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Kilic Z, Schweiger M, Moyer C, Shepherd D, Pressé S. Gene expression model inference from snapshot RNA data using Bayesian non-parametrics. NATURE COMPUTATIONAL SCIENCE 2023; 3:174-183. [PMID: 38125199 PMCID: PMC10732567 DOI: 10.1038/s43588-022-00392-0] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2022] [Accepted: 12/15/2022] [Indexed: 12/23/2023]
Abstract
Gene expression models, which are key towards understanding cellular regulatory response, underlie observations of single-cell transcriptional dynamics. Although RNA expression data encode information on gene expression models, existing computational frameworks do not perform simultaneous Bayesian inference of gene expression models and parameters from such data. Rather, gene expression models-composed of gene states, their connectivities and associated parameters-are currently deduced by pre-specifying gene state numbers and connectivity before learning associated rate parameters. Here we propose a method to learn full distributions over gene states, state connectivities and associated rate parameters, simultaneously and self-consistently from single-molecule RNA counts. We propagate noise from fluctuating RNA counts over models by treating models themselves as random variables. We achieve this within a Bayesian non-parametric paradigm. We demonstrate our method on the Escherichia coli lacZ pathway and the Saccharomyces cerevisiae STL1 pathway, and verify its robustness on synthetic data.
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Affiliation(s)
- Zeliha Kilic
- Department of Structural Biology, St. Jude Children’s Research Hospital, Memphis, TN, USA
- These authors contributed equally: Zeliha Kilic, Max Schweiger
| | - Max Schweiger
- Center for Biological Physics, ASU, Tempe, AZ, USA
- Department of Physics, ASU, Tempe, AZ, USA
- These authors contributed equally: Zeliha Kilic, Max Schweiger
| | - Camille Moyer
- Center for Biological Physics, ASU, Tempe, AZ, USA
- School of Mathematics and Statistical Sciences, ASU, Tempe, AZ, USA
| | - Douglas Shepherd
- Center for Biological Physics, ASU, Tempe, AZ, USA
- Department of Physics, ASU, Tempe, AZ, USA
| | - Steve Pressé
- Center for Biological Physics, ASU, Tempe, AZ, USA
- Department of Physics, ASU, Tempe, AZ, USA
- School of Molecular Sciences, ASU, Tempe, AZ, USA
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2
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Gorin G, Vastola JJ, Fang M, Pachter L. Interpretable and tractable models of transcriptional noise for the rational design of single-molecule quantification experiments. Nat Commun 2022; 13:7620. [PMID: 36494337 PMCID: PMC9734650 DOI: 10.1038/s41467-022-34857-7] [Citation(s) in RCA: 14] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/25/2022] [Accepted: 11/09/2022] [Indexed: 12/13/2022] Open
Abstract
The question of how cell-to-cell differences in transcription rate affect RNA count distributions is fundamental for understanding biological processes underlying transcription. Answering this question requires quantitative models that are both interpretable (describing concrete biophysical phenomena) and tractable (amenable to mathematical analysis). This enables the identification of experiments which best discriminate between competing hypotheses. As a proof of principle, we introduce a simple but flexible class of models involving a continuous stochastic transcription rate driving a discrete RNA transcription and splicing process, and compare and contrast two biologically plausible hypotheses about transcription rate variation. One assumes variation is due to DNA experiencing mechanical strain, while the other assumes it is due to regulator number fluctuations. We introduce a framework for numerically and analytically studying such models, and apply Bayesian model selection to identify candidate genes that show signatures of each model in single-cell transcriptomic data from mouse glutamatergic neurons.
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Affiliation(s)
- Gennady Gorin
- Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA, 91125, USA
| | - John J Vastola
- Department of Neurobiology, Harvard Medical School, Boston, MA, 02115, USA
| | - Meichen Fang
- Division of Biology and Biological Engineering, California Institute of Technology, Pasadena, CA, 91125, USA
| | - Lior Pachter
- Division of Biology and Biological Engineering, California Institute of Technology, Pasadena, CA, 91125, USA.
- Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, 91125, USA.
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3
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Solving the chemical master equation for monomolecular reaction systems and beyond: a Doi-Peliti path integral view. J Math Biol 2021; 83:48. [PMID: 34635944 DOI: 10.1007/s00285-021-01670-7] [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: 11/03/2019] [Revised: 09/01/2021] [Accepted: 09/09/2021] [Indexed: 10/20/2022]
Abstract
The chemical master equation (CME) is a fundamental description of interacting molecules commonly used to model chemical kinetics and noisy gene regulatory networks. Exact time-dependent solutions of the CME-which typically consists of infinitely many coupled differential equations-are rare, and are valuable for numerical benchmarking and getting intuition for the behavior of more complicated systems. Jahnke and Huisinga's landmark calculation of the exact time-dependent solution of the CME for monomolecular reaction systems is one of the most general analytic results known; however, it is hard to generalize, because it relies crucially on special properties of monomolecular reactions. In this paper, we rederive Jahnke and Huisinga's result on the time-dependent probability distribution and moments of monomolecular reaction systems using the Doi-Peliti path integral approach, which reduces solving the CME to evaluating many integrals. While the Doi-Peliti approach is less intuitive, it is also more mechanical, and hence easier to generalize. To illustrate how the Doi-Peliti approach can go beyond the method of Jahnke and Huisinga, we also find an explicit and exact time-dependent solution to a problem involving an autocatalytic reaction that Jahnke and Huisinga identified as not solvable using their method. Most interestingly, we are able to find a formal exact time-dependent solution for any CME whose list of reactions involves only zero and first order reactions, which may be the most general result currently known. This formal solution also yields a useful algorithm for efficiently computing numerical solutions to CMEs of this type.
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4
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Dinh T, Sidje RB. An adaptive solution to the chemical master equation using quantized tensor trains with sliding windows. Phys Biol 2020; 17:065014. [PMID: 32610302 DOI: 10.1088/1478-3975/aba1d2] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
Abstract
To cope with an extremely large or even infinite state space when solving the chemical master equation in biological problems, a potent strategy is to restrict to a finite state projection (FSP) and represent the transition matrix and probability vector in quantized tensor train (QTT) format, leading to savings in storage while retaining accuracy. In an earlier adaptive FSP-QTT algorithm, the multidimensional state space was downsized and kept in the form of a hyper rectangle that was updated when needed by selectively doubling some of its side dimensions. However, this could result in a much larger state space than necessary, with the effect of hampering both the execution time and stepping scheme. In this work, we improve the algorithm by enabling sliding windows that can dynamically slide, shrink or expand, with updates driven by a number of stochastic simulation algorithm trajectories. The ensuing state space is a considerably reduced hyper rectangle containing only the most probable states at each time step. Three numerical experiments of varying difficulty are performed to compare our approach with the original adaptive FSP-QTT algorithm.
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5
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Fox ZR, Neuert G, Munsky B. Optimal Design of Single-Cell Experiments within Temporally Fluctuating Environments. COMPLEXITY 2020; 2020:8536365. [PMID: 32982137 PMCID: PMC7515449 DOI: 10.1155/2020/8536365] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Modern biological experiments are becoming increasingly complex, and designing these experiments to yield the greatest possible quantitative insight is an open challenge. Increasingly, computational models of complex stochastic biological systems are being used to understand and predict biological behaviors or to infer biological parameters. Such quantitative analyses can also help to improve experiment designs for particular goals, such as to learn more about specific model mechanisms or to reduce prediction errors in certain situations. A classic approach to experiment design is to use the Fisher information matrix (FIM), which quantifies the expected information a particular experiment will reveal about model parameters. The Finite State Projection based FIM (FSP-FIM) was recently developed to compute the FIM for discrete stochastic gene regulatory systems, whose complex response distributions do not satisfy standard assumptions of Gaussian variations. In this work, we develop the FSP-FIM analysis for a stochastic model of stress response genes in S. cerevisae under time-varying MAPK induction. We verify this FSP-FIM analysis and use it to optimize the number of cells that should be quantified at particular times to learn as much as possible about the model parameters. We then extend the FSP-FIM approach to explore how different measurement times or genetic modifications help to minimize uncertainty in the sensing of extracellular environments, and we experimentally validate the FSP-FIM to rank single-cell experiments for their abilities to minimize estimation uncertainty of NaCl concentrations during yeast osmotic shock. This work demonstrates the potential of quantitative models to not only make sense of modern biological data sets, but to close the loop between quantitative modeling and experimental data collection.
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Affiliation(s)
- Zachary R Fox
- Inria Saclay Ile-de-France, Palaiseau 91120, France Institut Pasteur, USR 3756 IP CNRS Paris, 75015, France School of Biomedical Engineering, Colorado State University Fort Collins, CO 80523, USA
| | - Gregor Neuert
- Department of Molecular Physiology and Biophysics, School of Medicine, Vanderbilt University, Nashville, TN 37232, USA
| | - Brian Munsky
- Department of Chemical and Biological Engineering, Colorado State University Fort Collins, CO 80523, USA School of Biomedical Engineering, Colorado State University Fort Collins, CO 80523, USA
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6
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Chen M, Ahmadian M, Watson LT, Cao Y. Finding acceptable parameter regions of stochastic Hill functions for multisite phosphorylation mechanism. J Chem Phys 2020; 152:124108. [DOI: 10.1063/1.5143004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- M. Chen
- Department of Computer Science, Virginia Tech, Blacksburg, Virginia 24061, USA
| | - M. Ahmadian
- Department of Computer Science, Virginia Tech, Blacksburg, Virginia 24061, USA
| | - L. T. Watson
- Department of Computer Science, Virginia Tech, Blacksburg, Virginia 24061, USA
| | - Y. Cao
- Department of Computer Science, Virginia Tech, Blacksburg, Virginia 24061, USA
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7
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Catanach TA, Vo HD, Munsky B. BAYESIAN INFERENCE OF STOCHASTIC REACTION NETWORKS USING MULTIFIDELITY SEQUENTIAL TEMPERED MARKOV CHAIN MONTE CARLO. INTERNATIONAL JOURNAL FOR UNCERTAINTY QUANTIFICATION 2020; 10:515-542. [PMID: 34007522 PMCID: PMC8127724 DOI: 10.1615/int.j.uncertaintyquantification.2020033241] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
Stochastic reaction network models are often used to explain and predict the dynamics of gene regulation in single cells. These models usually involve several parameters, such as the kinetic rates of chemical reactions, that are not directly measurable and must be inferred from experimental data. Bayesian inference provides a rigorous probabilistic framework for identifying these parameters by finding a posterior parameter distribution that captures their uncertainty. Traditional computational methods for solving inference problems such as Markov Chain Monte Carlo methods based on classical Metropolis-Hastings algorithm involve numerous serial evaluations of the likelihood function, which in turn requires expensive forward solutions of the chemical master equation (CME). We propose an alternate approach based on a multifidelity extension of the Sequential Tempered Markov Chain Monte Carlo (ST-MCMC) sampler. This algorithm is built upon Sequential Monte Carlo and solves the Bayesian inference problem by decomposing it into a sequence of efficiently solved subproblems that gradually increase both model fidelity and the influence of the observed data. We reformulate the finite state projection (FSP) algorithm, a well-known method for solving the CME, to produce a hierarchy of surrogate master equations to be used in this multifidelity scheme. To determine the appropriate fidelity, we introduce a novel information-theoretic criteria that seeks to extract the most information about the ultimate Bayesian posterior from each model in the hierarchy without inducing significant bias. This novel sampling scheme is tested with high performance computing resources using biologically relevant problems.
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Affiliation(s)
- Thomas A. Catanach
- Sandia National Laboratories, Livermore, CA, 94550
- Address all correspondence to: Thomas A. Catanach,
| | - Huy D. Vo
- Dept. of Chemical and Biological Engr., Colorado State University, Fort Collins, CO, 80521
| | - Brian Munsky
- Dept. of Chemical and Biological Engr., Colorado State University, Fort Collins, CO, 80521
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8
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Reid BM, Sidje RB. Finite state projection for approximating the stationary solution to the chemical master equation using reaction rate equations. Math Biosci 2019; 316:108243. [PMID: 31449893 DOI: 10.1016/j.mbs.2019.108243] [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: 04/18/2019] [Revised: 08/21/2019] [Accepted: 08/22/2019] [Indexed: 10/26/2022]
Abstract
When modeling a physical system using a Markov chain, it is often instructive to compute its probability distribution at statistical equilibrium, thereby gaining insight into the stationary, or long-term, behavior of the system. Computing such a distribution directly is problematic when the state space of the system is large. Here, we look at the case of a chemical reaction system that models the dynamics of cellular processes, where it has become popular to constrain the computational burden by using a finite state projection, which aims only to capture the most likely states of the system, rather than every possible state. We propose an efficient method to further narrow these states to those that remain highly probable in the long run, after the transient behavior of the system has dissipated. Our strategy is to quickly estimate the local maxima of the stationary distribution using the reaction rate formulation, which is of considerably smaller size than the full-blown chemical master equation, and from there develop adaptive schemes to profile the distribution around the maxima. We include numerical tests that show the efficiency of our approach.
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Affiliation(s)
- Brandon M Reid
- Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487, USA.
| | - Roger B Sidje
- Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487, USA.
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9
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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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10
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Vo HD, Fox Z, Baetica A, Munsky B. Bayesian Estimation for Stochastic Gene Expression Using Multifidelity Models. J Phys Chem B 2019; 123:2217-2234. [PMID: 30777763 DOI: 10.1021/acs.jpcb.8b10946] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
The finite state projection (FSP) approach to solving the chemical master equation has enabled successful inference of discrete stochastic models to predict single-cell gene regulation dynamics. Unfortunately, the FSP approach is highly computationally intensive for all but the simplest models, an issue that is highly problematic when parameter inference and uncertainty quantification takes enormous numbers of parameter evaluations. To address this issue, we propose two new computational methods for the Bayesian inference of stochastic gene expression parameters given single-cell experiments. We formulate and verify an adaptive delayed acceptance Metropolis-Hastings (ADAMH) algorithm to utilize with reduced Krylov-basis projections of the FSP. We then introduce an extension of the ADAMH into a hybrid scheme that consists of an initial phase to construct a reduced model and a faster second phase to sample from the approximate posterior distribution determined by the constructed model. We test and compare both algorithms to an adaptive Metropolis algorithm with full FSP-based likelihood evaluations on three example models and simulated data to show that the new ADAMH variants achieve substantial speedup in comparison to the full FSP approach. By reducing the computational costs of parameter estimation, we expect the ADAMH approach to enable efficient data-driven estimation for more complex gene regulation models.
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Affiliation(s)
- Huy D Vo
- Department of Chemical and Biological Engineering , Colorado State University , Fort Collins , Colorado 80523 , United States
| | - Zachary Fox
- Keck Scholars, School of Biomedical Engineering , Colorado State University , Fort Collins , Colorado 80523 , United States
| | - Ania Baetica
- Department of Biochemistry and Biophysics , University of California San Francisco , San Francisco , California 94158 , United States
| | - Brian Munsky
- Department of Chemical and Biological Engineering , Colorado State University , Fort Collins , Colorado 80523 , United States.,Keck Scholars, School of Biomedical Engineering , Colorado State University , Fort Collins , Colorado 80523 , United States
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11
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Fox ZR, Munsky B. The finite state projection based Fisher information matrix approach to estimate information and optimize single-cell experiments. PLoS Comput Biol 2019; 15:e1006365. [PMID: 30645589 PMCID: PMC6355035 DOI: 10.1371/journal.pcbi.1006365] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/11/2018] [Revised: 01/31/2019] [Accepted: 12/02/2018] [Indexed: 12/31/2022] Open
Abstract
Modern optical imaging experiments not only measure single-cell and single-molecule dynamics with high precision, but they can also perturb the cellular environment in myriad controlled and novel settings. Techniques, such as single-molecule fluorescence in-situ hybridization, microfluidics, and optogenetics, have opened the door to a large number of potential experiments, which begs the question of how to choose the best possible experiment. The Fisher information matrix (FIM) estimates how well potential experiments will constrain model parameters and can be used to design optimal experiments. Here, we introduce the finite state projection (FSP) based FIM, which uses the formalism of the chemical master equation to derive and compute the FIM. The FSP-FIM makes no assumptions about the distribution shapes of single-cell data, and it does not require precise measurements of higher order moments of such distributions. We validate the FSP-FIM against well-known Fisher information results for the simple case of constitutive gene expression. We then use numerical simulations to demonstrate the use of the FSP-FIM to optimize the timing of single-cell experiments with more complex, non-Gaussian fluctuations. We validate optimal simulated experiments determined using the FSP-FIM with Monte-Carlo approaches and contrast these to experiment designs chosen by traditional analyses that assume Gaussian fluctuations or use the central limit theorem. By systematically designing experiments to use all of the measurable fluctuations, our method enables a key step to improve co-design of experiments and quantitative models.
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Affiliation(s)
- Zachary R Fox
- Keck Scholars, School of Biomedical Engineering, Colorado State University, Fort Collins, CO, USA
| | - Brian Munsky
- Keck Scholars, School of Biomedical Engineering, Colorado State University, Fort Collins, CO, USA
- Department of Chemical and Biological Engineering, Colorado State University, Fort Collins, CO, USA
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12
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Dowdy GR, Barton PI. Dynamic bounds on stochastic chemical kinetic systems using semidefinite programming. J Chem Phys 2018; 149:074103. [DOI: 10.1063/1.5029926] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/22/2023] Open
Affiliation(s)
- Garrett R. Dowdy
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - Paul I. Barton
- Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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13
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Strome B, Hsu IS, Li Cheong Man M, Zarin T, Nguyen Ba A, Moses AM. Short linear motifs in intrinsically disordered regions modulate HOG signaling capacity. BMC SYSTEMS BIOLOGY 2018; 12:75. [PMID: 29970070 PMCID: PMC6029073 DOI: 10.1186/s12918-018-0597-3] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/25/2017] [Accepted: 06/22/2018] [Indexed: 02/04/2023]
Abstract
Background The effort to characterize intrinsically disordered regions of signaling proteins is rapidly expanding. An important class of disordered interaction modules are ubiquitous and functionally diverse elements known as short linear motifs (SLiMs). Results To further examine the role of SLiMs in signal transduction, we used a previously devised bioinformatics method to predict evolutionarily conserved SLiMs within a well-characterized pathway in S. cerevisiae. Using a single cell, reporter-based flow cytometry assay in conjunction with a fluorescent reporter driven by a pathway-specific promoter, we quantitatively assessed pathway output via systematic deletions of individual motifs. We found that, when deleted, 34% (10/29) of predicted SLiMs displayed a significant decrease in pathway output, providing evidence that these motifs play a role in signal transduction. Assuming that mutations in SLiMs have quantitative effects on mechanisms of signaling, we show that perturbations of parameters in a previously published stochastic model of HOG signaling could reproduce the quantitative effects of 4 out of 7 mutations in previously unknown SLiMs. Conclusions Our study suggests that, even in well-characterized pathways, large numbers of functional elements remain undiscovered, and that challenges remain for application of systems biology models to interpret the effects of mutations in signaling pathways. Electronic supplementary material The online version of this article (10.1186/s12918-018-0597-3) contains supplementary material, which is available to authorized users.
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Affiliation(s)
- Bob Strome
- Department of Cell & Systems Biology, University of Toronto, 25 Willcocks Street, Toronto, ON, M5S 3B2, Canada
| | - Ian Shenyen Hsu
- Department of Cell & Systems Biology, University of Toronto, 25 Willcocks Street, Toronto, ON, M5S 3B2, Canada
| | - Mitchell Li Cheong Man
- Department of Cell & Systems Biology, University of Toronto, 25 Willcocks Street, Toronto, ON, M5S 3B2, Canada
| | - Taraneh Zarin
- Department of Cell & Systems Biology, University of Toronto, 25 Willcocks Street, Toronto, ON, M5S 3B2, Canada
| | - Alex Nguyen Ba
- Department of Cell & Systems Biology, University of Toronto, 25 Willcocks Street, Toronto, ON, M5S 3B2, Canada
| | - Alan M Moses
- Department of Cell & Systems Biology, University of Toronto, 25 Willcocks Street, Toronto, ON, M5S 3B2, Canada. .,Center for Analysis of Genome Evolution and Function, 25 Willcocks Street, Toronto, ON, M5S 3B2, Canada.
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14
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Abstract
Systems biology seeks to combine experiments with computation to predict biological behaviors. However, despite tremendous data and knowledge, biological models make less-accurate predictions compared with other fields. By analyzing single-cell, single-molecule measurements of mRNA during yeast stress response, we explore why and how the shapes of experimental distributions control prediction accuracy. We show how asymmetric data distributions with long tails cause standard modeling approaches to yield excellent fits but make meaningless predictions. We show how these biases arise from the violation of fundamental assumptions in standard modeling approaches. We demonstrate how advanced computational tools solve this dilemma and achieve predictive understanding of spatiotemporal mechanisms of transcription control including RNA polymerase initiation and elongation and mRNA accumulation, transport, and decay. Despite substantial experimental and computational efforts, mechanistic modeling remains more predictive in engineering than in systems biology. The reason for this discrepancy is not fully understood. One might argue that the randomness and complexity of biological systems are the main barriers to predictive understanding, but these issues are not unique to biology. Instead, we hypothesize that the specific shapes of rare single-molecule event distributions produce substantial yet overlooked challenges for biological models. We demonstrate why modern statistical tools to disentangle complexity and stochasticity, which assume normally distributed fluctuations or enormous datasets, do not apply to the discrete, positive, and nonsymmetric distributions that characterize mRNA fluctuations in single cells. As an example, we integrate single-molecule measurements and advanced computational analyses to explore mitogen-activated protein kinase induction of multiple stress response genes. Through systematic analyses of different metrics to compare the same model to the same data, we elucidate why standard modeling approaches yield nonpredictive models for single-cell gene regulation. We further explain how advanced tools recover precise, reproducible, and predictive understanding of transcription regulation mechanisms, including gene activation, polymerase initiation, elongation, mRNA accumulation, spatial transport, and decay.
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15
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Gupta A, Mikelson J, Khammash M. A finite state projection algorithm for the stationary solution of the chemical master equation. J Chem Phys 2018; 147:154101. [PMID: 29055349 DOI: 10.1063/1.5006484] [Citation(s) in RCA: 43] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
The chemical master equation (CME) is frequently used in systems biology to quantify the effects of stochastic fluctuations that arise due to biomolecular species with low copy numbers. The CME is a system of ordinary differential equations that describes the evolution of probability density for each population vector in the state-space of the stochastic reaction dynamics. For many examples of interest, this state-space is infinite, making it difficult to obtain exact solutions of the CME. To deal with this problem, the Finite State Projection (FSP) algorithm was developed by Munsky and Khammash [J. Chem. Phys. 124(4), 044104 (2006)], to provide approximate solutions to the CME by truncating the state-space. The FSP works well for finite time-periods but it cannot be used for estimating the stationary solutions of CMEs, which are often of interest in systems biology. The aim of this paper is to develop a version of FSP which we refer to as the stationary FSP (sFSP) that allows one to obtain accurate approximations of the stationary solutions of a CME by solving a finite linear-algebraic system that yields the stationary distribution of a continuous-time Markov chain over the truncated state-space. We derive bounds for the approximation error incurred by sFSP and we establish that under certain stability conditions, these errors can be made arbitrarily small by appropriately expanding the truncated state-space. We provide several examples to illustrate our sFSP method and demonstrate its efficiency in estimating the stationary distributions. In particular, we show that using a quantized tensor-train implementation of our sFSP method, problems admitting more than 100 × 106 states can be efficiently solved.
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Affiliation(s)
- Ankit Gupta
- Department of Biosystems Science and Engineering, ETH Zurich Mattenstrasse, 26 4058 Basel, Switzerland
| | - Jan Mikelson
- Department of Biosystems Science and Engineering, ETH Zurich Mattenstrasse, 26 4058 Basel, Switzerland
| | - Mustafa Khammash
- Department of Biosystems Science and Engineering, ETH Zurich Mattenstrasse, 26 4058 Basel, Switzerland
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16
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Abstract
In quantitative analyses of biological processes, one may use many different scales of models (e.g. spatial or non-spatial, deterministic or stochastic, time-varying or at steady-state) or many different approaches to match models to experimental data (e.g. model fitting or parameter uncertainty/sloppiness quantification with different experiment designs). These different analyses can lead to surprisingly different results, even when applied to the same data and the same model. We use a simplified gene regulation model to illustrate many of these concerns, especially for ODE analyses of deterministic processes, chemical master equation and finite state projection analyses of heterogeneous processes, and stochastic simulations. For each analysis, we employ Matlab and Python software to consider a time-dependent input signal (e.g. a kinase nuclear translocation) and several model hypotheses, along with simulated single-cell data. We illustrate different approaches (e.g. deterministic and stochastic) to identify the mechanisms and parameters of the same model from the same simulated data. For each approach, we explore how uncertainty in parameter space varies with respect to the chosen analysis approach or specific experiment design. We conclude with a discussion of how our simulated results relate to the integration of experimental and computational investigations to explore signal-activated gene expression models in yeast (Neuert et al 2013 Science 339 584-7) and human cells (Senecal et al 2014 Cell Rep. 8 75-83)5.
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Affiliation(s)
- Lisa Weber
- Department of Chemical and Biological Engineering, Colorado State University, Fort Collins, CO
| | - William Raymond
- Department of Chemical and Biological Engineering, Colorado State University, Fort Collins, CO
- School of Biomedical Engineering, Colorado State University, Fort Collins, CO
| | - Brian Munsky
- Department of Chemical and Biological Engineering, Colorado State University, Fort Collins, CO
- School of Biomedical Engineering, Colorado State University, Fort Collins, CO
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17
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Ceccato A, Frezzato D. Remarks on the chemical Fokker-Planck and Langevin equations: Nonphysical currents at equilibrium. J Chem Phys 2018; 148:064114. [DOI: 10.1063/1.5016158] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023] Open
Affiliation(s)
- Alessandro Ceccato
- Department of Chemical Sciences, University of Padova, Via Marzolo 1, I-35131 Padova, Italy
| | - Diego Frezzato
- Department of Chemical Sciences, University of Padova, Via Marzolo 1, I-35131 Padova, Italy
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Dinh KN, Sidje RB. An application of the Krylov-FSP-SSA method to parameter fitting with maximum likelihood. Phys Biol 2017; 14:065001. [PMID: 29098989 DOI: 10.1088/1478-3975/aa868a] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
Abstract
Monte Carlo methods such as the stochastic simulation algorithm (SSA) have traditionally been employed in gene regulation problems. However, there has been increasing interest to directly obtain the probability distribution of the molecules involved by solving the chemical master equation (CME). This requires addressing the curse of dimensionality that is inherent in most gene regulation problems. The finite state projection (FSP) seeks to address the challenge and there have been variants that further reduce the size of the projection or that accelerate the resulting matrix exponential. The Krylov-FSP-SSA variant has proved numerically efficient by combining, on one hand, the SSA to adaptively drive the FSP, and on the other hand, adaptive Krylov techniques to evaluate the matrix exponential. Here we apply this Krylov-FSP-SSA to a mutual inhibitory gene network synthetically engineered in Saccharomyces cerevisiae, in which bimodality arises. We show numerically that the approach can efficiently approximate the transient probability distribution, and this has important implications for parameter fitting, where the CME has to be solved for many different parameter sets. The fitting scheme amounts to an optimization problem of finding the parameter set so that the transient probability distributions fit the observations with maximum likelihood. We compare five optimization schemes for this difficult problem, thereby providing further insights into this approach of parameter estimation that is often applied to models in systems biology where there is a need to calibrate free parameters.
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Affiliation(s)
- Khanh N Dinh
- Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, United States of America
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Johnson R, Munsky B. The finite state projection approach to analyze dynamics of heterogeneous populations. Phys Biol 2017; 14:035002. [PMID: 28428446 DOI: 10.1088/1478-3975/aa6e5a] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
Abstract
Population modeling aims to capture and predict the dynamics of cell populations in constant or fluctuating environments. At the elementary level, population growth proceeds through sequential divisions of individual cells. Due to stochastic effects, populations of cells are inherently heterogeneous in phenotype, and some phenotypic variables have an effect on division or survival rates, as can be seen in partial drug resistance. Therefore, when modeling population dynamics where the control of growth and division is phenotype dependent, the corresponding model must take account of the underlying cellular heterogeneity. The finite state projection (FSP) approach has often been used to analyze the statistics of independent cells. Here, we extend the FSP analysis to explore the coupling of cell dynamics and biomolecule dynamics within a population. This extension allows a general framework with which to model the state occupations of a heterogeneous, isogenic population of dividing and expiring cells. The method is demonstrated with a simple model of cell-cycle progression, which we use to explore possible dynamics of drug resistance phenotypes in dividing cells. We use this method to show how stochastic single-cell behaviors affect population level efficacy of drug treatments, and we illustrate how slight modifications to treatment regimens may have dramatic effects on drug efficacy.
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Affiliation(s)
- Rob Johnson
- Department of Life Sciences, Centre for Integrative Systems Biology and Bioinformatics, Imperial College London, London SW7 2AZ, United Kingdom
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