Network representations and methods for the analysis of chemical and biochemical pathways

Physical Chemistry Models for Chemical Research in the XXth and XXIst Centuries

Thermodynamic hypotheses and models are the touchstone for chemical results, but the actual models based on time-invariance, which have performed efficiently in the development of chemistry, are nowadays invalid for the interpretation of the behavior of complex systems exhibiting nonlinear kinetics and with matter and energy exchange flows with the surroundings. Such fields of research will necessarily foment and drive the use of thermodynamic models based on the description of irreversibility at the macroscopic level, instead of the current models which are strongly anchored in microreversibility.

What makes a reaction network "chemical"?

BACKGROUND

Reaction networks (RNs) comprise a set X of species and a set [Formula: see text] of reactions [Formula: see text], each converting a multiset of educts [Formula: see text] into a multiset [Formula: see text] of products. RNs are equivalent to directed hypergraphs. However, not all RNs necessarily admit a chemical interpretation. Instead, they might contradict fundamental principles of physics such as the conservation of energy and mass or the reversibility of chemical reactions. The consequences of these necessary conditions for the stoichiometric matrix [Formula: see text] have been discussed extensively in the chemical literature. Here, we provide sufficient conditions for [Formula: see text] that guarantee the interpretation of RNs in terms of balanced sum formulas and structural formulas, respectively.

RESULTS

Chemically plausible RNs allow neither a perpetuum mobile, i.e., a "futile cycle" of reactions with non-vanishing energy production, nor the creation or annihilation of mass. Such RNs are said to be thermodynamically sound and conservative. For finite RNs, both conditions can be expressed equivalently as properties of the stoichiometric matrix [Formula: see text]. The first condition is vacuous for reversible networks, but it excludes irreversible futile cycles and-in a stricter sense-futile cycles that even contain an irreversible reaction. The second condition is equivalent to the existence of a strictly positive reaction invariant. It is also sufficient for the existence of a realization in terms of sum formulas, obeying conservation of "atoms". In particular, these realizations can be chosen such that any two species have distinct sum formulas, unless [Formula: see text] implies that they are "obligatory isomers". In terms of structural formulas, every compound is a labeled multigraph, in essence a Lewis formula, and reactions comprise only a rearrangement of bonds such that the total bond order is preserved. In particular, for every conservative RN, there exists a Lewis realization, in which any two compounds are realized by pairwisely distinct multigraphs. Finally, we show that, in general, there are infinitely many realizations for a given conservative RN.

CONCLUSIONS

"Chemical" RNs are directed hypergraphs with a stoichiometric matrix [Formula: see text] whose left kernel contains a strictly positive vector and whose right kernel does not contain a futile cycle involving an irreversible reaction. This simple characterization also provides a concise specification of random models for chemical RNs that additionally constrain [Formula: see text] by rank, sparsity, or distribution of the non-zero entries. Furthermore, it suggests several interesting avenues for future research, in particular, concerning alternative representations of reaction networks and infinite chemical universes.

Integrated Value of Influence: An Integrative Method for the Identification of the Most Influential Nodes within Networks

Biological systems are composed of highly complex networks, and decoding the functional significance of individual network components is critical for understanding healthy and diseased states. Several algorithms have been designed to identify the most influential regulatory points within a network. However, current methods do not address all the topological dimensions of a network or correct for inherent positional biases, which limits their applicability. To overcome this computational deficit, we undertook a statistical assessment of 200 real-world and simulated networks to decipher associations between centrality measures and developed an algorithm termed Integrated Value of Influence (IVI), which integrates the most important and commonly used network centrality measures in an unbiased way. When compared against 12 other contemporary influential node identification methods on ten different networks, the IVI algorithm outperformed all other assessed methods. Using this versatile method, network researchers can now identify the most influential network nodes.

A mechanism for hunchback promoters to readout morphogenetic positional information in less than a minute

Cell fate decisions in the fly embryo are rapid: hunchback genes decide in minutes whether nuclei follow the anterior/posterior developmental blueprint by reading out positional information in the Bicoid morphogen. This developmental system is a prototype of regulatory decision processes that combine speed and accuracy. Traditional arguments based on fixed-time sampling of Bicoid concentration indicate that an accurate readout is impossible within the experimental times. This raises the general issue of how speed-accuracy tradeoffs are achieved. Here, we compare fixed-time to on-the-fly decisions, based on comparing the likelihoods of anterior/posterior locations. We found that these more efficient schemes complete reliable cell fate decisions within the short embryological timescales. We discuss the influence of promoter architectures on decision times and error rates, present concrete examples that rapidly readout the morphogen, and predictions for new experiments. Lastly, we suggest a simple mechanism for RNA production and degradation that approximates the log-likelihood function.

Universal scaling across biochemical networks on Earth

The application of network science to biology has advanced our understanding of the metabolism of individual organisms and the organization of ecosystems but has scarcely been applied to life at a planetary scale. To characterize planetary-scale biochemistry, we constructed biochemical networks using a global database of 28,146 annotated genomes and metagenomes and 8658 cataloged biochemical reactions. We uncover scaling laws governing biochemical diversity and network structure shared across levels of organization from individuals to ecosystems, to the biosphere as a whole. Comparing real biochemical reaction networks to random reaction networks reveals that the observed biological scaling is not a product of chemistry alone but instead emerges due to the particular structure of selected reactions commonly participating in living processes. We show that the topology of biochemical networks for the three domains of life is quantitatively distinguishable, with >80% accuracy in predicting evolutionary domain based on biochemical network size and average topology. Together, our results point to a deeper level of organization in biochemical networks than what has been understood so far.

Analysis of stochastic bifurcations with phase portraits

We propose a method to obtain phase portraits for stochastic systems. Starting from the Fokker-Planck equation, we separate the dynamics into a convective and a diffusive part. We show that stable and unstable fixed points of the convective field correspond to maxima and minima of the stationary probability distribution if the probability current vanishes at these points. Stochastic phase portraits, which are vector plots of the convective field, therefore indicate the extrema of the stationary distribution and can be used to identify stochastic bifurcations that change the number and stability of these extrema. We show that limit cycles in stochastic phase portraits can indicate ridges of the probability distribution, and we identify a novel type of stochastic bifurcation, where the probability maximum moves to the edge of the system through a gap between the two nullclines of the convective field.

Energy-based analysis of biomolecular pathways

Decomposition of biomolecular reaction networks into pathways is a powerful approach to the analysis of metabolic and signalling networks. Current approaches based on analysis of the stoichiometric matrix reveal information about steady-state mass flows (reaction rates) through the network. In this work, we show how pathway analysis of biomolecular networks can be extended using an energy-based approach to provide information about energy flows through the network. This energy-based approach is developed using the engineering-inspired bond graph methodology to represent biomolecular reaction networks. The approach is introduced using glycolysis as an exemplar; and is then applied to analyse the efficiency of free energy transduction in a biomolecular cycle model of a transporter protein [sodium-glucose transport protein 1 (SGLT1)]. The overall aim of our work is to present a framework for modelling and analysis of biomolecular reactions and processes which considers energy flows and losses as well as mass transport.

Complex reaction networks in high temperature hydrocarbon chemistry

Complex network theory reveals novel insights into the chemical kinetics of high temperature hydrocarbon decomposition.

Metabolic network motifs can provide novel insights into evolution: The evolutionary origin of Eukaryotic organelles as a case study

Phylogenetic trees are typically constructed using genetic and genomic data, and provide robust evolutionary relationships of species from the genomic point of view. We present an application of network motif mining and analysis of metabolic pathways that when used in combination with phylogenetic trees can provide a more complete picture of evolution. By using distributions of three-node motifs as a proxy for metabolic similarity, we analyze the ancestral origin of Eukaryotic organelles from the metabolic point of view to illustrate the application of our motif mining and analysis network approach. Our analysis suggests that the hypothesis of an early proto-Eukaryote could be valid. It also suggests that a δ- or ϵ-Proteobacteria may have been the endosymbiotic partner that gave rise to modern mitochondria. Our evolutionary analysis needs to be extended by building metabolic network reconstructions of species from the phylum Crenarchaeota, which is considered to be a possible archaeal ancestor of the eukaryotic cell. In this paper, we also propose a methodology for constructing phylogenetic trees that incorporates metabolic network signatures to identify regions of genomically-estimated phylogenies that may be spurious. We find that results generated from our approach are consistent with a parallel phylogenetic analysis using the method of feature frequency profiles.

Thiazolidinones derived from dynamic systemic resolution of complex reversible-reaction networks

A complex dynamic system based on a network of multiple reversible reactions has been established. The network was applied to a dynamic systemic resolution protocol based on kinetically controlled lipase-catalyzed transformations. This resulted in the formation of cyclized products, where two thiazolidinone compounds were efficiently produced from a range of potential transformations.

The ecology of collective behavior

Networks of local interactions regulate biological systems. Ecological constraints set by resource distribution, operating costs, and the threat of rupture produce similar collective behavior in ants, cells, and gene transcription.

Similar patterns of interaction, such as network motifs and feedback loops, are used in many natural collective processes, probably because they have evolved independently under similar pressures. Here I consider how three environmental constraints may shape the evolution of collective behavior: the patchiness of resources, the operating costs of maintaining the interaction network that produces collective behavior, and the threat of rupture of the network. The ants are a large and successful taxon that have evolved in very diverse environments. Examples from ants provide a starting point for examining more generally the fit between the particular pattern of interaction that regulates activity, and the environment in which it functions.