Floquet theory: a useful tool for understanding nonequilibrium dynamics

Many ecological systems experience periodic variability. Theoretical investigation of population and community dynamics in periodic environments has been hampered by the lack of mathematical tools relative to equilibrium systems. Here, I describe one such mathematical tool that has been rarely used in the ecological literature but has widespread use: Floquet theory. Floquet theory is the study of the stability of linear periodic systems in continuous time. Floquet exponents/multipliers are analogous to the eigenvalues of Jacobian matrices of equilibrium points. In this paper, I describe the general theory, then give examples to illustrate some of its uses: it defines fitness of structured populations, it can be used for invasion criteria in models of competition, and it can test the stability of limit cycle solutions. I also provide computer code to calculate Floquet exponents and multipliers. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Optimal open-loop desynchronization of neural oscillator populations

Deep brain stimulation (DBS) is an increasingly used medical treatment for various neurological disorders. While its mechanisms are not fully understood, experimental evidence suggests that through application of periodic electrical stimulation DBS may act to desynchronize pathologically synchronized populations of neurons resulting desirable changes to a larger brain circuit. However, the underlying mathematical mechanisms by which periodic stimulation can engender desynchronization in a coupled population of neurons is not well understood. In this work, a reduced phase-amplitude reduction framework is used to characterize the desynchronizing influence of periodic stimulation on a population of coupled oscillators. Subsequently, optimal control theory allows for the design of periodic, open-loop stimuli with the capacity to destabilize completely synchronized solutions while simultaneously stabilizing rotating block solutions. This framework exploits system nonlinearities in order to strategically modify unstable Floquet exponents. In the limit of weak neural coupling, it is shown that this method only requires information about the phase response curves of the individual neurons. The effects of noise and heterogeneity are also considered and numerical results are presented. This framework could ultimately be used to inform the design of more efficient deep brain stimulation waveforms for the treatment of neurological disease.

     

Entrainment to Periodic Initiation and Transition Rates in a Computational Model for Gene Translation

Periodic oscillations play an important role in many biomedical systems. Proper functioning of biological systems that respond to periodic signals requires the ability to synchronize with the periodic excitation. For example, the sleep/wake cycle is a manifestation of an internal timing system that synchronizes to the solar day. In the terminology of systems theory, the biological system must entrain or phase-lock to the periodic excitation. Entrainment is also important in synthetic biology. For example, connecting several artificial biological systems that entrain to a common clock may lead to a well-functioning modular system. The cell-cycle is a periodic program that regulates DNA synthesis and cell division. Recent biological studies suggest that cell-cycle related genes entrain to this periodic program at the gene translation level, leading to periodically-varying protein levels of these genes. The ribosome flow model (RFM) is a deterministic model obtained via a mean-field approximation of a stochastic model from statistical physics that has been used to model numerous processes including ribosome flow along the mRNA. Here we analyze the RFM under the assumption that the initiation and/or transition rates vary periodically with a common period . We show that the ribosome distribution profile in the RFM entrains to this periodic excitation. In particular, the protein synthesis pattern converges to a unique periodic solution with period . To the best of our knowledge, this is the first proof of entrainment in a mathematical model for translation that encapsulates aspects such as initiation and termination rates, ribosomal movement and interactions, and non-homogeneous elongation speeds along the mRNA. Our results support the conjecture that periodic oscillations in tRNA levels and other factors related to the translation process can induce periodic oscillations in protein levels, and may suggest a new approach for re-engineering genetic systems to obtain a desired, periodic, protein synthesis rate.     

Synthetic biology: a new approach to study biological pattern formation

The principles and molecular mechanisms underlying biological pattern formation are difficult to elucidate in most cases due to the overwhelming physiologic complexity associated with the natural context. The understanding of a particular mechanism, not to speak of underlying universal principles, is difficult due to the diversity and uncertainty of the biological systems. Although current genetic and biochemical approaches have greatly advanced our understanding of pattern formation, the progress mainly relies on experimental phenotypes obtained from time-consuming studies of gain or loss of function mutants. It is prevailingly considered that synthetic biology will come to the application age, but more importantly synthetic biology can be used to understand the life. Using periodic stripe pattern formation as a paradigm, we discuss how to apply synthetic biology in understanding biological pattern formation and hereafter foster the applications like tissue engineering.     

Systems biology strategy to study lipotoxicity and the metabolic syndrome

Systems biology views and studies the biological systems in the context of complex interactions between their building blocks and processes. Given its multi-level complexity, metabolic syndrome (MetS) makes a strong case for adopting the systems biology approach. Despite many MetS traits being highly heritable, it is becoming evident that the genetic contribution to these traits is mediated via gene–gene and gene–environment interactions across several spatial and temporal scales, and that some of these traits such as lipotoxicity may even be a product of long-term dynamic changes of the underlying genetic and molecular networks. This presents several conceptual as well as methodological challenges and may demand a paradigm shift in how we study the undeniably strong genetic component of complex diseases such as MetS. The argument is made here that for adopting systems biology approaches to MetS an integrative framework is needed which glues the biological processes of MetS with specific physiological mechanisms and principles and that lipotoxicity is one such framework. The metabolic phenotypes, molecular and genetic networks can be modeled within the context of such integrative framework and the underlying physiology.     

Toward a conceptual framework for biology

Science progresses faster when researchers operate within an explicit framework of concepts and theories, but currently biology has no explicit, overarching conceptual framework and few general theories. The single general theory currently recognized is that of evolution, which was put forth by Charles Darwin 150 years ago. Recently, Scheiner and Willig (2008) explicated a similarly general theory of ecology. In this paper, using the theory of evolution as an exemplar, I discuss the nature of theory in biology and put forth an overarching theory, as well as new general theories for cells, organisms, and genetics. Along with the theories of evolution and ecology, these constitute a general conceptual framework for the biological sciences. This framework reveals linkages among the various parts of biology, makes explicit the assumptions behind more narrow theories and models, and provides new insights into the structures of biological theories. This framework can also be used as a teaching tool, moving the teaching of biology beyond the transference of a vast compendium of facts. My hope is that this essay will lead to a vigorous discussion and debate across all of biology about the nature and structure of its theories.     

Bayesian natural selection and the evolution of perceptual systems

In recent years, there has been much interest in characterizing statistical properties of natural stimuli in order to better understand the design of perceptual systems. A fruitful approach has been to compare the processing of natural stimuli in real perceptual systems with that of ideal observers derived within the framework of Bayesian statistical decision theory. While this form of optimization theory has provided a deeper understanding of the information contained in natural stimuli as well as of the computational principles employed in perceptual systems, it does not directly consider the process of natural selection, which is ultimately responsible for design. Here we propose a formal framework for analysing how the statistics of natural stimuli and the process of natural selection interact to determine the design of perceptual systems. The framework consists of two complementary components. The first is a maximum fitness ideal observer, a standard Bayesian ideal observer with a utility function appropriate for natural selection. The second component is a formal version of natural selection based upon Bayesian statistical decision theory. Maximum fitness ideal observers and Bayesian natural selection are demonstrated in several examples. We suggest that the Bayesian approach is appropriate not only for the study of perceptual systems but also for the study of many other systems in biology.     

A theoretical framework for defining some concepts in evolution.

We present a theoretical framework for biological evolution with the intention of giving precise mathematical definitions of some concepts in evolutionary biology such as fitness, evolutionary pressure, specialization and natural selection. In this framework, such concepts are identified with well-known mathematical terms within the theory of dynamical systems. We also discuss some more general implications in evolution; for instance, the fact that our model naturally exhibits a frequency spectrum of the type 1/f for low frequencies of evolutionary events.     

Causality and persistence in ecological systems: a nonparametric spectral granger causality approach

Abstract Directionality in coupling, defined as the linkage relating causes to their effects at a later time, can be used to explain the core dynamics of ecological systems by untangling direct and feedback relationships between the different components of the systems. Inferring causality from measured ecological variables sampled through time remains a formidable challenge further made difficult by the action of periodic drivers overlapping the natural dynamics of the system. Periodicity in the drivers can often mask the self-sustained oscillations originating from the autonomous dynamics. While linear and direct causal relationships are commonly addressed in the time domain, using the well-established machinery of Granger causality (G-causality), the presence of periodic forcing requires frequency-based statistics (e.g., the Fourier transform), able to distinguish coupling induced by oscillations in external drivers from genuine endogenous interactions. Recent nonparametric spectral extensions of G-causality to the frequency domain pave the way for the scale-by-scale decomposition of causality, which can improve our ability to link oscillatory behaviors of ecological networks to causal mechanisms. The performance of both spectral G-causality and its conditional extension for multivariate systems is explored in quantifying causal interactions within ecological networks. Through two case studies involving synthetic and actual time series, it is demonstrated that conditional G-causality outperforms standard G-causality in identifying causal links and their concomitant timescales.     

Engineering entrainment and adaptation in limit cycle systems

Periodic behavior is key to life and is observed in multiple instances and at multiple time scales in our metabolism, our natural environment, and our engineered environment. A natural way of modeling or generating periodic behavior is done by using oscillators, i.e., dynamical systems that exhibit limit cycle behavior. While there is extensive literature on methods to analyze such dynamical systems, much less work has been done on methods to synthesize an oscillator to exhibit some specific desired characteristics. The goal of this article is twofold: (1) to provide a framework for characterizing and designing oscillators and (2) to review how classes of well-known oscillators can be understood and related to this framework. The basis of the framework is to characterize oscillators in terms of their fundamental temporal and spatial behavior and in terms of properties that these two behaviors can be designed to exhibit. This focus on fundamental properties is important because it allows us to systematically compare a large variety of oscillators that might at first sight appear very different from each other. We identify several specifications that are useful for design, such as frequency-locking behavior, phase-locking behavior, and specific output signal shape. We also identify two classes of design methods by which these specifications can be met, namely offline methods and online methods. By relating these specifications to our framework and by presenting several examples of how oscillators have been designed in the literature, this article provides a useful methodology and toolbox for designing oscillators for a wide range of purposes. In particular, the focus on synthesis of limit cycle dynamical systems should be useful both for engineering and for computational modeling of physical or biological phenomena.     

The next step in biology: a periodic table?

Systems biology is an approach to explain the behaviour of a system in relation to its individual components. Synthetic biology uses key hierarchical and modular concepts of systems biology to engineer novel biological systems. In my opinion the next step in biology is to use molecule-to-phenotype data using these approaches and integrate them in the form a periodic table. A periodic table in biology would provide chassis to classify, systematize and compare diversity of component properties vis-a-vis system behaviour. Using periodic table it could be possible to compute higher- level interactions from component properties. This paper examines the concept of building a bio-periodic table using protein fold as the fundamental unit.     

The next step in biology: A periodic table?

Systems biology is an approach to explain the behaviour of a system in relation to its individual components. Synthetic biology uses key hierarchical and modular concepts of systems biology to engineer novel biological systems. In my opinion the next step in biology is to use molecule-to-phenotype data using these approaches and integrate them in the form a periodic table. A periodic table in biology would provide chassis to classify, systematize and compare diversity of component properties vis-a-vis system behaviour. Using periodic table it could be possible to compute higher-level interactions from component properties. This paper examines the concept of building a bio-periodic table using protein fold as the fundamental unit.     

Biological applications of the theory of birth-and-death processes

In this review, we discuss applications of the theory of birth-and-death processes to problems in biology, primarily, those of evolutionary genomics. The mathematical principles of the theory of these processes are briefly described. Birth-and-death processes, with some straightforward additions such as innovation, are a simple, natural and formal framework for modeling a vast variety of biological processes such as population dynamics, speciation, genome evolution, including growth of paralogous gene families and horizontal gene transfer and somatic evolution of cancers. We further describe how empirical data, e.g. distributions of paralogous gene family size, can be used to choose the model that best reflects the actual course of evolution among different versions of birth-death-and-innovation models. We conclude that birth-and-death processes, thanks to their mathematical transparency, flexibility and relevance to fundamental biological processes, are going to be an indispensable mathematical tool for the burgeoning field of systems biology.     

Extinction and permanence of one-prey multi-predators of Holling type II function response system with impulsive biological control

In this paper, one investigates the dynamic behaviors of one-prey multi-predator model with Holling type II functional response by introducing impulsive biological control strategy (periodic releasing natural enemies at different fixed time). By using Floquet theorem and small amplitude perturbation method, it is proved that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value and permanence condition is established via the method of comparison involving multiple Liapunov functions. It is shown that multi-predator impulsive control strategy is more effective than the classical and single one.     

Optimierung und ökonomisierung im kontext von evolutionstheorie und phylogenetischer rekonstruktion

The meaning of optimality and economy in phylogenetics and evolutionary biology is discussed. It can be shown that the prevailing concepts of optimality and economy are equivocal as they are not based on strict theoretical positions and as they have a variable meaning in different theoretical contexts. The ideas of optimality and economy can be considered to be identical with the expectation of a relatively simple order in a particular field of study. Although there exists no way of inferring one or several methods of solving scientific problems from the presupposed idea of economy and optimality, a lack of motivation for scientific investigations would result if the concepts of economy and optimality in nature were dropped. By reference to several examples, it is shown that the concepts of optimality and economy are only useful against the background of indispensable theories. If there is a shift from one theory to another, a restriction on the use of these concepts is necessary. Optimality and economy in the sense of operations research in engineering or economical sciences depend on the principle of minimum costs. Both theoretical concepts: technical efficiency in relation to the energy required to run a machine and profit maximation in an economical framework must be shown to be realistic assumptions. In the field of biology processes of optimization and economization are normally discussed under two different views:

1. The concept of economy is used in cases of functional adaptation when the organism makes good use of the building material which is available to fulfill one (or more) functions. The theoretical background must be seen in the energy-consuming aspect of the organism.
2. In evolutionary change and phylogeny ‘economization’ and ‘optimization’ are deduced from the evolutionary theory, and evolution is shown to produce a special kind of biological economy in biological systems (Bock & von Wahlert, 1965). The ‘Okonomie-Prinzip’ or ‘Lesrichtungskriterium’ points out the arguments needed to state a phylogenetic theory and to construct a dendrogram (Peters & Gutmann, 1971).

In every phylogenetic theory concerning the adaptational change in the evolving biological system an explanation for the function of all stages is required. Only those statements should be accepted as phylogenetic theories which are characterized by the demonstration of the process of economization in the functional relations of the evolving organism. The process of adaptation can be determined by the improved chance of some mutants to propagate their genetical information. In this process all functional systems in their interrelations — i.a. mutual dependence — and their relation with the environment add their functional efficiency to the information to be delivered to their progeny, because the more economical biological system in a certain environment will have a better chance to produce offspring. This outcome is affirmed by natural selection which works on all levels of the evolving biological systems (Gutmann & Peters 1973). Nevertheless a judgment about adaptation cannot be taken as a scale of measurement in the phylogenetic process. The conditions in the organism itself and in the environment or in the organic system alone can change in so profound a manner that the marginal conditions of the earlier stages of the process of adaptation are not the same as in the derived ones. During phylogenetic change of the evolving organism the selective strains are also continuously changing. As a consequence no state or invariant concept of economy can cover the different stages of the phylogenetic process. The pragmatical meaning of the theoretical consideration is substantiated by the example of the hydrostatic skeleton theory in which the chordates are derived from metameric worms with a fluid skeleton. Herrn Professor Dr. P. Dullemeijer sind die Verfasser für kritische Lektüre und wertovolle Hinweise zu Dank verpflichtet.     

Engineering microbial systems to explore ecological and evolutionary dynamics

A major goal of biological research is to provide a mechanistic understanding of diverse biological processes. To this end, synthetic biology offers a powerful approach, whereby biological questions can be addressed in a well-defined framework. By constructing simple gene circuits, such studies have generated new insights into the design principles of gene regulatory networks. Recently, this strategy has been applied to analyze ecological and evolutionary questions, where population-level interactions are critical. Here, we highlight recent development of such systems and discuss how they were used to address problems in ecology and evolutionary biology. As illustrated by these examples, synthetic ecosystems provide a unique platform to study ecological and evolutionary phenomena that are challenging to study in their natural contexts.     

Efficient information coding and degeneracy in the nervous system

Efficient information coding (EIC) is a universal biological framework rooted in the fundamental principle that system responses should match their natural stimulus statistics for maximizing environmental information. Quantitatively assessed through information theory, such adaptation to the environment occurs at all biological levels and timescales. The context dependence of environmental stimuli and the need for stable adaptations make EIC a daunting task. We argue that biological complexity is the principal architect that subserves deft execution of stable EIC. Complexity in a system is characterized by several functionally segregated subsystems that show a high degree of functional integration when they interact with each other. Complex biological systems manifest heterogeneities and degeneracy, wherein structurally different subsystems could interact to yield the same functional outcome. We argue that complex systems offer several choices that effectively implement EIC and homeostasis for each of the different contexts encountered by the system.     

Modelling the biological invasion of Carcinus maenas (the European green crab)

This paper proposes a system of integro-difference equations to model the spread of Carcinus maenas, commonly called the European green crab, that causes severe damage to coastal ecosystems. A model with juvenile and adult classes is first studied. Here, standard theory of monotone operators for integro-difference equations can be applied and yields explicit formulas for the asymptotic spreading speeds of the juvenile and adult crabs. A second model including an infected class is considered by introducing a castrating parasite Sacculina carcini as a biological control agent. The dynamics are complicated and simulations reveal the occurrence of periodic solutions and stacked fronts. In this case, only conjectures can be made for the asymptotic spreading speeds because of the lack of mathematical theory for non-monotone operators. This paper also emphasizes the need for mathematical studies of non-monotone operators in heterogeneous environments and the existence of stacked front solutions in biological invasion models.