Multi-modality in gene regulatory networks with slow promoter kinetics

High-throughput functional characterization of combinations of transcriptional activators and repressors

Despite growing knowledge of the functions of individual human transcriptional effector domains, much less is understood about how multiple effector domains within the same protein combine to regulate gene expression. Here, we measure transcriptional activity for 8,400 effector domain combinations by recruiting them to reporter genes in human cells. In our assay, weak and moderate activation domains synergize to drive strong gene expression, whereas combining strong activators often results in weaker activation. In contrast, repressors combine linearly and produce full gene silencing, and repressor domains often overpower activation domains. We use this information to build a synthetic transcription factor whose function can be tuned between repression and activation independent of recruitment to target genes by using a small-molecule drug. Altogether, we outline the basic principles of how effector domains combine to regulate gene expression and demonstrate their value in building precise and flexible synthetic biology tools. A record of this paper's transparent peer review process is included in the supplemental information.

Constrained Langevin approximation for the Togashi-Kaneko model of autocatalytic reactions

The Togashi Kaneko model (TK model) is a simple stochastic reaction network that displays discreteness-induced transitions between meta-stable patterns. Here we study a constrained Langevin approximation (CLA) of this model. This CLA, derived under the classical scaling, is an obliquely reflected diffusion process on the positive orthant and hence respects the constraint that chemical concentrations are never negative. We show that the CLA is a Feller process, is positive Harris recurrent and converges exponentially fast to the unique stationary distribution. We also characterize the stationary distribution and show that it has finite moments. In addition, we simulate both the TK model and its CLA in various dimensions. For example, we describe how the TK model switches between meta-stable patterns in dimension six. Our simulations suggest that, when the volume of the vessel in which all of the reactions that take place is large, the CLA is a good approximation of the TK model in terms of both the stationary distribution and the transition times between patterns.

Stochastic models of nucleosome dynamics reveal regulatory rules of stimulus-induced epigenome remodeling

The genomic positions of nucleosomes are a defining feature of the cell's epigenomic state, but signal-dependent transcription factors (SDTFs), upon activation, bind to specific genomic locations and modify nucleosome positioning. Here we leverage SDTFs as perturbation probes to learn about nucleosome dynamics in living cells. We develop Markov models of nucleosome dynamics and fit them to time course sequencing data of DNA accessibility. We find that (1) the dynamics of DNA unwrapping are significantly slower in cells than reported from cell-free experiments, (2) only models with cooperativity in wrapping and unwrapping fit the available data, (3) SDTF activity produces the highest eviction probability when its binding site is adjacent to but not on the nucleosome dyad, and (4) oscillatory SDTF activity results in high location variability. Our work uncovers the regulatory rules governing SDTF-induced nucleosome dynamics in live cells, which can predict chromatin accessibility alterations during inflammation at single-nucleosome resolution.

A robust method for designing multistable systems by embedding bistable subsystems

Although multistability is an important dynamic property of a wide range of complex systems, it is still a challenge to develop mathematical models for realising high order multistability using realistic regulatory mechanisms. To address this issue, we propose a robust method to develop multistable mathematical models by embedding bistable models together. Using the GATA1-GATA2-PU.1 module in hematopoiesis as the test system, we first develop a tristable model based on two bistable models without any high cooperative coefficients, and then modify the tristable model based on experimentally determined mechanisms. The modified model successfully realises four stable steady states and accurately reflects a recent experimental observation showing four transcriptional states. In addition, we develop a stochastic model, and stochastic simulations successfully realise the experimental observations in single cells. These results suggest that the proposed method is a general approach to develop mathematical models for realising multistability and heterogeneity in complex systems.

Competitive dCas9 binding as a mechanism for transcriptional control

Catalytically dead Cas9 (dCas9) is a programmable transcription factor that can be targeted to promoters through the design of small guide RNAs (sgRNAs), where it can function as an activator or repressor. Natural promoters use overlapping binding sites as a mechanism for signal integration, where the binding of one can block, displace, or augment the activity of the other. Here, we implemented this strategy in Escherichia coli using pairs of sgRNAs designed to repress and then derepress transcription through competitive binding. When designed to target a promoter, this led to 27-fold repression and complete derepression. This system was also capable of ratiometric input comparison over two orders of magnitude. Additionally, we used this mechanism for promoter sequence-independent control by adopting it for elongation control, achieving 8-fold repression and 4-fold derepression. This work demonstrates a new genetic control mechanism that could be used to build analog circuit or implement cis-regulatory logic on CRISPRi-targeted native genes.

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.

Derivation of stationary distributions of biochemical reaction networks via structure transformation

Long-term behaviors of biochemical reaction networks (BRNs) are described by steady states in deterministic models and stationary distributions in stochastic models. Unlike deterministic steady states, stationary distributions capturing inherent fluctuations of reactions are extremely difficult to derive analytically due to the curse of dimensionality. Here, we develop a method to derive analytic stationary distributions from deterministic steady states by transforming BRNs to have a special dynamic property, called complex balancing. Specifically, we merge nodes and edges of BRNs to match in- and out-flows of each node. This allows us to derive the stationary distributions of a large class of BRNs, including autophosphorylation networks of EGFR, PAK1, and Aurora B kinase and a genetic toggle switch. This reveals the unique properties of their stochastic dynamics such as robustness, sensitivity, and multi-modality. Importantly, we provide a user-friendly computational package, CASTANET, that automatically derives symbolic expressions of the stationary distributions of BRNs to understand their long-term stochasticity.

A mathematical model exhibiting the effect of DNA methylation on the stability boundary in cell-fate networks

Cell-fate networks are traditionally studied within the framework of gene regulatory networks. This paradigm considers only interactions of genes through expressed transcription factors and does not incorporate chromatin modification processes. This paper introduces a mathematical model that seamlessly combines gene regulatory networks and DNA methylation (DNAm), with the goal of quantitatively characterizing the contribution of epigenetic regulation to gene silencing. The 'Basin of Attraction percentage' is introduced as a metric to quantify gene silencing abilities. As a case study, a computational and theoretical analysis is carried out for a model of the pluripotent stem cell circuit as well as a simplified self-activating gene model. The results confirm that the methodology quantitatively captures the key role that DNAm plays in enhancing the stability of the silenced gene state.

Slack reactants: A state-space truncation framework to estimate quantitative behavior of the chemical master equation

State space truncation methods are widely used to approximate solutions of the chemical master equation. While most methods of this kind focus on truncating the state space directly, in this work, we propose modifying the underlying chemical reaction network by introducing slack reactants that indirectly truncate the state space. More specifically, slack reactants introduce an expanded chemical reaction network and impose a truncation scheme based on desired mass conservation laws. This network structure also allows us to prove inheritance of special properties of the original model, such as irreducibility and complex balancing. We use the network structure imposed by slack reactants to prove the convergence of the stationary distribution and first arrival times. We then provide examples comparing our method with the stationary finite state projection and finite buffer methods. Our slack reactant system appears to be more robust than some competing methods with respect to calculating first arrival times.

Stationary distributions of systems with discreteness-induced transitions

We provide a theoretical analysis of some autocatalytic reaction networks exhibiting the phenomenon of discreteness-induced transitions. The family of networks that we address includes the celebrated Togashi and Kaneko model. We prove positive recurrence, finiteness of all moments and geometric ergodicity of the models in the family. For some parameter values, we find the analytic expression for the stationary distribution and discuss the effect of volume scaling on the stationary behaviour of the chain. We find the exact critical value of the volume for which discreteness-induced transitions disappear.

Analysis of Single-Cell Gene Pair Coexpression Landscapes by Stochastic Kinetic Modeling Reveals Gene-Pair Interactions in Development

Single-cell transcriptomics is advancing discovery of the molecular determinants of cell identity, while spurring development of novel data analysis methods. Stochastic mathematical models of gene regulatory networks help unravel the dynamic, molecular mechanisms underlying cell-to-cell heterogeneity, and can thus aid interpretation of heterogeneous cell-states revealed by single-cell measurements. However, integrating stochastic gene network models with single cell data is challenging. Here, we present a method for analyzing single-cell gene-pair coexpression patterns, based on biophysical models of stochastic gene expression and interaction dynamics. We first developed a high-computational-throughput approach to stochastic modeling of gene-pair coexpression landscapes, based on numerical solution of gene network Master Equations. We then comprehensively catalogued coexpression patterns arising from tens of thousands of gene-gene interaction models with different biochemical kinetic parameters and regulatory interactions. From the computed landscapes, we obtain a low-dimensional "shape-space" describing distinct types of coexpression patterns. We applied the theoretical results to analysis of published single cell RNA sequencing data and uncovered complex dynamics of coexpression among gene pairs during embryonic development. Our approach provides a generalizable framework for inferring evolution of gene-gene interactions during critical cell-state transitions.

The Physics of Cellular Decision Making During Epithelial-Mesenchymal Transition

The epithelial-mesenchymal transition (EMT) is a process by which cells lose epithelial traits, such as cell-cell adhesion and apico-basal polarity, and acquire migratory and invasive traits. EMT is crucial to embryonic development and wound healing. Misregulated EMT has been implicated in processes associated with cancer aggressiveness, including metastasis. Recent experimental advances such as single-cell analysis and temporal phenotypic characterization have established that EMT is a multistable process wherein cells exhibit and switch among multiple phenotypic states. This is in contrast to the classical perception of EMT as leading to a binary choice. Mathematical modeling has been at the forefront of this transformation for the field, not only providing a conceptual framework to integrate and analyze experimental data, but also making testable predictions. In this article, we review the key features and characteristics of EMT dynamics, with a focus on the mathematical modeling approaches that have been instrumental to obtaining various useful insights.

Intracellular Energy Variability Modulates Cellular Decision-Making Capacity

Cells generate phenotypic diversity both during development and in response to stressful and changing environments, aiding survival. Functionally vital cell fate decisions from a range of phenotypic choices are made by regulatory networks, the dynamics of which rely on gene expression and hence depend on the cellular energy budget (and particularly ATP levels). However, despite pronounced cell-to-cell ATP differences observed across biological systems, the influence of energy availability on regulatory network dynamics is often overlooked as a cellular decision-making modulator, limiting our knowledge of how energy budgets affect cell behaviour. Here, we consider a mathematical model of a highly generalisable, ATP-dependent, decision-making regulatory network, and show that cell-to-cell ATP variability changes the sets of decisions a cell can make. Our model shows that increasing intracellular energy levels can increase the number of supported stable phenotypes, corresponding to increased decision-making capacity. Model cells with sub-threshold intracellular energy are limited to a singular phenotype, forcing the adoption of a specific cell fate. We suggest that energetic differences between cells may be an important consideration to help explain observed variability in cellular decision-making across biological systems.

Stochastic analysis of genetic feedback controllers to reprogram a pluripotency gene regulatory network

Cellular reprogramming is traditionally accomplished through an open loop (OL) control approach, wherein key transcription factors (TFs) are injected in cells to steer the state of the pluripotency (PL) gene regulatory network (GRN), as encoded by TFs concentrations, to the pluripotent state. Due to the OL nature of this approach, the concentration of TFs cannot be accurately controlled. Recently, a closed loop (CL) feedback control strategy was proposed to overcome this problem with promising theoretical results. However, previous analyses of the controller were based on deterministic models. It is well known that cellular systems are characterized by substantial stochasticity, especially when molecules are in low copy number as it is the case in reprogramming problems wherein the gene copy number is usually one or two. Hence, in this paper, we analyze the Chemical Master Equation (CME) for the reaction model of the PL GRN with and without the feedback controller. We computationally and analytically investigate the performance of the controller in biologically relevant parameter regimes where stochastic effects dictate system dynamics. Our results indicate that the feedback control approach still ensures reprogramming even when both the PL GRN and the controller are stochastic.

Stochastic multistationarity in a model of the hematopoietic stem cell differentiation network

A central issue in the analysis of multi-stable systems is that of controlling the relative size of the basins of attraction of alternative states through suitable choices of system parameters. We are interested here mainly in the stochastic version of this problem, that of shaping the stationary probability distribution of a Markov chain so that various alternative modes become more likely than others. Although many of our results are more general, we were motivated by an important biological question, that of cell differentiation. In the mathematical modeling of cell differentiation, it is common to think of internal states of cells (quanfitied by activation levels of certain genes) as determining the different cell types. Specifically, we study here the "PU.1/GATA-1 circuit" which is involved in the control of the development of mature blood cells from hematopoietic stem cells (HSCs). All mature, specialized blood cells have been shown to be derived from multipotent HSCs. Our first contribution is to introduce a rigorous chemical reaction network model of the PU.1/GATA-1 circuit, which incorporates current biological knowledge. We then find that the resulting ODE model of these biomolecular reactions is incapable of exhibiting multistability, contradicting the fact that differentiation networks have, by definition, alternative stable steady states. When considering instead the stochastic version of this chemical network, we analytically construct the stationary distribution, and are able to show that this distribution is indeed capable of admitting a multiplicity of modes. Finally, we study how a judicious choice of system parameters serves to bias the probabilities towards different stationary states. We remark that certain changes in system parameters can be physically implemented by a biological feedback mechanism; tuning this feedback gives extra degrees of freedom that allow one to assign higher likelihood to some cell types over others.