Scalable Rule-Based Modelling of Allosteric Proteins and Biochemical Networks
Figure 5
Cubic and quartic ternary complex models of a GPCR in our modelling framework.
The mapping between the cubic (A) and quartic (B) models shows how the two models are related. (A) A naive implementation of the cubic ternary complex model. The ANC-structure R has one allosteric component which transitions between a low-affinity, inactive (i) state and a high-affinity, active (a) state with the indicated equilibrium constant (in gray). LB and GB are binding sites for an extracellular ligand L (not shown) and an intracellular target G protein (not shown). In the corresponding cubic, 8-state transition diagram Kact is the unligated allosteric equilibrium constant, Ka and Kg are ligand affinities to the reference (inactive) state, and α and β are ratios of affinities. We parenthesize the cooperativity parameters δ and γ to indicate that these parameters of the cubic ternary complex model have to be added as ad hoc rules to the naïve implementation. (B) In our quartic ternary complex model, an ANC-structure R comprises two allosteric components: the extracellular domain ED transitions between low and high-affinity states (s and t); the intracellular domain ID transitions between inactive and active states (i and a). These transitions are reciprocally linked (dashed line) so each domain acts a modifier of the other with the interaction parameterized by Γ and Φ. The binding sites are allosterically coupled to both allosteric components, therefore each ligand “sees” 4 possible conformations of the receptor. In the quartic state-transition diagram KactG and KactL are the unligated allosteric equilibrium constants, Г is the regulatory factor linking the s↔t and i↔a transitions, Ka′ and Kg′ are ligand affinities to the reference state si, and α and β are ratios of ligand affinities of the subscripted state relative to the reference state. For clarity, we show only the unligated s↔t transition. (C) Rules for the cubic ternary complex model showing the rate and equilibrium constants for ligand and G protein binding. (D) A subset of the rules for the quartic ternary complex model shows the rate and equilibrium constants for ligand binding. A similar set of rules specifies rate and equilibrium constants for binding G protein (Figure 9 of Text S1).