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.     

Quasi-steady-state approximation for chemical reaction networks

 The parameter embedding leading to the quasi-steady-state approximation of Heinrich [9] is investigated within the theory of invariant manifolds of Fenichel [4] in order to clarify the essential assumptions needed for this reduction to a low dimensional system. In particular, the concept of pool-variables can be avoided in this generalized approach. Moreover, the dominating influence of the slow subnetwork over the complementary fast subnetwork is interpreted geometrically and in chemical terms and this can be seen as an “enslaving” of the fast subsystem by the slow subsystem. Finally, the results are applied to a system of slime mould communication [6, 7, 13] and to a maltose transport system [2, 3]. Received: 16 June 1997     

Parental teaching behavior in diverse cultural contexts

Humans, unlike any other species, have adapted to diverse environments across the globe due to cultural knowledge, skills, and practices. In early childhood, parent–child interactions play a pivotal role in cultural learning, but controversies about what constitutes teaching have stymied the systematic assessment of variation and similarities in parental teaching across cultures. We used a functional definition of teaching as behavior that evolved to facilitate learning in others and observed parents and their two-year-old child (N = 106) in a standardized setting (mealtime) in five diverse cultural contexts (rural: Brazil, Ecuador; urban: Argentina, Germany, Japan). Detailed coding of parental teaching (8520 teaching events in 1898 min of interactions) revealed that six teaching behaviors (prompts to do, prompts to stop, abstract communication of knowledge, demonstrations, providing choices, negative feedback) occurred frequently (> 5%) across contexts. At the same time, we found that the relative frequencies of these behaviors varied between contexts: Parents in rural contexts frequently used prompts towards their children. Parents in urban contexts often used abstract communication, demonstrations, and the provision of choices. We also identified nuanced differences between the urban samples regarding parental demonstrations (Japan) and the provision of choices (Germany). Our findings suggest that parents from all contexts mainly relied on a set of six teaching behaviors but that there was cross-cultural variation in how frequently behaviors occurred. Our study provides a cultural map of children's early learning experiences which lay the ontogenetic foundation for human cultural diversity.     

The flow topology of chemical reaction networks

Analysis of the flow topology of dynamic systems modelling chemical reaction networks leads to the following conclusions: (1) a connected chemical system with a negative feedback circuit containing all except one of the internal species (phase determining intermediates) can exhibit chaotic oscillations; (2) a chemical system in which the only negative feedback circuit contains all of the internal species can exhibit periodic oscillations; (3) a chemical system containing no negative feedback circuits cannot exhibit oscillatory behavior but only multiple steady states. In this analysis a negative feedback circuit refers to a circuit containing an odd number of inhibitory relationships between internal species.     

Concordant chemical reaction networks and the Species-Reaction Graph

In a recent paper it was shown that, for chemical reaction networks possessing a subtle structural property called concordance, dynamical behavior of a very circumscribed (and largely stable) kind is enforced, so long as the kinetics lies within the very broad and natural weakly monotonic class. In particular, multiple equilibria are precluded, as are degenerate positive equilibria. Moreover, under certain circumstances, also related to concordance, all real eigenvalues associated with a positive equilibrium are negative. Although concordance of a reaction network can be decided by readily available computational means, we show here that, when a nondegenerate network’s Species-Reaction Graph satisfies certain mild conditions, concordance and its dynamical consequences are ensured. These conditions are weaker than earlier ones invoked to establish kinetic system injectivity, which, in turn, is just one ramification of network concordance. Because the Species-Reaction Graph resembles pathway depictions often drawn by biochemists, results here expand the possibility of inferring significant dynamical information directly from standard biochemical reaction diagrams.     

Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates

This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence under the assumption that the rates of the reactions are monotone functions of the concentrations of their reactants. This is satisfied for, yet not restricted to, the most common choices of the reaction kinetics such as mass action, Michaelis-Menten and Hill kinetics. The key idea is to find an alternative representation under which the resulting system is monotone. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property. We also provide a global stability result for a more complicated example that describes a regulatory pathway of a prevalent signal transduction module, the MAPK cascade.     

Multiple equilibria in complex chemical reaction networks: extensions to entrapped species models

In two earlier papers, means were provided to decide the capacity of complex chemical reaction networks, taken with mass-action kinetics, to admit multiple equilibria in the context of the isothermal homogeneous continuous flow stirred tank reactor (CFSTR). In such a reactor, all species are deemed to be in the outflow, a fact which has an important bearing on the nature of the governing equations. In contrast, one can imagine CFSTR-like models of the cell in which certain large molecules (e.g., enzymes) remain entrapped within the cell, whereas smaller ones (e.g., metabolites) are free to diffuse through the cell boundary. Although such models bear a strong physical resemblance to the classical CFSTR picture, there are substantive differences in the corresponding mathematics. Without a presumption of mass-action kinetics, this research is intended to indicate a general way in which results about uniqueness of equilibria in the classical CFSTR context extend to entrapped species models.     

Inferring biomolecular regulatory networks from phase portraits of time-series expression profiles

Reverse engineering of biomolecular regulatory networks such as gene regulatory networks, protein interaction networks, and metabolic networks has received an increasing attention as more high-throughput time-series measurements become available. In spite of various approaches developed from this motivation, it still remains as a challenging subject to develop a new reverse engineering scheme that can effectively uncover the functional interaction structure of a biomolecular network from given time-series expression profiles (TSEPs). We propose a new reverse engineering scheme that makes use of phase portraits constructed by projection of every two TSEPs into respective phase planes. We introduce two measures of a slope index (SI) and a winding index (WI) to quantify the interaction properties embedded in the phase portrait. Based on the SI and WI, we can reconstruct the functional interaction network in a very efficient and systematic way with better inference results compared to previous approaches. By using the SI, we can also estimate the time-lag accompanied with the interaction between molecular components of a network.     

Prediction of occurrences of diverse chemical classes in the Asteraceae through artificial neural networks

The training and the application of a neural network system for the prediction of occurrences of secondary metabolites belonging to diverse chemical classes in the Asteraceae is described. From a database containing about 604 genera and 28,000 occurrences of secondary metabolites in the plant family, information was collected encompassing nine chemical classes and their respective occurrences for training of a multi-layer net using the back-propagation algorithm. The net supplied as output the presence or absence of the chemical classes as well as the number of compounds isolated from each taxon. The results provided by the net from the presence or absence of a chemical class showed a 89% hit rate; by excluding triterpenes from the analysis, only 5% of the genera studied exhibited errors greater than 10%.     

WIPER: Weighted in-Path Edge Ranking for biomolecular association networks

Background: In network biology researchers generate biomolecular networks with candidate genes or proteins experimentally-derived from high-throughput data and known biomolecular associations. Current bioinformatics research focuses on characterizing candidate genes/proteins, or nodes, with network characteristics, e.g., betweenness centrality. However, there have been few research reports to characterize and prioritize biomolecular associations (“edges”), which can represent gene regulatory events essential to biological processes.Method: We developed Weighted In-Path Edge Ranking (WIPER), a new computational algorithm which can help evaluate all biomolecular interactions/associations (“edges”) in a network model and generate a rank order of every edge based on their in-path traversal scores and statistical significance test result. To validate whether WIPER worked as we designed, we tested the algorithm on synthetic network models.Results: Our results showed WIPER can reliably discover both critical “well traversed in-path edges”, which are statistically more traversed than normal edges, and “peripheral in-path edges”, which are less traversed than normal edges. Compared with other simple measures such as betweenness centrality, WIPER provides better biological interpretations. In the case study of analyzing postanal pig hearts gene expression, WIPER highlighted new signaling pathways suggestive of cardiomyocyte regeneration and proliferation. In the case study of Alzheimer’s disease genetic disorder association, WIPER reports SRC:APP, AR:APP, APP:FYN, and APP:NES edges (gene-gene associations) both statistically and biologically important from PubMed co-citation.Conclusion: We believe that WIPER will become an essential software tool to help biologists discover and validate essential signaling/regulatory events from high-throughput biology data in the context of biological networks.Availability: The free WIPER API is described at discovery.informatics.uab.edu/wiper/     

Mining and state-space modeling and verification of sub-networks from large-scale biomolecular networks

Background  

Biomolecular networks dynamically respond to stimuli and implement cellular function. Understanding these dynamic changes is the key challenge for cell biologists. As biomolecular networks grow in size and complexity, the model of a biomolecular network must become more rigorous to keep track of all the components and their interactions. In general this presents the need for computer simulation to manipulate and understand the biomolecular network model.     

Discovery of biological networks from diverse functional genomic data

We have developed a general probabilistic system for query-based discovery of pathway-specific networks through integration of diverse genome-wide data. This framework was validated by accurately recovering known networks for 31 biological processes in Saccharomyces cerevisiae and experimentally verifying predictions for the process of chromosomal segregation. Our system, bioPIXIE, a public, comprehensive system for integration, analysis, and visualization of biological network predictions for S. cerevisiae, is freely accessible over the worldwide web.     

Substrate type determines metagenomic profiles from diverse chemical habitats

Environmental parameters drive phenotypic and genotypic frequency variations in microbial communities and thus control the extent and structure of microbial diversity. We tested the extent to which microbial community composition changes are controlled by shifting physiochemical properties within a hypersaline lagoon. We sequenced four sediment metagenomes from the Coorong, South Australia from samples which varied in salinity by 99 Practical Salinity Units (PSU), an order of magnitude in ammonia concentration and two orders of magnitude in microbial abundance. Despite the marked divergence in environmental parameters observed between samples, hierarchical clustering of taxonomic and metabolic profiles of these metagenomes showed striking similarity between the samples (>89%). Comparison of these profiles to those derived from a wide variety of publically available datasets demonstrated that the Coorong sediment metagenomes were similar to other sediment, soil, biofilm and microbial mat samples regardless of salinity (>85% similarity). Overall, clustering of solid substrate and water metagenomes into discrete similarity groups based on functional potential indicated that the dichotomy between water and solid matrices is a fundamental determinant of community microbial metabolism that is not masked by salinity, nutrient concentration or microbial abundance.     

Computing weakly reversible linearly conjugate chemical reaction networks with minimal deficiency

Mass-action kinetics is frequently used in systems biology to model the behavior of interacting chemical species. Many important dynamical properties are known to hold for such systems if their underlying networks are weakly reversible and have a low deficiency. In particular, the Deficiency Zero and Deficiency One Theorems guarantee strong regularity with regards to the number and stability of positive equilibrium states. It is also known that chemical reaction networks with distinct reaction structure can admit mass-action systems with the same qualitative dynamics. The theory of linear conjugacy encapsulates the cases where this relationship is captured by a linear transformation. In this paper, we propose a mixed-integer linear programming algorithm capable of determining the minimal deficiency weakly reversible reaction network which admits a mass-action system which is linearly conjugate to a given reaction network.     

Transformation of chemical energy in biomolecular systems with hydrogen bonds

The mechanism underlying the excitation of the hydrogen bond with ATP hydrolysis was considered. Coulomb interactions of the proton of the hydrogen bond A-H...B with the electrical field of the covalent bond of ADP-P were calculated. It was shown that the electrical field of the covalent bond of ADP-P excites oscillations of the proton in the complex with the hydrogen bond A-H...B and displaces it from the equilibrium towards the covalent bond. The distortion of the potential curve depends on a change in the length of the covalent bond of ADP-P. Adiabatic potentials U0 and UN of the ADP-P system were calculated, which correspond to the ground and excited states of the H-bond proton. It was found that as the length of the bond of ADP-P (rho) increases, the branches of the adiabatic potential U0(rho) and UN(rho) intersect. At the intersection point, the system can transit to the branch UN(rho), which can lead to a reduction of the barrier and a break of the covalent bond of ADP-P. Presumably, this mechanism is universal for processes of transformation of the chemical energy of ATP to the energy of excited hydrogen bond, a mechanism for the maintenance of heat balance and reduction of entropy in a living organism.