Laplacian Dynamics with Synthesis and Degradation

The linear framework: using graph theory to reveal the algebra and thermodynamics of biomolecular systems

The linear framework uses finite, directed graphs with labelled edges to model biomolecular systems. Graph vertices represent biochemical species or molecular states, edges represent reactions or transitions and labels represent rates. The graph yields a linear dynamics for molecular concentrations or state probabilities, with the graph Laplacian as the operator, and the labels encode the nonlinear interactions between system and environment. The labels can be specified by vertices of other graphs or by conservation laws or, when the environment consists of thermodynamic reservoirs, they may be constants. In the latter case, the graphs correspond to infinitesimal generators of Markov processes. The key advantage of the framework has been that steady states are determined as rational algebraic functions of the labels by the Matrix-Tree theorems of graph theory. When the system is at thermodynamic equilibrium, this prescription recovers equilibrium statistical mechanics but it continues to hold for non-equilibrium steady states. The framework goes beyond other graph-based approaches in treating the graph as a mathematical object, for which general theorems can be formulated that accommodate biomolecular complexity. It has been particularly effective at analysing enzyme-catalysed modification systems and input-output responses.

Allosteric conformational ensembles have unlimited capacity for integrating information

Integration of binding information by macromolecular entities is fundamental to cellular functionality. Recent work has shown that such integration cannot be explained by pairwise cooperativities, in which binding is modulated by binding at another site. Higher-order cooperativities (HOCs), in which binding is collectively modulated by multiple other binding events, appear to be necessary but an appropriate mechanism has been lacking. We show here that HOCs arise through allostery, in which effective cooperativity emerges indirectly from an ensemble of dynamically interchanging conformations. Conformational ensembles play important roles in many cellular processes but their integrative capabilities remain poorly understood. We show that sufficiently complex ensembles can implement any form of information integration achievable without energy expenditure, including all patterns of HOCs. Our results provide a rigorous biophysical foundation for analysing the integration of binding information through allostery. We discuss the implications for eukaryotic gene regulation, where complex conformational dynamics accompanies widespread information integration.

Gene Regulation in and out of Equilibrium

Determining whether and how a gene is transcribed are two of the central processes of life. The conceptual basis for understanding such gene regulation arose from pioneering biophysical studies in eubacteria. However, eukaryotic genomes exhibit vastly greater complexity, which raises questions not addressed by this bacterial paradigm. First, how is information integrated from many widely separated binding sites to determine how a gene is transcribed? Second, does the presence of multiple energy-expending mechanisms, which are absent from eubacterial genomes, indicate that eukaryotes are capable of improved forms of genetic information processing? An updated biophysical foundation is needed to answer such questions. We describe the linear framework, a graph-based approach to Markov processes, and show that it can accommodate many previous studies in the field. Under the assumption of thermodynamic equilibrium, we introduce a language of higher-order cooperativities and show how it can rigorously quantify gene regulatory properties suggested by experiment. We point out that fundamental limits to information processing arise at thermodynamic equilibrium and can only be bypassed through energy expenditure. Finally, we outline some of the mathematical challenges that must be overcome to construct an improved biophysical understanding of gene regulation.

Efficient manipulation and generation of Kirchhoff polynomials for the analysis of non-equilibrium biochemical reaction networks

Kirchhoff polynomials are central for deriving symbolic steady-state expressions of models whose dynamics are governed by linear diffusion on graphs. In biology, such models have been unified under a common linear framework subsuming studies across areas such as enzyme kinetics, G-protein coupled receptors, ion channels and gene regulation. Due to 'history dependence' away from thermodynamic equilibrium, these models suffer from a (super) exponential growth in the size of their symbolic steady-state expressions and, respectively, Kirchhoff polynomials. This algebraic explosion has limited applications of the linear framework. However, recent results on the graph-based prime factorization of Kirchhoff polynomials may help subdue the combinatorial complexity. By prime decomposing the graphs contained in an expression of Kirchhoff polynomials and identifying the graphs giving rise to equal polynomials, we formulate a coarse-grained variant of the expression suitable for symbolic simplification. We devise criteria to efficiently test the equality of Kirchhoff polynomials and propose two heuristic algorithms to explicitly generate individual Kirchhoff polynomials in a compressed form; they are inspired by algebraic simplifications but operate on the corresponding graphs. We illustrate the practicality of the developed theory and algorithms for a diverse set of graphs of different sizes and for non-equilibrium gene regulation analyses.

Steady-State Differential Dose Response in Biological Systems

In pharmacology and systems biology, it is a fundamental problem to determine how biological systems change their dose-response behavior upon perturbations. In particular, it is unclear how topologies, reactions, and parameters (differentially) affect the dose response. Because parameters are often unknown, systematic approaches should directly relate network structure and function. Here, we outline a procedure to compare general non-monotone dose-response curves and subsequently develop a comprehensive theory for differential dose responses of biochemical networks captured by non-equilibrium steady-state linear framework models. Although these models are amenable to analytical derivations of non-equilibrium steady states in principle, their size frequently increases (super) exponentially with model size. We extract general principles of differential responses based on a model's graph structure and thereby alleviate the combinatorial explosion. This allows us, for example, to determine reactions that affect differential responses, to identify classes of networks with equivalent differential, and to reject hypothetical models reliably without needing to know parameter values. We exemplify such applications for models of insulin signaling.