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A Multi-stage Representation of Cell Proliferation as a Markov Process

BULLETIN OF MATHEMATICAL BIOLOGY 2017年第12期79卷 2905-2928页

作者： Yates, Christian A. Ford, Matthew J. Mort, Richard L. Univ Bath Dept Math Sci Ctr Math Biol Bath BA2 7AY Avon England Univ Lancaster Div Biomed & Life Sci Fac Hlth & Med Furness Bldg Lancaster LA1 4YG England Univ Edinburgh Western Gen Hosp MRC Human Genet Unit MRC IGMM Edinburgh EH4 2XU Midlothian Scotland McGill Univ Dept Human Genet Rosalind & Morris Goodman Canc Res Ctr 1160 Pine Ave West Montreal PQ H3A 1A3 Canada

The stochastic simulation algorithm commonly known as gillespie's algorithm (originally derived for modelling well-mixed systems of chemical reactions) is now used ubiquitously in the modelling of biological processes in which stochastic effects play an important role. In well-mixed scenarios at the sub-cellular level it is often reasonable to assume that times between successive reaction/interaction events are exponentially distributed and can be appropriately modelled as a Markov process and hence simulated by the gillespie algorithm. However, gillespie's algorithm is routinely applied to model biological systems for which it was never intended. In particular, processes in which cell proliferation is important (e.g. embryonic development, cancer formation) should not be simulated naively using the gillespie algorithm since the history-dependent nature of the cell cycle breaks the Markov process. The variance in experimentally measured cell cycle times is far less than in an exponential cell cycle time distribution with the same mean. Here we suggest a method of modelling the cell cycle that restores the memoryless property to the system and is therefore consistent with simulation via the gillespie algorithm. By breaking the cell cycle into a number of independent exponentially distributed stages, we can restore the Markov property at the same time as more accurately approximating the appropriate cell cycle time distributions. The consequences of our revised mathematical model are explored analytically as far as possible. We demonstrate the importance of employing the correct cell cycle time distribution by recapitulating the results from two models incorporating cellular proliferation (one spatial and one non-spatial) and demonstrating that changing the cell cycle time distribution makes quantitative and qualitative differences to the outcome of the models. Our adaptation will allow modellers and experimentalists alike to appropriately represent cellular prolifera

关键词： Cell cycle Markovian representation Stochastic simulation gillespie algorithm Exponentially modified Erlang

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Mesoscopic behavior from microscopic Markov dynamics and its application to calcium release channels

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JOURNAL OF THEORETICAL BIOLOGY 2014年 343卷 102-112页

作者： Christian, Nils Skupin, Alexander Morante, Silvia Jansen, Karl Rossi, Giancarlo Ebenhoeh, Oliver Univ Aberdeen Dept Phys Aberdeen AB24 3UE Scotland Univ Luxembourg Luxembourg Ctr Syst Biomed L-4362 Esch Sur Alzette Luxembourg Inst Syst Biol Seattle WA 98109 USA Univ Roma Tor Vergata I-00133 Rome Italy Ist Nazl Fis Nucl Sez Roma I-00133 Rome Italy NIC DESY Zeuthen D-15738 Zeuthen Germany Helmholtz Ctr Potsdam GFZ German Res Ctr Geosci D-14473 Potsdam Germany Univ Dusseldorf Cluster Excellence Plant Sci CEPLAS D-40225 Dusseldorf Germany

A major challenge in biology is to understand how molecular processes determine phenotypic features. We address this fundamental problem in a class of model systems by developing a general mathematical framework that allows the calculation of mesoscopic properties from the knowledge of microscopic Markovian transition probabilities. We show how exact analytic formulae for the first and second moments of resident time distributions in mesostates can be derived from microscopic resident times and transition probabilities even for systems with a large number of microstates. We apply our formalism to models of the inositol trisphosphate receptor, which plays a key role in generating calcium signals triggering a wide variety of cellular responses. We demonstrate how experimentally accessible quantities, such as opening and closing times and the coefficient of variation of inter-spike intervals, and other, more elaborated, quantities can be analytically calculated from the underlying microscopic Markovian dynamics. A virtue of our approach is that we do not need to follow the detailed time evolution of the whole system, as we derive the relevant properties of its steady state without having to take into account the often extremely complicated transient features. We emphasize that our formulae fully agree with results obtained by stochastic simulations and approaches based on a full determination of the microscopic system's time evolution. We also illustrate how experiments can be devised to discriminate between alternative molecular models of the inositol trisphosphate receptor. The developed approach is applicable to any system described by a Markov process and, owing to the analytic nature of the resulting formulae, provides an easy way to characterize also rare events that are of particular importance to understand the intermittency properties of complex dynamic systems. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Calcium signaling IP3 receptor Emergent properties of mesostates gillespie algorithm

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Multiscale Stochastic Reaction-Diffusion algorithms Combining Markov Chain Models with Stochastic Partial Differential Equations

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BULLETIN OF MATHEMATICAL BIOLOGY 2019年第8期81卷 3185-3213页

作者： Kang, Hye-Won Erban, Radek Univ Maryland Baltimore Cty Dept Math & Stat 1000 Hilltop Circle Baltimore MD 21250 USA Univ Oxford Math Inst Radcliffe Observ Quarter Woodstock Rd Oxford OX2 6GG England

Two multiscale algorithms for stochastic simulations of reaction-diffusion processes are analysed. They are applicable to systems which include regions with significantly different concentrations of molecules. In both methods, a domain of interest is divided into two subsets where continuous-time Markov chain models and stochastic partial differential equations (SPDEs) are used, respectively. In the first algorithm, Markov chain (compartment-based) models are coupled with reaction-diffusion SPDEs by considering a pseudo-compartment (also called an overlap or handshaking region) in the SPDE part of the computational domain right next to the interface. In the second algorithm, no overlap region is used. Further extensions of both schemes are presented, including the case of an adaptively chosen boundary between different modelling approaches.

关键词： Stochastic reaction-diffusion systems Chemical reaction networks Markov chain gillespie algorithm Multiscale modelling Stochastic partial differential equations

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Chromatin computation: Epigenetic inheritance as a pattern reconstruction problem

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JOURNAL OF THEORETICAL BIOLOGY 2013年 336卷 61-74页

作者： Arnold, Christian Stadler, Peter F. Prohaska, Sonja J. Univ Leipzig Dept Comp Sci Computat EvoDevo Grp D-04107 Leipzig Germany Univ Leipzig Interdisciplinary Ctr Bioinformat D-04107 Leipzig Germany Univ Leipzig Dept Comp Sci Bioinformat Grp D-04107 Leipzig Germany Harvard Univ Dept Human Evolutionary Biol Cambridge MA 02138 USA Max Planck Inst Math Sci D-04103 Leipzig Germany Fraunhofer Inst Zelltherapie & Immunol D-04103 Leipzig Germany Univ Vienna Dept Theoret Chem A-1090 Vienna Austria Santa Fe Inst Santa Fe NM 87501 USA

Eukaryotic histones carry a diverse set of specific chemical modifications that accumulate over the life-time of a cell and have a crucial impact on the cell state in general and the transcriptional program in particular. Replication constitutes a dramatic disruption of the chromatin states that effectively amounts to partial erasure of stored information. To preserve its epigenetic state the cell reconstructs (at least part of) the histone modifications by means of processes that are still very poorly understood. A plausible hypothesis is that the different combinations of reader and writer domains in histone-modifying enzymes implement local rewriting rules that are capable of "recomputing" the desired parental modification patterns on the basis of the partial information contained in that half of the nucleosomes that predate replication. To test whether such a mechanism is theoretically feasible, we have developed a flexible stochastic simulation system (available at http://***/Software/StoChDyn) for studying the dynamics of histone modification states. The implementation is based on gillespie's approach, i.e., it models the master equation of a detailed chemical model. It is efficient enough to use an evolutionary algorithm to find patterns across multiple cell divisions with high accuracy. We found that it is easy to evolve a system of enzymes that can maintain a particular chromatin state roughly stable, even without explicit boundary elements separating differentially modified chromatin domains. However, the success of this task depends on several previously unanticipated factors, such as the length of the initial state, the specific pattern that should be maintained, the time between replications, and chemical parameters such as enzymatic binding and dissociation rates. All these factors also influence the accumulation of errors in the wake of cell divisions. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Chromatin complexity gillespie algorithm Evolutionary algorithm Epigenetic inheritance Histone modifications Chromatin computation

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Who is afraid of modelling time as a continuous variable?

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METHODS IN ECOLOGY AND EVOLUTION 2024年第10期15卷 1736-1756页

作者： Kokko, Hanna Johannes Gutenberg Univ Mainz Inst Organismal & Mol Evolut Mainz Germany Johannes Gutenberg Univ Mainz Inst Quant & Computat Biosci Mainz Germany

Most models of ecological and eco-evolutionary processes involve creating trajectories of something, be it population densities, average trait values, or environmental states, over time. This requires decision-making regarding how to represent the flow of time in models. Most ecologists are exposed to continuous-time models (typically in the form of ordinary differential equations) as part of their training, especially since the famous Lotka-Volterra predator-prey dynamics are formulated this way. However, few appear sufficiently well trained to produce their own work with continuous-time models and may lack exposure to the true versatility of available methods. Specifically, knowledge that discrete individuals can be modelled in continuous time using the gillespie algorithm is not as widespread as it should be. I will illustrate the flexibility of continous-time modelling methods such that researchers can make informed choices, and not resort to discretizing time as a 'default' without a clear biological motivation to do so. I provide three example-based tutorials: (1) a comparison of deterministic and stochastic dynamics of the Lotka-Volterra predator-prey model, (2) an evaluation of matelessness in a hypothetical insect population (and of selection to mate more often by either searching more efficiently or by shortening the 'time out' after each mating) and (3) within-season density dependence followed by a birth pulse leading to Beverton-Holt or Ricker dynamics depending on whether the deaths of conspecifics help reduce the mortality of others or not (compensatory mortality). I highlight properties of the exponential distribution that, while counter-intuitive, are good to know when deriving expected lifetime reproductive success or other similar quantities. I also give guidance on how to proceed if the so-called memorylessness assumption does not hold in a given situation, and show how continuous and discrete times can be freely mixed if the biological situation

关键词： Beverton-Holt model evolutionary dynamics gillespie algorithm mate searching population dynamics predator-prey dynamics Ricker model

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Stochastic model for tumor control probability: effects of cell cycle and (a) symmetric proliferation

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THEORETICAL BIOLOGY AND MEDICAL MODELLING 2014年第1期11卷 1-13页

作者： Dhawan, Andrew Kaveh, Kamran Kohandel, Mohammad Sivaloganathan, Sivabal Univ Waterloo Dept Math Waterloo ON N2L 3G1 Canada Queens Univ Sch Med Kingston ON K7L 3N6 Canada Fields Inst Ctr Math Med Toronto ON M5T 3J1 Canada

Background: Estimating the required dose in radiotherapy is of crucial importance since the administrated dose should be sufficient to eradicate the tumor and at the same time should inflict minimal damage on normal cells. The probability that a given dose and schedule of ionizing radiation eradicates all the tumor cells in a given tissue is called the tumor control probability (TCP), and is often used to compare various treatment strategies used in radiation therapy. Method: In this paper, we aim to investigate the effects of including cell-cycle phase on the TCP by analyzing a stochastic model of a tumor comprised of actively dividing cells and quiescent cells with different radiation sensitivities. Moreover, we use a novel numerical approach based on the method of characteristics for partial differential equations, validated by the gillespie algorithm, to compute the TCP as a function of time. Results: We derive an exact phase-diagram for the steady-state TCP of the model and show that at high, clinically-relevant doses of radiation, the distinction between active and quiescent tumor cells (i.e. accounting for cell-cycle effects) becomes of negligible importance in terms of its effect on the TCP curve. However, for very low doses of radiation, these proportions become significant determinants of the TCP. We also present the results of TCP as a function of time for different values of asymmetric division factor. Conclusion: We observe that our results differ from the results in the literature using similar existing models, even though similar parameters values are used, and the reasons for this are discussed.

关键词： Tumor control probability Cell cycle Mathematical modeling Stochastic birth-death process Method of characteristics gillespie algorithm

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Extending the Multi-level Method for the Simulation of Stochastic Biological Systems

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BULLETIN OF MATHEMATICAL BIOLOGY 2016年第8期78卷 1640-1677页

作者： Lester, Christopher Baker, Ruth E. Giles, Michael B. Yates, Christian A. Univ Oxford Math Inst Woodstock Rd Oxford OX2 6GG England Univ Bath Dept Math Sci Bath BA2 7AY Avon England

The multi-level method for discrete-state systems, first introduced by Anderson and Higham (SIAM Multiscale Model Simul 10(1):146-179, 2012), is a highly efficient simulation technique that can be used to elucidate statistical characteristics of biochemical reaction networks. A single point estimator is produced in a cost-effective manner by combining a number of estimators of differing accuracy in a telescoping sum, and, as such, the method has the potential to revolutionise the field of stochastic simulation. In this paper, we present several refinements of the multi-level method which render it easier to understand and implement, and also more efficient. Given the substantial and complex nature of the multi-level method, the first part of this work reviews existing literature, with the aim of providing a practical guide to the use of the multi-level method. The second part provides the means for a deft implementation of the technique and concludes with a discussion of a number of open problems.

关键词： Stochastic simulation Gene regulatory networks Multi-level gillespie algorithm Tau-leaping

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Effectiveness analysis of resistance and tolerance to infection

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GENETICS SELECTION EVOLUTION 2011年第1期43卷 1-10页

作者： Detilleux, Johann C. Univ Liege Quantitat Genet Grp Fac Vet Med Liege Belgium

Background: Tolerance and resistance provide animals with two distinct strategies to fight infectious pathogens and may exhibit different evolutionary dynamics. However, few studies have investigated these mechanisms in the case of animal diseases under commercial constraints. Methods: The paper proposes a method to simultaneously describe (1) the dynamics of transmission of a contagious pathogen between animals, (2) the growth and death of the pathogen within infected hosts and (3) the effects on their performances. The effectiveness of increasing individual levels of tolerance and resistance is evaluated by the number of infected animals and the performance at the population level. Results: The model is applied to a particular set of parameters and different combinations of values. Given these imputed values, it is shown that higher levels of individual tolerance should be more effective than increased levels of resistance in commercial populations. As a practical example, a method is proposed to measure levels of animal tolerance to bovine mastitis. Conclusions: The model provides a general framework and some tools to maximize health and performances of a population under infection. Limits and assumptions of the model are clearly identified so it can be improved for different epidemiological settings.

关键词： Mastitis Infected Host Bovine Mastitis gillespie algorithm Tolerant Host

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Efficient procedures for the numerical simulation of mid-size RNA kinetics

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algorithmS FOR MOLECULAR BIOLOGY 2012年第1期7卷 1-11页

作者： Aviram, Iddo Veltman, Ilia Churkin, Alexander Barash, Danny Ben Gurion Univ Negev Dept Comp Sci IL-84105 Beer Sheva Israel

Motivation: Methods for simulating the kinetic folding of RNAs by numerically solving the chemical master equation have been developed since the late 90's, notably the programs Kinfold and Treekin with Barriers that are available in the Vienna RNA package. Our goal is to formulate extensions to the algorithms used, starting from the gillespie algorithm, that will allow numerical simulations of mid-size (similar to 60-150 nt) RNA kinetics in some practical cases where numerous distributions of folding times are desired. These extensions can contribute to analyses and predictions of RNA folding in biologically significant problems. Results: By describing in a particular way the reduction of numerical simulations of RNA folding kinetics into the gillespie stochastic simulation algorithm for chemical reactions, it is possible to formulate extensions to the basic algorithm that will exploit memoization and parallelism for efficient computations. These can be used to advance forward from the small examples demonstrated to larger examples of biological interest. Software: The implementation that is described and used for the gillespie algorithm is freely available by contacting the authors, noting that the efficient procedures suggested may also be applicable along with Vienna's Kinfold.

关键词： Single Point Mutation Gibbs Free Energy Stochastic Simulation algorithm Chemical Master Equation gillespie algorithm

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Exploring the Molecular Origin of the High Selectivity of Multisubunit RNA Polymerases by Stochastic Kinetic Models

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INTERDISCIPLINARY SCIENCES-COMPUTATIONAL LIFE SCIENCES 2009年第2期1卷 91-98页

作者： Zhu, Rui de la Lande, Aurelien Zhang, Rui Salahub, Dennis R. Univ Calgary Dept Chem Calgary AB T2N 1N4 Canada Univ Calgary Inst Biocomplex & Informat Calgary AB T2N 1N4 Canada Univ Calgary Inst Sustainable Energy Environm & Econ Calgary AB T2N 1N4 Canada

RNA polymerases are molecular machines of great fidelity, which can recognize matched NTPs from unmatched NTPs and 2'-dNTPs. We investigated by a stochastic simulation algorithm the whole nucleotide addition cycle based on an event-driven model. This model allows us to examine possible molecular origins of the high fidelity of RNA polymerases. For unmatched NTP selectivity, the conclusions drawn from simulated elongation rates corroborate those derived from structural analysis. The presence of two conformations (Esite and pre-insertion site) for the incoming nucleotide before the polymerization reaction is sufficient to allow selectivity. Concerning sugar selectivity, our results indicate that selectivity is only achievable if slow chemical reactions occur for 2'-dNTP. These results can be used to understand recent experimental observations.

关键词： gene transcription multisubunit RNA polymerase nucleotide addition stochastic kinetics gillespie algorithm

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