Detecting selection for negative design in proteins through an improved model of the misfolded state

Abstract

Proteins that need to be structured in their native state must be stable both against the unfolded ensemble and against incorrectly folded (misfolded) conformations with low free energy. Positive design targets the first type of stability by strengthening native interactions. The second type of stability is achieved by destabilizing interactions that occur frequently in the misfolded ensemble, a strategy called negative design. Here, we investigate negative design adopting a statistical mechanical model of the misfolded ensemble, which improves the usual Gaussian approximation by taking into account the third moment of the energy distribution and contact correlations. Applying this model, we detect and quantify selection for negative design in most natural proteins, and we analytically design protein sequences that are stable both against unfolding and against misfolding. Proteins 2013; 81:1102–1112. © 2013 Wiley Periodicals, Inc.

Detecting Selection on Protein Stability through Statistical Mechanical Models of Folding and Evolution

“…We observe that the CFESs are positive, i.e. , native contacts that are short range are associated with energies that are, in general, higher than for long-range native contacts [24]. The same conclusion is found if we compare non-native contacts that are short range with non-native contacts that are short range.…”

“…It has been proposed that the misfolded state, consisting in the ensemble of compact, but wrongly folded conformations, is described by the Random Energy Model (REM) [20], which approximates the energy with a Gaussian random variable [21,22,23]. In a similar spirit, we can go beyond the REM by computing the free energy of the misfolded ensemble from the partition function of all possible compact contact matrices, obtaining the analytic approximation [24]: $\begin{array}{ll}{G}_{\text{misfolded}}& \equiv -Tlog\left(\text{∑}_C{e}^{-{\sum }_{i<j}{C}_{ij}U(A_i,A_j)/T+S(C)}\right)\hfill \\ & \approx \langle E\rangle -\frac{\langle {\left(E-\langle E\rangle \right)}^2\rangle }{2T}+\frac{\langle {\left(E-\langle E\rangle \right)}^3\rangle }{6T^2}-L{S}_{C}T\end{array}$where LS C is the logarithm of the number of compact contact matrices, 〈.〉 represents the average over the set of alternative compact contact matrices of L residues (To derive Equation (1), we write the sum over all contact matrices grouping those with the same number of contacts N C = ∑ i < j C ij , and we distinguish a homopolymer energy (all contact interactions equal to 〈 E 〉 / N C ) and a heteropolymer contribution, writing exp(− G / T ) = ∑ N C exp (− 〈 E 〉 / T + S ( N C )) ∑ C:N C exp(− βz ), with z = (∑ ij C ij U ij − 〈 E 〉/ N C and β = N C / T . We then approximate $log({\sum }_{C}exp(-\beta z)\approx S_C+log\langle exp(-\beta z)\rangle \approx \frac{1}{2}\langle {(\beta z)}^2\rangle -\frac{1}{6}\langle {(\beta z)}^3\rangle )$.…”

“…However, if the third moment is positive and the total conformational entropy, S C , is large, the conformational entropy is always positive, and the freezing transition does not take place. Instead, the specific heat vanishes at the critical temperature $T_c=\frac{\langle {(E-\langle E\rangle )}^3\rangle }{\langle {(E-\langle E\rangle )}^2\rangle }$ [24], and the misfolded ensemble has a second order phase transition reminiscent of a coil-globule E ransition. This model also shows that the freezing temperature is larger when the average contact energy is more negative, i.e.…”

“…Putting together these free energy estimates, we obtain the free energy difference between the native and the non-native states above the freezing temperature, Δ G ( A , C nat ) = G nat − G misfolded − G unfolded , as: $\begin{array}{cc}\mathrm{\Delta }G(A,C^{\text{nat}})& =TL(S_C+S_U)+\text{∑}_{i<j}false(C_{ij}^{\text{nat}}-\langle C_{ij}\rangle false)Ufalse(A_i,A_{j}false)\\ & +\text{∑}_{i<j,k<l}false(\langle C_{ij}\rangle -{\langle C_{ij}\rangle }^{2}false)\frac{Ufalse(A_i,A_{j}false)Ufalse(A_k,A_{l}false)}{2T}\\ & -\text{∑}_{i<j,k<l,m<n}{F}_{\text{italicijklmm}}\frac{Ufalse(A_i,A_{j}false)Ufalse(A_k,A_{l}false)Ufalse(A_m,A_{n}false)}{6T^2}\end{array}$where F ijklmm = 〈( C ij − 〈 C ij 〉 ( C kl − 〈 C kl 〉 ( C mn − 〈 C mn 〉)〉 [24]. The free energy depends on the mean contact frequency and on the correlation and skewness of the contacts.…”

“…To investigate selection on sequence composition, we divide amino acids into three classes: hydrophobic {L, I, V, F, Y, C, W, M}, polar and charged {D, E, G, S, N, K, R} and others {A, T, H, Q, P}, and we compared observed frequencies to a null model that takes into account that the sum of frequencies is one [24]. We found that selection favors two types of amino acid compositions: those in which a relatively large frequency of hydrophobic residues coexist with a large frequency of polar residues, and those in which both hydrophobic and polar residues are depleted.…”