Abstract:
The focus of this thesis is about statistical mechanics on heterogeneous random graphs, i.e. how this heterogeneity affects the cooperative behavior of model systems. It is not intended as a review on it, rather it is showed how this question emerges naturally and can give useful insights to specific instances. The first chapter is about the statistical mechanics of congestion in queuing networks. The second is devoted to the study of the glassy dynamics of facilitated spin models on disordered structures. In the third chapter, the presence of inverse phase transitions in tri-critical spin systems on heterogeneous random graphs is pointed out. Finally, the last chapter is on the role of volatility in the evolution of social networks. In the conclusions, a general insight about the interplay between structure and dynamics on heterogeneous random graphs is given. It is based on the different scaling of the transition point with the moments of the degree distribution for continuous and discontinuous transitions, respectively.

Abstract:
In this article the notion of metabolic turnover is revisited in the light of recent results of out-of-equilibrium thermodynamics. By means of Monte Carlo methods we perform an exact uniform sampling of the steady state fluxes in a genome scale metabolic network of E Coli from which we infer the metabolites turnover times. However the latter are inferred from net fluxes, and we argue that this approximation is not valid for enzymes working nearby thermodynamic equilibrium. We recalculate turnover times from total fluxes by performing an energy balance analysis of the network and recurring to the fluctuation theorem. We find in many cases values one of order of magnitude lower, implying a faster picture of intermediate metabolism.

Abstract:
In this work the Gardner problem of inferring interactions and fields for an Ising neural network from given patterns under a local stability hypothesis is addressed under a dual perspective. By means of duality arguments an integer linear system is defined whose solution space is the dual of the Gardner space and whose solutions represent mutually unstable patterns. We propose and discuss Monte Carlo methods in order to find and remove unstable patterns and uniformly sample the space of interactions thereafter. We illustrate the problem on a set of real data and perform ensemble calculation that shows how the emergence of phase dominated by unstable patterns can be triggered in a non-linear discontinuous way.

Abstract:
In this article the Gordan theorem is applied to the thermodynamics of a chemical reaction network at steady state. From a theoretical viewpoint it is equivalent to the Clausius formulation of the second law for the out of equilibrium steady states of chemical networks, i.e. it states that the exclusion (presence) of closed reactions loops makes possible (impossible) the definition of a thermodynamic potential and vice versa. On the computational side, it reveals that calculating reactions free energy and searching infeasible loops in flux states are dual problems whose solutions are alternatively inconsistent. The relevance of this result for applications is discussed with an example in the field of constraints-based modeling of cellular metabolism where it leads to efficient and scalable methods to afford the energy balance analysis.

Abstract:
We study how the volatility, node- or link-based, affects the evolution of social networks in simple models. The model describes the competition between order -- promoted by the efforts of agents to coordinate -- and disorder induced by volatility in the underlying social network. We find that when volatility affects mostly the decay of links, the model exhibit a sharp transition between an ordered phase with a dense network and a disordered phase with a sparse network. When volatility is mostly node-based, instead, only the symmetric (disordered) phase exists These two regimes are separated by a second order phase transition of unusual type, characterized by an order parameter critical exponent $\beta=0^+$. We argue that node volatility has the same effect in a broader class of models, and provide numerical evidence in this direction.

Abstract:
The integration of various types of genomic data into predictive models of biological networks is one of the main challenges currently faced by computational biology. Constraint-based models in particular play a key role in the attempt to obtain a quantitative understanding of cellular metabolism at genome scale. In essence, their goal is to frame the metabolic capabilities of an organism based on minimal assumptions that describe the steady states of the underlying reaction network via suitable stoichiometric constraints, specifically mass balance and energy balance (i.e. thermodynamic feasibility). The implementation of these requirements to generate viable configurations of reaction fluxes and/or to test given flux profiles for thermodynamic feasibility can however prove to be computationally intensive. We propose here a fast and scalable stoichiometry-based method to explore the Gibbs energy landscape of a biochemical network at steady state. The method is applied to the problem of reconstructing the Gibbs energy landscape underlying metabolic activity in the human red blood cell, and to that of identifying and removing thermodynamically infeasible reaction cycles in the Escherichia coli metabolic network (iAF1260). In the former case, we produce consistent predictions for chemical potentials (or log-concentrations) of intracellular metabolites; in the latter, we identify a restricted set of loops (23 in total) in the periplasmic and cytoplasmic core as the origin of thermodynamic infeasibility in a large sample () of flux configurations generated randomly and compatibly with the prior information available on reaction reversibility.

Abstract:
Within a fully microscopic setting, we derive a variational principle for the non-equilibrium steady states of chemical reaction networks, valid for time-scales over which chemical potentials can be taken to be slowly varying: at stationarity the system minimizes a global function of the reaction fluxes with the form of a Hopfield Hamiltonian with Hebbian couplings, that is explicitly seen to correspond to the rate of decay of entropy production over time. Guided by this analogy, we show that reaction networks can be formally re-cast as systems of interacting reactions that optimize the use of the available compounds by competing for substrates, akin to agents competing for a limited resource in an optimal allocation problem. As an illustration, we analyze the scenario that emerges in two simple cases: that of toy (random) reaction networks and that of a metabolic network model of the human red blood cell.

Abstract:
The stoichiometry of a metabolic network gives rise to a set of conservation laws for the aggregate level of specific pools of metabolites, which, on one hand, pose dynamical constraints that cross-link the variations of metabolite concentrations and, on the other, provide key insight into a cell's metabolic production capabilities. When the conserved quantity identifies with a chemical moiety, extracting all such conservation laws from the stoichiometry amounts to finding all non-negative integer solutions of a linear system, a programming problem known to be NP-hard. We present an efficient strategy to compute the complete set of integer conservation laws of a genome-scale stoichiometric matrix, also providing a certificate for correctness and maximality of the solution. Our method is deployed for the analysis of moiety conservation relationships in two large-scale reconstructions of the metabolism of the bacterium E. coli, in six tissue-specific human metabolic networks, and, finally, in the human reactome as a whole, revealing that bacterial metabolism could be evolutionarily designed to cover broader production spectra than human metabolism. Convergence to the full set of moiety conservation laws in each case is achieved in extremely reduced computing times. In addition, we uncover a scaling relation that links the size of the independent pool basis to the number of metabolites, for which we present an analytical explanation.

Abstract:
Networks of biochemical reactions, like cellular metabolic networks, are kept in non-equilibrium steady states by the exchange fluxes connecting them to the environment. In most cases, feasible flux configurations can be derived from minimal mass-balance assumptions upon prescribing in- and out-take fluxes. Here we consider the problem of inferring intracellular flux patterns from extracellular metabolite levels. Resorting to a thermodynamic out of equilibrium variational principle to describe the network at steady state, we show that the switch from fermentative to oxidative phenotypes in cells can be characterized in terms of the glucose, lactate, oxygen and carbon dioxide concentrations. Results obtained for an exactly solvable toy model are fully recovered for a large scale reconstruction of human catabolism. Finally we argue that, in spite of the many approximations involved in the theory, available data for several human cell types are well described by the predicted phenotypic map of the problem.

Abstract:
Cancer cells utilize large amounts of ATP to sustain growth, relying primarily on non-oxidative, fermentative pathways for its production. In many types of cancers this leads, even in the presence of oxygen, to the secretion of carbon equivalents (usually in the form of lactate) in the cell's surroundings, a feature known as the Warburg effect. While the molecular basis of this phenomenon are still to be elucidated, it is clear that the spilling of energy resources contributes to creating a peculiar microenvironment for tumors, possibly characterized by a degree of toxicity. This suggests that mechanisms for recycling the fermentation products (e.g. a lactate shuttle) may be active, effectively inducing a mutually beneficial metabolic coupling between aberrant and non-aberrant cells. Here we analyze this scenario through a large-scale in silico metabolic model of interacting human cells. By going beyond the cell-autonomous description, we show that elementary physico-chemical constraints indeed favor the establishment of such a coupling under very broad conditions. The characterization we obtained by tuning the aberrant cell's demand for ATP, amino-acids and fatty acids and/or the imbalance in nutrient partitioning provides quantitative support to the idea that synergistic multi-cell effects play a central role in cancer sustainment.