Investigation of routes and funnels in protein folding by free energy functional methods

Investigation of routes and funnels in protein folding by free energy functional methods

Department of Physics, University of California at San Diego, La Jolla, CA 92093-5003

Edited by Peter G. Wolynes, University of Illinois at Urbana–Champaign, Urbana, IL, and approved March 20, 2000 (received for review December 17, 1999)

Abstract

We use a free energy functional theory to elucidate general properties of heterogeneously ordering, fast folding proteins, and we test our conclusions with lattice simulations. We find that both structural and energetic heterogeneity can lower the free energy barrier to folding. Correlating stronger contact energies with entropically likely contacts of a given native structure lowers the barrier, and anticorrelating the energies has the reverse effect. Designing in relatively mild energetic heterogeneity can eliminate the barrier completely at the transition temperature. Sequences with native energies tuned to fold uniformly, as well as sequences tuned to fold reliably by a single or a few routes, are rare. Sequences with weak native energetic heterogeneity are more common; their folding kinetics is more strongly determined by properties of the native structure. Sequences with different distributions of stability throughout the protein may still be good folders to the same structure. A measure of folding route narrowness is introduced that correlates with rate and that can give information about the intrinsic biases in ordering arising from native topology. This theoretical framework allows us to investigate systematically the coupled effects of energy and topology in protein folding and to interpret recent experiments that investigate these effects.

Footnotes

↵ * To whom reprint requests should be addressed at: Department of Physics 0319, Urey Hall 7218, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0319. E-mail: plotkin{at}curio.ucsd.edu or jose{at}curio.ucsd.edu.
↵ a We treat only native couplings in detail, accounting for nonnative interactions as a uniform background field. Additionally, the correlation between contacts (i, j) is a function only of the overall order Q in our theory. This is analogous to the Hartree approximation in the one-electron theory of solids, where electrons mutually interact only through an averaged field; extensions of our theory to include correlation mediated by native structure may be examined within the density-functional framework and are a topic of future research. On the other hand, tests of the theory by simulation (Fig. 1) produce qualitatively the same results, so the conclusions are not affected by including correlations to any order.
↵ b Folding heterogeneity affects the free energy in three ways: (i) The number of folding routes to the native state decreases; this effect increases the folding barrier; (ii) the conformational entropy of polymer loops increases, because native cores with larger halo entropies are more strongly weighted. This decreases the folding barrier. (iii) making likely contacts stronger in energy lowers the thermal energy of partially native structures; this decreases the folding barrier.
↵ c This procedure is analogous to finding the most probable distribution of occupation numbers, and thus the thermodynamics, by maximizing the microcanonical entropy for a system of particles obeying a given occupation statistics; here, the effective particles (the contacts) obey Fermi–Dirac statistics; Eq. 7.
↵ d A similar derivation of the free energy for a uniform order parameter Q was calculated in ref. 10.
↵ e This approach assumes minimal frustration, in that native heterogeneity is retained explicitly, and nonnative heterogeneity is averaged over; phenomena specific to a particular set of nonnative energies, e.g., “off-pathway” intermediates, are smoothed over in this procedure.
↵ f Note that in Eq. 4, we explicitly include the trace over configurations at overall order Q. The Q _i that minimize F are the thermal values.
↵ g In the contact representation, the averaged bond occupation probabilities Q _i = 〈𝒬_i〉_TH are analogous to the averaged number density operator in an inhomogeneous fluid: 〈n(x)〉_TH = 〈Σ_iδ(x_i − x)〉_TH.
↵ h The value α = 1.37 gives the best fit to the lattice 27-mer data for the route entropy, whereas α ≅ 1.0 best fits the 27-mer free energy function. We generally use α ≅ 1.0, because the 27-mer is small; for larger systems, α is smaller: more polymer is buried, and thus it is more strongly constrained by surrounding contacts.
↵ i We avoid the word “pathway,” because several definitions exist in the literature; here a single route is unambiguously defined through the limit 𝒮_ROUTE → 0.
↵ j That is, if MQ contacts were made with probability 1, and M − MQ contacts were made with probability 0, then 〈(Q _i − Q)²〉_MAX = (1/M)(MQ(1 − Q)² + (M − MQ)Q ²) = Q(1 − Q). Thus ℛ(Q) is between 0 and 1.
↵ k That is, because all Q _i are only zero or one at any degree of nativeness, each successive bond added must always be the same one, so folding is then a random walk on the potential defined by that single route (chain entropy is still present). ℛ(Q) is in the spirit of a Debye–Waller factor applied to folding routes.
↵ l Corner, crankshaft, and end moves are allowed. Free energies and contact probabilities are obtained by equilibrium Monte Carlo sampling by using the histogram method (43). Sampling error is <5%.
↵ m We have expanded the route entropy Eq. 2 to second order in this expression for clarity; in deriving the results of the theory, the full expression is used.
↵ n We use a prime, because we actually look at the barrier peak along the Q coordinate.
Abbreviations:

TSE′,

ensemble of configurations at the free energy barrier peak