Cooperativity and the origins of rapid, single-exponential kinetics in protein folding

Abstract

The folding of naturally occurring, single-domain proteins is usually well described as a simple, single-exponential process lacking significant trapped states. Here we further explore the hypothesis that the smooth energy landscape this implies, and the rapid kinetics it engenders, arises due to the extraordinary thermodynamic cooperativity of protein folding. Studying Miyazawa-Jernigan lattice polymers, we find that, even under conditions where the folding energy landscape is relatively optimized (designed sequences folding at their temperature of maximum folding rate), the folding of protein-like heteropolymers is accelerated when their thermodynamic cooperativity is enhanced by enhancing the nonadditivity of their energy potentials. At lower temperatures, where kinetic traps presumably play a more significant role in defining folding rates, we observe still greater cooperativity-induced acceleration. Consistent with these observations, we find that the folding kinetics of our computational models more closely approximates single-exponential behavior as their cooperativity approaches optimal levels. These observations suggest that the rapid folding of naturally occurring proteins is, in part, a consequence of their remarkably cooperative folding.

Figure 1.

In this study we have used sequences folding into two topologies (top, numbered “1” and “2”), which are among the least and most complex topologies attainable for a maximally compact 48-residue lattice polymer. (Bottom) The bands perpendicular to the main diagonal in the contact maps (Saitoh et al. 1993), indicating contacts between one amino acid and its four successors, represent the structural equivalent of α-helices. β-sheets are represented as thick bands parallel or antiparallel to the diagonal.

Figure 2.

Enhancing the nonadditivity (i.e., cooperativity; see Wyman and Allen 1951) of the MJ energy potential enhances the thermodynamic cooperativity (Makhatadze and Privalov 1995) of MJ lattice polymer folding. As illustrated here using two representative sequences at their folding transition temperatures, T_f, the energy distributions (top row) and conformational distributions (bottom row) of our model polymers become more strongly biomodal (i.e., more thermodynamically cooperative; see, e.g., Makhatadze and Privalov 1995) as S is increased from 1 to S_opt. In particular, the population of fully native molecules (bottom row, Q = 1.0) is enhanced significantly at higher values of S.

Figure 3.

The folding of MJ lattice polymers is accelerated when the cooperativity of the MJ energy potential is enhanced. This effect holds for all six sequences we have investigated (adopting both low [top row] and high [bottom row] contact order structures). And while this acceleration is readily apparent even at the temperature of optimal folding (T_opt [left column]), it is significantly more pronounced at lower temperatures (T = T_opt − 0.03 [right column]), where kinetic traps might be expected to play a more significant role in defining kinetics. Above some optimal level of cooperativity (S_opt), however, folding decelerates with increasing cooperativity. This presumably occurs because further increases in S destabilize native elements in the folding transition state, slowing folding more than the destabilization of trapped states accelerates it.

Figure 4.

At the still lower temperature of T = T_opt − 0.05 (corresponding to an ∼16°C temperature drop for a protein with a T_opt of ∼40°C—e.g., the similarly sized fynSH3 domain [Plaxco et al. 1998b]), the accelerating effects of enhanced cooperativity are even more pronounced. Under these conditions, the folding rate of one of the sequences encoding topology 1 (left) increases by 5.7- to 6.9-fold as S increases from unity to S_opt. This is more than twice the maximum enhancement observed at T_opt and 2%–40% greater than the enhancement observed at T = (T_opt − 0.03). For a single sequence adopting the more complex topology 2 (right), we observe a 4.4-fold acceleration at this lower temperature, which is 80% greater than that observed at T_opt and 28% greater than that observed at T = (T_opt − 0.03).

Figure 5.

As the cooperativity increases to its optimal value (S_opt), the folding of MJ lattice polymers more closely approximates a single-exponential relaxation. Shown are the results for two representative sequences (sequence 1 of topology 1 and sequence 2 of topology 2) over a range of temperatures. While the folding of these sequences is generally well approximated as single exponential (because they were designed to fold efficiently), a distinct improvement in the single-exponential behavior of folding kinetics is nevertheless observed (from R² = 0.950–0.976 to R² > 0.988) when S reaches S_opt.

Figure 6.

The folding kinetics of MJ lattice polymers is strongly temperature-dependent. Shown is the dependence of the logarithmic folding rate, log₁₀(1/MFPT), on the simulation temperature, T, for sequence 1 of topology 1 and sequence 2 of topology 2, under conditions of no induced cooperativity (S = 1). The optimal folding temperature is the temperature at which folding is most rapid.