Global Stability of Enzymatic Chains of Full Reversible Michaelis-Menten Reactions

We consider a chain of metabolic reactions catalyzed by enzymes, of reversible Michaelis-Menten type with full dynamics, i.e. not reduced with any quasi-steady state approximations. We study the corresponding dynamical system and show its global stability if the equilibrium exists. If the system is open, the equilibrium may not exist. The main tool is monotone systems theory. Finally we study the implications of these results for the study of coupled genetic-metabolic systems.     

Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks

We consider stochastically modeled chemical reaction systems with mass-action kinetics and prove that a product-form stationary distribution exists for each closed, irreducible subset of the state space if an analogous deterministically modeled system with mass-action kinetics admits a complex balanced equilibrium. Feinberg’s deficiency zero theorem then implies that such a distribution exists so long as the corresponding chemical network is weakly reversible and has a deficiency of zero. The main parameter of the stationary distribution for the stochastically modeled system is a complex balanced equilibrium value for the corresponding deterministically modeled system. We also generalize our main result to some non-mass-action kinetics.     

Marginal distributions of genetic coalgebras

We consider the backward evolution of a particular type of Mendelian genetic system whose transition probabilities give place to the so-called coalgebras with genetic realization and describe the equilibrium states of such mathematical objects and therefore those of the genetic system. We exploit the relationship between the genetic coalgebras modeling the transference of the genetic inheritance and cubic stochastic matrices of type (1, 2) studying first the ergodicity of these matrices in terms of the stationary probability distributions of the bivariate Markov chains defined by their accompanying matrices. Then we apply the obtained results to describe the equilibrium states of coalgebras with genetic realization.     

Engineered internal noise stochastic resonator in gene network: a model study

Based on a genetic bistable switch model coupled with a gene oscillator model, we have constructed a mesoscopic stochastic model for the coupled synthetic gene network, and studied how internal noise would influence the oscillation of such a system. We found that the state-to-state transitions can occur if the internal noise is taken into account, and the performance of resulting oscillation can reach a maximum in a certain internal noise level, which indicates the occurrence of internal noise stochastic resonance (SR) and makes the coupled gene network work as a stochastic resonator. The potential role of such an effect on gene expression systems is also discussed.     

Complex metabolic network of 1,3-propanediol transport mechanisms and its system identification via biological robustness

The bioconversion of glycerol to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae (K. pneumoniae) can be characterized by an intricate metabolic network of interactions among biochemical fluxes, metabolic compounds, key enzymes and genetic regulation. Since there are some uncertain factors in the fermentation, especially the transport mechanisms of 1,3-PD across cell membrane, the metabolic network contains multiple possible metabolic systems. Considering the genetic regulation of dha regulon and inhibition of 3-hydroxypropionaldehyde to the growth of cells, we establish a 14-dimensional nonlinear hybrid dynamical system aiming to determine the most possible metabolic system and the corresponding optimal parameter. The existence, uniqueness and continuity of solutions are discussed. Taking the robustness index of the intracellular substances together as a performance index, a system identification model is proposed, in which 1,395 continuous variables and 90 discrete variables are involved. The identification problem is decomposed into two subproblems and a parallel particle swarm optimization procedure is constructed to solve them. Numerical results show that it is most possible that 1,3-PD passes the cell membrane by active transport coupled with passive diffusion.     

CAN REINFORCEMENT OCCUR WITH A LEARNED TRAIT?

We use birdsong as a case study to ask whether reinforcement can occur via the spread of a genetically determined female preference for a socially inherited (learned) male trait. We envision secondary contact between two neighboring populations with different song dialects. An individual's ability to learn song is confined by a genetic predisposition: if predispositions are strong, there will be no phenotypic overlap in song between populations, whereas weak predispositions allow phenotypic overlap, or "mixed" song. To determine if reinforcement has occurred, we consider if an allele for within-population female mating preference, based on song, can spread, and whether population specific songs can concurrently be maintained at equilibrium. We model several scenarios, including costs to mating preferences, mating preferences in hybrids, and hybrids having the ability to learn pure songs. We find that when weak predispositions are fixed within a population reinforcement based on song cannot occur. However, when some individuals have strong predispositions, restricting phenotypic overlap between populations in the trait, reinforcement is only slightly inhibited from a purely genetic model. Generalizing beyond the example of song, we conclude that socially learned signals will tend to prohibit reinforcement, but it may still occur if some individuals acquire trait phenotypes genetically.     

Epigenetic inheritance, genetic assimilation and speciation.

Epigenetic inheritance systems enable the environmentally induced phenotypes to be transmitted between generations. Jablonka and Lamb (1991, 1995) proposed that these systems have a substantial role during speciation. They argued that divergence of isolated populations may be first triggered by the accumulation of (heritable) phenotypic differences that are later followed and strengthened by genetic changes. The plausibility of this idea is examined in this paper. At first, we discuss the "exploratory" behaviour of an epigenetic inheritance system on a one peak adaptive landscape. If a quantitative trait is far from the optimum, then it is advantageous to induce heritable phenotypic variation. Conversely, if the genotypes get closer to the peak, it is more favorable to canalize the phenotypic expression of the character. This process would lead to genetic assimilation. Next we show that the divergence of heritable epigenetic marks acts to reduce or to eliminate the genetic barrier between two adaptive peaks. Therefore, an epigenetic inheritance system can increase the probability of transition from one adaptive state to another. Peak shift might be initiated by (i) slight changes in the inducing environment or by (ii) genetic drift of the genes controlling epigenetic variability. Remarkably, drift-induced transition is facilitated even if phenotypic variation is not heritable. A corollary of our thesis is that evolution can proceed through suboptimal phenotypic states, without passing through a deep adaptive valley of the genotype. We also consider the consequences of this finding on the dynamics and mode of reproductive isolation.     

Global dynamics of two-compartment models for cell production systems with regulatory mechanisms

We present a global stability analysis of two-compartment models of a hierarchical cell production system with a nonlinear regulatory feedback loop. The models describe cell differentiation processes with the stem cell division rate or the self-renewal fraction regulated by the number of mature cells. The two-compartment systems constitute a basic version of the multicompartment models proposed recently by Marciniak-Czochra and collaborators [25] to investigate the dynamics of the hematopoietic system. Using global stability analysis, we compare different regulatory mechanisms. For both models, we show that there exists a unique positive equilibrium that is globally asymptotically stable if and only if the respective reproduction numbers exceed one. The proof is based on constructing Lyapunov functions, which are appropriate to handle the specific nonlinearities of the model. Additionally, we propose a new model to test biological hypothesis on the regulation of the fraction of differentiating cells. We show that such regulatory mechanism is incapable of maintaining homeostasis and leads to unbounded cell growth. Potential biological implications are discussed.     

Hopf bifurcation and bursting synchronization in an excitable systems with chemical delayed coupling

In this paper we consider the Hopf bifurcation and synchronization in the two coupled Hindmarsh–Rose excitable systems with chemical coupling and time-delay. We surveyed the conditions for Hopf bifurcations by means of dynamical bifurcation analysis and numerical simulation. The results show that the coupled excitable systems with no delay have supercritical Hopf bifurcation, while the delayed system undergoes Hopf bifurcations at critical time delays when coupling strength lies in a particular region. We also investigated the effect of the delay on the transition of bursting synchronization in the coupled system. The results are helpful for us to better understand the dynamical properties of excitable systems and the biological mechanism of information encoding and cognitive activity.     

Mathematical modelling for intra-specific brood-parasitism: coexistence between parasite and non-parasite

In this paper, we consider an aspect of the intra-specific brood-parasitism with a mathematical modelling. As in case of moorhen Gallinula chloropus, the case dealt with in this paper in such that just a part of the whole population has the parasitising behaviour against the individuals belonging to another part of non-parasite subpopulation that does not have such behaviour. Analysing the expected fitness gain from the brood-parasitism, we consider the condition in order that parasite individuals coexist with non-parasite ones within a population. From the mathematical modelling analysis, it is shown that the stable equilibrium frequency of parasite individuals within a population, if exists, depends on the difference among individuals in terms of the individual quality reflected to the survival probability of bred offsprings.     

Genetic versus phenotypic models of selection: can genetics be neglected in a long-term perspective?

Game theoretical concepts in evolutionary biology have been criticized by populations geneticists, because they neglect such crucial aspects as the mating system or the mode of inheritance. In fact, the dynamics of natural selection does not necessarily lead to a fitness maximum or an ESS if genetic constraints are taken into account. Yet, it may be premature to conclude that game theoretical concepts do not have a dynamical justification. The new paradigm of long-term evolution postulates that genetic constraints, which may be dominant in a short-term perspective, will in the long run disappera in the face of the ongoing influx of mutations. Two basic results (see Hammerstein; this issue) seem to reconcile the dynamical approach of long-term population genetics with the static approach of evolutionary game theory: (1) only populations at local fitness optima (Nash strategies) can be long-term stable; and (2) in monomorphic populations, evolutionary stability is necessary and sufficient to ensure long-term dynamic stability. The present paper has a double purpose. On the one hand, it is demonstrated by fairly general arguments that the scope of the results mentioned above extends to non-linear frequency dependent selection, to multiple loci, and to quite general mating systems. On the other hand, some limitations of the theory of long-term evolution will also be stressed: (1) there is little hope for a game theoretical characterization of stability in polymorphic populations; (2) many interesting systems do not admit long-term stable equilibria; and (3) even if a long-term stable equilibrium exists, it is not at all clear whether and how it is attainable by a series of gene substition events.     

Stochastic predation exposes prey to predator pits and local extinction

Understanding how predators affect prey populations is a fundamental goal for ecologists and wildlife managers. A well-known example of regulation by predators is the predator pit, where two alternative stable states exist and prey can be held at a low density equilibrium by predation if they are unable to pass the threshold needed to attain a high density equilibrium. While empirical evidence for predator pits exists, deterministic models of predator–prey dynamics with realistic parameters suggest they should not occur in these systems. Because stochasticity can fundamentally change the dynamics of deterministic models, we investigated if incorporating stochasticity in predation rates would change the dynamics of deterministic models and allow predator pits to emerge. Based on realistic parameters from an elk–wolf system, we found predator pits were predicted only when stochasticity was included in the model. Predator pits emerged in systems with highly stochastic predation and high carrying capacities, but as carrying capacity decreased, low density equilibria with a high likelihood of extinction became more prevalent. We found that incorporating stochasticity is essential to fully understand alternative stable states in ecological systems, and due to the interaction between top–down and bottom–up effects on prey populations, habitat management and predator control could help prey to be resilient to predation stochasticity.     

Prey dispersal and predator persistence

To understand how patchiness influences population dynamics of a tri-trophic interaction, a tractable model is formulated in terms of differential equations. Motivated by the structure of systems such as plants, phytophagous mites and predatory mites, the model takes dispersal into account at the middle trophic level. The effect of dispersal for the middle level in a tri-trophic system could be either stabilising or destabilising since the middle level acts both as prey and as predator. First a simple model with logistic growth for the lowest level is formulated. A model with logistic growth for the lowest level and instantaneous dispersal has a globally stable three-species equilibrium, if this equilibrium exists. Addition of a middle level dispersal phase of non-negligible duration influences equilibrium stability. In the absence of the top trophic level a limit cycle can occur, caused by the delay that exists in the reaction of the middle level to the changes in the lowest level. With low predator efficiency, it is also possible to have an unstable three-species equilibrium. So addition of a middle level dispersal phase of non-negligible duration can work destabilising. Next the persistence of the third trophic level is studied. Even when the three-species equilibrium exists, the third trophic level need not be persistent. A two-species limit cycle can keep its stability when a three-species equilibrium exists; the system is then bistable. It is argued that such a bistability can offer an alternative explanation for pesticide-induced outbreaks of spider mites and failure of predator introduction.     

The effects of human movement on the persistence of vector-borne diseases

With the recent resurgence of vector-borne diseases due to urbanization and development there is an urgent need to understand the dynamics of vector-borne diseases in rapidly changing urban environments. For example, many empirical studies have produced the disturbing finding that diseases continue to persist in modern city centers with zero or low rates of transmission. We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian models that mimic human commuting behavior and (ii) Eulerian models that mimic human migration. We determine the basic reproduction number R₀ for both modeling approaches. We show that for both approaches that if the disease-free equilibrium is stable (R₀<1) then it is globally stable and if the disease-free equilibrium is unstable (R₀>1) then there exists a unique positive (endemic) equilibrium that is globally stable among positive solutions. Finally, we prove in general that Lagrangian and Eulerian modeling approaches are not equivalent. The modeling approaches presented provide a framework to explore spatial vector-borne disease dynamics and control in heterogeneous environments. As an example, we consider two patches in which the disease dies out in both patches when there is no movement between them. Numerical simulations demonstrate that the disease becomes endemic in both patches when humans move between the two patches.     

Mutation in evolutionary games can increase average fitness at equilibrium

We study game dynamical interactions between two strategies, A and B, and analyse whether the average fitness of the population at equilibrium can be increased by adding mutation from A to B. Classifying all two by two games with payoff matrix [(a,b),(c,d)], we show that mutation from A to B enhances the average fitness of the whole population (i) if both a and d are less than (b + c)/2 and (ii) if c is less than b. Furthermore, we study conditions for maximizing the productivity of strategy A, and we analyse the effect of mutations in both directions. Depending on the biological system, a mutation in an evolutionary game can be interpreted as a genetic alteration, a cellular differentiation, a change in gene expression, an accidental or deliberate modification in cultural transmission, or a learning error. In a cultural context, our results indicate that the equilibrium payoff of the population can be increased if players sometimes choose the strategy with lower payoff. In a genetic context, we have shown that for frequency-dependent selection mutation can enhance the average fitness of the population at equilibrium.     

Inbreeding depression and heterosis in a subdivided population: influence of the mating system

We investigate the joint effects of gene flow and selfing on the level of inbreeding depression, heterosis and genetic load in a subdivided population at equilibrium. Low gene flow reduces inbreeding depression and substantially increases heterosis. However, in highly self-fertilizing populations, inbreeding depression is independent of the amount of gene flow. When migration occurs via pollen, consanguinity of the reproductive system could have a negative influence on subpopulation persistence, in contrast to the case of isolated populations. However, with only seed migration, genetic load and heterosis depend mildly on the mating system. From an evolutionary point of view, we reach two main conclusions: first, outcrossing is selected for if gene flow is low; second, intermediate levels of gene flow could promote mixed mating systems, especially when migration occurs through pollen.     

A multilocus-multiallele analysis of frequency-dependent selection induced by intraspecific competition

The study of the mechanisms that maintain genetic variation has a long history in population genetics. We analyze a multilocus-multiallele model of frequency- and density-dependent selection in a large randomly mating population. The number of loci and the number of alleles per locus are arbitrary. The n loci are assumed to contribute additively to a quantitative character under stabilizing or directional selection as well as under frequency-dependent selection caused by intraspecific competition. We assume the strength of stabilizing selection to be weak, whereas the strength of frequency dependence may be arbitrary. Density-dependence is induced by population regulation. Our main result is a characterization of the equilibrium structure and its stability properties in terms of all parameters. It turns out that no equilibrium exists with more than two alleles segregating per locus. We give necessary and sufficient conditions on the strength of frequency dependence to ensure the maintenance of multilocus polymorphism. We also give explicit formulas on the number of polymorphic loci maintained at equilibrium. These results are based on the assumption that selection is sufficiently weak compared with recombination, so that linkage equilibrium can be assumed. If additionally the population size is assumed to be constant, we prove that the dynamics of the model form a generalized gradient system. For the model in its general form we are able to derive necessary and sufficient conditions for the stability of the monomorphic equilibria. Furthermore, we briefly analyze a special symmetric two-locus two-allele model for a constant population size but allowing for linkage disequilibrium. Finally, we analyze a single diallelic locus with dominance to illustrate the complications that can occur if the assumption of additivity is relaxed.     

一个设立海洋保护区的捕鱼模型的生物经济分析

考虑一个设立海洋保护区的捕鱼模型．从生物学和经济学两个角度研究海洋保护区的作用．给出所研究系统的生物和经济平衡点以及平衡点的局部稳定性,不稳定性,全局稳定性的条件．研究结果说明设立海洋保护区对于保护海洋资源是一种有效途径,保护区内的鱼群即使在对非保护区进行连续捕捞的情形下也能够保持在一个适当的水平．进一步研究了最优收获策略,并对所得结果的生物经济解释予以说明．