Conformational Temperature Characterizing the Folding of a Protein

The time sequences of the molecular dynamics simulation for the folding process of a protein is analyzed with the inherent structure landscape which focuses on the configurational dynamics of the system. Time-dependent energy and entropy for inherent structures are introduced, and from these quantities a conformational temperature is defined. The conformational temperature follows the time evolution of a slow relaxation process and reaches the bath temperature when the system is equilibrated. We show that the nonequilibrium system is described by two temperatures, one for fast vibration and the other for slow configurational relaxation, while the equilibrium system is described by one temperature. The proposed formalism is applicable widely for systems with many metastable states.

Figure 1

Time evolution of mean inherent structure energy $U_{\mathrm{IS}}$ and (inset) mean vibrational energy $U_v$. In the case $0.55T_f>T_d$, the folding process reaches thermal equilibrium around $t=2\times {10}^7$, whereas in $0.34T_f<T_d$ it does not within the computation time. $U_v$ is in the equilibrium value which is slightly larger than the harmonic approximation $3/2k_B{N}_{r}T$ (indicated as 0.51 and 0.83 for $T/T_f=0.34$ and $0.55$, respectively). $N_r$ is the number of residues.

Figure 2

Time evolution of the probability distribution $P(e_{\alpha },t,T)$ for the inherent structure. The horizontal axis is inherent structure energy $e_{\alpha }$ scaled by $k_B{T}_f$. $T=0.34T_f$ and $0.55T_f$ correspond to $T<T_d$ and $T>T_d$, respectively. The right bottom figure shows the equilibrium distribution at $T=0.55T_f$. At $t=0$, the protein has been cooled to $T$.

Figure 3

Time evolution in the plot on $S_{\mathrm{IS}}$ vs $U_{\mathrm{IS}}$ for $t>{10}^6$. $T/T_f=0.34$ and 0.55. The line is the relation between $S_{\mathrm{IS}}$ and $U_{\mathrm{IS}}$ in thermal equilibrium. Higher temperatures correspond to the right-upper part of the line.

Figure 4

(a) Time evolution of $T_{\mathrm{cnf}}\mathbf{(}t,(T_1+T_2)/2\mathbf{)}$ in Eq. (6) estimated by $\{S_{\mathrm{IS}}(t,T_2)-S_{\mathrm{IS}}(t,T_1)\}/(T_2-T_1)$ and $\{U_{\mathrm{IS}}(t,T_2)-U_{\mathrm{IS}}(t,T_1)\}/(T_2-T_1)$. Here $(T_1,T_2)=(0.34T_f,0.41T_f)$, $(0.41T_f,0.48T_f)$, and $(0.55T_f,0.59T_f)$. The dotted horizontal line is $(T_1+T_2)/2$ for $(0.55T_f,0.59T_f)$, which is expected to be a convergence line of $T_{\mathrm{cnf}}(t,0.57T_f)$. (b) Confirmation of the thermodynamic relation $T_{\mathrm{cnf}}=T$ in the thermal equilibrium condition.

Figure 5

(a) $U_{\mathrm{cnf}}(t,T)$ vs $U_{\mathrm{IS}}(t,T)$. Here $T=(T_1+T_2)/2$, with $(T_1,T_2)=(0.34T_f,0.41T_f)$, $(0.41T_f,0.48T_f)$, and $(0.55T_f,0.59T_f)$.