Deconstructing thermodynamic parameters of a coupled system from site-specific observables

Abstract

Cooperative interactions mediate information transfer between structural domains of a protein molecule and are major determinants of protein function and modulation. The prevalent theories to understand the thermodynamic origins of cooperativity have been developed to reproduce the complex behavior of a global thermodynamic observable such as ligand binding or enzyme activity. However, in most cases the measurement of a single global observable cannot uniquely define all the terms that fully describe the energetics of the system. Here we establish a theoretical groundwork for analyzing protein thermodynamics using site-specific information. Our treatment involves extracting a site-specific parameter (defined as χ value) associated with a structural unit. We demonstrate that, under limiting conditions, the χ value is related to the direct interaction terms associated with the structural unit under observation and its intrinsic activation energy. We also introduce a site-specific interaction energy term (χ(diff)) that is a function of the direct interaction energy of that site with every other site in the system. When combined with site-directed mutagenesis and other molecular level perturbations, analyses of χ values of site-specific observables may provide valuable insights into protein thermodynamics and structure.

Fig. 1.

Coupled model for a three-particle system. A system of three particles, 1, 2, and 3, each of which can exist in two conformations: i₀ (resting state of i) and i₁ (activated state of i). Vertical double-arrowed lines indicate the intrinsic activation constants of the particles. All three particles are thermodynamically linked to each other via four state-dependent pairwise coupling factors, as described in the text. These are depicted by the horizontal and diagonal lines connecting the two particle microstates.

Fig. 2.

Simulations of the ln ε vs. V plots. (A) The plot of ln ε vs. V for a three-particle system in the presence (▪) or absence (+) of interactions between the three particles. The extrapolation of the linear segments at low and high voltages to the V = 0 axis (vertical dashed line) gives the respective χ values. Difference between these intercepts (χ values) is the χ^diff parameter. (B) The ln ε vs. V for a three-particle system where χ^diff for a specific particle is zero (▪). Although the χ^diff is zero, perturbation of the intrinsic equilibrium constant of another particle alters ln ε vs. V plot (+). (C–F) ln ε₁ vs. V plots for the DI voltage sensor (particle 1) of the model of the voltage-dependent sodium channel (Fig. S1) in response to perturbation of different parameters of the model. The intrinsic chemical activation constant of particles 1, (C), and 5, (E), were varied over 6 orders of magnitude: (∘), 0 (+), 3(▪) (i = 1 or 5). Interaction between resting states of particles 1 and 5, θ₁₀₅₀ (D), and that between resting states of particles 2 and 5, θ₂₀₅₀ (F), were varied over 6 orders of magnitude: log θ_i050 = -3 (∘), 0 (+), 3(▪) (i = 1 or 2). Vertical dashed line corresponds to the voltage axis at 0 mV. Extrapolating the linear segments of the traces (at high and low voltages) to this axis gives the corresponding χ values.