Sedimentation Patterns of Rapidly Reversible Protein Interactions

The transport behavior of macromolecular mixtures with rapidly reversible complex formation is of great interest in the study of protein interactions by many different methods. Complicated transport patterns arise even for simple bimolecular reactions, when all species exhibit different migration velocities. Although partial differential equations are available to describe the spatial and temporal evolution of the interacting system given particular initial conditions, a general overview of the phase behavior of the systems in parameter space has not yet been reported. In the case of sedimentation of two-component mixtures, this study presents simple analytical solutions that solve the underlying equations in the diffusion-free limit previously subject to Gilbert-Jenkins theory. The new expressions describe, with high precision, the average sedimentation coefficients and composition of each boundary, which allow the examination of features of the whole parameter space at once, and may be used for experimental design and robust analysis of experimental boundary patterns to derive the stoichiometry and affinity of the complex. This study finds previously unrecognized features, including a phase transition between boundary patterns. The model reveals that the time-average velocities of all components in the reaction mixture must match—a condition that suggests an intuitive physical picture of an effective particle of the coupled cosedimentation of an interacting system. Adding to the existing numerical solutions of the relevant partial differential equations, the effective particle model provides physical insights into the relationships of the parameters that govern sedimentation patterns.     

Linking the allee effect, sexual reproduction, and temperature-dependent sex determination via spatial dynamics

We develop a spatially explicit, two-sex, individual-based model (IBM) and a derived spatially homogeneous model (SHM) to describe the Allee effect due to scarcity of mating possibilities at low population sizes or densities. The SHM, based on coupled difference equations, represents the first spatially homogeneous approach to this phenomenon, which differentiates between sexes and relies only on measurable population parameters. The IBM reinforces the findings of the SHM by adopting more realistic mate search strategies of diffusive movement and active search. Both models are characterized by a hyperbolic-shaped extinction boundary in the male-female state space, which contrasts with a linear boundary in one-dimensional models of the Allee effect. We examine how the position of the extinction boundary depends on population demography (primary sex ratio, reproduction and mortality probabilities) and adopted mate search strategies. The investigation of different phases in the IBM dynamics emphasizes the differences between local and global densities and shows the importance of scale when assessing the Allee effect. To demonstrate the potential application of our models, we combine the SHM and available data to predict the impact of environmental temperature changes on two turtle species with temperature-dependent sex determination.     

Minimum domains for spatial patterns in a class of reaction diffusion equations

We study a general class of scalar reaction/interacting population diffusion equations in two space dimensions: convective terms, due to wind, are included. We consider boundary conditions which include a measure of the hostility to the species in the exterior of the domain. The main point of the paper is to obtain estimates for the minimum domain size which can sustain spatially heterogeneous structures and indicate the type of patterns which could appear.     

Spatial distribution of competing populations

We consider spatial distributions of two competing and diffusing populations whose habitats are partly overlapping. As a model, certain reaction—diffusion equations are used in the finite and in the infinite regions with Dirichlet boundary conditions. On the assumption of extremely different diffusive rates of the two species, it is verified, by the use of singular perturbation techniques, that the slowly diffusing species can survive in some subregions, although the species with the greater diffusive rate rapidly occupies the region at the initial stage, and that coexistence of two populations is realized, reducing the effect of interspecific competition by spatial segregation. It will be also shown that the size of the region where the slowly moving population can survive exhibits a markedly qualitative change, depending on the values of some parameters, and that the population can extends the distribution infinitely, when the parameters satisfy a certain condition.     

Quantum-mechanical theory of CD for infinite helical polymers

Quantum-mechanical equations are derived that are particularly well suited to actual computations of the CD for helical polymers. They make use of cyclic boundary conditions and helical symmetry, so that only two matrices with a size equal to the number of transitions considered need be diagonalized. The final equations are expressed directly in terms of monomer properties and helical parameters to invite the same input as earlier calculations, and are given as a rotational strength times a shape function for ease of comparison with the earlier work. The shape of the helix term is expressed as a derivative with respect to ω and depends on the distance between monomers along the helix axis. Other terms involving two electric transition dipoles depend on the distance from the helix axis to the transition center. These equations are directly comparable to the classical equations derived for cyclic boundary conditions and helical symmetry. We present an outline of the derivation and enough intermediate steps to clarify how the equations arise.     

Dispersal and competition models for plants

New models for seed dispersal and competition between plant species are formulated and analyzed. The models are integrodifference equations, discrete in time and continuous in space, and have applications to annual and perennial species. The spread or invasion of a single plant species into a geographic region is investigated by studying the travelling wave solutions of these equations. Travelling wave solutions are shown to exist in the single-species models and are compared numerically. The asymptotic wave speed is calculated for various parameter values. The single-species integrodifference equations are extended to a model for two competing annual plants. Competition in the two-species model is based on a difference equation model developed by Pakes and Maller [26]. The two-species model with competition and dispersal yields a system of integrodifference equations. The effects of competition on the travelling wave solutions of invading plant species is investigated numerically.     

G-Jitter Induced Magnetohydrodynamics Flow of Nanofluid with Constant Convective Thermal and Solutal Boundary Conditions

Taking into account the effect of constant convective thermal and mass boundary conditions, we present numerical solution of the 2-D laminar g-jitter mixed convective boundary layer flow of water-based nanofluids. The governing transport equations are converted into non-similar equations using suitable transformations, before being solved numerically by an implicit finite difference method with quasi-linearization technique. The skin friction decreases with time, buoyancy ratio, and thermophoresis parameters while it increases with frequency, mixed convection and Brownian motion parameters. Heat transfer rate decreases with time, Brownian motion, thermophoresis and diffusion-convection parameters while it increases with the Reynolds number, frequency, mixed convection, buoyancy ratio and conduction-convection parameters. Mass transfer rate decreases with time, frequency, thermophoresis, conduction-convection parameters while it increases with mixed convection, buoyancy ratio, diffusion-convection and Brownian motion parameters. To the best of our knowledge, this is the first paper on this topic and hence the results are new. We believe that the results will be useful in designing and operating thermal fluids systems for space materials processing. Special cases of the results have been compared with published results and an excellent agreement is found.     

On the cable theory of nerve conduction

Conduction of an impulse in the nonmyelinated nerve fiber is treated quantitatively by considering it as a direct consequence of the coexistence of two structurally distinct regions, resting and active, in the fiber. The profile of the electrical potential change induced in the vicinity of the boundary between the two regions is analyzed by using the cable equations. Simple mathematical formulae relating the conduction velocity to the electrical parameters of the fiber are derived from the symmetry of the potential profile at the boundary. The factors that determine the conduction velocity in the myelinated nerve fiber are reexamined.     

Dynamic aspects of diversity

Abstract. Ecological diversity is viewed as a measure of information, as the variety of messages compounded with the number of individuals belonging to different species. Two limit situations of diversity are indicated: a chemostat-like system tending to become monospecific, and a Noah's ark or museum situation with an infinite number of species each represented by just one specimen. In a dynamic approach, two differential equations for S, the number of species, and N, the number of individuals, can be considered. A new index of diversity is proposed based on these two equations. The concept of diversity spectrum is proposed for a series of diversity measurements in samples of increasing size and related to the concept of minimal area. Diversity can also be measured on non-taxonomic categories, e.g. size classes. The contrast between the two boundary types, limes convergens (ecotone) and limes divergens (ecocline), is emphasized. The nature of the difference between them is compared with that of tension zones between fluids in contact. Differences between ecosystems can be characterized through differences in the ratio P/B (Production/Biomass) and a boundary situation is imagined including two systems with P/B different values and different diversity levels. Some examples are presented from communities in streams.     

Dynamical effects of nonlocal interactions in discrete-time growth-dispersal models with logistic-type nonlinearities

The paper is devoted to the study of discrete time and continuous space models with nonlocal resource competition and periodic boundary conditions. We consider generalizations of logistic and Ricker's equations as intraspecific resource competition models with symmetric nonlocal dispersal and interaction terms. Both interaction and dispersal are modeled using convolution integrals, each of which has a parameter describing the range of nonlocality. It is shown that the spatially homogeneous equilibrium of these models becomes unstable for some kernel functions and parameter values by performing a linear stability analysis. To be able to further analyze the behavior of solutions to the models near the stability boundary, weakly nonlinear analysis, a well-known method for continuous time systems, is employed. We obtain Stuart–Landau type equations and give their parameters in terms of Fourier transforms of the kernels. This analysis allows us to study the change in amplitudes of the solutions with respect to ranges of nonlocalities of two symmetric kernel functions. Our calculations indicate that supercritical bifurcations occur near stability boundary for uniform kernel functions. We also verify these results numerically for both models.     

A parametric method for cumulative incidence modeling with a new four-parameter log-logistic distribution

Background

The invasion of a new species into an established ecosystem can be directly compared to the steps involved in cancer metastasis. Cancer must grow in a primary site, extravasate and survive in the circulation to then intravasate into target organ (invasive species survival in transport). Cancer cells often lay dormant at their metastatic site for a long period of time (lag period for invasive species) before proliferating (invasive spread). Proliferation in the new site has an impact on the target organ microenvironment (ecological impact) and eventually the human host (biosphere impact).

Results

Tilman has described mathematical equations for the competition between invasive species in a structured habitat. These equations were adapted to study the invasion of cancer cells into the bone marrow microenvironment as a structured habitat. A large proportion of solid tumor metastases are bone metastases, known to usurp hematopoietic stem cells (HSC) homing pathways to establish footholds in the bone marrow. This required accounting for the fact that this is the natural home of hematopoietic stem cells and that they already occupy this structured space. The adapted Tilman model of invasion dynamics is especially valuable for modeling the lag period or dormancy of cancer cells.

Conclusions

The Tilman equations for modeling the invasion of two species into a defined space have been modified to study the invasion of cancer cells into the bone marrow microenvironment. These modified equations allow a more flexible way to model the space competition between the two cell species. The ability to model initial density, metastatic seeding into the bone marrow and growth once the cells are present, and movement of cells out of the bone marrow niche and apoptosis of cells are all aspects of the adapted equations. These equations are currently being applied to clinical data sets for verification and further refinement of the models.     

Instability of non-constant equilibrium solutions of a system of competition-diffusion equations

The system of interaction-diffusion equations describing competition between two species is investigated. By using a version of the Perron-Frobenius theorem of positive matrices generalized to function spaces, it is proved that any non-constant equilibrium solution of the system is unstable both under Neumann boundary conditions (for the rectangular parallelepiped domain) and under periodic conditions. It is conjectured that this result extends to convex domains, and that the simple interaction-diffusion model cannot explain spatially segregated distributions of two competing species in such domains.     

Animal exchange ratios: an alternative point of view

An alternative interpretation is provided of the concepts of carrying capacity and exchange ratios, particularly suitable for game animal species, based on management models for a given area of rangeland or pasture. It involves modelling animal population dynamics as discrete‐time logistic equations. Carrying capacity is then generated endogenously using rainfall as a proxy. The model interaction parameters, also generated endogenously, represent the animal exchange ratios. Because these two parameters are generated endogenously, this approach takes into account all the animals' habitat requirements (food, cover, water and space) simultaneously, unlike other approaches that tend to consider food requirements only. This makes the approach amenable to multi‐species situations. It also captures the ecological definition of population growth models where the realized rather than the theoretical carrying capacity is determined endogenously.     

Coexistence phenomena and global bifurcation structure in a chemostat-like model with species-dependent diffusion rates

We study the competition of two species for a single resource in a chemostat. In the simplest space-homogeneous situation, it is known that only one species survives, namely the best competitor. In order to exhibit coexistence phenomena, where the two competitors are able to survive, we consider a space dependent situation: we assume that the two species and the resource follow a diffusion process in space, on top of the competition process. Besides, and in order to consider the most general case, we assume each population is associated with a distinct diffusion constant. This is a key difficulty in our analysis: the specific (and classical) case where all diffusion constants are equal, leads to a particular conservation law, which in turn allows to eliminate the resource in the equations, a fact that considerably simplifies the analysis and the qualitative phenomena. Using the global bifurcation theory, we prove that the underlying 2-species, stationary, diffusive, chemostat-like model, does possess coexistence solutions, where both species survive. On top of that, we identify the domain, in the space of the relevant bifurcation parameters, for which the system does have coexistence solutions.     

Unsteady Boundary Layer Flow and Heat Transfer of a Casson Fluid past an Oscillating Vertical Plate with Newtonian Heating

In this paper, the heat transfer effect on the unsteady boundary layer flow of a Casson fluid past an infinite oscillating vertical plate with Newtonian heating is investigated. The governing equations are transformed to a systems of linear partial differential equations using appropriate non-dimensional variables. The resulting equations are solved analytically by using the Laplace transform method and the expressions for velocity and temperature are obtained. They satisfy all imposed initial and boundary conditions and reduce to some well-known solutions for Newtonian fluids. Numerical results for velocity, temperature, skin friction and Nusselt number are shown in various graphs and discussed for embedded flow parameters. It is found that velocity decreases as Casson parameters increases and thermal boundary layer thickness increases with increasing Newtonian heating parameter.     

The volterra competition equations with resource - Independent growth coefficients and discussion on their biological and biophysical implications

Analysis of the biophysical conditions for a correct application of the Volterra Competition Equations with resource-independent coefficients reveals the following:The traditional, mathematical formalism with the two equations representing two straight lines at the condition of zero growth applies.As a directly resource-limited situation does not permit for stable equilibrium (One-line or K-system, Walker [18]; Ceiling model, Pollard [11]), the combined equilibrium density represented by the intersect of the two lines (Two-line system or S-system; Equilibrium model; lit. ref. see above) is by necessity smaller than the carrying capacity of shared resources would permit. The physical determinant is density in space as a result of behavioural/physiological interaction between organisms. The traditional inequality conditions then mean that coexistence of species sharing the same space is possible provided density-dependent reduction of growth relative to the intrinsic growth rate is more effective within species than between species. The distance of the intersect from the line linking the specific population densities of single species on the axes is a measure for the overlap in the use of space by the two species; thus, overlap of the spatial niche results in stability.This biophysical system allows for coexistence of species in identical space/resource niches.Food-jealous behaviour is a direct function of density in space; it is only indirectly and inconsistently influenced by resource availability.     

Dynamical analysis of density-dependent selection in a discrete one-island migration model

A system of non-linear difference equations is used to model the effects of density-dependent selection and migration in a population characterized by two alleles at a single gene locus. Results for the existence and stability of polymorphic equilibria are established. Properties for a genetically important class of equilibria associated with complete dominance in fitness are described. The birth of an unusual chaotic attractor is also illustrated. This attractor is produced when migration causes chaotic dynamics on a boundary of phase space to bifurcate into the interior of phase space, resulting in bistable genetic polymorphic behavior.     

Stable oscillations in mathematical models of biological control systems

Summary Oscillations in a class of piecewise linear (PL) equations which have been proposed to model biological control systems are considered. The flows in phase space determined by the PL equations can be classified by a directed graph, called a state transition diagram, on anN-cube. Each vertex of theN-cube corresponds to an orthant in phase space and each edge corresponds to an open boundary between neighboring orthants. If the state transition diagram contains a certain configuration called a cyclic attractor, then we prove that for the associated PL equation, all trajectories in the regions of phase space corresponding to the cyclic attractor either (i) approach a unique stable limit cycle attractor, or (ii) approach the origin, in the limitt→∞. An algebraic criterion is given to distinguish the two cases. Equations which can be used to model feedback inhibition are introduced to illustrate the techniques.     

Consequences of symbiosis for food web dynamics

Basic Lotka-Volterra type models in which mutualism (a type of symbiosis where the two populations benefit both) is taken into account, may give unbounded solutions. We exclude such behaviour using explicit mass balances and study the consequences of symbiosis for the long-term dynamic behaviour of a three species system, two prey and one predator species in the chemostat. We compose a theoretical food web where a predator feeds on two prey species that have a symbiotic relationships. In addition to a species-specific resource, the two prey populations consume the products of the partner population as well. In turn, a common predator forages on these prey populations. The temporal change in the biomass and the nutrient densities in the reactor is described by ordinary differential equations (ODE). Since products are recycled, the dynamics of these abiotic materials must be taken into account as well, and they are described by odes in a similar way as the abiotic nutrients. We use numerical bifurcation analysis to assess the long-term dynamic behaviour for varying degrees of symbiosis. Attractors can be equilibria, limit cycles and chaotic attractors depending on the control parameters of the chemostat reactor. These control parameters that can be experimentally manipulated are the nutrient density of the inflow medium and the dilution rate. Bifurcation diagrams for the three species web with a facultative symbiotic association between the two prey populations, are similar to that of a bi-trophic food chain; nutrient enrichment leads to oscillatory behaviour. Predation combined with obligatory symbiotic prey-interactions has a stabilizing effect, that is, there is stable coexistence in a larger part of the parameter space than for a bi-trophic food chain. However, combined with a large growth rate of the predator, the food web can persist only in a relatively small region of the parameter space. Then, two zero-pair bifurcation points are the organizing centers. In each of these points, in addition to a tangent, transcritical and Hopf bifurcation a global heteroclinic bifurcation is emanating. This heteroclinic cycle connects two saddle equilibria where the predator is absent. Under parameter variation the period of the stable limit cycle goes to infinity and the cycle tends to the heteroclinic cycle. At this global bifurcation point this cycle breaks and the boundary of the basin of attraction disappears abruptly because the separatrix disappears together with the cycle. As a result, it becomes possible that a stable two-nutrient–two-prey population system becomes unstable by invasion of a predator and eventually the predator goes extinct together with the two prey populations, that is, the complete food web is destroyed. This is a form of over-exploitation by the predator population of the two symbiotic prey populations. When obligatory symbiotic prey-interactions are modelled with Liebigs minimum law, where growth is limited by the most limiting resource, more complicated types of bifurcations are found. This results from the fact that the Jacobian matrix changes discontinuously with respect to a varying parameter when another resource becomes most limiting.Revised version: 21 July 2003