Effect of local thermal fluctuations on folding kinetics: A study from the perspective of nonextensive statistical mechanics

The search through the proteins conformational space is thought as an early independent stage of the folding process, governed mainly by the hydrophobic effect. Because of the nanoscopic size of proteins, we assume that the effects of local thermal fluctuations work like folding assistants, managed by the nonextensive parameter $q.$ Using a 27-mer heteropolymer on a cubic lattice, we obtained—by Monte Carlo simulations—kinetic and thermodynamic amounts (such as the characteristic folding time and the native stability) as a function of temperature $T$ and $q$ for a few distinct native targets. We found that for each native structure, at a specific system temperature $T,$ there exists an optimum $q^{*}$ that minimizes the folding characteristic time ${\tau }_{\min }$; for $T=1,$ it is found that $q^{*}$ lies in the interval $1.15\pm 0.05,$ even for native structures presenting significantly different topological complexities. The distribution of ${\tau }_{\min }$ obtained for specific $q>1$ (nonextensive approach) and temperature $T$ can be fully reproduced for $q=1$ (Boltzmann approach), but only at higher temperatures $T{}^{\prime }>T$. However, assuming that the complete set of proteins of each organism is optimized to work in a narrow range of temperature, we conclude that—for the present problem—the two approaches, namely, $(T,q>1)$ and $(T>T^{\prime },q=1),$ cannot be equivalent; it is not a simple matter of reparametrization. Finally, by associating the nonextensive parameter $q$ with the instantaneous degree of compactness of the globule, $q$ becomes a dynamic variable, self-adjusted along the simulation. The results obtained through the $q$-variable approach are utterly consistent with those obtained by using a target-tuned parameter $q^{*}.$ However, in the former approach, $q$ is automatically adjusted by the chain conformational evolution, eliminating the need to seek for a specific optimized value of $q$ for each case. Besides, using the $q$-variable approach, different target structures are promptly characterized by inherent distributions of $q$, which reflect the overall complexity of their corresponding native topologies and energy landscapes.

Figure 1

Functional difference between the generalized and the ordinary exponential function: ${\Delta }_q\left(x\right)={\exp }_q(-x)-\exp (-x)$, with $q\gtrsim 1,$ and between two ordinary exponential functions with the argument of one of them rescaled: ${\Delta }_a\left(x\right)=\exp (-ax)-\exp (-x),$ with $a\lesssim 1$; the parameter $a$ represents a small increment in the system temperature. Their profiles are similar, but, for $a$ and $q$ near to $1,$ the maximum of ${\Delta }_a\left(x\right)$ and ${\Delta }_q\left(x\right)$ occur at about $x_a=1$ and $x_q=2$, respectively. In both cases, $x_a$ and $x_q$ increase very slowly when $a$ and $q$ depart from 1.

Figure 2

The $\tau \text{vs}T$ “U shape”: Folding characteristic time $\tau$ as a function of the translated temperature ${\mathcal{T}}_q=T-T_q^{*}$ for several $(q,T)$ combinations. $T_q^{*}$ is the temperature in which $\tau ={\tau }_{\min }$ for the specified value of $q$. For each $q$, the temperature range was covered by increments $\Delta T=0.05.$ All curves behave very similarly around ${\mathcal{T}}_q=0$.

Figure 3

Folding time distributions $\Phi$ for structure ID 866. The lognormal distribution for $q=1.1$ and $T=1.0$ (larger open circles) was fitted using two peaks Gaussian [continuous thicker line, amplitude: ($A_1;A_2)=(1310\pm 40;430\pm 70)$; width ($w_1;$ $w_2)=(0.49\pm 0.07;0.8\pm 0.3)$, and average $({\mu }_1;{\mu }_2)=(2.91\pm 0.09;5.0\pm 0.2)$]. For $q=1.0$ the folding time behavior is shown for several temperatures (fitted by basic spline functions). In particular, the folding time distributions for $(q,T)=(1.0,1.25)$ and $(1.1,1.0)$ are practically the same, confirming that the generalized Boltzmann factor $(q>1)$ at temperature $T$ corresponds to the conventional Boltzmann factor $(q=1)$ with a certain increase in $T$. In the region of the smaller folding times $\left[\ln \right(t)<2.5]$ and for temperatures in the interval $1.0\le T<1.5$, all curves coalesce, including the case $(q,T)=(1.1,1.0)$. At $T=1.0$ and $q=1$ (open gray circles) the system approaches the glassy regime, and manifestation of ergodic difficulties is evident. At temperature $T=1.5$ the distribution has the lowest domain, that is, $0<\ln \left(t\right)<5.5$ (open squares), and inasmuch as $T$ increases from this point, the peak of the distribution moves again toward larger $t$, as illustrated for $T=1.9$ (full squares).

Figure 4

The native stability. The fraction of time in which the chain stays in the native state along the simulation as a function of the temperature $T$, namely, ${\sigma }_q\left(T\right),$ is shown for $q=1$ and $q=1.1$. In the inset, the native stability [actually $\Delta G/\left(k_{B}T\right)$] is shown as a function of the temperature. Note that at $(q;T)=(1;1.25)$ and $(q;T)=(1.1;1),$ the native stability is the same: $\Delta G/\left(k_{B}T\right)=0.25,$ as emphasized by dotted lines. This result (for structure ID 866) is a direct consequence of the coalescence of the distribution of folding time $\Phi$ for the same two pairs $(q;T)$ used here (as shown in Fig. 3, small full and larger open circles).

Figure 5

Average normalized distribution $\rho$ of $q$ for four targets; 600 runs were used in each case. For the structure ID 1128 $\rho$ is clearly monomodal, but all other cases present a multimodal behavior. Their distribution averages $\langle q\rangle$ differ only slightly, namely, $\langle q\rangle \simeq 1.15$ for IDs 1128 and 866, and $\langle q\rangle =1.17$ for IDs 868 and 36335, but their distinct shapes reflect better the topological and energetic landscape complexity. Note the congruence between the values $\langle q\rangle$ (about 1.15) and the optimal values of $q$ in Table (ranging from 1.1 to 1.2).

Figure 6

3D configurations and corresponding contact maps for the four target structure. Each structure is labeled by its ID number (1128, 866, 868, and 36335) and the corresponding Contact Order. Chain segments that constitute folding unfavorable patterns are highlighted by gray planes (see text).