space are ill-defined and are problematic when at the boundaries of the state space or when copy numbers are small. By re-defining the derivative and divergence operators based on the discrete nature of reactions, we introduce new formulations of discrete fluxes. Our flux model fully accounts for the discreetness of both the state space and the jump processes of reactions. The reactional discrete flux satisfies the continuity equation and describes the behavior of the system evolving along directions of reactions. The species discrete flux directly describes the dynamic behavior in the state space of the reactants such as the transfer of probability mass. With the relationship between these two fluxes specified, we show how to construct time-evolving and steady-state global flow-maps of probability flux and velocity in the directions of every species at every microstate and how they are related to the outflow and inflow of probability fluxes when tracing out reaction trajectories. We also describe how to impose proper conditions enabling exact quantification of flux and velocity in the boundary regions, without the difficulty of enforcing artificial reflecting conditions. We illustrate the computation of probability flux and velocity using three model systems, namely, the birth-death process, the bistable Schlögl model, and the oscillating Schnakenberg model.

PRODIGEN: visualizing the probability landscape of stochastic gene regulatory networks in state and time space.

BACKGROUND: Visualizing the complex probability landscape of stochastic gene regulatory networks can further biologists' understanding of phenotypic behavior associated with specific genes. RESULTS: We present PRODIGEN (PRObability DIstribution of GEne Networks), a web-based visual analysis tool for the systematic exploration of probability distributions over simulation time and state space in such networks. PRODIGEN was designed in collaboration with bioinformaticians who research stochastic gene networks. The analysis tool combines in a novel way existing, expanded, and new visual encodings to capture the time-varying characteristics of probability distributions: spaghetti plots over one dimensional projection, heatmaps of distributions over 2D projections, enhanced with overlaid time curves to display temporal changes, and novel individual glyphs of state information corresponding to particular peaks. CONCLUSIONS: We demonstrate the effectiveness of the tool through two case studies on the computed probabilistic landscape of a gene regulatory network and of a toggle-switch network. Domain expert feedback indicates that our visual approach can help biologists: 1) visualize probabilities of stable states, 2) explore the temporal probability distributions, and 3) discover small peaks in the probability landscape that have potential relation to specific diseases.

ACCURATE CHEMICAL MASTER EQUATION SOLUTION USING MULTI-FINITE BUFFERS.

The discrete chemical master equation (dCME) provides a fundamental framework for studying stochasticity in mesoscopic networks. Because of the multi-scale nature of many networks where reaction rates have large disparity, directly solving dCMEs is intractable due to the exploding size of the state space. It is important to truncate the state space effectively with quantified errors, so accurate solutions can be computed. It is also important to know if all major probabilistic peaks have been computed. Here we introduce the Accurate CME (ACME) algorithm for obtaining direct solutions to dCMEs. With multi-finite buffers for reducing the state space by O(n!), exact steady-state and time-evolving network probability landscapes can be computed. We further describe a theoretical framework of aggregating microstates into a smaller number of macrostates by decomposing a network into independent aggregated birth and death processes, and give an a priori method for rapidly determining steady-state truncation errors. The maximal sizes of the finite buffers for a given error tolerance can also be pre-computed without costly trial solutions of dCMEs. We show exactly computed probability landscapes of three multi-scale networks, namely, a 6-node toggle switch, 11-node phage-lambda epigenetic circuit, and 16-node MAPK cascade network, the latter two with no known solutions. We also show how probabilities of rare events can be computed from first-passage times, another class of unsolved problems challenging for simulation-based techniques due to large separations in time scales. Overall, the ACME method enables accurate and efficient solutions of the dCME for a large class of networks.

State Space Truncation with Quantified Errors for Accurate Solutions to Discrete Chemical Master Equation.

The discrete chemical master equation (dCME) provides a general framework for studying stochasticity in mesoscopic reaction networks. Since its direct solution rapidly becomes intractable due to the increasing size of the state space, truncation of the state space is necessary for solving most dCMEs. It is therefore important to assess the consequences of state space truncations so errors can be quantified and minimized. Here we describe a novel method for state space truncation. By partitioning a reaction network into multiple molecular equivalence groups (MEGs), we truncate the state space by limiting the total molecular copy numbers in each MEG. We further describe a theoretical framework for analysis of the truncation error in the steady-state probability landscape using reflecting boundaries. By aggregating the state space based on the usage of a MEG and constructing an aggregated Markov process, we show that the truncation error of a MEG can be asymptotically bounded by the probability of states on the reflecting boundary of the MEG. Furthermore, truncating states of an arbitrary MEG will not undermine the estimated error of truncating any other MEGs. We then provide an overall error estimate for networks with multiple MEGs. To rapidly determine the appropriate size of an arbitrary MEG, we also introduce an a priori method to estimate the upper bound of its truncation error. This a priori estimate can be rapidly computed from reaction rates of the network, without the need of costly trial solutions of the dCME. As examples, we show results of applying our methods to the four stochastic networks of (1) the birth and death model, (2) the single gene expression model, (3) the genetic toggle switch model, and (4) the phage lambda bistable epigenetic switch model. We demonstrate how truncation errors and steady-state probability landscapes can be computed using different sizes of the MEG(s) and how the results validate our theories. Overall, the novel state space truncation and error analysis methods developed here can be used to ensure accurate direct solutions to the dCME for a large number of stochastic networks.

Mechanisms of stochastic focusing and defocusing in biological reaction networks: insight from accurate chemical master equation (ACME) solutions.

Stochasticity plays important roles in regulation of biochemical reaction networks when the copy numbers of molecular species are small. Studies based on Stochastic Simulation Algorithm (SSA) has shown that a basic reaction system can display stochastic focusing (SF) by increasing the sensitivity of the network as a result of the signal noise. Although SSA has been widely used to study stochastic networks, it is ineffective in examining rare events and this becomes a significant issue when the tails of probability distributions are relevant as is the case of SF. Here we use the ACME method to solve the exact solution of the discrete Chemical Master Equations and to study a network where SF was reported. We showed that the level of SF depends on the degree of the fluctuations of signal molecule. We discovered that signaling noise under certain conditions in the same reaction network can lead to a decrease in the system sensitivities, thus the network can experience stochastic defocusing. These results highlight the fundamental role of stochasticity in biological reaction networks and the need for exact computation of probability landscape of the molecules in the system.

Multiscale Modeling of Cellular Epigenetic States: Stochasticity in Molecular Networks, Chromatin Folding in Cell Nuclei, and Tissue Pattern Formation of Cells.

Genome sequences provide the overall genetic blueprint of cells, but cells possessing the same genome can exhibit diverse phenotypes. There is a multitude of mechanisms controlling cellular epigenetic states and that dictate the behavior of cells. Among these, networks of interacting molecules, often under stochastic control, depending on the specific wirings of molecular components and the physiological conditions, can have a different landscape of cellular states. In addition, chromosome folding in three-dimensional space provides another important control mechanism for selective activation and repression of gene expression. Fully differentiated cells with different properties grow, divide, and interact through mechanical forces and communicate through signal transduction, resulting in the formation of complex tissue patterns. Developing quantitative models to study these multi-scale phenomena and to identify opportunities for improving human health requires development of theoretical models, algorithms, and computational tools. Here we review recent progress made in these important directions.

Exact computation of probability landscape of stochastic networks of Single Input and Coupled Toggle Switch Modules.

Gene regulatory networks depict the interactions between genes, proteins, and other components of the cell. These interactions often are stochastic that can influence behavior of the cells. Discrete Chemical Master Equation (dCME) provides a general framework for understanding the stochastic nature of these networks. However solving dCME is challenging due to the enormous state space, one effective approach is to study the behavior of individual modules of the stochastic network. Here we used the finite buffer dCME method and directly calculated the exact steady state probability landscape for the two stochastic networks of Single Input and Coupled Toggle Switch Modules. The first example is a switch network consisting of three genes, and the second example is a double switching network consisting of four coupled genes. Our results show complex switching behavior of these networks can be quantified.