Estimating a predator-prey dynamical model with the parameter cascades method

Summary .   Ordinary differential equations (ODEs) are widely used in ecology to describe the dynamical behavior of systems of interacting populations. However, systems of ODEs rarely provide quantitative solutions that are close to real field observations or experimental data because natural systems are subject to environmental and demographic noise and ecologists are often uncertain about the correct parameterization. In this article we introduce "parameter cascades" as an improved method to estimate ODE parameters such that the corresponding ODE solutions fit the real data well. This method is based on the modified penalized smoothing with the penalty defined by ODEs and a generalization of profiled estimation, which leads to fast estimation and good precision for ODE parameters from noisy data. This method is applied to a set of ODEs originally developed to describe an experimental predator–prey system that undergoes oscillatory dynamics. The new parameterization considerably improves the fit of the ODE model to the experimental data sets. At the same time, our method reveals that important structural assumptions that underlie the original ODE model are essentially correct. The mathematical formulations of the two nonlinear interaction terms (functional responses) that link the ODEs in the predator–prey model are validated by estimating the functional responses nonparametrically from the real data. We suggest two major applications of "parameter cascades" to ecological modeling: It can be used to estimate parameters when original data are noisy, missing, or when no reliable priori estimates are available; it can help to validate the structural soundness of the mathematical modeling approach.     

Robust estimation for ordinary differential equation models

Applied scientists often like to use ordinary differential equations (ODEs) to model complex dynamic processes that arise in biology, engineering, medicine, and many other areas. It is interesting but challenging to estimate ODE parameters from noisy data, especially when the data have some outliers. We propose a robust method to address this problem. The dynamic process is represented with a nonparametric function, which is a linear combination of basis functions. The nonparametric function is estimated by a robust penalized smoothing method. The penalty term is defined with the parametric ODE model, which controls the roughness of the nonparametric function and maintains the fidelity of the nonparametric function to the ODE model. The basis coefficients and ODE parameters are estimated in two nested levels of optimization. The coefficient estimates are treated as an implicit function of ODE parameters, which enables one to derive the analytic gradients for optimization using the implicit function theorem. Simulation studies show that the robust method gives satisfactory estimates for the ODE parameters from noisy data with outliers. The robust method is demonstrated by estimating a predator-prey ODE model from real ecological data.     

Efficient solution of ordinary differential equations modeling electrical activity in cardiac cells

The contraction of the heart is preceded and caused by a cellular electro-chemical reaction, causing an electrical field to be generated. Performing realistic computer simulations of this process involves solving a set of partial differential equations, as well as a large number of ordinary differential equations (ODEs) characterizing the reactive behavior of the cardiac tissue. Experiments have shown that the solution of the ODEs contribute significantly to the total work of a simulation, and there is thus a strong need to utilize efficient solution methods for this part of the problem. This paper presents how an efficient implicit Runge-Kutta method may be adapted to solve a complicated cardiac cell model consisting of 31 ODEs, and how this solver may be coupled to a set of PDE solvers to provide complete simulations of the electrical activity.     

Successional state dynamics: A novel approach to modeling nonequilibrium foodweb dynamics

Communities and ecosystems are often far from equilibrium, but our understanding of nonequilibrium dynamics has been hampered by a paucity of analytical tools. Here I describe a novel approach to modeling seasonally forced food webs, called “successional state dynamics” (SSD). It is applicable to communities where species dynamics are fast relative to the external forcing, such as plankton and other microbes, diseases, and some insect communities. The approach treats succession as a series of state transitions driven by both the internal dynamics of species interactions and external forcing. First, I motivate the approach with numerical solutions of a seasonally forced predator-prey model. Second, I describe how to set up and analyze an SSD model. Finally, I apply the techniques to three additional models of two-species interactions: resource competition (r-K selection), facilitation, and flip-flop competition (where the competitive hierarchy alternates over time). This approach allows easy and thorough exploration of how dynamics depend on the environmental forcing regime, and uncovers unexpected phenomena such as multiple stable annual trajectories and year-to-year irregularity in successional trajectories (chaos).     

Small Sample Correction for the Variance of GEE Estimators

When clustered multinomial responses are fit using the generalized logistic link, Morel (1989) introduced a small sample correction in the Taylor series based estimator of the covariance matrix of the parameter estimates. The correction reduces the bias of the Type I error rates in small samples and guarantees positive definiteness of the estimated variance‐covariance matrix. It is well known that small sample bias in the use of the Delta method persists in any application of the Generalized Estimating Equations (GEE) methodology. In this article, we extend the correction originally suggested for the generalized logistic link, to other link functions and distributions, when parameters are estimated by GEE. In a Monte Carlo study with correlated data generated under different sampling schemes, the small sample correction has been shown to be effective in reducing the Type I error rates when the number of clusters is relatively small.     

Partial equilibrium approximations in apoptosis. I. The intracellular-signaling subsystem

Apoptosis is one of the most basic biological processes. In apoptosis, tens of species are involved in many biochemical reactions with times scales of widely differing orders of magnitude. By the law of mass action, the process is mathematically described with a large and stiff system of ODEs (ordinary differential equations). The goal of this work is to simplify such systems of ODEs with the PEA (partial equilibrium approximation) method. In doing so, we propose a general framework of the PEA method together with some conditions, under which the PEA method can be justified rigorously. The main condition is the principle of detailed balance for fast reactions as a whole and the framework provides some meaningful physical insights of the full chemical kinetics. With the justified method as a tool, we simplify the Fas-signaling pathway model due to Hua et al. [6] under the empirical assumption that nine reactions therein can be well regarded as relatively fast. This paper reports our simplification, together with numerical results which confirm the reliability of both our simplified model and the empirical assumption.     

Estimating parameters for generalized mass action models using constraint propagation

As modern molecular biology moves towards the analysis of biological systems as opposed to their individual components, the need for appropriate mathematical and computational techniques for understanding the dynamics and structure of such systems is becoming more pressing. For example, the modeling of biochemical systems using ordinary differential equations (ODEs) based on high-throughput, time-dense profiles is becoming more common-place, which is necessitating the development of improved techniques to estimate model parameters from such data. Due to the high dimensionality of this estimation problem, straight-forward optimization strategies rarely produce correct parameter values, and hence current methods tend to utilize genetic/evolutionary algorithms to perform non-linear parameter fitting. Here, we describe a completely deterministic approach, which is based on interval analysis. This allows us to examine entire sets of parameters, and thus to exhaust the global search within a finite number of steps. In particular, we show how our method may be applied to a generic class of ODEs used for modeling biochemical systems called Generalized Mass Action Models (GMAs). In addition, we show that for GMAs our method is amenable to the technique in interval arithmetic called constraint propagation, which allows great improvement of its efficiency. To illustrate the applicability of our method we apply it to some networks of biochemical reactions appearing in the literature, showing in particular that, in addition to estimating system parameters in the absence of noise, our method may also be used to recover the topology of these networks.     

Developmental selection and self-organization

Developmental selection is the differential survival and proliferation of developmental units, such as cellular lineages. This type of internal selection has been proposed as an explanation for diverse examples of self-organization, from the wiring of brains to the formation of pores on leaf surfaces. A general understanding of developmental selection has been slowed by failure to understand its relationship to familiar forms of genetical selection and evolution. I show the formal analogies between models of developmental selection and genetical selection. The general method I outline for the analysis of selective systems partitions self-organizing selective systems into generative rules that create variation and selective filters that move the population toward a target design. The method also emphasizes aggregate statistical measures of evolving systems, such as the covariance between particular traits and fitness. The identification of useful aggregate measures is a crucial step in the analysis of selective systems. I apply these concepts to a model of self-organization in ant colonies.     

Application of a generalized MWC model for the mathematical simulation of metabolic pathways regulated by allosteric enzymes

In our effort to elucidate the systems biology of the model organism, Escherichia coli, we have developed a mathematical model that simulates the allosteric regulation for threonine biosynthesis pathway starting from aspartate. To achieve this goal, we used kMech, a Cellerator language extension that describes enzyme mechanisms for the mathematical modeling of metabolic pathways. These mechanisms are converted by Cellerator into ordinary differential equations (ODEs) solvable by Mathematica. In this paper, we describe a more flexible model in Cellerator, which generalizes the Monod, Wyman, Changeux (MWC) model for enzyme allosteric regulation to allow for multiple substrate, activator and inhibitor binding sites. Furthermore, we have developed a model that describes the behavior of the bifunctional allosteric enzyme aspartate kinase I-homoserine dehydrogenase I (AKI-HDHI). This model predicts the partition of enzyme activities in the steady state which paves the way for a more generalized prediction of the behavior of bifunctional enzymes.     

植被系统中植物与环境因子相互作用的动态模拟

本文在综合考虑现有植被发展动态模型的基础上提出包含植被,环境因子,人类或其它外力干扰三者间相互作用情况下的植被系统一般动态模型,并从控制论的角度阐述了植被与环境因子间的耦合,反馈作用。最后以线性逼近和微分拟合的方法求取模型的局部解,进而作出外延预测。实例分析表明,模型预测对实际观测有较好的跟踪性能。     

Disturbances prevent stem size-density distributions in natural forests from following scaling relationships

Enquist and Niklas propose that trees in natural forests have invariant size-density distributions (SDDs) that scale as a −2 power of stem diameter, although early studies described such distributions using negative exponential functions. Using New Zealand and ‘global’ data sets, we demonstrate that neither type of function accurately describes the SDD over the entire diameter range. Instead, scaling functions provide the best fit to smaller stems, while negative exponential functions provide the best fit to larger stems. We argue that these patterns are consistent with competition shaping the small-stem phase and exogenous disturbance shaping the large-stem phase. Mortality rates, estimated from repeat measurements on 1546 New Zealand plots, fell precipitously with stem size until 18 cm but remained constant after that, consistent with our arguments. Even in the small-stem phase, where SDDs were best described by scaling functions, the scaling exponents were not invariantly −2, but differed significantly from this value in both the ‘global’ and New Zealand data sets, and varied through time in the New Zealand data set.     

Model of the impulse activity of a neuron receiving a stationary impulse imput]

The stationary model of spike activity of a single neuron is considered. This model exponential decay and constant threshold can be described as discontinuous one. The problem of calculation of Laplas' transform the probability density function (PDF) of interspike interval (ISI) can be reduced to the boundary problem for the usual differential equation. For the exponential PDF of the magnitude of the PSP the Laplas' transform of ISI PDF is expressed by special functions. The mean and standard deviation of ISI for different combinations of parameters were plotted with the aid of computer. The experiment without intracellular recoding, allowing to estimate the values of certain physiological parameters of a neuron on the basis of the model is proposed.