Nonlinear elasticity, proteinquakes, and the energy landscapes of functional transitions in proteins

Abstract

Large-scale motions of biomolecules involve linear elastic deformations along low-frequency normal modes, but for function nonlinearity is essential. In addition, unlike macroscopic machines, biological machines can locally break and then reassemble during function. We present a model for global structural transformations, such as allostery, that involve large-scale motion and possible partial unfolding, illustrating the method with the conformational transition of adenylate kinase. Structural deformation between open and closed states occurs via low-frequency modes on separate reactant and product surfaces, switching from one state to the other when energetically favorable. The switching model is the most straightforward anharmonic interpolation, which allows the barrier for a process to be estimated from a linear normal mode calculation, which by itself cannot be used for activated events. Local unfolding, or cracking, occurs in regions where the elastic stress becomes too high during the transition. Cracking leads to a counterintuitive catalytic effect of added denaturant on allosteric enzyme function. It also leads to unusual relationships between equilibrium constant and rate like those seen recently in single-molecule experiments of motor proteins.

Fig. 1.

Comparison of energy surfaces with different models. When the energy surface is perfectly harmonic as a function of Cartesian coordinates and the conformational change path is linear, the energy surface along the conformational change path can be derived from the normal mode frequency of mode 1 at the initial state (solid line). However, the energy surface along the normal mode 1 (×) computed explicitly with the full anharmonic Tirion potential yields a much higher energy than the harmonic approximation owing to nonlinearities. The energy surface along the nonlinear conformational change path generated by using an iterative method with one mode (○) agrees with the harmonic approximation quite well up to 4 Å of rmsd but exceeds the harmonic result beyond this point. The energy using three modes to connect initial and final states (□) is also shown.

Fig. 2.

Residue strain energy of structures along nonlinear conformational change path. (a) The change of the strain energy localized in individual residues as the structure is deformed is shown. The rmsd of each structure from the open structure is indicated on the right. Residues in blue have no strain energy while red residues have high strain energy. The maximum strain (indicated in red) is 0.5 kcal/mol. (b) The residue strain energy of the structure after 15 steps of iteration is shown in a 2D plot. The secondary structure of this protein is indicated on the top of the plot. (c) The structure after 15 steps of iteration is shown along with the residue strain energy. The residues again are colored according to the strain energy; blue corresponding to no strain and red residues corresponding to high strain energy as before. c was prepared with vmd (43) and raster3d (44).

Fig. 3.

The energy profiles for open and closed states with and without cracking. Calculations of free energy profiles without cracking and with cracking for two values of and 0.1 kcal/mol are shown. The threshold of the closed state, , is set so that the open and the closed states have the same energy at the totally unfolded state. The strain energy computed from the open form is shown in black. The one computed from the closed form is shown in red for free energy change of ΔG_eq = 0 kcal/mol and is shown in blue for ΔG_eq = –3 kcal/mol. Results computed without allowing cracking are shown as solid lines, and the broken lines correspond to a threshold for cracking of . The dotted lines use a cracking threshold of .

Fig. 4.

The transition-state barrier dependence on the reaction driving force. Calculations of how the transition-state barrier, ΔG*, depends on the reaction driving force, (–ΔG_eq), both without cracking (a) and with cracking (c) are shown. The corresponding energy surfaces at the different driving forces (–ΔG_eq) without cracking (b) and with cracking (d) are also shown. A quadratic curvature is observed for the fully elastic model without cracking. Including the cracking effect makes the barrier dependence on driving force linear for a very large range of driving forces.

Fig. 5.

The transition-state barrier dependence on the cracking threshold. Shown is how the transition-state barrier, ΔG*, depends on the cracking threshold, . A driving force of ΔG_eq =–3 kcal/mol is used. The data (○) are fitted by a hyperbolic relation, . From the slope of the line, the Tafel coefficient is estimated as 47 at and 16 at . The experimentally determined urea dependence of the stability, m = ∂ΔG_D–N/∂[urea] = 2.9 kcal/mol per M (31), here M is the molar concentration of urea, corresponds to a change of cracking threshold with the value m/N_res ≈0.014 kcal/mol per M, where the total number of residues N_res = 214. Combining these, the urea dependence of the transition barrier ∂ΔG*/∂[urea] is ≈0.65 kcal/mol per M if and ≈0.22 kcal/mol per M if is used.