External noise and feedback regulation: steady-state statistics of auto-regulatory genetic network

Quantifying the Stability of Coupled Genetic and Epigenetic Switches With Variational Methods

The Waddington landscape provides an intuitive metaphor to view development as a ball rolling down the hill, with distinct phenotypes as basins and differentiation pathways as valleys. Since, at a molecular level, cell differentiation arises from interactions among the genes, a mathematical definition for the Waddington landscape can, in principle, be obtained by studying the gene regulatory networks. For eukaryotes, gene regulation is inextricably and intimately linked to histone modifications. However, the impact of such modifications on both landscape topography and stability of attractor states is not fully understood. In this work, we introduced a minimal kinetic model for gene regulation that combines the impact of both histone modifications and transcription factors. We further developed an approximation scheme based on variational principles to solve the corresponding master equation in a second quantized framework. By analyzing the steady-state solutions at various parameter regimes, we found that histone modification kinetics can significantly alter the behavior of a genetic network, resulting in qualitative changes in gene expression profiles. The emerging epigenetic landscape captures the delicate interplay between transcription factors and histone modifications in driving cell-fate decisions.

Moment dynamics for gene regulation with rational rate laws

This aim of this paper is mainly to investigate the performance of two typical moment closure schemes in gene regulatory master equations of rational rate laws. When the reaction rate is polynomial, the error bounds between the authentic and approximate moments obtained by schemes of Gaussian moment closure and log-normal moment closure are explicitly given. When the reaction rate is not polynomial, it is shown that the two schemes both behave well in the absence of active-inactive state switch, but in the presence of active-inactive state switch the log-normal closure scheme is far superior to the Gaussian closure scheme in capturing the asymptotic ensemble statistics. Moreover, the accuracy of the log-normal closure method is further confirmed by steady-state analytic results and the conditional Gaussian closure method. It is also disclosed that optimal negative feedback exists in suppressing protein noise in the presence of the on-off switch control.

Transitions in the cell-fate induction induced by colored noise associated with the inductive stimulus

The cell-fate induction based on the saddle-node bifurcation is undoubtedly a very important concept in developmental biology, which provides a possible mechanism to explain the intrinsic irreversibility in the developmental process. In this paper, the effect of a colored noise, which is associated with the inductive stimulus, on the saddle-node landscape of cell-fate induction is investigated, especially, the effect of the change of correlation time of colored noise on cell-fate induction. The main results show clearly that the change of correlation time of colored noise could induce the transitions of the system. This implies that the colored noise associated with inductive stimulus may have a profound effect on the saddle-node bifurcation landscape of cell-fate induction. This will also help us to understand more deeply the role of cell-fate induction in developmental biology.

Emergent dynamics of coordinated cells with time delays in a tissue

In this article, we investigate the emergence of tissue dynamics with time delays of diffusion. Such emergent dynamics, describing the tissue homeostasis, usually correspond to particular tissue functions, which are attracting a tremendous amount of attention from both communities of mathematical modeling and systems biology. Specifically, in addition to the within-cell genome dynamics and the diffusion among the cells, we consider several types of time delays of diffusion present in the coordinated cells. We establish several generalized versions of the "monotonicity condition" (MC), whose traditional version [I. Rajapakse and S. Smale, Proc. Natl. Acad. Sci. U.S.A. 114, 1462-1467 (2017)] guaranteed the stability of the equilibrium in a system of coordinated cells without time delay. Indeed, we find that one generalized MC we establish still guarantees the stability of the time-delayed system's equilibrium, which corresponds to a formation of tissue functions depending primarily on individual genome dynamics but less on interacting structures and time delays of diffusion. We also find that, when the generalized MC is further relaxed, the system is able to sustain periodic oscillations, whose periods are verified to have delicate dependence with the selected time delays. These produced oscillations usually represent realistic behaviors of "alive" cells. We use several representative examples to demonstrate the usefulness of the established analytical conditions to the understanding of the emergent dynamics observed in computational models and in real systems as well.

Population density approach for discrete mRNA distributions in generalized switching models for stochastic gene expression

We present a generalization of a population density approach for modeling and analysis of stochastic gene expression. In the model, the gene of interest fluctuates stochastically between an inactive state, in which transcription cannot occur, and an active state, in which discrete transcription events occur; and the individual mRNA molecules are degraded stochastically in an independent manner. This sort of model in simplest form with exponential dwell times has been used to explain experimental estimates of the discrete distribution of random mRNA copy number. In our generalization, the random dwell times in the inactive and active states, T_{0} and T_{1}, respectively, are independent random variables drawn from any specified distributions. Consequently, the probability per unit time of switching out of a state depends on the time since entering that state. Our method exploits a connection between the fully discrete random process and a related continuous process. We present numerical methods for computing steady-state mRNA distributions and an analytical derivation of the mRNA autocovariance function. We find that empirical estimates of the steady-state mRNA probability mass function from Monte Carlo simulations of laboratory data do not allow one to distinguish between underlying models with exponential and nonexponential dwell times in some relevant parameter regimes. However, in these parameter regimes and where the autocovariance function has negative lobes, the autocovariance function disambiguates the two types of models. Our results strongly suggest that temporal data beyond the autocovariance function is required in general to characterize gene switching.

Distinct noise-controlling roles of multiple negative feedback mechanisms in a prokaryotic operon system

Molecular fluctuations are known to affect dynamics of cellular systems in important ways. Studies aimed at understanding how molecular systems of certain regulatory architectures control noise therefore become essential. The interplay between feedback regulation and noise has been previously explored for cellular networks governed by a single negative feedback loop. However, similar issues within networks consisting of more complex regulatory structures remain elusive. The authors investigate how negative feedback loops manage noise within a biochemical cascade concurrently governed by multiple negative feedback loops, using the prokaryotic tryptophan (trp) operon system in Escherechia coli as the model system. To the authors knowledge, this is the first study of noise in the trp operon system. They show that the loops in the trp operon system possess distinct, even opposing, noise-controlling effects despite their seemingly analogous feedback structures. The enzyme inhibition loop, although controlling the last reaction of the cascade, was found to suppress noise not only for the tryptophan output but also for other upstream components. In contrast, the Repression (Rep) loop enhances noise for all systems components. Attenuation (Att) poses intermediate effects by attenuating noise for the upstream components but promoting noise for components downstream of its target. Regarding noise at the output tryptophan, Rep and Att can be categorised as noise-enhancing loops whereas Enzyme Inhibition as a noise-reducing loop. These findings suggest novel implications in how cellular systems with multiple feedback mechanisms control noise. [Includes supplementary material].

Global robust power-rate stability of delayed genetic regulatory networks with noise perturbations

In this paper, by using the Lyapunov method, Itô's differential formula and linear matrix inequality (LMI) approach, the global robust power-rate stability in mean square is discussed for genetic regulatory networks with unbounded time-varying delay, noise perturbations and parameter uncertainties. Sufficient conditions are given to ensure the robust power-rate stability (in mean square) of the genetic regulatory networks. Meanwhile, the criteria ensuring global power-rate stability in mean square are a byproduct of the criteria guaranteeing global robust power-rate stability in mean square. The obtained conditions are derived in terms of linear matrix inequalities (LMIs) which are easy to be verified via the LMI toolbox. An illustrative example is given to show the effectiveness of the obtained result.

Propagation of extrinsic perturbation in a negatively auto-regulated pathway

The apparent precision of the output in multi-step biochemical pathways in the face of external and intrinsic perturbations is non-obvious and conceptually difficult. Using a simple three-step negatively auto-regulated model pathway, we show that the effect of perturbation at different steps of the pathway and its transmission through the network is dependent on the context (i.e., the position) of the particular reaction step in relation to the topology of the regulatory network, stoichiometry of reactions, type of nonlinearity involved in the reactions and also on the intrinsic dynamical state of the pathway variables. We delineate the qualitative and quantitative changes in the pathway dynamics for constant ('bias') and random external perturbations acting on the pathway steps locally or globally to all steps. We show that constant perturbation induces qualitative change in dynamics, whereas random fluctuations cause significant quantitative variations in the concentrations of the different variables. Thus, the dynamic response of multi-step biochemical pathways to external perturbation depends on their biochemical, topological and dynamical features.