A model with asymptotically stable dynamics for the network of Drosophila gap genes

A model of the expression of gap genes during the early development of Drosophila embryo has been studied. The parameter values for the model have been obtained by fitting the solutions to experimental patterns under an additional assumption of the asymptotic stability of solutions at large times. The patterns at the beginning of gastrulation in these solutions are very close to the actual attractor in the model. It is shown that these solutions have a better stability, or robustness, to perturbations of concentrations and parameter values in the model.     

A unified approach to the Richards-model family for use in growth analyses: Why we need only two model forms

This paper advances a unified approach to the modeling of sigmoid organismal growth. There are numerous studies on growth, and there have been several proposals and applications of candidate models. Still, a lack of interpretation of the parameter values persists and, consequently, differences in growth patterns have riddled this field. A candidate regression model as a tool should be able to assess and compare growth-curve shapes, systematically and precisely. The Richards models constitute a useful family of growth models that amongst a multitude of parameterizations, re-parameterizations and special cases, include familiar models such as the negative exponential, the logistic, the Bertalanffy and the Gompertz. We have reviewed and systemized this family of models. We demonstrate that two specific parameterizations (or re-parameterizations) of the Richards model are able to substitute, and thus to unify all other forms and models. This unified-Richards model (with its two forms) constitutes a powerful tool for an interpretation of important characteristics of observed growth patterns, namely, [I] maximum (relative) growth rate (i.e., slope at inflection), [II] age at maximum growth rate (i.e., time at inflection), [III] relative mass or length at maximum growth rate (i.e., relative value at an inflection), [IV] value at age zero (i.e., birth, hatching or germination), and [V] asymptotic value (i.e., adult weight or length). These five parameters can characterize uniquely any sigmoid-growth data. To date most studies only compare what is referred to as the “growth-rate constant” or simply “growth rate” (k). This parameter can be interpreted as neither relative nor actual growth rate, but only as a parameter that affects the slope at inflection. We fitted the unified-Richards and five other candidate models to six artificial data sets, generated from the same models, and made a comparison based on the corrected Akaike’s Information Criterion (AICc). The outcome may in part be the result of the random generation of data points. Still, in conclusion, the unified-Richards model performed consistently well for all data sets, despite the penalty imposed by the AICc.     

Assessment of uncertainty in functional-structural plant models

Background and Aims

Constructing functional–structural plant models (FSPMs) is a valuable method for examining how physiology and morphology interact in determining plant processes. However, such models always have uncertainty concerned with whether model components have been selected and represented effectively, with the number of model outputs simulated and with the quality of data used in assessment. We provide a procedure for defining uncertainty of an FSPM and how this uncertainty can be reduced.

Methods

An important characteristic of FSPMs is that typically they calculate many variables. These can be variables that the model is designed to predict and also variables that give indications of how the model functions. Together these variables are used as criteria in a method of multi-criteria assessment. Expected ranges are defined and an evolutionary computation algorithm searches for model parameters that achieve criteria within these ranges. Typically, different combinations of model parameter values provide solutions achieving different combinations of variables within their specified ranges. We show how these solutions define a Pareto Frontier that can inform about the functioning of the model.

Key Results

The method of multi-criteria assessment is applied to development of BRANCHPRO, an FSPM for foliage reiteration on old-growth branches of Pseudotsuga menziesii. A geometric model utilizing probabilities for bud growth is developed into a causal explanation for the pattern of reiteration found on these branches and how this pattern may contribute to the longevity of this species.

Conclusions

FSPMs should be assessed by their ability to simulate multiple criteria simultaneously. When different combinations of parameter values achieve different groups of assessment criteria effectively a Pareto Frontier can be calculated and used to define the sources of model uncertainty.     

On the choice of outlet boundary conditions for patient-specific analysis of aortic flow using computational fluid dynamics

Boundary conditions (BCs) are an essential part in computational fluid dynamics (CFD) simulations of blood flow in large arteries. Although several studies have investigated the influence of BCs on predicted flow patterns and hemodynamic wall parameters in various arterial models, there is a lack of comprehensive assessment of outlet BCs for patient-specific analysis of aortic flow. In this study, five different sets of outlet BCs were tested and compared using a subject-specific model of a normal aorta. Phase-contrast magnetic resonance imaging (PC-MRI) was performed on the same subject and velocity profiles extracted from the in vivo measurements were used as the inlet boundary condition. Computational results obtained with different outlet BCs were assessed in terms of their agreement with the PC-MRI velocity data and key hemodynamic parameters, such as pressure and flow waveforms and wall shear stress related indices. Our results showed that the best overall performance was achieved by using a well-tuned three-element Windkessel model at all model outlets, which not only gave a good agreement with in vivo flow data, but also produced physiological pressure waveforms and values. On the other hand, opening outlet BCs with zero pressure at multiple outlets failed to reproduce any physiologically relevant flow and pressure features.     

Effect of landscape heterogeneity on termite tunnel pattern: Simulation study

Although the environment in which termites live is very heterogeneous, most experimental studies on the termite tunnel patterns have been conducted on homogeneous sand substrates. In order to explore how the heterogeneity affects tunnel patterns, I developed an agent-based model to simulate termite tunneling behavior at the individual level. In this model, grid space consists of easy and difficult areas for tunneling. Heterogeneity, H, was defined as the degree of the mixture of the two areas. The tunnel patterns formed by changing the number of termites, N, and H were quantitatively characterized by territory circularity and the territory area. These patterns were categorized into two groups, one with a small territory area and high circularity (group 1) and the other with a large area and low circularity (group 2). Considering the previous study that the termite populations with high N values have high territorial scalability, it can be said that the territories belonging to group 2 have higher foraging abilities and viability than those belonging to group 1. The simulation results showed that the tunnel patterns generated for small N and high H belonged to group 2. This implies that the heterogeneity can make a positive contribution to the expansion of the foraging area by effectively focusing the foraging energy of a termite population. I briefly discussed the mechanism of this positive role and the limitations of this simulation study. In addition, I discussed issues that need to be resolved in the near future to overcome the limitations.     

Time-dependent solutions of the spatially implicit neutral model of biodiversity

Previous research into the neutral theory of biodiversity has focused mainly on equilibrium solutions rather than time-dependent solutions. Understanding the time-dependent solutions is essential for applying neutral theory to ecosystems in which time-dependent processes, such as succession and invasion, are driving the dynamics. Time-dependent solutions also facilitate tests against data that are stronger than those based on static equilibrium patterns. Here I investigate the time-dependent solutions of the classic spatially implicit neutral model, in which a small local community is coupled to a much larger metacommunity through immigration. I present explicit general formulas for the eigenvalues, left eigenvectors and right eigenvectors of the models’s transition matrix. The time-dependent solutions can then be expressed in terms of these eigenvalues and eigenvectors. Some of these results are translated directly from existing results for the classic Moran model of population genetics (the Moran model is equivalent to the spatially implicit neutral model after a reparameterization); others of the results are new. I demonstrate that the asymptotic time-dependent solution corresponding to just these first two eigenvectors can be a good approximation to the full time-dependent solution. I also demonstrate the feasibility of a partial eigendecomposition of the transition matrix, which facilitates direct application of the results to a biologically relevant example in which a newly invading species is initially present in the metacommunity but absent from the local community.     

Co-existent activity patterns in inhibitory neuronal networks with short-term synaptic depression

A network of two neurons mutually coupled through inhibitory synapses that display short-term synaptic depression is considered. We show that synaptic depression expands the number of possible activity patterns that the network can display and allows for co-existence of different patterns. Specifically, the network supports different types of n-m anti-phase firing patterns, where one neuron fires n spikes followed by the other neuron firing m spikes. When maximal synaptic conductances are identical, n-n anti-phase firing patterns are obtained and there are conductance intervals over which different pairs of these solutions co-exist. The multitude of n-m anti-phase patterns and their co-existence are not found when the synapses are non-depressing. Geometric singular perturbation methods for dynamical systems are applied to the original eight-dimensional model system to derive a set of one-dimensional conditions for the existence and co-existence of different anti-phase solutions. The generality and validity of these conditions are demonstrated through numerical simulations utilizing the Hodgkin-Huxley and Morris-Lecar neuronal models.     

Effects of density dependent sex allocation on the dynamics of a simultaneous hermaphroditic population: Modelling and analysis

In this work we present a mathematical model describing the dynamics of a population where sex allocation remains flexible throughout adult life and so can be adjusted to current environmental conditions. We consider that the fractions of immature individuals acquiring male and female sexual roles are density dependent through nonlinear functions of a weighted total population size. The main goal of this work is to understand the role of life-history parameters on the stabilization or destabilization of the population dynamics.The model turns out to be a nonlinear discrete model which is analysed by studying the existence of fixed points as well as their stability conditions in terms of model parameters. The existence of more complex asymptotic behaviours of system solutions is shown by means of numerical simulations.Females have larger fertility rate than males. On the other hand, increasing population density favours immature individuals adopting the male role. A positive equilibrium of the system exists whenever fertility and survival rates of one of the sexual roles, if shared by all adults, allow population growing while the opposite happens with the other sexual role. In terms of the female inherent net reproductive number, η_F, it is shown that the positive equilibria are stable when η_F is larger and closed to 1 while for larger values of η_F a certain asymptotic assumption on the investment rate in the female function implies that the population density is permanent. Depending on the other parameters values, the asymptotic behaviour of solutions becomes more complex, even chaotic. In this setting the stabilization/destabilization effects of the abruptness rate in density dependence, of the survival rates and of the competition coefficients are analysed.     

Collective colony growth is optimized by branching pattern formation in Pseudomonas aeruginosa

Branching pattern formation is common in many microbes. Extensive studies have focused on addressing how such patterns emerge from local cell–cell and cell–environment interactions. However, little is known about whether and to what extent these patterns play a physiological role. Here, we consider the colonization of bacteria as an optimization problem to find the colony patterns that maximize colony growth efficiency under different environmental conditions. We demonstrate that Pseudomonas aeruginosa colonies develop branching patterns with characteristics comparable to the prediction of modeling; for example, colonies form thin branches in a nutrient‐poor environment. Hence, the formation of branching patterns represents an optimal strategy for the growth of Pseudomonas aeruginosa colonies. The quantitative relationship between colony patterns and growth conditions enables us to develop a coarse‐grained model to predict diverse colony patterns under more complex conditions, which we validated experimentally. Our results offer new insights into branching pattern formation as a problem‐solving social behavior in microbes and enable fast and accurate predictions of complex spatial patterns in branching colonies.     

Modelling the effect of temperature variation on the seasonal dynamics of Ixodes ricinus tick populations

Seasonal variation in temperature is known to drive annual patterns of tick activity and can influence the dynamics of tick-borne diseases. An age-structured model of the dynamics of Ixodes ricinus populations was developed to explore how changes in average temperature and different levels of temperature variability affect seasonal patterns of tick activity and the transmission of tick-borne diseases. The model produced seasonal patterns of tick emergence that are consistent with those observed throughout Great Britain. Varying average temperature across a continuous spectrum produced a systematic pattern in the times of peak emergence of questing ticks which depends on cumulative temperature over the year. Examination of the effects of between-year stochastic temperature variation on this pattern indicated that peak emergence times are more strongly affected by temperature stochasticity at certain levels of average temperature. Finally the model was extended to give a simple representation of the dynamics of a tick-borne disease. A threshold level of annual cumulative temperature was identified at which disease persistence is sensitive to stochastic temperature variation. In conclusion, the effect of changing patterns of temperature variation on the dynamics of I. ricinus ticks and the diseases they transmit may depend on the cumulative temperature over the year and will therefore vary across different locations. The results also indicate that diapause mechanisms have an important influence on seasonal patterns of tick activity and require further study.     

The evolution of resource partitioning in a multidimensional resource space

The evolution of resource partitioning in a multidimensional resource space is studied for two and three competing species. Optimal patterns of resource partitioning are determined by simultaneously maximizing the fitness of each species with respect to its own niche position, conditional on the positions of all other species. We find that there are only a finite number of possible solutions, and several of these may be optimal simultaneously. Some solutions of the three-species model involve partitioning along more resource axes than any solutions of the two-species model. The results are related to empirical resource partitioning phenomena in Anolis lizard populations.     

Adjust quality scores from alignment and improve sequencing accuracy

In shotgun sequencing, statistical reconstruction of a consensus from alignment requires a model of measurement error. Churchill and Waterman proposed one such model and an expectation–maximization (EM) algorithm to estimate sequencing error rates for each assembly matrix. Ewing and Green defined Phred quality scores for base-calling from sequencing traces by training a model on a large amount of data. However, sample preparations and sequencing machines may work under different conditions in practice and therefore quality scores need to be adjusted. Moreover, the information given by quality scores is incomplete in the sense that they do not describe error patterns. We observe that each nucleotide base has its specific error pattern that varies across the range of quality values. We develop models of measurement error for shotgun sequencing by combining the two perspectives above. We propose a logistic model taking quality scores as covariates. The model is trained by a procedure combining an EM algorithm and model selection techniques. The training results in calibration of quality values and leads to a more accurate construction of consensus. Besides Phred scores obtained from ABI sequencers, we apply the same technique to calibrate quality values that come along with Beckman sequencers.     

Dynamic phenomena arising from an extended Core Group model

In order to obtain a reasonably accurate model for the spread of a particular infectious disease through a population, it may be necessary for this model to possess some degree of structural complexity. Many such models have, in recent years, been found to exhibit a phenomenon known as backward bifurcation, which generally implies the existence of two subcritical endemic equilibria. It is often possible to refine these models yet further, and we investigate here the influence such a refinement may have on the dynamic behaviour of a system in the region of the parameter space near R₀=1.We consider a natural extension to a so-called Core Group model for the spread of a sexually transmitted disease, arguing that this may in fact give rise to a more realistic model. From the deterministic viewpoint we study the possible shapes of the resulting bifurcation diagrams and the associated stability patterns. Stochastic versions of both the original and the extended models are also developed so that the probability of extinction and time to extinction may be examined, allowing us to gain further insights into the complex system dynamics near R₀=1. A number of interesting phenomena are observed, for which heuristic explanations are provided.     

Diffusion mechanism for phyllotaxis: theoretical physico-chemical and computer study

The mechanism for leaf position determination by the diffusion of an inhibitor has been studied in relation to the geometry of leaf positions. A computer model has been constructed for the inhibitor-diffusion process on a cellular cylindrical surface. The behavior of the model has been analyzed mathematically. The main results are: (a) that our model generates most of the phyllotactic patterns observed in nature; and (b) that restraints have been found for permissible values of diffusion rates and decay rates of the hypothetical inhibitor.     

Feedback models allowing estimation of thresholds for self-promoting body weight gain

Objective: Most people maintain almost constant body weight over long time with varying physical activity and food intake. This indicates the existence of a regulation that works well for most individuals. Yet some people develop obesity, indicating that this regulation sometimes fails. The difference between the two situations is typically an energy imbalance of about 1% over a long period of time.Theory: Weight gain increases basal metabolic rate. Weight gain is often associated with a decrease in physical activity, although not to such an extent that it prevents an increase in total energy expenditure and energy intake. Dependent on the precise balance between these effects of weight gain, they may make the body weight unstable and tend to further promote weight gain. With the aim of identifying the thresholds beyond which such self-promoting weight gain may take place, we develop a simple mathematical model of the body as an energy-consuming machine in which the changes in physical activity and food intake are described as feedback effects in addition to the effect of the weight gain on basal metabolic rate. The feedback parameters of the model may differ between individuals and only in some cases do they take values that make weight gain self-promoting.Results: We determine the quantitative conditions under which body weight gain becomes self-promoting. We find that these conditions can easily be met, and that they are so small that they are not observable with currently available techniques. This phenomenon encourages emphasis on even minor changes in food intake and physical activity to abate or stop weight gain.     

The mono-exponential pattern of phosphocreatine recovery after muscle exercise is a particular case of a more complex behaviour

A mathematical model is proposed showing that the mono-exponential recovery of phosphocreatine (PCr) after exercise is an approximation of a more complex pattern, which is identified by a second-order differential equation. The model predicts the possibility of three different patterns of PCr recovery: bi-exponential, oscillatory damped, and critically damped; the mono-exponential pattern being a particular case of the functions which are solutions of the differential equation. The model was tested on a sample of recovery data from 50 volunteers, checking whether the recovery patterns predicted by the model lead to a significant improvement of fit (IF) compared with the mono-exponential pattern. Results show that the IF is linked to pH. Bi-exponential solutions showed an IF in the pH range 6.65-6.85, and the oscillatory solutions at pH>6.9. Critically damped solutions displayed a poor IF. Oscillation frequencies found in the oscillatory recoveries increase at increasing pH. These results show that pH has a pivotal role on the pattern of PCr recovery and implications on the regulation of oxidative phosphorylation are discussed.     

Selecting an appropriate genetic evaluation model for selection in a developing dairy sector

This study aimed to identify genetic evaluation models (GEM) to accurately select cattle for milk production when only limited data are available. It is based on a data set from the Pakistani Sahiwal progeny testing programme which includes records from five government herds, each consisting of 100 to 350 animals, with lactation records dating back to 1968. Different types of GEM were compared, namely: (1) multivariate v. repeatability model when using the first three lactations, (2) an animal v. a sire model, (3) different fixed effects models to account for effects such as herd, year and season; and (4) fitting a model with genetic parameters fixed v. estimating the genetic parameters as part of the model fitting process. Two methods were used for the comparison of models. The first method used simulated data based on the Pakistani progeny testing system and compared estimated breeding values with true breeding values. The second method used cross-validation to determine the best model in subsets of actual Australian herd-recorded data. Subsets were chosen to reflect the Pakistani data in terms of herd size and number of herds. Based on the simulation and the cross-validation method, the multivariate animal model using fixed genetic parameters was generally the superior GEM, but problems arise in determining suitable values for fixing the parameters. Using mean square error of prediction, the best fixed effects structure could not be conclusively determined. The simulation method indicated the simplest fixed effects structure to be superior whereas in contrast, the cross-validation method on actual data concluded that the most complex one was the best. In conclusion it is difficult to propose a universally best GEM that can be used in any data set of this size. However, some general recommendations are that it is more appropriate to estimate the genetic parameters when evaluating for selection purposes, the animal model was superior to the sire model and that in the Pakistani situation the repeatability model is more suitable than a multivariate.     

Searching for generality in the patterns of parasite abundance and distribution: Ectoparasites of a South African rodent, Rhabdomys pumilio

We studied abundance and distribution of seven ectoparasite species (fleas Chiastopsylla rossi and Dynopsyllus ellobius, a louse Polyplax arvicanthis, mites Androlaelaps fahrenholzi and Laelaps giganteus and two ticks Haemaphysalis elliptica and Hyalomma truncatum) exploiting the same populations of the rodent host Rhabdomys pumilio in South Africa. We considered three general patterns of abundance and distribution, namely (i) aggregated distribution of parasites amongst individual hosts; (ii) positive relationships between mean parasite abundance and their prevalence; and (iii) applicability of a simple epidemiological model based on mean parasite abundance and its variance to predict the observed patterns of prevalence. Our aims were to evaluate the relative role of host- versus parasite-associated factors by looking at similarity amongst different parasites in these patterns. In general, all parasites demonstrated strong similarity in each of the three patterns of abundance and distribution. However, the strength of these patterns differed amongst parasite species. We conclude that these patterns are driven mainly by hosts, but differences are caused by differences between various life-history traits of parasite species. Our results support the idea that general laws apply to parasite population ecology.