Hidden structure in protein energy landscapes

Inherent structure theory is used to discover strong connections between simple characteristics of protein structure and the energy landscape of a Gō model. The potential energies and vibrational free energies of inherent structures are highly correlated, and both reflect simple measures of networks of native contacts. These connections have important consequences for models of protein dynamics and thermodynamics.

Figure 1

Schematic illustration of GB1 created using RASMOL [16].

Figure 2

Estimated probabilities of inherent structures with energy $e_{\alpha }$ at the collapse temperature $T_{\theta }$.

Figure 3

(Color online) Vibrational free energies $F_v$ of inherent structures vs their potential energies $e_{\alpha }$ (black points). The dependence is well modeled by Eq. (4) with $e_c=88.4k_B{T}_{\theta }$ (piecewise-linear red line). The contribution from the highest 1/3 of the eigenvalues is constant (green points following $y=0$).

Figure 4

Eigenvalue spectrum of an arbitrary inherent structure before (open circles) and after (solid circles) increasing the bond-distance spring constant $K_b$ by a factor of 10. The low-frequency spectra are indistinguishable.

Figure 5

Density of inherent structures ${\Omega }_{IS}\left(e_{\alpha }\right)$. The knee at $e_r=47.4k_B{T}_{\theta }$ is due to a change in stress, and the plateau beginning at roughly $e_c=88.4k_B{T}_{\theta }$ is due to a change in rigidity; both are understood in terms of the network of native contacts (Figs. 6, 7).

Figure 6

(Color online) Potential energy $e_{\alpha }$ vs number of broken contacts, ${\overline{n}}_{\alpha }$. The dependence is well modeled by Eq. (8) (red line). The energy required to break a native contact is approximately equal to the binding energy $ϵ$, with differences due to stress in the structure. The knee corresponds to a change in stress at ${\overline{n}}_r=60$, where $e_{\alpha }\left({\overline{n}}_r\right)=59.6k_B{T}_{\theta }$.

Figure 7

Number of residues, $m_{\alpha }$, for which all native contacts are broken vs the number of broken contacts, ${\overline{n}}_{\alpha }$. The dependence is well modeled by Eq. (9) (black line). Note that $m_{\alpha }=55$ (close to the expected value of 56) at ${\overline{n}}_{\alpha }=207$.

Figure 8

Comparison of vibrational free energies calculated from Eq. (10) ($y$ axis) and estimated using inherent structure theory ($x$ axis).