Scalable Rule-Based Modelling of Allosteric Proteins and Biochemical Networks
Figure 1
The Allosteric Network Compiler – modelling elements and methodological flowchart.
(A) Example structures. Each structure has a name (underlined) and comprises a set of named components. Hierarchical components (triangles) represent part or all of a biomolecule and contain, as denoted by arrows, one or more interaction sites (circles). Left: The structure X represents a simple ligand with a single binding site (circle with horizontal bar). Centre-left: The structure A represents a generic, divalent allosteric adaptor protein. The adaptor's hierarchical component is allosteric (indicated by a tilde) and transitions between low (R) and high-affinity (T) conformational states. The dashed lines indicate that each binding site acts as a modifier for the allosteric transition, with each interaction parameterized by the indicated Φ-value, and that ligands can distinguish each conformation. Centre-right: The structure R is a simplified model of the nicotinic aceltylcholine receptor (nAChR), following Edelstein et al. [67] but without desensitized states. The allosteric component transitions between closed (C) and open (O) states. Right: The structure K is a model of a mitogen activated protein kinase (MAPK) with two activating phosphorylation sites (circles with vertical bar and a grey dot as a placeholder for the state) and a catalytic site (circle with cross). The allosteric component transitions between inactive (I) and active (A) states. Both the phosphorylation sites and the catalytic site are modifiers of the allosteric transition: each successive phosphorylation biases the equilibrium of the enzyme towards the active state by a regulatory factor ΓY1 or ΓY2. Each of these interactions is also parameterized by a distinct Φ-value. (B) Example rules. A pair of binding rules for the adaptor A and the ligand X specify the association and dissociation rates of AX with X when AX is in the R and T states, a similar pair (not shown) specifies the rates for AY and Y, and we define the affinities KRX and KTX implied by the rates (in gray, e.g. KRX = kfRX/kbRX). A covalent modification rule for the kinase K acting on an unphosphorylated (open dot) downstream target Y follows the Michaelis-Menten mechanism for enzyme-substrate interactions and yields a phosphorylated substrate (filled dot). (C) Methodological flowchart. In a model of the adaptor protein A and its ligands X and Y (Figure 7 of Text S1), the rules state that both ligands bind with higher affinity to the T state of the adaptor. This model is compiled by ANC to generate a reaction network where horizontal transitions correspond to conformational changes, vertical transitions correspond to binding the ligand X, and transitions into the page represent binding the ligand Y. KRT is the allosteric equilibrium constant, while the regulatory factors ΓX and ΓY are the differential affinity of the ligands to each conformation of A and are calculated by ANC using the rate constants given in the rules (e.g. ΓX = KTX/KRX). The reaction network is converted into ordinary differential equations by Facile and these are simulated in Matlab to compute the output response of the system (bound AY vs. X, with ATOT = 1, YTOT = 1, KRT = 10−3 KRX = 0.1, KTX = 10, KRY = 0.01, KTY = 100, arbitrary units).