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

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.

Figure 1

The effects of heterogeneity in contact probability (increased, Top to Bottom) on barrier height F^‡, folding temperature T_F, and ordering heterogeneity are summarized here; plots are for simulations of a 27-mer lattice Gō model (yellow) to the same native structure (given in ref. 21), and for the analytic theory in the text (red). The simulation results make no assumptions as to the nature of the configurational entropy; the theoretical results use the approximate state function of Eq. 3, along with a cutoff used for the shorter loops so the bond entropy loss for each loop is always ≤0 (the same loop length distribution is used as in the lattice structure. In the top row, energies are tuned for both simulation and theory to fully symmetrize the funnel: Q_i(ɛ_i^★) = Q; second row: energies are then relaxed for the simulation results so they are all equal: ɛ_i = ɛ̄; energies in the theory are relaxed the same way until a comparable T_F is achieved; third row: energies are then further tuned to a distribution ɛ_i ≅ ɛ_i^o that kills the barrier (there are many such distributions; all that is necessary is sufficient contact heterogeneity). The top three rows are funneled folding mechanisms with many routes to the native structure. Bottom row: energies are tuned to induce a single or a few specific routes for folding. All the while, the energies are constrained to sum to E_N: Σ_iɛ_i = E_N. The free energy profile F(Q) (in units of ɛ̄) is plotted in the left column at the folding transition temperature T_F, which is given. The next column shows the distribution of thermodynamic contact probabilities Q_i(Q^‡) ≡ φ′ at the barrier peak [we use the notation φ′, because this is a thermodynamic rather than kinetic measurement; however, for well-designed proteins, the two are strongly correlated with coefficient ≈0.85 (42)]. Only simulation results are shown to keep the figure easy to read; the theory gives φ′ distributions within ∼10%, as may be inferred from their similar route measures. The next column shows the route measure ℛ(Q) of Eq. 5 and gives the dispersion in native energies required to induce the scenario of that row [ℛ(0, 1) = 0/0 is undefined and so is omitted from the simulation plots; it is defined in the theory through the limits Q → 0,1]. The right column shows schematically the different folding routes as heterogeneity is increased; from a maximum number of routes through Q^† to essentially just one route. Top row: In the uniformly ordering funnel, we can see first that P(φ′) is a δ function, and ℛ(Q^‡) = 0 (cf. Eq. 5), so ordering at the transition state (or barrier peak Q^†) is essentially homogeneous. The number of routes through the bottleneck (cf. Eq. 2) is maximized, as schematically drawn (Right). Branches are drawn in the routes to illustrate the minimum of ℛ(Q) at Q^‡. The free energy barrier is maximized (Eq. 10), thus the stability of the native state at fixed temperature and native energy is maximized, and so the folding temperature T_F at fixed native energy is maximized. T_F in the simulation is defined as the temperature where the native state (Q = 1) is occupied 50% of the time. In the theory, at T_F the probability for Q ≥ 0.8 is 0.5. A very large dispersion in energies is required to induce this scenario. Some contact energies are nearly zero; others are several times stronger than the average. Second row: In the uniform native energy funnel, the barrier height is roughly halved while hardly changing T_F for the following reason. In a Gō model, as the contact energies are relaxed from {ɛ_i^★} to a uniform value ɛ_i = ɛ̄, the energy of the transition state is essentially constant: initially the energy is Σ_iQ_i^★(Q^†)ɛ_i^★ = Q Σ_i ɛ_i^★ = QE_N and as the contact energies are relaxed to a uniform value Σ_iQ_iɛ̄ = ɛ̄Σ_iQ_i = QE_N once again. However, the transition-state entropy increases and obtains its maximal value when ɛ_i = ɛ̄, because then all microstates at Q^† are equally probable, because the probability of occupying a microstate is p_i ∼ exp(−E_i(Q^†)/T) = exp(−QE_N/T)/Z = 1/Ω(Q^†). The thermal entropy −Σ_ip_ilogp_i then equals the configurational entropy log Ω(Q^†) (its largest possible value). Thus, as contact energies are relaxed from ɛ_i^★ where they are anticorrelated to their loop lengths (more negative energies tend to be required for longer loops to have equal free energies) to ɛ̄ where they are uncorrelated to their loop lengths, the barrier initially decreases because the total entropy of the bottleneck increases (drawn schematically on the right); i.e., increases in polymer halo entropy are more important than decreases in route entropy. The system is still sufficiently two state that T_F is hardly changed. P(φ′) is broad, indicating inhomogeneity in the transition state, solely in this scenario because of the topology of the native structure: all contacts are equivalent energetically. Routing is more pronounced when ɛ_i = ɛ̄, ℛ(Q) is a measure of the intrinsic fluctuations in order because of the natural inhomogeneity present in the native structure. Different structures will have different profiles, and it will be interesting to see how this measure of structure couples with thermodynamics and kinetics of folding. Loops and dead ends in the schematic drawings are used to illustrate local decreases and increases in ℛ(Q); these fluctuations are captured by the theory only when the routing becomes pronounced (bottom row). The solid curves presented for the theory are shown for a reduction in T_F comparable to the simulations. There is still some energetic heterogeneity present, as indicated. When ɛ_i = ɛ̄ in the theory (dashed curves), the fluctuations in Q_i are somewhat larger than the simulation values, and the entropic heterogeneity is sufficient to kill the barrier—the free energy is downhill at T_F ≅ 0.5ɛ̄. The free energy barrier results from a cancellation of large terms and is significantly more sensitive than intensive parameters such as route measure ℛ(Q). Third row: In approaching the zero-barrier funnel scenario for the simulation, the energies are further perturbed and now begin to anticorrelate with contact probability (and tend to correlate with loop length); i.e., more probable contacts (which tend to have shorter loops) have stronger energies. For the theory, not as much heterogeneity is required. Contact energies are still correlated with formation probability, as indicated by the signs in parentheses. The free energy barrier continues to decrease until some set of energies {ɛ_i^o}, where the barrier at T_F vanishes entirely. All the while, the transition temperature T_F decreases only ∼10%, so that slowing of dynamics (as T_F approaches T_G) would not be a major factor. At this point, the φ′ distribution at the barrier position Q^‡(ɛ̄) is essentially bimodal, but the distribution at Q^‡({ɛ_i^o}) (Inset) is less so because of transition state drift towards lower Q values (the Hammond effect). A relatively small amount of energetic heterogeneity is needed to kill the barrier at T_F. There are still many routes to the native state, because ℛ(Q^‡) ≈ 0.3 − 0.4, but some contacts are fully formed in the transition state (some φ′ ≅ 1). Bottom row: As the energies continue to be perturbed to values that cause folding to occur by a single dominant route rather than a funnel mechanism, folding becomes strongly downhill at the transition temperature, which drops more sharply towards T_G: to induce a single pathway here, T_F must be decreased to about 1/4 the putative estimate of T_G (about T_F({ɛ̄})/1.6; see ref. 9). In this scenario, the actual shape of the free energy profile depends strongly on which route the system is tuned to; nonnative interactions not included here become important. Contact participation at the barrier is essentially one or zero, and the route measure at the barrier is essentially one. The entropy at the bottleneck is relatively small (the halo entropy of a single native core). The energetic heterogeneity necessary to achieve this scenario is again very large, comparable to what is needed to achieve a uniform funnel.