Statistical Model To Decipher Protein Folding/Unfolding at a Local Scale

Abstract

Protein folding/unfolding can be analyzed experimentally at a local scale by monitoring the physical properties of local probes as a function of the temperature, for example, the distance between fluorophores or the values of chemical shifts of backbone atoms. Here, the analytical Lifson-Roig model for the helix-coil transition is modified to analyze local thermal unfolding of the fast-folder W protein of bacteriophage lambda (gpW) simulated by all-atom molecular dynamics (MD) simulations in explicit solvent at 15 different temperatures. The protein structure is described by the coarse-grained dihedral angles (γ) and bond angles (θ) built between successive C-C virtual bonds. Each (γ,θ) pair serves as a local probe of protein unfolding. Local native/non-native states are defined for each pair of (γ,θ) angles by analyzing the free-energy landscapes Δ G(γ,θ) computed from MD trajectories. The three local elementary equilibrium constants of the model are extracted for each (γ,θ) pair along the sequence from MD simulations, and the model predictions are compared to the MD data. Using only the local equilibrium constants as an input, we show that the local denaturation curves computed from the model partition function fit their MD simulated counterparts in a satisfying manner without any adjustment. In the model and MD simulations, gpW unfolds gradually between 320 and 340 K, with an average native percentage decreasing from 0.8 (320 K) to 0.2 (340 K). In the prism of the model, there is no stable structure at the local scale in this 20 K unfolding temperature range. The enthalpy change upon local unfolding computed from the model and from MD trajectories suggests that the unfolded state between 320 and 340 K corresponds to a dynamical equilibrium between a large ensemble of constantly changing structures. The present results confirm the downhill unfolding of gpW, which does not obey a two-state global folding/unfolding model, and shed light on the interpretation of local denaturation curves.

Curvature and Torsion of Protein Main Chain as Local Order Parameters of Protein Unfolding

“…The variation ⟨f i ⟩ as a function of the temperature compares very well with the variation of the global nativeness computed from the average variation of the CS measured as a function of the temperature. 3 Application of a two-state model to the global unfolding denaturation curve ⟨f i ⟩(T) is meaningless as intermediate states are significantly populated in the thermal unfolding of gpW between the fully native and fully unfolded states as expected for a downhill folder and shown in ref 3.…”

“…2 In a previous work, we defined the local nativeness of a protein from the free-energy landscape of coarse-grained angles (CGA) built on the sole Cartesian coordinates of the C α atoms. 3 These local nativeness-based CGA were successfully included in an Ising type-model to describe unfolding of the model gpW protein. 3 The CGA fully characterized the protein fold.…”

“…3 These local nativeness-based CGA were successfully included in an Ising type-model to describe unfolding of the model gpW protein. 3 The CGA fully characterized the protein fold. 4 Assuming a constant virtual bond length between C α atoms of successive residues, a chain of N amino acids is fully characterized by N − 3 CGA: the torsion angles γ n , built from the positions of C n−1 α , C n α , C n+1 α , and C n+2 α , and the bond angles θ n , built from C n−1 α , C n α , and C n+1 α , with n = 2 to N − 2.…”

“…From a theoretical point of view, the concept of protein local nativeness backs to the early statistical models of the helix–coil transition where each residue occurs in only two local states . In a previous work, we defined the local nativeness of a protein from the free-energy landscape of coarse-grained angles (CGA) built on the sole Cartesian coordinates of the C α atoms . These local nativeness-based CGA were successfully included in an Ising type-model to describe unfolding of the model gpW protein .…”

Adaptive Steered Path Integral Molecular Dynamics

“…Molecular dynamics (MD) simulation has become a practical tool widely used in modern chemical research. [1][2][3][4][5][6] Although atomic detailed statistics can be extracted from atomistic simulations, the high dimensionality of practical complex systems leads to the existence of a huge number of (free-)energy minima separated by high (free-)energy barriers, which poses problems for the unbiased simulation technique. [7][8][9][10][11][12] As a result, the ergodic condition assumed in basic statistical mechanics is difficult to achieve, which triggers deviations of the time-averaged simulation data from the real expectation or ensemble average.…”