1
|
Stochastic approximations of higher-molecular by bi-molecular reactions. J Math Biol 2023; 86:28. [PMID: 36637496 PMCID: PMC9839831 DOI: 10.1007/s00285-022-01848-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/30/2020] [Revised: 09/24/2022] [Accepted: 11/22/2022] [Indexed: 01/14/2023]
Abstract
Reactions involving three or more reactants, called higher-molecular reactions, play an important role in mathematical modelling in systems and synthetic biology. In particular, such reactions underpin a variety of important bio-dynamical phenomena, such as multi-stability/multi-modality, oscillations, bifurcations, and noise-induced effects. However, as opposed to reactions involving at most two reactants, called bi-molecular reactions, higher-molecular reactions are biochemically improbable. To bridge the gap, in this paper we put forward an algorithm for systematically approximating arbitrary higher-molecular reactions with bi-molecular ones, while preserving the underlying stochastic dynamics. Properties of the algorithm and convergence are established via singular perturbation theory. The algorithm is applied to a variety of higher-molecular biochemical networks, and is shown to play an important role in synthetic biology.
Collapse
|
2
|
Eilertsen J, Schnell S. On the Validity of the Stochastic Quasi-Steady-State Approximation in Open Enzyme Catalyzed Reactions: Timescale Separation or Singular Perturbation? Bull Math Biol 2021; 84:7. [PMID: 34825985 PMCID: PMC8768927 DOI: 10.1007/s11538-021-00966-5] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/18/2021] [Accepted: 10/26/2021] [Indexed: 10/19/2022]
Abstract
The quasi-steady-state approximation is widely used to develop simplified deterministic or stochastic models of enzyme catalyzed reactions. In deterministic models, the quasi-steady-state approximation can be mathematically justified from singular perturbation theory. For several closed enzymatic reactions, the homologous extension of the quasi-steady-state approximation to the stochastic regime, known as the stochastic quasi-steady-state approximation, has been shown to be accurate under the analogous conditions that permit the quasi-steady-state reduction in the deterministic counterpart. However, it was recently demonstrated that the extension of the stochastic quasi-steady-state approximation to an open Michaelis-Menten reaction mechanism is only valid under a condition that is far more restrictive than the qualifier that ensures the validity of its corresponding deterministic quasi-steady-state approximation. In this paper, we suggest a possible explanation for this discrepancy from the lens of geometric singular perturbation theory. In so doing, we illustrate a misconception in the application of the quasi-steady-state approximation: timescale separation does not imply singular perturbation.
Collapse
Affiliation(s)
- Justin Eilertsen
- Department of Molecular & Integrative Physiology, University of Michigan Medical School, Ann Arbor, MI, 48109, USA
- Mathematical Reviews, American Mathematical Society, 416 4th Street, Ann Arbor, MI, 48103, USA
| | - Santiago Schnell
- Department of Molecular & Integrative Physiology, University of Michigan Medical School, Ann Arbor, MI, 48109, USA.
- Department of Computational Medicine & Bioinformatics, University of Michigan Medical School, Ann Arbor, MI, 48109, USA.
- Department of Biological Sciences, and Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, 46556, USA.
| |
Collapse
|
3
|
Song YM, Hong H, Kim JK. Universally valid reduction of multiscale stochastic biochemical systems using simple non-elementary propensities. PLoS Comput Biol 2021; 17:e1008952. [PMID: 34662330 PMCID: PMC8562860 DOI: 10.1371/journal.pcbi.1008952] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/26/2021] [Revised: 11/02/2021] [Accepted: 10/04/2021] [Indexed: 11/19/2022] Open
Abstract
Biochemical systems consist of numerous elementary reactions governed by the law of mass action. However, experimentally characterizing all the elementary reactions is nearly impossible. Thus, over a century, their deterministic models that typically contain rapid reversible bindings have been simplified with non-elementary reaction functions (e.g., Michaelis-Menten and Morrison equations). Although the non-elementary reaction functions are derived by applying the quasi-steady-state approximation (QSSA) to deterministic systems, they have also been widely used to derive propensities for stochastic simulations due to computational efficiency and simplicity. However, the validity condition for this heuristic approach has not been identified even for the reversible binding between molecules, such as protein-DNA, enzyme-substrate, and receptor-ligand, which is the basis for living cells. Here, we find that the non-elementary propensities based on the deterministic total QSSA can accurately capture the stochastic dynamics of the reversible binding in general. However, serious errors occur when reactant molecules with similar levels tightly bind, unlike deterministic systems. In that case, the non-elementary propensities distort the stochastic dynamics of a bistable switch in the cell cycle and an oscillator in the circadian clock. Accordingly, we derive alternative non-elementary propensities with the stochastic low-state QSSA, developed in this study. This provides a universally valid framework for simplifying multiscale stochastic biochemical systems with rapid reversible bindings, critical for efficient stochastic simulations of cell signaling and gene regulation. To facilitate the framework, we provide a user-friendly open-source computational package, ASSISTER, that automatically performs the present framework.
Collapse
Affiliation(s)
- Yun Min Song
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea
- Biomedical Mathematics Group, Institute for Basic Science, Daejeon, Republic of Korea
| | - Hyukpyo Hong
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea
- Biomedical Mathematics Group, Institute for Basic Science, Daejeon, Republic of Korea
| | - Jae Kyoung Kim
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea
- Biomedical Mathematics Group, Institute for Basic Science, Daejeon, Republic of Korea
| |
Collapse
|
4
|
Johnson ME, Chen A, Faeder JR, Henning P, Moraru II, Meier-Schellersheim M, Murphy RF, Prüstel T, Theriot JA, Uhrmacher AM. Quantifying the roles of space and stochasticity in computer simulations for cell biology and cellular biochemistry. Mol Biol Cell 2021; 32:186-210. [PMID: 33237849 PMCID: PMC8120688 DOI: 10.1091/mbc.e20-08-0530] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2020] [Revised: 10/13/2020] [Accepted: 11/17/2020] [Indexed: 12/29/2022] Open
Abstract
Most of the fascinating phenomena studied in cell biology emerge from interactions among highly organized multimolecular structures embedded into complex and frequently dynamic cellular morphologies. For the exploration of such systems, computer simulation has proved to be an invaluable tool, and many researchers in this field have developed sophisticated computational models for application to specific cell biological questions. However, it is often difficult to reconcile conflicting computational results that use different approaches to describe the same phenomenon. To address this issue systematically, we have defined a series of computational test cases ranging from very simple to moderately complex, varying key features of dimensionality, reaction type, reaction speed, crowding, and cell size. We then quantified how explicit spatial and/or stochastic implementations alter outcomes, even when all methods use the same reaction network, rates, and concentrations. For simple cases, we generally find minor differences in solutions of the same problem. However, we observe increasing discordance as the effects of localization, dimensionality reduction, and irreversible enzymatic reactions are combined. We discuss the strengths and limitations of commonly used computational approaches for exploring cell biological questions and provide a framework for decision making by researchers developing new models. As computational power and speed continue to increase at a remarkable rate, the dream of a fully comprehensive computational model of a living cell may be drawing closer to reality, but our analysis demonstrates that it will be crucial to evaluate the accuracy of such models critically and systematically.
Collapse
Affiliation(s)
- M. E. Johnson
- Thomas C. Jenkins Department of Biophysics, Johns Hopkins University, Baltimore, MD, 21218
| | - A. Chen
- Thomas C. Jenkins Department of Biophysics, Johns Hopkins University, Baltimore, MD, 21218
| | - J. R. Faeder
- Department of Computational and Systems Biology, University of Pittsburgh School of Medicine, Pittsburgh, PA, 15260
| | - P. Henning
- Institute for Visual and Analytic Computing, University of Rostock, 18055 Rostock, Germany
| | - I. I. Moraru
- Department of Cell Biology, Center for Cell Analysis and Modeling, University of Connecticut Health Center, Farmington, CT 06030
| | - M. Meier-Schellersheim
- Laboratory of Immune System Biology, National Institute of Allergy and Infectious Diseases, National Institutes of Health, Bethesda, MD 20892
| | - R. F. Murphy
- Computational Biology Department, Department of Biological Sciences, Department of Biomedical Engineering, Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA 15289
| | - T. Prüstel
- Laboratory of Immune System Biology, National Institute of Allergy and Infectious Diseases, National Institutes of Health, Bethesda, MD 20892
| | - J. A. Theriot
- Department of Biology and Howard Hughes Medical Institute, University of Washington, Seattle, WA 98195
| | - A. M. Uhrmacher
- Institute for Visual and Analytic Computing, University of Rostock, 18055 Rostock, Germany
| |
Collapse
|
5
|
Kim JK, Tyson JJ. Misuse of the Michaelis-Menten rate law for protein interaction networks and its remedy. PLoS Comput Biol 2020; 16:e1008258. [PMID: 33090989 PMCID: PMC7581366 DOI: 10.1371/journal.pcbi.1008258] [Citation(s) in RCA: 29] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022] Open
Abstract
For over a century, the Michaelis-Menten (MM) rate law has been used to describe the rates of enzyme-catalyzed reactions and gene expression. Despite the ubiquity of the MM rate law, it accurately captures the dynamics of underlying biochemical reactions only so long as it is applied under the right condition, namely, that the substrate is in large excess over the enzyme-substrate complex. Unfortunately, in circumstances where its validity condition is not satisfied, especially so in protein interaction networks, the MM rate law has frequently been misused. In this review, we illustrate how inappropriate use of the MM rate law distorts the dynamics of the system, provides mistaken estimates of parameter values, and makes false predictions of dynamical features such as ultrasensitivity, bistability, and oscillations. We describe how these problems can be resolved with a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA). Furthermore, we show that the tQSSA can be used for accurate stochastic simulations at a lower computational cost than using the full set of mass-action rate laws. This review describes how to use quasi-steady state approximations in the right context, to prevent drawing erroneous conclusions from in silico simulations.
Collapse
Affiliation(s)
- Jae Kyoung Kim
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea
| | - John J. Tyson
- Department of Biological Sciences, Virginia Tech, Blacksburg, Virginia, United States of America
- Division of Systems Biology, Virginia Tech, Blacksburg, Virginia, United States of America
| |
Collapse
|
6
|
Herath N, Del Vecchio D. Reduced linear noise approximation for biochemical reaction networks with time-scale separation: The stochastic tQSSA+. J Chem Phys 2018. [DOI: 10.1063/1.5012752] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022] Open
Affiliation(s)
- Narmada Herath
- Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
| | - Domitilla Del Vecchio
- Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
| |
Collapse
|
7
|
Late-Arriving Signals Contribute Less to Cell-Fate Decisions. Biophys J 2017; 113:2110-2120. [PMID: 29117533 DOI: 10.1016/j.bpj.2017.09.012] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2017] [Revised: 08/14/2017] [Accepted: 09/05/2017] [Indexed: 11/24/2022] Open
Abstract
Gene regulatory networks are largely responsible for cellular decision-making. These networks sense diverse external signals and respond by adjusting gene expression, enabling cells to reach environment-dependent decisions crucial for their survival or reproduction. However, information-carrying signals may arrive at variable times. Besides the intrinsic strength of these signals, their arrival time (timing) may also carry information about the environment and can influence cellular decision-making in ways that are poorly understood. For example, it is unclear how the timing of individual phage infections affects the lysis-lysogeny decision of bacteriophage λ despite variable infection times being likely in the wild and even in laboratory conditions. In this work, we combine mathematical modeling with experimentation to address this question. We develop an experimentally testable theory, which reveals that late-infecting phages contribute less to cellular decision-making. This implies that infection delays lower the probability of lysogeny compared to simultaneous infections. Furthermore, we show that infection delays reduce lysogenization by providing insufficient CII for threshold crossing during the critical decision-making period. We find evidence for a cutoff time after which subsequent infections cannot influence the cellular decision. We derive an intuitive formula that approximates the probability of lysogeny for variable infection times by a time-weighted average of probabilities for simultaneous infections. We validate these theoretical predictions experimentally. Similar concepts and simplifying modeling approaches may help elucidate the mechanisms underlying other cellular decisions.
Collapse
|
8
|
Kim JK, Sontag ED. Reduction of multiscale stochastic biochemical reaction networks using exact moment derivation. PLoS Comput Biol 2017; 13:e1005571. [PMID: 28582397 PMCID: PMC5481150 DOI: 10.1371/journal.pcbi.1005571] [Citation(s) in RCA: 16] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/13/2017] [Revised: 06/22/2017] [Accepted: 05/12/2017] [Indexed: 12/14/2022] Open
Abstract
Biochemical reaction networks (BRNs) in a cell frequently consist of reactions with disparate timescales. The stochastic simulations of such multiscale BRNs are prohibitively slow due to high computational cost for the simulations of fast reactions. One way to resolve this problem uses the fact that fast species regulated by fast reactions quickly equilibrate to their stationary distribution while slow species are unlikely to be changed. Thus, on a slow timescale, fast species can be replaced by their quasi-steady state (QSS): their stationary conditional expectation values for given slow species. As the QSS are determined solely by the state of slow species, such replacement leads to a reduced model, where fast species are eliminated. However, it is challenging to derive the QSS in the presence of nonlinear reactions. While various approximation schemes for the QSS have been developed, they often lead to considerable errors. Here, we propose two classes of multiscale BRNs which can be reduced by deriving an exact QSS rather than approximations. Specifically, if fast species constitute either a feedforward network or a complex balanced network, the reduced model based on the exact QSS can be derived. Such BRNs are frequently observed in a cell as the feedforward network is one of fundamental motifs of gene or protein regulatory networks. Furthermore, complex balanced networks also include various types of fast reversible bindings such as bindings between transcriptional factors and gene regulatory sites. The reduced models based on exact QSS, which can be calculated by the computational packages provided in this work, accurately approximate the slow scale dynamics of the original full model with much lower computational cost.
Collapse
Affiliation(s)
- Jae Kyoung Kim
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, Korea
- * E-mail: (JKK); , (EDS)
| | - Eduardo D. Sontag
- Department of Mathematics and Center for Quantitative Biology, Rutgers University, New Brunswick, New Jersey, United States of America
- * E-mail: (JKK); , (EDS)
| |
Collapse
|
9
|
Stochastic neutral modelling of the Gut Microbiota's relative species abundance from next generation sequencing data. BMC Bioinformatics 2016; 17 Suppl 2:16. [PMID: 26821617 PMCID: PMC4959352 DOI: 10.1186/s12859-015-0858-8] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/25/2022] Open
Abstract
Background Interest in understanding the mechanisms that lead to a particular composition of the Gut Microbiota is highly increasing, due to the relationship between this ecosystem and the host health state. Particularly relevant is the study of the Relative Species Abundance (RSA) distribution, that is a component of biodiversity and measures the number of species having a given number of individuals. It is the universal behaviour of RSA that induced many ecologists to look for theoretical explanations. In particular, a simple stochastic neutral model was proposed by Volkov et al. relying on population dynamics and was proved to fit the coral-reefs and rain forests RSA. Our aim is to ascertain if this model also describes the Microbiota RSA and if it can help in explaining the Microbiota plasticity. Results We analyzed 16S rRNA sequencing data sampled from the Microbiota of three different animal species by Jeraldo et al. Through a clustering procedure (UCLUST), we built the Operational Taxonomic Units. These correspond to bacterial species considered at a given phylogenetic level defined by the similarity threshold used in the clustering procedure. The RSAs, plotted in the form of Preston plot, were fitted with Volkov’s model. The model fits well the Microbiota RSA, except in the tail region, that shows a deviation from the neutrality assumption. Looking at the model parameters we were able to discriminate between different animal species, giving also a biological explanation. Moreover, the biodiversity estimator obtained by Volkov’s model also differentiates the animal species and is in good agreement with the first and second order Hill’s numbers, that are common evenness indexes simply based on the fraction of individuals per species. Conclusions We conclude that the neutrality assumption is a good approximation for the Microbiota dynamics and the observation that Volkov’s model works for this ecosystem is a further proof of the RSA universality. Moreover, the ability to separate different animals with the model parameters and biodiversity number are promising results if we think about future applications on human data, in which the Microbiota composition and biodiversity are in close relationships with a variety of diseases and life-styles.
Collapse
|
10
|
Dovzhenok AA, Baek M, Lim S, Hong CI. Mathematical modeling and validation of glucose compensation of the neurospora circadian clock. Biophys J 2016; 108:1830-1839. [PMID: 25863073 DOI: 10.1016/j.bpj.2015.01.043] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/12/2014] [Revised: 12/05/2014] [Accepted: 01/09/2015] [Indexed: 11/26/2022] Open
Abstract
Autonomous circadian oscillations arise from transcriptional-translational feedback loops of core clock components. The period of a circadian oscillator is relatively insensitive to changes in nutrients (e.g., glucose), which is referred to as "nutrient compensation". Recently, a transcription repressor, CSP-1, was identified as a component of the circadian system in Neurospora crassa. The transcription of csp-1 is under the circadian regulation. Intriguingly, CSP-1 represses the circadian transcription factor, WC-1, forming a negative feedback loop that can influence the core oscillator. This feedback mechanism is suggested to maintain the circadian period in a wide range of glucose concentrations. In this report, we constructed a mathematical model of the Neurospora circadian clock incorporating the above WC-1/CSP-1 feedback loop, and investigated molecular mechanisms of glucose compensation. Our model shows that glucose compensation exists within a narrow range of parameter space where the activation rates of csp-1 and wc-1 are balanced with each other, and simulates loss of glucose compensation in csp-1 mutants. More importantly, we experimentally validated rhythmic oscillations of the wc-1 gene expression and loss of glucose compensation in the wc-1(ov) mutant as predicted in the model. Furthermore, our stochastic simulations demonstrate that the CSP-1-dependent negative feedback loop functions in glucose compensation, but does not enhance the overall robustness of oscillations against molecular noise. Our work highlights predictive modeling of circadian clock machinery and experimental validations employing Neurospora and brings a deeper understanding of molecular mechanisms of glucose compensation.
Collapse
Affiliation(s)
- Andrey A Dovzhenok
- Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio
| | - Mokryun Baek
- Department of Molecular and Cellular Physiology, University of Cincinnati College of Medicine, Cincinnati, Ohio
| | - Sookkyung Lim
- Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio
| | - Christian I Hong
- Department of Molecular and Cellular Physiology, University of Cincinnati College of Medicine, Cincinnati, Ohio.
| |
Collapse
|
11
|
Kim JK, Josić K, Bennett MR. The relationship between stochastic and deterministic quasi-steady state approximations. BMC SYSTEMS BIOLOGY 2015; 9:87. [PMID: 26597159 PMCID: PMC4657384 DOI: 10.1186/s12918-015-0218-3] [Citation(s) in RCA: 38] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 04/06/2015] [Accepted: 10/06/2015] [Indexed: 09/03/2023]
Abstract
Background The quasi steady-state approximation (QSSA) is frequently used to reduce deterministic models of biochemical networks. The resulting equations provide a simplified description of the network in terms of non-elementary reaction functions (e.g. Hill functions). Such deterministic reductions are frequently a basis for heuristic stochastic models in which non-elementary reaction functions are used to define reaction propensities. Despite their popularity, it remains unclear when such stochastic reductions are valid. It is frequently assumed that the stochastic reduction can be trusted whenever its deterministic counterpart is accurate. However, a number of recent examples show that this is not necessarily the case. Results Here we explain the origin of these discrepancies, and demonstrate a clear relationship between the accuracy of the deterministic and the stochastic QSSA for examples widely used in biological systems. With an analysis of a two-state promoter model, and numerical simulations for a variety of other models, we find that the stochastic QSSA is accurate whenever its deterministic counterpart provides an accurate approximation over a range of initial conditions which cover the likely fluctuations from the quasi steady-state (QSS). We conjecture that this relationship provides a simple and computationally inexpensive way to test the accuracy of reduced stochastic models using deterministic simulations. Conclusions The stochastic QSSA is one of the most popular multi-scale stochastic simulation methods. While the use of QSSA, and the resulting non-elementary functions has been justified in the deterministic case, it is not clear when their stochastic counterparts are accurate. In this study, we show how the accuracy of the stochastic QSSA can be tested using their deterministic counterparts providing a concrete method to test when non-elementary rate functions can be used in stochastic simulations. Electronic supplementary material The online version of this article (doi:10.1186/s12918-015-0218-3) contains supplementary material, which is available to authorized users.
Collapse
Affiliation(s)
- Jae Kyoung Kim
- Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 291 Daehak-ro Yuseong-gu, Daejeon, 305-701, Korea. .,Mathematical Biosciences Institute, The Ohio State University, 1735 Neil Avenue, OH 43210, Columbus, USA.
| | - Krešimir Josić
- Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, TX 77204-3008, USA. .,Department of Biology and Biochemistry, University of Houston, 4800 Calhoun Rd, Houston, TX 77204-3008, USA.
| | - Matthew R Bennett
- Department of Biosciences, Rice University, 6100 Main St, Houston, 77005-1892, TX, USA. .,Department of Bioengineering, Rice University, 6100 Main St, Houston, TX 77005-1892, USA.
| |
Collapse
|
12
|
Kim JK, Josić K, Bennett MR. The validity of quasi-steady-state approximations in discrete stochastic simulations. Biophys J 2015; 107:783-793. [PMID: 25099817 DOI: 10.1016/j.bpj.2014.06.012] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2014] [Revised: 05/28/2014] [Accepted: 06/06/2014] [Indexed: 10/25/2022] Open
Abstract
In biochemical networks, reactions often occur on disparate timescales and can be characterized as either fast or slow. The quasi-steady-state approximation (QSSA) utilizes timescale separation to project models of biochemical networks onto lower-dimensional slow manifolds. As a result, fast elementary reactions are not modeled explicitly, and their effect is captured by nonelementary reaction-rate functions (e.g., Hill functions). The accuracy of the QSSA applied to deterministic systems depends on how well timescales are separated. Recently, it has been proposed to use the nonelementary rate functions obtained via the deterministic QSSA to define propensity functions in stochastic simulations of biochemical networks. In this approach, termed the stochastic QSSA, fast reactions that are part of nonelementary reactions are not simulated, greatly reducing computation time. However, it is unclear when the stochastic QSSA provides an accurate approximation of the original stochastic simulation. We show that, unlike the deterministic QSSA, the validity of the stochastic QSSA does not follow from timescale separation alone, but also depends on the sensitivity of the nonelementary reaction rate functions to changes in the slow species. The stochastic QSSA becomes more accurate when this sensitivity is small. Different types of QSSAs result in nonelementary functions with different sensitivities, and the total QSSA results in less sensitive functions than the standard or the prefactor QSSA. We prove that, as a result, the stochastic QSSA becomes more accurate when nonelementary reaction functions are obtained using the total QSSA. Our work provides an apparently novel condition for the validity of the QSSA in stochastic simulations of biochemical reaction networks with disparate timescales.
Collapse
Affiliation(s)
- Jae Kyoung Kim
- Mathematical Biosciences Institute, The Ohio State University, Columbus, Ohio
| | - Krešimir Josić
- Department of Mathematics, University of Houston, Houston, Texas; Department of Biology and Biochemistry, University of Houston, Houston, Texas.
| | - Matthew R Bennett
- Department of Biochemistry & Cell Biology, Rice University, Houston, Texas; Institute of Biosciences and Bioengineering, Rice University, Houston, Texas.
| |
Collapse
|
13
|
Wu S, Fu J, Petzold LR. Adaptive deployment of model reductions for tau-leaping simulation. J Chem Phys 2015; 142:204108. [PMID: 26026435 PMCID: PMC4449353 DOI: 10.1063/1.4921638] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2015] [Accepted: 05/13/2015] [Indexed: 11/14/2022] Open
Abstract
Multiple time scales in cellular chemical reaction systems often render the tau-leaping algorithm inefficient. Various model reductions have been proposed to accelerate tau-leaping simulations. However, these are often identified and deployed manually, requiring expert knowledge. This is time-consuming and prone to error. In previous work, we proposed a methodology for automatic identification and validation of model reduction opportunities for tau-leaping simulation. Here, we show how the model reductions can be automatically and adaptively deployed during the time course of a simulation. For multiscale systems, this can result in substantial speedups.
Collapse
Affiliation(s)
- Sheng Wu
- Department of Computer Science, University of California Santa Barbara, Santa Barbara, California 93106, USA
| | - Jin Fu
- Department of Computer Science, University of California Santa Barbara, Santa Barbara, California 93106, USA
| | - Linda R Petzold
- Department of Computer Science, University of California Santa Barbara, Santa Barbara, California 93106, USA
| |
Collapse
|
14
|
Hochendoner P, Ogle C, Mather WH. A queueing approach to multi-site enzyme kinetics. Interface Focus 2014; 4:20130077. [PMID: 24904740 DOI: 10.1098/rsfs.2013.0077] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Multi-site enzymes, defined as where multiple substrate molecules can bind simultaneously to the same enzyme molecule, play a key role in a number of biological networks, with the Escherichia coli protease ClpXP a well-studied example. These enzymes can form a low latency 'waiting line' of substrate to the enzyme's catalytic core, such that the enzyme molecule can continue to collect substrate even when the catalytic core is occupied. To understand multi-site enzyme kinetics, we study a discrete stochastic model that includes a single catalytic core fed by a fixed number of substrate binding sites. A natural queueing systems analogy is found to provide substantial insight into the dynamics of the model. From this, we derive exact results for the probability distribution of the enzyme configuration and for the distribution of substrate departure times in the case of identical but distinguishable classes of substrate molecules. Comments are also provided for the case when different classes of substrate molecules are not processed identically.
Collapse
Affiliation(s)
- Philip Hochendoner
- Department of Physics , Virginia Polytechnic Institute and State University , Blacksburg, VA 24061 , USA
| | - Curtis Ogle
- Department of Physics , Virginia Polytechnic Institute and State University , Blacksburg, VA 24061 , USA
| | - William H Mather
- Department of Physics , Virginia Polytechnic Institute and State University , Blacksburg, VA 24061 , USA ; Department of Biology , Virginia Polytechnic Institute and State University , Blacksburg, VA 24061 , USA
| |
Collapse
|
15
|
Agarwal A, Adams R, Castellani GC, Shouval HZ. Erratum: “On the precision of quasi steady state assumptions in stochastic dynamics” [J. Chem. Phys. 137, 044105 (2012)]. J Chem Phys 2014. [DOI: 10.1063/1.4861248] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
|
16
|
Thomas P, Grima R, Straube AV. Rigorous elimination of fast stochastic variables from the linear noise approximation using projection operators. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:041110. [PMID: 23214532 DOI: 10.1103/physreve.86.041110] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2012] [Indexed: 06/01/2023]
Abstract
The linear noise approximation (LNA) offers a simple means by which one can study intrinsic noise in monostable biochemical networks. Using simple physical arguments, we have recently introduced the slow-scale LNA (ssLNA), which is a reduced version of the LNA under conditions of timescale separation. In this paper we present the first rigorous derivation of the ssLNA using the projection operator technique and show that the ssLNA follows uniquely from the standard LNA under the same conditions of timescale separation as those required for the deterministic quasi-steady-state approximation. We also show that the large molecule number limit of several common stochastic model reduction techniques under timescale separation conditions constitutes a special case of the ssLNA.
Collapse
Affiliation(s)
- Philipp Thomas
- Department of Physics, Humboldt University of Berlin, Newtonstr. 15, D-12489 Berlin, Germany.
| | | | | |
Collapse
|