REPETITIVE SHALLOW DIVES POSE DECOMPRESSION RISK IN DEEP-DIVING BEAKED WHALES

The impact of naval sonar on beaked whales is of increasing concern. In recent years the presence of gas and fat embolism consistent with decompression sickness (DCS) has been reported through postmortem analyses on beaked whales that stranded in connection with naval sonar exercises. In the present study, we use basic principles of diving physiology to model nitrogen tension and bubble growth in several tissue compartments during normal diving behavior and for several hypothetical dive profiles to assess the risk of DCS. Assuming that normal diving does not cause nitrogen tensions in excess of those shown to be safe for odontocetes, the modeling indicates that repetitive shallow dives, perhaps as a consequence of an extended avoidance reaction to sonar sound, can indeed pose a risk for DCS and that this risk should increase with the duration of the response. If the model is correct, then limiting the duration of sonar exposure to minimize the duration of any avoidance reaction therefore has the potential to reduce the risk of DCS.     

Mathematical model of gas bubble evolution in a straight tube.

Deep sea divers suffer from decompression sickness (DCS) when their rate of ascent to the surface is too rapid. When the ambient pressure drops, inert gas bubbles may form in blood vessels and tissues. The evolution of a gas bubble in a rigid tube filled with slowly moving fluid, intended to simulate a bubble in a blood vessel, is studied by solving a coupled system of fluid-flow and gas transport equations. The governing equations for the fluid motion are solved using two techniques: an analytical method appropriate for small nondeformable spherical bubbles, and the boundary element method for deformable bubbles of arbitrary size, given an applied steady flow rate. A steady convection-diffusion equation is then solved numerically to determine the concentration of gas. The bubble volume, or equivalently the gas mass inside the bubble for a constant bubble pressure, is adjusted over time according to the mass flux at the bubble surface. Using a quasi-steady approximation, the evolution of a gas bubble in a tube is obtained. Results show that convection increases the gas pressure gradient at the bubble surface, hence increasing the rate of bubble evolution. Comparing with the result for a single gas bubble in an infinite tissue, the rate of evolution in a tube is approximately twice as fast. Surface tension is also shown to have a significant effect. These findings may have important implications for our understanding of the mechanisms of inert gas bubbles in the circulation underlying decompression sickness.     

A Likelihood Approach to Populations Samples of Microsatellite Alleles

This paper presents a likelihood approach to population samples of microsatellite alleles. A Markov chain recursion method previously published by GRIFFITHS and TAVARE is applied to estimate the likelihood function under different models of microsatellite evolution. The method presented can be applied to estimate a fundamental population genetics parameter θ as well as parameters of the mutational model. The new likelihood estimator provides a better estimator of θ in terms of the mean square error than previous approaches. Furthermore, it is demonstrated how the method may easily be applied to test models of microsatellite evolution. In particular it is shown how to compare a one-step model of microsatellite evolution to a multi-step model by a likelihood ratio test.     

Predicting risk of decompression sickness in humans from outcomes in sheep.

In animals, the response to decompression scales as a power of species body mass. Consequently, decompression sickness (DCS) risk in humans should be well predicted from an animal model with a body mass comparable to humans. No-stop decompression outcomes in compressed air and nitrogen-oxygen dives with sheep (n = 394 dives, 14.5% DCS) and humans (n = 463 dives, 4.5% DCS) were used with linear-exponential, probabilistic modeling to test this hypothesis. Scaling the response parameters of this model between species (without accounting for body mass), while estimating tissue-compartment kinetic parameters from combined human and sheep data, predicts combined risk better, based on log likelihood, than do separate sheep and human models, a combined model without scaling, and a kinetic-scaled model. These findings provide a practical tool for estimating DCS risk in humans from outcomes in sheep, especially in decompression profiles too risky to test with humans. This model supports the hypothesis that species of similar body mass have similar DCS risk.     

An ensemble of models of the acute inflammatory response to bacterial lipopolysaccharide in rats: results from parameter space reduction

In previous work, we developed an 8-state nonlinear dynamic model of the acute inflammatory response, including activated phagocytic cells, pro- and anti-inflammatory cytokines, and tissue damage, and calibrated it to data on cytokines from endotoxemic rats. In the interest of parsimony, the present work employed parametric sensitivity and local identifiability analysis to establish a core set of parameters predominantly responsible for variability in model solutions. Parameter optimization, facilitated by varying only those parameters belonging to this core set, was used to identify an ensemble of parameter vectors, each representing an acceptable local optimum in terms of fit to experimental data. Individual models within this ensemble, characterized by their different parameter values, showed similar cytokine but diverse tissue damage behavior. A cluster analysis of the ensemble of models showed the existence of a continuum of acceptable models, characterized by compensatory mechanisms and parameter changes. We calculated the direct correlations between the core set of model parameters and identified three mechanisms responsible for the conversion of the diverse damage time courses to similar cytokine behavior in these models. Given that tissue damage level could be an indicator of the likelihood of mortality, our findings suggest that similar cytokine dynamics could be associated with very different mortality outcomes, depending on the balance of certain inflammatory elements.     

Specifics of Gas Bubble Formation and Growth in Tissues during Decompression after Saturation Dives

It is shown that the decompression schedules after saturation diving to the depth of 30 m designed to hold the nitrogen supersaturation for the most “slow” tissues at the acceptable levels is significantly shorter than the decompression schedules with zero supersaturation of these tissues with nitrogen and all dissolved gases. Equality of the risk for decompression sickness (DCS) onset during this decompression schedule to the risk of DCS onset under non-stop ascent to the surface after saturation diving to the depth of 6.1 m indicates that the effect of the high ambient pressure decreases the density of gas bubble seeds in tissues and the growth rate of their total volume. The DCS symptoms in the experienced divers under dangerous decompression profiles not appear due to the lower density of gas bubble seeds in their tissues relatively to the average level inherent to the many of humans.     

Predicting the time of occurrence of decompression sickness.

Probabilistic models and maximum likelihood estimation have been used to predict the occurrence of decompression sickness (DCS). We indicate a means of extending the maximum likelihood parameter estimation procedure to make use of knowledge of the time at which DCS occurs. Two models were compared in fitting a data set of nearly 1,000 exposures, in which greater than 50 cases of DCS have known times of symptom onset. The additional information provided by the time at which DCS occurred gave us better estimates of model parameters. It was also possible to discriminate between good models, which predict both the occurrence of DCS and the time at which symptoms occur, and poorer models, which may predict only the overall occurrence. The refined models may be useful in new applications for customizing decompression strategies during complex dives involving various times at several different depths. Conditional probabilities of DCS for such dives may be reckoned as the dive is taking place and the decompression strategy adjusted to circumstance. Some of the mechanistic implications and the assumptions needed for safe application of decompression strategies on the basis of conditional probabilities are discussed.     

Estimating Mass Properties of Dinosaurs Using Laser Imaging and 3D Computer Modelling

Body mass reconstructions of extinct vertebrates are most robust when complete to near-complete skeletons allow the reconstruction of either physical or digital models. Digital models are most efficient in terms of time and cost, and provide the facility to infinitely modify model properties non-destructively, such that sensitivity analyses can be conducted to quantify the effect of the many unknown parameters involved in reconstructions of extinct animals. In this study we use laser scanning (LiDAR) and computer modelling methods to create a range of 3D mass models of five specimens of non-avian dinosaur; two near-complete specimens of Tyrannosaurus rex, the most complete specimens of Acrocanthosaurus atokensis and Strutiomimum sedens, and a near-complete skeleton of a sub-adult Edmontosaurus annectens. LiDAR scanning allows a full mounted skeleton to be imaged resulting in a detailed 3D model in which each bone retains its spatial position and articulation. This provides a high resolution skeletal framework around which the body cavity and internal organs such as lungs and air sacs can be reconstructed. This has allowed calculation of body segment masses, centres of mass and moments or inertia for each animal. However, any soft tissue reconstruction of an extinct taxon inevitably represents a best estimate model with an unknown level of accuracy. We have therefore conducted an extensive sensitivity analysis in which the volumes of body segments and respiratory organs were varied in an attempt to constrain the likely maximum plausible range of mass parameters for each animal. Our results provide wide ranges in actual mass and inertial values, emphasizing the high level of uncertainty inevitable in such reconstructions. However, our sensitivity analysis consistently places the centre of mass well below and in front of hip joint in each animal, regardless of the chosen combination of body and respiratory structure volumes. These results emphasize that future biomechanical assessments of extinct taxa should be preceded by a detailed investigation of the plausible range of mass properties, in which sensitivity analyses are used to identify a suite of possible values to be tested as inputs in analytical models.     

Periodicity in an epidemic model with a generalized non-linear incidence

We develop and analyze a simple SIV epidemic model including susceptible, infected and perfectly vaccinated classes, with a generalized non-linear incidence rate subject only to a few general conditions. These conditions are satisfied by many models appearing in the literature. The detailed dynamics analysis of the model, using the Poincaré index theory, shows that non-linearity of the incidence rate leads to vital dynamics, such as bistability and periodicity, without seasonal forcing or being cyclic. Furthermore, it is shown that the basic reproductive number is independent of the functional form of the non-linear incidence rate. Under certain, well-defined conditions, the model undergoes a Hopf bifurcation. Using the normal form of the model, the first Lyapunov coefficient is computed to determine the various types of Hopf bifurcation the model undergoes. These general results are applied to two examples: unbounded and saturated contact rates; in both cases, forward or backward Hopf bifurcations occur for two distinct values of the contact parameter. It is also shown that the model may undergo a subcritical Hopf bifurcation leading to the appearance of two concentric limit cycles. The results are illustrated by numerical simulations with realistic model parameters estimated for some infectious diseases of childhood.     

Bridging the gap between structural and lattice models: a parameterization of energy transfer and trapping in Photosystem I

In the absence of an accurate structural model, the excited state dynamics of energy-transferring systems are often modeled using lattice models. To demonstrate the validity and other potential merits of such an approach we present the results of the modeling of the energy transfer and trapping in Photosystem I based upon the 2.5 A structural model, and show that these results can be reproduced in terms of a lattice model with only a few parameters. It has recently been shown that at room temperature the dynamics of a hypothetical Photosystem I particle, not containing any red chlorophylls (chls), are characterized by a longest (trapping) lifetime of 18 ps. The structure-based modeling of the dynamics of this particle yields an almost linear relationship between the possible values of the intrinsic charge-separation time at P700, 1/gamma, and the average single-site lifetime in the antenna, tauss. Lattice-based modeling, using the approach of a perturbed two-level model, reproduces this linear relation between tauss and 1/gamma. Moreover, this approach results in a value of the (modified) structure-function corresponding to a structure exhibiting a mixture of the characteristics of both a square and a cubic lattice, consistent with the structural model. These findings demonstrate that the lattice model describes the dynamics of the system appropriately. In the lattice model, the total trapping time is the sum of the delivery time to the reaction center and the time needed to quench the excitation after delivery. For the literature value of tauss=150 fs, both these times contribute almost equally to the total trapping time of 18 ps, indicating that the system is neither transfer- nor trap-limited. The value of approximately 9 ps for the delivery time is basically equal to the excitation-transfer time from the bulk chls to the red chls in Synechococcus elongatus, indicating that energy transfer from the bulk to the reaction center and to the red chls are competing processes. These results are consistent with low-temperature time-resolved and steady-state fluorescence measurements. We conclude that lattice models can be used to describe the global energy-transfer properties in complex chromophore networks, with the advantage that such models deal with only a few global, intuitive parameters rather than the many microscopic parameters obtained in structure-based modeling.     

Rapid evolution of a consumer stoichiometric trait destabilizes consumer–producer dynamics

Recent studies have shown that adaptive evolution can be rapid enough to affect contemporary ecological dynamics in nature (i.e. ‘rapid evolution’). These studies tend to focus on trait functions relating to interspecific interactions; however, the importance of rapid evolution of stoichiometric traits has been relatively overlooked. Various traits can affect the balance of elements (carbon, nitrogen, and phosphorus) of organisms, and rapid evolution of such stoichiometric traits will not only alter population and community dynamics but also influence ecosystem functions such as nutrient cycling. Multiple environmental changes may exert a selection pressure leading to adaptation of stoichiometrically important traits, such as an organism's growth rate. In this paper, we use theoretical approaches to explore the connections between rapid evolution and ecological stoichiometry at both the population and ecosystem level. First, we incorporate rapid evolution into an ecological stoichiometry model to investigate the effects of rapid evolution of a consumer's stoichiometric phosphorus:carbon ratio on consumer–producer population dynamics. We took two complementary approaches, an asexual clonal genotype model and a quantitative genetic model. Next, we extended these models to explicitly track nutrients in order to evaluate the effect of rapid evolution at the ecosystem level. Our model results indicate rapid evolution of the consumer stoichiometric trait can cause complex dynamics where rapid evolution destabilizes population dynamics and rescues the consumer population from extinction (evolutionary rescue). The model results also show that rapid evolution may influence the level of nutrients available in the environment and the flux of nutrients across trophic levels. Our study represents an important step for theoretical integration of rapid evolution and ecological stoichiometry.     

Molecular Dynamic Simulation of Transmembrane Pore Growth

A molecular dynamic approach was applied for simulation of dynamics of pore formation and growth in a phospholipid bilayer in the presence of an external electric field. Processing the simulation results permitted recovery of the kinetic coefficients used in the Einstein–Smoluchowski equation describing the dynamics of pore evolution. Two different models of the bilayer membrane were considered: membrane consisting of POPC and POPE lipids. The simulations permitted us to find nonempirical values of the pore energy parameters, which are compared with empirical values. It was found that the parameters are sensitive to membrane type.     

Branching tree model with fractal vascular resistance explains fractal perfusion heterogeneity

Perfusion heterogeneities in organs such as the heart obey a power law as a function of scale, a behavior termed "fractal." An explanation of why vascular systems produce such a specific perfusion pattern is still lacking. An intuitive branching tree model is presented that reveals how this behavior can be generated as a consequence of scale-independent branching asymmetry and fractal vessel resistance. Comparison of computer simulations to experimental data from the sheep heart shows that the values of the two free model parameters are realistic. Branching asymmetry within the model is defined by the relative tissue volume being fed by each branch. Vessel ordering for fractal analysis of morphology based on fed or drained tissue volumes is preferable to the commonly used Strahler system, which is shown to depend on branching asymmetry. Recently, noninvasive imaging techniques such as PET and MRI have been used to measure perfusion heterogeneity. The model allows a physiological interpretation of the measured fractal parameters, which could in turn be used to characterize vascular morphology and function.     

Periodic dynamics in a two-stage Allee effect model are driven by tension between stage equilibria

Single species difference population models can show complex dynamics such as periodicity and chaos under certain circumstances, but usually only when rates of intrinsic population growth or other life history parameter are unrealistically high. Single species models with Allee effects (positive density dependence at low density) have also been shown to exhibit complex dynamics when combined with over-compensatory density dependence or a narrow fertility window. Here we present a simple two-stage model with Allee effects which shows large amplitude periodic fluctuations for some initial conditions, without these requirements. Periodicity arises out of a tension between the critical equilibrium of each stage, i.e. when the initial population vector is such that the adult stage is above the critical value, while the juvenile stage is below the critical value. Within this area of parameter space, the range of initial conditions giving rise to periodic dynamics is driven mainly by adult mortality rates. Periodic dynamics become more important as adult mortality increases up to a certain point, after which periodic dynamics are replaced by extinction. This model has more realistic life history parameter values than most 'chaotic' models. Conditions for periodic dynamics might arise in some marine species which are exploited (high adult mortality) leading to recruitment limitation (low juvenile density) and might be an additional source of extinction risk.     

Insights into the behaviour of systems biology models from dynamic sensitivity and identifiability analysis: a case study of an NF-kappaB signalling pathway

Mathematical modelling offers a variety of useful techniques to help in understanding the intrinsic behaviour of complex signal transduction networks. From the system engineering point of view, the dynamics of metabolic and signal transduction models can always be described by nonlinear ordinary differential equations (ODEs) following mass balance principles. Based on the state-space formulation, many methods from the area of automatic control can conveniently be applied to the modelling, analysis and design of cell networks. In the present study, dynamic sensitivity analysis is performed on a model of the IkappaB-NF-kappaB signal pathway system. Univariate analysis of the Euclidean-form overall sensitivities shows that only 8 out of the 64 parameters in the model have major influence on the nuclear NF-kappaB oscillations. The sensitivity matrix is then used to address correlation analysis, identifiability assessment and measurement set selection within the framework of least squares estimation and multivariate analysis. It is shown that certain pairs of parameters are exactly or highly correlated to each other in terms of their effects on the measured variables. The experimental design strategy provides guidance on which proteins should best be considered for measurement such that the unknown parameters can be estimated with the best statistical precision. The whole analysis scheme we describe provides efficient parameter estimation techniques for complex cell networks.     

ON THE EVOLUTION OF PHENOTYPIC PLASTICITY IN A SPATIALLY HETEROGENEOUS ENVIRONMENT

A genetic model for the dynamics of a quantitative trait is analyzed in terms of gene frequencies, linkage disequilibria, and environmental effects on the trait. In a randomly mating population, at each generation progeny move to niches where they are subject to weak Gaussian selection on the trait, with different fitness levels in the different niches. Initially, the variability of the trait is due to additive loci with heterozygous homeostasis. The evolution of plasticity is then described in terms of the invasion of the population by genetic modifiers that may epistatically affect the trait, its optimum in each niche, the strengths of selection, and other parameters characteristic of the niches. We show that the evolution of trait means within niches depends on the overall evolution in the whole system, and in general, optimum phenotypic values are not attained. The reaction norm and genotype-environment interaction may evolve even if the only effects of the modifier are on individual rates of dispersal, or on fitness effects resulting from the different environments in the different niches; this evolution does not require that the modifier affect parameters that influence the values of the trait. It is conjectured that in the least frequently reached niches with low fitness levels, the deviations from the trait optima should be larger than those in more commonly experienced and less stringent niches. Our analysis makes explicit the different contribution of between- and within-niche effects on the evolutionary dynamics of phenotypic plasticity in heterogeneous environments.     

Lattice Boltzmann simulations of the time evolution of living multicellular systems

Embryonic tissues and multicellular aggregates of adult cells mimic the behavior of highly viscous liquids. The liquid analogy helps to understand morphogenetic phenomena, such as cell sorting and tissue fusion, observed in developmental biology and tissue engineering. Tissue fusion is vital in tissue printing, an emergent technique based on computer-controlled deposition of tissue fragments and biocompatible materials. Computer simulations proved useful in predicting post-printing shape changes of tissue constructs. The simulation methods available to date, however, are unable to describe the time evolution of living systems made of millions of cells. The Lattice Boltzmann (LB) approach allows the implementation of interaction forces between the constituents of the system and yields time evolution in terms of distribution functions. With tissue engineering applications in mind, we have developed a finite difference Lattice Boltzmann model of a multicellular system and applied it to simulate the sidewise fusion of two contiguous cylinders made of cohesive cells and embedded in a medium (hydrogel). We have identified a biologically relevant range of model parameters. The proposed LB model may be extended to describe the time evolution of more complex multicellular structures such as sheets or tubes produced by tissue printing.     

Understanding the overdispersed molecular clock

Rates of molecular evolution at some protein-encoding loci are more irregular than expected under a simple neutral model of molecular evolution. This pattern of excessive irregularity in protein substitutions is often called the "overdispersed molecular clock" and is characterized by an index of dispersion, R(T) > 1. Assuming infinite sites, no recombination model of the gene R(T) is given for a general stationary model of molecular evolution. R(T) is shown to be affected by only three things: fluctuations that occur on a very slow time scale, advantageous or deleterious mutations, and interactions between mutations. In the absence of interactions, advantageous mutations are shown to lower R(T); deleterious mutations are shown to raise it. Previously described models for the overdispersed molecular clock are analyzed in terms of this work as are a few very simple new models. A model of deleterious mutations is shown to be sufficient to explain the observed values of R(T). Our current best estimates of R(T) suggest that either most mutations are deleterious or some key population parameter changes on a very slow time scale. No other interpretations seem plausible. Finally, a comment is made on how R(T) might be used to distinguish selective sweeps from background selection.     

Stability in simple grazing models: effects of explicit functions

A class of simple models of a grazed pasture is based on the assumption that the dynamics of green biomass can be expressed as a balance between rates of growth and consumption, both of which are functions only of the total amount of green biomass present. Graphical analysis, using only the general shape of these functions, has shown that the pasture may be either continuously or discontinuously stable under grazing. Further results require explicit specification of these functions. Four growth functions and four consumption functions are considered here and their mathematical and biological properties discussed. The stability properties of the 16 models resulting from combinations of these functions are compared by plotting the isoclines of zero pasture growth, and by obtaining expressions in terms of the parameters for some critical points and quantities. Both methods show considerable similarity between the models in their qualitative behaviour and in the role of certain parameters in determining critical values. There are substantial differences in quantitative predictions of critical values and particularly of the boundary between continuous and discontinuous stability. In the absence of an ungrazable residual, some models are discontinuously stable in the whole parameter space, others only in a limited region of it-albeit the region in which most real systems may be.