Non-linear qualitative signal processing for biological systems: application to the algal growth in bioreactors

We present in this paper a qualitative method to validate and monitor the structure of a non-linear model with respect to experimental data, under some hypotheses. This method is broadly independent of the analytical formulation of the model, and depends only on the qualitative structure (the signs of the Jacobian matrix). The temporal sequences of the extrema of a filtered experimental signal are compared with the transitions allowed by a graph. In particular, we show that the usual moving average of the outputs follows this transition graph. We apply this method to compare models of algal growth in a bioreactor with experimental data.     

Dynamic heterogeneity in life histories

Longitudinal data on natural populations have been analysed using multistage models in which survival depends on reproductive stage, and individuals change stages according to a Markov chain. These models are special cases of stage-structured population models. We show that stage-structured models generate dynamic heterogeneity: life-history differences produced by stochastic stratum dynamics. We characterize dynamic heterogeneity in a range of species across taxa by properties of the Markov chain: the entropy, which describes the extent of heterogeneity, and the subdominant eigenvalue, which describes the persistence of reproductive success during the life of an individual. Trajectories of reproductive stage determine survivorship, and we analyse the variance in lifespan within and between trajectories of reproductive stage. We show how stage-structured models can be used to predict realized distributions of lifetime reproductive success. Dynamic heterogeneity contrasts with fixed heterogeneity: unobserved differences that generate variation between life histories. We show by an example that observed distributions of lifetime reproductive success are often consistent with the claim that little or no fixed heterogeneity influences this trait. We propose that dynamic heterogeneity provides a 'neutral' model for assessing the possible role of unobserved 'quality' differences between individuals. We discuss fitness for dynamic life histories, and the implications of dynamic heterogeneity for the evolution of life histories and senescence.     

Experimental testing of dynamic energy budget models

1. Dynamic energy budget (DEB) models describing the allocation of assimilate to the competing processes of growth, reproduction and maintenance in individual organisms have been applied to a variety of species with some success. There are two contrasting model formulations based on dynamic allocation rules that have been widely used (net production and net assimilation formulations). However, the predictions of these two classes of DEB models are not easily distinguished on the basis of simple growth and fecundity data.
2. It is shown that different assumptions incorporated in the rules determining allocation to growth and reproduction in two classes of commonly applied DEB models predict qualitatively distinct patterns for an easily measured variable, cumulative reproduction by the time an individual reaches an arbitrary size.
3. A comparison with experimental data from Daphnia pulex reveals that, in their simplest form, neither model predicts the observed qualitative pattern of reproduction, despite the fact that both formulations capture basic growth features.
4. An examination of more elaborate versions of the two models, in which the allocation rules are modified to account for brief periods of starvation experienced in the laboratory cultures, reveals that a version of the net production model can predict the qualitative pattern seen for cumulative eggs as a function of mass in D. pulex . The analysis leads to new predictions which can be easily tested with further laboratory experiments.     

Relating dispersal and range expansion of California sea otters

Linking dispersal and range expansion of invasive species has long challenged theoretical and quantitative ecologists. Subtle differences in dispersal can yield large differences in geographic spread, with speeds ranging from constant to rapidly increasing. We developed a stage-structured integrodifference equation (IDE) model of the California sea otter range expansion that occurred between 1914 and 1986. The non-spatial model, a linear matrix population model, was coupled to a suite of candidate dispersal kernels to form stage-structured IDEs. Demographic and dispersal parameters were estimated independent of range expansion data. Using a single dispersal parameter, alpha, we examined how well these stage-structured IDEs related small scale demographic and dispersal processes with geographic population expansion. The parameter alpha was estimated by fitting the kernels to dispersal data and by fitting the IDE model to range expansion data. For all kernels, the alpha estimate from range expansion data fell within the 95% confidence intervals of the alpha estimate from dispersal data. The IDE models with exponentially bounded kernels predicted invasion velocities that were captured within the 95% confidence bounds on the observed northbound invasion velocity. However, the exponentially bounded kernels yielded range expansions that were in poor qualitative agreement with range expansion data. An IDE model with fat (exponentially unbounded) tails and accelerating spatial spread yielded the best qualitative match. This model explained 94% and 97% of the variation in northbound and southbound range expansions when fit to range expansion data. These otters may have been fat-tailed accelerating invaders or they may have followed a piece-wise linear spread first over kelp forests and then over sandy habitats. Further, habitat-specific dispersal data could resolve these explanations.     

Modeling vital rates improves estimation of population projection matrices

Population projection matrices are commonly used by ecologists and managers to analyze the dynamics of stage-structured populations. Building projection matrices from data requires estimating transition rates among stages, a task that often entails estimating many parameters with few data. Consequently, large sampling variability in the estimated transition rates increases the uncertainty in the estimated matrix and quantities derived from it, such as the population multiplication rate and sensitivities of matrix elements. Here, we propose a strategy to avoid overparameterized matrix models. This strategy involves fitting models to the vital rates that determine matrix elements, evaluating both these models and ones that estimate matrix elements individually with model selection via information criteria, and averaging competing models with multimodel averaging. We illustrate this idea with data from a population of Silene acaulis (Caryophyllaceae), and conduct a simulation to investigate the statistical properties of the matrices estimated in this way. The simulation shows that compared with estimating matrix elements individually, building population projection matrices by fitting and averaging models of vital-rate estimates can reduce the statistical error in the population projection matrix and quantities derived from it.     

Thermodynamic strategies for stabilizing intermediate states of proteins

This paper presents three theorems pertaining to thermodynamic properties of the intermediate (e.g., molten globule) state of proteins exhibiting such a conformation in the presence of GuHCl or urea. The theorems are proved for the three-state case using the denaturant binding model and the linear extrapolation model; their utility is illustrated via applications to examples in the literature. Theorem One states that the denaturant activity that maximizes the population of a partly folded conformation is at any temperature independent of the Gibbs free energy difference between the intermediate and native states. This result holds for both models of protein–denaturant interaction. The second theorem claims that the population maximum is independent, of the denaturant association constant for the denaturant binding model. Theorem Three, which also applies to both models considered here, states that at the temperatures corresponding to the extrema in the population of the intermediate, the enthalpy change of the intermediate is equal to the excess enthalpy function, an experimentally accessible quantity. In the absence of denaturant, the enthalpy change of the intermediate state at the population extrema can be written as a function of the thermodynamic parameters of the unfolded state alone. These results, which can be applied to systems of any number of states under certain conditions, should aid in the optimization of conditions employed for experimental studies of partly organized states of proteins. © 1994 John Wiley & Sons, Inc.     

The interplay between chromosome stability and cell cycle control explored through gene–gene interaction and computational simulation

Chromosome stability models are usually qualitative models derived from molecular-genetic mechanisms for DNA repair, DNA synthesis, and cell division. While qualitative models are informative, they are also challenging to reformulate as precise quantitative models. In this report we explore how (A) laboratory experiments, (B) quantitative simulation, and (C) seriation algorithms can inform models of chromosome stability. Laboratory experiments were used to identify 19 genes that when over-expressed cause chromosome instability in the yeast Saccharomyces cerevisiae. To better understand the molecular mechanisms by which these genes act, we explored their genetic interactions with 18 deletion mutations known to cause chromosome instability. Quantitative simulations based on a mathematical model of the cell cycle were used to predict the consequences of several genetic interactions. These simulations lead us to suspect that the chromosome instability genes cause cell-cycle perturbations. Cell-cycle involvement was confirmed using a seriation algorithm, which was used to analyze the genetic interaction matrix to reveal an underlying cyclical pattern. The seriation algorithm searched over 10¹⁴ possible arrangements of rows and columns to find one optimal arrangement, which correctly reflects events during cell cycle phases. To conclude, we illustrate how the molecular mechanisms behind these cell cycle events are consistent with established molecular interaction maps.     

Analysis of affected sib pairs, with covariates--with and without constraints

Covariate models have previously been developed as an extension to affected-sib-pair methods in which the covariate effects are jointly estimated with the degree of excess allele sharing. These models can estimate the differences in sib-pair allele sharing that are associated with measurable environment or genes. When there are no covariates, the pattern of identical-by-descent allele sharing in affected sib pairs is expected to fall within a small triangular region of the potential parameter space, under most genetic models. By restriction of the estimated allele sharing to this triangle, improved power is obtained in tests for genetic linkage. When the affected-sib-pair model is generalized to allow for covariates that affect allele sharing, however, new constraints and new methods for the application of constraints are required. Three generalized constraint methods are proposed and evaluated by use of simulated data. The results compare the power of the different methods, with and without covariates, for a single-gene model with age-dependent onset and for quantitative and qualitative gene-environment and gene-gene interaction models. Covariates can improve the power to detect linkage and can be particularly valuable when there are qualitative gene-environment interactions. In most situations, the best strategy is to assume that there is no dominance variance and to obtain constrained estimates for covariate models under this assumption.     

Path PPI: an integrated dataset of human pathways and protein-protein interactions

Integration of pathway and protein-protein interaction(PPI) data can provide more information that could lead to new biological insights. PPIs are usually represented by a simple binary model, whereas pathways are represented by more complicated models. We developed a series of rules for transforming protein interactions from pathway to binary model, and the protein interactions from seven pathway databases, including PID, Bio Carta, Reactome, Net Path, INOH, SPIKE and KEGG, were transformed based on these rules. These pathway-derived binary protein interactions were integrated with PPIs from other five PPI databases including HPRD, Int Act, Bio GRID, MINT and DIP, to develop integrated dataset(named Path PPI). More detailed interaction type and modification information on protein interactions can be preserved in Path PPI than other existing datasets. Comparison analysis results indicate that most of the interaction overlaps values(OAB) among these pathway databases were less than 5%, and these databases must be used conjunctively. The Path PPI data was provided at http://proteomeview. hupo.org.cn/Path PPI/Path PPI.html.     

Age or Stage Structure?

Discrete stage-structured density-dependent and discrete age-structured density-dependent population models are considered. Regarding the former, we prove that the model at hand is permanent (i.e., that the population will neither go extinct nor exhibit explosive oscillations) and given density dependent fecundity terms we also show that species with delayed semelparous life histories tend to be more stable than species which possess precocious semelparous life histories. Moreover, our findings together with results obtained from other stage-structured models seem to illustrate a fairly general ecological principle, namely that iteroparous species are more stable than semelparous species. Our analysis of various age-structured models does not necessarily support the conclusions above. In fact, species with precocious life histories now appear to possess better stability properties than species with delayed life histories, especially in the iteroparous case. We also show that there are dynamical outcomes from semelparous age-structured models which we are not able to capture in corresponding stage-structured cases. Finally, both age- and stage-structured population models may generate periodic dynamics of low period (either exact or approximate). The important prerequisite is to assume density-dependent survival probabilities.     

Permanence of single-species stage-structured models

In this paper, we consider population survival by using single-species stage-structured models. As a criterion of population survival, we employ the mathematical notation of permanence. Permanence of stage-structured models has already been studied by Cushing (1998). We generalize his result of permanence, and obtain a condition which guarantees that population survives. The condition is applicable to a wide class of stage-structured models. In particular, we apply our results to the Neubert-Caswell model, which is a typical stage-structured model, and obtain a condition for population survival of the model.The research is partially supported by the Ministry of Education, Science and Culture, Japan, under Grant in Aid for Scientific Research (A) 13304006.     

Mating,births, and transitions: a flexible two-sex matrix model for evolutionary demography

Models of sexually-reproducing populations that consider only a single sex cannot capture the effects of sex-specific demographic differences and mate availability. We present a new framework for two-sex demographic models that implements and extends the birth-matrix mating-rule approach of Pollak. The model is a continuous-time matrix model that explicitly includes the processes of mating (which is nonlinear but homogeneous), offspring production, and demographic transitions and survival. The resulting nonlinear model converges to exponential growth with an equilibrium population composition. The model can incorporate age- or stage-structured life histories and flexible mating functions. As an example, we apply the model to analyze the effects of mating strategies (polygamy or monogamy, and mated unions composed of males and females, of variable duration) on the response to sex-biased harvesting. The combination of demographic complexity with the interaction of the sexes can have major population dynamic effects and can change the outcome of evolution on sex-related characters.     

Models of HERG gating

HERG (Kv11.1, KCNH2) is a voltage-gated potassium channel with unique gating characteristics. HERG has fast voltage-dependent inactivation, relatively slow deactivation, and fast recovery from inactivation. This combination of gating kinetics makes study of HERG difficult without using mathematical models. Several HERG models have been developed, with fundamentally different organization. HERG is the molecular basis of I_Kr, which plays a critical role in repolarization. We programmed and compared five distinct HERG models. HERG gating cannot be adequately replicated using Hodgkin-Huxley type formulation. Using Markov models, a five-state model is required with three closed, one open, and one inactivated state, and a voltage-independent step between some of the closed states. A fundamental difference between models is the presence/absence of a transition directly from the proximal closed state to the inactivated state. The only models that effectively reproduce HERG data have no direct closed-inactivated transition, or have a closed-inactivated transition that is effectively zero compared to the closed-open transition, rendering the closed-inactivation transition superfluous. Our single-channel model demonstrates that channels can inactivate without conducting with a flickering or bursting open-state. The various models have qualitative and quantitative differences that are critical to accurate predictions of HERG behavior during repolarization, tachycardia, and premature depolarizations.     

Mathematical prediction of core body temperature from environment,activity, and clothing: The heat strain decision aid (HSDA)

Physiological models provide useful summaries of complex interrelated regulatory functions. These can often be reduced to simple input requirements and simple predictions for pragmatic applications. This paper demonstrates this modeling efficiency by tracing the development of one such simple model, the Heat Strain Decision Aid (HSDA), originally developed to address Army needs. The HSDA, which derives from the Givoni-Goldman equilibrium body core temperature prediction model, uses 16 inputs from four elements: individual characteristics, physical activity, clothing biophysics, and environmental conditions. These inputs are used to mathematically predict core temperature (T_c) rise over time and can estimate water turnover from sweat loss. Based on a history of military applications such as derivation of training and mission planning tools, we conclude that the HSDA model is a robust integration of physiological rules that can guide a variety of useful predictions. The HSDA model is limited to generalized predictions of thermal strain and does not provide individualized predictions that could be obtained from physiological sensor data-driven predictive models. This fully transparent physiological model should be improved and extended with new findings and new challenging scenarios.     

A Two-Ellipsoid Model Based Upon A Gaussian Overlap Potential

Abstract

The two-ellipsoid model (TEM) is proposed as a versatile single-site model which can be used in the study of liquid crystal phases. This TEM uses two ellipsoids to describe a molecule, one ellipsoid for the geometry and the other for the interaction strengths of the molecule. The present TEM can mimic asymmetric interactions of a liquid crystal molecule by separating the center of the interaction ellipsoid from that of the geometry ellipsoid. The potential energy surfaces of the present TEMs compare favorably with those of the corresponding Gay-Berne and the site–site models.

Monte Carlo simulations with 320 particles are performed for a symmetric interaction TEM and an asymmetric interaction TEM. The asymmetric interaction TEM displays a slightly higher transition temperature than the symmetric interaction TEM indicating that asymmetric interactions can be a driving force in a phase transition. Radial and cylindrical distribution functions of two models in the isotropic phase are similar, but those in the nematic phase are quite different.     

Development and regeneration of the retinotectal map in goldfish: a computational study

We present a simple computational model to study the interplay of activity-dependent and intrinsic processes thought to be involved in the formation of topographic neural projections. Our model consists of two input layers which project to one target layer. The connections between layers are described by a set of synaptic weights. These weights develop according to three interacting developmental rules: (i) an intrinsic fibre-target interaction which generates chemospecific adhesion between afferent fibres and target cells; (ii) an intrinsic fibre-fibre interaction which generates mutual selective adhesion between the afferent fibres; and (iii) an activity-dependent fibre-fibre interaction which implements Hebbian learning. Additionally, constraints are imposed to keep synaptic weights finite. The model is applied to a set of eleven experiments on the regeneration of the retinotectal projection in goldfish. We find that the model is able to reproduce the outcome of an unprecedented range of experiments with the same set of model parameters, including details of the size of receptive and projective fields. We expect this mathematical framework to be a useful tool for the analysis of developmental processes in general. <br>     

Large amplification in stage-structured models: Arnol’d tongues revisited

The coexistence of periodic and point attractors has been confirmed for a range of stage-structured discrete time models. The periodic attractor cycles have large amplitude, with the populations cycling between extremely low and surprisingly high values when compared to the equilibrium level. In this situation a stable state can be shocked by noise of sufficient strength into a state of high volatility. We found that the source of these large amplitude cycles are Arnold tongues, special regions of parameter space where the system exhibits periodic behaviour. Most of these tongues lie entirely in that part of parameter space where the system is unstable, but there are exceptions and these exceptions are the tongues that lead to attractor coexistence. Similarity in the geometry of Arnold tongues over the range of models considered might suggest that this is a common feature of stage-structured models but in the absence of proof this can only be a useful working hypothesis. The analysis shows that although large amplitude cycles might exist mathematically they might not be accessible biologically if biological constraints, such as non-negativity of population densities and vital rates, are imposed. Accessibility is found to be highly sensitive to model structure even though the mathematical structure is not. This highlights the danger of drawing biological conclusions from particular models. Having a comprehensive view of the different mechanisms by which periodic states can arise in families of discrete time models is important in the debate on whether the causes of periodicity in particular ecological systems are intrinsic, environmental or trophic. This paper is a contribution to that continuing debate.     

Protein symmetry and the co-operative ligand-binding behaviour predicted by allosteric Koshland models

It is shown that the nearest-neighbour interaction two-conformation allosteric models of Koshland, Nemethy & Filmer (1966) predict binding curves with a centre of symmetry when the protein is also symmetrical and induced-fit is assumed. When nonexclusive binding to both conformations is assumed, the models predict that the family of homotropic binding curves obtained by varying the heterotropic ligand has a centre of symmetry. It is argued that the symmetry or asymmetry of binding curves is the main experimentally verifiable prediction of allosteric models insofar as they are models of interaction between protein subunits.Symmetry in a binding curve greatly simplifies the analysis of cooperative behaviour. The co-operative features possible with a symmetric binding curve for a four-site protein are analysed. The sign of co-operativity may either be uniform or change twice as saturation increases; the conditions for the various possibilities are given. For example, in terms of the intrinsic binding constants per site A^∥₁, A2, etc. the necessary and sufficient condition for positive macroscopic co-operativity over the whole symmetric binding curve is A^∥₁≤ A2, A ^∥₁ ≤ A3 which should be contrasted with the obvious A^∥₁ ≤ Al, AZ ≤A^∥₃ (positive microscopic co-operativity) which is only a sufficient but not a necessary condition. A symmetric curve may have one or three, but no more, extrema of the “Hill coefficient” h. For three extrema a change of sign of microscopic (but not necessarily macroscopic) co-operativity is necessary but not sufficient. In the case where there are off-centre maxima of h, then h < 2 everywhere on the curve.The Koshland models predict qualitative and quantitative restrictions on the forms of binding curves additional to that of symmetry. In tetrameric induced fit models, negative co-operativity in the mid-region of the curve and positive co-operativity in the outside regions is possible, but not the opposite, and three extrema of h are possible with uniform positive but not with uniform negative co-operativity.Thus by recognising the importance of symmetry it has been possible to describe and categorise all the co-operativity behaviour possible with the most plausible Koshland tetrameric models. Several experimental examples of probable non-exclusive binding to proteins and enzymes are discussed, and it is shown how the symmetry point of view illuminates their interpretation.     

Bayesian inference for identifying interaction rules in moving animal groups

The emergence of similar collective patterns from different self-propelled particle models of animal groups points to a restricted set of "universal" classes for these patterns. While universality is interesting, it is often the fine details of animal interactions that are of biological importance. Universality thus presents a challenge to inferring such interactions from macroscopic group dynamics since these can be consistent with many underlying interaction models. We present a Bayesian framework for learning animal interaction rules from fine scale recordings of animal movements in swarms. We apply these techniques to the inverse problem of inferring interaction rules from simulation models, showing that parameters can often be inferred from a small number of observations. Our methodology allows us to quantify our confidence in parameter fitting. For example, we show that attraction and alignment terms can be reliably estimated when animals are milling in a torus shape, while interaction radius cannot be reliably measured in such a situation. We assess the importance of rate of data collection and show how to test different models, such as topological and metric neighbourhood models. Taken together our results both inform the design of experiments on animal interactions and suggest how these data should be best analysed.