1
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Hughes FA, Barr AR, Thomas P. Patterns of interdivision time correlations reveal hidden cell cycle factors. eLife 2022; 11:e80927. [PMID: 36377847 PMCID: PMC9822260 DOI: 10.7554/elife.80927] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/09/2022] [Accepted: 11/14/2022] [Indexed: 11/16/2022] Open
Abstract
The time taken for cells to complete a round of cell division is a stochastic process controlled, in part, by intracellular factors. These factors can be inherited across cellular generations which gives rise to, often non-intuitive, correlation patterns in cell cycle timing between cells of different family relationships on lineage trees. Here, we formulate a framework of hidden inherited factors affecting the cell cycle that unifies known cell cycle control models and reveals three distinct interdivision time correlation patterns: aperiodic, alternator, and oscillator. We use Bayesian inference with single-cell datasets of cell division in bacteria, mammalian and cancer cells, to identify the inheritance motifs that underlie these datasets. From our inference, we find that interdivision time correlation patterns do not identify a single cell cycle model but generally admit a broad posterior distribution of possible mechanisms. Despite this unidentifiability, we observe that the inferred patterns reveal interpretable inheritance dynamics and hidden rhythmicity of cell cycle factors. This reveals that cell cycle factors are commonly driven by circadian rhythms, but their period may differ in cancer. Our quantitative analysis thus reveals that correlation patterns are an emergent phenomenon that impact cell proliferation and these patterns may be altered in disease.
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Affiliation(s)
- Fern A Hughes
- Department of Mathematics, Imperial College LondonLondonUnited Kingdom
- MRC London Institute of Medical SciencesLondonUnited Kingdom
| | - Alexis R Barr
- MRC London Institute of Medical SciencesLondonUnited Kingdom
- Institute of Clinical Sciences, Imperial College LondonLondonUnited Kingdom
| | - Philipp Thomas
- Department of Mathematics, Imperial College LondonLondonUnited Kingdom
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2
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Xiong L, Garfinkel A. A common pathway to cancer: Oncogenic mutations abolish p53 oscillations. PROGRESS IN BIOPHYSICS AND MOLECULAR BIOLOGY 2022; 174:28-40. [PMID: 35752348 DOI: 10.1016/j.pbiomolbio.2022.06.002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2021] [Revised: 06/13/2022] [Accepted: 06/20/2022] [Indexed: 06/15/2023]
Abstract
The tumor suppressor p53 oscillates in response to DNA double-strand breaks, a behavior that has been suggested to be essential to its anti-cancer function. Nearly all human cancers have genetic alterations in the p53 pathway; a number of these alterations have been shown to be oncogenic by experiment. These alterations include somatic mutations and copy number variations as well as germline polymorphisms. Intriguingly, they exhibit a mixed pattern of interactions in tumors, such as co-occurrence, mutual exclusivity, and paradoxically, mutual antagonism. Using a differential equation model of p53-Mdm2 dynamics, we employ Hopf bifurcation analysis to show that these alterations have a common mode of action, to abolish the oscillatory competence of p53, thereby, we suggest, impairing its tumor suppressive function. In this analysis, diverse genetic alterations, widely associated with human cancers clinically, have a unified mechanistic explanation of their role in oncogenesis.
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Affiliation(s)
- Lingyun Xiong
- Department of Stem Cell Biology and Regenerative Medicine, University of Southern California, Los Angeles, CA 90007 USA; Department of Quantitative and Computational Biology, University of Southern California, Los Angeles, CA, 90007, USA; Ludwig Institute for Cancer Research, University of Oxford, Oxford, OX3 7DQ, UK
| | - Alan Garfinkel
- Departments of Medicine (Cardiology) and Integrative Biology and Physiology, University of California, Los Angeles, CA, 90095, USA; Newton-Abraham Visiting Professor (2019-2020), Lincoln College and Department of Computer Science, University of Oxford, Oxford, OX1 3DR, UK.
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3
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Abstract
The invention of the Fourier integral in the 19th century laid the foundation for modern spectral analysis methods. This integral decomposes a temporal signal into its frequency components, providing deep insights into its generating process. While this idea has precipitated several scientific and technological advances, its impact has been fairly limited in cell biology, largely due to the difficulties in connecting the underlying noisy intracellular networks to the frequency content of observed single-cell trajectories. Here we develop a spectral theory and computational methodologies tailored specifically to the computation and analysis of frequency spectra of noisy intracellular networks. Specifically, we develop a method to compute the frequency spectrum for general nonlinear networks, and for linear networks we present a decomposition that expresses the frequency spectrum in terms of its sources. Several examples are presented to illustrate how our results provide frequency-based methods for the design and analysis of noisy intracellular networks.
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Affiliation(s)
- Ankit Gupta
- Department of Biosystems Science and Engineering, ETH Zürich, Mattenstrasse 26, 4058, Basel, Switzerland
| | - Mustafa Khammash
- Department of Biosystems Science and Engineering, ETH Zürich, Mattenstrasse 26, 4058, Basel, Switzerland.
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4
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Dey S, Singh A. Diverse role of decoys on emergence and precision of oscillations in a biomolecular clock. Biophys J 2021; 120:5564-5574. [PMID: 34774502 PMCID: PMC8715246 DOI: 10.1016/j.bpj.2021.11.013] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2021] [Revised: 08/11/2021] [Accepted: 11/05/2021] [Indexed: 01/14/2023] Open
Abstract
Biomolecular clocks are key drivers of oscillatory dynamics in diverse biological processes including cell-cycle regulation, circadian rhythms, and pattern formation during development. A minimal clock implementation is based on the classical Goodwin oscillator, in which a repressor protein inhibits its own synthesis via time-delayed negative feedback. Clock motifs, however, do not exist in isolation; its components are open to interacting with the complex environment inside cells. For example, there are ubiquitous high-affinity binding sites along the genome, known as decoys, where transcription factors such as repressor proteins can potentially interact. This binding affects the availability of transcription factors and has often been ignored in theoretical studies. How does such genomic decoy binding impact the clock's robustness and precision? To address this question, we systematically analyze deterministic and stochastic models of the Goodwin oscillator in the presence of reversible binding of the repressor to a finite number of decoy sites. Our analysis reveals that the relative stability of decoy-bound repressors compared to the free repressor plays distinct roles on the emergence and precision of oscillations. Interestingly, active degradation of the bound repressor can induce sustained oscillations that are otherwise absent without decoys. In contrast, decoy abundances can kill oscillation dynamics if the bound repressor is protected from degradation. Taking into account low copy-number fluctuations in clock components, we show that the degradation of the bound repressors enhances precision by attenuating noise in both the amplitude and period of oscillations. Overall, these results highlight the versatile role of otherwise hidden decoys in shaping the stochastic dynamics of biological clocks and emphasize the importance of synthetic decoys in designing robust clocks.
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Affiliation(s)
- Supravat Dey
- Department of Electrical and Computer Engineering, University of Delaware, Newark, Delaware,Corresponding author
| | - Abhyudai Singh
- Department of Electrical and Computer Engineering, University of Delaware, Newark, Delaware,Department of Biomedical Engineering, University of Delaware, Newark, Delaware
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5
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Straube AV, Winkelmann S, Schütte C, Höfling F. Stochastic pH Oscillations in a Model of the Urea-Urease Reaction Confined to Lipid Vesicles. J Phys Chem Lett 2021; 12:9888-9893. [PMID: 34609862 DOI: 10.1021/acs.jpclett.1c03016] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
The urea-urease clock reaction is a pH switch from acid to basic that can turn into a pH oscillator if it occurs inside a suitable open reactor. We numerically study the confinement of the reaction to lipid vesicles, which permit the exchange with an external reservoir by differential transport, enabling the recovery of the pH level and yielding a constant supply of urea molecules. For microscopically small vesicles, the discreteness of the number of molecules requires a stochastic treatment of the reaction dynamics. Our analysis shows that intrinsic noise induces a significant statistical variation of the oscillation period, which increases as the vesicles become smaller. The mean period, however, is found to be remarkably robust for vesicle sizes down to approximately 200 nm, but the periodicity of the rhythm is gradually destroyed for smaller vesicles. The observed oscillations are explained as a canard-like limit cycle that differs from the wide class of conventional feedback oscillators.
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Affiliation(s)
| | | | - Christof Schütte
- Zuse Institute Berlin, Takustraße 7, 14195 Berlin, Germany
- Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany
| | - Felix Höfling
- Zuse Institute Berlin, Takustraße 7, 14195 Berlin, Germany
- Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany
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6
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Cieri F, Zhuang X, Caldwell JZK, Cordes D. Brain Entropy During Aging Through a Free Energy Principle Approach. Front Hum Neurosci 2021; 15:647513. [PMID: 33828471 PMCID: PMC8019811 DOI: 10.3389/fnhum.2021.647513] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/30/2020] [Accepted: 02/25/2021] [Indexed: 02/01/2023] Open
Abstract
Neural complexity and brain entropy (BEN) have gained greater interest in recent years. The dynamics of neural signals and their relations with information processing continue to be investigated through different measures in a variety of noteworthy studies. The BEN of spontaneous neural activity decreases during states of reduced consciousness. This evidence has been showed in primary consciousness states, such as psychedelic states, under the name of "the entropic brain hypothesis." In this manuscript we propose an extension of this hypothesis to physiological and pathological aging. We review this particular facet of the complexity of the brain, mentioning studies that have investigated BEN in primary consciousness states, and extending this view to the field of neuroaging with a focus on resting-state functional Magnetic Resonance Imaging. We first introduce historic and conceptual ideas about entropy and neural complexity, treating the mindbrain as a complex nonlinear dynamic adaptive system, in light of the free energy principle. Then, we review the studies in this field, analyzing the idea that the aim of the neurocognitive system is to maintain a dynamic state of balance between order and chaos, both in terms of dynamics of neural signals and functional connectivity. In our exploration we will review studies both on acute psychedelic states and more chronic psychotic states and traits, such as those in schizophrenia, in order to show the increase of entropy in those states. Then we extend our exploration to physiological and pathological aging, where BEN is reduced. Finally, we propose an interpretation of these results, defining a general trend of BEN in primary states and cognitive aging.
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Affiliation(s)
- Filippo Cieri
- Department of Neurology, Cleveland Clinic Lou Ruvo Center for Brain Health, Las Vegas, NV, United States
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7
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Niemann JH, Winkelmann S, Wolf S, Schütte C. Agent-based modeling: Population limits and large timescales. CHAOS (WOODBURY, N.Y.) 2021; 31:033140. [PMID: 33810764 DOI: 10.1063/5.0031373] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/30/2020] [Accepted: 02/28/2021] [Indexed: 06/12/2023]
Abstract
Modeling, simulation, and analysis of interacting agent systems is a broad field of research, with existing approaches reaching from informal descriptions of interaction dynamics to more formal, mathematical models. In this paper, we study agent-based models (ABMs) given as continuous-time stochastic processes and their pathwise approximation by ordinary and stochastic differential equations (SDEs) for medium to large populations. By means of an appropriately adapted transfer operator approach, we study the behavior of the ABM process on long time scales. We show that, under certain conditions, the transfer operator approach allows us to bridge the gap between the pathwise results for large populations on finite timescales, i.e., the SDE limit model, and approaches built to study dynamical behavior on long time scales like large deviation theory. The latter provides a rigorous analysis of rare events including the associated asymptotic rates on timescales that scale exponentially with the population size. We demonstrate that it is possible to reveal metastable structures and timescales of rare events of the ABM process by finite-length trajectories of the SDE process for large enough populations. This approach has the potential to drastically reduce computational effort for the analysis of ABMs.
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Affiliation(s)
- J-H Niemann
- Zuse Institute Berlin, Berlin 14195, Germany
| | | | - S Wolf
- Department of Mathematics and Computer Science, Freie Universität Berlin, Berlin 14195, Germany
| | - C Schütte
- Zuse Institute Berlin, Berlin 14195, Germany
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8
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Identifying a stochastic clock network with light entrainment for single cells of Neurospora crassa. Sci Rep 2020; 10:15168. [PMID: 32938998 PMCID: PMC7495483 DOI: 10.1038/s41598-020-72213-1] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/21/2020] [Accepted: 08/25/2020] [Indexed: 11/09/2022] Open
Abstract
Stochastic networks for the clock were identified by ensemble methods using genetic algorithms that captured the amplitude and period variation in single cell oscillators of Neurospora crassa. The genetic algorithms were at least an order of magnitude faster than ensemble methods using parallel tempering and appeared to provide a globally optimum solution from a random start in the initial guess of model parameters (i.e., rate constants and initial counts of molecules in a cell). The resulting goodness of fit [Formula: see text] was roughly halved versus solutions produced by ensemble methods using parallel tempering, and the resulting [Formula: see text] per data point was only [Formula: see text] = 2,708.05/953 = 2.84. The fitted model ensemble was robust to variation in proxies for "cell size". The fitted neutral models without cellular communication between single cells isolated by microfluidics provided evidence for only one Stochastic Resonance at one common level of stochastic intracellular noise across days from 6 to 36 h of light/dark (L/D) or in a D/D experiment. When the light-driven phase synchronization was strong as measured by the Kuramoto (K), there was degradation in the single cell oscillations away from the stochastic resonance. The rate constants for the stochastic clock network are consistent with those determined on a macroscopic scale of 107 cells.
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9
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Jia C, Grima R. Small protein number effects in stochastic models of autoregulated bursty gene expression. J Chem Phys 2020; 152:084115. [DOI: 10.1063/1.5144578] [Citation(s) in RCA: 23] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
Affiliation(s)
- Chen Jia
- Division of Applied and Computational Mathematics, Beijing Computational Science Research Center, Beijing 100193, China
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
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10
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Abstract
The stochastic dynamical behaviors of an elementary reaction system can be investigated by the chemical Langevin equation (CLE). However, most of the reactions in engineering belong to complex reactions. It is not appropriate to describe the random evolution process of a complex chemical reaction system by directly using CLE because the foundation of the deviation of CLE is the equilibrium equation of the number of molecules in elementary reaction systems. In the study, the chemical Langevin equation for complex reactions (CLE-CR) is proposed based on the random process theory by introducing the extent of reactions to express the reaction rates of complex reactions. The reaction rates of complex reactions are regarded as some random variables following Poisson distribution. To illustrate the essential consistency of CLE-CR and CLE, the physical meaning of the propensity function in CLE is comprehensively discussed. A numerical example from chemical engineering is employed to demonstrate the effectiveness of CLE-CR and the solving procedure. The results show that CLE-CR can be conveniently applied into engineering to investigate the stochastic dynamical behaviors of complex reaction systems, giving the probabilistic information of the concentration evolution of chemical constituents.
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Affiliation(s)
- Tao Li
- School of Environment and Architecture , University of Shanghai for Science and Technology , Shanghai 200093 , China
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11
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Song S, Yang GS, Park SJ, Hong S, Kim JH, Sung J. Frequency spectrum of chemical fluctuation: A probe of reaction mechanism and dynamics. PLoS Comput Biol 2019; 15:e1007356. [PMID: 31525182 PMCID: PMC6762214 DOI: 10.1371/journal.pcbi.1007356] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2019] [Revised: 09/26/2019] [Accepted: 08/22/2019] [Indexed: 11/18/2022] Open
Abstract
Even in the steady-state, the number of biomolecules in living cells fluctuates dynamically, and the frequency spectrum of this chemical fluctuation carries valuable information about the dynamics of the reactions creating these biomolecules. Recent advances in single-cell techniques enable direct monitoring of the time-traces of the protein number in each cell; however, it is not yet clear how the stochastic dynamics of these time-traces is related to the reaction mechanism and dynamics. Here, we derive a rigorous relation between the frequency-spectrum of the product number fluctuation and the reaction mechanism and dynamics, starting from a generalized master equation. This relation enables us to analyze the time-traces of the protein number and extract information about dynamics of mRNA number and transcriptional regulation, which cannot be directly observed by current experimental techniques. We demonstrate our frequency spectrum analysis of protein number fluctuation, using the gene network model of luciferase expression under the control of the Bmal 1a promoter in mouse fibroblast cells. We also discuss how the dynamic heterogeneity of transcription and translation rates affects the frequency-spectra of the mRNA and protein number.
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Affiliation(s)
- Sanggeun Song
- Center for Chemical Dynamics in Living Cells, Chung-Ang University, Seoul, Korea
- Department of Chemistry, Chung-Ang University, Seoul, Korea
| | - Gil-Suk Yang
- Center for Chemical Dynamics in Living Cells, Chung-Ang University, Seoul, Korea
- Department of Chemistry, Chung-Ang University, Seoul, Korea
| | - Seong Jun Park
- Center for Chemical Dynamics in Living Cells, Chung-Ang University, Seoul, Korea
- Department of Chemistry, Chung-Ang University, Seoul, Korea
| | - Sungguan Hong
- Department of Chemistry, Chung-Ang University, Seoul, Korea
- * E-mail: (SH); (JHK); (JS)
| | - Ji-Hyun Kim
- Center for Chemical Dynamics in Living Cells, Chung-Ang University, Seoul, Korea
- * E-mail: (SH); (JHK); (JS)
| | - Jaeyoung Sung
- Center for Chemical Dynamics in Living Cells, Chung-Ang University, Seoul, Korea
- Department of Chemistry, Chung-Ang University, Seoul, Korea
- * E-mail: (SH); (JHK); (JS)
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12
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Lötstedt P. The Linear Noise Approximation for Spatially Dependent Biochemical Networks. Bull Math Biol 2019; 81:2873-2901. [PMID: 29644520 PMCID: PMC6677697 DOI: 10.1007/s11538-018-0428-0] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/09/2018] [Accepted: 03/29/2018] [Indexed: 10/26/2022]
Abstract
An algorithm for computing the linear noise approximation (LNA) of the reaction-diffusion master equation (RDME) is developed and tested. The RDME is often used as a model for biochemical reaction networks. The LNA is derived for a general discretization of the spatial domain of the problem. If M is the number of chemical species in the network and N is the number of nodes in the discretization in space, then the computational work to determine approximations of the mean and the covariances of the probability distributions is proportional to [Formula: see text] in a straightforward implementation. In our LNA algorithm, the work is proportional to [Formula: see text]. Since N usually is larger than M, this is a significant reduction. The accuracy of the approximation in the algorithm is estimated analytically and evaluated in numerical experiments.
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Affiliation(s)
- Per Lötstedt
- Division of Scientific Computing, Department of Information Technology, Uppsala University, SE-75105, Uppsala, Sweden.
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13
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Keizer EM, Bastian B, Smith RW, Grima R, Fleck C. Extending the linear-noise approximation to biochemical systems influenced by intrinsic noise and slow lognormally distributed extrinsic noise. Phys Rev E 2019; 99:052417. [PMID: 31212540 DOI: 10.1103/physreve.99.052417] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2018] [Indexed: 06/09/2023]
Abstract
It is well known that the kinetics of an intracellular biochemical network is stochastic. This is due to intrinsic noise arising from the random timing of biochemical reactions in the network as well as due to extrinsic noise stemming from the interaction of unknown molecular components with the network and from the cell's changing environment. While there are many methods to study the effect of intrinsic noise on the system dynamics, few exist to study the influence of both types of noise. Here we show how one can extend the conventional linear-noise approximation to allow for the rapid evaluation of the molecule numbers statistics of a biochemical network influenced by intrinsic noise and by slow lognormally distributed extrinsic noise. The theory is applied to simple models of gene regulatory networks and its validity confirmed by comparison with exact stochastic simulations. In particular, we consider three important biological examples. First, we investigate how extrinsic noise modifies the dependence of the variance of the molecule number fluctuations on the rate constants. Second, we show how the mutual information between input and output of a network motif is affected by extrinsic noise. And third, we study the robustness of the ubiquitously found feed-forward loop motifs when subjected to extrinsic noise.
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Affiliation(s)
- Emma M Keizer
- Laboratory of Systems & Synthetic Biology, Wageningen UR, Wageningen, The Netherlands
| | - Björn Bastian
- Institut für Ionenphysik und Angewandte Physik, Universität Innsbruck, Innsbruck, Austria
| | - Robert W Smith
- Laboratory of Systems & Synthetic Biology, Wageningen UR, Wageningen, The Netherlands
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
| | - Christian Fleck
- Laboratory of Systems & Synthetic Biology, Wageningen UR, Wageningen, The Netherlands
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14
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Vet S, Vandervelde A, Gelens L. Excitable dynamics through toxin-induced mRNA cleavage in bacteria. PLoS One 2019; 14:e0212288. [PMID: 30794601 PMCID: PMC6386449 DOI: 10.1371/journal.pone.0212288] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/08/2018] [Accepted: 01/30/2019] [Indexed: 11/19/2022] Open
Abstract
Toxin-antitoxin (TA) systems in bacteria and archaea are small genetic elements consisting of the genes coding for an intracellular toxin and an antitoxin that can neutralize this toxin. In various cases, the toxins cleave the mRNA. In this theoretical work we use deterministic and stochastic modeling to explain how toxin-induced cleavage of mRNA in TA systems can lead to excitability, allowing large transient spikes in toxin levels to be triggered. By using a simplified network where secondary complex formation and transcriptional regulation are not included, we show that a two-dimensional, deterministic model captures the origin of such toxin excitations. Moreover, it allows to increase our understanding by examining the dynamics in the phase plane. By systematically comparing the deterministic results with Gillespie simulations we demonstrate that even though the real TA system is intrinsically stochastic, toxin excitations can be accurately described deterministically. A bifurcation analysis of the system shows that the excitable behavior is due to a nearby Hopf bifurcation in the parameter space, where the system becomes oscillatory. The influence of stress is modeled by varying the degradation rate of the antitoxin and the translation rate of the toxin. We find that stress increases the frequency of toxin excitations. The inclusion of secondary complex formation and transcriptional regulation does not fundamentally change the mechanism of toxin excitations. Finally, we show that including growth rate suppression and translational inhibition can lead to longer excitations, and even cause excitations in cases when the system would otherwise be non-excitable. To conclude, the deterministic model used in this work provides a simple and intuitive explanation of toxin excitations in TA systems.
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Affiliation(s)
- Stefan Vet
- Applied Physics Research Group, Vrije Universiteit Brussel (VUB), Brussels, Belgium
- Interuniversity Institute of Bioinformatics in Brussels (IB2), VUB-ULB, Brussels, Belgium
- Unité de Chronobiologie théorique, Université Libre de Bruxelles (ULB), Brussels, Belgium
| | | | - Lendert Gelens
- Applied Physics Research Group, Vrije Universiteit Brussel (VUB), Brussels, Belgium
- Laboratory of Dynamics in Biological Systems, KU Leuven, Leuven, Belgium
- * E-mail:
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15
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Comparison of Deterministic and Stochastic Regime in a Model for Cdc42 Oscillations in Fission Yeast. Bull Math Biol 2019; 81:1268-1302. [PMID: 30756233 DOI: 10.1007/s11538-019-00573-5] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2018] [Accepted: 01/29/2019] [Indexed: 01/13/2023]
Abstract
Oscillations occur in a wide variety of essential cellular processes, such as cell cycle progression, circadian clocks and calcium signaling in response to stimuli. It remains unclear how intrinsic stochasticity can influence these oscillatory systems. Here, we focus on oscillations of Cdc42 GTPase in fission yeast. We extend our previous deterministic model by Xu and Jilkine to construct a stochastic model, focusing on the fast diffusion case. We use SSA (Gillespie's algorithm) to numerically explore the low copy number regime in this model, and use analytical techniques to study the long-time behavior of the stochastic model and compare it to the equilibria of its deterministic counterpart. Numerical solutions suggest noisy limit cycles exist in the parameter regime in which the deterministic system converges to a stable limit cycle, and quasi-cycles exist in the parameter regime where the deterministic model has a damped oscillation. Near an infinite period bifurcation point, the deterministic model has a sustained oscillation, while stochastic trajectories start with an oscillatory mode and tend to approach deterministic steady states. In the low copy number regime, metastable transitions from oscillatory to steady behavior occur in the stochastic model. Our work contributes to the understanding of how stochastic chemical kinetics can affect a finite-dimensional dynamical system, and destabilize a deterministic steady state leading to oscillations.
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16
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Caranica C, Al-Omari A, Deng Z, Griffith J, Nilsen R, Mao L, Arnold J, Schüttler HB. Ensemble methods for stochastic networks with special reference to the biological clock of Neurospora crassa. PLoS One 2018; 13:e0196435. [PMID: 29768444 PMCID: PMC5955539 DOI: 10.1371/journal.pone.0196435] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2018] [Accepted: 04/12/2018] [Indexed: 11/18/2022] Open
Abstract
A major challenge in systems biology is to infer the parameters of regulatory networks that operate in a noisy environment, such as in a single cell. In a stochastic regime it is hard to distinguish noise from the real signal and to infer the noise contribution to the dynamical behavior. When the genetic network displays oscillatory dynamics, it is even harder to infer the parameters that produce the oscillations. To address this issue we introduce a new estimation method built on a combination of stochastic simulations, mass action kinetics and ensemble network simulations in which we match the average periodogram and phase of the model to that of the data. The method is relatively fast (compared to Metropolis-Hastings Monte Carlo Methods), easy to parallelize, applicable to large oscillatory networks and large (~2000 cells) single cell expression data sets, and it quantifies the noise impact on the observed dynamics. Standard errors of estimated rate coefficients are typically two orders of magnitude smaller than the mean from single cell experiments with on the order of ~1000 cells. We also provide a method to assess the goodness of fit of the stochastic network using the Hilbert phase of single cells. An analysis of phase departures from the null model with no communication between cells is consistent with a hypothesis of Stochastic Resonance describing single cell oscillators. Stochastic Resonance provides a physical mechanism whereby intracellular noise plays a positive role in establishing oscillatory behavior, but may require model parameters, such as rate coefficients, that differ substantially from those extracted at the macroscopic level from measurements on populations of millions of communicating, synchronized cells.
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Affiliation(s)
- C. Caranica
- Department of Statistics, University of Georgia, Athens, Georgia
| | - A. Al-Omari
- Department of Biomedical Systems and Informatics Engineering, Yarmouk University, Irbid, Jordan
| | - Z. Deng
- School of Electrical and Computer Engineering, College of Engineering, University of Georgia, Athens, Georgia
| | - J. Griffith
- Genetics Department, University of Georgia, Athens, Georgia
- College of Agricultural and Environmental Sciences, University of Georgia, Athens, Georgia
| | - R. Nilsen
- Genetics Department, University of Georgia, Athens, Georgia
| | - L. Mao
- School of Electrical and Computer Engineering, College of Engineering, University of Georgia, Athens, Georgia
| | - J. Arnold
- Genetics Department, University of Georgia, Athens, Georgia
- * E-mail:
| | - H.-B. Schüttler
- Department of Physics and Astronomy, University of Georgia, Athens, Georgia
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17
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Information Theoretical Study of Cross-Talk Mediated Signal Transduction in MAPK Pathways. ENTROPY 2017. [DOI: 10.3390/e19090469] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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18
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Phillips NE, Manning C, Papalopulu N, Rattray M. Identifying stochastic oscillations in single-cell live imaging time series using Gaussian processes. PLoS Comput Biol 2017; 13:e1005479. [PMID: 28493880 PMCID: PMC5444866 DOI: 10.1371/journal.pcbi.1005479] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2016] [Revised: 05/25/2017] [Accepted: 03/24/2017] [Indexed: 12/05/2022] Open
Abstract
Multiple biological processes are driven by oscillatory gene expression at different time scales. Pulsatile dynamics are thought to be widespread, and single-cell live imaging of gene expression has lead to a surge of dynamic, possibly oscillatory, data for different gene networks. However, the regulation of gene expression at the level of an individual cell involves reactions between finite numbers of molecules, and this can result in inherent randomness in expression dynamics, which blurs the boundaries between aperiodic fluctuations and noisy oscillators. This underlies a new challenge to the experimentalist because neither intuition nor pre-existing methods work well for identifying oscillatory activity in noisy biological time series. Thus, there is an acute need for an objective statistical method for classifying whether an experimentally derived noisy time series is periodic. Here, we present a new data analysis method that combines mechanistic stochastic modelling with the powerful methods of non-parametric regression with Gaussian processes. Our method can distinguish oscillatory gene expression from random fluctuations of non-oscillatory expression in single-cell time series, despite peak-to-peak variability in period and amplitude of single-cell oscillations. We show that our method outperforms the Lomb-Scargle periodogram in successfully classifying cells as oscillatory or non-oscillatory in data simulated from a simple genetic oscillator model and in experimental data. Analysis of bioluminescent live-cell imaging shows a significantly greater number of oscillatory cells when luciferase is driven by a Hes1 promoter (10/19), which has previously been reported to oscillate, than the constitutive MoMuLV 5’ LTR (MMLV) promoter (0/25). The method can be applied to data from any gene network to both quantify the proportion of oscillating cells within a population and to measure the period and quality of oscillations. It is publicly available as a MATLAB package. Technological advances now allow us to observe gene expression in real-time at a single-cell level. In a wide variety of biological contexts this new data has revealed that gene expression is highly dynamic and possibly oscillatory. It is thought that periodic gene expression may be useful for keeping track of time and space, as well as transmitting information about signalling cues. Classifying a time series as periodic from single cell data is difficult because it is necessary to distinguish whether peaks and troughs are generated from an underlying oscillator or whether they are aperiodic fluctuations. To this end, we present a novel tool to classify live-cell data as oscillatory or non-oscillatory that accounts for inherent biological noise. We first demonstrate that the method outperforms a competing scheme in classifying computationally simulated single-cell data, and we subsequently analyse live-cell imaging time series. Our method is able to successfully detect oscillations in a known genetic oscillator, but it classifies data from a constitutively expressed gene as aperiodic. The method forms a basis for discovering new gene expression oscillators and quantifying how oscillatory activity alters in response to changes in cell fate and environmental or genetic perturbations.
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Affiliation(s)
- Nick E. Phillips
- Faculty of Biology, Medicine and Health, University of Manchester, Manchester, United Kingdom
| | - Cerys Manning
- Faculty of Biology, Medicine and Health, University of Manchester, Manchester, United Kingdom
| | - Nancy Papalopulu
- Faculty of Biology, Medicine and Health, University of Manchester, Manchester, United Kingdom
- * E-mail: (NP); (MR)
| | - Magnus Rattray
- Faculty of Biology, Medicine and Health, University of Manchester, Manchester, United Kingdom
- * E-mail: (NP); (MR)
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19
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Falk J, Mendler M, Drossel B. A minimal model of burst-noise induced bistability. PLoS One 2017; 12:e0176410. [PMID: 28448638 PMCID: PMC5407650 DOI: 10.1371/journal.pone.0176410] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/21/2016] [Accepted: 04/09/2017] [Indexed: 11/18/2022] Open
Abstract
We investigate the influence of intrinsic noise on stable states of a one-dimensional dynamical system that shows in its deterministic version a saddle-node bifurcation between monostable and bistable behaviour. The system is a modified version of the Schlögl model, which is a chemical reaction system with only one type of molecule. The strength of the intrinsic noise is varied without changing the deterministic description by introducing bursts in the autocatalytic production step. We study the transitions between monostable and bistable behavior in this system by evaluating the number of maxima of the stationary probability distribution. We find that changing the size of bursts can destroy and even induce saddle-node bifurcations. This means that a bursty production of molecules can qualitatively change the dynamics of a chemical reaction system even when the deterministic description remains unchanged.
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Affiliation(s)
- Johannes Falk
- Institut für Festkörperphysik, Technische Universität Darmstadt, Hochschulstr. 6, 64289 Darmstadt, Germany
- * E-mail:
| | - Marc Mendler
- Institut für Festkörperphysik, Technische Universität Darmstadt, Hochschulstr. 6, 64289 Darmstadt, Germany
| | - Barbara Drossel
- Institut für Festkörperphysik, Technische Universität Darmstadt, Hochschulstr. 6, 64289 Darmstadt, Germany
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20
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Guisoni N, Monteoliva D, Diambra L. Promoters Architecture-Based Mechanism for Noise-Induced Oscillations in a Single-Gene Circuit. PLoS One 2016; 11:e0151086. [PMID: 26958852 PMCID: PMC4784906 DOI: 10.1371/journal.pone.0151086] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2016] [Accepted: 02/23/2016] [Indexed: 12/20/2022] Open
Abstract
It is well known that single-gene circuits with negative feedback loop can lead to oscillatory gene expression when they operate with time delay. In order to generate these oscillations many processes can contribute to properly timing such delay. Here we show that the time delay coming from the transitions between internal states of the cis-regulatory system (CRS) can drive sustained oscillations in an auto-repressive single-gene circuit operating in a small volume like a cell. We found that the cooperative binding of repressor molecules is not mandatory for a oscillatory behavior if there are enough binding sites in the CRS. These oscillations depend on an adequate balance between the CRS kinetic, and the synthesis/degradation rates of repressor molecules. This finding suggest that the multi-site CRS architecture can play a key role for oscillatory behavior of gene expression. Finally, our results can also help to synthetic biologists on the design of the promoters architecture for new genetic oscillatory circuits.
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Affiliation(s)
- N. Guisoni
- Instituto de Física de Líquidos y Sistemas Biológicos, Universidad Nacional de La Plata, La Plata, Argentina
| | - D. Monteoliva
- Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, La Plata, Argentina
| | - L. Diambra
- Centro Regional de Estudios Genómicos, Universidad Nacional de La Plata, La Plata, Argentina
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21
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Smith S, Cianci C, Grima R. Model reduction for stochastic chemical systems with abundant species. J Chem Phys 2015; 143:214105. [DOI: 10.1063/1.4936394] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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22
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Abel JH, Widmer LA, John PCS, Stelling J, Doyle FJ. A Coupled Stochastic Model Explains Differences in Cry Knockout Behavior. ACTA ACUST UNITED AC 2015. [DOI: 10.1109/lls.2015.2439498] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022]
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23
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Maity AK, Chaudhury P, Banik SK. Role of relaxation time scale in noisy signal transduction. PLoS One 2015; 10:e0123242. [PMID: 25955500 PMCID: PMC4425683 DOI: 10.1371/journal.pone.0123242] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2014] [Accepted: 02/28/2015] [Indexed: 11/18/2022] Open
Abstract
Intra-cellular fluctuations, mainly triggered by gene expression, are an inevitable phenomenon observed in living cells. It influences generation of phenotypic diversity in genetically identical cells. Such variation of cellular components is beneficial in some contexts but detrimental in others. To quantify the fluctuations in a gene product, we undertake an analytical scheme for studying few naturally abundant linear as well as branched chain network motifs. We solve the Langevin equations associated with each motif under the purview of linear noise approximation and derive the expressions for Fano factor and mutual information in close analytical form. Both quantifiable expressions exclusively depend on the relaxation time (decay rate constant) and steady state population of the network components. We investigate the effect of relaxation time constraints on Fano factor and mutual information to indentify a time scale domain where a network can recognize the fluctuations associated with the input signal more reliably. We also show how input population affects both quantities. We extend our calculation to long chain linear motif and show that with increasing chain length, the Fano factor value increases but the mutual information processing capability decreases. In this type of motif, the intermediate components act as a noise filter that tune up input fluctuations and maintain optimum fluctuations in the output. For branched chain motifs, both quantities vary within a large scale due to their network architecture and facilitate survival of living system in diverse environmental conditions.
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Affiliation(s)
| | | | - Suman K Banik
- Department of Chemistry, Bose Institute, Kolkata, India
- * E-mail:
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24
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Duncan A, Liao S, Vejchodský T, Erban R, Grima R. Noise-induced multistability in chemical systems: Discrete versus continuum modeling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:042111. [PMID: 25974443 DOI: 10.1103/physreve.91.042111] [Citation(s) in RCA: 32] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2014] [Indexed: 06/04/2023]
Abstract
The noisy dynamics of chemical systems is commonly studied using either the chemical master equation (CME) or the chemical Fokker-Planck equation (CFPE). The latter is a continuum approximation of the discrete CME approach. It has recently been shown that for a particular system, the CFPE captures noise-induced multistability predicted by the CME. This phenomenon involves the CME's marginal probability distribution changing from unimodal to multimodal as the system size decreases below a critical value. We here show that the CFPE does not always capture noise-induced multistability. In particular we find simple chemical systems for which the CME predicts noise-induced multistability, whereas the CFPE predicts monostability for all system sizes.
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Affiliation(s)
- Andrew Duncan
- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
| | - Shuohao Liao
- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
| | - Tomáš Vejchodský
- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
- Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-115 67, Czech Republic
| | - Radek Erban
- Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom
| | - Ramon Grima
- School of Biological Sciences, Kings Buildings, Mayfield Road, University of Edinburgh, EH9 3JF, United Kingdom
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25
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Greenwood PE, McDonnell MD, Ward LM. Dynamics of Gamma Bursts in Local Field Potentials. Neural Comput 2015; 27:74-103. [DOI: 10.1162/neco_a_00688] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/02/2023]
Abstract
In this letter, we provide a stochastic analysis of, and supporting simulation data for, a stochastic model of the generation of gamma bursts in local field potential (LFP) recordings by interacting populations of excitatory and inhibitory neurons. Our interest is in behavior near a fixed point of the stochastic dynamics of the model. We apply a recent limit theorem of stochastic dynamics to probe into details of this local behavior, obtaining several new results. We show that the stochastic model can be written in terms of a rotation multiplied by a two-dimensional standard Ornstein-Uhlenbeck (OU) process. Viewing the rewritten process in terms of phase and amplitude processes, we are able to proceed further in analysis. We demonstrate that gamma bursts arise in the model as excursions of the modulus of the OU process. The associated pair of stochastic phase and amplitude processes satisfies their own pair of stochastic differential equations, which indicates that large phase slips occur between gamma bursts. This behavior is mirrored in LFP data simulated from the original model. These results suggest that the rewritten model is a valid representation of the behavior near the fixed point for a wide class of models of oscillatory neural processes.
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Affiliation(s)
- Priscilla E. Greenwood
- Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z4, Canada
| | - Mark D. McDonnell
- Computational and Theoretical Neuroscience Laboratory, Institute for Telecommunications Research, University of South Australia, Mawson Lakes, SA 5001, Australia
| | - Lawrence M. Ward
- Department of Psychology and Brain Research Centre, University of British Columbia, Vancouver, BC V6T 1Z4 Canada
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26
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Woller A, Gonze D, Erneux T. The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment. Phys Biol 2014; 11:045002. [PMID: 25075916 DOI: 10.1088/1478-3975/11/4/045002] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Abstract
The three-variable Goodwin oscillator is a minimal model demonstrating the emergence of oscillations in simple biochemical feedback systems. As a prototypical oscillator, this model was extensively studied from a theoretical point of view and applied to various biological systems, including circadian clocks. Here, we reexamine this model, derive analytically the amplitude equation near the Hopf bifurcation and investigate the effect of a periodic modulation of the oscillator. In particular, we compare the entrainment performance when the free oscillator displays either self-sustained or damped oscillations. We discuss the results in the context of circadian oscillators.
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Affiliation(s)
- Aurore Woller
- Unité Mixte de Recherche 1011, Université Lille 2, INSERM, Institut Pasteur de Lille, Lille, France. Laboratoire de Physique des Lasers, Atomes, Molécules, Unité Mixte de Recherche 8523, Université Lille 1, CNRS, Villeneuve d'Ascq, France
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27
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Brett T, Galla T. Gaussian approximations for stochastic systems with delay: Chemical Langevin equation and application to a Brusselator system. J Chem Phys 2014; 140:124112. [DOI: 10.1063/1.4867786] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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28
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Mattei TA. Unveiling complexity: non-linear and fractal analysis in neuroscience and cognitive psychology. Front Comput Neurosci 2014; 8:17. [PMID: 24600384 PMCID: PMC3930866 DOI: 10.3389/fncom.2014.00017] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Key Words] [Track Full Text] [Download PDF] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/05/2014] [Accepted: 02/05/2014] [Indexed: 11/13/2022] Open
Affiliation(s)
- Tobias A Mattei
- Department of Neurological Surgery, The Ohio State University Medical Center Columbus, OH, USA
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