Reproduction Number and Asymptotic Stability for the Dynamics of a Honey Bee Colony with Continuous Age Structure.

A system of partial differential equations is derived as a model for the dynamics of a honey bee colony with a continuous age distribution, and the system is then extended to include the effects of a simplified infectious disease. In the disease-free case, we analytically derive the equilibrium age distribution within the colony and propose a novel approach for determining the global asymptotic stability of a reduced model. Furthermore, we present a method for determining the basic reproduction number [Formula: see text] of the infection; the method can be applied to other age-structured disease models with interacting susceptible classes. The results of asymptotic stability indicate that a honey bee colony suffering losses will recover naturally so long as the cause of the losses is removed before the colony collapses. Our expression for [Formula: see text] has potential uses in the tracking and control of an infectious disease within a bee colony.

Evolutionary rescue on genotypic fitness landscapes.

Populations facing adverse environments, novel pathogens or invasive competitors may be destined to extinction if they are unable to adapt rapidly. Quantitative predictions of the probability of survival through adaptation, evolutionary rescue, have been previously developed for one of the most natural and well-studied mappings from an organism's traits to its fitness, Fisher's geometric model (FGM). While FGM assumes that all possible trait values are accessible via mutation, in many applications only a finite set of rescue mutations will be available, such as mutations conferring resistance to a parasite, predator or toxin. We predict the probability of evolutionary rescue, via de novo mutation, when this underlying genetic structure is included. We find that rescue probability is always reduced when its genetic basis is taken into account. Unlike other known features of the genotypic FGM, however, the probability of rescue increases monotonically with the number of available mutations and approaches the behaviour of the classical FGM as the number of available mutations approaches infinity.

Self-tolerance and autoimmunity in a regulatory T cell model.

The class of immunosuppressive lymphocytes known as regulatory T cells (Tregs) has been identified as a key component in preventing autoimmune diseases. Although Tregs have been incorporated previously in mathematical models of autoimmunity, we take a novel approach which emphasizes the importance of professional antigen presenting cells (pAPCs). We examine three possible mechanisms of Treg action (each in isolation) through ordinary differential equation (ODE) models. The immune response against a particular autoantigen is suppressed both by Tregs specific for that antigen and by Tregs of arbitrary specificities, through their action on either maturing or already mature pAPCs or on autoreactive effector T cells. In this deterministic approach, we find that qualitative long-term behaviour is predicted by the basic reproductive ratio R(0) for each system. When R(0)<1, only the trivial equilibrium exists and is stable; when R(0)>1, this equilibrium loses its stability and a stable non-trivial equilibrium appears. We interpret the absence of self-damaging populations at the trivial equilibrium to imply a state of self-tolerance, and their presence at the non-trivial equilibrium to imply a state of chronic autoimmunity. Irrespective of mechanism, our model predicts that Tregs specific for the autoantigen in question play no role in the system's qualitative long-term behaviour, but have quantitative effects that could potentially reduce an autoimmune response to sub-clinical levels. Our results also suggest an important role for Tregs of arbitrary specificities in modulating the qualitative outcome. A stochastic treatment of the same model demonstrates that the probability of developing a chronic autoimmune response increases with the initial exposure to self antigen or autoreactive effector T cells. The three different mechanisms we consider, while leading to a number of similar predictions, also exhibit key differences in both transient dynamics (ODE approach) and the probability of chronic autoimmunity (stochastic approach).

Mutual information without the influence of phylogeny or entropy dramatically improves residue contact prediction.

MOTIVATION: Compensating alterations during the evolution of protein families give rise to coevolving positions that contain important structural and functional information. However, a high background composed of random noise and phylogenetic components interferes with the identification of coevolving positions. RESULTS: We have developed a rapid, simple and general method based on information theory that accurately estimates the level of background mutual information for each pair of positions in a given protein family. Removal of this background results in a metric, MIp, that correctly identifies substantially more coevolving positions in protein families than any existing method. A significant fraction of these positions coevolve strongly with one or only a few positions. The vast majority of such position pairs are in contact in representative structures. The identification of strongly coevolving position pairs can be used to impose significant structural limitations and should be an important additional constraint for ab initio protein folding. AVAILABILITY: Alignments and program files can be found in the Supplementary Information.

Fixation probability for lytic viruses: the attachment-lysis model.

The fixation probability of a beneficial mutation is extremely sensitive to assumptions regarding the organism's life history. In this article we compute the fixation probability using a life-history model for lytic viruses, a key model organism in experimental studies of adaptation. The model assumes that attachment times are exponentially distributed, but that the lysis time, the time between attachment and host cell lysis, is constant. We assume that the growth of the wild-type viral population is controlled by periodic sampling (population bottlenecks) and also include the possibility that clearance may occur at a constant rate, for example, through washout in a chemostat. We then compute the fixation probability for mutations that increase the attachment rate, decrease the lysis time, increase the burst size, or reduce the probability of clearance. The fixation probability of these four types of beneficial mutations can be vastly different and depends critically on the time between population bottlenecks. We also explore mutations that affect lysis time, assuming that the burst size is constrained by the lysis time, for experimental protocols that sample either free phage or free phage and artificially lysed infected cells. In all cases we predict that the fixation probability of beneficial alleles is remarkably sensitive to the time between population bottlenecks.

Hepatic fibrosis in schistosomiasis: egg granulomas secrete fibroblast stimulating factor in vitro.

Cytosol extracts and culture supernatants of isolated egg granulomas obtained from livers of mice with Schistosoma mansoni infection stimulated fibroblasts to incorporate tritiated thymidine and to proliferate in vitro. This finding suggests that hepatic granulomas may play a role in regulating hepatic fibrosis in Schistosoma mansoni infections.

Collagenase production by lymphokine-activated macrophages.

Macrophages incubated with products (lymphokines) secreted by stimulated spleen cells produced collagenase. Active lymphokines were obtained both from mitogen- and antigen-stimulated lymphocytes. These observations suggest that the degradation of collagen in chronic inflammatory lesions may be caused by macrophage collagenase.

Benzodiazepine receptor-mediated chemotaxis of human monocytes.

Benzodiazepines, which are widely prescribed for their antianxiety effects, are shown to be potent stimulators of human monocyte chemotaxis. The chemotactic effects of benzodiazepine receptor agonists were blocked by the peripheral benzodiazepine receptor antagonist PK-11195, suggesting that these effects are mediated by the peripheral-type benzodiazepine receptor. Diazepam was also active in inducing chemotaxis. Binding studies on purified monocytes revealed high-affinity peripheral benzodiazepine receptors, and the displacement potencies of various benzodiazepines correlated with their relative potencies in mediating chemotaxis. The demonstration of functional benzodiazepine receptors on human monocytes, together with recent evidence of receptor-mediated monocyte chemotaxis by other psychoactive peptides (such as opiate peptides), suggests a biochemical substrate for psychosomatic communication.

Mechanism of suppression of cell-mediated immunity by measles virus.

The mechanisms underlying the profound suppression of cell-mediated immunity (CMI) accompanying measles are unclear. Interleukin-12 (IL-12), derived principally from monocytes and macrophages, is critical for the generation of CMI. Measles virus (MV) infection of primary human monocytes specifically down-regulated IL-12 production. Cross-linking of CD46, a complement regulatory protein that is the cellular receptor for MV, with antibody or with the complement activation product C3b similarly inhibited monocyte IL-12 production, providing a plausible mechanism for MV-induced immunosuppression. CD46 provides a regulatory link between the complement system and cellular immune responses.

The impact of host-cell dynamics on the fixation probability for lytic viruses.

Lytic viruses are obligate parasites whose population dynamics are necessarily coupled to the dynamics of their host-cell population. The adaptation rate of these viruses has attracted considerable scientific interest, as they are a key model organism in experimental evolution. Nevertheless, to date mathematical models of experimental evolution have largely ignored the host-cell population. In this paper we incorporate two important features of host-cell dynamics-the possibility of clearance or death of an infected cell before lysis, and the possibility of changing host-cell density-into previous models for the fixation probability of lytic viruses. We compute the fixation probabilities of rare alleles that confer reproductive benefit through either an increase in attachment rate or burst size, or a reduction in lysis time. We find that host-cell clearance significantly reduces the fixation probabilities of all types of beneficial mutations, having the largest impact on mutations which reduce the lysis time, but has only modest effects on the pattern of fixation probabilities previously observed. We further predict that exponential growth of the host-cell population preferentially selects for mutations that affect burst size or lysis time, and exacerbates the sensitive dependence of fixation probabilities on the time between population bottlenecks. Even when burst size and lysis time are constrained to vary together, our results suggest that lytic viruses should readily adapt to optimize these traits to the timing between population bottlenecks.

Fixation probabilities depend on life history: fecundity, generation time and survival in a burst-death model.

The burst-death model has been developed to describe the life history of organisms with variable generation times and a burst of a fixed number of offspring. The model also includes an optional constant clearance rate, such as washout from a chemostat, and the possibility of sustained periods of population growth followed by severe bottlenecks, as in serial passaging. In this model, a beneficial mutation can either increase the burst rate or the burst size, or reduce the clearance rate, thus increasing survival. In this article we examine the effects of these three possible mechanisms on both the Malthusian fitness and the fixation probability of the lineage. We find that equivalent relative increases in the burst rate or burst size confer equivalent increases in the Malthusian fitness of a lineage, whereas increasing survival typically has a more moderate effect on Malthusian fitness. In contrast, for beneficial mutations that confer the same increase in fitness, mutations that increase survival are the most likely to fix, followed by mutations that increase the burst rate. Mutations that increase the burst size are the least likely to fix. These results imply that mutant lineages with the highest Malthusian fitness are not, in many cases, the most likely to escape extinction.

Functionally important residues in the peptidyl-prolyl isomerase Pin1 revealed by unigenic evolution.

Pin1 is a phosphorylation-dependent member of the parvulin family of peptidyl-prolyl isomerases exhibiting functional conservation between yeast and man. To perform an unbiased analysis of the regions of Pin1 essential for its functions, we generated libraries of randomly mutated forms of the human Pin1 cDNA and identified functional Pin1 alleles by their ability to complement the Pin1 homolog Ess1 in Saccharomyces cerevisiae. We isolated an extensive collection of functional mutant Pin1 clones harboring a total of 356 amino acid substitutions. Surprisingly, many residues previously thought to be critical in Pin1 were found to be altered in this collection of functional mutants. In fact, only 17 residues were completely conserved in these mutants and in Pin1 sequences from other eukaryotic organisms, with only two of these conserved residues located within the WW domain of Pin1. Examination of invariant residues provided new insights regarding a phosphate-binding loop that distinguishes a phosphorylation-dependent peptidyl-prolyl isomerase such as Pin1 from other parvulins. In addition, these studies led to an investigation of residues involved in catalysis including C113 that was previously implicated as the catalytic nucleophile. We demonstrate that substitution of C113 with D does not compromise Pin1 function in vivo nor does this substitution abolish catalytic activity in purified recombinant Pin1. These findings are consistent with the prospect that the function of residue 113 may not be that of a nucleophile, thus raising questions about the model of nucleophilic catalysis. Accordingly, an alternative catalytic mechanism for Pin1 is postulated.

Fixation probabilities when generation times are variable: the burst death model.

Estimating the fixation probability of a beneficial mutation has a rich history in theoretical population genetics. Typically, to attain mathematical tractability, we assume that generation times are fixed, while the number of offspring per individual is stochastic. However, fixation probabilities are extremely sensitive to these assumptions regarding life history. In this article, we compute the fixation probability for a "burst-death" life-history model. The model assumes that generation times are exponentially distributed, but the number of offspring per individual is constant. We estimate the fixation probability for populations of constant size and for populations that grow exponentially between periodic population bottlenecks. We find that the fixation probability is, in general, substantially lower in the burst-death model than in classical models. We also note striking qualitative differences between the fates of beneficial mutations that increase burst size and mutations that increase the burst rate. In particular, once the burst size is sufficiently large relative to the wild type, the burst-death model predicts that fixation probability depends only on burst rate.

The fixation probability of beneficial mutations.

The fixation probability, the probability that the frequency of a particular allele in a population will ultimately reach unity, is one of the cornerstones of population genetics. In this review, we give a brief historical overview of mathematical approaches used to estimate the fixation probability of beneficial alleles. We then focus on more recent work that has relaxed some of the key assumptions in these early papers, providing estimates that have wider applicability to both natural and laboratory settings. In the final section, we address the possibility of future work that might bridge the gap between theoretical results to date and results that might realistically be applied to the experimental evolution of microbial populations. Our aim is to highlight the concrete, testable predictions that have arisen from the theoretical literature, with the intention of further motivating the invaluable interplay between theory and experiment.

Estimating the optimal bottleneck ratio for experimental evolution: the burst-death model.

Population bottlenecks are ubiquitous in nature, and are an inherent feature of the experimental protocol for many laboratory selection experiments. These bottlenecks can have profound effects on the rate and trajectory of evolution. In particular, there is a trade-off between sampling the population too frequently and imposing infrequent, but more severe, bottlenecks. In this paper we consider the effects of population bottlenecks, assuming a burst-death model for the life history of the organism under study. This model assumes that generation times are exponentially distributed and that at each generation, individuals in the population have a fixed number of offspring. The model also allows for a constant death rate between bottlenecks. We use this model to estimate the optimal bottleneck ratio, that is, the fraction of the population that should be sampled at each bottleneck in order to maximize the probability that beneficial mutations occur and are not lost. We find that the optimal ratio is roughly constant with respect to many of the model parameters, and that sampling about 20% of the population will maximize the rate of adaptation.

Soluble surface proteins from Helicobacter pylori activate monocytes/macrophages by lipopolysaccharide-independent mechanism.

The inflammatory lesions associated with Helicobacter pylori gastritis and duodenitis contain large numbers of mononuclear cells. The close proximity of H. pylori to gastric mucosa suggests that the organism interacts with mononuclear cells, thereby modulating the inflammatory response. To investigate the role of monocytes/macrophages in this response, we examined the effect of whole H. pylori bacteria, H. pylori surface proteins, and H. pylori lipopolysaccharide (LPS) on purified human monocytes. Whole H. pylori and the extracted LPS induced expression of the monocyte surface antigen HLA-DR and interleukin-2 receptors, production of the inflammatory cytokines interleukin 1 and tumor necrosis factor (peptide and messenger RNA), and secretion of the reactive oxygen intermediate superoxide anion. Since H. pylori in vivo does not invade mucosal tissue, we determined whether soluble constituents of the bacteria could activate monocytes. Soluble H. pylori surface proteins, which are enriched for urease and do not contain LPS, stimulated phenotypic, transcriptional, and functional changes consistent with highly activated monocytes. These findings indicate that H. pylori is capable of activating human monocytes by an LPS-independent as well as an LPS-dependent mechanism. H. pylori activation of resident lamina propria macrophages and monocytes trafficking through the mucosa, leading to the secretion of increased amounts of inflammatory cytokines and reactive oxygen intermediates, could play an important role in mediating the inflammatory response associated with H. pylori gastritis and duodenitis.

Secretory leukocyte protease inhibitor suppresses the production of monocyte prostaglandin H synthase-2, prostaglandin E2, and matrix metalloproteinases.

Secretory leukocyte protease inhibitor (SLPI) is a serine protease inhibitor found in fluids lining mucosal surfaces. In addition to its primary function as an antiprotease, SLPI may also influence cellular functions associated with enzyme synthesis and retroviral infection. In this study, SLPI was examined for its effect on signaling events involved in the production of matrix metalloproteinases (MMPs) by monocytes. Addition of SLPI before stimulation with concanavalin A or LPS resulted in a significant inhibition of monocyte prostaglandin H synthase-2 (PGHS-2), a pivotal enzyme in the PGE2-cAMP dependent pathway of monocyte MMP synthesis. Suppression of PGHS-2 was detected with 0.1 microg/ml of SLPI with a substantial inhibition at 1 and 10 micro/ml. Attenuation of PGHS-2 by SLPI was accompanied by decreased production of PGE2 resulting in the suppression of interstitial collagenase (MMP-1) and gelatinase B (MMP-9) that was reversed by PGE2 or Bt2cAMP. The inhibitory effect of SLPI was largely independent of its antiprotease activity because SLPI muteins, with significantly lower antiprotease activity, also suppressed the induction of PGHS-2 and MMPs. The inhibitory effects of SLPI did not involve the modulation of monokine production since TNF-alpha and IL-10 were unaffected. These findings demonstrate that SLPI also functions as a potent antiinflammatory agent by interfering with the signal transduction pathway leading to monocyte MMP production.

Expression of interleukin 2 receptors by monocytes from patients with acquired immunodeficiency syndrome and induction of monocyte interleukin 2 receptors by human immunodeficiency virus in vitro.

A population of circulating mononuclear cells from patients with AIDS was identified which expressed interleukin 2 receptors (IL-2R). By dual-fluorescence flow microfluorometry, the patients' IL-2R+ cells were further identified as Leu M3+ monocytes (29.4 +/- 5.2% of the Leu M3+ cells were IL-2R+, n = 15), whereas Leu M3+ monocytes from normal subjects were IL-2R negative (2.0 +/- 0.42%; P less than 0.001). By Northern analysis, monocytes from AIDS patients, but not control subjects, constitutively expressed steady-state levels of IL-2R mRNA. Functionally, the IL-2R+ monocytes were capable of depleting IL-2 from culture supernatants, suggesting a mechanism for the reduced IL-2 levels commonly seen in AIDS patients. IL-2R+ monocytes also expressed increased levels of surface HLA-DR which may favor monocyte T-cell interactions and the transmission of human immunodeficiency virus (HIV). In additional studies, normal monocytes were infected with a macrophage-tropic HIV isolate in vitro and monitored for IL-2R and HLA-DR expression. Within 24-48 h after exposure to HIV in vitro, but before evidence of productive infection, greater than 25% of the monocytes became IL-2R+ with increasing numbers of IL-2R+ cells and HLA-DR levels through day 6. These early signaling effects of HIV could be mimicked by adding purified HIV envelope glycoprotein gp120 to the monocytes. This stimulation of monocytes before or independent of productive infection of the cells by HIV is consistent with in vivo observations of activated and/or abnormal functions by monocytes that do not appear to be infected with HIV in AIDS patients.

Dynamics of recurrent viral infection.

In chronic viral infection, low levels of viral replication and infectious particle production are maintained over long periods, punctuated by brief bursts of high viral production and release. We apply well-established principles of modelling virus dynamics to the study of chronic viral infection, demonstrating that a model which incorporates the distinct contributions of cytotoxic T lymphocytes (CTLs) and antibodies exhibits long periods of quiescence followed by brief bursts of viral production. This suggests that for recurrent viral infections, no special mechanism or exogenous trigger is necessary to provoke an episode of reactivation; rather, the system may naturally cycle through recurrent episodes at intervals which can be many years long. We also find that exogenous factors which cause small fluctuations in the natural course of the infection can trigger a recurrent episode. Our model predicts that longer periods between recurrences are associated with more severe viral episodes. Four factors move the system towards less frequent, more severe episodes: decreased viral infectivity, decreased CTL efficacy, decreased memory T cell response and increased antibody efficacy.

Age structure is critical to the population dynamics and survival of honeybee colonies.

Age structure is an important feature of the division of labour within honeybee colonies, but its effects on colony dynamics have rarely been explored. We present a model of a honeybee colony that incorporates this key feature, and use this model to explore the effects of both winter and disease on the fate of the colony. The model offers a novel explanation for the frequently observed phenomenon of 'spring dwindle', which emerges as a natural consequence of the age-structured dynamics. Furthermore, the results indicate that a model taking age structure into account markedly affects the predicted timing and severity of disease within a bee colony. The timing of the onset of disease with respect to the changing seasons may also have a substantial impact on the fate of a honeybee colony. Finally, simulations predict that an infection may persist in a honeybee colony over several years, with effects that compound over time. Thus, the ultimate collapse of the colony may be the result of events several years past.