Effects of the DNA state fluctuation on single-cell dynamics of self-regulating gene

Stochastic epigenetic dynamics of gene switching

Epigenetic modifications of histones crucially affect eukaryotic gene activity, while the epigenetic histone state is largely determined by the binding of specific factors such as the transcription factors (TFs) to DNA. Here, the way in which the TFs and the histone state are dynamically correlated is not obvious when the TF synthesis is regulated by the histone state. This type of feedback regulatory relation is ubiquitous in gene networks to determine cell fate in differentiation and other cell transformations. To gain insights into such dynamical feedback regulations, we theoretically analyze a model of epigenetic gene switching by extending the Doi-Peliti operator formalism of reaction kinetics to the problem of coupled molecular processes. Spin-1 and spin-1/2 coherent-state representations are introduced to describe stochastic reactions of histones and binding or unbinding of TFs in a unified way, which provides a concise view of the effects of timescale difference among these molecular processes; even in the case that binding or unbinding of TFs to or from DNA is adiabatically fast, the slow nonadiabatic histone dynamics gives rise to a distinct circular flow of the probability flux around basins in the landscape of the gene state distribution, which leads to hysteresis in gene switching. In contrast to the general belief that the change in the amount of TF precedes the histone state change, flux drives histones to be modified prior to the change in the amount of TF in self-regulating circuits. Flux-landscape analyses shed light on the nonlinear nonadiabatic mechanism of epigenetic cell fate decision making.

Effects of Collective Histone State Dynamics on Epigenetic Landscape and Kinetics of Cell Reprogramming

Cell reprogramming is a process of transitions from differentiated to pluripotent cell states via transient intermediate states. Within the epigenetic landscape framework, such a process is regarded as a sequence of transitions among basins on the landscape; therefore, theoretical construction of a model landscape which exhibits experimentally consistent dynamics can provide clues to understanding epigenetic mechanism of reprogramming. We propose a minimal gene-network model of the landscape, in which each gene is regulated by an integrated mechanism of transcription-factor binding/unbinding and the collective chemical modification of histones. We show that the slow collective variation of many histones around each gene locus alters topology of the landscape and significantly affects transition dynamics between basins. Differentiation and reprogramming follow different transition pathways on the calculated landscape, which should be verified experimentally via single-cell pursuit of the reprogramming process. Effects of modulation in collective histone state kinetics on transition dynamics and pathway are examined in search for an efficient protocol of reprogramming.

The fluctuation-dissipation theorem for stochastic kinetics--implications on genetic regulations

The Fluctuation-Dissipation theorem (FDT) connects the "memory" in the fluctuation in equilibrium to the response of a system after a perturbation, which has been a fundamental ground in many branches of physics. When viewing a cell as a stochastic biochemical system, the cell's response under a perturbation is related to its intrinsic steady-state correlation functions via the FDT, a theorem we derived and present in this work. FDT allows us to use the noise to derive dynamic response and infer dynamic properties in the system. We tested FDT's validity with gene regulation models and found that it is limited to the linear response. For an indirect regulation pathway where unknown components may exist, FDT still works within the linear response region. Thus, FDT may be used for systems with partial knowledge, and it is potentially possible to identify the existence of unobserved components. With FDT, the dynamic response can be composed of steady-state measurements without the complete detailed knowledge for the regulation or kinetics. The response function derived can give important insights into the dynamics and time scales of the system.

Quantifying Waddington landscapes and paths of non-adiabatic cell fate decisions for differentiation, reprogramming and transdifferentiation

Cellular differentiation, reprogramming and transdifferentiation are determined by underlying gene regulatory networks. Non-adiabatic regulation via slow binding/unbinding to the gene can be important in these cell fate decision-making processes. Based on a stem cell core gene network, we uncovered the stem cell developmental landscape. As the binding/unbinding speed decreases, the landscape topography changes from bistable attractors of stem and differentiated states to more attractors of stem and other different cell states as well as substates. Non-adiabaticity leads to more differentiated cell types and provides a natural explanation for the heterogeneity observed in the experiments. We quantified Waddington landscapes with two possible cell fate decision mechanisms by changing the regulation strength or regulation timescale (non-adiabaticity). Transition rates correlate with landscape topography through barrier heights between different states and quantitatively determine global stability. We found the optimal speeds of these cell fate decision-making processes. We quantified biological paths and predict that differentiation and reprogramming go through an intermediate state (IM1), whereas transdifferentiation goes through another intermediate state (IM2). Some predictions are confirmed by recent experimental studies.

Eddy current and coupled landscapes for nonadiabatic and nonequilibrium complex system dynamics

Physical and biological systems are often involved with coupled processes of different time scales. In the system with electronic and atomic motions, for example, the interplay between the atomic motion along the same energy landscape and the electronic hopping between different landscapes is critical: the system behavior largely depends on whether the intralandscape motion is slower (adiabatic) or faster (nonadiabatic) than the interlandscape hopping. For general nonequilibrium dynamics where Hamiltonian or energy function is unknown a priori, the challenge is how to extend the concepts of the intra- and interlandscape dynamics. In this paper we establish a theoretical framework for describing global nonequilibrium and nonadiabatic complex system dynamics by transforming the coupled landscapes into a single landscape but with additional dimensions. On this single landscape, dynamics is driven by gradient of the potential landscape, which is closely related to the steady-state probability distribution of the enlarged dimensions, and the probability flux, which has a curl nature. Through an example of a self-regulating gene circuit, we show that the curl flux has dramatic effects on gene regulatory dynamics. The curl flux and landscape framework developed here are easy to visualize and can be used to guide further investigation of physical and biological nonequilibrium systems.

EFFECT OF NON-CODING RNA ON BISTABILITY AND OSCILLATIONS IN THE mRNA-PROTEIN INTERPLAY

The feedbacks between the mRNA and protein synthesis may result in kinetic bistability and oscillations. Two generic models predicting bistability include, respectively, a gene with positive regulation of the mRNA production by protein and two genes with mutual suppression of the mRNA production due to negative regulation of the gene transcription by protein. The simplest model predicting oscillations describes a gene with negative regulation of the mRNA production by protein formed via mRNA translation and a few steps of conversion. We complement these models by the steps of non-coding RNA (ncRNA) formation and ncRNA-mRNA association and degradation. With this extension, the bistability can often be observed as well. Without and with ncRNA, the biochemistry behind the steady states may be different. In the latter case, for example, ncRNA may control the mRNA population in the situations when this population is relatively small, and one can observe a switch in the mRNA, protein and ncRNA populations. Our analysis of oscillatory kinetics of the mRNA-protein interplay shows that with ncRNA the oscillations may be observed in a wider range of parameters and the amplitude of oscillations may be larger.

A perturbation analysis of rate theory of self-regulating genes and signaling networks

A thorough kinetic analysis of the rate theory for stochastic self-regulating gene networks is presented. The chemical master equation kinetic model in terms of a coupled birth-death process is deconstructed into several simpler kinetic modules. We formulate and improve upon the rate theory of self-regulating genes in terms of perturbation theory. We propose a simple five-state scheme as a faithful caricature that elucidates the full kinetics including the "resonance phenomenon" discovered by Walczak et al. [Proc. Natl. Acad. Sci. U.S.A. 102, 18926 (2005)]. The same analysis can be readily applied to other biochemical networks such as phosphorylation signaling with fluctuating kinase activity. Generalization of the present approach can be included in multiple time-scale numerical computations for large biochemical networks.

Model of gene transcription including the return of a RNA polymerase to the beginning of a transcriptional cycle

The gene transcription occurs via the RNA polymerase (RNAP) recruitment on the DNA promoter sequence, formation of a locally open DNA chain, promoter escape, steps of the RNA synthesis, and RNA and RNAP release after reading the final DNA base. Just after the end of the RNA synthesis, RNAP surrounds the closed DNA chain and may diffuse along DNA, desorb, or reach the promoter and start the RNA-synthesis cycle again. We present a generic kinetic model taking the latter steps into account and show analytically and by Monte Carlo simulations that it predicts transcriptional bursts even in the absence of explicit regulation of the transcription by master proteins.

Stable, precise, and reproducible patterning of bicoid and hunchback molecules in the early Drosophila embryo

Precise patterning of morphogen molecules and their accurate reading out are of key importance in embryonic development. Recent experiments have visualized distributions of proteins in developing embryos and shown that the gradient of concentration of Bicoid morphogen in Drosophila embryos is established rapidly after fertilization and remains stable through syncytial mitoses. This stable Bicoid gradient is read out in a precise way to distribute Hunchback with small fluctuations in each embryo and in a reproducible way, with small embryo-to-embryo fluctuation. The mechanisms of such stable, precise, and reproducible patterning through noisy cellular processes, however, still remain mysterious. To address these issues, here we develop the one- and three-dimensional stochastic models of the early Drosophila embryo. The simulated results show that the fluctuation in expression of the hunchback gene is dominated by the random arrival of Bicoid at the hunchback enhancer. Slow diffusion of Hunchback protein, however, averages out this intense fluctuation, leading to the precise patterning of distribution of Hunchback without loss of sharpness of the boundary of its distribution. The coordinated rates of diffusion and transport of input Bicoid and output Hunchback play decisive roles in suppressing fluctuations arising from the dynamical structure change in embryos and those arising from the random diffusion of molecules, and give rise to the stable, precise, and reproducible patterning of Bicoid and Hunchback distributions.

Stochasticity of gene products from transcriptional pulsing

Transcriptional pulsing has been observed in both prokaryotes and eukaryotes and plays a crucial role in cell-to-cell variability of protein and mRNA numbers. An important issue is how the time constants associated with episodes of transcriptional bursting and mRNA and protein degradation rates lead to different cellular mRNA and protein distributions, starting from the transient regime leading to the steady state. We address this by deriving and then investigating the exact time-dependent solution of the master equation for a transcriptional pulsing model of mRNA distributions. We find a plethora of results. We show that, among others, bimodal and long-tailed (power-law) distributions occur in the steady state as the rate constants are varied over biologically significant time scales. Since steady state may not be reached experimentally we present results for the time evolution of the distributions. Because cellular behavior is determined by proteins, we also investigate the effect of the different mRNA distributions on the corresponding protein distributions using numerical simulations.

Elimination of fast variables in chemical Langevin equations

Internal and external fluctuations are ubiquitous in cellular signaling processes. Because biochemical reactions often evolve on disparate time scales, mathematical perturbation techniques can be invoked to reduce the complexity of stochastic models. Previous work in this area has focused on direct treatment of the master equation. However, eliminating fast variables in the chemical Langevin equation is also an important problem. We show how to solve this problem by utilizing a partial equilibrium assumption. Our technique is applied to a simple birth-death-dimerization process and a more involved gene regulation network, demonstrating great computational efficiency. Excellent agreement is found with results computed from exact stochastic simulations. We compare our approach with existing reduction schemes and discuss avenues for future improvement.