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

Abstract

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.

Thermodynamic Bounds on the Range and Sensitivity of Covalent Switching

“…The framework was introduced in [17]. A recent review [28] discusses the material needed here and should be consulted for more details and background.…”

“…In previous work, post-translational modification systems, like the covalent modification cycle of Figure 1(a), were modeled as interacting systems of linear framework graphs [9,48]. This approach is also reviewed in [28], which may be consulted for more details. An important feature of this approach is that enzyme reaction mechanisms can be substantially more general than the conventional Michaelis-Menten mechanism in Eq.…”

Time-keeping and decision-making in living cells: Part II

“…Their models, based on the well-known cyclin-dependent kinase (CDK) control system ( figure 1 d ), account for both clock-like CDK oscillations during early embryonic cell divisions and switch-like CDK-arrested states (checkpoints) during somatic cell cycles. Lastly, Nam et al [ 9 ] review a graph-based approach to biochemical reaction dynamics, called ‘the linear framework’ ( figure 1 e ). In this approach, the nonlinear kinetics of a network of biochemical reactions is decomposed into a coupled set of graphs, each of which has linear dynamics, and the steady states of the network can be expressed as rational algebraic functions of the parameters.…”

“…MPF = M-phase promoting factor (a CDK); APC/C = anaphase promoting complex; Wee1 and Gwl are protein kinases, Cdc25 and B55 are protein phosphatases; ENSA is a stoichiometric inhibitor. ( e ) A linear framework graph for an enzyme-catalysed reaction; from Nam et al [ 9 ]. The phosphatase F removes a phosphate group from S 1 , via the complexes Y 5 = F : S 1 and Y 6 = F : S 0 , to release the unphosphorylated form S 0 .…”

Flexibility and sensitivity in gene regulation out of equilibrium

“…Given their simplicity, equivalents of the system in Fig. 31 1A have enjoyed earlier study in guises such as enzymatic 32 control (12); remodeling of nucleosomes (5); and other settings 33 in transcription (13,14). In this work, we use tools from 34 graph theory (15, 16) to explore the full space of transcrip-35 tional steady-state outputs available for this system under 36 different energetic drives, compared to equilibrium control.…”