Stochastic hybrid systems for studying biochemical processes

Singh, Abhyudai; Hespanha, João P.

doi:10.1098/rsta.2010.0211

Cited by 126 publications

(98 citation statements)

References 40 publications

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“…The review by Singh and Hespanha (2010) [31] shows some generalized approaches to model biochemical processes. For example Galpin et al (2008) [13] developed a modelling framework based on process algebra to describe the behaviour of hybrid biological systems.…”

Section: Intricate Modelsmentioning

confidence: 99%

“…Reviews dedicated to some specific applications of hybrid models can also be found, for example in biological networks modelling [12], in cancer modelling [10,21,30,40], and more specifically tumour growth [24,29], tumour immunology [41], brain cancer [42] or angiogenesis [16]. Many special issues or reviews are also dedicated to the more specific field of hybrid systems [2,17,31,39] which can be considered as a subclass of hybrid models mainly (but not only) related to the field of automatic control.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hybrid Modelling in Biology: a Classification Review

Stéphanou

Volpert

2015

Math. Model. Nat. Phenom.

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Abstract. This paper presents a general review on hybrid modelling which is about to become ubiquitous in biological and medical modelling. Hybrid modelling is classically defined as the coupling of a continuous approach with a discrete one, in order to model a complex phenomenon that cannot be described in a standard homogeneous way mainly due to its inherent multiscale nature. In fact, hybrid modelling can be more than that since any types of coupled formalisms qualify as being hybrid. This review first presents the evolution and current context of this modelling approach. It then proposes a classification of the models through three different types that relate to the nature and level of coupling of the formalisms used.

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Section: Intricate Modelsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hybrid Modelling in Biology: a Classification Review

Stéphanou

Volpert

2015

Math. Model. Nat. Phenom.

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“…Typical examples of such discontinuities are threshold-triggered firing in neurons (Izhikevich 2010), on-off switching of gene expression by a transcription factor (Imura et al 2010;Perkins et al 2010;Singh & Hespanha 2010), division in cells (Chen et al 2004;Battogtokh et al 2006;Osborne et al 2010) and certain types of chronotherapy for prostate cancer (Guo et al 2008;Ideta et al 2008;Shimada & Aihara 2008;Tanaka et al 2008Tanaka et al , 2010Hirata et al 2010;Suzuki et al 2010). Hence, the second aim of this Theme Issue is to discuss recent studies which apply hybrid dynamical systems to the fields of biology and medicine, through the nine articles with a variety of topics; this should stimulate applications of hybrid systems modelling to other research areas as well.…”

Section: Nonlinear Dynamics and Mathematical Modellingmentioning

confidence: 99%

Theory of hybrid dynamical systems and its applications to biological and medical systems

Aihara

Suzuki

2010

Phil. Trans. R. Soc. A.

104

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In this introductory article, we survey the contents of this Theme Issue. This Theme Issue deals with a fertile region of hybrid dynamical systems that are characterized by the coexistence of continuous and discrete dynamics. It is now well known that there exist many hybrid dynamical systems with discontinuities such as impact, switching, friction and sliding. The first aim of this Issue is to discuss recent developments in understanding nonlinear dynamics of hybrid dynamical systems in the two main theoretical fields of dynamical systems theory and control systems theory. A combined study of the hybrid systems dynamics in the two theoretical fields might contribute to a more comprehensive understanding of hybrid dynamical systems. In addition, mathematical modelling by hybrid dynamical systems is particularly important for understanding the nonlinear dynamics of biological and medical systems as they have many discontinuities such as threshold-triggered firing in neurons, on-off switching of gene expression by a transcription factor, division in cells and certain types of chronotherapy for prostate cancer. Hence, the second aim is to discuss recent applications of hybrid dynamical systems in biology and medicine. Thus, this Issue is not only general to serve as a survey of recent progress in hybrid systems theory but also specific to introduce interesting and stimulating applications of hybrid systems in biology and medicine. As the introduction to the topics in this Theme Issue, we provide a brief history of nonlinear dynamics and mathematical modelling, different mathematical models of hybrid dynamical systems, the relationship between dynamical systems theory and control systems theory, examples of complex behaviour in a simple neuron model and its variants, applications of hybrid dynamical systems in biology and medicine as a road map of articles in this Theme Issue and future directions of hybrid systems modelling.Keywords: hybrid dynamical systems; mathematical modelling; biological systems; medical systems; nonlinear dynamics; control Nonlinear dynamics and mathematical modellingThe history of natural science can be traced back to Isaac Newton in the seventeenth century. He formulated equations relating force and motion and co-invented calculus. This seminal work inaugurated the field of dynamical systems. His laws of force and motion are also the origin of mathematical modelling of dynamical phenomena in this world. Because calculus generally hates discontinuities and instabilities, most classical studies of dynamical systems and mathematical modelling concentrated on solutions that were smooth and stable. The universe was thus conceived of as a precision machine with smoothness and stability; based on the initial conditions, one could accurately predict future behaviour such as that of planets orbiting the Sun or that of a pendulum clock. However, in the 1960s and 1970s, 'catastrophe theory' (Thom 1975) and 'chaos theory' (Li & Yorke 1975;May 1976) were formulated as special branches of dynamical syst...

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“…A number of numerical methods have been developed to approximate the solution of the CME for more complex systems, including methods that approximate the CME by describing the probability distribution in terms of its moments. [4][5][6][7][8][9][10][11] When only the mean (the first moment) is taken into account, the moment expansion reduces to the MAK description. In the linear noise approximation (LNA), the CME is approximated by taking into account the mean (the first moment), and the variance and covariance (second central moments) of the distribution, whereby the second central moments are decoupled form the mean.…”

Section: Introductionmentioning

confidence: 99%

A general moment expansion method for stochastic kinetic models

Ale

Kirk

Stumpf

2013

The Journal of Chemical Physics

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Moment approximation methods are gaining increasing attention for their use in the approximation of the stochastic kinetics of chemical reaction systems. In this paper we derive a general moment expansion method for any type of propensities and which allows expansion up to any number of moments. For some chemical reaction systems, more than two moments are necessary to describe the dynamic properties of the system, which the linear noise approximation is unable to provide. Moreover, also for systems for which the mean does not have a strong dependence on higher order moments, moment approximation methods give information about higher order moments of the underlying probability distribution. We demonstrate the method using a dimerisation reaction, Michaelis-Menten kinetics and a model of an oscillating p53 system. We show that for the dimerisation reaction and MichaelisMenten enzyme kinetics system higher order moments have limited influence on the estimation of the mean, while for the p53 system, the solution for the mean can require several moments to converge to the average obtained from many stochastic simulations. We also find that agreement between lower order moments does not guarantee that higher moments will agree. Compared to stochastic simulations, our approach is numerically highly efficient at capturing the behaviour of stochastic systems in terms of the average and higher moments, and we provide expressions for the computational cost for different system sizes and orders of approximation. We show how the moment expansion method can be employed to efficiently quantify parameter sensitivity. Finally we investigate the effects of using too few moments on parameter estimation, and provide guidance on how to estimate if the distribution can be accurately approximated using only a few moments.

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Stochastic hybrid systems for studying biochemical processes

Cited by 126 publications

References 40 publications

Hybrid Modelling in Biology: a Classification Review

Hybrid Modelling in Biology: a Classification Review

Theory of hybrid dynamical systems and its applications to biological and medical systems

A general moment expansion method for stochastic kinetic models

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