Analyses of simulations of three-dimensional lattice proteins in comparison with a simplified statistical mechanical model of protein folding

Folding and unfolding simulations of three-dimensional lattice proteins were analyzed using a simplified statistical mechanical model in which their amino acid sequences and native conformations were incorporated explicitly. Using this statistical mechanical model, under the assumption that only interactions between amino acid residues within a local structure in a native state are considered, the partition function of the system can be calculated for a given native conformation without any adjustable parameter. The simulations were carried out for two different native conformations, for each of which two foldable amino acid sequences were considered. The native and non-native contacts between amino acid residues occurring in the simulations were examined in detail and compared with the results derived from the theoretical model. The equilibrium thermodynamic quantities (free energy, enthalpy, entropy, and the probability of each amino acid residue being in the native state) at various temperatures obtained from the simulations and the theoretical model were also examined in order to characterize the folding processes that depend on the native conformations and the amino acid sequences. Finally, the free energy landscapes were discussed based on these analyses.

Figure 1

Two native conformations of 3D cubic lattice proteins (top) and their contact maps (bottom). Two foldable amino acid sequences were designed for each conformation by Shakhnovich and co-workers. Proteins $a1$ and $a2$ [18] are folded into conformation (a) and $b1$ and $b2$ [19] into (b) (see text for their amino acid sequences). In the contact maps, the amino acid residue pairs in contact in the native conformations are indicated by an open circle. The total number of contacts $\left(N_{\text{total}}\right)$ in these native conformations is 40 for proteins $a1$ and $a2$ and 57 for proteins $b1$ and $b2$.

Figure 2

(Color online) Two snapshots of the conformations generated in the simulation of protein $a1$ illustrate the existence of several local structures and various types of amino acid residue contacts. The local structures are the enclosed regions indicated by the dashed lines in the molecular models, and the triangular regions in the triangle maps correspond to them. The others are the random-coil regions. Top: A conformation with both native contact (NC) and non-native contact (non-NC) pairs. The NC pairs are classified into the following four classes: (1) intra-$N$ (the amino acid residue pairs 7-10 and 18-21), (2) $N\text{−}N$ (19-32), (3) $N\text{−}D$ (2-7), and (4) $D\text{−}D$ (no such pair found). In this case, the non-NC pairs are 16-23, 17-22, 21-26, and 22-25. In the $A\text{−}W$ model, only the interaction energies for the intra-$N$ are considered while the other interactions are neglected. Bottom: A conformation only with intra-$N$ (2-7, 3-6, 18-21, and 27-30 in this case).

Figure 3

Equilibrium transition curves ${\theta }_h^{\mathrm{th}}\left(T\right)$ and ${\theta }_h^{\mathrm{sim}}\left(T\right)$ obtained from the $A\text{−}W$ model (theory) and the simulations (simulation), and the normalized average enthalpies ${\theta }_{\mathrm{NC}}^{\mathrm{sim}}\left(T\right)$, ${\theta }_{\mathrm{intra}\text{−}N}^{\mathrm{sim}}\left(T\right)$, ${\theta }_{N\text{−}N}^{\mathrm{sim}}\left(T\right)$, ${\theta }_{N\text{−}D}^{\mathrm{sim}}\left(T\right)$, and ${\theta }_{D\text{−}D}^{\mathrm{sim}}\left(T\right)$ for all native contacts (total NC) and for the four types of native contacts, i.e., intra-$N$, $N\text{−}N$, $N\text{−}D$, and $D\text{−}D$, respectively, obtained from the simulations for the proteins $a1$, $a2$, $b1$, and $b2$ as functions of temperature $T$. The difference between ${\theta }_h^{\mathrm{sim}}\left(T\right)$ and ${\theta }_{\mathrm{NC}}^{\mathrm{sim}}\left(T\right)$ indicates the normalized average enthalpy for the non-NC $\left[{\theta }_{\mathrm{non}\text{−}\mathrm{NC}}^{\mathrm{sim}}\left(T\right)\right]$.

Figure 4

Probability $p_i$ of an amino acid residue $i$ being in a native state obtained from the simulations and the $A\text{−}W$ model at several temperatures for proteins $a1$, $a2$, $b1$, and $b2$. The temperatures are shown near the corresponding lines. The turns and extended regions in the native conformations are indicated by symbols ● and -, respectively (see also Fig. 1).

Figure 5

Thermodynamic properties $H\left(\eta \right)$, $-TS\left(\eta \right)$, and $F\left(\eta \right)$ (in dimensionless $k_{\mathrm{B}}T$ unit) as functions of the progress variable $\eta$ (fraction of the number of amino acid residues in the native state) obtained from the simulations and the $A\text{−}W$ model for proteins $a1$, $a2$, $b1$, and $b2$. The thin solid, thick solid, and dotted lines are the profiles for $T=0.24$, 0.22, and 0.21 for $a1$; $T=0.28$, 0.26, and 0.22 for $a2$; and $T=0.21$, 0.20, and 0.18 for $b1$ and $b2$, respectively.

Figure 6

(Color online) Free-energy $F(Q,C)$ (in dimensionless $k_{\mathrm{B}}T$ unit) plotted against the two characteristics of folding $Q$ (fraction of the number of native contacts) and $C$ (fraction of the number of both native and non-native contacts) obtained from the simulations for proteins $a1$, $a2$, $b1$, and $b2$ at the following temperatures that correspond to ${\theta }_h^{\mathrm{sim}}=0.6$: $T=0.22$ for $a1$, $T=0.26$ for $a2$, and $T=0.20$ for $b1$ and $b2$. For the native conformation, $(Q,C)=(1,1)$. The light blue and pink points indicate the local minima $\partial F∕\partial C=0$ and $\partial F∕\partial Q=0$, respectively. The yellow points show the coincident points of both the local minima. These points indicate the most probable equilibrium paths of the folding. In the inset boxes, free energies $F\left(Q\right)$ are plotted only as a function of $Q$. The swelling at $Q=0.9$ may be a lattice artifact due to the very small number of conformations [2].