Geometry of nonequilibrium reaction networks

Building on Kirchhoff's treatment of electrical circuits, Hill and Schnakenberg - among others - proposed a celebrated theory for the thermodynamics of Markov processes and linear biochemical networks that exploited tools from graph theory to build fundamental nonequilibrium observables. However, such simple geometrical interpretation does not carry through for arbitrary chemical reaction networks because reactions can be many-to-many and are thus represented by a hypergraph, rather than a graph. Here we generalize some of the geometric intuitions behind the Hill-Schnakenberg approach to arbitrary reaction networks. In particular, we give simple procedures to build bases of cycles (encoding stationary nonequilibrium behavior) and cocycles (encoding relaxation), to interpret them in terms of circulations and gradients, and to use them to properly project nonequilibrium observables onto the relevant subspaces. We develop the theory for chemical reaction networks endowed with mass-action kinetics and enrich the description with insights from the corresponding stochastic models. Finally, basing on the linear regime assumption, we deploy the formalism to propose a reconstruction algorithm for metabolic networks consistent with Kirchhoff's Voltage and Current Laws.Comment: 36 pages, 10 figure

Structural conditions for saddle-node bifurcations in chemical reaction networks

Motivated by investigating multistationarity in biochemical systems, we address saddle-node bifurcations for chemical reaction networks endowed with general kinetics. At positive equilibria, we identify structural network conditions that guarantee the bifurcation behavior, and we develop a method to identify the proper bifurcation parameters. As a relevant example, we explicitly provide such bifurcation parameters for Michaelis-Menten and Hill kinetics. Examples of applications include reversible feedback cycles, the central carbon metabolism of Escherichia coli, and autocatalytic networks.Comment: 32 page

Thermodynamic patterns of life: emergent phenomena in reaction networks

Reaction networks are an important tool for the analysis of complex chemical reaction systems. They help us understand systems ranging from specific metabolisms to planetary atmospheres. This thesis develops methods for the analysis of living systems by using reaction networks with a focus on the inclusion of thermodynamic properties. New methods for more realistic artificial chemistries are developed using thermodynamic constraints. A model of evolvable artificial ecosystems is created to understand the effect of evolution and life on the flow of matter and energy through the system. To investigate general thermodynamic properties of large-scale reaction networks, artificial reaction networks are created with a simple scheme for deriving thermodynamically consistent reaction rates. Linear and nonlinear networks using four different complex network models are simulated to their non-equilibrium steady state for various boundary fluxes. Increasing the flow through nonlinear networks shows to increases the number of cycles and leads to a narrower distribution of chemical potentials. In the context of finding signs of life by detecting chemical disequilibrium, a photochemical model of the modern atmosphere and a model of the Archean atmosphere are compared. Calculating the reaction pathways that are most relevant for explaining their reaction network's steady state with a new method allows for the detection of topological differences between the two models. Pathways of the modern Earth atmosphere are simpler (less reactions) and contain fewer cycles than their Archean counterparts. To model the influence of life on reaction pathways, an artificial ecosystem model is developed. Evolution of the reaction networks entails an evolution of reaction pathways towards simplicity, thus indicating that the presence of pronounced, relatively simple pathways in real systems is a consequence of an evolutionary mechanism

Irreversible thermodynamics of open chemical networks I: Emergent cycles and broken conservation laws

In this and a companion paper we outline a general framework for the thermodynamic description of open chemical reaction networks, with special regard to metabolic networks regulating cellular physiology and biochemical functions. We first introduce closed networks "in a box", whose thermodynamics is subjected to strict physical constraints: the mass-action law, elementarity of processes, and detailed balance. We further digress on the role of solvents and on the seemingly unacknowledged property of network independence of free energy landscapes. We then open the system by assuming that the concentrations of certain substrate species (the chemostats) are fixed, whether because promptly regulated by the environment via contact with reservoirs, or because nearly constant in a time window. As a result, the system is driven out of equilibrium. A rich algebraic and topological structure ensues in the network of internal species: Emergent irreversible cycles are associated to nonvanishing affinities, whose symmetries are dictated by the breakage of conservation laws. These central results are resumed in the relation $a + b = s^Y$ between the number of fundamental affinities $a$, that of broken conservation laws $b$ and the number of chemostats $s^Y$. We decompose the steady state entropy production rate in terms of fundamental fluxes and affinities in the spirit of Schnakenberg's theory of network thermodynamics, paving the way for the forthcoming treatment of the linear regime, of efficiency and tight coupling, of free energy transduction and of thermodynamic constraints for network reconstruction.Comment: 18 page

Thermodynamics of accuracy in kinetic proofreading: Dissipation and efficiency trade-offs

The high accuracy exhibited by biological information transcription processes is due to kinetic proofreading, i.e., by a mechanism which reduces the error rate of the information-handling process by driving it out of equilibrium. We provide a consistent thermodynamic description of enzyme-assisted assembly processes involving competing substrates, in a Master Equation framework. We introduce and evaluate a measure of the efficiency based on rigorous non-equilibrium inequalities. The performance of several proofreading models are thus analyzed and the related time, dissipation and efficiency vs. error trade-offs exhibited for different discrimination regimes. We finally introduce and analyze in the same framework a simple model which takes into account correlations between consecutive enzyme-assisted assembly steps. This work highlights the relevance of the distinction between energetic and kinetic discrimination regimes in enzyme-substrate interactions.Comment: IOP Class, 20 pages, 9 figure

Transient fluctuation theorems for the currents and initial equilibrium ensembles

We prove a transient fluctuation theorem for the currents for continuous-time Markov jump processes with stationary rates, generalizing an asymptotic result by Andrieux and Gaspard [J. Stat. Phys. 127, 107 (2007)] to finite times. The result is based on a graph theoretical decomposition in cycle currents and an additional set of tidal currents that characterize the transient relaxation regime. The tidal term can then be removed by a preferred choice of a suitable initial equilibrium ensemble, a result that provides the general theory for the fluctuation theorem without ensemble quantities recently addressed in [Phys. Rev. E 89, 052119 (2014)]. As an example we study the reaction network of a simple stochastic chemical engine, and finally we digress on general properties of fluctuation relations for more complex chemical reaction networks.Comment: 19 pages, 2 figures. Sign error corrected in Eq.(50) and followin