Memory but no suppression in low-dimensional symmetric idiotypic networks

We present a new symmetric model of the idiotypic immune network. The model specifies clones of B-lymphocytes and incorporates: (1) influx and decay of cells; (2) symmetric stimulatory and inhibitory idiotypic interactions; (3) an explicit affinity parameter (matrix); (4) external (i.e. non-idiotypic) antigens. Suppression is the dominant interaction, i.e. strong idiotypic interactions are always suppressive. This precludes reciprocal stimulation of large clones and thus infinite proliferation. Idiotypic interactions first evoke proliferation, this enlarges the clones, and may in turn evoke suppression. We investigate the effect of idiotypic interactions on normal proliferative immune responses to antigens (e.g. viruses). A 2-D, i.e. two clone, network has a maximum of three stable equilibria: the virgin state and two asymmetric immune states. The immune states only exist if the affinity of the idiotypic interaction is high enough. Stimulation with antigen leads to a switch from the virgin state to the corresponding immune state. The network therefore remembers antigens, i.e. it accounts for immunity/memory by switching beteen multiple stable states. 3-D systems have, depending on the affinities, 9 qualitatively different states. Most of these also account for memory by state switching. Our idiotypic network however fails to account for the control of proliferation, e.g. suppression of excessive proliferation. In symmetric networks, the proliferating clones suppress their anti-idiotypic suppressors long before the latter can suppress the former. The absence of proliferation control violates the general assumption that idiotypic interactions play an important role in immune regulation. We therefore test the robustness of these results by abandoning our assumption that proliferation occurs before suppression. We thus define an “escape from suppression” model, i.e. in the “virgin” state idiotypic interactions are now suppressive. This system erratically accounts for memory and never for suppression. We conclude that our “absence of suppression from idiotypic interactions” does not hinge upon our “proliferation before suppression” assumption.     

Impact of fear effect on the growth of prey in a predator-prey interaction model

Several field data and experiments on a terrestrial vertebrates exhibited that the fear of predators would cause a substantial variability of prey demography. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. Based on the experimental evidence, we proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling type-II functional response. We investigate all the biologically feasible equilibrium points, and their stability is analyzed in terms of the model parameters. Our mathematical analysis exhibits that for strong anti-predator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. Our model system undergoes Hopf bifurcation by considering the birth rate r₀ as a bifurcation parameter. For larger prey birth rate, we investigate the transition to a stable coexisting equilibrium state, with oscillatory approach to this equilibrium state, indicating that the greatest characteristic eigenvalues are actually a pair of imaginary eigenvalues with real part negative, which is increasing for r₀. We obtained the conditions for the occurrence of Hopf bifurcation and conditions governing the direction of Hopf bifurcation, which imply that the prey birth rate will not only influence the occurrence of Hopf bifurcation but also alter the direction of Hopf bifurcation. We identify the parameter regions associated with the extinct equilibria, predator-free equilibria and coexisting equilibria with respect to prey birth rate, predator mortality rates. Fear can stabilize the predator-prey system at an interior steady state, where all the species can exists together, or it can create the oscillatory coexistence of all the populations. We performed some numerical simulations to investigate the relationship between the effects of fear and other biologically related parameters (including growth/decay rate of prey/predator), which exhibit the impact that fear can have in prey-predator system. Our numerical illustrations also demonstrate that the prey become less sensitive to perceive the risk of predation with increasing prey growth rate or increasing predators decay rate.     

Equilibrium properties of a multi-locus, haploid-selection, symmetric-viability model

Under haploid selection, a multi-locus, diallelic, two-niche Levene (1953) model is studied. Viability coefficients with symmetrically opposing directional selection in each niche are assumed, and with a further simplification that the most and least favored haplotype in each niche shares no alleles in common, and that the selection coefficients monotonically increase or decrease with the number of alleles shared. This model always admits a fully polymorphic symmetric equilibrium, which may or may not be stable.We show that a stable symmetric equilibrium can become unstable via either a supercritical or subcritical pitchfork bifurcation. In the supercritical bifurcation, the symmetric equilibrium bifurcates to a pair of stable fully polymorphic asymmetric equilibria; in the subcritical bifurcation, the symmetric equilibrium bifurcates to a pair of unstable fully polymorphic asymmetric equilibria, which then connect to either another pair of stable fully polymorphic asymmetric equilibria through saddle-node bifurcations, or to a pair of monomorphic equilibria through transcritical bifurcations. As many as three fully polymorphic stable equilibria can coexist, and jump bifurcations can occur between these equilibria when model parameters are varied.In our Levene model, increasing recombination can act to either increase or decrease the genetic diversity of a population. By generating more hybrid offspring from the mating of purebreds, recombination can act to increase genetic diversity provided the symmetric equilibrium remains stable. But by destabilizing the symmetric equilibrium, recombination can ultimately act to decrease genetic diversity.     

Unreasonable implications of reasonable idiotypic network assumptions

We first analyse a simple symmetric model of the idiotypic network. In the model idiotypic interactions regulate B cell proliferation. Three non-idiotypic processes are incorporated: (1) influx of newborn cells; (2) turnover of cells: (3) antigen. Antigen also regulates proliferation. A model of 2 B cell populations has 3 stable equilibria: one virgin, two immune. The twodimensional system thus remembers antigens, i.e. accounts for immunity. By contrast, if an idiotypic clone proliferates (in response to antigen), its anti-idiotypic partner is unable to control this. Symmetric idiotypic networks thus fail to account for proliferation regulation. In high-D networks we run into two problems. Firstly, if the network accounts for memory, idiotypic activation always propagates very deeply into the network. This is very unrealistic, but is an implication of the “realistic” assumption that it should be easier to activate all cells of a small virgin clone than to maintain the activation of all cells of a large (immune) clone. Secondly, graph theory teaches us that if the (random) network connectance exceeds a threshold level of one interaction per clone, most clones are interconnected. We show that this theory is also applicable to immune networks based on complementary matching idiotypes. The combination of the first “percolation” result with the “interconnectancr” result means that the first stimulation of the network with antigen should eventually affect most of the clones. We think this is unreasonable. Another threshold property of the network connectivity is the existence of a virgin state. A gradual increase in network connectance eliminates the virgin state and thus causes an abrupt change in network behaviour. In contrast to weakly connected systems, highly connected networks display autonomous activity and are unresponsive to external antigens. Similar differences between neonatal and adult networks have been described by experimentalists. The robustness of these results is tested with a network in which idiotypic inactivation of a clone occurs more generally than activation. Such “long-range inhibition” is known to promote pattern formation. However, in our model it fails to reduce the percolation, and additionally, generates semi-chaotic behaviour. In our network, the inhibition of a clone that is inhibiting can alter this clone into a clone that is activating. Hence “long-range inhibition” implies “long-range activation”, and idiotypic activation fails to remain localized. We next complicate this model by incorporating antibody production. Although this “antibody” model statically accounts for the same set of equilibrium points, it dynamically fails to account for state switching (i.e. memory). The switching behaviour is disturbed by the autonomous slow decay of the (long-lived) antibodies. After antigenic triggering the system now performs complex cyclic behaviour. Finally, it is suggested that (idiotypic) formation of antibody complexes can play only a secondary role in the network. In conclusion, our results cast doubt on the functional role of a profound idiotypic network. The network fails to account for proliferation regulation, and if it accounts for memory phenomena, it “explodes” upon the first encounter with antigen due to extensive percolation.     

How tumors escape immune destruction and what we can do about it

There is strong circumstantial evidence that tumor progression in cancer patients is controlled by the immune system. As will be detailed below, this conclusion is based on observations that tumor progression is often associated with secretion of immune suppressive factors and/or downregulation of MHC class I antigen presentation functions. The inference is that tumors must have elaborated strategies to circumvent an apparently effective immune response. Importantly, a tumor-specific immune response cannot be detected in most individuals. While this failure is in part technical, it also suggests that the magnitude of the immune responses to which tumors have to respond is low. This raises the concern, which is the underlying theme of this commentary, that a more robust immune response elicited by deliberate vaccination will exacerbate the rate of immune escape and nullify the potential benefits of immune-based therapies. Received: 20 March 1999 / Accepted: 3 May 1999     

Impacts of Foraging Facilitation Among Predators on Predator-prey Dynamics

Whereas impacts of predator interference on predator-prey dynamics have received considerable attention, the “inverse” process—foraging facilitation among predators—have not been explored yet. Here we show, via mathematical models, that impacts of foraging facilitation on predator-prey dynamics depend on the way this process is modeled. In particular, foraging facilitation destabilizes predator-prey dynamics when it affects the encounter rate between predators and prey. By contrast, it might have a stabilizing effect if the predator handling time of prey is affected. Foraging facilitation is an Allee effect mechanism among predators and we show that for many parameters, it gives rise to a demographic Allee effect or a critical predator density in need to be crossed for predators to persist. We explore also the effects of predator interference, to make the picture “symmetric” and complete. Predator interference is shown to stabilize predator-prey dynamics once its strength is not too high, and thus corroborates results of others. On the other hand, there is a wide range of model parameters for which predator interference gives rise to three co-occurring co-existence equilibria. Such a multi-equilibrial regime is rather robust as we observe it for all the functional response types we explore. This is a previously unreported phenomenon which we show cannot occur for the Beddington–DeAngelis functional response. An interesting topic for future research thus might be to seek for general conditions on predator functional responses that would produce multiple co-existence equilibria in a predator-prey model.     

G-CSF Control of Neutrophils Dynamics in the Blood

White blood cell neutrophil is a key component in the fast initial immune response against bacterial and fungal infections. Granulocyte colony stimulating factor (G-CSF) which is naturally produced in the body, is known to control the neutrophils production in the bone marrow and the neutrophils delivery into the blood. In oncological practice, G-CSF injections are widely used to treat neutropenia (dangerously low levels of neutrophils in the blood) and to prevent the infectious complications that often follow chemotherapy. However, the accurate dynamics of G-CSF neutrophil interaction has not been fully determined and no general scheme exists for an optimal G-CSF application in neutropenia. Here we develop a two-dimensional ordinary differential equation model for the G-CSF—neutrophil dynamics in the blood. The model is built axiomatically by first formally defining from the biology the expected properties of the model, and then deducing the dynamic behavior of the resulting system. The resulting model is structurally stable, and its dynamical features are independent of the precise form of the various rate functions. Choosing a specific form for these functions, three complementary parameter estimation procedures for one clinical (training) data set are utilized. The fully parameterized model (6 parameters) provides adequate predictions for several additional clinical data sets on time scales of several days. We briefly discuss the utility of this relatively simple and robust model in several clinical conditions. Dedicated to Lee Segel who guided us to apply mathematics for the benefit of mankind—a teacher, a colleague, a friend. L.A. Segel passed away on 31 January 2005.     

Temporal patterns in immune responses to a range of microbial insults (Tenebrio molitor)

Much work has elucidated the pathways and mechanisms involved in the production of insect immune effector systems. However, the temporal nature of these responses with respect to different immune insults is less well understood. This study investigated the magnitude and temporal variation in phenoloxidase and antimicrobial activity in the mealworm beetle Tenebrio molitor in response to a number of different synthetic and real immune elicitors. We found that antimicrobial activity in haemolymph increased rapidly during the first 48h after a challenge and was maintained at high levels for at least 14 days. There was no difference in the magnitude of responses to live or dead Escherichia coli or Bacillus subtilis. While peptidoglylcan also elicited a long-lasting antimicrobial response, the response to LPS was short lived. There was no long-lasting upregulation of phenoloxidase activity, suggesting that this immune effector system is not involved in the management of microbial infections over a long time scale.     

Ca2+ signaling in injured in situ endothelium of rat aorta

The inner wall of excised rat aorta was scraped by a microelectrode and Ca(2+) signals were investigated by fluorescence microscopy in endothelial cells (ECs) directly coupled with injured cells. The injury caused an immediate increase in the intracellular Ca(2+) concentration ([Ca(2+)](i)), followed by a long-lasting decay phase due to Ca(2+) influx from extracellular space. The immediate response was mainly due to activation of purinergic receptors, as shown by the effect of P(2X) and P(2Y) receptors agonists and antagonists, such as suramin, alpha,beta-MeATP, MRS-2179 and 2-MeSAMP. Inhibition of store-operated Ca(2+) influx did not affect either the peak response or the decay phase. Furthermore, the latter was: (i) insensitive to phospholipase C inhibition, (ii) sensitive to the gap junction blockers, palmitoleic acid, heptanol, octanol and oleamide, and (iii) sensitive to La(3+) and Ni(2+), but not to Gd(3+). Finally, ethidium bromide or Lucifer Yellow did not enter ECs facing the scraped area. These results suggest that endothelium scraping: (i) causes a short-lasting stimulation of healthy ECs by extracellular nucleotides released from damaged cells and (ii) uncouples the hemichannels of the ECs facing the injury site; these hemichannels do not fully close and allow a long-lasting Ca(2+) entry.     

From network-to-antibody robustness in a bio-inspired immune system

Behavioural robustness at antibody and immune network level is discussed. The robustness of the immune response that drives an autonomous mobile robot is examined with two computational experiments in the autonomous mobile robots trajectory generation context in unknown environments. The immune response is met based on the immune network metaphor for different low-level behaviours coordination. These behaviours are activated when a robot sense the appropriate conditions in the environment in relation to the network current state. Results are obtained over a case study in computer simulation as well as in laboratory experiments with a Khepera II microrobot. In this work, we develop a set of tests where such an immune response is externally perturbed at network or low-level behavioural modules to analyse the robust capacity of the system to unexpected perturbations. Emergence of robust behaviour and high-level immune response relates to the coupling between behavioural modules that are selectively engaged with the environment based on immune response. Experimental evidence leads discussions on a dynamical systems perspective of behavioural robustness in artificial immune systems that goes beyond the isolated immune network response.     

A Dynamical Model of Lipoprotein Metabolism

We present a dynamical model of lipoprotein metabolism derived by combining a cascading process in the blood stream and cellular level regulatory dynamics. We analyse the existence and stability of equilibria and show that this low-dimensional, nonlinear model exhibits bistability between a low and a high cholesterol state. A sensitivity analysis indicates that the intracellular concentration of cholesterol is robust to parametric variations while the plasma cholesterol can vary widely. We show how the dynamical response to time-dependent inputs can be used to diagnose the state of the system. We also establish the connection between parameters in the system and medical and genetic conditions.     

Transient dynamics and early diagnostics in infectious disease

To date, mathematical models of the dynamics of infectious disease have consistently focused on understanding the long-term behavior of the interacting components, where the steady state solutions are paramount. However for most acute infections, the long-term behavior of the pathogen population is of little importance to the host and population health. We introduce the notion of transient pathology, where the short-term dynamics of interaction between the immune system and pathogens is the principal focus. We identify the amplifying effect of the absence of a fully operative immune system on the pathogenesis of the initial inoculum, and its implication for the acute severity of the infection. We then formalize the underlying dynamics, and derive two measures of transient pathogenicity: the peak of infection (maximum pathogenic load) and the time to peak of infection, both crucial to understanding the early dynamics of infection and its consequences for early intervention. Received: 25 January 2000 / Revised version: 30 November 2000 / Published online: 12 October 2001     

Decay rates of attractive and repellent pheromones in an ant foraging trail network

Pharaoh’s ants (Monomorium pharaonis) use at least three types of foraging trail pheromone: a long-lasting attractive pheromone and two short-lived pheromones, one attractive and one repellent. We measured the decay rates of the behavioural response of ant workers at a trail bifurcation to trail substrate marked with either repellent or attractive short-lived pheromones. Our results show that the repellent pheromone effect lasts more than twice as long as the attractive pheromone effect (78 min versus 33 min). Although the effects of these two pheromones decay at approximately the same rate, the initial effect of the repellent pheromone on branch choice is almost twice that of the attractive pheromone (48% versus 25% above control). We hypothesise that the two pheromones have complementary but distinct roles, with the repellent pheromone specifically directing ants at bifurcations, while the attractive pheromone guides ants along the entire trail. Received 15 November 2007; revised 7 March 2008; accepted 18 March 2008.     

Immune networks modeled by replicator equations

In order to evaluate the role of idiotypic networks in the operation of the immune system a number of mathematical models have been formulated. Here we examine a class of B-cell models in which cell proliferation is governed by a non-negative, unimodal, symmetric response function f(h), where the field h summarizes the effect of the network on a single clone. We show that by transforming into relative concentrations, the B-cell network equations can be brought into a form that closely resembles the replicator equation. We then show that when the total number of clones in a network is conserved, the dynamics of the network can be represented by the dynamics of a replicator equation. The number of equilibria and their stability are then characterized using methods developed for the study of second-order replicator equations. Analogies with standard Lotka-Volterra equations are also indicated. A particularly interesting result of our analysis is the fact that even though the immune network equations are not second-order, the number and stability of their equilibria can be obtained by a superposition of second-order replicator systems. As a consequence, the problem of finding all of the equilibrium points of the nonlinear network equations can be reduced to solving linear equations.     

Equilibria and dynamics of selection at a diallelic autosomal locus in the Nagylaki-Crow continuous model of a monoecious population

Let birth rates and death rates be constant, birth rates positive, fertilities additive, and each birth rate not larger than twice any other birth rate. Global convergence to equilibria is proved for the model in the title. There is at most one polymorphic equilibrium or there are a continuum of equilibria. The phase portraits are given. If there is a polymorphic equilibrium, then the largest negatively invariant set in the state space is a continuous curve connecting the two fixation equilibria. This curve coincides with the Hardy-Weinberg manifold exactly when the death rate is additive. Disregarding extinction, the polymorphic equilibria are the same for the continuous model as for the corresponding discrete model exactly when the death rate is additive.     

Short-term stress experienced at time of immunization induces a long-lasting increase in immunologic memory

It would be extremely beneficial if one could harness natural, endogenous, health-promoting defense mechanisms to fight disease and restore health. The psychophysiological stress response is the most underappreciated of nature's survival mechanisms. We show that acute stress experienced before primary immunization induces a long-lasting increase in immunity. Compared with controls, mice restrained for 2.5 h before primary immunization with keyhole limpet hemocyanin (KLH) show a significantly enhanced immune response when reexposed to KLH 9 mo later. This immunoenhancement is mediated by an increase in numbers of memory and effector helper T cells in sentinel lymph nodes at the time of primary immunization. Further analyses show that the early stress-induced increase in T cell memory may stimulate the robust increase in infiltrating lymphocyte and macrophage numbers observed months later at a novel site of antigen reexposure. Enhanced leukocyte infiltration may be driven by increased levels of the type 1 cytokines, IL-2 and IFN-gamma, and TNF-alpha, observed at the site of antigen reexposure in animals that had been stressed at the time of primary immunization. In contrast, no differences were observed in type 2 cytokines, IL-4 or IL-5. Given the importance of inducing long-lasting increases in immunologic memory during vaccination, we suggest that the neuroendocrine stress response is nature's adjuvant that could be psychologically and/or pharmacologically manipulated to safely increase vaccine efficacy. These studies introduce the novel concept that a psychophysiological stress response is nature's fundamental survival mechanism that could be therapeutically harnessed to augment immune function during vaccination, wound healing, or infection.     

Long-term treatment effects in chronic myeloid leukemia

We propose and analyze a simplified version of a partial differential equation (PDE) model for chronic myeloid leukemia (CML) derived from an agent-based model proposed by Roeder et al. This model describes the proliferation and differentiation of leukemic stem cells in the bone marrow and the effect of the drug Imatinib on these cells. We first simplify the PDE model by noting that most of the dynamics occurs in a subspace of the original 2D state space. Then we determine the dominant eigenvalue of the corresponding linearized system that controls the long-term behavior of solutions. We mathematically show a non-monotonous dependence of the dominant eigenvalue with respect to treatment dose, with the existence of a unique minimal negative eigenvalue. In terms of CML treatment, this shows that there is a unique dose that maximizes the decay rate of the CML tumor load over long time scales. Moreover this unique dose is lower than the dose that maximizes the initial tumor load decay. Numerical simulations of the full model confirm that this phenomenon is not an artifact of the simplification. Therefore, while optimal asymptotic dosage might not be the best one at short time scales, our results raise interesting perspectives in terms of strategies for achieving and improving long-term deep response.

     

Excitable Population Dynamics,Biological Control Failure,and Spatiotemporal Pattern Formation in a Model Ecosystem

Biological control has been attracting an increasing attention over the last two decades as an environmentally friendly alternative to the more traditional chemical-based control. In this paper, we address robustness of the biological control strategy with respect to fluctuations in the controlling species density. Specifically, we consider a pest being kept under control by its predator. The predator response is assumed to be of Holling type III, which makes the system’s kinetics “excitable.” The system is studied by means of mathematical modeling and extensive numerical simulations. We show that the system response to perturbations in the predator density can be completely different in spatial and non-spatial systems. In the nonspatial system, an overcritical perturbation of the population density results in a pest outbreak that will eventually decay with time, which can be regarded as a success of the biological control strategy. However, in the spatial system, a similar perturbation can drive the system into a self-sustained regime of spatiotemporal pattern formation with a high pest density, which is clearly a biological control failure. We then identify the parameter range where the biological control can still be successful and describe the corresponding regime of the system dynamics. Finally, we identify the main scenarios of the system response to the population density perturbations and reveal the corresponding structure of the parameter space of the system. A. Morozov is on leave from Shirshov Institute of Oceanology, Russian Academy of Science, Nakhimovsky Prosp. 36, Moscow 117218, Russia.     

Darwinian aspects of molecular evolution at sublinear propagation rates

Mean fitness is non-decreasing in the symmetry sector of the frequency trajectory followed in competitive replication at sublinear propagation rates (parabolic time course). This sector contains the pairwise symmetric distribution of species frequencies and its neighboring states, and represents at least half the possible states of an evolving sublinear system. States in the non-symmetry sector produce a negative rate of change in mean fitness. The heterogeneous steady state attained in a finite sublinear system is destabilized by formation of a variant with above-threshold fitness. Evolution in the post-steady-state interval elevates the fitness threshold for coexistence. Contrary to the proposition that ‘parabolic growth invariably results in the survival of all competing species’, only species with sufficient fitness to avoid subthreshold fitness survive. An erratum to this article is available at .