Predicting variability in biological control of a plant-pathogen system using stochastic models.

A stochastic model for the dynamics of a plant-pathogen interaction is developed and fitted to observations of the fungal pathogen Rhizoctonia solani (Kühn) in radish (Raphanus sativus L.), in both the presence and absence of the antagonistic fungus Trichoderma viride (Pers ex Gray). The model incorporates parameters for primary and secondary infection mechanisms and for characterizing the time-varying susceptibility of the host population. A parameter likelihood is developed and used to fit the model to data from microcosm experiments. It is shown that the stochastic model accounts well for observed variability both within and between treatments. Moreover, it enables us to describe the time evolution of the probability distribution for the variability among replicate epidemics in terms of the underlying epidemiological parameters for primary and secondary infection and decay in susceptibility. Consideration of profile likelihoods for each parameter provides strong evidence that T. viride mainly affects primary infection. By using the stochastic model to study the dependence of the probability distribution of disease levels on the primary infection rate we are therefore able to predict the effectiveness of a widely used biological control agent.     

Quantitative analysis and model simplification of an epidemic model with primary and secondary infection

Models of particular epidemiological systems can rapidly become complicated by biological detail which can obscure their essential features and behaviour. In general, we wish to retain only those components and processes that contribute to the dynamics of the system. In this paper, we apply asymptotic techniques to an SEI-type model with primary and secondary infection in order to reduce it to a much simpler form. This allows the identification of parameter groupings discriminating between regions of contrasting dynamics and leads to simple approximations for the model’s transient behaviour. These can be used to follow the evolution of the developing infection process. The techniques examined in this paper will be applicable to a large number of similar models.     

Accounting for uncertainty in model-based prevalence estimation: paratuberculosis control in dairy herds

ABSTRACT: BACKGROUND: A common approach to the application of epidemiological models is to determine a single (point estimate) parameterisation using the information available in the literature. However, in many cases there is considerable uncertainty about parameter values, reflecting both the incomplete nature of current knowledge and natural variation, for example between farms. Furthermore model outcomes may be highly sensitive to different parameter values. Paratuberculosis is an infection for which many of the key parameter values are poorly understood and highly variable, and for such infections there is a need to develop and apply statistical techniques which make maximal use of available data. RESULTS: A technique based on Latin hypercube sampling combined with a novel reweighting method was developed which enables parameter uncertainty and variability to be incorporated into a model-based framework for estimation of prevalence. The method was evaluated by applying it to a simulation of paratuberculosis in dairy herds which combines a continuous time stochastic algorithm with model features such as within herd variability in disease development and shedding, which have not been previously explored in paratuberculosis models. Generated sample parameter combinations were assigned a weight, determined by quantifying the model's resultant ability to reproduce prevalence data. Once these weights are generated the model can be used to evaluate other scenarios such as control options. To illustrate the utility of this approach these reweighted model outputs were used to compare standard test and cull control strategies both individually and in combination with simple husbandry practices that aim to reduce infection rates. CONCLUSIONS: The technique developed has been shown to be applicable to a complex model incorporating realistic control options. For models where parameters are not well known or subject to significant variability, the reweighting scheme allowed estimated distributions of parameter values to be combined with additional sources of information, such as that available from prevalence distributions, resulting in outputs which implicitly handle variation and uncertainty. This methodology allows for more robust predictions from modelling approaches by allowing for parameter uncertainty and combining different sources of information, and is thus expected to be useful in application to a large number of disease systems.     

Improving evolutionary models of protein interaction networks

MOTIVATION: Theoretical models of biological networks are valuable tools in evolutionary inference. Theoretical models based on gene duplication and divergence provide biologically plausible evolutionary mechanics. Similarities found between empirical networks and their theoretically generated counterpart are considered evidence of the role modeled mechanics play in biological evolution. However, the method by which these models are parameterized can lead to questions about the validity of the inferences. Selecting parameter values in order to produce a particular topological value obfuscates the possibility that the model may produce a similar topology for a large range of parameter values. Alternately, a model may produce a large range of topologies, allowing (incorrect) parameter values to produce a valid topology from an otherwise flawed model. In order to lend biological credence to the modeled evolutionary mechanics, parameter values should be derived from the empirical data. Furthermore, recent work indicates that the timing and fate of gene duplications are critical to proper derivation of these parameters. RESULTS: We present a methodology for deriving evolutionary rates from empirical data that is used to parameterize duplication and divergence models of protein interaction network evolution. Our method avoids shortcomings of previous methods, which failed to consider the effect of subsequent duplications. From our parameter values, we find that concurrent and existing existing duplication and divergence models are insufficient for modeling protein interaction network evolution. We introduce a model enhancement based on heritable interaction sites on the surface of a protein and find that it more closely reflects the high clustering found in the empirical network.     

Comparative study of parameter sensitivity analyses of the TCR-activated Erk-MAPK signalling pathway

Parameter estimation is a major challenge for mathematical modelling of biological systems. Given the uncertainties associated with model parameters, it is important to understand how sensitive the model output is to variations in parameter values. A local sensitivity analysis determines the model sensitivity to parameter variations over a localised region around the nominal parameter values, whereas a global sensitivity analysis (GSA) investigates the sensitivity over the entire parameter space. Using a T-cell receptor-activated Erk-MAPK signalling pathway model as an example, the authors present a comparative study of a variety of different sensitivity analysis techniques. These techniques include: local sensitivity analysis, existing GSA methods of partial rank correlation coefficient, Sobol's, extended Fourier amplitude sensitivity test, as well as a weighted average of local sensitivities and a new GSA method to extract global parameter sensitivities from a parameter identification routine. Results of this study revealed critical reactions in the signalling pathway and their impact on the signalling dynamics and provided insights into embedded regulatory mechanisms such as feedback loops in the pathway. From this study, a recommendation emerges for a general sensitivity analysis strategy to efficiently and reliably infer quantitative, dynamic as well as topological properties from systems biology models.     

On the Emergence of Biology from Chemistry: A Discontinuist Perspective from the Point of View of Stability and Regulation

In this paper we argue that molecular evolution, and the evolution of prebiotic and early biological systems are qualitatively different processes, in which a crucial role is played respectively by structural stability and by dynamical mechanisms of regulation and integration. These different features entail also distinct modalities of interaction between system and environment that need to be taken into consideration when discussing molecular and biological evolution and selection.     

Bi-trophic mathematical model for pest adaptation to a resistant crop

A version of the Lotka-Volterra predator-prey model with logistic crop growth is modified to explore the rate of adaptation of a herbivore to a pest-resistant crop. This provides a phenotypic model for the evolution of resistance in a population comprising three different pest types each defined by differing parameter values for respiration rate and crop palatability. Expressions estimating the rates of increase of the fitter pest types are obtained as a function of the food qualities, and respiration and mortality rates. Potential strategies for delaying the rate of adaptation with regard to the expressions derived above, via the use of pest-susceptible refugia and natural enemies, are discussed. Although the model is formulated as one in which a single gene is the factor conferring resistance it can be interpreted and used independently of this.     

A dynamic mathematical model to clarify signaling circuitry underlying programmed cell death control in Arabidopsis disease resistance

Plant cells undergo programmed cell death in response to invading pathogens. This cell death limits the spread of the infection and triggers whole plant antimicrobial and immune responses. The signaling network connecting molecular recognition of pathogens to these responses is a prime target for manipulation in genetic engineering strategies designed to improve crop plant disease resistance. Moreover, as alterations to metabolism can be misinterpreted as pathogen infection, successful plant metabolic engineering will ultimately depend on controlling these signaling pathways to avoid inadvertent activation of cell death. Programmed cell death resulting from infection of Arabidopsis thaliana with Pseudomonas syringae bacterial pathogens was chosen as a model system. Signaling circuitry hypotheses in this model system were tested by construction of a differential-equations-based mathematical model. Model-based simulations of time evolution of signaling components matched experimental measurements of programmed cell death and associated signaling components obtained in a companion study. Simulation of systems-level consequences of mutations used in laboratory studies led to two major improvements in understanding of signaling circuitry: (1) Simulations supported experimental evidence that a negative feedback loop in salicylic acid biosynthesis postulated by others does not exist. (2) Simulations showed that a second negative regulatory circuit for which there was strong experimental support did not affect one of two pathways leading to programmed cell death. Simulations also generated testable predictions to guide future experiments. Additional testable hypotheses were generated by results of individually varying each model parameter over 2 orders of magnitude that predicted biologically important changes to system dynamics. These predictions will be tested in future laboratory studies designed to further elucidate the signaling network control structure.     

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.     

Selection between multiple periodic regimes in a biochemical system: complex dynamic behaviour resolved by use of one-dimensional maps

We analyse a model biochemical system in which two autocatalytic enzyme reactions are coupled in series, in conditions where multiple stable periodic regimes coexist for the same set of parameter values. We determine how the periodic regimes are reached from different initial conditions. The structure of the attraction basins is generally simple in the case of two coexisting limit cycles (birhythmicity). This structure and the associated behaviour may, however, become highly complex. In particular, the system exhibits enhanced sensitivity to initial conditions when the boundaries of the attraction basins are fractal. In the latter case, it becomes difficult to predict the evolution towards either one of two limit cycles, a phenomenon known as final state sensitivity. We show how these complex phenomena can be explained in a unified and simple manner by means of one-dimensional return maps derived from the time evolution of the model and from fifth degree polynomial equations. We suggest experimental tests of the sensitivity to initial conditions in chemical systems presenting birhythmicity. The physiological significance of the results is discussed with respect to the sensitivity of regulatory systems admitting multiple stable biological rhythms.     

INDISIM-YEAST: an individual-based simulator on a website for experimenting and investigating diverse dynamics of yeast populations in liquid media

Simulation modelling can be used to capture and mimic real-world microbial systems that, unlike the real-world, can then be experimented upon as a new kind of experimental milieu. Individual-based models, in which individuals interact dynamically with each other as structural elements in the model world, exemplify this view of simulation modelling. These models are more difficult to analyze, understand and communicate than traditional analytical models. It is good practice to provide executable versions that perform simulation results. INDISIM-YEAST, developed to deal with yeast populations in liquid media, models the evolution of a set of yeasts by setting up rules of behavior for each individual cell according to its own biological regulations and characteristics. The aim of this work is to develop and present a website from which INDISIM-YEAST is accessible, and how to carry out yeast simulations to further the skills associated with the use of this individual-based simulator. A good and useful way to analyze this yeast simulator is to experiment and explore the manner in which it reacts to changes in parameter values, initial conditions or assumptions. The application results in a very versatile program that could be used in controlled simulation experiments via the Internet.     

Parameter Estimation and Model Selection in Computational Biology

A central challenge in computational modeling of biological systems is the determination of the model parameters. Typically, only a fraction of the parameters (such as kinetic rate constants) are experimentally measured, while the rest are often fitted. The fitting process is usually based on experimental time course measurements of observables, which are used to assign parameter values that minimize some measure of the error between these measurements and the corresponding model prediction. The measurements, which can come from immunoblotting assays, fluorescent markers, etc., tend to be very noisy and taken at a limited number of time points. In this work we present a new approach to the problem of parameter selection of biological models. We show how one can use a dynamic recursive estimator, known as extended Kalman filter, to arrive at estimates of the model parameters. The proposed method follows. First, we use a variation of the Kalman filter that is particularly well suited to biological applications to obtain a first guess for the unknown parameters. Secondly, we employ an a posteriori identifiability test to check the reliability of the estimates. Finally, we solve an optimization problem to refine the first guess in case it should not be accurate enough. The final estimates are guaranteed to be statistically consistent with the measurements. Furthermore, we show how the same tools can be used to discriminate among alternate models of the same biological process. We demonstrate these ideas by applying our methods to two examples, namely a model of the heat shock response in E. coli, and a model of a synthetic gene regulation system. The methods presented are quite general and may be applied to a wide class of biological systems where noisy measurements are used for parameter estimation or model selection.     

Linking dynamical and population genetic models of persistent viral infection

This article develops a theoretical framework to link dynamical and population genetic models of persistent viral infection. This linkage is useful because, while the dynamical and population genetic theories have developed independently, the biological processes they describe are completely interrelated. Parameters of the dynamical models are important determinants of evolutionary processes such as natural selection and genetic drift. We develop analytical methods, based on coupled differential equations and Markov chain theory, to predict the accumulation of genetic diversity within the viral population as a function of dynamical parameters. These methods are first applied to the standard model of viral dynamics and then generalized to consider the infection of multiple host cell types by the viral population. Each cell type is characterized by specific parameter values. Inclusion of multiple cell types increases the likelihood of persistent infection and can increase the amount of genetic diversity within the viral population. However, the overall rate of gene sequence evolution may actually be reduced.     

Parametric energy conversion--a possible, universal approach to bioenergetics in biological structures.

It is shown that the mechanism of parametric energy conversion—a non-linear phenomenon which is known to occur in all branches of physics—may play a fundamental role in energy conversion in biological structures. Parametric energy conversion means pumping of energy through the variation of an energy storing quantity (a parameter). In biological systems the energy storing parameter is the membrane itself, the structure and composition of which is varied by proceeding structure bound biochemical reactions. The principle of parametric energy conversion is introduced into a molecular kinetical model and three coupled differential equations are derived, which interconnect chemical, electrical and mechanical energy in biological structures. It is shown that they describe parametric pumping of energy. It is a particular mechanism, which is also found in the physical phenomenon of Bethenod. The mechanism is tested with the derivation and explanation of various important bioenergetical functions as special cases of parametric energy conversion, of ATP synthesis, the pumping of ions and molecules during active transport, the excitability of nerve membranes and the dynamics of oscillatory muscles. A new interpretation of the connection of structure and function in striated muscles is also derived and signal transformation in receptors discussed. It is suggested that parametric energy conversion may be the uniform basis of energy conversion in biological structures and that the path of bioenergetic evolution might have essentially followed the line marked by the characteristic properties of this flexible mechanism. The parametric hypothesis offers an elegant ordering scheme and reasonable explanation for evolution and function of a large variety of important bioenergetic mechanisms. In order to handle the intricate mechanism properly it would be necessary to give up the conventional, intuitive way of formulating and understanding biochemical mechanisms and to develop a new dimension of chemical thinking.     

A population biological approach to the collective dynamics of countries undergoing demographic transition

Birth rates have been declining in higher-income countries since the middle of the 19th century. A growing number of other countries have entered this demographic transition to lower fertility, as socioeconomic development continues. Analyses of this demographic transition vary widely, but most analyze individual populations in isolation from others, and most come from fields outside the biological sciences. Here, we develop a population biological model of population dynamics in higher-income countries. Individual countries evolve through density-regulated growth, where gradual evolution toward higher population densities boosts productivity (and hence socioeconomic growth) through economics of agglomeration and scale, in turn reducing birth rates. The exchange of technology and capital between countries can further boost productivity gains in any given country, thus contributing to its demographic transition. As a result, countries can down-regulate one another's population growth through mutual improvements in productivity. The model is fitted to time series data on population size, GDP per capita, and birth rates for the United States, the United Kingdom, and France. The metapopulation dynamics are also characterized across a range of parameter values close to the fitted values. This work may help advance population biological approaches to understanding the implications of the fertility demographic transition for modern human populations. This is relevant to developing long-term predictions of the earth's total population size, which must be based upon a model that incorporates underlying mechanisms.     

Dynamic Stabilization in the PU1-GATA1 Circuit Using a Model with Time-Dependent Kinetic Change

The PU.1 and GATA1 genes play an important role in the differentiation of blood stem cells. The protein levels expressed by these genes are thought to be regulated by a self-excitatory feedback loop for each gene and a cross-inhibitory feedback loop between the two genes. A mathematical model that captures the dynamical interaction between these two genes reveals that constant levels of self-excitation and cross-inhibition allow the most self-exciting or cross-inhibiting gene to dominate the system. However, since biological systems rarely exist in an unchanging equilibrium, we modeled this gene circuit using discrete time-dependent changes in the parameters in lieu of steady state parameters. These time-dependent parameters lead to new phenomena, including the development of new limit cycles and basins of attraction. These phenomena are not present in models using constant parameter values. Our findings suggest that even small perturbations in the PU.1 and GATA1 feedback loops may substantially alter the gene expression and therefore the cell phenotype. These time-dependent parameter models may also have implications for other gene systems and provide new ways to understand the mechanisms of cellular differentiation.     

Mathematical modeling of evolution of horizontally transferred genes

We describe a stochastic birth-and-death model of evolution of horizontally transferred genes in microbial populations. The model is a generalization of the stochastic model described by Berg and Kurland and includes five parameters: the rate of mutational inactivation, selection coefficient, invasion rate (i.e., rate of arrival of a novel sequence from outside of the recipient population), within-population horizontal transmission ("infection") rate, and population size. The model of Berg and Kurland included four parameters, namely, mutational inactivation, selection coefficient, population size, and "infection." However, the effect of "infection" was disregarded in the interpretation of the results, and the overall conclusion was that horizontally acquired sequences can be fixed in a population only when they confer a substantial selective advantage onto the recipient and therefore are subject to strong positive selection. Analysis of the present model in different domains of parameter values shows that, as long as the rate of within-population horizontal transmission is comparable to the mutational inactivation rate and there is even a low rate of invasion, horizontally acquired sequences can be fixed in the population or at least persist for a long time in a substantial fraction of individuals in the population even when they are neutral or slightly deleterious. The available biological data strongly suggest that intense within-population and even between-populations gene flows are realistic for at least some prokaryotic species and environments. Therefore, our modeling results are compatible with the notion of a pivotal role of horizontal gene transfer in the evolution of prokaryotes.     

Effects of crop species richness on pest-natural enemy systems based on an experimental model system using a microlandscape

The relationship between crop richness and predator-prey interactions as they relate to pest-natural enemy systems is a very important topic in ecology and greatly affects biological control services. The effects of crop arrangement on predator-prey interactions have received much attention as the basis for pest population management. To explore the internal mechanisms and factors driving the relationship between crop richness and pest population management, we designed an experimental model system of a microlandscape that included 50 plots and five treatments. Each treatment had 10 repetitions in each year from 2007 to 2010. The results showed that the biomass of pests and their natural enemies increased with increasing crop biomass and decreased with decreasing crop biomass; however, the effects of plant biomass on the pest and natural enemy biomass were not significant. The relationship between adjacent trophic levels was significant (such as pests and their natural enemies or crops and pests), whereas non-adjacent trophic levels (crops and natural enemies) did not significantly interact with each other. The ratio of natural enemy/pest biomass was the highest in the areas of four crop species that had the best biological control service. Having either low or high crop species richness did not enhance the pest population management service and lead to loss of biological control. Although the resource concentration hypothesis was not well supported by our results, high crop species richness could suppress the pest population, indicating that crop species richness could enhance biological control services. These results could be applied in habitat management aimed at biological control, provide the theoretical basis for agricultural landscape design, and also suggest new methods for integrated pest management.     

Data-based identifiability analysis of non-linear dynamical models

MOTIVATION: Mathematical modelling of biological systems is becoming a standard approach to investigate complex dynamic, non-linear interaction mechanisms in cellular processes. However, models may comprise non-identifiable parameters which cannot be unambiguously determined. Non-identifiability manifests itself in functionally related parameters, which are difficult to detect. RESULTS: We present the method of mean optimal transformations, a non-parametric bootstrap-based algorithm for identifiability testing, capable of identifying linear and non-linear relations of arbitrarily many parameters, regardless of model size or complexity. This is performed with use of optimal transformations, estimated using the alternating conditional expectation algorithm (ACE). An initial guess or prior knowledge concerning the underlying relation of the parameters is not required. Independent, and hence identifiable parameters are determined as well. The quality of data at disposal is included in our approach, i.e. the non-linear model is fitted to data and estimated parameter values are investigated with respect to functional relations. We exemplify our approach on a realistic dynamical model and demonstrate that the variability of estimated parameter values decreases from 81 to 1% after detection and fixation of structural non-identifiabilities.