Growth with regulation in random environment

The diffusion model for a population subject to Malthusian growth is generalized to include regulation effects. This is done by incorporating a logarithmic term in the regulation function in a way to obtain, in the absence of noise, an S-shaped growth law retaining the qualitative features of the logistic growth curve. The growth phenomenon is modeled as a diffusion process whose transition p.d.f. is obtained in closed form. Its steady state behavior turns out to be described by the lognormal distribution. The expected values and the mode of the transition p.d.f. are calculated, and it is proved that their time course is also represented by monotonically increasing functions asymptotically approaching saturation values. The first passage time problem is then considered. The Laplace transform of the first passage time p.d.f. is obtained for arbitrary thresholds and is used to calculate the expected value of the first passage time. The inverse Laplace transform is then determined for a threshold equal to the saturation value attained by the population size in the absence of random components. The probability of absorption for an arbitrary barrier is finally calculated as the limit of the absorption probability in a two-barrier problem.     

Computer Simulation of Radial Immunodiffusion: I. Selection of an Algorithm for the Diffusion Process

Theories of diffusion with chemical reaction are reviewed as to their contributions toward developing an algorithm needed for computer simulation of immunodiffusion. The Spiers-Augustin moving sink and the Engelberg stationary sink theories show how the antibody-antigen reaction can be incorporated into boundary conditions of the free diffusion differential equations. For this, a stoichiometric precipitate was assumed and the location of precipitin lines could be predicted. The Hill simultaneous linear adsorption theory provides a mathematical device for including another special type of antibody-antigen reaction in antigen excess regions of the gel. It permits an explanation for the lowered antigen diffusion coefficient, observed in the Oudin arrangement of single linear diffusion, but does not enable prediction of the location of precipitin lines. The most promising mathematical approach for a general solution is implied in the Augustin alternating cycle theory. This assumes the immunodiffusion process can be evaluated by alternating computation cycles: free diffusion without chemical reaction and chemical reaction without diffusion. The algorithm for the free diffusion update cycle, extended to both linear and radial geometries, is given in detail since it was based on gross flow rather than more conventional expressions in terms of net flow. Limitations on the numerical integration process using this algorithm are illustrated for free diffusion from a cylindrical well.     

A membrane-carrier mechanism for generating new sources and sinks within gradient-controlled tissues—Its application to regeneration in Oncopeltus

A model is presented that can, in principle, generate new sources and sinks within an existing gradient in the concentration of a morphogen. The novel and crucial feature of the model is that morphogens are transported between cells by membrane-based carrier molecules and not by diffusion. A further aspect of the model is the presence of a second substance within each cell whose concentration is uniform over the tissue; this molecule binds to but is not transported by the carrier and is therefore a competitive inhibitor of the morphogen. The concentration of free inhibitor in a cell determines its fate: if at any time it exceeds some threshold, that cell becomes a morphogen source; if it falls below a second threshold, the cell becomes a sink; in between them, the cell shows no special properties. Provided that differences in morphogen concentration between adjacent cells are not too great, the mechanism is indistinguishable from a normal, diffusion gradient. Examination of the kinetics of the system over a one-dimensional line of cells, however, shows that any stable morphogen difference leads to a carrier imbalance and to a change in the degree of inhibitor binding. If this difference is sufficiently great and if there is morphogen homostasis in each cell, then the free inhibitor concentration in the high morphogen cell may exceed the higher threshold causing it to become a source while the low morphogen cell becomes a sink.A numerical example of the mechanism is given and the results calculated for two-dimensional cellular arrays on either side of a morphogen discontinuity. The predictions match the observations of Wright & Lawrence (1981a, b) on Oncopeltus. These authors showed that, if pieces of epidermis from sufficiently different positions were grafted together in vivo, an ectopic boundary would form with regions of reversed polarity on either side of the join. The ability of the model to explain the regeneration and axial graft observations on hydra is also discussed and some experiments that might test the model are put forward. It is suggested that the significance of the membrane-carrier mechanism in vivo is twofold: first, to interpret the basic segmentation mechanism in embryogenesis by turning its morphogen discontinuities into source-sink pairs and so generating actual boundaries; second, to act as a homeostatic mechanism in later development, thus ensuring the maintenance of boundaries.     

Kinetic analysis of the attachment of a biological particle to a surface by macromolecular binding

Motivated by the problem of microbial deposition, a dynamic model is developed for the attachment of a Brownian particle to a surface mediated by colloidal forces as well as macromolecular bridging. The model predicts the attachment probability of the particle to the surface based upon the free energy as a function of fluctuating bond number and separation distance from the surface. From this model, the mean first-passage time approach is used to predict the mean time required for the particle moving from the unattached state to the attached state based on the properties of the binding macromolecules. This approach provides an analytical approximation for mean transition time from the secondary energy minimum as well as the attachment rate constant for the general case where neither binding nor particle diffusion are necessarily rate-limiting.     

Passage time,resilience, and structure of compartmental models

The relation between the graphical structure and two measures of stability of a compartmental model is studied. The two are resilience as measured by deviation from equilibrium throughout the history of the system and mean first passage time from a central compartment back to itself. A number of examples are studied and a theory developed for models whose graphical structure is in the form of an advanced rosette. The resilience, known to increase with complexity, is shown to increase as well with uneveness in length of food chains. The mean first passage time, on the other hand, depends only on the number of food chains and their average length.     

Experimental evidence that source genetic variation drives pathogen emergence

A pathogen can readily mutate to infect new host types, but this does not guarantee successful establishment in the new habitat. What factors, then, dictate emergence success? One possibility is that the pathogen population cannot sustain itself on the new host type (i.e. host is a sink), but migration from a source population allows adaptive sustainability and eventual emergence by delivering beneficial mutations sampled from the source''s standing genetic variation. This idea is relevant regardless of whether the sink host is truly novel (host shift) or whether the sink is an existing or related, similar host population thriving under conditions unfavourable to pathogen persistence (range expansion). We predicted that sink adaptation should occur faster under range expansion than during a host shift owing to the effects of source genetic variation on pathogen adaptability in the sink. Under range expansion, source migration should benefit emergence in the sink because selection acting on source and sink populations is likely to be congruent. By contrast, during host shifts, source migration is likely to disrupt emergence in the sink owing to uncorrelated selection or performance tradeoffs across host types. We tested this hypothesis by evolving bacteriophage populations on novel host bacteria under sink conditions, while manipulating emergence via host shift versus range expansion. Controls examined sink adaptation when unevolved founding genotypes served as migrants. As predicted, adaptability was fastest under range expansion, and controls did not adapt. Large, similar and similarly timed increases in fitness were observed in the host-shift populations, despite declines in mean fitness of immigrants through time. These results suggest that source populations are the origin of mutations that drive adaptive emergence at the edge of a pathogen''s ecological or geographical range.     

New parameter relationships determined via stochastic ordering for spike activity in a reversal potential neural model

Purpose of this work is to study the dependence of interspike interval distribution on the model parameters when use is made of the Feller diffusion process to describe the subthreshold membrane potential of a neuron. To this aim we make use of a new approach, namely the ordering of first passage times. The functional dependence among the model parameters (e.g, membrane time constant, reversal potential, etc.) resulting from the ordering criteria employed and from the study of mean trajectory plots is analyzed into detail for four different scenario.     

The statistics of calcium-mediated focal excitations on a one-dimensional cable

It is well known that various cardiac arrhythmias are initiated by an ill-timed excitation that originates from a focal region of the heart. However, up to now, it is not known what governs the timing, location, and morphology of these focal excitations. Recent studies have shown that these excitations can be caused by abnormalities in the calcium (Ca) cycling system. However, the cause-and-effect relationships linking subcellular Ca dynamics and focal activity in cardiac tissue is not completely understood. In this article, we present a minimal model of Ca-mediated focal excitations in cardiac tissue. This model accounts for the stochastic nature of spontaneous Ca release on a one-dimensional cable of cardiac cells. Using this model, we show that the timing of focal excitations is equivalent to a first passage time problem in a spatially extended system. In particular, we find that for a short cable the mean first passage time increases exponentially with the number of cells in tissue, and is critically dependent on the ratio of inward to outward currents near the threshold for an action potential. For long cables excitations occurs due to ectopic foci that occur on a length scale determined by the minimum length of tissue that can induce an action potential. Furthermore, we find that for long cables the mean first passage time decreases as a power law in the number cells. These results provide precise criteria for the occurrence of focal excitations in cardiac tissue, and will serve as a guide to determine the propensity of Ca-mediated triggered arrhythmias in the heart.     

A core set of metabolite sink/source ratios indicative for plant organ productivity in Lotus japonicus

Plant growth is an important process in physiological as well as ecological respect and a number of metabolic parameters (elemental ratios as well as steady-state levels of individual metabolites) have been demonstrated to reflect this process on the whole plant level. Since plant growth is highly localized and is the result of a complex interplay of metabolic activities in sink and source organs, we propose that ratios in metabolite levels of sink and source organs are particularly well suited to characterize this process. To demonstrate such a connection, we studied organ-specific metabolite ratios from Lotus japonicus treated with mineral nutrients, salt stress or arbuscular mycorrhizal fungi. The plants were displaying a wide range of biomass and of flower/biomass ratios. In the analysis of our data we looked for correlations between shifts in sink/source metabolite ratios and plant productivity (biomass accumulated at the time of harvest). In addition we correlated shifts in metabolite ratios comparing competing generative and vegetative sink organs with shifts in productivity of the two organs (changes in flower/biomass ratios). In our analyses we observed clear shifts of carbohydrates and of compounds connected to nitrogen metabolism in favour of sink organs of particularly high productivity. These shifts were in agreement with general differences in metabolite steady-state levels when comparing sink and source organs. Our findings suggest that differentiation of sink and source organs during sampling for metabolomic experiments substantially increases the amount of information obtained from such experiments.     

Persistence times of populations with large random fluctuations

The growth of populations which undergo large random fluctuations can be modelled with stochastic differential equations involving Poisson processes. The problem of determining the persistence time is that of finding the time of first passage to some small critical population size. We consider in detail a simple model of logistic growth with additive Poisson disasters of fixed magnitude. The expectation and variability of the persistence time are obtained as solutions of singular differential-difference equations. The dependence of the persistence time of a colonizing species on the parameters of the model is discussed. The model may also be viewed as random harvesting with fixed quotas and a comparison is made between the mean extinction time and those for deterministic models.     

Source-sink landscape theory and its ecological significance

Exploring the relatiouships between landscape pattern and ecological processes is the key topic of landscape ecology,for which,a large number of indices as well as landscape pattern analysis model were developed.However,one problem faced by landscape ecologists is that it is hard to link the landscape indices with a specific ecological process.Linking landscape pattern and ecological processes has become a challenge for landscape ecologists."Source" and "sink" are common concepts used in air pollution research,by which the movement direction and pattern of different pollutants in air can be clearly identified.In fact,for any ecological process,the research can be considered as a balance between the source and the sink in space.Thus,the concepts of "source" and "sink" could be implemented to the research of landscape pattern and ecological processes.In this paper,a theory of sourcesink landscape was proposed,which include:(1) In the research of landscape pattern and ecological process,all landscape types can be divided into two groups,"source"landscape and "sink" landscape."Source" landscape contributes positively to the ecological process,while "sink" landscape is unhelpful to the ecological process.(2) Both landscapes are recognized with regard to the specific ecological process."Source" landscape in a target ecological process may change into a "sink"landscape as in another ecological process.Therefore,the ecological process should be determined before "source"or "sink" landscape were defined.(3) The key point to distinguish "source" landscape from "sink" landscape is to quantify the effect of landscape on ecological process.The positive effect is made by "source" landscape,and the negative effect by "sink" landscape.(4) For the same ecological process,the contribution of "source" landscapes may vary,and it is the same to the "sink"landscapes.It is required to determine the weight of each landscape type on ecological processes.(5) The sourcesink principle can be applied to non-point source pollution control,biologic diversity protection,urban heat island effect mitigation,etc.However,the landscape evaluation models need to be calibrated respectively,because different ecological processes correspond with different source-sink landscapes and evaluation models for the different study areas.This theory is helpful to further study landscape pattern and ecological process,and offers a basis for new landscape index design.     

Sieve element unloading: cellular pathway, mechanism and control

The transport and distribution of phloem – mobile solutes is predominantly determined by transport processes located at the sink end of the source – transport – sink system. Transport across the sieve element boundary, sieve element unloading, is the first of a series of sink transport processes. Unloading of solutes from the sieve elements may follow an apo- or symplastic route. It is speculated that the unloading pathway is integrated with sink function and that apoplastic unloading is restricted to situations in which movement through the symplast is not compatible with sink function. These situations include axial transport and storage of osmotically active solutes against concentration and turgor gradients between the sieve elements and sink cells. Coupled with alteration in sink function, the cellular pathway of unloading can switch in stems and possibly other sinks. Experimental systems and approaches used to elucidate the mechanism of sieve element unloading are reviewed. Unloading fluxes to the apoplast can largely be accounted for by membrane diffusion in axial sinks. However, the higher fluxes in storage sinks suggests dependence on some form of facilitated transport. Proton sucrose symport is assessed to be a possible mechanism for facilitated efflux of solutes across the sieve element plasma membrane to the sink apoplast. Unloading through the symplast may occur by diffusion or mass flow. The latter mechanism serves to dissipate phloem water and hence prevent the potential elevation of sieve element turgor that would otherwise slow phloem import into the sink. The possibility of energised plasmodesmatal transport is raised. Sieve element unloading must be integrated with subsequent compartmentation and metabolism of the unloaded solute. Solute levels are an obvious basis for control of sieve element unloading, but are found to offer limited scope for a mass action mechanism. Apoplastic, cellular pathway, sieve element, solute transport, symplastic. Translated into a turgor signal, solute levels could regulate the rate of unloading, metabolism and compartmentation forming part of a turgor homeostat irrespective of the pathway of unloading.     

Source–sink dynamics: how sinks affect demography of sources

Models of source–sink population dynamics have to make assumptions about whether, and eventually how, demographic parameters in source habitats are dependent on the demography in sink habitats. However, the empirical basis for making such assumptions has been weak. Here we report a study on experimental root vole populations, where estimates of demographic parameters were contrasted between source patches in source–sink (treatment) and source–source systems (control). In the presence of a sink patch (simulated by a pulsed removal of immigrants), source‐patch populations failed to increase over the breeding season, mainly due to a high spatially density‐dependent dispersal rate from source to sink patches. The per capita recruitment rate was almost two times higher in source–sink than in the source–source systems, but this did not compensate for the loss rate due to dispersal from source to sink patches. Sex ratio in the source–sink systems became less female biased, probably as a result of an enhanced frequency of dispersal movements in females. Good knowledge of the degree of density‐and habitat‐dependent dispersal is critical for predicting the dynamics of source–sink populations.     

Modulation of lateral transport of membrane components by spatial variations in diffusivity and solubility.

The effect of spatially varying diffusivity and solubility on the efficiency of intramembrane transport is investigated by obtaining solutions to the generalized lateral diffusion equation in which both the diffusion coefficient, D(r), and the partition coefficient, K(r), are functions of position. The mean-time-to-capture by a sink, tc, of particles diffusing in a plane is obtained analytically for the case of a sink surrounded by gradients in D(r) and K(r) with radially symmetrical geometry. It is shown that for particles originating at random locations, tc is shortened dramatically, if in an annular region around the sink, D and K are significantly greater than in the remainder of the plane. Similarly, a viscous boundary layer surrounding a sink is demonstrated to represent a significant barrier for diffusing particles. To investigate more complex geometries, a finite difference numerical integration method is used and is shown to provide comparable results for tc with modest computational power. The same method is used to calculate the tc for particles originating at a source that is joined to the sink by a channel. The increase in the rate with which particles travel from a source to a sink when they are joined by a high diffusivity and/or solubility channel is illustrated by several numerical examples and by graphical representations that show the equilibrium particle density (and hence the effective particle flow) in the presence of different sink, source, and channel combinations. These results are discussed in terms of fluidity domains and other membrane heterogeneities.     

Amino acid accumulation in sink and source tissues of Coleus blumei Benth. during salinity stress

The purpose of this study was to analyse the accumulation of amino acid in source and sink tissues of variegated Coleus blumei Benth. leaves during an extended exposure to salinity stress. The imposed stress resulted in a reduction in shoot height and leaf size, as well as a reduction in total protein and nitrogen content in both the sink and source tissues. At the same time, accumulation of low molecular weight nitrogen-containing compounds in Coleus leaves was observed, which peaked within the first 10 d of exposure to salinity, and then declined, but remained slightly elevated for the remainder of the study. A number of amino acids were found to accumulate in both the sink and source tissues, including arginine, asparagine, and serine. A larger proportion of asparagine and less arginine was observed in the sink tissue than the source tissue of the salinity-stressed plants. This difference may reflect the mobility of these compounds in the phloem. No proline was found to accumulate in either the source or sink tissue during the exposure to salinity. From the pulse-chase labelling of stressed Coleus leaves it can deduced that some of the observed accumulation of amino acids and amides observed is due to de novo synthesis and not simply the result of protein degradation.     

Diffusion model of tumor vascularization and growth

A diffusion model of tumor growth, vascularization and necrosis is used to analyze experimental data describing the temporal changes in tumor cell and blood vessel radial distributions in a host-tissue field transplanted with a fibrosarcoma. The experimental results showed a peak density of vessels occurring at the advancing migration front of the tumor and a decline in the vessel surface area at the tumor center with time. The peak density of tumor cells shifts away from the tumor center with time. These dynamic changes can be explained by a mathematical model which views the process as one of diffusion and proliferation in time and space. Coupled diffusion equations with nonlinear source and sink terms describe the proliferation, death, and migration of tumor cells and vascular surface area. The concept of an angiogenic factor elaborated by tumor cells is incorporated.     

A method for establishing stable concentration gradients in agar suitable for studying chemotaxis on a solid surface.

A simple technique has been developed for establishing stable gradients of a substance in agar. The technique involves the creation of a spherically symmetric concentration profile in which concentration varies inversely with the distance from the source and is independent of the diffusion coefficient of the substance. It has been shown that the gradients established with this technique are stable for at least 190 h. and, on a theoretical basis, they can be kept stable for more than 1000 h. Time-variant gradients can also be established, if desired, using the same system and limiting either the source or the agar sink. It must be emphasized that a stable gradient cannot be obtained by using a shallow agar layer as a sink. The use of such conditions (e.g. the agar in a standard petri dish) can result only in time-variant gradients. The solution to the diffusion equation in a spherically symmetric system establishes the expected concentration profile, the basis for adjusting it, and the parameters that control the behavior of the system. Some useful applications for examining chemotaxis on a solid surface as well as possible further developments are discussed.     

A model of pollinator‐mediated gene flow between plant populations with numerical solutions for bumblebees pollinating oilseed rape

We present a model that predicts the level of gene flow mediated by animal pollinators from a source population to a sink population. The model requires specification of three elements: (1) the paternity that originates from a single flower, the paternity shadow; (2) the mean number of flowers that pollinators visit during stays in the sink population, the residence; (3) the proportion of pollinators arriving at the sink that carry pollen from the source population. Provided that pollinators visit enough flowers in the sink to exhaust the paternity shadows from the source, the general results are that gene flow is inversely proportional to the mean pollinator residence in the sink population, and is proportional to the fraction of pollinators arriving with pollen from the source. These results are used to propose explanations for two of the widely observed patterns in gene flow among plant populations. Numerical solutions to the model are derived using experimentally determined values of elements (1) and (2) that represent bumblebees, Bombus spp., visiting agricultural fields of oilseed rape, Brassica napus L. In B. napus, the paternity shadow attenuates rapidly over approximately 20 recipient flowers. Mean bumblebee residences in the fields studied varied between 490 and 720 flowers. In the absence of a direct measurement of element (3), we calculated the maximum level of bumblebee‐mediated gene flow by assuming that all bees arrived at the sink saturated with pollen from extrinsic sources. In this case, the model predicts that bumblebee‐mediated gene flow accounted for between 0.1% and 0.5% of the progeny in the agricultural fields studied. A likelihood analysis of our observations is unable to reveal convincingly the proportion of bees arriving at the sink via a source population, but the literature suggests that bumblebees have high site fidelity, which implies that bee‐mediated gene flow may be substantially less than our estimated maximum. We consider the role of various factors, including wind pollination, in accounting for the differences between the model's predictions and the generally higher levels of gene flow observed in previous studies of oilseed rape.     

The Ornstein-Uhlenbeck process as a model for neuronal activity

Mean and variance of the first passage time through a constant boundary for the Ornstein-Uhlenbeck process are determined by a straight-forward differentiation of the Laplace transform of the first passage time probability density function. The results of some numerical computations are discussed to shed some light on the input-output behavior of a formal neuron whose dynamics is modeled by a diffusion process of Ornstein-Uhlenbeck type.Work supported in part by the Group for Mathematical Information Science (GNIM) of the National Council for Research     

MIGRATION ENHANCES ADAPTATION IN BACTERIOPHAGE POPULATIONS EVOLVING IN ECOLOGICAL SINKS

Migration between populations can be a major evolutionary force. However, some disagreement exists as to precisely how migration affects population adaptation. Some theories emphasize the inhibitory effects of gene flow between locally adapted populations, whereas others propose that migration can enhance adaptation. Migration has also been theorized to rescue sink populations from extinction. In our experiments, we serially passaged bacteriophage Φ6 host range mutants under sink conditions on a novel host while manipulating the source and number of migrants into these experimental populations. Migrants from two sources were used: mutant Φ6 phage able to infect a novel host (treatment) and wild‐type Φ6 phage unable to infect a novel host (control). We used quadratic regressions to determine the relationship between the number of migrants per passage and the absolute fitnesses of experimental populations following 30 passages. Our results showed that migration from a control population had no effect on absolute fitnesses of our serially passaged populations following 30 passages. By contrast, the relationship between migrants per passage and absolute fitnesses for populations receiving migrants able to infect the novel host was best described by an upwardly concave curve. These results suggest that intermediate levels of migration can have favorable impacts on evolutionary adaptation.