Dynamical properties of Discrete Reaction Networks

Reaction networks are commonly used to model the dynamics of populations subject to transformations that follow an imposed stoichiometry. This paper focuses on the efficient characterisation of dynamical properties of Discrete Reaction Networks (DRNs). DRNs can be seen as modeling the underlying discrete nondeterministic transitions of stochastic models of reaction networks. In that sense, a proof of non-reachability in a given DRN has immediate implications for any concrete stochastic model based on that DRN, independent of the choice of kinetic laws and constants. Moreover, if we assume that stochastic kinetic rates are given by the mass-action law (or any other kinetic law that gives non-vanishing probability to each reaction if the required number of interacting substrates is present), then reachability properties are equivalent in the two settings. The analysis of two types of global dynamical properties of DRNs is addressed: irreducibility, i.e., the ability to reach any discrete state from any other state; and recurrence, i.e., the ability to return to any initial state. Our results consider both the verification of such properties when species are present in a large copy number, and in the general case. The necessary and sufficient conditions obtained involve algebraic conditions on the network reactions which in most cases can be verified using linear programming. Finally, the relationship of DRN irreducibility and recurrence with dynamical properties of stochastic and continuous models of reaction networks is discussed.     

Coupling oscillations and switches in genetic networks

Switches (bistability) and oscillations (limit cycle) are omnipresent in biological networks. Synthetic genetic networks producing bistability and oscillations have been designed and constructed experimentally. However, in real biological systems, regulatory circuits are usually interconnected and the dynamics of those complex networks is often richer than the dynamics of simple modules. Here we couple the genetic Toggle switch and the Repressilator, two prototypic systems exhibiting bistability and oscillations, respectively. We study two types of coupling. In the first type, the bistable switch is under the control of the oscillator. Numerical simulation of this system allows us to determine the conditions under which a periodic switch between the two stable steady states of the Toggle switch occurs. In addition we show how birhythmicity characterized by the coexistence of two stable small-amplitude limit cycles, can easily be obtained in the system. In the second type of coupling, the oscillator is placed under the control of the Toggleswitch. Numerical simulation of this system shows that this construction could for example be exploited to generate a permanent transition from a stable steady state to self-sustained oscillations (and vice versa) after a transient external perturbation. Those results thus describe qualitative dynamical behaviors that can be generated through the coupling of two simple network modules. These results differ from the dynamical properties resulting from interlocked feedback loops systems in which a given variable is involved at the same time in both positive and negative feedbacks. Finally the models described here may be of interest in synthetic biology, as they give hints on how the coupling should be designed to get the required properties.     

Intermediates and Generic Convergence to Equilibria

Known graphical conditions for the generic and global convergence to equilibria of the dynamical system arising from a reaction network are shown to be invariant under the so-called successive removal of intermediates, a systematic procedure to simplify the network, making the graphical conditions considerably easier to check.     

Multistationarity in mass action networks with applications to ERK activation

Ordinary Differential Equations (ODEs) are an important tool in many areas of Quantitative Biology. For many ODE systems multistationarity (i.e. the existence of at least two positive steady states) is a desired feature. In general establishing multistationarity is a difficult task as realistic biological models are large in terms of states and (unknown) parameters and in most cases poorly parameterized (because of noisy measurement data of few components, a very small number of data points and only a limited number of repetitions). For mass action networks establishing multistationarity hence is equivalent to establishing the existence of at least two positive solutions of a large polynomial system with unknown coefficients. For mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix, we present necessary and sufficient conditions for multistationarity that take the form of linear inequality systems. Solutions of these inequality systems define pairs of steady states and parameter values. We also present a sufficient condition to identify networks where the aforementioned conditions hold. To show the applicability of our results we analyse an ODE system that is defined by the mass action network describing the extracellular signal-regulated kinase (ERK) cascade (i.e. ERK-activation).     

Structural Instability and Emergence of the Biodiversity

In the framework of population dynamics, we start from the logistic equation describing the evolution of one species with limited food supply. A split device allows us to consider the population as two sub-populations x and y evolving analogously. The dynamical system has a one-parameter family of equilibria which is structurally unstable. Then small perturbations of the system (describing functional or ethological differentiations between the sub-species) lead in general to a new system involving a fast and a slow dynamics with a finite number of equilibria. In simple situations where the differentiation is clearly either an advantage or an inconvenience for one of the subspecies, the stable equilibrium amounts to extinction of the disadvantaged subspecies (elementary Darwinism). Oppositely more complex differentiations (involving both advantages and inconveniences) often lead to stable equilibria with well-defined non zero proportions of the sub-populations (preservation of the biodiversity). Other examples are concerned with symbiosis-like differentiations, leading to preservation, whereas the opposite case (mutual nuisances) has an unstable equilibrium and lead to extinction of one or the other subspecies according to the initial conditions. The case of a scission into three subspecies is more rich in consequences. In certain cases, predator-prey relations lead to auto-organization phenomena with stable diversity-preserving diversity. Cases of instability are also possible, leading to orbits tending towards a poly-cycle.This implies some kind of pseudo-extinction: this amounts to “pseudo-periodic-like” orbits with “pseudo-periods” larger and larger, tending to infinity; each pseudo-period contains parts where one of the sub-populations practically vanish. Other non-linear perturbations lead to stable orbits.     

Balance of Interactions Determines Optimal Survival in Multi-Species Communities

We consider a multi-species community modelled as a complex network of populations, where the links are given by a random asymmetric connectivity matrix J, with fraction 1 − C of zero entries, where C reflects the over-all connectivity of the system. The non-zero elements of J are drawn from a Gaussian distribution with mean μ and standard deviation σ. The signs of the elements J _ij reflect the nature of density-dependent interactions, such as predatory-prey, mutualism or competition, and their magnitudes reflect the strength of the interaction. In this study we try to uncover the broad features of the inter-species interactions that determine the global robustness of this network, as indicated by the average number of active nodes (i.e. non-extinct species) in the network, and the total population, reflecting the biomass yield. We find that the network transitions from a completely extinct system to one where all nodes are active, as the mean interaction strength goes from negative to positive, with the transition getting sharper for increasing C and decreasing σ. We also find that the total population, displays distinct non-monotonic scaling behaviour with respect to the product μC, implying that survival is dependent not merely on the number of links, but rather on the combination of the sparseness of the connectivity matrix and the net interaction strength. Interestingly, in an intermediate window of positive μC, the total population is maximal, indicating that too little or too much positive interactions is detrimental to survival. Rather, the total population levels are optimal when the network has intermediate net positive connection strengths. At the local level we observe marked qualitative changes in dynamical patterns, ranging from anti-phase clusters of period 2 cycles and chaotic bands, to fixed points, under the variation of mean μ of the interaction strengths. We also study the correlation between synchronization and survival, and find that synchronization does not necessarily lead to extinction. Lastly, we propose an effective low dimensional map to capture the behavior of the entire network, and this provides a broad understanding of the interplay of the local dynamical patterns and the global robustness trends in the network.     

La Dynamique Posturale Non-Linéaire. Partie I : bases et fondements

The different sensorial systems involved in the postural control form a super system preserving the static and dynamical equilibrium of a standing human being, motionless or moving. Now, the human posture captors are endowed with conditions that are sufficient to constitute a dynamical system. Then, we can show that: (i) generic properties of dynamical systems may be applied to the postural system, (ii) we may benefit from conceptual justifications to postural interactions experienced at a clinical level. Moreover, concerning the issue of the multi-causal chain in posturology, it is possible to infer and characterize a certain number of generic properties relative to these postural interactions. From this theoretical grounding, based on the knowledge of dynamical systems, it appears licit to introduce the concept of Non-linear Postural Dynamics (NPD). In a first part of this paper, the bases and foundations of NPD are presented, while the second part deals with different possible modes of NPD. Indeed, because the postural system is comparable to the structure of a dynamical system, it turns out that the former, natural seat of non-linear interaction dynamics, exhibits four dynamical modes and only four, « solutions » to the whole postural system. These four modes are as follows: (1) the quasi-linear mode, (2) the stationary mode, (3) the non-linear convergent mode, and (4) the chaotic mode. The present paper is devoted to the definition and articulation of these postural dynamic's modes. A codification of the clinical elements of their recognition should constitute a logically indispensable sequel. However, by their very constitutive principles themselves, some of these modes call for a functional multi-disciplinary approach. As a consequence, in the search for therapeutic protocols, not blindly but rather built on the interplay of various medical fields relative to posture and its physio-dynamical equilibrium, it is necessary to identify in a first step, and as surely as possible, the postural mode in which the patient stands.     

Bistability and oscillations in chemical reaction networks

Bifurcation theory is one of the most widely used approaches for analysis of dynamical behaviour of chemical and biochemical reaction networks. Some of the interesting qualitative behaviour that are analyzed are oscillations and bistability (a situation where a system has at least two coexisting stable equilibria). Both phenomena have been identified as central features of many biological and biochemical systems. This paper, using the theory of stoichiometric network analysis (SNA) and notions from algebraic geometry, presents sufficient conditions for a reaction network to display bifurcations associated with these phenomena. The advantage of these conditions is that they impose fewer algebraic conditions on model parameters than conditions associated with standard bifurcation theorems. To derive the new conditions, a coordinate transformation will be made that will guarantee the existence of branches of positive equilibria in the system. This is particularly useful in mathematical biology, where only positive variable values are considered to be meaningful. The first part of the paper will be an extended introduction to SNA and algebraic geometry-related methods which are used in the coordinate transformation and set up of the theorems. In the second part of the paper we will focus on the derivation of bifurcation conditions using SNA and algebraic geometry. Conditions will be derived for three bifurcations: the saddle-node bifurcation, a simple branching point, both linked to bistability, and a simple Hopf bifurcation. The latter is linked to oscillatory behaviour. The conditions derived are sufficient and they extend earlier results from stoichiometric network analysis as can be found in (Aguda and Clarke in J Chem Phys 87:3461–3470, 1987; Clarke and Jiang in J Chem Phys 99:4464–4476, 1993; Gatermann et al. in J Symb Comput 40:1361–1382, 2005). In these papers some necessary conditions for two of these bifurcations were given. A set of examples will illustrate that algebraic conditions arising from given sufficient bifurcation conditions are not more difficult to interpret nor harder to calculate than those arising from necessary bifurcation conditions. Hence an increasing amount of information is gained at no extra computational cost. The theory can also be used in a second step for a systematic bifurcation analysis of larger reaction networks. We have added a dedication of the paper to K. Gatermann.     

Nash equilibria for an evolutionary language game

We study an evolutionary language game that describes how signals become associated with meaning. In our context, a language, L, is described by two matrices: the P matrix contains the probabilities that for a speaker certain objects are associated with certain signals, while the Q matrix contains the probabilities that for a listener certain signals are associated with certain objects. We define the payoff in our evolutionary language game as the total amount of information exchanged between two individuals. We give a formal classification of all languages, L(P, Q), describing the conditions for Nash equilibria and evolutionarily stable strategies (ESS). We describe an algorithm for generating all languages that are Nash equilibria. Finally, we show that starting from any random language, there exists an evolutionary trajectory using selection and neutral drift that ends up with a strategy that is a strict Nash equilibrium (or very close to a strict Nash equilibrium). Received: 1 March 2000 / Published online: 3 August 2000     

Effects of density dependent sex allocation on the dynamics of a simultaneous hermaphroditic population: Modelling and analysis

In this work we present a mathematical model describing the dynamics of a population where sex allocation remains flexible throughout adult life and so can be adjusted to current environmental conditions. We consider that the fractions of immature individuals acquiring male and female sexual roles are density dependent through nonlinear functions of a weighted total population size. The main goal of this work is to understand the role of life-history parameters on the stabilization or destabilization of the population dynamics.The model turns out to be a nonlinear discrete model which is analysed by studying the existence of fixed points as well as their stability conditions in terms of model parameters. The existence of more complex asymptotic behaviours of system solutions is shown by means of numerical simulations.Females have larger fertility rate than males. On the other hand, increasing population density favours immature individuals adopting the male role. A positive equilibrium of the system exists whenever fertility and survival rates of one of the sexual roles, if shared by all adults, allow population growing while the opposite happens with the other sexual role. In terms of the female inherent net reproductive number, η_F, it is shown that the positive equilibria are stable when η_F is larger and closed to 1 while for larger values of η_F a certain asymptotic assumption on the investment rate in the female function implies that the population density is permanent. Depending on the other parameters values, the asymptotic behaviour of solutions becomes more complex, even chaotic. In this setting the stabilization/destabilization effects of the abruptness rate in density dependence, of the survival rates and of the competition coefficients are analysed.     

Parent-offspring conflict that is not limited by degree of kinship

Models are presented showing that, under certain conditions, when multiple offspring are reared simultaneously, the magnitude of offspring selfishness is independent of offspring-offspring genetic relatedness. When the cost (θ) of the selfish offspring's behavior approaches zero, alleles causing offspring selfishness spread as long as $\text{c}\text{b}$ is positive.In general, simultaneous offspring are predicted to behave more selfishly than sequential offspring.Stable equilibria between alleles for selfish and non-selfish behavior are predicted for certain selfish alleles.     

Community congruence of plants,invertebrates and birds in natural and constructed shallow open-water wetlands: Do we need to monitor multiple assemblages?

Biomonitoring is a common means of evaluating wetlands. It is based on the premise that the community composition of one taxonomic group is indicative of overall biology and the underlying environmental conditions at a wetland. To be a good bioindicator, there must be adequate concordance between the indicator group and other biotic assemblages. Otherwise, multi-assemblage monitoring is necessary to glean a complete picture of wetland condition. In 32 sites ranging from reference wetlands to stormwater retention ponds, we evaluated concordance in community composition among the six most commonly monitored wetland assemblages: waterfowl, wetland dependent songbirds, aquatic macroinvertebrates, and plants in the wet meadow, emergent, and open-water vegetation zones. We also assessed agreement in environmental correlates among these six assemblages and investigated the impact of human disturbance on cross-assemblage concordance. We found that cross-assemblage concordance was positive (p < 0.03 in 14 of 15 pair-wise comparisons, p = 0.06 in 15th case), but relatively low (Mantel R values 0.11–0.37), suggesting that the assemblages are mediocre surrogates for one another. Yet, we found very strong agreement among environmental correlates of the six assemblages, especially along the first axis of assemblage-specific ordinations (mean Spearman rho = 0.923), indicating that despite low concordance, the six assemblages are likely responding to the same environmental gradients. Thus, while a single assemblage may not serve as a surrogate for the other assemblages, it should yield an adequate estimate of underlying environmental conditions and the degree of disturbance. Most important among the environmental correlates were sediment and water nutrient levels, shoreline slope, and the size of wet meadow and emergent vegetation zones. Perhaps most interestingly, we found that the strength of cross-assemblage concordance was greatest in reference wetlands and was lower (p ≤ 0.05) in constructed wetlands. This implies that cross-assemblage concordance present in undisturbed sites may not persist in disturbed wetlands where several of these cross-assemblage relationships deteriorate. Furthermore, a general change in cross-assemblage concordance may itself be indicative of human disturbance in wetlands.     

Multiparametric bifurcations of an epidemiological model with strong Allee effect

In this paper we completely study bifurcations of an epidemic model with five parameters introduced by Hilker et al. (Am Nat 173:72–88, 2009), which describes the joint interplay of a strong Allee effect and infectious diseases in a single population. Existence of multiple positive equilibria and all kinds of bifurcation are examined as well as related dynamical behavior. It is shown that the model undergoes a series of bifurcations such as saddle-node bifurcation, pitchfork bifurcation, Bogdanov–Takens bifurcation, degenerate Hopf bifurcation of codimension two and degenerate elliptic type Bogdanov–Takens bifurcation of codimension three. Respective bifurcation surfaces in five-dimensional parameter spaces and related dynamical behavior are obtained. These theoretical conclusions confirm their numerical simulations and conjectures by Hilker et al., and reveal some new bifurcation phenomena which are not observed in Hilker et al. (Am Nat 173:72–88, 2009). The rich and complicated dynamics exhibit that the model is very sensitive to parameter perturbations, which has important implications for disease control of endangered species.     

DNA condensation with polyamines I. Spectroscopic studies

The addition of polyamines with three or four positive charges to very dilute solutions of phage T7 DNA leads to a co-operative condensation. The reaction is very rapid and the DNA remains in the B-form as characterized by circular dichroism. The particles which are formed are roughly the size of a phage particle when they are prepared for electron microscopy. This aspect is discussed more completely in the accompanying paper (Chattoraj et al., 1978).Most of the experiments were performed at low ionic strength (roughly 0.002 m) with the triamine, spermidine. The reaction also occurs in 0.15 m-sodium chloride but here the experiments are accompanied by slow irreversible effects which are evidently due to aggregation since they are accompanied by a commensurate increase in turbidity. Consequently, most of the experiments have been done under the reversible low ionic strength conditions.Neither Mg²⁺ nor the diamine putrescine produce the reaction at concentrations similar to those found in bacterial cells. The tetramer spermine, on the other hand, which is not found in bacterial cells, is a very strong condensation agent in the μm region. The spermidine analog, bis-(3-aminopropyl)amine is very similar in behavior to spermidine.The role which polyamines might play in the condensation of DNA in phage heads is discussed.     

S-Adenosyl homocysteine and DNA ends stimulate promiscuous nuclease activities in the Type III restriction endonuclease EcoPI

In the absence of the methyl donor S-adenosyl methionine and under certain permissive reaction conditions, EcoPI shows non-specific endonuclease activity. We show here that the cofactor analogue S-adenosyl homocysteine promotes this promiscuous DNA cleavage. Additionally, an extensive exonuclease-like processing of the DNA is also observed that can even result in digestion of non-specific DNA in trans. We suggest a model for how DNA communication events initiating from non-specific sites, and in particular free DNA ends, could produce the observed cleavage patterns.     

Site-specific recombination Xis-independent excisive recombination of bacteriophage lambda

The integration of bacteriophage lambda into the Escherichia coli chromosome depends on the phage-encoded Int protein; prophage excision depends on Int and a second phage function, Xis. Limited excisive recombination has been observed in vivo with certain xis mutants, suggesting that Int may be able to carry out excision without Xis.We report here that purified Int protein carries out lambda site-specific excisive recombination in vitro in the absence of Xis. This reaction requires host factors derived from a non-lysogenic E. coli strain and is influenced strongly by ionic strength. Excision in the absence of Xis occurs slowly at low salt concentrations (40 mm-NaCl) and very little excision occurs at high salt concentrations (100 mm-NaCl). In the presence of Xis, excisive recombination proceeds rapidly at both low and high ionic strengths. These observations are consistent with previous experiments that suggested the partial dispensability of Xis for excision.     

Analysis of volume regulation in an epithelial cell model

An epithelial cell is modeled as a single compartment, bounded by apical and basolateral cell membranes, and containing two nonelectrolyte solute species, nominally NaCl and KCl. Membrane transport of these species may be metabolically driven, or it may follow the transmembrane concentration gradients, either singly (a channel) or jointly (a cotransporter). To represent the effect of stretch-activated channels or shrinkage-activated cotransporters, the membrane permeabilities and cotransport coefficients are permitted to be functions of cell volume. When this epithelium is considered as a dynamical system, conditions are indicated which guarantee the uniqueness and stability of equilibria. Experimentally, many epithelial cells can regulate their volume, and such volume regulatory capability is defined for this model. It is clearly distinct from dynamical stability of the equilibrium and requires more stringent conditions on the volume-dependent permeabilities and cotransporters. For a previously developed model of the toad urinary bladder (Strieteret al., 1990,J. gen. Physiol. 96, 319–344) the uniqueness and stability of its equilibria are indicated. The analysis also demonstrates that under some conditions a second stable equilibrium may appear, along with a saddle-node bifurcation. This is illustrated numerically in a modified model of the epithelium of the thick ascending limb of Henle.     

The pyrite iron cycle catalyzed by Acidithiobacillus ferrooxidans

In this paper, we study a model of the biotic pyrite iron cycle catalyzed by bacteria Acidithiobacillus ferrooxidans, in mining activity sites waste dumps. Chemical reactions, reaction rates and the population growth model are mostly taken from the existing literature. Analysis of the corresponding dynamical system shows the existence of up to four non-trivial steady state solutions. The stability of the equilibria solutions is determined, finding up to two coexisting stable solutions. Two Hopf bifurcations and a region of parameter space in which there are stable periodic solutions are found. In addition, we find a homoclinic bifurcation which gives rise to a stable periodic orbit of large period. The existence of these stable oscillatory solutions gives a possible explanation for the oscillations of bacteria concentration and pH for the iron cycle, described in Jaynes et al. (Water Resour Res 20:233–242, 1984).     

Stabilizing patterning in the Drosophila segment polarity network by selecting models in silico

The segmentation of Drosophila is a prime model to study spatial patterning during embryogenesis. The spatial expression of segment polarity genes results from a complex network of interacting proteins whose expression products are maintained after successful segmentation. This prompted us to investigate the stability and robustness of this process using a dynamical model for the segmentation network based on Boolean states. The model consists of intra-cellular as well as inter-cellular interactions between adjacent cells in one spatial dimension. We quantify the robustness of the dynamical segmentation process by a systematic analysis of mutations. Our starting point consists in a previous Boolean model for Drosophila segmentation. We define mathematically the notion of dynamical robustness and show that the proposed model exhibits limited robustness in gene expression under perturbations. We applied in silico evolution (mutation and selection) and discover two classes of modified gene networks that have a more robust spatial expression pattern. We verified that the enhanced robustness of the two new models is maintained in differential equations models. By comparing the predicted model with experiments on mutated flies, we then discuss the two types of enhanced models. Drosophila patterning can be explained by modelling the underlying network of interacting genes. Here we demonstrate that simple dynamical considerations and in silico evolution can enhance the model to robustly express the expected pattern, helping to elucidate the role of further interactions.     

Comprehensive Logic Based Analyses of Toll-Like Receptor 4 Signal Transduction Pathway

Among the 13 TLRs in the vertebrate systems, only TLR4 utilizes both Myeloid differentiation factor 88 (MyD88) and Toll/Interleukin-1 receptor (TIR)-domain-containing adapter interferon-β-inducing Factor (TRIF) adaptors to transduce signals triggering host-protective immune responses. Earlier studies on the pathway combined various experimental data in the form of one comprehensive map of TLR signaling. But in the absence of adequate kinetic parameters quantitative mathematical models that reveal emerging systems level properties and dynamic inter-regulation among the kinases/phosphatases of the TLR4 network are not yet available. So, here we used reaction stoichiometry-based and parameter independent logical modeling formalism to build the TLR4 signaling network model that captured the feedback regulations, interdependencies between signaling kinases and phosphatases and the outcome of simulated infections. The analyses of the TLR4 signaling network revealed 360 feedback loops, 157 negative and 203 positive; of which, 334 loops had the phosphatase PP1 as an essential component. The network elements'' interdependency (positive or negative dependencies) in perturbation conditions such as the phosphatase knockout conditions revealed interdependencies between the dual-specific phosphatases MKP-1 and MKP-3 and the kinases in MAPK modules and the role of PP2A in the auto-regulation of Calmodulin kinase-II. Our simulations under the specific kinase or phosphatase gene-deficiency or inhibition conditions corroborated with several previously reported experimental data. The simulations to mimic Yersinia pestis and E. coli infections identified the key perturbation in the network and potential drug targets. Thus, our analyses of TLR4 signaling highlights the role of phosphatases as key regulatory factors in determining the global interdependencies among the network elements; uncovers novel signaling connections; identifies potential drug targets for infections.