Understanding bursting oscillations as periodic slow passages through bifurcation and limit points

We consider a biochemical system consisting of two allosteric enzyme reactions coupled in series. The system has been modeled by Decroly and Goldbeter (J. Theor. Biol. 124, 219 (1987)) and is described by three coupled, first-order, nonlinear, differential equations. Bursting oscillations correspond to a succession of alternating active and silent phases. The active phase is characterized by rapid oscillations while the silent phase is a period of quiescence. We propose an asymptotic analysis of the differential equations which is based on the limit of large allosteric constants. This analysis allows us to construct a time-periodic bursting solution. This solution is jumping periodically between a slowly varying steady state and a slowly varying oscillatory state. Each jump follows a slow passage through a bifurcation or limit point which we analyze in detail. Of particular interest is the slow passage through a supercritical Hopf bifurcation. The transition is from an oscillatory solution to a steady state solution. We show that the transition is delayed considerably and characterize this delay by estimating the amplitude of the oscillations at the Hopf bifurcation point.     

Spatial distribution of major bacterial species and different volatile fatty acids in a two-phase anaerobic biofilm reactor with PVA gel beads as bio-carrier

Abstract

Conventional completely mixed anaerobic treatment systems limit the chances of the different species of bacteria to spatially group together according to their mutual cooperation and as a result, show a lower efficiency and vulnerability towards shock situations. It is interesting to know about the stratification of the different bacterial species participating in the degradation process and the intermediates that they produce. In this study, we established and optimized a two-phase anaerobic packed bed biofilm reactor system (AnPBR) with porous PVA gel beads used as bio-carriers and ran the reactor system in a steady state to observe the VFAs produced along with the microbial diversity of the predominant species at different stages of the reactor system. We observed that acetate and butyrate were the predominant intermediate VFAs while concentrations of other VFAs such that propionic acid were low. Acetobacterium and Clostridium were found to be the most abundant bacterial species in acidogenic reactor while methanogenic reactor was highly enriched with Methanobacterium and Methanosarcina. Apart from the above, syntrophic populations such as Syntrophobactor wolinii were also observed to be dominant in both the reactors – especially towards the end of acidogenic reactor and the initial part of the methanogenic reactor.     

Stability Analysis of a Model of Interaction Between the Immune System and Cancer Cells in Chronic Myelogenous Leukemia

We describe here a simple model for the interaction between leukemic cells and the autologous immune response in chronic phase chronic myelogenous leukemia (CML). This model is a simplified version of the model we proposed in Clapp et al. (Cancer Res 75:4053–4062, 2015). Our simplification is based on the observation that certain key characteristics of the dynamics of CML can be captured with a three-compartment model: two for the leukemic cells (stem cells and mature cells) and one for the immune response. We characterize the existence of steady states and their stability for generic forms of immunosuppressive effects of leukemic cells. We provide a complete co-dimension one bifurcation analysis. Our results show how clinical response to tyrosine kinase inhibitors treatment is compatible with the existence of a stable low disease, treatment-free steady state.     

The Sleep Cycle Modelled as a Cortical Phase Transition

We present a mean-field model of the cortex that attempts to describe the gross changes in brain electrical activity for the cycles of natural sleep. We incorporate within the model two major sleep modulatory effects: slow changes in both synaptic efficiency and in neuron resting voltage caused by the ∼90-min cycling in acetylcholine, together with even slower changes in resting voltage caused by gradual elimination during sleep of somnogens (fatigue agents) such as adenosine. We argue that the change from slow-wave sleep (SWS) to rapid-eye-movement (REM) sleep can be understood as a first-order phase transition from a low-firing, coherent state to a high-firing, desychronized cortical state. We show that the model predictions for changes in EEG power, spectral distribution, and correlation time at the SWS-to-REM transition are consistent not only with those observed in clinical recordings of a sleeping human subject, but also with the on-cortex EEG patterns recently reported by Destexhe et al. [J. Neurosci. 19(11), (1999) 4595–4608] for the sleeping cat.     

研究报告：果蝇习得性无助影响空间位置记忆

目的 习得性无助（learned helplessness）是指动物经历了不可逃避性刺激后而产生的无助状态。它对随后实验动物的运动能力、生理状态等多方面都会产生影响,也是人类抑郁症的有效动物模型之一。目前已在多种动物模型中建立了研究习得性无助神经机制的行为实验范式,包括黑腹果蝇（Drosophila melanogaster）。之前的研究发现,经历过不可逃避高温惩罚的果蝇在活跃程度等方面有显著改变。然而,果蝇的习得性无助状态是否也影响学习认知能力,目前仍然未知。本文对果蝇空间位置学习能力是否受到习得性无助状态影响进行了研究。方法 此研究中,我们完善实验范式,分别检测果蝇在新范式中的空间位置学习行为和习得性无助行为,以及经过长时程随机刺激后果蝇空间位置学习能力的变化。结果 野生型果蝇可在此范式中展现空间位置学习能力与习得性无助状态,而经过长时程不可逃避刺激训练的果蝇,空间位置记忆有显著下降。结论 此研究完善了果蝇习得性无助行为实验范式,实验结果表明习得性无助对果蝇空间位置记忆存在影响。这将对进一步加深对动物习得性无助行为的理解,进而其发生机制研究起到推进作用。     

Indirect Genetic Effects and the Dynamics of Social Interactions

BackgroundIndirect genetic effects (IGEs) occur when genes expressed in one individual alter the expression of traits in social partners. Previous studies focused on the evolutionary consequences and evolutionary dynamics of IGEs, using equilibrium solutions to predict phenotypes in subsequent generations. However, whether or not such steady states may be reached may depend on the dynamics of interactions themselves.ResultsIn our study, we focus on the dynamics of social interactions and indirect genetic effects and investigate how they modify phenotypes over time. Unlike previous IGE studies, we do not analyse evolutionary dynamics; rather we consider within-individual phenotypic changes, also referred to as phenotypic plasticity. We analyse iterative interactions, when individuals interact in a series of discontinuous events, and investigate the stability of steady state solutions and the dependence on model parameters, such as population size, strength, and the nature of interactions. We show that for interactions where a feedback loop occurs, the possible parameter space of interaction strength is fairly limited, affecting the evolutionary consequences of IGEs. We discuss the implications of our results for current IGE model predictions and their limitations.     

Modeling the adaptive immune response in HBV infection

The aim of this work is to investigate a new mathematical model that describes the interactions between Hepatitis B virus (HBV), liver cells (hepatocytes), and the adaptive immune response. The qualitative analysis of this as cytotoxic T lymphocytes (CTL) cells and the antibodies. These outcomes are (1) a disease free steady state, which its local stability is characterized as usual by R ₀ < 1, (2) and the existence of four endemic steady states when R ₀ > 1. The local stability of these steady states depends on functions of R ₀. Our study shows that although we give conditions of stability of these steady states, not all conditions are feasible. This rules out the local stability of two steady states. The conditions of stability of the two other steady states (which represent the complete failure of the adaptive immunity and the persistence of the disease) are formulated based on the domination of CTL cells response or the antibody response.     

Indirect inactivation of tyrosinase in its action on 4-tert-butylphenol

Abstract

Under anaerobic conditions, the o-diphenol 4-tert-butylcatechol (TBC) irreversibly inactivates met and deoxytyrosinase enzymatic forms of tyrosinase. However, the monophenol 4-tert-butylphenol (TBF) protects the enzyme from this inactivation. Under aerobic conditions, the enzyme suffers suicide inactivation when it acts on TBC. We suggest that TBF does not directly cause the suicide inactivation of the enzyme in the hydroxylase activity, but that the o-diphenol, which is necessary for the system to reach the steady state, is responsible for the process. Therefore, monophenols do not induce the suicide inactivation of tyrosinase in its hydroxylase activity, and there is a great difference between the monophenols that give rise to unstable o-quinones such as L-tyrosine, which rapidly accumulate L-dopa in the medium and those like TBF, after oxidation, give rise to a very stable o-quinone.     

A mathematical model for quorum sensing in Pseudomonas aeruginosa

The bacteria Pseudomonas aeruginosa use the size and density of their colonies to regulate the production of a large variety of substances, including toxins. This phenomenon, called quorum sensing, apparently enables colonies to grow to sufficient size undetected by the immune system of the host organism. In this paper, we present a mathematical model of quorum sensing in P. aeruginosa that is based on the known biochemistry of regulation of the autoinducer that is crucial to this signalling mechanism. Using this model we show that quorum sensing works because of a biochemical switch between two stable steady solutions, one with low levels of autoinducer and one with high levels of autoinducer.     

Dissipative Irreversibility from Nosé's Reversible Mechanics

Abstract

Nose's Hamiltonian mechanics makes possible the efficient simulation of irreversible flows of mass, momentum and energy. Such flows illustrate the paradox that reversible microscopic equations of motion underlie the irreversible behavior described by the second law of thermodynamics. This generic behavior of molecular many-body systems is illustrated here for the simplest possible system, with only one degree of freedom: a one-body Frenkel-Kontorova model for isothermal electronic conduction. This model system, described by Nosé-Hoover Hamiltonian dynamics, exhibits several interesting features: (1) deterministic and reversible equations of motion; (2) Lyapunov instability, with phase-space offsets increasing exponentially with time; (3) limit cycles; (4) dissipative conversion of work (potential energy) into heat (kinetic energy): and (5) phase-space contraction, a characteristic feature of steady irreversible flows. The model is particularly instructive in illustrating and explaining a paradox associated with steady-state statistical mechanics: the Gibbs entropy of a nonequilibrium steady state decreases continuously to minus infinity.     

Understanding drug resistance for monotherapy treatment of HIV infection

The purpose of this study was to investigate strategies in the monotherapy treatment of HIV infection in the presence of drug-resistant (mutant) strains. A mathematical system is developed to model resistance in HIV chemotherapy. It includes the key players in the immune response to HIV infection: virus and both uninfected CD4⁺ and infected CD4⁺ T-cell populations. We model the latent and progressive stages of the disease, and then introduce monotherapy treatment. The model is a system of differential equations describing the interaction of two distinct classes of HIV—drug-sensitive (wild type) and drug-resistant (mutant)—with lymphocytes in the peripheral blood. We then introduce chemotherapy effects. In the absence of treatment, the model produces the three types of qualitative clinical behavior—anuninfected steady state, andinfected steady state (latency), andprogression to AIDS. Simulation of treatment is provided for monotherapy, during theprogression to AIDS state, in the consideration of resistance effects. Treatment benefit is based on an increase or retention in CD4⁺ T-cell counts together with a low viral titer. We explore the following treatment approaches: an antiviral drug which reduces viral infectivity that is administered early—when the CD4⁺ T-cell count is ≥300/mm³, and late—when the CD4⁺ T-cell count is less than 300/mm³. We compare all results with data. When treatment is initiated during the progression to AIDS state, treatment prevents T-cell collapse, but gradually loses effectiveness due to drug resistance. We hypothesize that it is the careful balance of mutant and wild-type HIV strains which provides the greatest prolonged benefit from treatment. This is best achieved when treatment is initiated when the CD4⁺ T-cell counts are greater than 250/mm³, but less than 400/mm³ in this model (i.e. not too early, not too late). These results are supported by clinical data. The work is novel in that it is the first model to accurately simultate data before, during and after monotherapy treatment. Our model also provides insight into recent clinical results, as well as suggests plausible guidelines for clinical testing in the monotherapy of HIV infection.     

Window automata analysis of population dynamics in the immune system

A formalism based on window automata is proposed as a method to analyse complex population dynamics. The method is applied to a model of the immune network (Weisbuch, G.et al., 1990.J. theor. Biol. 146, 483–499), and used to predict which attractor the system reaches after antigenic stimulation, as a function of the parameters. The attractors of the dynamics are interpreted in terms of immune conditions such as vaccination or tolerance. Scaling laws that define the regimes in the parameter space corresponding to the specific attractor reached under antigenic stimulation are derived.     

Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth

Chronic hepatitis B virus (HBV) infection is a major cause of human suffering, and a number of mathematical models have examined within-host dynamics of the disease. Most previous HBV infection models have assumed that: (a) hepatocytes regenerate at a constant rate from a source outside the liver; and/or (b) the infection takes place via a mass action process. Assumption (a) contradicts experimental data showing that healthy hepatocytes proliferate at a rate that depends on current liver size relative to some equilibrium mass, while assumption (b) produces a problematic basic reproduction number. Here we replace the constant infusion of healthy hepatocytes with a logistic growth term and the mass action infection term by a standard incidence function; these modifications enrich the dynamics of a well-studied model of HBV pathogenesis. In particular, in addition to disease free and endemic steady states, the system also allows a stable periodic orbit and a steady state at the origin. Since the system is not differentiable at the origin, we use a ratio-dependent transformation to show that there is a region in parameter space where the origin is globally stable. When the basic reproduction number, R ₀, is less than 1, the disease free steady state is stable. When R ₀ > 1 the system can either converge to the chronic steady state, experience sustained oscillations, or approach the origin. We characterize parameter regions for all three situations, identify a Hopf and a homoclinic bifurcation point, and show how they depend on the basic reproduction number and the intrinsic growth rate of hepatocytes.     

Fitness versus longevity in age-structured population dynamics

 We examine the dynamics of an age-structured population model in which the life expectancy of an offspring may be mutated with respect to that of the parent. While the total population of the system always reaches a steady state, the fitness and age characteristics exhibit counter-intuitive behavior as a function of the mutational bias. By analytical and numerical study of the underlying rate equations, we show that if deleterious mutations are favored, the average fitness of the population reaches a steady state, while the average population age is a decreasing function of the average fitness. When advantageous mutations are favored, the average population fitness grows linearly with time t, while the average age is independent of the average fitness. For no mutational bias, the average fitness grows as $t^{2/3}$. Received: 21 December 1999 / Revised version: 31 October 2001 / Published online: 14 March 2002     

A TNF‐p100 pathway subverts noncanonical NF‐κB signaling in inflamed secondary lymphoid organs

Lymphotoxin‐beta receptor (LTβR) present on stromal cells engages the noncanonical NF‐κB pathway to mediate RelB‐dependent expressions of homeostatic chemokines, which direct steady‐state ingress of naïve lymphocytes to secondary lymphoid organs (SLOs). In this pathway, NIK promotes partial proteolysis of p100 into p52 that induces nuclear translocation of the RelB NF‐κB heterodimers. Microbial infections often deplete homeostatic chemokines; it is thought that infection‐inflicted destruction of stromal cells results in the downregulation of these chemokines. Whether inflammation per se also regulates these processes remains unclear. We show that TNF accumulated upon non‐infectious immunization of mice similarly downregulates the expressions of these chemokines and consequently diminishes the ingress of naïve lymphocytes in inflamed SLOs. Mechanistically, TNF inactivated NIK in LTβR‐stimulated cells and induced the synthesis of Nfkb2 mRNA encoding p100; these together potently accumulated unprocessed p100, which attenuated the RelB activity as inhibitory IκBδ. Finally, a lack of p100 alleviated these TNF‐mediated inhibitions in inflamed SLOs of immunized Nfkb2^?/? mice. In sum, we reveal that an inhibitory TNF‐p100 pathway modulates the adaptive compartment during immune responses.     

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     

A reaction–diffusion malaria model with incubation period in the vector population

Malaria is one of the most important parasitic infections in humans and more than two billion people are at risk every year. To understand how the spatial heterogeneity and extrinsic incubation period (EIP) of the parasite within the mosquito affect the dynamics of malaria epidemiology, we propose a nonlocal and time-delayed reaction–diffusion model. We then define the basic reproduction ratio R₀{\mathcal{R}_0} and show that R₀{\mathcal{R}_0} serves as a threshold parameter that predicts whether malaria will spread. Furthermore, a sufficient condition is obtained to guarantee that the disease will stabilize at a positive steady state eventually in the case where all the parameters are spatially independent. Numerically, we show that the use of the spatially averaged system may highly underestimate the malaria risk. The spatially heterogeneous framework in this paper can be used to design the spatial allocation of control resources.     

Viral dynamics model with CTL immune response incorporating antiretroviral therapy

We present two HIV models that include the CTL immune response, antiretroviral therapy and a full logistic growth term for uninfected $\text{ CD4}^+$ T-cells. The difference between the two models lies in the inclusion or omission of a loss term in the free virus equation. We obtain critical conditions for the existence of one, two or three steady states, and analyze the stability of these steady states. Through numerical simulation we find substantial differences in the reproduction numbers and the behaviour at the infected steady state between the two models, for certain parameter sets. We explore the effect of varying the combination drug efficacy on model behaviour, and the possibility of reconstituting the CTL immune response through antiretroviral therapy. Furthermore, we employ Latin hypercube sampling to investigate the existence of multiple infected equilibria.     

Age- and density-dependent population dynamics in static and variable environments

We consider a general model of a single-species population with age- and density-dependent per capita birth and death rates. In a static environment we show that if the per capita death rate is independent of age, then the local stability of any stationary state is guaranteed by the requirement that, in the region of the steady state, the density dependence of the birth rate should be negative and that of the death rate positive. In a variable environment we show that, provided the system is locally stable, small environmental fluctuations will give rise to small age structure and population fluctuations which are related to the driving environmental fluctuations by a simple “transfer function.” We illustrate our general theory by examining a model with a per capita death rate which is age and density independent and a per capita birth rate which is zero up to some threshold age a₀, adopts a finite density-dependent value up to a maximum age a_o + α, and is zero thereafter. We conclude from this model that resonance due specifically to single-species age-structure effects will only be of practical importance in populations whose members have a life cycle consisting of a long immature phase followed by a short burst of intense reproductive effort (α a_o).