1
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Jia C, Grima R. Holimap: an accurate and efficient method for solving stochastic gene network dynamics. Nat Commun 2024; 15:6557. [PMID: 39095346 PMCID: PMC11297302 DOI: 10.1038/s41467-024-50716-z] [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: 03/02/2024] [Accepted: 07/13/2024] [Indexed: 08/04/2024] Open
Abstract
Gene-gene interactions are crucial to the control of sub-cellular processes but our understanding of their stochastic dynamics is hindered by the lack of simulation methods that can accurately and efficiently predict how the distributions of gene product numbers vary across parameter space. To overcome these difficulties, here we present Holimap (high-order linear-mapping approximation), an approach that approximates the protein or mRNA number distributions of a complex gene regulatory network by the distributions of a much simpler reaction system. We demonstrate Holimap's computational advantages over conventional methods by applying it to predict the stochastic time-dependent dynamics of various gene networks, including transcriptional networks ranging from simple autoregulatory loops to complex randomly connected networks, post-transcriptional networks, and post-translational networks. Holimap is ideally suited to study how the intricate network of gene-gene interactions results in precise coordination and control of gene expression.
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Affiliation(s)
- Chen Jia
- Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing, China
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, UK.
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2
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Jo H, Hong H, Hwang HJ, Chang W, Kim JK. Density physics-informed neural networks reveal sources of cell heterogeneity in signal transduction. PATTERNS (NEW YORK, N.Y.) 2024; 5:100899. [PMID: 38370126 PMCID: PMC10873160 DOI: 10.1016/j.patter.2023.100899] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/08/2023] [Revised: 11/05/2023] [Accepted: 11/24/2023] [Indexed: 02/20/2024]
Abstract
The transduction time between signal initiation and final response provides valuable information on the underlying signaling pathway, including its speed and precision. Furthermore, multi-modality in a transduction-time distribution indicates that the response is regulated by multiple pathways with different transduction speeds. Here, we developed a method called density physics-informed neural networks (Density-PINNs) to infer the transduction-time distribution from measurable final stress response time traces. We applied Density-PINNs to single-cell gene expression data from sixteen promoters regulated by unknown pathways in response to antibiotic stresses. We found that promoters with slower signaling initiation and transduction exhibit larger cell-to-cell heterogeneity in response intensity. However, this heterogeneity was greatly reduced when the response was regulated by slow and fast pathways together. This suggests a strategy for identifying effective signaling pathways for consistent cellular responses to disease treatments. Density-PINNs can also be applied to understand other time delay systems, including infectious diseases.
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Affiliation(s)
- Hyeontae Jo
- Biomedical Mathematics Group, Pioneer Research Center for Mathematical and Computational Sciences, Institute for Basic Science, Daejeon 34126, Republic of Korea
| | - Hyukpyo Hong
- Biomedical Mathematics Group, Pioneer Research Center for Mathematical and Computational Sciences, Institute for Basic Science, Daejeon 34126, Republic of Korea
- Department of Mathematical Sciences, KAIST, Daejeon 34141, Republic of Korea
| | - Hyung Ju Hwang
- Department of Mathematics, Pohang University of Science and Technology, Pohang 37673, Republic of Korea
| | - Won Chang
- Division of Statistics and Data Science, University of Cincinnati, Cincinnati, OH 45221, USA
| | - Jae Kyoung Kim
- Biomedical Mathematics Group, Pioneer Research Center for Mathematical and Computational Sciences, Institute for Basic Science, Daejeon 34126, Republic of Korea
- Department of Mathematical Sciences, KAIST, Daejeon 34141, Republic of Korea
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3
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Goetz A, Akl H, Dixit P. The ability to sense the environment is heterogeneously distributed in cell populations. eLife 2024; 12:RP87747. [PMID: 38293960 PMCID: PMC10942581 DOI: 10.7554/elife.87747] [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] [Indexed: 02/01/2024] Open
Abstract
Channel capacity of signaling networks quantifies their fidelity in sensing extracellular inputs. Low estimates of channel capacities for several mammalian signaling networks suggest that cells can barely detect the presence/absence of environmental signals. However, given the extensive heterogeneity and temporal stability of cell state variables, we hypothesize that the sensing ability itself may depend on the state of the cells. In this work, we present an information-theoretic framework to quantify the distribution of sensing abilities from single-cell data. Using data on two mammalian pathways, we show that sensing abilities are widely distributed in the population and most cells achieve better resolution of inputs compared to an 'average cell'. We verify these predictions using live-cell imaging data on the IGFR/FoxO pathway. Importantly, we identify cell state variables that correlate with cells' sensing abilities. This information-theoretic framework will significantly improve our understanding of how cells sense in their environment.
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Affiliation(s)
- Andrew Goetz
- Department of Biomedical Engineering, Yale UniversityNew HavenUnited States
| | - Hoda Akl
- Department of Physics, University of FloridaGainesvilleUnited States
| | - Purushottam Dixit
- Department of Biomedical Engineering, Yale UniversityNew HavenUnited States
- Systems Biology Institute, Yale UniversityWest HavenUnited States
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4
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Wagner V, Strässer R, Allgöwer F, Radde N. A Provably Convergent Control Closure Scheme for the Method of Moments of the Chemical Master Equation. J Chem Theory Comput 2023; 19:9049-9059. [PMID: 38051675 DOI: 10.1021/acs.jctc.3c00548] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/07/2023]
Abstract
In this article, we introduce a novel moment closure scheme based on concepts from model predictive control (MPC) to accurately describe the time evolution of the statistical moments of the solution of the chemical master equation (CME). The method of moments, a set of ordinary differential equations frequently used to calculate the first nm moments, is generally not closed since lower-order moments depend on higher-order moments. To overcome this limitation, we interpret the moment equations as a nonlinear dynamical system, where the first nm moments serve as states, and the closing moments serve as the control input. We demonstrate the efficacy of our approach using three example systems and show that it outperforms existing closure schemes. For polynomial systems, which encompass all mass-action systems, we provide probability bounds for the error between true and estimated moment trajectories. We achieve this by combining the convergence properties of a priori moment estimates from stochastic simulations with guarantees for nonlinear reference tracking MPC. Our proposed method offers an effective solution to accurately predict the time evolution of moments of the CME, which has wide-ranging implications for many fields, including biology, chemistry, and engineering.
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Affiliation(s)
- Vincent Wagner
- University of Stuttgart, Institute for Stochastics and Applications, 70550 Stuttgart, Germany
| | - Robin Strässer
- University of Stuttgart, Institute for Systems Theory and Automatic Control, 70550 Stuttgart, Germany
| | - Frank Allgöwer
- University of Stuttgart, Institute for Systems Theory and Automatic Control, 70550 Stuttgart, Germany
| | - Nicole Radde
- University of Stuttgart, Institute for Stochastics and Applications, 70550 Stuttgart, Germany
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5
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Sinzger-D’Angelo M, Startceva S, Koeppl H. Bye bye, linearity, bye: quantification of the mean for linear CRNs in a random environment. J Math Biol 2023; 87:43. [PMID: 37573263 PMCID: PMC10423146 DOI: 10.1007/s00285-023-01973-x] [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: 08/26/2022] [Revised: 07/04/2023] [Accepted: 07/22/2023] [Indexed: 08/14/2023]
Abstract
Molecular reactions within a cell are inherently stochastic, and cells often differ in morphological properties or interact with a heterogeneous environment. Consequently, cell populations exhibit heterogeneity both due to these intrinsic and extrinsic causes. Although state-of-the-art studies that focus on dissecting this heterogeneity use single-cell measurements, the bulk data that shows only the mean expression levels is still in routine use. The fingerprint of the heterogeneity is present also in bulk data, despite being hidden from direct measurement. In particular, this heterogeneity can affect the mean expression levels via bimolecular interactions with low-abundant environment species. We make this statement rigorous for the class of linear reaction systems that are embedded in a discrete state Markov environment. The analytic expression that we provide for the stationary mean depends on the reaction rate constants of the linear subsystem, as well as the generator and stationary distribution of the Markov environment. We demonstrate the effect of the environment on the stationary mean. Namely, we show how the heterogeneous case deviates from the quasi-steady state (Q.SS) case when the embedded system is fast compared to the environment.
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Affiliation(s)
- Mark Sinzger-D’Angelo
- Electrical Engineering and Information Technology, Technische Universität Darmstadt, Darmstadt, Germany
- Centre for Synthetic Biology, Technische Universität Darmstadt, Darmstadt, Germany
| | - Sofia Startceva
- Electrical Engineering and Information Technology, Technische Universität Darmstadt, Darmstadt, Germany
- Centre for Synthetic Biology, Technische Universität Darmstadt, Darmstadt, Germany
| | - Heinz Koeppl
- Electrical Engineering and Information Technology, Technische Universität Darmstadt, Darmstadt, Germany
- Centre for Synthetic Biology, Technische Universität Darmstadt, Darmstadt, Germany
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6
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Fletcher A, Wunderlich Z, Enciso G. Shadow enhancers mediate trade-offs between transcriptional noise and fidelity. PLoS Comput Biol 2023; 19:e1011071. [PMID: 37205714 DOI: 10.1371/journal.pcbi.1011071] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2022] [Accepted: 04/03/2023] [Indexed: 05/21/2023] Open
Abstract
Enhancers are stretches of regulatory DNA that bind transcription factors (TFs) and regulate the expression of a target gene. Shadow enhancers are two or more enhancers that regulate the same target gene in space and time and are associated with most animal developmental genes. These multi-enhancer systems can drive more consistent transcription than single enhancer systems. Nevertheless, it remains unclear why shadow enhancer TF binding sites are distributed across multiple enhancers rather than within a single large enhancer. Here, we use a computational approach to study systems with varying numbers of TF binding sites and enhancers. We employ chemical reaction networks with stochastic dynamics to determine the trends in transcriptional noise and fidelity, two key performance objectives of enhancers. This reveals that while additive shadow enhancers do not differ in noise and fidelity from their single enhancer counterparts, sub- and superadditive shadow enhancers have noise and fidelity trade-offs not available to single enhancers. We also use our computational approach to compare the duplication and splitting of a single enhancer as mechanisms for the generation of shadow enhancers and find that the duplication of enhancers can decrease noise and increase fidelity, although at the metabolic cost of increased RNA production. A saturation mechanism for enhancer interactions similarly improves on both of these metrics. Taken together, this work highlights that shadow enhancer systems may exist for several reasons: genetic drift or the tuning of key functions of enhancers, including transcription fidelity, noise and output.
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Affiliation(s)
- Alvaro Fletcher
- Mathematical, Computational, and Systems Biology, University of California, Irvine, Irvine, CA, United States of America
| | - Zeba Wunderlich
- Department of Biology, Boston University, Boston, MA, United States of America
- Biological Design Center, Boston University, Boston, MA, United States of America
| | - German Enciso
- Department of Mathematics, University of California, Irvine, Irvine, CA, United States of America
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7
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Glastad RC, Johnston IG. Mitochondrial network structure controls cell-to-cell mtDNA variability generated by cell divisions. PLoS Comput Biol 2023; 19:e1010953. [PMID: 36952562 PMCID: PMC10072490 DOI: 10.1371/journal.pcbi.1010953] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2022] [Revised: 04/04/2023] [Accepted: 02/15/2023] [Indexed: 03/25/2023] Open
Abstract
Mitochondria are highly dynamic organelles, containing vital populations of mitochondrial DNA (mtDNA) distributed throughout the cell. Mitochondria form diverse physical structures in different cells, from cell-wide reticulated networks to fragmented individual organelles. These physical structures are known to influence the genetic makeup of mtDNA populations between cell divisions, but their influence on the inheritance of mtDNA at divisions remains less understood. Here, we use statistical and computational models of mtDNA content inside and outside the reticulated network to quantify how mitochondrial network structure can control the variances of inherited mtDNA copy number and mutant load. We assess the use of moment-based approximations to describe heteroplasmy variance and identify several cases where such an approach has shortcomings. We show that biased inclusion of one mtDNA type in the network can substantially increase heteroplasmy variance (acting as a genetic bottleneck), and controlled distribution of network mass and mtDNA through the cell can conversely reduce heteroplasmy variance below a binomial inheritance picture. Network structure also allows the generation of heteroplasmy variance while controlling copy number inheritance to sub-binomial levels, reconciling several observations from the experimental literature. Overall, different network structures and mtDNA arrangements within them can control the variances of key variables to suit a palette of different inheritance priorities.
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Affiliation(s)
| | - Iain G. Johnston
- Department of Mathematics, University of Bergen, Bergen, Norway
- Computational Biology Unit, University of Bergen, Bergen, Norway
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8
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Galstyan V, Saakian DB. Quantifying the stochasticity of policy parameters in reinforcement learning problems. Phys Rev E 2023; 107:034112. [PMID: 37072940 DOI: 10.1103/physreve.107.034112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/07/2022] [Accepted: 02/16/2023] [Indexed: 04/20/2023]
Abstract
The stochastic dynamics of reinforcement learning is studied using a master equation formalism. We consider two different problems-Q learning for a two-agent game and the multiarmed bandit problem with policy gradient as the learning method. The master equation is constructed by introducing a probability distribution over continuous policy parameters or over both continuous policy parameters and discrete state variables (a more advanced case). We use a version of the moment closure approximation to solve for the stochastic dynamics of the models. Our method gives accurate estimates for the mean and the (co)variance of policy variables. For the case of the two-agent game, we find that the variance terms are finite at steady state and derive a system of algebraic equations for computing them directly.
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Affiliation(s)
- Vahe Galstyan
- AMOLF, Science Park 104, 1098 XG Amsterdam, Netherlands
- A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, 2 Alikhanian Brothers Street, Yerevan 375036, Armenia
| | - David B Saakian
- A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, 2 Alikhanian Brothers Street, Yerevan 375036, Armenia
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9
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Zhao H, Sarpeshkar R, Mandal S. A Compact and Power-Efficient Noise Generator for Stochastic Simulations. IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS. I, REGULAR PAPERS : A PUBLICATION OF THE IEEE CIRCUITS AND SYSTEMS SOCIETY 2023; 70:3-16. [PMID: 39157673 PMCID: PMC11329235 DOI: 10.1109/tcsi.2022.3199561] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 08/20/2024]
Abstract
This paper describes an adaptive noise generator circuit suitable for on-chip simulations of stochastic chemical kinetics. The circuit uses amplified BJT white noise and adaptive low-pass filtering to emulate the power spectrum and autocorrelation of random telegraph signals (RTS) with Poisson-distributed level transitions. A current-mode implementation in the AMS 0.35 μm BiCMOS process shows excellent agreement with theoretical results from the Gillespie stochastic simulation algorithm over a 60 dB range in mean current levels (modeling molecule count numbers). The circuit has an estimated layout area of 0.032 mm2 and typically consumes 400 μA, which are 73% and 50% less, respectively, than prior implementations. Moreover, it does not require any off-chip capacitors. Experimental results from a discrete board-level implementation of the circuit are in good agreement with theoretical predictions.
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Affiliation(s)
- Haixiang Zhao
- Dept. of Electrical & Computer Engineering, University of Florida, Gainesville, FL 32611, USA
| | - Rahul Sarpeshkar
- Engineering departments of Engineering, Physics, Microbiology&Immunology, and Molecular and Systems Biology, Dartmouth College, Hanover, NH 03755, USA
| | - Soumyajit Mandal
- Instrumentation Division, Brookhaven NationalLaboratory, Upton, NY 11973, USA
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10
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García-Tejera R, Schumacher L, Grima R. Regulation of stem cell dynamics through volume exclusion. Proc Math Phys Eng Sci 2022. [DOI: 10.1098/rspa.2022.0376] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022] Open
Abstract
The maintenance and regeneration of adult tissues rely on the self-renewal of stem cells. Regeneration without over-proliferation requires precise regulation of the stem cell proliferation and differentiation rates. The nature of such regulatory mechanisms in different tissues, and how to incorporate them in models of stem cell population dynamics, is incompletely understood. The critical birth-death (CBD) process is widely used to model stem cell populations, capturing key phenomena, such as scaling laws in clone size distributions. However, the CBD process neglects regulatory mechanisms. Here, we propose the birth-death process with volume exclusion (vBD), a variation of the birth-death process that considers crowding effects, such as may arise due to limited space in a stem cell niche. While the deterministic rate equations predict a single non-trivial attracting steady state, the master equation predicts extinction and transient distributions of stem cell numbers with three possible behaviours: long-lived quasi-steady state (QSS), and short-lived bimodal or unimodal distributions. In all cases, we approximate solutions to the vBD master equation using a renormalized system-size expansion, QSS approximation and the Wentzel–Kramers–Brillouin method. Our study suggests that the size distribution of a stem cell population bears signatures that are useful to detect negative feedback mediated via volume exclusion.
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Affiliation(s)
- Rodrigo García-Tejera
- Centre for Regenerative Medicine, University of Edinburgh, 5 Little France Dr, Edinburgh EH16 4UU, UK
- School of Biological Sciences, University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JF, UK
| | - Linus Schumacher
- Centre for Regenerative Medicine, University of Edinburgh, 5 Little France Dr, Edinburgh EH16 4UU, UK
- School of Biological Sciences, University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JF, UK
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JF, UK
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11
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Wagner V, Castellaz B, Oesting M, Radde N. Quasi-Entropy Closure: a fast and reliable approach to close the moment equations of the Chemical Master Equation. Bioinformatics 2022; 38:4352-4359. [PMID: 35916726 DOI: 10.1093/bioinformatics/btac501] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/30/2021] [Revised: 05/13/2022] [Accepted: 08/01/2022] [Indexed: 12/24/2022] Open
Abstract
MOTIVATION The Chemical Master Equation is a stochastic approach to describe the evolution of a (bio)chemical reaction system. Its solution is a time-dependent probability distribution on all possible configurations of the system. As this number is typically large, the Master Equation is often practically unsolvable. The Method of Moments reduces the system to the evolution of a few moments, which are described by ordinary differential equations. Those equations are not closed, since lower order moments generally depend on higher order moments. Various closure schemes have been suggested to solve this problem. Two major problems with these approaches are first that they are open loop systems, which can diverge from the true solution, and second, some of them are computationally expensive. RESULTS Here we introduce Quasi-Entropy Closure, a moment-closure scheme for the Method of Moments. It estimates higher order moments by reconstructing the distribution that minimizes the distance to a uniform distribution subject to lower order moment constraints. Quasi-Entropy Closure can be regarded as an advancement of Zero-Information Closure, which similarly maximizes the information entropy. Results show that both approaches outperform truncation schemes. Quasi-Entropy Closure is computationally much faster than Zero-Information Closure, although both methods consider solutions on the space of configurations and hence do not completely overcome the curse of dimensionality. In addition, our scheme includes a plausibility check for the existence of a distribution satisfying a given set of moments on the feasible set of configurations. All results are evaluated on different benchmark problems. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.
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Affiliation(s)
- Vincent Wagner
- Institute for Systems Theory and Automatic Control, University of Stuttgart, 70569 Stuttgart, Germany
| | - Benjamin Castellaz
- Institute for Systems Theory and Automatic Control, University of Stuttgart, 70569 Stuttgart, Germany
| | - Marco Oesting
- Stuttgart Center for Simulation Science, University of Stuttgart, 70569 Stuttgart, Germany.,Institute for Stochastics and Applications, University of Stuttgart, 70569 Stuttgart, Germany
| | - Nicole Radde
- Institute for Systems Theory and Automatic Control, University of Stuttgart, 70569 Stuttgart, Germany.,Stuttgart Center for Simulation Science, University of Stuttgart, 70569 Stuttgart, Germany
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12
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Öcal K, Gutmann MU, Sanguinetti G, Grima R. Inference and uncertainty quantification of stochastic gene expression via synthetic models. J R Soc Interface 2022; 19:20220153. [PMID: 35858045 PMCID: PMC9277240 DOI: 10.1098/rsif.2022.0153] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2022] [Accepted: 06/21/2022] [Indexed: 12/26/2022] Open
Abstract
Estimating uncertainty in model predictions is a central task in quantitative biology. Biological models at the single-cell level are intrinsically stochastic and nonlinear, creating formidable challenges for their statistical estimation which inevitably has to rely on approximations that trade accuracy for tractability. Despite intensive interest, a sweet spot in this trade-off has not been found yet. We propose a flexible procedure for uncertainty quantification in a wide class of reaction networks describing stochastic gene expression including those with feedback. The method is based on creating a tractable coarse-graining of the model that is learned from simulations, a synthetic model, to approximate the likelihood function. We demonstrate that synthetic models can substantially outperform state-of-the-art approaches on a number of non-trivial systems and datasets, yielding an accurate and computationally viable solution to uncertainty quantification in stochastic models of gene expression.
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Affiliation(s)
- Kaan Öcal
- School of Informatics, University of Edinburgh, Edinburgh EH9 3JH, UK
- School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JH, UK
| | | | - Guido Sanguinetti
- Scuola Internazionale Superiore di Studi Avanzati, 34136 Trieste, Italy
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JH, UK
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13
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Lunz D, Bonnans JF, Ruess J. Revisiting moment-closure methods with heterogeneous multiscale population models. Math Biosci 2022; 350:108866. [PMID: 35753520 DOI: 10.1016/j.mbs.2022.108866] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/14/2021] [Revised: 04/10/2022] [Accepted: 06/08/2022] [Indexed: 11/29/2022]
Abstract
Stochastic chemical kinetics at the single-cell level give rise to heterogeneous populations of cells even when all individuals are genetically identical. This heterogeneity can lead to nonuniform behaviour within populations, including different growth characteristics, cell-fate dynamics, and response to stimuli. Ultimately, these diverse behaviours lead to intricate population dynamics that are inherently multiscale: the population composition evolves based on population-level processes that interact with stochastically distributed single-cell states. Therefore, descriptions that account for this heterogeneity are essential to accurately model and control chemical processes. However, for real-world systems such models are computationally expensive to simulate, which can make optimisation problems, such as optimal control or parameter inference, prohibitively challenging. Here, we consider a class of multiscale population models that incorporate population-level mechanisms while remaining faithful to the underlying stochasticity at the single-cell level and the interplay between these two scales. To address the complexity, we study an order-reduction approximations based on the distribution moments. Since previous moment-closure work has focused on the single-cell kinetics, extending these techniques to populations models prompts us to revisit old observations as well as tackle new challenges. In this extended multiscale context, we encounter the previously established observation that the simplest closure techniques can lead to non-physical system trajectories. Despite their poor performance in some systems, we provide an example where these simple closures outperform more sophisticated closure methods in accurately, efficiently, and robustly solving the problem of optimal control of bioproduction in a microbial consortium model.
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Affiliation(s)
- Davin Lunz
- Inria Paris, 2 rue Simone Iff, 75012 Paris, France; Institut Pasteur, 28 rue du Docteur Roux, 75015 Paris, France.
| | - J Frédéric Bonnans
- Université Paris-Saclay, CNRS, CentraleSupélec, Inria, Laboratory of signals and systems, 91190, Gif-sur-Yvette, France
| | - Jakob Ruess
- Inria Paris, 2 rue Simone Iff, 75012 Paris, France; Institut Pasteur, 28 rue du Docteur Roux, 75015 Paris, France
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14
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Wang Y, Zhao J, Park HJ, Zhou D. Effect of dedifferentiation on noise propagation in cellular hierarchy. Phys Rev E 2022; 105:054409. [PMID: 35706189 DOI: 10.1103/physreve.105.054409] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/26/2021] [Accepted: 04/25/2022] [Indexed: 06/15/2023]
Abstract
Many fast renewing tissues have a hierarchical structure. Tissue-specific stem cells are at the root of this cellular hierarchy, which give rive to a whole range of specialized cells via cellular differentiation. However, increasing evidence shows that the hierarchical structure can be broken due to cellular dedifferentiation in which cells at differentiated stages can revert to the stem cell stage. Dedifferentiation has significant impacts on many aspects of hierarchical tissues. Here we investigate the effect of dedifferentiation on noise propagation by developing a stochastic model composed of different cell types. The moment equations are derived, via which we systematically investigate how the noise in the cell number is changed by dedifferentiation. Our results suggest that dedifferentiation have different effects on the noises in the numbers of stem cells and nonstem cells. Specifically, the noise in the number of stem cells is significantly reduced by increasing dedifferentiation probability. Due to the dual effect of dedifferentiation on nonstem cells, however, more complex changes could happen to the noise in the number of nonstem cells by increasing dedifferentiation probability. Furthermore, it is found that even though dedifferentiation could turn part of the noise propagation process into a noise-amplifying step, it is very unlikely to turn the entire process into a noise-amplifying cascade.
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Affiliation(s)
- Yuman Wang
- School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China
| | - Jintong Zhao
- School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China
| | - Hye Jin Park
- Asia Pacific Center for Theoretical Physics, Pohang 37673, Republic of Korea
- Department of Physics, Inha University, Incheon 22212, Republic of Korea
| | - Da Zhou
- School of Mathematical Sciences, Xiamen University, Xiamen 361005, People's Republic of China
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15
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Kim CA, Ricke ND, Van Voorhis T. Machine learning dynamic correlation in chemical kinetics. J Chem Phys 2021; 155:144107. [PMID: 34654306 DOI: 10.1063/5.0065874] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/17/2022] Open
Abstract
Lattice models are a useful tool to simulate the kinetics of surface reactions. Since it is expensive to propagate the probabilities of the entire lattice configurations, it is practical to consider the occupation probabilities of a typical site or a cluster of sites instead. This amounts to a moment closure approximation of the chemical master equation. Unfortunately, simple closures, such as the mean-field and the pair approximation (PA), exhibit weaknesses in systems with significant long-range correlation. In this paper, we show that machine learning (ML) can be used to construct accurate moment closures in chemical kinetics using the lattice Lotka-Volterra model as a model system. We trained feedforward neural networks on kinetic Monte Carlo (KMC) results at select values of rate constants and initial conditions. Given the same level of input as PA, the ML moment closure (MLMC) gave accurate predictions of the instantaneous three-site occupation probabilities. Solving the kinetic equations in conjunction with MLMC gave drastic improvements in the simulated dynamics and descriptions of the dynamical regimes throughout the parameter space. In this way, MLMC is a promising tool to interpolate KMC simulations or construct pretrained closures that would enable researchers to extract useful insight at a fraction of the computational cost.
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Affiliation(s)
- Changhae Andrew Kim
- Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Nathan D Ricke
- Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Troy Van Voorhis
- Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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16
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Abstract
The interaction between a consumer (such as, a predator or a parasitoid) and a resource (such as, a prey or a host) forms an integral motif in ecological food webs, and has been modeled since the early 20th century starting from the seminal work of Lotka and Volterra. While the Lotka-Volterra predator-prey model predicts a neutrally stable equilibrium with oscillating population densities, a density-dependent predator attack rate is known to stabilize the equilibrium. Here, we consider a stochastic formulation of the Lotka-Volterra model where the prey's reproduction rate is a random process, and the predator's attack rate depends on both the prey and predator population densities. Analysis shows that increasing the sensitivity of the attack rate to the prey density attenuates the magnitude of stochastic fluctuations in the population densities. In contrast, these fluctuations vary non-monotonically with the sensitivity of the attack rate to the predator density with an optimal level of sensitivity minimizing the magnitude of fluctuations. Interestingly, our systematic study of the predator-prey correlations reveals distinct signatures depending on the form of the density-dependent attack rate. In summary, stochastic dynamics of nonlinear Lotka-Volterra models can be harnessed to infer density-dependent mechanisms regulating predator-prey interactions. Moreover, these mechanisms can have contrasting consequences on population density fluctuations, with predator-dependent attack rates amplifying stochasticity, while prey-dependent attack rates countering to buffer fluctuations.
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Affiliation(s)
- Abhyudai Singh
- Departments of Electrical and Computer Engineering, Biomedical Engineering and Mathematical Sciences, University of Delaware, Newark, DE, United States of America
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17
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Kurdyaeva T, Milias-Argeitis A. Uncertainty propagation for deterministic models of biochemical networks using moment equations and the extended Kalman filter. J R Soc Interface 2021; 18:20210331. [PMID: 34343452 DOI: 10.1098/rsif.2021.0331] [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] [Indexed: 12/28/2022] Open
Abstract
Differential equation models of biochemical networks are frequently associated with a large degree of uncertainty in parameters and/or initial conditions. However, estimating the impact of this uncertainty on model predictions via Monte Carlo simulation is computationally demanding. A more efficient approach could be to track a system of low-order statistical moments of the state. Unfortunately, when the underlying model is nonlinear, the system of moment equations is infinite-dimensional and cannot be solved without a moment closure approximation which may introduce bias in the moment dynamics. Here, we present a new method to study the time evolution of the desired moments for nonlinear systems with polynomial rate laws. Our approach is based on solving a system of low-order moment equations by substituting the higher-order moments with Monte Carlo-based estimates from a small number of simulations, and using an extended Kalman filter to counteract Monte Carlo noise. Our algorithm provides more accurate and robust results compared to traditional Monte Carlo and moment closure techniques, and we expect that it will be widely useful for the quantification of uncertainty in biochemical model predictions.
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Affiliation(s)
- Tamara Kurdyaeva
- Molecular Systems Biology, Groningen Biomolecular Sciences & Biotechnology Institute, University of Groningen, Groningen, Netherlands
| | - Andreas Milias-Argeitis
- Molecular Systems Biology, Groningen Biomolecular Sciences & Biotechnology Institute, University of Groningen, Groningen, Netherlands
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18
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Waizmann T, Bortolussi L, Vandin A, Tribastone M. Improved estimations of stochastic chemical kinetics by finite-state expansion. Proc Math Phys Eng Sci 2021. [DOI: 10.1098/rspa.2020.0964] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant. The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive, several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics. Here we propose finite-state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations and multi-stability.
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Affiliation(s)
| | - Luca Bortolussi
- Department of Mathematics and Geosciences, University of Trieste, Trieste 34127, Italy
| | - Andrea Vandin
- Sant’Anna School of Advanced Studies, Pisa 56127, Italy
- Department of Applied Mathematics and Computer Science, DTU Technical University of Denmark, Kgs. Lyngby 2800, Denmark
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19
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Sukys A, Grima R. MomentClosure.jl: automated moment closure approximations in Julia. Bioinformatics 2021; 38:289-290. [PMID: 34170295 PMCID: PMC8696096 DOI: 10.1093/bioinformatics/btab469] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/12/2021] [Revised: 06/15/2021] [Accepted: 06/22/2021] [Indexed: 02/03/2023] Open
Abstract
SUMMARY MomentClosure.jl is a Julia package providing automated derivation of the time-evolution equations of the moments of molecule numbers for virtually any chemical reaction network using a wide range of moment closure approximations. It extends the capabilities of modelling stochastic biochemical systems in Julia and can be particularly useful when exact analytic solutions of the chemical master equation are unavailable and when Monte Carlo simulations are computationally expensive. AVAILABILITY AND IMPLEMENTATION MomentClosure.jl is freely accessible under the MIT licence. Source code and documentation are available at https://github.com/augustinas1/MomentClosure.jl.
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Affiliation(s)
| | - Ramon Grima
- To whom correspondence should be addressed. or
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20
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Kang YM, Liu RN. Moment dynamics for gene regulation with rational rate laws. Phys Rev E 2020; 102:042407. [PMID: 33212610 DOI: 10.1103/physreve.102.042407] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2020] [Accepted: 09/09/2020] [Indexed: 06/11/2023]
Abstract
This aim of this paper is mainly to investigate the performance of two typical moment closure schemes in gene regulatory master equations of rational rate laws. When the reaction rate is polynomial, the error bounds between the authentic and approximate moments obtained by schemes of Gaussian moment closure and log-normal moment closure are explicitly given. When the reaction rate is not polynomial, it is shown that the two schemes both behave well in the absence of active-inactive state switch, but in the presence of active-inactive state switch the log-normal closure scheme is far superior to the Gaussian closure scheme in capturing the asymptotic ensemble statistics. Moreover, the accuracy of the log-normal closure method is further confirmed by steady-state analytic results and the conditional Gaussian closure method. It is also disclosed that optimal negative feedback exists in suppressing protein noise in the presence of the on-off switch control.
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Affiliation(s)
- Yan-Mei Kang
- School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China
| | - Ruo-Nan Liu
- School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China
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21
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Blaber S, Sivak DA. Optimal control of protein copy number. Phys Rev E 2020; 101:022118. [PMID: 32168689 DOI: 10.1103/physreve.101.022118] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/03/2018] [Accepted: 01/16/2020] [Indexed: 11/07/2022]
Abstract
Cell-cell communication is often achieved by secreted signaling molecules that bind membrane-bound receptors. A common class of such receptors are G-protein coupled receptors, where extracellular binding induces changes on the membrane affinity near the receptor for certain cytosolic proteins, effectively altering their chemical potential. We analyze the minimum-dissipation schedules for dynamically changing chemical potential to induce steady-state changes in protein copy-number distributions, and illustrate with analytic solutions for linear chemical reaction networks. Protocols that change chemical potential on biologically relevant timescales are experimentally accessible using optogenetic manipulations, and our framework provides nontrivial predictions about functional dynamical cell-cell interactions.
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Affiliation(s)
- Steven Blaber
- Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada.,Department of Physics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
| | - David A Sivak
- Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada
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22
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Lecca P, Re A. A Reaction-Based Model of the State Space of Chemical Reaction Systems Enables Efficient Simulations. IEEE/ACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS 2020; 17:469-482. [PMID: 30676973 DOI: 10.1109/tcbb.2019.2894699] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
The choice of the state space representation of a system can turn into a prominent advantage or burden in any endeavour to mathematically model dynamical systems since it entails the analytical tractability of the related modelling formalism and the efficiency of the numerical computation. The Reaction-Based Model (RBM) of the state space, which is presented in this article, is a novel formalization of the kinetics of a system of interacting molecules. According to our representation, the state Sμ of a system of M reactions and N molecular species, is identified with the occurrence of the reaction Rμ ( μ = 1, ..., M). The transition between any two states Sμ and Sν is modelled as a first-order reaction Sμ → Sν and described by mass action-like equation for the partial time derivative of the variables P(Sμ, t) and P(Sν, t), which denote the probabilities that the system lies in the two states, respectively. The rate equations for the state probabilities are coupled with those for the abundance of molecular species. Altogether, the rate equations along with the specification of the initial conditions define the Cauchy problem whose solution describes the time-evolution of the system. The RBM has been applied to a typical severely stiff biological case study. The numerical solutions of the system's dynamics turned out to be computationally more efficient and in agreement with the results of the stochastic and hybrid stochastic/deterministic simulation algorithms.
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23
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Pucci F, Rooman M. Deciphering noise amplification and reduction in open chemical reaction networks. J R Soc Interface 2019; 15:20180805. [PMID: 30958227 DOI: 10.1098/rsif.2018.0805] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
The impact of fluctuations on the dynamical behaviour of complex biological systems is a longstanding issue, whose understanding would elucidate how evolutionary pressure tends to modulate intrinsic noise. Using the Itō stochastic differential equation formalism, we performed analytic and numerical analyses of model systems containing different molecular species in contact with the environment and interacting with each other through mass-action kinetics. For networks of zero deficiency, which admit a detailed- or complex-balanced steady state, all molecular species are uncorrelated and their Fano factors are Poissonian. Systems of higher deficiency have non-equilibrium steady states and non-zero reaction fluxes flowing between the complexes. When they model homo-oligomerization, the noise on each species is reduced when the flux flows from the oligomers of lowest to highest degree, and amplified otherwise. In the case of hetero-oligomerization systems, only the noise on the highest-degree species shows this behaviour.
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Affiliation(s)
- Fabrizio Pucci
- 2 Department of BioModeling, BioInformatics and BioProcesses, Université Libre de Bruxelles , 50 Roosevelt Ave, 1050 Brussels , Belgium
| | - Marianne Rooman
- 1 Department of Theoretical Physics, BioInformatics and BioProcesses, Université Libre de Bruxelles , 50 Roosevelt Ave, 1050 Brussels , Belgium.,2 Department of BioModeling, BioInformatics and BioProcesses, Université Libre de Bruxelles , 50 Roosevelt Ave, 1050 Brussels , Belgium
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24
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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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25
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Smith S, Grima R. Spatial Stochastic Intracellular Kinetics: A Review of Modelling Approaches. Bull Math Biol 2019; 81:2960-3009. [PMID: 29785521 PMCID: PMC6677717 DOI: 10.1007/s11538-018-0443-1] [Citation(s) in RCA: 27] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/14/2018] [Accepted: 05/03/2018] [Indexed: 01/22/2023]
Abstract
Models of chemical kinetics that incorporate both stochasticity and diffusion are an increasingly common tool for studying biology. The variety of competing models is vast, but two stand out by virtue of their popularity: the reaction-diffusion master equation and Brownian dynamics. In this review, we critically address a number of open questions surrounding these models: How can they be justified physically? How do they relate to each other? How do they fit into the wider landscape of chemical models, ranging from the rate equations to molecular dynamics? This review assumes no prior knowledge of modelling chemical kinetics and should be accessible to a wide range of readers.
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Affiliation(s)
- Stephen Smith
- School of Biological Sciences, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JR, Scotland, UK
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Mayfield Road, Edinburgh, EH9 3JR, Scotland, UK.
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26
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Kuntz J, Thomas P, Stan GB, Barahona M. Bounding the stationary distributions of the chemical master equation via mathematical programming. J Chem Phys 2019; 151:034109. [PMID: 31325941 DOI: 10.1063/1.5100670] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
Abstract
The stochastic dynamics of biochemical networks are usually modeled with the chemical master equation (CME). The stationary distributions of CMEs are seldom solvable analytically, and numerical methods typically produce estimates with uncontrolled errors. Here, we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy. First, we use semidefinite programming to compute increasingly tighter upper and lower bounds on the moments of the stationary distributions for networks with rational propensities. Second, we use these moment bounds to formulate linear programs that yield convergent upper and lower bounds on the stationary distributions themselves, their marginals, and stationary averages. The bounds obtained also provide a computational test for the uniqueness of the distribution. In the unique case, the bounds form an approximation of the stationary distribution with a computable bound on its error. In the nonunique case, our approach yields converging approximations of the ergodic distributions. We illustrate our methodology through several biochemical examples taken from the literature: Schlögl's model for a chemical bifurcation, a two-dimensional toggle switch, a model for bursty gene expression, and a dimerization model with multiple stationary distributions.
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Affiliation(s)
- Juan Kuntz
- Department of Mathematics, Imperial College London, London, United Kingdom
| | - Philipp Thomas
- Department of Mathematics, Imperial College London, London, United Kingdom
| | - Guy-Bart Stan
- Department of Bioengineering, Imperial College London, London, United Kingdom
| | - Mauricio Barahona
- Department of Mathematics, Imperial College London, London, United Kingdom
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27
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Beentjes CHL, Baker RE. Uniformization techniques for stochastic simulation of chemical reaction networks. J Chem Phys 2019; 150:154107. [PMID: 31005097 DOI: 10.1063/1.5081043] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Abstract
This work considers the method of uniformization for continuous-time Markov chains in the context of chemical reaction networks. Previous work in the literature has shown that uniformization can be beneficial in the context of time-inhomogeneous models, such as chemical reaction networks incorporating extrinsic noise. This paper lays focus on the understanding of uniformization from the viewpoint of sample paths of chemical reaction networks. In particular, an efficient pathwise stochastic simulation algorithm for time-homogeneous models is presented which is complexity-wise equal to Gillespie's direct method. This new approach therefore enlarges the class of problems for which the uniformization approach forms a computationally attractive choice. Furthermore, as a new application of the uniformization method, we provide a novel variance reduction method for (raw) moment estimators of chemical reaction networks based upon the combination of stratification and uniformization.
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Affiliation(s)
| | - Ruth E Baker
- Mathematical Institute, University of Oxford, Oxford, United Kingdom
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28
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Cao Z, Grima R. Accuracy of parameter estimation for auto-regulatory transcriptional feedback loops from noisy data. J R Soc Interface 2019; 16:20180967. [PMID: 30940028 PMCID: PMC6505555 DOI: 10.1098/rsif.2018.0967] [Citation(s) in RCA: 26] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/09/2023] Open
Abstract
Bayesian and non-Bayesian moment-based inference methods are commonly used to estimate the parameters defining stochastic models of gene regulatory networks from noisy single cell or population snapshot data. However, a systematic investigation of the accuracy of the predictions of these methods remains missing. Here, we present the results of such a study using synthetic noisy data of a negative auto-regulatory transcriptional feedback loop, one of the most common building blocks of complex gene regulatory networks. We study the error in parameter estimation as a function of (i) number of cells in each sample; (ii) the number of time points; (iii) the highest-order moment of protein fluctuations used for inference; (iv) the moment-closure method used for likelihood approximation. We find that for sample sizes typical of flow cytometry experiments, parameter estimation by maximizing the likelihood is as accurate as using Bayesian methods but with a much reduced computational time. We also show that the choice of moment-closure method is the crucial factor determining the maximum achievable accuracy of moment-based inference methods. Common likelihood approximation methods based on the linear noise approximation or the zero cumulants closure perform poorly for feedback loops with large protein-DNA binding rates or large protein bursts; this is exacerbated for highly heterogeneous cell populations. By contrast, approximating the likelihood using the linear-mapping approximation or conditional derivative matching leads to highly accurate parameter estimates for a wide range of conditions.
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29
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Muñoz-Cobo JL, Berna C. Chemical Kinetics Roots and Methods to Obtain the Probability Distribution Function Evolution of Reactants and Products in Chemical Networks Governed by a Master Equation. ENTROPY 2019; 21:e21020181. [PMID: 33266897 PMCID: PMC7514663 DOI: 10.3390/e21020181] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Received: 01/24/2019] [Accepted: 02/11/2019] [Indexed: 11/16/2022]
Abstract
In this paper first, we review the physical root bases of chemical reaction networks as a Markov process in multidimensional vector space. Then we study the chemical reactions from a microscopic point of view, to obtain the expression for the propensities for the different reactions that can happen in the network. These chemical propensities, at a given time, depend on the system state at that time, and do not depend on the state at an earlier time indicating that we are dealing with Markov processes. Then the Chemical Master Equation (CME) is deduced for an arbitrary chemical network from a probability balance and it is expressed in terms of the reaction propensities. This CME governs the dynamics of the chemical system. Due to the difficulty to solve this equation two methods are studied, the first one is the probability generating function method or z-transform, which permits to obtain the evolution of the factorial moment of the system with time in an easiest way or after some manipulation the evolution of the polynomial moments. The second method studied is the expansion of the CME in terms of an order parameter (system volume). In this case we study first the expansion of the CME using the propensities obtained previously and splitting the molecular concentration into a deterministic part and a random part. An expression in terms of multinomial coefficients is obtained for the evolution of the probability of the random part. Then we study how to reconstruct the probability distribution from the moments using the maximum entropy principle. Finally, the previous methods are applied to simple chemical networks and the consistency of these methods is studied.
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Affiliation(s)
- José-Luis Muñoz-Cobo
- Department of Chemical and Nuclear Engineering, Universitat Politècnica de València, 46022 Valencia, Spain
- Instituto Universitario de Ingeniería Energética, Universitat Politècnica de València, 46022 Valencia, Spain
- Correspondence: ; Tel.: +34-96-387-7631
| | - Cesar Berna
- Instituto Universitario de Ingeniería Energética, Universitat Politècnica de València, 46022 Valencia, Spain
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30
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Vu TV, Hasegawa Y. An algebraic method to calculate parameter regions for constrained steady-state distribution in stochastic reaction networks. CHAOS (WOODBURY, N.Y.) 2019; 29:023123. [PMID: 30823706 DOI: 10.1063/1.5047579] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/09/2018] [Accepted: 01/25/2019] [Indexed: 06/09/2023]
Abstract
Steady state is an essential concept in reaction networks. Its stability reflects fundamental characteristics of several biological phenomena such as cellular signal transduction and gene expression. Because biochemical reactions occur at the cellular level, they are affected by unavoidable fluctuations. Although several methods have been proposed to detect and analyze the stability of steady states for deterministic models, these methods cannot be applied to stochastic reaction networks. In this paper, we propose an algorithm based on algebraic computations to calculate parameter regions for constrained steady-state distribution of stochastic reaction networks, in which the means and variances satisfy some given inequality constraints. To evaluate our proposed method, we perform computer simulations for three typical chemical reactions and demonstrate that the results obtained with our method are consistent with the simulation results.
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Affiliation(s)
- Tan Van Vu
- Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
| | - Yoshihiko Hasegawa
- Department of Information and Communication Engineering, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo 113-8656, Japan
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31
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Thomas P. Intrinsic and extrinsic noise of gene expression in lineage trees. Sci Rep 2019; 9:474. [PMID: 30679440 PMCID: PMC6345792 DOI: 10.1038/s41598-018-35927-x] [Citation(s) in RCA: 33] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/01/2018] [Accepted: 11/08/2018] [Indexed: 12/30/2022] Open
Abstract
Cell-to-cell heterogeneity is driven by stochasticity in intracellular reactions and the population dynamics. While these sources are usually studied separately, we develop an agent-based framework that accounts for both factors while tracking every single cell of a growing population. Apart from the common intrinsic variability, the framework also predicts extrinsic noise without the need to introduce fluctuating rate constants. Instead, extrinsic fluctuations are explained by cell cycle fluctuations and differences in cell age. We provide explicit formulas to quantify mean molecule numbers, intrinsic and extrinsic noise statistics in two-colour experiments. We find that these statistics differ significantly depending on the experimental setup used to observe the cells. We illustrate this fact using (i) averages over an isolated cell lineage tracked over many generations as observed in the mother machine, (ii) population snapshots with known cell ages as recorded in time-lapse microscopy, and (iii) snapshots with unknown cell ages as measured from static images or flow cytometry. Applying the method to models of stochastic gene expression and feedback regulation elucidates that isolated lineages, as compared to snapshot data, can significantly overestimate the mean number of molecules, overestimate extrinsic noise but underestimate intrinsic noise and have qualitatively different sensitivities to cell cycle fluctuations.
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Affiliation(s)
- Philipp Thomas
- Department of Mathematics, Imperial College London, London, SW7 2AZ, UK.
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32
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Pájaro M, Otero-Muras I, Vázquez C, Alonso AA. SELANSI: a toolbox for simulation of stochastic gene regulatory networks. Bioinformatics 2018; 34:893-895. [PMID: 29040384 PMCID: PMC6030881 DOI: 10.1093/bioinformatics/btx645] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2017] [Accepted: 10/10/2017] [Indexed: 02/04/2023] Open
Abstract
Motivation Gene regulation is inherently stochastic. In many applications concerning Systems and Synthetic Biology such as the reverse engineering and the de novo design of genetic circuits, stochastic effects (yet potentially crucial) are often neglected due to the high computational cost of stochastic simulations. With advances in these fields there is an increasing need of tools providing accurate approximations of the stochastic dynamics of gene regulatory networks (GRNs) with reduced computational effort. Results This work presents SELANSI (SEmi-LAgrangian SImulation of GRNs), a software toolbox for the simulation of stochastic multidimensional gene regulatory networks. SELANSI exploits intrinsic structural properties of gene regulatory networks to accurately approximate the corresponding Chemical Master Equation with a partial integral differential equation that is solved by a semi-lagrangian method with high efficiency. Networks under consideration might involve multiple genes with self and cross regulations, in which genes can be regulated by different transcription factors. Moreover, the validity of the method is not restricted to a particular type of kinetics. The tool offers total flexibility regarding network topology, kinetics and parameterization, as well as simulation options. Availability and implementation SELANSI runs under the MATLAB environment, and is available under GPLv3 license at https://sites.google.com/view/selansi.
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Affiliation(s)
- Manuel Pájaro
- BioProcess Engineering Group, IIM-CSIC, Spanish National Research Council, 36208 Vigo, Spain
| | - Irene Otero-Muras
- BioProcess Engineering Group, IIM-CSIC, Spanish National Research Council, 36208 Vigo, Spain
| | - Carlos Vázquez
- Department of Mathematics, University of A Coruña Campus Elviña s/n, 15071 A Coruña, Spain
| | - Antonio A Alonso
- BioProcess Engineering Group, IIM-CSIC, Spanish National Research Council, 36208 Vigo, Spain
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33
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Solving Stochastic Reaction Networks with Maximum Entropy Lagrange Multipliers. ENTROPY 2018; 20:e20090700. [PMID: 33265789 PMCID: PMC7513230 DOI: 10.3390/e20090700] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/03/2018] [Revised: 09/05/2018] [Accepted: 09/06/2018] [Indexed: 11/16/2022]
Abstract
The time evolution of stochastic reaction networks can be modeled with the chemical master equation of the probability distribution. Alternatively, the numerical problem can be reformulated in terms of probability moment equations. Herein we present a new alternative method for numerically solving the time evolution of stochastic reaction networks. Based on the assumption that the entropy of the reaction network is maximum, Lagrange multipliers are introduced. The proposed method derives equations that model the time derivatives of these Lagrange multipliers. We present detailed steps to transform moment equations to Lagrange multiplier equations. In order to demonstrate the method, we present examples of non-linear stochastic reaction networks of varying degrees of complexity, including multistable and oscillatory systems. We find that the new approach is as accurate and significantly more efficient than Gillespie’s original exact algorithm for systems with small number of interacting species. This work is a step towards solving stochastic reaction networks accurately and efficiently.
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34
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Rule M, Sanguinetti G. Autoregressive Point Processes as Latent State-Space Models: A Moment-Closure Approach to Fluctuations and Autocorrelations. Neural Comput 2018; 30:2757-2780. [PMID: 30148704 DOI: 10.1162/neco_a_01121] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/04/2022]
Abstract
Modeling and interpreting spike train data is a task of central importance in computational neuroscience, with significant translational implications. Two popular classes of data-driven models for this task are autoregressive point-process generalized linear models (PPGLM) and latent state-space models (SSM) with point-process observations. In this letter, we derive a mathematical connection between these two classes of models. By introducing an auxiliary history process, we represent exactly a PPGLM in terms of a latent, infinite-dimensional dynamical system, which can then be mapped onto an SSM by basis function projections and moment closure. This representation provides a new perspective on widely used methods for modeling spike data and also suggests novel algorithmic approaches to fitting such models. We illustrate our results on a phasic bursting neuron model, showing that our proposed approach provides an accurate and efficient way to capture neural dynamics.
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Affiliation(s)
- Michael Rule
- Institute for Adaptive and Neural Computation, School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, U.K.
| | - Guido Sanguinetti
- Institute for Adaptive and Neural Computation, School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, U.K.
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35
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Pucci F, Rooman M. Insights into noise modulation in oligomerization systems of increasing complexity. Phys Rev E 2018; 98:012137. [PMID: 30110836 DOI: 10.1103/physreve.98.012137] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/03/2017] [Indexed: 11/07/2022]
Abstract
Understanding under which conditions the increase of systems complexity is evolutionarily advantageous, and how this trend is related to the modulation of the intrinsic noise, are fascinating issues of utmost importance for synthetic and systems biology. To get insights into these matters, we analyzed a series of chemical reaction networks with different topologies and complexity, described by mass-action kinetics. We showed, analytically and numerically, that the global level of fluctuations at the steady state, measured by the sum over all species of the Fano factors of the number of molecules, is directly related to the network's deficiency. For zero-deficiency systems, this sum is constant and equal to the rank of the network. For higher deficiencies, additional terms appear in the Fano factor sum, which are proportional to the net reaction fluxes between the molecular complexes. We showed that the system's global intrinsic noise is reduced when all fluxes flow from lower to higher degree oligomers, or equivalently, towards the species of higher complexity, whereas it is amplified when the fluxes are directed towards lower complexity species.
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Affiliation(s)
- Fabrizio Pucci
- Department of BioModeling, BioInformatics & BioProcesses, Université Libre de Bruxelles, Roosevelt Avenue 50, B-1050 Brussels, Belgium and Department of Theoretical Physics, Université Libre de Bruxelles, Triumph Boulevard, B-1050 Brussels, Belgium
| | - Marianne Rooman
- Department of BioModeling, BioInformatics & BioProcesses, Université Libre de Bruxelles, Roosevelt Avenue 50, B-1050 Brussels, Belgium and Department of Theoretical Physics, Université Libre de Bruxelles, Triumph Boulevard, B-1050 Brussels, Belgium
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36
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Kazeroonian A, Theis FJ, Hasenauer J. A scalable moment-closure approximation for large-scale biochemical reaction networks. Bioinformatics 2018; 33:i293-i300. [PMID: 28881983 PMCID: PMC5870627 DOI: 10.1093/bioinformatics/btx249] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Motivation Stochastic molecular processes are a leading cause of cell-to-cell variability. Their dynamics are often described by continuous-time discrete-state Markov chains and simulated using stochastic simulation algorithms. As these stochastic simulations are computationally demanding, ordinary differential equation models for the dynamics of the statistical moments have been developed. The number of state variables of these approximating models, however, grows at least quadratically with the number of biochemical species. This limits their application to small- and medium-sized processes. Results In this article, we present a scalable moment-closure approximation (sMA) for the simulation of statistical moments of large-scale stochastic processes. The sMA exploits the structure of the biochemical reaction network to reduce the covariance matrix. We prove that sMA yields approximating models whose number of state variables depends predominantly on local properties, i.e. the average node degree of the reaction network, instead of the overall network size. The resulting complexity reduction is assessed by studying a range of medium- and large-scale biochemical reaction networks. To evaluate the approximation accuracy and the improvement in computational efficiency, we study models for JAK2/STAT5 signalling and NF κ B signalling. Our method is applicable to generic biochemical reaction networks and we provide an implementation, including an SBML interface, which renders the sMA easily accessible. Availability and implementation The sMA is implemented in the open-source MATLAB toolbox CERENA and is available from https://github.com/CERENADevelopers/CERENA . Contact jan.hasenauer@helmholtz-muenchen.de or atefeh.kazeroonian@tum.de. Supplementary information Supplementary data are available at Bioinformatics online.
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Affiliation(s)
- Atefeh Kazeroonian
- Institute of Computational Biology, Helmholtz Zentrum München - German Research Center for Environmental Health, Neuherberg, Germany.,Department of Mathematics, Technische Universität München, Garching, Germany.,Institut für Medizinische Mikrobiologie, Immunologie und Hygiene, Fakultät für Medizin, Technische Universität München, München, Germany
| | - Fabian J Theis
- Institute of Computational Biology, Helmholtz Zentrum München - German Research Center for Environmental Health, Neuherberg, Germany.,Department of Mathematics, Technische Universität München, Garching, Germany
| | - Jan Hasenauer
- Institute of Computational Biology, Helmholtz Zentrum München - German Research Center for Environmental Health, Neuherberg, Germany.,Department of Mathematics, Technische Universität München, Garching, Germany
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37
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Bronstein L, Koeppl H. A variational approach to moment-closure approximations for the kinetics of biomolecular reaction networks. J Chem Phys 2018; 148:014105. [DOI: 10.1063/1.5003892] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Affiliation(s)
- Leo Bronstein
- Department of Electrical Engineering and Information Technology, Technische Universität Darmstadt, 64283 Darmstadt, Germany
| | - Heinz Koeppl
- Department of Electrical Engineering and Information Technology, Technische Universität Darmstadt, 64283 Darmstadt, Germany
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38
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Smith E, Krishnamurthy S. Flows, scaling, and the control of moment hierarchies for stochastic chemical reaction networks. Phys Rev E 2017; 96:062102. [PMID: 29335680 PMCID: PMC5765883 DOI: 10.1103/physreve.96.062102] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Abstract
Stochastic chemical reaction networks (CRNs) are complex systems that combine the features of concurrent transformation of multiple variables in each elementary reaction event and nonlinear relations between states and their rates of change. Most general results concerning CRNs are limited to restricted cases where a topological characteristic known as deficiency takes a value 0 or 1, implying uniqueness and positivity of steady states and surprising, low-information forms for their associated probability distributions. Here we derive equations of motion for fluctuation moments at all orders for stochastic CRNs at general deficiency. We show, for the standard base case of proportional sampling without replacement (which underlies the mass-action rate law), that the generator of the stochastic process acts on the hierarchy of factorial moments with a finite representation. Whereas simulation of high-order moments for many-particle systems is costly, this representation reduces the solution of moment hierarchies to a complexity comparable to solving a heat equation. At steady states, moment hierarchies for finite CRNs interpolate between low-order and high-order scaling regimes, which may be approximated separately by distributions similar to those for deficiency-zero networks and connected through matched asymptotic expansions. In CRNs with multiple stable or metastable steady states, boundedness of high-order moments provides the starting condition for recursive solution downward to low-order moments, reversing the order usually used to solve moment hierarchies. A basis for a subset of network flows defined by having the same mean-regressing property as the flows in deficiency-zero networks gives the leading contribution to low-order moments in CRNs at general deficiency, in a 1/n expansion in large particle numbers. Our results give a physical picture of the different informational roles of mean-regressing and non-mean-regressing flows and clarify the dynamical meaning of deficiency not only for first-moment conditions but for all orders in fluctuations.
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Affiliation(s)
- Eric Smith
- Earth-Life Science Institute, Tokyo Institute of Technology, 2-12-1-IE-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan; Department of Biology, Georgia Institute of Technology, 310 Ferst Drive NW, Atlanta, Georgia 30332, USA; Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA; and Ronin Institute, 127 Haddon Place, Montclair, New Jersey 07043, USA
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39
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Schnoerr D, Cseke B, Grima R, Sanguinetti G. Efficient Low-Order Approximation of First-Passage Time Distributions. PHYSICAL REVIEW LETTERS 2017; 119:210601. [PMID: 29219406 DOI: 10.1103/physrevlett.119.210601] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2017] [Indexed: 06/07/2023]
Abstract
We consider the problem of computing first-passage time distributions for reaction processes modeled by master equations. We show that this generally intractable class of problems is equivalent to a sequential Bayesian inference problem for an auxiliary observation process. The solution can be approximated efficiently by solving a closed set of coupled ordinary differential equations (for the low-order moments of the process) whose size scales with the number of species. We apply it to an epidemic model and a trimerization process and show good agreement with stochastic simulations.
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Affiliation(s)
- David Schnoerr
- School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, United Kingdom
| | - Botond Cseke
- Microsoft Research, Cambridge CB1 2FB, United Kingdom
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh EH9 3JH, United Kingdom
| | - Guido Sanguinetti
- School of Informatics, University of Edinburgh, Edinburgh EH8 9AB, United Kingdom
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40
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Krishnamurthy S, Smith E. Solving moment hierarchies for chemical reaction networks. JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL 2017; 50:425002. [PMID: 29333197 PMCID: PMC5764546 DOI: 10.1088/1751-8121/aa89d0] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
The study of chemical reaction networks (CRN's) is a very active field. Earlier well-known results (Feinberg 1987 Chem. Enc. Sci. 42 2229, Anderson et al 2010 Bull. Math. Biol. 72 1947) identify a topological quantity called deficiency, for any CRN, which, when exactly equal to zero, leads to a unique factorized steady-state for these networks. No results exist however for the steady states of non-zero-deficiency networks. In this paper, we show how to write the full moment-hierarchy for any non-zero-deficiency CRN obeying mass-action kinetics, in terms of equations for the factorial moments. Using these, we can recursively predict values for lower moments from higher moments, reversing the procedure usually used to solve moment hierarchies. We show, for nontrivial examples, that in this manner we can predict any moment of interest, for CRN's with non-zero deficiency and non-factorizable steady states.
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Affiliation(s)
| | - Eric Smith
- Earth-Life Science Institute, Tokyo Institute of Technology, 2-12-1-IE-1 Ookayama, Meguro-ku, Tokyo 152-8550, Japan
- Department of Biology, Georgia Institute of Technology, 310 Ferst Drive NW, Atlanta, GA 30332, United States of America
- Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, United States of America
- Ronin Institute, 127 Haddon Place, Montclair, NJ 07043, United States of America
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41
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Lakatos E, Stumpf MPH. Control mechanisms for stochastic biochemical systems via computation of reachable sets. ROYAL SOCIETY OPEN SCIENCE 2017; 4:160790. [PMID: 28878957 PMCID: PMC5579072 DOI: 10.1098/rsos.160790] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 05/15/2017] [Accepted: 07/21/2017] [Indexed: 06/07/2023]
Abstract
Controlling the behaviour of cells by rationally guiding molecular processes is an overarching aim of much of synthetic biology. Molecular processes, however, are notoriously noisy and frequently nonlinear. We present an approach to studying the impact of control measures on motifs of molecular interactions that addresses the problems faced in many biological systems: stochasticity, parameter uncertainty and nonlinearity. We show that our reachability analysis formalism can describe the potential behaviour of biological (naturally evolved as well as engineered) systems, and provides a set of bounds on their dynamics at the level of population statistics: for example, we can obtain the possible ranges of means and variances of mRNA and protein expression levels, even in the presence of uncertainty about model parameters.
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Affiliation(s)
- Eszter Lakatos
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Biosciences, Imperial College London, London SW7 2AZ, UK
| | - Michael P. H. Stumpf
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Biosciences, Imperial College London, London SW7 2AZ, UK
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42
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Bravi B, Sollich P. Statistical physics approaches to subnetwork dynamics in biochemical systems. Phys Biol 2017; 14:045010. [PMID: 28510539 DOI: 10.1088/1478-3975/aa7363] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
Abstract
We apply a Gaussian variational approximation to model reduction in large biochemical networks of unary and binary reactions. We focus on a small subset of variables (subnetwork) of interest, e.g. because they are accessible experimentally, embedded in a larger network (bulk). The key goal is to write dynamical equations reduced to the subnetwork but still retaining the effects of the bulk. As a result, the subnetwork-reduced dynamics contains a memory term and an extrinsic noise term with non-trivial temporal correlations. We first derive expressions for this memory and noise in the linearized (Gaussian) dynamics and then use a perturbative power expansion to obtain first order nonlinear corrections. For the case of vanishing intrinsic noise, our description is explicitly shown to be equivalent to projection methods up to quadratic terms, but it is applicable also in the presence of stochastic fluctuations in the original dynamics. An example from the epidermal growth factor receptor signalling pathway is provided to probe the increased prediction accuracy and computational efficiency of our method.
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Affiliation(s)
- B Bravi
- Current affiliation: Institute of Theoretical Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland
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43
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Ghusinga KR, Vargas-Garcia CA, Lamperski A, Singh A. Exact lower and upper bounds on stationary moments in stochastic biochemical systems. Phys Biol 2017; 14:04LT01. [PMID: 28661893 DOI: 10.1088/1478-3975/aa75c6] [Citation(s) in RCA: 46] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
In the stochastic description of biochemical reaction systems, the time evolution of statistical moments for species population counts is described by a linear dynamical system. However, except for some ideal cases (such as zero- and first-order reaction kinetics), the moment dynamics is underdetermined as lower-order moments depend upon higher-order moments. Here, we propose a novel method to find exact lower and upper bounds on stationary moments for a given arbitrary system of biochemical reactions. The method exploits the fact that statistical moments of any positive-valued random variable must satisfy some constraints that are compactly represented through the positive semidefiniteness of moment matrices. Our analysis shows that solving moment equations at steady state in conjunction with constraints on moment matrices provides exact lower and upper bounds on the moments. These results are illustrated by three different examples-the commonly used logistic growth model, stochastic gene expression with auto-regulation and an activator-repressor gene network motif. Interestingly, in all cases the accuracy of the bounds is shown to improve as moment equations are expanded to include higher-order moments. Our results provide avenues for development of approximation methods that provide explicit bounds on moments for nonlinear stochastic systems that are otherwise analytically intractable.
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Affiliation(s)
- Khem Raj Ghusinga
- Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, United States of America
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44
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Weber MF, Frey E. Master equations and the theory of stochastic path integrals. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2017; 80:046601. [PMID: 28306551 DOI: 10.1088/1361-6633/aa5ae2] [Citation(s) in RCA: 31] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon expanding the forward and the backward path integrals around stationary paths, we then discuss and extend a recent method for the computation of rare event probabilities. Besides, we also derive path integral representations for processes with continuous state spaces whose forward and backward master equations admit Kramers-Moyal expansions. A truncation of the backward expansion at the level of a diffusion approximation recovers a classic path integral representation of the (backward) Fokker-Planck equation. One can rewrite this path integral in terms of an Onsager-Machlup function and, for purely diffusive Brownian motion, it simplifies to the path integral of Wiener. To make this review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory and do not assume any knowledge of the latter. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit.
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Affiliation(s)
- Markus F Weber
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 München, Germany
| | - Erwin Frey
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 München, Germany
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45
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Zhang J, Nie Q, Zhou T. A moment-convergence method for stochastic analysis of biochemical reaction networks. J Chem Phys 2017; 144:194109. [PMID: 27208938 DOI: 10.1063/1.4950767] [Citation(s) in RCA: 26] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Traditional moment-closure methods need to assume that high-order cumulants of a probability distribution approximate to zero. However, this strong assumption is not satisfied for many biochemical reaction networks. Here, we introduce convergent moments (defined in mathematics as the coefficients in the Taylor expansion of the probability-generating function at some point) to overcome this drawback of the moment-closure methods. As such, we develop a new analysis method for stochastic chemical kinetics. This method provides an accurate approximation for the master probability equation (MPE). In particular, the connection between low-order convergent moments and rate constants can be more easily derived in terms of explicit and analytical forms, allowing insights that would be difficult to obtain through direct simulation or manipulation of the MPE. In addition, it provides an accurate and efficient way to compute steady-state or transient probability distribution, avoiding the algorithmic difficulty associated with stiffness of the MPE due to large differences in sizes of rate constants. Applications of the method to several systems reveal nontrivial stochastic mechanisms of gene expression dynamics, e.g., intrinsic fluctuations can induce transient bimodality and amplify transient signals, and slow switching between promoter states can increase fluctuations in spatially heterogeneous signals. The overall approach has broad applications in modeling, analysis, and computation of complex biochemical networks with intrinsic noise.
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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
| | - Qing Nie
- Department of Mathematics, University of California at Irvine, Irvine, California 92697, USA
| | - Tianshou Zhou
- School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, People's Republic of China
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46
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Lück A, Wolf V. Generalized method of moments for estimating parameters of stochastic reaction networks. BMC SYSTEMS BIOLOGY 2016; 10:98. [PMID: 27769280 PMCID: PMC5073941 DOI: 10.1186/s12918-016-0342-8] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/06/2016] [Accepted: 10/11/2016] [Indexed: 11/10/2022]
Abstract
BACKGROUND Discrete-state stochastic models have become a well-established approach to describe biochemical reaction networks that are influenced by the inherent randomness of cellular events. In the last years several methods for accurately approximating the statistical moments of such models have become very popular since they allow an efficient analysis of complex networks. RESULTS We propose a generalized method of moments approach for inferring the parameters of reaction networks based on a sophisticated matching of the statistical moments of the corresponding stochastic model and the sample moments of population snapshot data. The proposed parameter estimation method exploits recently developed moment-based approximations and provides estimators with desirable statistical properties when a large number of samples is available. We demonstrate the usefulness and efficiency of the inference method on two case studies. CONCLUSIONS The generalized method of moments provides accurate and fast estimations of unknown parameters of reaction networks. The accuracy increases when also moments of order higher than two are considered. In addition, the variance of the estimator decreases, when more samples are given or when higher order moments are included.
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Affiliation(s)
- Alexander Lück
- Department of Computer Science, Saarland University, Campus E 13, Saarbrücken, 66123, Germany
| | - Verena Wolf
- Department of Computer Science, Saarland University, Campus E 13, Saarbrücken, 66123, Germany.
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47
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Fan S, Geissmann Q, Lakatos E, Lukauskas S, Ale A, Babtie AC, Kirk PDW, Stumpf MPH. MEANS: python package for Moment Expansion Approximation, iNference and Simulation. Bioinformatics 2016; 32:2863-5. [PMID: 27153663 PMCID: PMC5018365 DOI: 10.1093/bioinformatics/btw229] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2015] [Revised: 03/23/2016] [Accepted: 04/21/2016] [Indexed: 11/14/2022] Open
Abstract
MOTIVATION Many biochemical systems require stochastic descriptions. Unfortunately these can only be solved for the simplest cases and their direct simulation can become prohibitively expensive, precluding thorough analysis. As an alternative, moment closure approximation methods generate equations for the time-evolution of the system's moments and apply a closure ansatz to obtain a closed set of differential equations; that can become the basis for the deterministic analysis of the moments of the outputs of stochastic systems. RESULTS We present a free, user-friendly tool implementing an efficient moment expansion approximation with parametric closures that integrates well with the IPython interactive environment. Our package enables the analysis of complex stochastic systems without any constraints on the number of species and moments studied and the type of rate laws in the system. In addition to the approximation method our package provides numerous tools to help non-expert users in stochastic analysis. AVAILABILITY AND IMPLEMENTATION https://github.com/theosysbio/means CONTACTS m.stumpf@imperial.ac.uk or e.lakatos13@imperial.ac.uk SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.
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Affiliation(s)
- Sisi Fan
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
| | - Quentin Geissmann
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
| | - Eszter Lakatos
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
| | - Saulius Lukauskas
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
| | - Angelique Ale
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
| | - Ann C Babtie
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
| | | | - Michael P H Stumpf
- Centre for Integrative Systems Biology and Bioinformatics, Department of Life Sciences, Imperial College London, London SW7 2AZ, UK
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48
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Fröhlich F, Thomas P, Kazeroonian A, Theis FJ, Grima R, Hasenauer J. Inference for Stochastic Chemical Kinetics Using Moment Equations and System Size Expansion. PLoS Comput Biol 2016; 12:e1005030. [PMID: 27447730 PMCID: PMC4957800 DOI: 10.1371/journal.pcbi.1005030] [Citation(s) in RCA: 46] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/28/2015] [Accepted: 06/23/2016] [Indexed: 11/18/2022] Open
Abstract
Quantitative mechanistic models are valuable tools for disentangling biochemical pathways and for achieving a comprehensive understanding of biological systems. However, to be quantitative the parameters of these models have to be estimated from experimental data. In the presence of significant stochastic fluctuations this is a challenging task as stochastic simulations are usually too time-consuming and a macroscopic description using reaction rate equations (RREs) is no longer accurate. In this manuscript, we therefore consider moment-closure approximation (MA) and the system size expansion (SSE), which approximate the statistical moments of stochastic processes and tend to be more precise than macroscopic descriptions. We introduce gradient-based parameter optimization methods and uncertainty analysis methods for MA and SSE. Efficiency and reliability of the methods are assessed using simulation examples as well as by an application to data for Epo-induced JAK/STAT signaling. The application revealed that even if merely population-average data are available, MA and SSE improve parameter identifiability in comparison to RRE. Furthermore, the simulation examples revealed that the resulting estimates are more reliable for an intermediate volume regime. In this regime the estimation error is reduced and we propose methods to determine the regime boundaries. These results illustrate that inference using MA and SSE is feasible and possesses a high sensitivity. In this manuscript, we introduce efficient methods for parameter estimation for stochastic processes. The stochasticity of chemical reactions can influence the average behavior of the considered system. For some biological systems, a microscopic, stochastic description is computationally intractable but a macroscopic, deterministic description too inaccurate. This inaccuracy manifests itself in an error in parameter estimates, which impede the predictive power of the proposed model. Until now, no rigorous analysis on the magnitude of the estimation error exists. We show by means of two simulation examples that using mesoscopic descriptions based on the system size expansions and moment-closure approximations can reduce this estimation error compared to inference using a macroscopic description. This reduction is most pronounced in an intermediate volume regime where the influence of stochasticity on the average behavior is moderately strong. For the JAK/STAT pathway where experimental data is available, we show that one parameter that was not structurally identifiable when using a macroscopic description becomes structurally identifiable when using a mesoscopic description for parameter estimation.
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Affiliation(s)
- Fabian Fröhlich
- Helmholtz Zentrum München - German Research Center for Environmental Health, Institute of Computational Biology, Neuherberg, Germany
- Technische Universität München, Center for Mathematics, Chair of Mathematical Modeling of Biological Systems, Garching, Germany
| | - Philipp Thomas
- Department of Mathematics, Imperial College London, London, United Kingdom
| | - Atefeh Kazeroonian
- Helmholtz Zentrum München - German Research Center for Environmental Health, Institute of Computational Biology, Neuherberg, Germany
- Technische Universität München, Center for Mathematics, Chair of Mathematical Modeling of Biological Systems, Garching, Germany
| | - Fabian J. Theis
- Helmholtz Zentrum München - German Research Center for Environmental Health, Institute of Computational Biology, Neuherberg, Germany
- Technische Universität München, Center for Mathematics, Chair of Mathematical Modeling of Biological Systems, Garching, Germany
| | - Ramon Grima
- School of Biological Sciences, University of Edinburgh, Edinburgh, United Kingdom
- * E-mail: (RG); (JH)
| | - Jan Hasenauer
- Helmholtz Zentrum München - German Research Center for Environmental Health, Institute of Computational Biology, Neuherberg, Germany
- Technische Universität München, Center for Mathematics, Chair of Mathematical Modeling of Biological Systems, Garching, Germany
- * E-mail: (RG); (JH)
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Kazeroonian A, Fröhlich F, Raue A, Theis FJ, Hasenauer J. CERENA: ChEmical REaction Network Analyzer--A Toolbox for the Simulation and Analysis of Stochastic Chemical Kinetics. PLoS One 2016; 11:e0146732. [PMID: 26807911 PMCID: PMC4726759 DOI: 10.1371/journal.pone.0146732] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2015] [Accepted: 12/21/2015] [Indexed: 01/27/2023] Open
Abstract
Gene expression, signal transduction and many other cellular processes are subject to stochastic fluctuations. The analysis of these stochastic chemical kinetics is important for understanding cell-to-cell variability and its functional implications, but it is also challenging. A multitude of exact and approximate descriptions of stochastic chemical kinetics have been developed, however, tools to automatically generate the descriptions and compare their accuracy and computational efficiency are missing. In this manuscript we introduced CERENA, a toolbox for the analysis of stochastic chemical kinetics using Approximations of the Chemical Master Equation solution statistics. CERENA implements stochastic simulation algorithms and the finite state projection for microscopic descriptions of processes, the system size expansion and moment equations for meso- and macroscopic descriptions, as well as the novel conditional moment equations for a hybrid description. This unique collection of descriptions in a single toolbox facilitates the selection of appropriate modeling approaches. Unlike other software packages, the implementation of CERENA is completely general and allows, e.g., for time-dependent propensities and non-mass action kinetics. By providing SBML import, symbolic model generation and simulation using MEX-files, CERENA is user-friendly and computationally efficient. The availability of forward and adjoint sensitivity analyses allows for further studies such as parameter estimation and uncertainty analysis. The MATLAB code implementing CERENA is freely available from http://cerenadevelopers.github.io/CERENA/.
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Affiliation(s)
- Atefeh Kazeroonian
- Institute of Computational Biology, Helmholtz Zentrum München—German Research Center for Environmental Health, Neuherberg, Germany
- Department of Mathematics, Chair of Mathematical Modeling of Biological Systems, Technische Universität München, Garching, Germany
| | - Fabian Fröhlich
- Institute of Computational Biology, Helmholtz Zentrum München—German Research Center for Environmental Health, Neuherberg, Germany
- Department of Mathematics, Chair of Mathematical Modeling of Biological Systems, Technische Universität München, Garching, Germany
| | - Andreas Raue
- Merrimack Pharmaceuticals Inc., Discovery Devision, Cambridge, MA 02139, United States of America
| | - Fabian J. Theis
- Institute of Computational Biology, Helmholtz Zentrum München—German Research Center for Environmental Health, Neuherberg, Germany
- Department of Mathematics, Chair of Mathematical Modeling of Biological Systems, Technische Universität München, Garching, Germany
| | - Jan Hasenauer
- Institute of Computational Biology, Helmholtz Zentrum München—German Research Center for Environmental Health, Neuherberg, Germany
- Department of Mathematics, Chair of Mathematical Modeling of Biological Systems, Technische Universität München, Garching, Germany
- * E-mail:
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González-Vargas AM, Cinquemani E, Ferrari-Trecate G. Validation methods for population models of gene expression dynamics. ACTA ACUST UNITED AC 2016. [DOI: 10.1016/j.ifacol.2016.12.112] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023]
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