Nonlinear Dynamics in Biological Systems
01 Aug 2017-
TL;DR: This relatively slim volume is effectively the proceedings of the First BCAM Workshop on Nonlinear Dynamics in Biological Systems, held at the Basque Center of Applied Mathematics (BCAM), 19-20 June 2014, and can only deal with some selected aspects of this huge subject area.
Abstract: Nature is inherently nonlinear, at every level, and the physicist’s comfortably familiar linear approximation can only rarely be applied in practice. So applications of physics to biology must always be mindful of the resultant effects, which can often be counter-intuitive. Provided that nonlinearity it is taken properly into account, however, physics has a great deal to say about biology. Indeed, there are many biological phenomena that can only be understood in the context of physics, e.g. the beating of the heart, cardio-respiratory synchronisation, cell synchronisation, calcium signalling in cells, and the dynamics of protein folding. Practical applications of nonlinear dynamics methods in physiology, either already existing or quickly coming into view, include for example cochlear amplifiers, early diagnosis of autism spectrum disorder, identification of melanoma, and measurement of depth of anæsthesia. This relatively slim volume is effectively the proceedings of the First BCAM Workshop on Nonlinear Dynamics in Biological Systems, held at the Basque Center of Applied Mathematics (BCAM), 19-20 June 2014. Obviously, it can only deal with some selected aspects of this huge subject area – a major tome would be needed to do full justice to the title. There are three chapters at the subcellular level, dealing respectively with evolving RNA replicators, analysis of synthesised nucleic acid pools, and logic gates in transcriptional regulation. The fourth chapter is more on the cellular level, about pattern formation on cellular membranes; and the last two chapters are basically at the whole-organ level, on the mathematical modeling of the heart and on the origins of the cardiac alternans phenomenon where there is a beat-to-beat alternation in the strength of the heart’s contraction. Nonlinear dynamics is, of course, also relevant at the whole whole-organism and species levels, but these topics were seemingly not represented at the conference. So the book is effectively a collection of six roughly 20-page reviews, on quite a wide range of topics, but linked by the common theme of nonlinear dynamics in biology, and pulled together by the Editors’ explanatory Preface. They are mostly well-written, I enjoyed reading them, and I particularly liked the chapter on transcriptional regulation, where I felt that Till Frank and co-authors had made a real effort to address readers not already closely involved in their field.
Citations
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TL;DR: It is demonstrated that variation in Fgf8 expression has a nonlinear relationship to phenotypic variation, predicting levels of robustness among genotypes, and embedded features of development explain robustness differences.
Abstract: Robustness to perturbation is a fundamental feature of complex organisms. Mutations are the raw material for evolution, yet robustness to their effects is required for species survival. The mechanisms that produce robustness are poorly understood. Nonlinearities are a ubiquitous feature of development that may link variation in development to phenotypic robustness. Here, we manipulate the gene dosage of a signaling molecule, Fgf8, a critical regulator of vertebrate development. We demonstrate that variation in Fgf8 expression has a nonlinear relationship to phenotypic variation, predicting levels of robustness among genotypes. Differences in robustness are not due to gene expression variance or dysregulation, but emerge from the nonlinearity of the genotype-phenotype curve. In this instance, embedded features of development explain robustness differences. How such features vary in natural populations and relate to genetic variation are key questions for unraveling the origin and evolvability of this feature of organismal development.
74 citations
TL;DR: This study presents some fourth-order operator splitting methods, having real and complex coefficients, and compares their performance when applied to the Niederer benchmark as well as a variant with a stiffer cell model.
10 citations
01 Jan 2018
TL;DR: A survey of operator-splitting methods for the numerical solution of differential equations is provided in this article, where the authors focus on splitting methods with order higher than two that, according to the Sheng-Suzuki theorem, require backward time integration and historically have been considered unstable for solving deterministic parabolic systems.
Abstract: The bidomain and monodomain models are among the most widely used mathematical models to describe cardiac electrophysiology. They take the form of multi-scale reaction-diffusion partial differential equations that couple the dynamic behaviour on the cellular scale with that on the tissue scale. The systems of differential equations associated with these models are large and strongly non-linear, but they also have a distinct structure due to their multi-scale nature. For these reasons, numerical solutions to these systems are often found via operator-splitting methods. In this chapter, we provide a survey of operator-splitting methods for the numerical solution of differential equations. In particular, we focus on splitting methods with order higher than two that, according to the Sheng–Suzuki theorem, require backward time integration and historically have been considered unstable for solving deterministic parabolic systems. We demonstrate the stability of operator-splitting methods of up to order four to solve the bidomain and monodomain models on several examples arising in the field of cardiovascular modeling.
4 citations
References
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TL;DR: It is demonstrated that variation in Fgf8 expression has a nonlinear relationship to phenotypic variation, predicting levels of robustness among genotypes, and embedded features of development explain robustness differences.
Abstract: Robustness to perturbation is a fundamental feature of complex organisms. Mutations are the raw material for evolution, yet robustness to their effects is required for species survival. The mechanisms that produce robustness are poorly understood. Nonlinearities are a ubiquitous feature of development that may link variation in development to phenotypic robustness. Here, we manipulate the gene dosage of a signaling molecule, Fgf8, a critical regulator of vertebrate development. We demonstrate that variation in Fgf8 expression has a nonlinear relationship to phenotypic variation, predicting levels of robustness among genotypes. Differences in robustness are not due to gene expression variance or dysregulation, but emerge from the nonlinearity of the genotype-phenotype curve. In this instance, embedded features of development explain robustness differences. How such features vary in natural populations and relate to genetic variation are key questions for unraveling the origin and evolvability of this feature of organismal development.
74 citations
TL;DR: This study presents some fourth-order operator splitting methods, having real and complex coefficients, and compares their performance when applied to the Niederer benchmark as well as a variant with a stiffer cell model.
10 citations
01 Jan 2018
TL;DR: A survey of operator-splitting methods for the numerical solution of differential equations is provided in this article, where the authors focus on splitting methods with order higher than two that, according to the Sheng-Suzuki theorem, require backward time integration and historically have been considered unstable for solving deterministic parabolic systems.
Abstract: The bidomain and monodomain models are among the most widely used mathematical models to describe cardiac electrophysiology. They take the form of multi-scale reaction-diffusion partial differential equations that couple the dynamic behaviour on the cellular scale with that on the tissue scale. The systems of differential equations associated with these models are large and strongly non-linear, but they also have a distinct structure due to their multi-scale nature. For these reasons, numerical solutions to these systems are often found via operator-splitting methods. In this chapter, we provide a survey of operator-splitting methods for the numerical solution of differential equations. In particular, we focus on splitting methods with order higher than two that, according to the Sheng–Suzuki theorem, require backward time integration and historically have been considered unstable for solving deterministic parabolic systems. We demonstrate the stability of operator-splitting methods of up to order four to solve the bidomain and monodomain models on several examples arising in the field of cardiovascular modeling.
4 citations