1
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Bressloff PC. Truncated stochastically switching processes. Phys Rev E 2024; 109:024103. [PMID: 38491685 DOI: 10.1103/physreve.109.024103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2023] [Accepted: 01/09/2024] [Indexed: 03/18/2024]
Abstract
There are a large variety of hybrid stochastic systems that couple a continuous process with some form of stochastic switching mechanism. In many cases the system switches between different discrete internal states according to a finite-state Markov chain, and the continuous dynamics depends on the current internal state. The resulting hybrid stochastic differential equation (hSDE) could describe the evolution of a neuron's membrane potential, the concentration of proteins synthesized by a gene network, or the position of an active particle. Another major class of switching system is a search process with stochastic resetting, where the position of a diffusing or active particle is reset to a fixed position at a random sequence of times. In this case the system switches between a search phase and a reset phase, where the latter may be instantaneous. In this paper, we investigate how the behavior of a stochastically switching system is modified when the maximum number of switching (or reset) events in a given time interval is fixed. This is motivated by the idea that each time the system switches there is an additive energy cost. We first show that in the case of an hSDE, restricting the number of switching events is equivalent to truncating a Volterra series expansion of the particle propagator. Such a truncation significantly modifies the moments of the resulting renormalized propagator. We then investigate how restricting the number of reset events affects the diffusive search for an absorbing target. In particular, truncating a Volterra series expansion of the survival probability, we calculate the splitting probabilities and conditional MFPTs for the particle to be absorbed by the target or exceed a given number of resets, respectively.
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
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2
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Wang X, Li Y, Jia C. Poisson representation: a bridge between discrete and continuous models of stochastic gene regulatory networks. J R Soc Interface 2023; 20:20230467. [PMID: 38016635 PMCID: PMC10684348 DOI: 10.1098/rsif.2023.0467] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/11/2023] [Accepted: 11/08/2023] [Indexed: 11/30/2023] Open
Abstract
Stochastic gene expression dynamics can be modelled either discretely or continuously. Previous studies have shown that the mRNA or protein number distributions of some simple discrete and continuous gene expression models are related by Gardiner's Poisson representation. Here, we systematically investigate the Poisson representation in complex stochastic gene regulatory networks. We show that when the gene of interest is unregulated, the discrete and continuous descriptions of stochastic gene expression are always related by the Poisson representation, no matter how complex the model is. This generalizes the results obtained in Dattani & Barahona (Dattani & Barahona 2017 J. R. Soc. Interface 14, 20160833 (doi:10.1098/rsif.2016.0833)). In addition, using a simple counter-example, we find that the Poisson representation in general fails to link the two descriptions when the gene is regulated. However, for a general stochastic gene regulatory network, we demonstrate that the discrete and continuous models are approximately related by the Poisson representation in the limit of large protein numbers. These theoretical results are further applied to analytically solve many complex gene expression models whose exact distributions are previously unknown.
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Affiliation(s)
- Xinyu Wang
- Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China
| | - Youming Li
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, People’s Republic of China
| | - Chen Jia
- Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China
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3
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Das AK. Stochastic gene transcription with non-competitive transcription regulatory architecture. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2022; 45:61. [PMID: 35831727 DOI: 10.1140/epje/s10189-022-00213-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2022] [Accepted: 06/22/2022] [Indexed: 06/15/2023]
Abstract
The transcription factors, such as activators and repressors, can interact with the promoter of gene either in a competitive or non-competitive way. In this paper, we construct a stochastic model with non-competitive transcriptional regulatory architecture and develop an analytical theory that re-establishes the experimental results with an improved data fitting. The analytical expressions in the theory allow us to study the nature of the system corresponding to any of its parameters and hence, enable us to find out the factors that govern the regulation of gene expression for that architecture. We notice that, along with transcriptional reinitiation and repressors, there are other parameters that can control the noisiness of this network. We also observe that, the Fano factor (at mRNA level) varies from sub-Poissonian regime to super-Poissonian regime. In addition to the aforementioned properties, we observe some anomalous characteristics of the Fano factor (at mRNA level) and that of the variance of protein at lower activator concentrations in the presence of repressor molecules. This model is useful to understand the architecture of interactions which may buffer the stochasticity inherent to gene transcription.
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4
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Emergent phenomena in living systems: A statistical mechanical perspective. J Biosci 2022. [DOI: 10.1007/s12038-021-00247-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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5
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Hsieh CC, Li CE, Shu CC. Modulating the frequency of switching between multiple DNA states to qualitatively and quantitatively control the protein distribution. J Taiwan Inst Chem Eng 2022. [DOI: 10.1016/j.jtice.2022.104330] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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6
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Zhang Z, Deng Q, Wang Z, Chen Y, Zhou T. Exact results for queuing models of stochastic transcription with memory and crosstalk. Phys Rev E 2021; 103:062414. [PMID: 34271765 DOI: 10.1103/physreve.103.062414] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2021] [Accepted: 06/03/2021] [Indexed: 11/07/2022]
Abstract
Gene transcription is a complex multistep biochemical process, which can create memory between individual reaction events. On the other hand, many inducible genes, when activated by external cues, are often coregulated by several competitive pathways with crosstalk. This raises an unexplored question: how do molecular memory and crosstalk together affect gene expressions? To address this question, we introduce a queuing model of stochastic transcription, where two crossing signaling pathways are used to direct gene activation in response to external signals and memory functions to model multistep reaction processes involved in transcription. We first establish, based on the total probability principle, the chemical master equation for this queuing model, and then we derive, based on the binomial moment approach, exact expressions for statistical quantities (including distributions) of mRNA, which provide insights into the roles of crosstalk and memory in controlling the mRNA level and noise. We find that molecular memory of gene activation decreases the mRNA level but increases the mRNA noise, and double activation pathways always reduce the mRNA noise in contrast to a single pathway. In addition, we find that molecular memory can make the mRNA bimodality disappear.
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Affiliation(s)
- Zhenquan Zhang
- Guangdong Province Key Laboratory of Computational Science, School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
| | - Qiqi Deng
- Guangdong Province Key Laboratory of Computational Science, School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
| | - Zihao Wang
- Guangdong Province Key Laboratory of Computational Science, School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
| | - Yiren Chen
- College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China
| | - Tianshou Zhou
- Guangdong Province Key Laboratory of Computational Science, School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
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7
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Bressloff PC. First-passage processes and the target-based accumulation of resources. Phys Rev E 2021; 103:012101. [PMID: 33601498 DOI: 10.1103/physreve.103.012101] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/26/2020] [Accepted: 12/16/2020] [Indexed: 12/26/2022]
Abstract
A random search for one or more targets in a bounded domain occurs widely in nature, with examples ranging from animal foraging to the transport of vesicles within cells. Most theoretical studies take a searcher-centric viewpoint, focusing on the first passage time (FTP) problem to find a target. This single search-and-capture event then triggers a downstream process or provides the searcher with some resource such as food. In this paper we take a target-centric viewpoint by considering the accumulation of resources in one or more targets due to multiple rounds of search-and-capture events combined with resource degradation; whenever a searcher finds a target it delivers a resource packet to the target, after which it escapes and returns to its initial position. The searcher is then resupplied with cargo and a new search process is initiated after a random delay. It has previously been shown how queuing theory can be used to derive general expressions for the steady-state mean and variance of the resulting resource distributions. Here we apply the theory to some classical FPT problems involving diffusion in simple geometries with absorbing boundaries, including concentric spheres, wedge domains, and branching networks. In each case, we determine how the resulting Fano factor depends on the degradation rate, the delay distribution, and various geometric parameters. We thus establish that the Fano factor can deviate significantly from Poisson statistics and exhibits a nontrivial dependence on model parameters, including nonmonotonicity and crossover behavior. This indicates the nontrivial nature of the higher-order statistics of resource accumulation.
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, University of Utah 155 South 1400 East, Salt Lake City, Utah 84112, USA
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8
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Hortsch SK, Kremling A. Stochastic Models for Studying the Role of Cellular Noise and Heterogeneity. SYSTEMS MEDICINE 2021. [DOI: 10.1016/b978-0-12-801238-3.11466-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022] Open
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9
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Fluctuating-rate model with multiple gene states. J Math Biol 2020; 81:1099-1141. [PMID: 33000313 DOI: 10.1007/s00285-020-01538-2] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/20/2019] [Revised: 08/28/2020] [Indexed: 10/23/2022]
Abstract
Multiple phenotypic states of single cells often co-exist in the presence of positive feedbacks. Stochastic gene-state switchings and low copy numbers of proteins in single cells cause considerable fluctuations. The chemical master equation (CME) is a powerful tool that describes the dynamics of single cells, but it may be overly complicated. Among many simplified models, a fluctuating-rate (FR) model has been proposed recently to approximate the full CME model in the realistic intermediate region of gene-state switchings. However, only the scenario with two gene states has been carefully analysed. In this paper, we generalise the FR model to the case with multiple gene states, in which the mathematical derivation becomes more complicated. The leading order of fluctuations around each phenotypic state, as well as the transition rates between phenotypic states, in the intermediate gene-state switching region is characterized by the rate function of the stationary distribution of the FR model in the Freidlin-Wentzell-type large deviation principle (LDP). Under certain reasonable assumptions, we show that the derivative of the rate function is equal to the unique nontrivial solution of a dominant generalised eigenvalue problem, leading to a new numerical algorithm for obtaining the LDP rate function directly. Furthermore, we prove the Lyapunov property of the rate function for the corresponding deterministic mean-field dynamics. Finally, through a tristable example, we show that the local fluctuations (the asymptotic variance of the stationary distribution at each phenotypic state) in the intermediate and rapid regions of gene-state switchings are different. Finally, a tri-stable example is constructed to illustrate the validity of our theory.
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10
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Ali Al-Radhawi M, Del Vecchio D, Sontag ED. Multi-modality in gene regulatory networks with slow promoter kinetics. PLoS Comput Biol 2019; 15:e1006784. [PMID: 30779734 PMCID: PMC6396950 DOI: 10.1371/journal.pcbi.1006784] [Citation(s) in RCA: 20] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2018] [Revised: 03/01/2019] [Accepted: 01/14/2019] [Indexed: 12/27/2022] Open
Abstract
Phenotypical variability in the absence of genetic variation often reflects complex energetic landscapes associated with underlying gene regulatory networks (GRNs). In this view, different phenotypes are associated with alternative states of complex nonlinear systems: stable attractors in deterministic models or modes of stationary distributions in stochastic descriptions. We provide theoretical and practical characterizations of these landscapes, specifically focusing on stochastic Slow Promoter Kinetics (SPK), a time scale relevant when transcription factor binding and unbinding are affected by epigenetic processes like DNA methylation and chromatin remodeling. In this case, largely unexplored except for numerical simulations, adiabatic approximations of promoter kinetics are not appropriate. In contrast to the existing literature, we provide rigorous analytic characterizations of multiple modes. A general formal approach gives insight into the influence of parameters and the prediction of how changes in GRN wiring, for example through mutations or artificial interventions, impact the possible number, location, and likelihood of alternative states. We adapt tools from the mathematical field of singular perturbation theory to represent stationary distributions of Chemical Master Equations for GRNs as mixtures of Poisson distributions and obtain explicit formulas for the locations and probabilities of metastable states as a function of the parameters describing the system. As illustrations, the theory is used to tease out the role of cooperative binding in stochastic models in comparison to deterministic models, and applications are given to various model systems, such as toggle switches in isolation or in communicating populations, a synthetic oscillator, and a trans-differentiation network.
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Affiliation(s)
- M. Ali Al-Radhawi
- Departments of Electrical and Computer Engineering and of Bioengineering, Northeastern University, Boston, Massachusetts, United States of America
| | - Domitilla Del Vecchio
- Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, United States of America
| | - Eduardo D. Sontag
- Departments of Electrical and Computer Engineering and of Bioengineering, Northeastern University, Boston, Massachusetts, United States of America
- Laboratory of Systems Pharmacology, Harvard Medical School, Boston, Massachusetts, United States of America
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11
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Bressloff PC, MacLaurin J. Synchronization of stochastic hybrid oscillators driven by a common switching environment. CHAOS (WOODBURY, N.Y.) 2018; 28:123123. [PMID: 30599535 DOI: 10.1063/1.5054795] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/03/2018] [Accepted: 11/30/2018] [Indexed: 06/09/2023]
Abstract
Many systems in biology, physics, and chemistry can be modeled through ordinary differential equations (ODEs), which are piecewise smooth, but switch between different states according to a Markov jump process. In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we suppose that this limit ODE supports a stable limit cycle. We demonstrate that a set of such oscillators can synchronize when they are uncoupled, but they share the same switching Markov jump process. The latter is taken to represent the effect of a common randomly switching environment. We determine the leading order of the Lyapunov coefficient governing the rate of decay of the phase difference in the fast switching limit. The analysis bears some similarities to the classical analysis of synchronization of stochastic oscillators subject to common white noise. However, the discrete nature of the Markov jump process raises some difficulties: in fact, we find that the Lyapunov coefficient from the quasi-steady-state approximation differs from the Lyapunov coefficient one obtains from a second order perturbation expansion in the waiting time between jumps. Finally, we demonstrate synchronization numerically in the radial isochron clock model and show that the latter Lyapunov exponent is more accurate.
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA
| | - James MacLaurin
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA
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12
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Bressloff PC, Maclaurin JN. Stochastic Hybrid Systems in Cellular Neuroscience. JOURNAL OF MATHEMATICAL NEUROSCIENCE 2018; 8:12. [PMID: 30136005 PMCID: PMC6104574 DOI: 10.1186/s13408-018-0067-7] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/09/2018] [Accepted: 08/05/2018] [Indexed: 06/08/2023]
Abstract
We review recent work on the theory and applications of stochastic hybrid systems in cellular neuroscience. A stochastic hybrid system or piecewise deterministic Markov process involves the coupling between a piecewise deterministic differential equation and a time-homogeneous Markov chain on some discrete space. The latter typically represents some random switching process. We begin by summarizing the basic theory of stochastic hybrid systems, including various approximation schemes in the fast switching (weak noise) limit. In subsequent sections, we consider various applications of stochastic hybrid systems, including stochastic ion channels and membrane voltage fluctuations, stochastic gap junctions and diffusion in randomly switching environments, and intracellular transport in axons and dendrites. Finally, we describe recent work on phase reduction methods for stochastic hybrid limit cycle oscillators.
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13
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Bressloff PC, MacLaurin J. A variational method for analyzing limit cycle oscillations in stochastic hybrid systems. CHAOS (WOODBURY, N.Y.) 2018; 28:063105. [PMID: 29960393 DOI: 10.1063/1.5027077] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
Many systems in biology can be modeled through ordinary differential equations, which are piece-wise continuous, and switch between different states according to a Markov jump process known as a stochastic hybrid system or piecewise deterministic Markov process (PDMP). In the fast switching limit, the dynamics converges to a deterministic ODE. In this paper, we develop a phase reduction method for stochastic hybrid systems that support a stable limit cycle in the deterministic limit. A classic example is the Morris-Lecar model of a neuron, where the switching Markov process is the number of open ion channels and the continuous process is the membrane voltage. We outline a variational principle for the phase reduction, yielding an exact analytic expression for the resulting phase dynamics. We demonstrate that this decomposition is accurate over timescales that are exponential in the switching rate ϵ-1. That is, we show that for a constant C, the probability that the expected time to leave an O(a) neighborhood of the limit cycle is less than T scales as T exp (-Ca/ϵ).
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA
| | - James MacLaurin
- Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA
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14
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Ge H, Wu P, Qian H, Xie XS. Relatively slow stochastic gene-state switching in the presence of positive feedback significantly broadens the region of bimodality through stabilizing the uninduced phenotypic state. PLoS Comput Biol 2018. [PMID: 29529037 PMCID: PMC5864076 DOI: 10.1371/journal.pcbi.1006051] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022] Open
Abstract
Within an isogenic population, even in the same extracellular environment, individual cells can exhibit various phenotypic states. The exact role of stochastic gene-state switching regulating the transition among these phenotypic states in a single cell is not fully understood, especially in the presence of positive feedback. Recent high-precision single-cell measurements showed that, at least in bacteria, switching in gene states is slow relative to the typical rates of active transcription and translation. Hence using the lac operon as an archetype, in such a region of operon-state switching, we present a fluctuating-rate model for this classical gene regulation module, incorporating the more realistic operon-state switching mechanism that was recently elucidated. We found that the positive feedback mechanism induces bistability (referred to as deterministic bistability), and that the parameter range for its occurrence is significantly broadened by stochastic operon-state switching. We further show that in the absence of positive feedback, operon-state switching must be extremely slow to trigger bistability by itself. However, in the presence of positive feedback, which stabilizes the induced state, the relatively slow operon-state switching kinetics within the physiological region are sufficient to stabilize the uninduced state, together generating a broadened parameter region of bistability (referred to as stochastic bistability). We illustrate the opposite phenotype-transition rate dependence upon the operon-state switching rates in the two types of bistability, with the aid of a recently proposed rate formula for fluctuating-rate models. The rate formula also predicts a maximal transition rate in the intermediate region of operon-state switching, which is validated by numerical simulations in our model. Overall, our findings suggest a biological function of transcriptional “variations” among genetically identical cells, for the emergence of bistability and transition between phenotypic states. Identifying the mechanism underlying the coexistence of multiple stable phenotypic states has been a challenging scientific problem for more than half a century, and an appropriate mathematical model at the single-cell level is also in high demand. Single-cell measurements conducted in the past ten years have shown that gene-state switching is slow relative to the typical rates of active transcription and translation; hence the recently proposed fluctuating-rate model is a good candidate for describing the single-cell dynamics. We use the classic gene regulation module of the lac operon as an archetype and build a specific fluctuating-rate model based on the recently identified operon-state switching mechanism. This model is analyzed to dissect the interplay between positive feedback and the stochastic switching of gene states in the emergence of bistability/multistablity and the transition between phenotypic states. We show that relatively slow operon-state switching stabilizes the uninduced state and that the positive feedback stabilizes the induced state. Thus, the parameter range for bistability is significantly broadened. In addition, recently proposed landscape theory and rate formula predict opposite phenotype-transition rate dependence on operon-state switching rates for the two types of bistability.
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Affiliation(s)
- Hao Ge
- Biodynamic Optical Imaging Center (BIOPIC), Peking University, Beijing, P.R.China
- Beijing International Center for Mathematical Research (BICMR), Peking University, Beijing, P.R.China
- * E-mail: (HG); (XSX)
| | - Pingping Wu
- School of Mathematical Sciences and Centre for Computational Systems Biology, Fudan University, Shanghai, P.R.China
| | - Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington, United States of America
| | - Xiaoliang Sunney Xie
- Biodynamic Optical Imaging Center (BIOPIC), Peking University, Beijing, P.R.China
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts, United States of America
- * E-mail: (HG); (XSX)
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15
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Jia C, Qian H, Chen M, Zhang MQ. Relaxation rates of gene expression kinetics reveal the feedback signs of autoregulatory gene networks. J Chem Phys 2018. [DOI: 10.1063/1.5009749] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Affiliation(s)
- Chen Jia
- Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75080, USA
| | - Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
| | - Min Chen
- Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75080, USA
| | - Michael Q. Zhang
- Department of Biological Sciences, Center for Systems Biology, University of Texas at Dallas, Richardson, Texas 75080, USA
- MOE Key Lab and Division of Bioinformatics, CSSB, TNLIST, Tsinghua University, Beijing 100084, China
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16
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Das D, Dey S, Brewster RC, Choubey S. Effect of transcription factor resource sharing on gene expression noise. PLoS Comput Biol 2017; 13:e1005491. [PMID: 28414750 PMCID: PMC5411101 DOI: 10.1371/journal.pcbi.1005491] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/30/2016] [Revised: 05/01/2017] [Accepted: 03/31/2017] [Indexed: 12/31/2022] Open
Abstract
Gene expression is intrinsically a stochastic (noisy) process with important implications for cellular functions. Deciphering the underlying mechanisms of gene expression noise remains one of the key challenges of regulatory biology. Theoretical models of transcription often incorporate the kinetics of how transcription factors (TFs) interact with a single promoter to impact gene expression noise. However, inside single cells multiple identical gene copies as well as additional binding sites can compete for a limiting pool of TFs. Here we develop a simple kinetic model of transcription, which explicitly incorporates this interplay between TF copy number and its binding sites. We show that TF sharing enhances noise in mRNA distribution across an isogenic population of cells. Moreover, when a single gene copy shares it’s TFs with multiple competitor sites, the mRNA variance as a function of the mean remains unaltered by their presence. Hence, all the data for variance as a function of mean expression collapse onto a single master curve independent of the strength and number of competitor sites. However, this result does not hold true when the competition stems from multiple copies of the same gene. Therefore, although previous studies showed that the mean expression follows a universal master curve, our findings suggest that different scenarios of competition bear distinct signatures at the level of variance. Intriguingly, the introduction of competitor sites can transform a unimodal mRNA distribution into a multimodal distribution. These results demonstrate the impact of limited availability of TF resource on the regulation of noise in gene expression. Genetically identical cells, even when they are exposed to the same environmental conditions, display incredible diversity. Gene expression noise is attributed to be a key source of this phenotypic diversity. Transcriptional dynamics is a dominant source of expression noise. Although scores of theoretical and experimental studies have explored how noise is regulated at the level of transcription, most of them focus on the gene specific, cis regulatory elements, such as the number of transcription factor (TF) binding sites, their binding strength, etc. However, how the global properties of transcription, such as the limited availability of TFs impact noise in gene expression remains rather elusive. Here we build a theoretical model that incorporates the effect of limiting TF pool on gene expression noise. We find that competition between genes for TFs leads to enhanced variability in mRNA copy number across an isogenic population. Moreover, for gene copies sharing TFs with other competitor sites, mRNA variance as a function of the mean shows distinct imprints for one gene copy and multiple gene copies respectively. This stands in sharp contrast to the universal behavior found in mean expression irrespective of the different scenarios of competition. An interesting feature of competition is that introduction of competitor sites can transform a unimodal mRNA distribution into a multimodal distribution, which could lead to phenotypic variability.
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Affiliation(s)
- Dipjyoti Das
- Department of Molecular, Cellular and Developmental Biology, Yale University, New Haven, Connecticut, United States of America
| | - Supravat Dey
- Laboratoire Charles Coulomb, Université de Montpellier and CNRS, Montpellier, France
| | - Robert C. Brewster
- Program in Systems Biology, University of Massachusetts Medical School, Worcester, Massachusetts, United States of America
- Department of Microbiology and Physiological Systems, University of Massachusetts Medical School, Worcester, Massachusetts, United States of America
- * E-mail: (RCB); (SC)
| | - Sandeep Choubey
- FAS Center for Systems Biology, Harvard University, Cambridge, Massachusetts, United States of America
- Department of Molecular and Cellular Biology, Harvard University, Cambridge, Massachusetts, United States of America
- * E-mail: (RCB); (SC)
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17
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Bressloff PC. Stochastic Liouville equation for particles driven by dichotomous environmental noise. Phys Rev E 2017; 95:012124. [PMID: 28208333 DOI: 10.1103/physreve.95.012124] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2016] [Indexed: 11/06/2022]
Abstract
We analyze the stochastic dynamics of a large population of noninteracting particles driven by a global environmental input in the form of a dichotomous Markov noise process (DMNP). The population density of particle states evolves according to a stochastic Liouville equation with respect to different realizations of the DMNP. We then exploit the connection with previous work on diffusion in randomly switching environments, in order to derive moment equations for the distribution of solutions to the stochastic Liouville equation. We illustrate the theory by considering two simple examples of dichotomous flows, a velocity jump process and a two-state gene regulatory network. In both cases we show how the global environmental input induces statistical correlations between different realizations of the population density.
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Affiliation(s)
- Paul C Bressloff
- Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112, USA
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18
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Macromolecular Crowding Regulates the Gene Expression Profile by Limiting Diffusion. PLoS Comput Biol 2016; 12:e1005122. [PMID: 27893768 PMCID: PMC5125560 DOI: 10.1371/journal.pcbi.1005122] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2016] [Accepted: 08/26/2016] [Indexed: 01/24/2023] Open
Abstract
We seek to elucidate the role of macromolecular crowding in transcription and translation. It is well known that stochasticity in gene expression can lead to differential gene expression and heterogeneity in a cell population. Recent experimental observations by Tan et al. have improved our understanding of the functional role of macromolecular crowding. It can be inferred from their observations that macromolecular crowding can lead to robustness in gene expression, resulting in a more homogeneous cell population. We introduce a spatial stochastic model to provide insight into this process. Our results show that macromolecular crowding reduces noise (as measured by the kurtosis of the mRNA distribution) in a cell population by limiting the diffusion of transcription factors (i.e. removing the unstable intermediate states), and that crowding by large molecules reduces noise more efficiently than crowding by small molecules. Finally, our simulation results provide evidence that the local variation in chromatin density as well as the total volume exclusion of the chromatin in the nucleus can induce a homogenous cell population. The cellular nucleus is packed with macromolecules such as DNAs and proteins, which leaves limited space for other molecules to move around. Recent experimental results by C. Tan et al. have shown that macromolecular crowding can regulate gene expression, resulting in a more homogenous cell population. We introduce a computational model to uncover the mechanism by which macromolecular crowding functions. Our results suggest that macromolecular crowding limits the diffusion of the transcription factors and attenuates the transcriptional bursting, which leads to a more homogenous cell population. Regulation of gene expression noise by macromolecules depends on the size of the crowders, i.e. larger macromolecules can reduce the noise more effectively than smaller macromolecules. We also demonstrate that local variation of chromatin density can affect the noise of gene expression. This shows the importance of the chromatin structure in gene expression regulation.
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Abstract
Molecular noise in gene regulatory networks has two intrinsic components, one part being due to fluctuations caused by the birth and death of protein or mRNA molecules which are often present in small numbers and the other part arising from gene state switching, a single molecule event. Stochastic dynamics of gene regulatory circuits appears to be largely responsible for bifurcations into a set of multi-attractor states that encode different cell phenotypes. The interplay of dichotomous single molecule gene noise with the nonlinear architecture of genetic networks generates rich and complex phenomena. In this paper, we elaborate on an approximate framework that leads to simple hybrid multi-scale schemes well suited for the quantitative exploration of the steady state properties of large-scale cellular genetic circuits. Through a path sum based analysis of trajectory statistics, we elucidate the connection of these hybrid schemes to the underlying master equation and provide a rigorous justification for using dichotomous noise based models to study genetic networks. Numerical simulations of circuit models reveal that the contribution of the genetic noise of single molecule origin to the total noise is significant for a wide range of kinetic regimes.
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Affiliation(s)
- Davit A Potoyan
- Department of Chemistry and Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA
| | - Peter G Wolynes
- Department of Chemistry and Center for Theoretical Biological Physics, Rice University, Houston, Texas 77005, USA
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de Franciscis S, Caravagna G, Mauri G, d’Onofrio A. Gene switching rate determines response to extrinsic perturbations in the self-activation transcriptional network motif. Sci Rep 2016; 6:26980. [PMID: 27256916 PMCID: PMC4891709 DOI: 10.1038/srep26980] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2015] [Accepted: 05/11/2016] [Indexed: 01/01/2023] Open
Abstract
Gene switching dynamics is a major source of randomness in genetic networks, also in the case of large concentrations of the transcription factors. In this work, we consider a common network motif - the positive feedback of a transcription factor on its own synthesis - and assess its response to extrinsic noises perturbing gene deactivation in a variety of settings where the network might operate. These settings are representative of distinct cellular types, abundance of transcription factors and ratio between gene switching and protein synthesis rates. By investigating noise-induced transitions among the different network operative states, our results suggest that gene switching rates are key parameters to shape network response to external perturbations, and that such response depends on the particular biological setting, i.e. the characteristic time scales and protein abundance. These results might have implications on our understanding of irreversible transitions for noise-related phenomena such as cellular differentiation. In addition these evidences suggest to adopt the appropriate mathematical model of the network in order to analyze the system consistently to the reference biological setting.
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Affiliation(s)
| | - Giulio Caravagna
- Università degli Studi di Milano-Bicocca, Dipartimento di Informatica, Sistemistica e Comunicazione, Milano, Italy
- School of Informatics, University of Edinburgh, Edinburgh, UK
| | - Giancarlo Mauri
- Università degli Studi di Milano-Bicocca, Dipartimento di Informatica, Sistemistica e Comunicazione, Milano, Italy
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van Neerven SM, Tieken M, Vermeulen L, Bijlsma MF. Bidirectional interconversion of stem and non-stem cancer cell populations: A reassessment of theoretical models for tumor heterogeneity. Mol Cell Oncol 2015; 3:e1098791. [PMID: 27308617 PMCID: PMC4905404 DOI: 10.1080/23723556.2015.1098791] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/31/2015] [Revised: 09/18/2015] [Accepted: 09/18/2015] [Indexed: 02/07/2023]
Abstract
Resolving the origin of intratumor heterogeneity has proven to be one of the central challenges in cancer research during recent years. Two theoretical models explaining the emergence of intratumor heterogeneity have come to dominate cancer biology literature: the clonal evolution model and the hierarchical/cancer stem cell model. Recently, a plastic model that combines elements of both the clonal and the hierarchical model has gained traction. Basically, this model proposes that cancer stem cells engage in bidirectional interconversion with non-stem cells, thereby providing the missing link between the 2 conventional models. Confirming bidirectional interconversion as a hallmark of cancer is a crucial step in understanding tumor heterogeneity and has important therapeutic implications. In this review, current methodologies and theoretical and empirical evidence regarding bidirectional interconversion will be discussed.
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Affiliation(s)
- Sanne M van Neerven
- Laboratory for Experimental Oncology and Radiobiology, Center for Experimental and Molecular Medicine, Academic Medical Center , Amsterdam, The Netherlands
| | - Mathijs Tieken
- Laboratory for Experimental Oncology and Radiobiology, Center for Experimental and Molecular Medicine, Academic Medical Center , Amsterdam, The Netherlands
| | - Louis Vermeulen
- Laboratory for Experimental Oncology and Radiobiology, Center for Experimental and Molecular Medicine, Academic Medical Center , Amsterdam, The Netherlands
| | - Maarten F Bijlsma
- Laboratory for Experimental Oncology and Radiobiology, Center for Experimental and Molecular Medicine, Academic Medical Center , Amsterdam, The Netherlands
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23
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Benaïm M, Le Borgne S, Malrieu F, Zitt PA. Qualitative properties of certain piecewise deterministic Markov processes. ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES 2015. [DOI: 10.1214/14-aihp619] [Citation(s) in RCA: 52] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Ge H, Qian H, Xie XS. Stochastic phenotype transition of a single cell in an intermediate region of gene state switching. PHYSICAL REVIEW LETTERS 2015; 114:078101. [PMID: 25763973 DOI: 10.1103/physrevlett.114.078101] [Citation(s) in RCA: 41] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/03/2014] [Indexed: 06/04/2023]
Abstract
Multiple phenotypic states often arise in a single cell with different gene-expression states that undergo transcription regulation with positive feedback. Recent experiments show that, at least in E.coli, the gene state switching can be neither extremely slow nor exceedingly rapid as many previous theoretical treatments assumed. Rather, it is in the intermediate region which is difficult to handle mathematically. Under this condition, from a full chemical-master-equation description we derive a model in which the protein copy number, for a given gene state, follows a deterministic mean-field description while the protein-synthesis rates fluctuate due to stochastic gene state switching. The simplified kinetics yields a nonequilibrium landscape function, which, similar to the energy function for equilibrium fluctuation, provides the leading orders of fluctuations around each phenotypic state, as well as the transition rates between the two phenotypic states. This rate formula is analogous to Kramers' theory for chemical reactions. The resulting behaviors are significantly different from the two limiting cases studied previously.
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Affiliation(s)
- Hao Ge
- Biodynamic Optical Imaging Center (BIOPIC), Peking University, Beijing 100871, People's Republic of China
- Beijing International Center for Mathematical Research (BICMR), Peking University, Beijing 100871, People's Republic of China
| | - Hong Qian
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
| | - X Sunney Xie
- Biodynamic Optical Imaging Center (BIOPIC), Peking University, Beijing 100871, People's Republic of China
- Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, USA
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25
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Zhang J, Zhou T. Promoter-mediated transcriptional dynamics. Biophys J 2014; 106:479-88. [PMID: 24461023 DOI: 10.1016/j.bpj.2013.12.011] [Citation(s) in RCA: 48] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2013] [Revised: 12/06/2013] [Accepted: 12/09/2013] [Indexed: 11/24/2022] Open
Abstract
Genes in eukaryotic cells are typically regulated by complex promoters containing multiple binding sites for a variety of transcription factors, but how promoter dynamics affect transcriptional dynamics has remained poorly understood. In this study, we analyze gene models at the transcriptional regulation level, which incorporate the complexity of promoter structure (PS) defined as transcriptional exits (i.e., ON states of the promoter) and the transition pattern (described by a matrix consisting of transition rates among promoter activity states). We show that multiple exits of transcription are the essential origin of generating multimodal distributions of mRNA, but promoters with the same transition pattern can lead to multimodality of different modes, depending on the regulation of transcriptional factors. In turn, for similar mRNA distributions in the models, the mean ON or OFF time distributions may exhibit different characteristics, thus providing the supplemental information on PS. In addition, we demonstrate that the transcriptional noise can be characterized by a nonlinear function of mean ON and OFF times. These results not only reveal essential characteristics of promoter-mediated transcriptional dynamics but also provide signatures useful for inferring PS based on characteristics of transcriptional outputs.
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Affiliation(s)
- Jiajun Zhang
- Guangdong Province Key Laboratory of Computational Science, School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, People's Republic of China
| | - Tianshou Zhou
- Guangdong Province Key Laboratory of Computational Science, School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, People's Republic of China.
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26
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Zhang J, Nie Q, He M, Zhou T. An effective method for computing the noise in biochemical networks. J Chem Phys 2013; 138:084106. [PMID: 23464139 DOI: 10.1063/1.4792444] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023] Open
Abstract
We present a simple yet effective method, which is based on power series expansion, for computing exact binomial moments that can be in turn used to compute steady-state probability distributions as well as the noise in linear or nonlinear biochemical reaction networks. When the method is applied to representative reaction networks such as the ON-OFF models of gene expression, gene models of promoter progression, gene auto-regulatory models, and common signaling motifs, the exact formulae for computing the intensities of noise in the species of interest or steady-state distributions are analytically given. Interestingly, we find that positive (negative) feedback does not enlarge (reduce) noise as claimed in previous works but has a counter-intuitive effect and that the multi-OFF (or ON) mechanism always attenuates the noise in contrast to the common ON-OFF mechanism and can modulate the noise to the lowest level independently of the mRNA mean. Except for its power in deriving analytical expressions for distributions and noise, our method is programmable and has apparent advantages in reducing computational cost.
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Affiliation(s)
- Jiajun Zhang
- School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, People's Republic of China
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27
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Gregorio-Godoy P, Míguez DG. Synthetic approaches to study transcriptional networks and noise in mammalian systems. IET Syst Biol 2013; 7:11-7. [PMID: 23848051 DOI: 10.1049/iet-syb.2012.0026] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022] Open
Abstract
Synthetic biology aims to build new functional organisms and to rationally re-design existing ones by applying the engineering principle of modularity. Apart from building new life forms to perform technical applications, the approach of synthetic biology is useful to dissect complex biological phenomena into simple and easy to understand synthetic modules. Synthetic gene networks have been successfully implemented in prokaryotes and lower eukaryotes, with recent approaches moving ahead towards the mammalian environment. However, synthetic circuits in higher eukaryotes present a more challenging scenario, since its reliability is compromised because of the strong stochastic nature of transcription. Here, the authors review recent approaches that take advantage of the noisy response of synthetic regulatory circuits to learn key features of the complex machinery that orchestrates transcription in higher eukaryotes. Understanding the causes and consequences of biological noise will allow us to design more reliable mammalian synthetic circuits with revolutionary medical applications.
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Affiliation(s)
- Paula Gregorio-Godoy
- Facultad de Ciencias, Departamento de Física de la Materia Condensada e Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, 28049 Madrid, Spain
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28
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Bose I, Ghosh S. Origins of binary gene expression in post-transcriptional regulation by microRNAs. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2012; 35:102. [PMID: 23064825 DOI: 10.1140/epje/i2012-12102-2] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/13/2012] [Accepted: 09/28/2012] [Indexed: 06/01/2023]
Abstract
MicroRNA-mediated regulation of gene expression is characterised by some distinctive features that set it apart from unregulated and transcription-factor-regulated gene expression. Recently, a mathematical model has been proposed to describe the dynamics of post-transcriptional regulation by microRNAs. The model explains the observations made in single-cell experiments quite well. In this paper, we introduce some additional features into the model and consider two specific cases. In the first case, a noncooperative positive feedback loop is included in the transcriptional regulation of the target gene expression. In the second case, a stochastic version of the original model is considered in which there are random transitions between the inactive and active expression states of the gene. In the first case we show that bistability is possible in a parameter regime, due to the presence of a nonlinear protein decay term in the gene expression dynamics. In the second case, we derive the conditions for obtaining stochastic binary gene expression. We find that this type of gene expression is more favourable in the case of regulation by microRNAs as compared to the case of unregulated gene expression. The theoretical predictions relating to binary gene expression are experimentally testable.
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Affiliation(s)
- I Bose
- Department of Physics, Bose Institute, A. P. C. Road, 93/1, 700009, Kolkata, India.
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Stinchcombe AR, Peskin CS, Tranchina D. Population density approach for discrete mRNA distributions in generalized switching models for stochastic gene expression. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:061919. [PMID: 23005139 DOI: 10.1103/physreve.85.061919] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2012] [Indexed: 06/01/2023]
Abstract
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.
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Affiliation(s)
- Adam R Stinchcombe
- Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, USA.
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30
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Analytical distribution and tunability of noise in a model of promoter progress. Biophys J 2012; 102:1247-57. [PMID: 22455907 DOI: 10.1016/j.bpj.2012.02.001] [Citation(s) in RCA: 52] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2011] [Revised: 01/31/2012] [Accepted: 02/02/2012] [Indexed: 12/27/2022] Open
Abstract
Chromatin template (CT), which accumulates over time until the promoter becomes active, determines upstream dynamics of transcription, but how upstream sequential steps impact downstream dynamics qualitatively and quantitatively is unclear. Here, we analyze a stochastic gene model with a simple yet typical CT that contains one active state and several inactive states of the promoter. We derive the analytical expressions for the noise in mRNA probability distributions governed by master equations. The derived results extend previous work by including the effects of promoter progress on variability and bimodality. Specifically, given a CT for transcription, we analytically demonstrate that inactive phases of the promoter can modulate the noise intensity to the minimum independently of the mean expression of mRNA. If one new inactive state is added to the CT, then the resulting noise will be reduced, implying that the multi-off mechanism plays a role of attenuating the noise. In contrast to the simple on-off mechanism, the multi-off mechanism can also narrow bimodal regions in a certain parameter plane and obscure two peaks, explaining why bimodal distributions are rarely observed in experiments. Our results provide insight into the role of promoter progress in determining the level of cell-to-cell variability in gene expression.
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ZHDANOV VLADIMIRP. EFFECT OF NON-CODING RNA ON BISTABILITY AND OSCILLATIONS IN THE mRNA-PROTEIN INTERPLAY. ACTA ACUST UNITED AC 2011. [DOI: 10.1142/s1793048010001159] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
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.
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Affiliation(s)
- VLADIMIR P. ZHDANOV
- Department of Applied Physics, Chalmers University of Technology,-41296 Göteborg, Sweden
- Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia
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Bistability versus bimodal distributions in gene regulatory processes from population balance. PLoS Comput Biol 2011; 7:e1002140. [PMID: 21901083 PMCID: PMC3161895 DOI: 10.1371/journal.pcbi.1002140] [Citation(s) in RCA: 42] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2011] [Accepted: 06/14/2011] [Indexed: 11/18/2022] Open
Abstract
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.
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Elgart V, Jia T, Fenley AT, Kulkarni R. Connecting protein and mRNA burst distributions for stochastic models of gene expression. Phys Biol 2011; 8:046001. [PMID: 21490380 DOI: 10.1088/1478-3975/8/4/046001] [Citation(s) in RCA: 30] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
The intrinsic stochasticity of gene expression can lead to large variability in protein levels for genetically identical cells. Such variability in protein levels can arise from infrequent synthesis of mRNAs which in turn give rise to bursts of protein expression. Protein expression occurring in bursts has indeed been observed experimentally and recent studies have also found evidence for transcriptional bursting, i.e. production of mRNAs in bursts. Given that there are distinct experimental techniques for quantifying the noise at different stages of gene expression, it is of interest to derive analytical results connecting experimental observations at different levels. In this work, we consider stochastic models of gene expression for which mRNA and protein production occurs in independent bursts. For such models, we derive analytical expressions connecting protein and mRNA burst distributions which show how the functional form of the mRNA burst distribution can be inferred from the protein burst distribution. Additionally, if gene expression is repressed such that observed protein bursts arise only from single mRNAs, we show how observations of protein burst distributions (repressed and unrepressed) can be used to completely determine the mRNA burst distribution. Assuming independent contributions from individual bursts, we derive analytical expressions connecting means and variances for burst and steady-state protein distributions. Finally, we validate our general analytical results by considering a specific reaction scheme involving regulation of protein bursts by small RNAs. For a range of parameters, we derive analytical expressions for regulated protein distributions that are validated using stochastic simulations. The analytical results obtained in this work can thus serve as useful inputs for a broad range of studies focusing on stochasticity in gene expression.
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Affiliation(s)
- Vlad Elgart
- Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA.
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Weber M, Buceta J. Noise regulation by quorum sensing in low mRNA copy number systems. BMC SYSTEMS BIOLOGY 2011; 5:11. [PMID: 21251314 PMCID: PMC3037314 DOI: 10.1186/1752-0509-5-11] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 07/28/2010] [Accepted: 01/20/2011] [Indexed: 01/27/2023]
Abstract
Background Cells must face the ubiquitous presence of noise at the level of signaling molecules. The latter constitutes a major challenge for the regulation of cellular functions including communication processes. In the context of prokaryotic communication, the so-called quorum sensing (QS) mechanism relies on small diffusive molecules that are produced and detected by cells. This poses the intriguing question of how bacteria cope with the fluctuations for setting up a reliable information exchange. Results We present a stochastic model of gene expression that accounts for the main biochemical processes that describe the QS mechanism close to its activation threshold. Within that framework we study, both numerically and analytically, the role that diffusion plays in the regulation of the dynamics and the fluctuations of signaling molecules. In addition, we unveil the contribution of different sources of noise, intrinsic and transcriptional, in the QS mechanism. Conclusions The interplay between noisy sources and the communication process produces a repertoire of dynamics that depends on the diffusion rate. Importantly, the total noise shows a non-monotonic behavior as a function of the diffusion rate. QS systems seems to avoid values of the diffusion that maximize the total noise. These results point towards the direction that bacteria have adapted their communication mechanisms in order to improve the signal-to-noise ratio.
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Affiliation(s)
- Marc Weber
- Computer Simulation and Modelling (Co.S.Mo.) Lab, Parc Científic de Barcelona, C/Baldiri Reixac 10-12, Barcelona 08028, Spain
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Dong D, Shao X, Deng N, Zhang Z. Gene expression variations are predictive for stochastic noise. Nucleic Acids Res 2010; 39:403-13. [PMID: 20860999 PMCID: PMC3025572 DOI: 10.1093/nar/gkq844] [Citation(s) in RCA: 28] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/25/2022] Open
Abstract
Fluctuations in protein abundance among single cells are primarily due to the inherent stochasticity in transcription and translation processes, such stochasticity can often confer phenotypic heterogeneity among isogenic cells. It has been proposed that expression noise can be triggered as an adaptation to environmental stresses and genetic perturbations, and as a mechanism to facilitate gene expression evolution. Thus, elucidating the relationship between expression noise, measured at the single-cell level, and expression variation, measured on population of cells, can improve our understanding on the variability and evolvability of gene expression. Here, we showed that noise levels are significantly correlated with conditional expression variations. We further demonstrated that expression variations are highly predictive for noise level, especially in TATA-box containing genes. Our results suggest that expression variabilities can serve as a proxy for noise level, suggesting that these two properties share the same underlining mechanism, e.g. chromatin regulation. Our work paves the way for the study of stochastic noise in other single-cell organisms.
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Affiliation(s)
- Dong Dong
- Donnelly Centre for Cellular and Biomolecular Research, University of Toronto, Toronto, ON, Canada.
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36
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Jia T, Kulkarni RV. Post-transcriptional regulation of noise in protein distributions during gene expression. PHYSICAL REVIEW LETTERS 2010; 105:018101. [PMID: 20867481 DOI: 10.1103/physrevlett.105.018101] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/04/2010] [Indexed: 05/29/2023]
Abstract
The intrinsic stochasticity of gene expression can lead to a large variability of protein levels across a population of cells. Variability (or noise) in protein distributions can be modulated by cellular mechanisms of gene regulation; in particular, there is considerable interest in understanding the role of post-transcriptional regulation. To address this issue, we propose and analyze a stochastic model for post-transcriptional regulation of gene expression. The analytical solution of the model provides insight into the effects of different mechanisms of post-transcriptional regulation on the noise in protein distributions. The results obtained also demonstrate how different sources of intrinsic noise in gene expression can be discriminated based on observations of regulated protein distributions.
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Affiliation(s)
- Tao Jia
- Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA.
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Proulx SR, Smiley MW. The evolutionary origins of gene regulation. JOURNAL OF EXPERIMENTAL ZOOLOGY PART B-MOLECULAR AND DEVELOPMENTAL EVOLUTION 2010; 314:327-40. [PMID: 20095005 DOI: 10.1002/jez.b.21339] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
Abstract
One way that organisms cope with constantly changing physical and biological conditions is by regulating the expression of genes and thereby altering protein production. Clearly, altering the protein production to match the environmental demands can be adaptive, but there may be evolutionary barriers to the transition from constitutive expression to regulated expression. In particular, down-regulating a gene when it is not needed means that there will necessarily be a delay in protein production when the protein is up-regulated in the future. We develop a model of simple gene regulation in response to randomly changing environmental conditions. We calculate the long-term behavior of gene expression and determine the fitness consequences of changes in the gene regulation. We then embed this model into a population genetic framework in order to determine the conditions that allow populations to evolve environment-specific transcription rates. The population genetic model follows the evolutionary transition from constitutive expression to regulated expression. There are three distinct possible evolutionary outcomes. The gene may be stuck in the always on position, the gene may first evolve to an intermediate constitutive expression level and then evolve regulation, or regulation can evolve directly from the ancestral state in a smooth fashion. Regulation is most likely to evolve when the costs of mis-expression are low and the transcript decay rate is high. This suggests that genes that have less severe reductions in fitness when mis-expressed are more likely to initially evolve regulation.
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Affiliation(s)
- Stephen R Proulx
- Ecology, Evolution and Marine Biology Department, UC Santa Barbara, Santa Barbara, California 93106-9620, USA.
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Karmakar R. Conversion of graded to binary response in an activator-repressor system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:021905. [PMID: 20365593 DOI: 10.1103/physreve.81.021905] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2009] [Indexed: 05/29/2023]
Abstract
Appropriate regulation of gene expression is essential to ensure that protein synthesis occurs in a selective manner. The control of transcription is the most dominant type of regulation mediated by a complex of molecules such as transcription factors. In general, regulatory molecules are of two types: activator and repressor. Activators promote the initiation of transcription whereas repressors inhibit transcription. In many cases, they regulate the gene transcription on binding the promoter mutually exclusively and the observed gene expression response is either graded or binary. In experiments, the gene expression response is quantified by the amount of proteins produced on varying the concentration of an external inducer molecules in the cell. In this paper, we study a gene regulatory network where activators and repressors both bind the same promoter mutually exclusively. The network is modeled by assuming that the gene can be in three possible states: repressed, unregulated, and active. An exact analytical expression for the steady-state probability distribution of protein levels is then derived. The exact result helps to explain the experimental observations that in the presence of activator molecules the response is graded at all inducer levels, whereas in the presence of both activator and repressor molecules, the response is graded at low and high inducer levels and binary at an intermediate inducer level.
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Affiliation(s)
- Rajesh Karmakar
- Department of Physics, A.K.P.C. Mahavidyalaya, Subhasnagar, Bengai, Hooghly-712 611, India.
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Power-laws in interferon-B mRNA distribution in virus-infected dendritic cells. Biophys J 2009; 97:1984-9. [PMID: 19804729 DOI: 10.1016/j.bpj.2009.05.067] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2009] [Revised: 05/20/2009] [Accepted: 05/26/2009] [Indexed: 11/21/2022] Open
Abstract
Interferon-beta (IFNB1) mRNA shows very large cell-to-cell variability in primary human dendritic cells infected by Newcastle disease virus, with copy numbers varying from a few to several thousands. Analysis of data from the direct measurement of the expression of this gene in its natural chromatin environment in primary human cells shows that the distribution of mRNA across cells follows a power law with an exponent close to -1, and thus encompasses a range of variation much more extensive than a Gaussian. We also investigate the single cell levels of IFNB1 mRNA induced by infection with Texas influenza A mutant viruses, which vary in their capacity to inhibit the signaling pathways responsible for activation of this gene. Here as well we observe power-law behavior for the distribution of IFNB1 mRNA, albeit over a truncated range of values, with exponents similar to the one for cells infected by Newcastle disease virus. We propose a model of stochastic enhanceosome and preinitiation complex formation that incorporates transcriptional pulsing. Analytical and numerical results show good agreement with the observed power laws, and thus support the existence of transcriptional pulsing of an unmodified, intact gene regulated by a natural stimulus.
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Kierzek AM, Zhou L, Wanner BL. Stochastic kinetic model of two component system signalling reveals all-or-none, graded and mixed mode stochastic switching responses. MOLECULAR BIOSYSTEMS 2009; 6:531-42. [PMID: 20174681 DOI: 10.1039/b906951h] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
Two-component systems (TCSs) are prevalent signal transduction systems in bacteria that control innumerable adaptive responses to environmental cues and host-pathogen interactions. We constructed a detailed stochastic kinetic model of two component signalling based on published data. Our model has been validated with flow cytometry data and used to examine reporter gene expression in response to extracellular signal strength. The model shows that, depending on the actual kinetic parameters, TCSs exhibit all-or-none, graded or mixed mode responses. In accordance with other studies, positively autoregulated TCSs exhibit all-or-none responses. Unexpectedly, our model revealed that TCSs lacking a positive feedback loop exhibit not only graded but also mixed mode responses, in which variation of the signal strength alters the level of gene expression in induced cells while the regulated gene continues to be expressed at the basal level in a substantial fraction of cells. The graded response of the TCS changes to mixed mode response by an increase of the translation initiation rate of the histidine kinase. Thus, a TCS is an evolvable design pattern capable of implementing deterministic regulation and stochastic switches associated with both graded and threshold responses. This has implications for understanding the emergence of population diversity in pathogenic bacteria and the design of genetic circuits in synthetic biology applications. The model is available in systems biology markup language (SBML) and systems biology graphical notation (SBGN) formats and can be used as a component of large-scale biochemical reaction network models.
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Affiliation(s)
- Andrzej M Kierzek
- Faculty of Health and Medical Sciences, University of Surrey, Guildford, Surrey, UK GU2 7XH.
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Zhdanov VP. Model of gene transcription including the return of a RNA polymerase to the beginning of a transcriptional cycle. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:051925. [PMID: 20365024 DOI: 10.1103/physreve.80.051925] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2009] [Revised: 10/10/2009] [Indexed: 05/29/2023]
Abstract
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.
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Affiliation(s)
- Vladimir P Zhdanov
- Division of Biological Physics, Department of Applied Physics, Chalmers University of Technology, S-41296 Göteborg, Sweden.
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Smiley MW, Proulx SR. Gene expression dynamics in randomly varying environments. J Math Biol 2009; 61:231-251. [PMID: 19756606 DOI: 10.1007/s00285-009-0298-z] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2008] [Revised: 08/21/2009] [Indexed: 01/01/2023]
Abstract
A simple model of gene regulation in response to stochastically changing environmental conditions is developed and analyzed. The model consists of a differential equation driven by a continuous time 2-state Markov process. The density function of the resulting process converges to a beta distribution. We show that the moments converge to their stationary values exponentially in time. Simulations of a two-stage process where protein production depends on mRNA concentrations are also presented demonstrating that protein concentration tracks the environment whenever the rate of protein turnover is larger than the rate of environmental change. Single-celled organisms are therefore expected to have relatively high mRNA and protein turnover rates for genes that respond to environmental fluctuations.
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Affiliation(s)
- Michael W Smiley
- Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA, 50011, USA
| | - Stephen R Proulx
- Ecology Evolution and Marine Biology, University of California, Santa Barbara, Santa Barbara, CA, 93106-9620, USA.
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Crudu A, Debussche A, Radulescu O. Hybrid stochastic simplifications for multiscale gene networks. BMC SYSTEMS BIOLOGY 2009; 3:89. [PMID: 19735554 PMCID: PMC2761401 DOI: 10.1186/1752-0509-3-89] [Citation(s) in RCA: 75] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/03/2009] [Accepted: 09/07/2009] [Indexed: 01/06/2023]
Abstract
BACKGROUND Stochastic simulation of gene networks by Markov processes has important applications in molecular biology. The complexity of exact simulation algorithms scales with the number of discrete jumps to be performed. Approximate schemes reduce the computational time by reducing the number of simulated discrete events. Also, answering important questions about the relation between network topology and intrinsic noise generation and propagation should be based on general mathematical results. These general results are difficult to obtain for exact models. RESULTS We propose a unified framework for hybrid simplifications of Markov models of multiscale stochastic gene networks dynamics. We discuss several possible hybrid simplifications, and provide algorithms to obtain them from pure jump processes. In hybrid simplifications, some components are discrete and evolve by jumps, while other components are continuous. Hybrid simplifications are obtained by partial Kramers-Moyal expansion [1-3] which is equivalent to the application of the central limit theorem to a sub-model. By averaging and variable aggregation we drastically reduce simulation time and eliminate non-critical reactions. Hybrid and averaged simplifications can be used for more effective simulation algorithms and for obtaining general design principles relating noise to topology and time scales. The simplified models reproduce with good accuracy the stochastic properties of the gene networks, including waiting times in intermittence phenomena, fluctuation amplitudes and stationary distributions. The methods are illustrated on several gene network examples. CONCLUSION Hybrid simplifications can be used for onion-like (multi-layered) approaches to multi-scale biochemical systems, in which various descriptions are used at various scales. Sets of discrete and continuous variables are treated with different methods and are coupled together in a physically justified approach.
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Affiliation(s)
- Alina Crudu
- IRMAR UMR CNRS 6625 Université de Rennes1, Campus de Beaulieu, 35042 Rennes, France.
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Iyer-Biswas S, Hayot F, Jayaprakash C. Stochasticity of gene products from transcriptional pulsing. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:031911. [PMID: 19391975 DOI: 10.1103/physreve.79.031911] [Citation(s) in RCA: 111] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/28/2008] [Indexed: 05/27/2023]
Abstract
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.
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Affiliation(s)
- Srividya Iyer-Biswas
- Department of Physics, Ohio State University, Woodruff Avenue, Columbus, Ohio 43210, USA.
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Abstract
Gene expression is significantly stochastic making modeling of genetic networks challenging. We present an approximation that allows the calculation of not only the mean and variance, but also the distribution of protein numbers. We assume that proteins decay substantially more slowly than their mRNA and confirm that many genes satisfy this relation by using high-throughput data from budding yeast. For a two-stage model of gene expression, with transcription and translation as first-order reactions, we calculate the protein distribution for all times greater than several mRNA lifetimes and thus qualitatively predict the distribution of times for protein levels to first cross an arbitrary threshold. If in addition the fluctuates between inactive and active states, we can find the steady-state protein distribution, which can be bimodal if fluctuations of the promoter are slow. We show that our assumptions imply that protein synthesis occurs in geometrically distributed bursts and allows mRNA to be eliminated from a master equation description. In general, we find that protein distributions are asymmetric and may be poorly characterized by their mean and variance. Through maximum likelihood methods, our expressions should therefore allow more quantitative comparisons with experimental data. More generally, we introduce a technique to derive a simpler, effective dynamics for a stochastic system by eliminating a fast variable.
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Abstract
Gene expression is a fundamentally stochastic process, with randomness in transcription and translation leading to cell-to-cell variations in mRNA and protein levels. This variation appears in organisms ranging from microbes to metazoans, and its characteristics depend both on the biophysical parameters governing gene expression and on gene network structure. Stochastic gene expression has important consequences for cellular function, being beneficial in some contexts and harmful in others. These situations include the stress response, metabolism, development, the cell cycle, circadian rhythms, and aging.
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Affiliation(s)
- Arjun Raj
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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Pandey AK, Raman S, Proff R, Joshi S, Kang CM, Rubin EJ, Husson RN, Sassetti CM. Nitrile-inducible gene expression in mycobacteria. Tuberculosis (Edinb) 2008; 89:12-6. [PMID: 18801704 DOI: 10.1016/j.tube.2008.07.007] [Citation(s) in RCA: 58] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/06/2007] [Revised: 06/19/2008] [Accepted: 07/08/2008] [Indexed: 01/30/2023]
Abstract
The ability to ectopically control gene expression is a fundamental tool for the study of bacterial physiology and pathogenesis. While many efficient inducible expression systems are available for Gram-negative bacteria, few are useful in phylogenetically distant organisms, such as mycobacteria. We have adapted a highly-inducible regulon of Rhodococcus rhodochrous to artificially regulate gene expression in both rapidly-growing environmental mycobacteria and slow-growing pathogens, such as Mycobacterium tuberculosis. We demonstrate that this artificial regulatory circuit behaves as a bistable switch, which can be manipulated regardless of growth phase in vitro, and during intracellular growth in macrophages. High-level overexpression is also possible, facilitating biochemical and structural studies of mycobacterial proteins produced in their native host.
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Affiliation(s)
- Amit K Pandey
- Department of Molecular Genetics and Microbiology, University of Massachusetts Medical School, Worcester, MA 01655, USA
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Abstract
A single gene, regulating its own expression via a positive feedback loop, constitutes a common motif in gene regulatory networks and signalling cascades. Recent experiments on the development of competence in the bacterial population B. subtilis show that the autoregulatory genetic module by itself can give rise to two types of cellular states. The states correspond to the low and high expression states of the master regulator ComK. The high expression state is attained when the ComK protein level exceeds a threshold value leading to a full activation of the autostimulatory loop. Stochasticity in gene expression drives the transitions between the two stable states. In this paper, we explain the appearance of bimodal protein distributions in B. subtilis cell population in the framework of three possible scenarios. In two of the cases, bistability provides the basis for binary gene expression. In the third case, the system is monostable in a deterministic description and stochasticity in gene expression is solely responsible for the appearance of the two expression states.
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van Zon JS, Morelli MJ, Tănase-Nicola S, ten Wolde PR. Diffusion of transcription factors can drastically enhance the noise in gene expression. Biophys J 2006; 91:4350-67. [PMID: 17012327 PMCID: PMC1779939 DOI: 10.1529/biophysj.106.086157] [Citation(s) in RCA: 117] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/29/2006] [Accepted: 09/06/2006] [Indexed: 11/18/2022] Open
Abstract
We study by Green's Function Reaction Dynamics the effect of the diffusive motion of repressor molecules on the noise in mRNA and protein levels for a gene that is under the control of a repressor. We find that spatial fluctuations due to diffusion can drastically enhance the noise in gene expression. After dissociation from the operator, a repressor can rapidly rebind to the DNA. Our results show that the rebinding trajectories are so short that, on this timescale, the RNA polymerase (RNAP) cannot effectively compete with the repressor for binding to the promoter. As a result, a dissociated repressor molecule will on average rebind many times, before it eventually diffuses away. These rebindings thus lower the effective dissociation rate, and this increases the noise in gene expression. Another consequence of the timescale separation between repressor rebinding and RNAP association is that the effect of spatial fluctuations can be described by a well-stirred, zero-dimensional, model by renormalizing the reaction rates for repressor-DNA (un) binding. Our results thus support the use of well-stirred, zero-dimensional models for describing noise in gene expression. We also show that for a fixed repressor strength, the noise due to diffusion can be minimized by increasing the number of repressors or by decreasing the rate of the open complex formation. Lastly, our results emphasize that power spectra are a highly useful tool for studying the propagation of noise through the different stages of gene expression.
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Affiliation(s)
- Jeroen S van Zon
- Division of Physics and Astronomy, Vrije Universiteit, Amsterdam, The Netherlands
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Abstract
We consider a stochastic model of transcription factor (TF)-regulated gene expression. The model describes two genes, gene A and gene B, which synthesize the TFs and the target gene proteins, respectively. We show through analytic calculations that the TF fluctuations have a significant effect on the distribution of the target gene protein levels when the mean TF level falls in the highest sensitive region of the dose-response curve. We further study the effect of reducing the copy number of gene A from two to one. The enhanced TF fluctuations yield results different from those in the deterministic case. The probability that the target gene protein level exceeds a threshold value is calculated with the knowledge of the probability density functions associated with the TF and target gene protein levels. Numerical simulation results for a more detailed stochastic model are shown to be in agreement with those obtained through analytic calculations. The relevance of these results in the context of the genetic disorder haploinsufficiency is pointed out. Some experimental observations on the haploinsufficiency of the tumour suppressor gene, Nkx 3.1, are explained with the help of the stochastic model of TF-regulated gene expression.
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