arXiv subjects

Dimitri Loutchko

Publications and source records attributed to Dimitri Loutchko.

15 recordsLinked to original sources

arXiv2026

Convex Analysis of Relaxation Dynamics in Chemical Reaction Networks and Generalized Gradient Flows

We obtain bounds on the Kullback--Leibler divergence to equilibrium for mass-action chemical reaction networks (CRNs) with equilibrium. The associated decay rates are characterized in terms of the singular values of the stoichiometric matrix, convexity parameters, and time-integrated activities via deformed-exponential-type functions. We further extend these bounds within a generalized gradient flow framework. We highlight the biological relevance of this framework: the resulting bounds apply to quasi-steady-state regimes, where long transients and plateau-like behavior are common and functionally important. We illustrate the framework using a catalytic CRN exhibiting plateaus, where the bounds capture slow relaxation induced by local convexity and provide a bound-based approach to quantifying relaxation in CRNs.

q-bio.MN↗

arXiv2025

Information geometry of perturbed gradient flow systems on hypergraphs: A perspective towards nonequilibrium physics

This article serves to concisely review the link between gradient flow systems on hypergraphs and information geometry which has been established within the last five years. Gradient flow systems describe a wealth of physical phenomena and provide powerful analytical technquies which are based on the variational energy-dissipation principle. Modern nonequilbrium physics has complemented this classical principle with thermodynamic uncertaintly relations, speed limits, entropy production rate decompositions, and many more. In this article, we formulate these modern principles within the framework of perturbed gradient flow systems on hypergraphs. In particular, we discuss the geometry induced by the Bregman divergence, the physical implications of dual foliations, as well as the corresponding infinitesimal Riemannian geometry for gradient flow systems. Through the geometrical perspective, we are naturally led to new concepts such as moduli spaces for perturbed gradient flow systems and thermodynamical area which is crucial for understanding speed limits. We hope to encourage the readers working in either of the two fields to further expand on and foster the interaction between the two fields.

cond-mat.stat-mech↗

arXiv2025

Information geometry of chemical reaction networks: Cramer-Rao bound and absolute sensitivity revisited

Information geometry is based on classical Legendre duality but allows to incorporate additional structure such as algebraic constraints and Bregman divergence functions. It is naturally suited, and has been successfully used, to describe the thermodynamics of chemical reaction networks (CRNs) based on the Legendre duality between concentration and potential spaces, where algebraic constraints are enforced by the stoichiometry. In this article, the Riemannian geometrical aspects of the theory are explored. It is shown that duality between concentration and potential spaces and the natural parametrizations of equilibrium subspace are isometries, which leads to a multivariate Cramer-Rao bound through the comparison of two Riemannian metric tensors. In the subsequent part, the theory is applied to the recently introduced concept of absolute sensitivity. Using the Riemannian geometric tools, it is proven that the absolute sensitivity is a projection operator onto the tangent bundle of the equilibrium manifold. A linear algebraic characterization and explicit results on first order corrections to the thermodynamics of ideal solutions are provided. Finally, the theory is applied to the IDHKP-IDH glyoxylate bypass regulation system. The novelty of the theory is that it is applicable to CRNs with non-ideal thermodynamical behavior, which are prevalent in highly crowded cellular environments due to various interactions between the chemicals. Indeed, the analyzed example shows remarkable behavior ranging from hypersensitivity to negative-self regulations. These are effects which usually require strongly nonlinear reaction kinetics. However, here, they are obtained by tuning thermodynamical interactions providing a complementary, and physically well-founded, viewpoint on such phenomena.

q-bio.MN↗

arXiv2024

Transitions and Thermodynamics on Species Graphs of Chemical Reaction Networks

Chemical reaction networks (CRNs) exhibit complex dynamics governed by their underlying network structure. In this paper, we propose a novel approach to study the dynamics of CRNs by representing them on species graphs (S-graphs). By scaling concentrations by conservation laws, we obtain a graph representation of transitions compatible with the S-graph, which allows us to treat the dynamics in CRNs as transitions between chemicals. We also define thermodynamic-like quantities on the S-graph from the introduced transitions and investigate their properties, including the relationship between specieswise forces, activities, and conventional thermodynamic quantities. Remarkably, we demonstrate that this formulation can be developed for a class of irreversible CRNs, while for reversible CRNs, it is related to conventional thermodynamic quantities associated with reactions. The behavior of these specieswise quantities is numerically validated using an oscillating system (Brusselator). Our work provides a novel methodology for studying dynamics on S-graphs, paving the way for a deeper understanding of the intricate interplay between the structure and dynamics of chemical reaction networks.

q-bio.MN↗

arXiv2024

Cramer-Rao bound and absolute sensitivity in chemical reaction networks

Chemical reaction networks (CRN) comprise an important class of models to understand biological functions such as cellular information processing, the robustness and control of metabolic pathways, circadian rhythms, and many more. However, any CRN describing a certain function does not act in isolation but is a part of a much larger network and as such is constantly subject to external changes. In [Shinar, Alon, and Feinberg. "Sensitivity and robustness in chemical reaction networks." SIAM J App Math (2009): 977-998.], the responses of CRN to changes in the linear conserved quantities, called sensitivities, were studied in and the question of how to construct absolute, i.e., basis-independent, sensitivities was raised. In this article, by applying information geometric methods, such a construction is provided. The idea is to track how concentration changes in a particular chemical propagate to changes of all concentrations within a steady state. This is encoded in the matrix of absolute sensitivities. As the main technical tool, a multivariate Cramer-Rao bound for CRN is proven, which is based on the the analogy between quasi-thermostatic steady states and the exponential family of probability distributions. This leads to a linear algebraic characterization of the matrix of absolute sensitivities for quasi-thermostatic CRN. As an example, the core module of the IDHKP-IDH glyoxylate bypass regulation system is analyzed analytically and numerically by extensive random sampling of the concentration space. The experimentally known findings for the robustness of the IDH enzyme are confirmed and a hidden symmetry at the level of distributions is revealed, providing a blueprint for the analysis of the robustness properties of CRNs.

q-bio.MN↗

arXiv2023

The Geometry of Thermodynamic Uncertainty Relations in Chemical Reaction Networks

Recently, Hessian geometry - an extension of information geometry - has emerged as a framework to naturally connect the geometries appearing in the theory of chemical reaction networks (CRN) to their inherent thermodynamical and kinetic properties. This framework is used in this letter to derive multivariate thermodynamic uncertainty relations (TUR) for CRN. The matrices featured in the TUR are shown to be representations of Riemmanian metric tensors, whereby one tensor characterizes the pseudo entropy production rate and the other the current fluctuations. It is shown that the latter tensor is a restriction of the former one to a linear subspace in the flux tangent space. Therefore, in addition to clarifying the geometric origin of TUR in CRN, the Hessian geometric setup yields a characterization of the error term in the TUR as the norm of a linear subspace component of the flux vector and thus characterizes the fluxes where TUR become equalities.

cond-mat.stat-mech↗

arXiv2022

Information Geometry of Dynamics on Graphs and Hypergraphs

We introduce a new information-geometric structure associated with the dynamics on discrete objects such as graphs and hypergraphs. The presented setup consists of two dually flat structures built on the vertex and edge spaces, respectively. The former is the conventional duality between density and potential, e.g., the probability density and its logarithmic form induced by a convex thermodynamic function. The latter is the duality between flux and force induced by a convex and symmetric dissipation function, which drives the dynamics of the density. These two are connected topologically by the homological algebraic relation induced by the underlying discrete objects. The generalized gradient flow in this doubly dual flat structure is an extension of the gradient flows on Riemannian manifolds, which include Markov jump processes and nonlinear chemical reaction dynamics as well as the natural gradient and mirror descent. The information-geometric projections on this doubly dual flat structure lead to information-geometric extensions of the Helmholtz-Hodge decomposition and the Otto structure in $L^{2}$ Wasserstein geometry. The structure can be extended to non-gradient nonequilibrium flows, from which we also obtain the induced dually flat structure on cycle spaces. This abstract but general framework can extend the applicability of information geometry to various problems of linear and nonlinear dynamics.

cs.IT↗

arXiv2022

An algebraic characterization of self-generating chemical reaction networks using semigroup models

The ability of a chemical reaction network to generate itself by catalyzed reactions from constantly present environmental food sources is considered a fundamental property in origin-of-life research. Based on Kaufmann's autocatalytic sets, Hordijk and Steel have constructed the versatile formalism of catalytic reaction systems (CRS) to model and to analyze such self-generating networks, which they named reflexively autocatalytic and food generated (RAF). Previously, it was established that the subsequent and simultaenous catalytic functions of the chemicals of a CRS give rise to an algebraic structure, termed a semigroup model. The semigroup model allows to naturally consider the function of any subset of chemicals on the whole CRS. This gives rise to a generative dynamics by iteratively applying the function of a subset to the externally supplied food set. The fixed point of this dynamics yields the maximal self-generating set of chemicals. Moreover, the lattice of all functionally closed self-generating sets of chemicals is discussed and a structure theorem for this lattice is proven. It is also shown that a CRS which contains self-generating sets of chemicals cannot be nilpotent and thus a useful link to the combinatorial theory of finite semigroups is established. The main technical tool introduced and utilized in this work is the representation of the semigroup elements as decorated rooted trees, allowing to translate the generation of chemicals from a given set of resources into the semigroup language.

q-bio.MN↗

arXiv2022

Geometry of Nonequilibrium Chemical Reaction Networks and Generalized Entropy Production Decompositions

We derive the Hessian geometric structure of nonequilibrium chemical reaction networks (CRN) on the flux and force spaces induced by the Legendre duality of convex dissipation functions and characterize their dynamics as a generalized flow. With this structure, we can extend theories of nonequilibrium systems with quadratic dissipation functions to more general ones with nonquadratic ones, which are pivotal for studying chemical reaction networks. By applying generalized notions of orthogonality in Hessian geometry to chemical reaction networks, we obtain two generalized decompositions of the entropy production rate, each of which captures gradient-flow and minimum-dissipation aspects in nonequilibrium dynamics.

physics.chem-ph↗

arXiv2022

Riemannian Geometry of Optimal Driving and Thermodynamic Length and its Application to Chemical Reaction Networks

It is known that the trajectory of an endoreversibly driven system with minimal dissipation is a geodesic on the equilibrium state space. Thereby, the state space is equipped with the Riemannian metric given by the Hessian of the free energy function, known as Fisher information metric. However, the derivations given until now require both the system and the driving reservoir to be in local equilibrium. In the present work, we rederive the framework for chemical reaction networks and thereby enhance its scope of applicability to the nonequilibrium situation. Moreover, because our results are derived without restrictive assumptions, we are able to discuss phenomena that could not been seen previously. We introduce a suitable weighted Fisher information metric on the space of chemical concentrations and show that it characterizes the dissipation caused by diffusive driving, with arbitrary diffusion rate constants. This allows us to consider driving far from equilibrium. As the main result, we show that the isometric embedding of a steady state manifold into the concentration space yields a lower bound for the dissipation when the system is driven along the manifold. We give an analytic expression for this bound and for the corresponding geodesic, and thereby are able to dissect the contributions from the driving kinetics and from thermodynamics. Finally, we discuss in detail the application to quasi-thermostatic steady states.

cond-mat.stat-mech↗

arXiv2022

Chemical Thermodynamics for Growing Systems

We consider growing open chemical reaction systems (CRSs), in which autocatalytic chemical reactions are encapsulated in a finite volume and its size can change in conjunction with the reactions. The thermodynamics of growing CRSs is indispensable for understanding biological cells and designing protocells by clarifying the physical conditions and costs for their growing states. In this work, we establish a thermodynamic theory of growing CRSs by extending the Hessian geometric structure of non-growing CRSs. The theory provides the environmental conditions to determine the fate of the growing CRSs; growth, shrinking or equilibration. We also identify thermodynamic constraints; one to restrict the possible states of the growing CRSs and the other to further limit the region where a nonequilibrium steady growing state can exist. Moreover, we evaluate the entropy production rate in the steady growing state. The growing nonequilibrium state has its origin in the extensivity of thermodynamics, which is different from the conventional nonequilibrium states with constant volume. These results are derived from general thermodynamic considerations without assuming any specific thermodynamic potentials or reaction kinetics; i.e., they are obtained based solely on the second law of thermodynamics.

cond-mat.stat-mech↗

arXiv2021

Kinetic Derivation of the Hessian Geometric Structure in Chemical Reaction Systems

The theory of chemical kinetics form the basis to describe the dynamics of chemical systems. Owing to physical and thermodynamic constraints, chemical reaction systems possess various structures, which can be utilized to characterize important physical properties of the systems. In this work, we reveal the Hessian geometry which underlies chemical reaction systems and demonstrate how it originates from the interplay of stoichiometric and thermodynamic constraints. Our derivation is based on kinetics, we assume the law of mass action and characterize the equilibrium states by the detailed balance condition. The obtained geometric structure is then related to thermodynamics via the Hessian geometry appearing in a pure thermodynamic derivation. We demonstrate, based on the fact that both equilibrium and complex balanced states form toric varieties, how the Hessian geometric framework can be extended to nonequilibrium complex balanced steady states. We conclude that Hessian geometry provides a natural framework to capture the thermodynamic aspects of chemical reaction kinetics.

physics.bio-ph↗

arXiv2021

A Hessian Geometric Structure of Chemical Thermodynamic Systems with Stoichiometric Constraints

We establish a Hessian geometric structure in chemical thermodynamics which describes chemical reaction networks (CRNs) with equilibrium states. In our setup, the ideal gas assumption and mass action kinetics are not required. The existence and uniqueness condition of the equilibrium state is derived by using the Legendre duality inherent to the Hessian structure. The entropy production during a relaxation to the equilibrium state can be evaluated by the Bregman divergence. Furthermore, the equilibrium state is characterized by four distinct minimization problems of the divergence, which are obtained from the generalized Pythagorean theorem originating in the dual flatness. For the ideal gas case, we confirm that our existence and uniqueness condition implies Birch's theorem, and that the entropy production represented by the divergence coincides with the generalized Kullback-Leibler divergence. In addition, under mass action kinetics, our general framework reproduces the local detailed balance condition.

cond-mat.stat-mech↗

arXiv2019

Algebraic Coarse-Graining of Biochemical Reaction Networks

Biological systems exhibit processes on a wide range of time and length scales. This work demonstrates that models, wherein the interaction between system constituents is captured by algebraic operations, inherently allow for successive coarse-graining operations through quotients of the algebra. Thereby, the class of model is retained and all possible coarse-graining operations are encoded in the lattice of congruences of the model. We analyze a class of algebraic models generated by the subsequent and simultaneous catalytic functions of chemicals within a reaction network. Our ansatz yields coarse-graining operations that cover the network with local functional patches and delete the information about the environment, and complementary operations that resolve only the large-scale functional structure of the network. Finally, we present a geometric interpretation of the algebraic models through an analogy with classical models on vector fields. We then use the geometric framework to show how a coarse-graining of the algebraic model naturally leads to a coarse-graining of the state-space. The framework developed here is aimed at the study of the functional structure of cellular reaction networks spanning a wide range of scales.

q-bio.MN↗

arXiv2019

Semigroup models for biochemical reaction networks

The catalytic reaction system (CRS) formalism by Hordijk and Steel is a versatile method to model autocatalytic biochemical reaction networks. It is particularly suited, and has been widely used, to study self-sustainment and self-generation properties. Its distinguishing feature is the explicit assignment of a catalytic function to chemicals that are part of the system. In this work, it is shown that the subsequent and simultaneous catalytic functions gives rise to an algebraic structure of a semigroup with the additional compatible operation of idempotent addition and a partial order. The aim of this article is to demonstrate that such semigroup models are a natural setup to describe and analyze self-sustaining CRS. The basic algebraic properties of the models are established and the notion of the function of any set of chemicals on the whole CRS is made precise. This leads to a natural discrete dynamics on the network, which results from iteratively considering the self-action on a set of chemicals by its own function. The fixed points of this dynamics are proven to correspond to self-sustaining sets of chemicals, which are functionally closed. Finally, as the main application, a theorem on the maximal self-sustaining set and a structure theorem on the lattice of functionally closed self-sustaining sets of chemicals are proven.

q-bio.MN↗