An optimization framework of biological dynamical systems

Different biological dynamics are often described by different mathematical equations. On the other hand, some mathematical models describe many biological dynamics universally. Here, we focus on three biological dynamics: the Lotka-Volterra equation, the Hopfield neural networks, and the replicator equation. We describe these three dynamical models using a single optimization framework, which is constructed with employing the Riemannian geometry. Then, we show that the optimization structures of these dynamics are identical, and the differences among the three dynamics are only in the constraints of the optimization. From this perspective, we discuss the unified view for biological dynamics. We also discuss the plausible categorizations, the fundamental nature, and the efficient modeling of the biological dynamics, which arise from the optimization perspective of the dynamical systems.     

Population mixing and the adaptive divergence of quantitative traits in discrete populations: a theoretical framework for empirical tests

Empirical tests for the importance of population mixing in constraining adaptive divergence have not been well grounded in theory for quantitative traits in spatially discrete populations. We develop quantitative-genetic models to examine the equilibrium difference between two populations that are experiencing different selective regimes and exchanging individuals. These models demonstrate that adaptive divergence is negatively correlated with the rate of population mixing (m, most strongly so when m is low), positively correlated with the difference in phenotypic optima between populations, and positively correlated with the amount of additive genetic variance (G, most strongly so when G is low). The approach to equilibrium is quite rapid (fewer than 50 generations for two populations to evolve 90% of the distance to equilibrium) when either heritability or mixing are not too low (h2 > 0.2 or m > 0.05). The theory can be used to aid empirical tests that: (1) compare observed divergence to that predicted using estimates of population mixing, additive genetic variance/covariance, and selection; (2) test for a negative correlation between population mixing and adaptive divergence across multiple independent population pairs; and (3) experimentally manipulate the rate of mixing. Application of the first two of these approaches to data from two well-studied natural systems suggests that population mixing has constrained adaptive divergence for color patterns in Lake Erie water snakes (Nerodia sipedon), but not for trophic traits in sympatric pairs of benthic and limnetic stickleback (Gasterosteus aculeatus). The theoretical framework we outline should provide an improved basis for future empirical tests of the role of population mixing in adaptive divergence.     

Reverse engineering discrete dynamical systems from data sets with random input vectors.

Recently a new algorithm for reverse engineering of biochemical networks was developed by Laubenbacher and Stigler. It is based on methods from computational algebra and finds most parsimonious models for a given data set. We derive mathematically rigorous estimates for the expected amount of data needed by this algorithm to find the correct model. In particular, we demonstrate that for one type of input parameter (graded term orders), the expected data requirements scale polynomially with the number n of chemicals in the network, while for another type of input parameters (randomly chosen lex orders) this number scales exponentially in n. We also show that, for a modification of the algorithm, the expected data requirements scale as the logarithm of n.     

Healthcare systems and motivation

Despite the fact that most American physicians, at least until around the 1970s, stood in the way of developing a universal healthcare system, most are generally not happy with the current state of healthcare--or its lack thereof--today. The primary reasons for this general unhappiness are that insurance companies and managed care have successfully conspired to remove much of the physician's autonomy (via imposed time constraints, burdensome paperwork, the time-consuming chore of having to defend going against stringent treatment algorithms that are often inappropriate for some patients) and the satisfaction of knowing their patients. Few physicians in managed care organizations (MCOs) are able to practice without constant and blindly algorithmic interference concerning the diagnostic tests and therapeutic interventions they order. As copayments have increased, they often find that patients, even though "covered," cannot afford the therapy they deem necessary. While physicians expect to earn sufficient to pay back their not insignificant educational debts, provide their children with help through college, and assure retirements sufficient for themselves and their spouses, these should not be considered unreasonable expectations. Most physicians today do favor universal healthcare -- to the point of having included such language in their various professional codes of ethics (which, perversely enough, bioethicists as a group have failed to do). Contrary to the claims of our colleagues, Altom and Churchill, physicians seem to be genuinely frustrated as to what else they can do to change the current inequitable system.     

Trophic interaction modifications: an empirical and theoretical framework

Consumer–resource interactions are often influenced by other species in the community. At present these ‘trophic interaction modifications’ are rarely included in ecological models despite demonstrations that they can drive system dynamics. Here, we advocate and extend an approach that has the potential to unite and represent this key group of non‐trophic interactions by emphasising the change to trophic interactions induced by modifying species. We highlight the opportunities this approach brings in comparison to frameworks that coerce trophic interaction modifications into pairwise relationships. To establish common frames of reference and explore the value of the approach, we set out a range of metrics for the ‘strength’ of an interaction modification which incorporate increasing levels of contextual information about the system. Through demonstrations in three‐species model systems, we establish that these metrics capture complimentary aspects of interaction modifications. We show how the approach can be used in a range of empirical contexts; we identify as specific gaps in current understanding experiments with multiple levels of modifier species and the distributions of modifications in networks. The trophic interaction modification approach we propose can motivate and unite empirical and theoretical studies of system dynamics, providing a route to confront ecological complexity.     

Post-fire ecological succession: A theoretical modeling framework

Fire is considered as an extreme disturbance in Mediterranean grasslands or shrublands as it often brings about many sudden changes in the vegetation structure, composition, and diversity patterns. In addition, it creates opportunities for exotic plant species to establish successfully in foreign habitat, and to outperform dominating native species. Monitoring and simulating post-fire successional changes, therefore, are essential tasks to efficiently restore native grasslands or shrublands. In this paper, we develop a theoretical framework for simulating fire-induced successional changes, mainly for Mediterranean vegetation, based on a three-level hierarchy of successional causes. Within this proposed framework, fire effects are considered by associating it with the number of burned sites open-up and specific changes at the burned sites relative to unburned sites. Three distinct site-specific neighborhoods are constructed; changes within each neighborhood allow sequential replacement of plant species by another plant species with greater maximum size, age and lower maximum growth rates and dispersal abilities. The proposed framework can be used to develop a spatially explicit individual-based model which will be useful for monitoring and predicting successional changes and hence for restoring native grasslands or shrublands.     

Steady-states of receptor-ligand dynamics: a theoretical framework

This paper studies aspects of the dynamics of a conventional mechanism of ligand-receptor interactions, with a focus on the stability and location of steady-states. A theoretical framework is developed, which is based upon the rich and deep formalism of irreducible biochemical networks. When represented in this manner, the mass action kinetics of biochemical processes can be clearly seen in terms of their component biochemical interactions, their kinetic rate constants, and the stoichiometry for the system. A minimal parametrization is provided for models for two- or multi-state receptor interaction with ligand, and an "affinity quotient" is introduced, which allows an elegant classification of ligands into agonists, neutral agonists, and inverse agonists.     

Post-fire ecological succession: A theoretical modeling framework

The effects of starvation on larval growth, survival, and metamorphosis of Manila clam Ruditapes philippinarum at the temperature of 19.6–21.6 °C, the salinity of 34‰ and pH of 8.0 were investigated from May 18 to July 18, 2006. In this study, the early, middle and late umbo-veliger larvae with the shell lengths of 100, 140, and 190 μm were subject to temporary food deprivation for up to 4.5, 20, and 25d at 0.5, 4, 5d intervals, followed by refeeding for the remaining of a 24, 20, 25d period, respectively. The results suggested that the larvae should have shown considerable tolerance to starvation due to their endogenous and exterior nutrition material, for larvae and time to the point-of-no-return (PNR: the threshold point during starvation after which larvae could no longer metamorphose even if food is provided) were calculated to be 4.25, 17.54, and 22.17d. As the starvation period prolonged, the mean shell length of larvae starved got close to constants at 1.5, 4, and 15d after starvation, which were different for larvae at different stages when starvation began, survival of larvae decreased, and was lower in treatments starved earlier in development than those starved later, for the early, middle and late umbo-veliger larvae, after 4.5, 20 and 25d of starvation period, few larvaes were alive. After starvation period, the alive larvaes were able to metamorphose and had a capability of compensatory growth when refeeding was given. Starvation not only affected metamorphosis rate, but also caused the delay in the time to metamorphosis and the decrease in the metamorphosed sizes. For example, for the continuously-fed larvae, duration to metamorphosis was 20.7d, for larvae with a size of 100-μm starved for up to 4d, larvae with a size of 140-μm starved for up to 16d, larvae with a size of 190-μm starved for up to 20d, duration to metamorphosis were 29.7, 31.7, and 37.7d, the delay in duration to metamorphosis were 9, 11, and 17d, respectively. Furthermore, importance of nutrition material for maintaining larval survival during starvation and the compensatory growth on larvae at the same feeding time were discussed.     

Ventilatory support: a dynamical systems approach

Misunderstanding of the dynamical behavior of the ventilatory system, especially under assisted ventilation, may explain the problems encountered in ventilatory support monitoring. Proportional assist ventilation (PAV) that theoretically gives a breath by breath assistance presents instability with high levels of assistance. We have constructed a mathematical model of interactions between three objects: the central respiratory pattern generator modelled by a modified Van der Pol oscillator, the mechanical respiratory system which is the passive part of the system and a controlled ventilator that follows its own law. The dynamical study of our model shows the existence of two crucial behaviors, i.e. oscillations and damping, depending on only two parameters, namely the time constant of the mechanical respiratory system and a cumulative interaction index. The same result is observed in simulations of spontaneous breathing as well as of PAV. In this last case, increasing assistance leads first to an increase of the tidal volume (V_T), a further increase in assistance inducing a decrease in V_T, ending in damping of the whole system to an attractive fixed point. We conclude that instabilities observed in PAV may be explained by the different possible dynamical behaviors of the system rather than changes in mechanical characteristics of the respiratory system.     

Biocomplexity: adaptive behavior in complex stochastic dynamical systems

Existing methods of complexity research are capable of describing certain specifics of bio systems over a given narrow range of parameters but often they cannot account for the initial emergence of complex biological systems, their evolution, state changes and sometimes-abrupt state transitions. Chaos tools have the potential of reaching to the essential driving mechanisms that organize matter into living substances. Our basic thesis is that while established chaos tools are useful in describing complexity in physical systems, they lack the power of grasping the essence of the complexity of life. This thesis illustrates sensory perception of vertebrates and the operation of the vertebrate brain. The study of complexity, at the level of biological systems, cannot be completed by the analytical tools, which have been developed for non-living systems. We propose a new approach to chaos research that has the potential of characterizing biological complexity. Our study is biologically motivated and solidly based in the biodynamics of higher brain function. Our biocomplexity model has the following features, (1) it is high-dimensional, but the dimensionality is not rigid, rather it changes dynamically; (2) it is not autonomous and continuously interacts and communicates with individual environments that are selected by the model from the infinitely complex world; (3) as a result, it is adaptive and modifies its internal organization in response to environmental factors by changing them to meet its own goals; (4) it is a distributed object that evolves both in space and time towards goals that is continually re-shaping in the light of cumulative experience stored in memory; (5) it is driven and stabilized by noise of internal origin through self-organizing dynamics. The resulting theory of stochastic dynamical systems is a mathematical field at the interface of dynamical system theory and stochastic differential equations. This paper outlines several possible avenues to analyze these systems. Of special interest are input-induced and noise-generated, or spontaneous state-transitions and related stability issues.     

A Bayesian framework for parameter estimation in dynamical models

Mathematical models in biology are powerful tools for the study and exploration of complex dynamics. Nevertheless, bringing theoretical results to an agreement with experimental observations involves acknowledging a great deal of uncertainty intrinsic to our theoretical representation of a real system. Proper handling of such uncertainties is key to the successful usage of models to predict experimental or field observations. This problem has been addressed over the years by many tools for model calibration and parameter estimation. In this article we present a general framework for uncertainty analysis and parameter estimation that is designed to handle uncertainties associated with the modeling of dynamic biological systems while remaining agnostic as to the type of model used. We apply the framework to fit an SIR-like influenza transmission model to 7 years of incidence data in three European countries: Belgium, the Netherlands and Portugal.     

Consistency principle in biological dynamical systems

We propose a principle of consistency between different hierarchical levels of biological systems. Given a consistency between molecule replication and cell reproduction, universal statistical laws on cellular chemical abundances are derived and confirmed experimentally. They include a power law distribution of gene expressions, a lognormal distribution of cellular chemical abundances over cells, and embedding of the power law into the network connectivity distribution. Second, given a consistency between genotype and phenotype, a general relationship between phenotype fluctuations by genetic variation and isogenic phenotypic fluctuation by developmental noise is derived. Third, we discuss the chaos mechanism for stem cell differentiation with autonomous regulation, resulting from a consistency between cell reproduction and growth of the cell ensemble.

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Integrated pest management models and their dynamical behaviour

Two impulsive models of integrated pest management (IPM) strategies are proposed, one with fixed intervention times and the other with these unfixed. The first model allows natural enemies to survive but under some conditions may lead to extinction of the pest. We use a simple prey-dependent consumption model with fixed impulsive effects and show that there exists a globally stable pesteradication periodic solution when the impulsive period is less than certain critical values. The effects of pest resistance to pesticides are also studied. The second model is constructed in the light of IPM practice such that when the pest population reaches the economic injury level (EIL), a combination of biological, cultural, and chemical tactics that reduce pests to tolerable levels is invoked. Using analytical methods, we show that there exists an orbitally asymptotically stable periodic solution with a maximum value no larger than the given Economic Threshold (ET). The complete expression for this periodic solution is given and the ET is evaluated for given parameters.We also show that in some cases control costs can be reduced by replacing IPM interventions at unfixed times with periodic interventions. Further, we show that small perturbations of the system do not affect the existence and stability of the periodic solution. Thus, we provide the first demonstration using mathematical models that an IPM strategy is more effective than classical control methods.