Chemical frustration in the protein folding landscape: grand canonical ensemble simulations of cytochrome c

Abstract

A grand canonical formalism is developed to combine discrete simulations for chemically distinct species in equilibrium. Each simulation is based on a perturbed funneled landscape. The formalism is illustrated using the alkaline-induced transitions of cytochrome c as observed by FTIR spectroscopy and with various other experimental approaches. The grand canonical simulation method accounts for the acid/base chemistry of deprotonation, the inorganic chemistry of heme ligation and misligation, and the minimally frustrated folding energy landscape, thus elucidating the physics of protein folding involved with an acid/base titration of a protein. The formalism combines simulations for each of the relevant chemical species, varying by protonation and ligation states. In contrast to models based on perfectly funneled energy landscapes that contain only contacts found in the native structure, this study introduces "chemical frustration" from deprotonation and misligation that gives rise to many intermediates at alkaline pH. While the nature of these intermediates cannot be easily inferred from available experimental data, this study provides specific structural details of these intermediates, thus extending our understanding of how cytochrome c changes with an increase in pH. The results demonstrate the importance of chemical frustration for understanding biomolecular energy landscapes.

Figure 1

Crystal structure of cytochrome c colored by the folding units: N- and C-terminal helices (blue), 60's helix (green), two stranded beta sheet (yellow), Met80 loop (red), and the omega loop (grey). H-bonds included into the model are shown with blue lines.

Figure 2

A scheme for combining energy landscapes for folding individual chemical species with acid/base and coordination chemistry. In the above scheme, each chemical species is represented by its own folding funnel of varying depths. The pH 7 folding funnel has an energy minima at the native state (N), but as the pH or ligation conditions change, the energy landscape is perturbed giving rise to slightly different funnels with distinct minima such as at a misligated state (M₁ or M₂). The grand canonical partition function contains a sum over each chemical species (s). The folding landscape contributions are clearly separated from the terms containing chemical potentials that give rise to the chemical equilibria between species. For more information on the grand canonical partition function see the Methods section.

Figure 3

Free energy curves calculated using the grand canonical partition function as a function of the reaction coordinate Q at specific pH values of 6.8, 8.4, 10.0, 11.6, 13.2, and 14.0. The curves are obtained from simulations using the pure-funnel (left column), the electrostatic/collapse (middle column), and the H-bonding models (right column).

Figure 4

Fractional concentrations of the most stable chemical species as a function of pH for (A) the pure-funnel model, (B) the electrostatic/collapse model, and (C) the H-bonding model. The native (), Lys73-misligated (), Lys79-misligated (), Lys72-misligated (), unfolded and water/lysine-misligated (), OH-misligated lysine/tyrosine deprotonated (), and OHmisligated lysine/tyrosine/arginine deprotonated (○) states are shown. (D) The probability distribution of states inferred from FTIR studies of semi-synthetic protein with carbon-deuterium labeled residues.

Figure 5

Free energy curves of the most stable chemical species as a function of the reaction coordinate Q at specific pH values of 6.8, 8.4, 10.0, 11.6, 13.2, and 14.0. The chemical species depicted are native state (red), Lys73-misligated (blue), Lys79-misligated (magenta), Lys72-misligated (green), hydroxide-misligated lysine/tyrosine deprotonated (cyan), and hydroxide-misligated lysine/tyrosine/arginine deprotonated (yellow). The free energy curves are calculated using the grand canonical partition function based on pure-funnel model simulations.

Figure 6

Free energy curves as a function of the reaction coordinate Q for the five most stable lysine-misligated intermediates from simulations using (A) the pure-funnel, (B) the electro-static/collapse, and (C) the H-bonding models. The most stable lysine intermediates are Lys53 (), Lys55 (), Lys72 (), Lys73 (), and Lys79 ().

Figure 7

(A) NMR structure of a Lys79Ala Lys73-misligated cytochrome c (PDB code 1LMS). Representative structures of the Lys73-misligated cytochrome c taken from the free energy minima of the (B) pure-funnel, (C) electrostatic/collapse, and (D) H-bonding simulations.

Figure 8

(A) The ensemble average of the number of contacts ($<c>={\displaystyle \underset{c}{\Sigma }}\mathit{cP}\left(c\right)$) per residue at pH 6.0 (red), pH 10.8 (green), and pH 14.0 (blue) calculated using the entire structural ensemble at those pH values. The lines across the top indicate the folding units as defined by hydrogen exchange experiments and simulation. (B) The value of < c > averaged over each folding unit. (C) The average of the number of contacts per residue calculated for individual species: native (red), Tyr48-misligated (green), Tyr67-misligated (blue), and hydroxide ligated and lysine/tyrosine deprotonated (magenta). Each plot is generated using the H-bonding model simulations. Similar results were calculated for the electrostatic/collapse and pure-funnel models.

Figure 9

(A) Probability of finding 10 contacts, as found in the native state, with Leu68 calculated using the pure-funnel (squares), electrostatic/collapse (triangles), and H-bonding models (circles). (B) Probability of finding 6 (red), 4 (blue), or 2 (green) contacts with Met80 calculated using the pure-funnel model. (C) Fractional concentrations of the folded (red), high pH solvent exposed (green), and intermediate (blue) signals observed for Met80 by FTIR. (D-F) Probability of the number of contacts (c) with Lys72, Lys73, and Lys79 respectively calculated using the H-bonding model. The curves plotted are for pH 6.0 (red), pH 9.2 (green), pH 10.0 (blue), pH 11.6 (magenta), and pH 14.0 (teal). The complete set of probability distributions of the number of contacts for all models is shown in the supporting information.