Identification of nonlinear systems using random impulse train inputs

Nonlinear systems that require discrete inputs can be characterized by using random impulse train (Poisson process) inputs. The method is analagous to the Wiener method for continuous input systems, where Gaussian white-noise is the input. In place of the Wiener functional expansion for the output of a continuous input system, a new series for discrete input systems is created by making certain restrictions on the integrals in a Volterra series. The kernels in the new series differ from the Wiener kernels, but also serve to identify a system and are simpler to compute. For systems whose impulse responses vary in amplitude but maintain a similar shape, one argument may be held fixed in each kernel. This simplifies the identification problem. As a test of the theory presented, the output of a hypothetical second order nonlinear system in response to a random impulse train stimulus was computer simulated. Kernels calculated from the simulated data agreed with theoretical predictions. The Poisson impulse train method is applicable to any system whose input can be delivered in discrete pulses. It is particularly suited to neuronal synaptic systems when the pattern of input nerve impulses can be made random.     

Detection of cellular rhythms and global stability within interlocked feedback systems

In this paper, we show how to detect cellular rhythm and its global stability by extending the techniques from the recently developed theory of monotone systems. We establish theoretical results for globally asymptotic stability with consideration of delay by a discrete map. The relationship between positive, negative elements and delay in a general class of interlocked feedback networks can be understood in a system level. Moreover, the correspondence of attractors between a network and its reduced map is obtained and can be used to detect cellular rhythm, and further control the dynamics of the network. We show that global cellular rhythms can always be obtained, thereby enhancing robustness against perturbations of initial conditions and avoiding chaotic oscillations or complete abolishment of oscillations. In this paper, we focus on analyzing the circadian oscillator in Drosophila as an example to detect the occurrence of cellular rhythm and its global stability.     

Evolutionary optimization with data collocation for reverse engineering of biological networks

MOTIVATION: Modern experimental biology is moving away from analyses of single elements to whole-organism measurements. Such measured time-course data contain a wealth of information about the structure and dynamic of the pathway or network. The dynamic modeling of the whole systems is formulated as a reverse problem that requires a well-suited mathematical model and a very efficient computational method to identify the model structure and parameters. Numerical integration for differential equations and finding global parameter values are still two major challenges in this field of the parameter estimation of nonlinear dynamic biological systems. RESULTS: We compare three techniques of parameter estimation for nonlinear dynamic biological systems. In the proposed scheme, the modified collocation method is applied to convert the differential equations to the system of algebraic equations. The observed time-course data are then substituted into the algebraic system equations to decouple system interactions in order to obtain the approximate model profiles. Hybrid differential evolution (HDE) with population size of five is able to find a global solution. The method is not only suited for parameter estimation but also can be applied for structure identification. The solution obtained by HDE is then used as the starting point for a local search method to yield the refined estimates.     

Stochastic physics,complex systems and biology

In complex systems, the interplay between nonlinear and stochastic dynamics, e.g., J. Monod’s necessity and chance, gives rise to an evolutionary process in Darwinian sense, in terms of discrete jumps among attractors, with punctuated equilibria, spontaneous random “mutations” and “adaptations”. On an evolutionary time scale it produces sustainable diversity among individuals in a homogeneous population rather than convergence as usually predicted by a deterministic dynamics. The emergent discrete states in such a system, i.e., attractors, have natural robustness against both internal and external perturbations. Phenotypic states of a biological cell, a mesoscopic nonlinear stochastic open biochemical system, could be understood through such a perspective.     

Finding complex oscillatory phenomena in biochemical systems. An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.     

Hybrid semi-parametric mathematical systems: bridging the gap between systems biology and process engineering

Systems biology is an integrative science that aims at the global characterization of biological systems. Huge amounts of data regarding gene expression, proteins activity and metabolite concentrations are collected by designing systematic genetic or environmental perturbations. Then the challenge is to integrate such data in a global model in order to provide a global picture of the cell. The analysis of these data is largely dominated by nonparametric modelling tools. In contrast, classical bioprocess engineering has been primarily founded on first principles models, but it has systematically overlooked the details of the embedded biological system. The full complexity of biological systems is currently assumed by systems biology and this knowledge can now be taken by engineers to decide how to optimally design and operate their processes. This paper discusses possible methodologies for the integration of systems biology and bioprocess engineering with emphasis on applications involving animal cell cultures. At the mathematical systems level, the discussion is focused on hybrid semi-parametric systems as a way to bridge systems biology and bioprocess engineering.     

Time lags in ecological systems.

A model representing ten species in a four trophic level community is constructed by using Volterra's equations including time lags, and is solved numerically for some values of the parameters. Classical discrete and continuous time lags yield similar results; simulating with discrete time lags thus appears useful. Parasitic versus predatory structures are compared by measuring the amplitudes of oscillations and the time taken to settle down to equilibrium, and give the following qualitative conclusions for the model.An increase in the carrying capacity of a trophic level increases the destabilizing influence of time lag in that level, this increase is more marked in predatory structures. Time lags appear to cause more violent oscillations in species strongly linked to a predatory system than in weakly linked species. The oscillatory period increases in predatory systems as the time lag is moved to higher levels, while it decreases in parasitic systems.     

Biological populations obeying difference equations: stable points, stable cycles, and chaos.

For biological populations with nonoverlapping generations, population growth takes place in discrete time steps and is described by difference equations. Some of the simplest such nonlinear difference equations can exhibit a remarkable spectrum of dynamical behavior, from stable equilibrium points, to stable cyclic oscillations between two population points, to stable cycles with four points, then eight, 16, etc., points, through to a chaotic regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but bounded population fluctuations, can occur. This rich dynamical structure is overlooked in conventional linearized stability analyses; its existence in the simplest and fully deterministic nonlinear (“density dependent”) difference equations is a fact of considerable mathematical and ecological interest.     

Statistical inference for multi-pathogen systems

There is growing interest in understanding the nature and consequences of interactions among infectious agents. Pathogen interactions can be operational at different scales, either within a co-infected host or in host populations where they co-circulate, and can be either cooperative or competitive. The detection of interactions among pathogens has typically involved the study of synchrony in the oscillations of the protagonists, but as we show here, phase association provides an unreliable dynamical fingerprint for this task. We assess the capacity of a likelihood-based inference framework to accurately detect and quantify the presence and nature of pathogen interactions on the basis of realistic amounts and kinds of simulated data. We show that when epidemiological and demographic processes are well understood, noisy time series data can contain sufficient information to allow correct inference of interactions in multi-pathogen systems. The inference power is dependent on the strength and time-course of the underlying mechanism: stronger and longer-lasting interactions are more easily and more precisely quantified. We examine the limitations of our approach to stochastic temporal variation, under-reporting, and over-aggregation of data. We propose that likelihood shows promise as a basis for detection and quantification of the effects of pathogen interactions and the determination of their (competitive or cooperative) nature on the basis of population-level time-series data.     

Damped oscillating processes in biological and biochemical systems

Damped nonlinear oscillations in biological and biochemical systems are investigated by the extended Krylov-Bogoliubov-Mitropolskii (KBM) method. A review on the extension made by Popov to the KBM method is given and also further improvements are presented. Applications are made to models of oscillating chemical reactions (Lefever and Nicolis, 1971), FitzHugh (1961) equations, and population dynamics (Gatto and Rinaldi, 1977). Comparison to damped oscillating physical and engineering systems is made.     

Towards resolving the paradox of enrichment: the impact of zooplankton vertical migrations on plankton systems stability

Eutrophication, often resulting from human activity, is a serious threat to aquatic communities. Theoretical analysis of this phenomenon, based on conceptual mathematical models, leads to controversial predictions known as Rosenzweig's paradox of enrichment. At the same time, field observations demonstrate that real plankton communities exhibit various mechanisms of self-regulation which can buffer negative effects of enrichment. In this paper, we study potential effects of zooplankton vertical migration on stability of plankton systems functioning. We consider an intrinsically unstable plankton model, which is characterized by an unlimited phytoplankton multiplication and population oscillations of increasing amplitude, and investigate whether vertical migrations of zooplankton can stabilize such a system at low plankton densities. By means of developing two different models accounting for different ecological situations, e.g. deep waters and shallow waters, we show that vertical migrations of zooplankton can result in stabilization of eutrophic plankton systems. Thus, we show that this mechanism, rarely taken into account in models of plankton dynamics, may be important for resolving the paradox of enrichment in plankton communities.     

Development of nerve connections under the control of neurotrophic factors: parallels with consumer-resource systems in population biology

The development of connections between neurons and their target cells involves competition between axons for target-derived neurotrophic factors. Although the notion of competition is commonly used in neurobiology, the process is not well understood, and only a few formal models exist. In population biology, in contrast, the concept of competition is well developed and has been studied by means of many formal models of consumer-resource systems. Here we show that a recently formulated model of axonal competition can be rewritten as a general consumer-resource system. This allows neurobiological phenomena to be interpreted in population biological terms and, conversely, results from population biology to be applied to neurobiology. Using findings from population biology, we have studied two extensions of our axonal competition model. In the first extension, the spatial dimension of the target is explicitly taken into account. We show that distance between axons on their target mitigates competition and permits the coexistence of axons. The model can account for the fact that in many types of neurons a positive correlation exists between the size of the dendritic tree and the number of innervating axons surviving into adulthood. In the second extension, axons are allowed to respond to more than one neurotrophic factor. We show that this permits competitive exclusion among axons of one type, while at the same time there is coexistence with axons of another type innervating the same target. The model offers an explanation for the innervation pattern found on cerebellar Purkinje cells, where climbing fibres compete with each other until only a single one remains, which coexists with parallel fibre input to the same Purkinje cell.     

Computer simulation of cell renewal systems]

This paper deals with a block diagram which describes the structure of the control of erythropoiesis by negative feedback loops. The model is transformed into an adequate simulation program using a special block-oriented programming language called ASIM (Analoge SIMulation). For both normal and diseased states of the blood forming process the dynamic responses of the erythrocytes and reticulocytes are simulated by a digital computer analysis. The computer simulation includes different forms of anemia caused by parameter variations as well as by structural alterations. Rising oscillations are obtained too in multiloop control systems containing complex paths with minor loops, which for example take into account the erythrocytic chalones. The described model shows that rising oscillations, that are unstable control loops, can be produced by changing the control-loop structure as well as by parameter changes. In case of malignant disorders such failures becoming effective in oscillations are discussed as parts of disturbed homoeostasis. The results of these studies obtained by simulation should especially stimulate scientists working in the fields of biology and medicine to new test series to verify the proposed hypothesis.     

Enrichment can damp population cycles: a balance of inflexible and flexible interactions

Destabilization of one predator–one prey systems with an increase in nutrient input has been viewed as a paradox. We report that enrichment can damp population cycles by a food‐web structure that balances inflexible and flexible interaction links (i.e. specialist and generalist predators). We modeled six predator–prey systems involving three or four species in which the predators practice optimal foraging based on prey profitability determined by handling time. In all models, the balance of interaction links simultaneously decreased the amplitude of population oscillations and increased the minimum density with increasing enrichment, leading to a potential theoretical resolution of the paradox of enrichment in non‐equilibrium dynamics. The stabilization mechanism was common to all of the models. Important previous studies on the stability of food webs have also demonstrated that a balance of interaction strengths stabilizes systems, suggesting a general rule of ecosystem stability.     

Dynamic optimization of distributed biological systems using robust and efficient numerical techniques

ABSTRACT: BACKGROUND: Systems biology allows the analysis of biological systems behavior under different conditions through in silico experimentation. The possibility of perturbing biological systems in different manners calls for the design of perturbations to achieve particular goals. Examples would include, the design of a chemical stimulation to maximize the amplitude of a given cellular signal or to achieve a desired pattern in pattern formation systems, etc. Such design problems can be mathematically formulated as dynamic optimization problems which are particularly challenging when the system is described by partial differential equations. This work addresses the numerical solution of such dynamic optimization problems for spatially distributed biological systems. The usual nonlinear and large scale nature of the mathematical models related to this class of systems and the presence of constraints on the optimization problems, impose a number of difficulties, such as the presence of suboptimal solutions, which call for robust and efficient numerical techniques. RESULTS: Here, the use of a control vector parameterization approach combined with efficient and robust hybrid global optimization methods and a reduced order model methodology is proposed. The capabilities of this strategy are illustrated considering the solution of a two challenging problems: bacterial chemotaxis and the FitzHugh-Nagumo model. CONCLUSIONS: In the process of chemotaxis the objective was to efficiently compute the time-varying optimal concentration of chemotractant in one of the spatial boundaries in order to achieve predefined cell distribution profiles. Results are in agreement with those previously published in the literature. The FitzHugh-Nagumo problem is also efficiently solved and it illustrates very well how dynamic optimization may be used to force a system to evolve from an undesired to a desired pattern with a reduced number of actuators. The presented methodology can be used for the efficient dynamic optimization of generic distributed biological systems.     

Integrating Biosystem Models Using Waveform Relaxation

Modelling in systems biology often involves the integration of component models into larger composite models. How to do this systematically and efficiently is a significant challenge: coupling of components can be unidirectional or bidirectional, and of variable strengths. We adapt the waveform relaxation (WR) method for parallel computation of ODEs as a general methodology for computing systems of linked submodels. Four test cases are presented: (i) a cascade of unidirectionally and bidirectionally coupled harmonic oscillators, (ii) deterministic and stochastic simulations of calcium oscillations, (iii) single cell calcium oscillations showing complex behaviour such as periodic and chaotic bursting, and (iv) a multicellular calcium model for a cell plate of hepatocytes. We conclude that WR provides a flexible means to deal with multitime-scale computation and model heterogeneity. Global solutions over time can be captured independently of the solution techniques for the individual components, which may be distributed in different computing environments.     

Preventing extinction and outbreaks in chaotic populations

Interactions in ecological communities are inherently nonlinear and can lead to complex population dynamics including irregular fluctuations induced by chaos. Chaotic population dynamics can exhibit violent oscillations with extremely small or large population abundances that might cause extinction and recurrent outbreaks, respectively. We present a simple method that can guide management efforts to prevent crashes, peaks, or any other undesirable state. At the same time, the irregularity of the dynamics can be preserved when chaos is desirable for the population. The control scheme is easy to implement because it relies on time series information only. The method is illustrated by two examples: control of crashes in the Ricker map and control of outbreaks in a stage-structured model of the flour beetle Tribolium. It turns out to be effective even with few available data and in the presence of noise, as is typical for ecological settings.     

Simulation of chemical oscillations in membranes

A computer simulation model has been developed to follow chemical oscillations in a membrane for immobilized enzyme systems. It is a discrete particle type model which follows the spatial and temporal fluctuations of the concentrations in a reaction involving two substrates. The parameters can be readily varied to allow dissipative structures to result from the sustained nonlinear reaction kinetics and to determine which parameters cause damping of the oscillations. The nature of the diffusion mechanism allows extension to more than one dimension.     

Symbiont genomics,our new tangled bank

Microbial symbionts inhabit the soma and surfaces of most multicellular species and instigate both beneficial and harmful infections. Despite their ubiquity, we are only beginning to resolve major patterns of symbiont ecology and evolution. Here, we summarize the history, current progress, and projected future of the study of microbial symbiont evolution throughout the tree of life. We focus on the recent surge of data that whole-genome sequencing has introduced into the field, in particular the links that are now being made between symbiotic lifestyle and molecular evolution. Post-genomic and systems biology approaches are also emerging as powerful techniques to investigate host–microbe interactions, both at the molecular level of the species interface and at the global scale. In parallel, next-generation sequencing technologies are allowing new questions to be addressed by providing access to population genomic data, as well as the much larger genomes of microbial eukaryotic symbionts and hosts. Throughout we describe the questions that these techniques are tackling and we conclude by listing a series of unanswered questions in microbial symbiosis that can potentially be addressed with the new technologies.