Structural Determinant of Protein Designability

Here we present an approximate analytical theory for the relationship between a protein structure’s contact matrix and the shape of its energy spectrum in amino acid sequence space. We demonstrate a dependence of the number of sequences of low energy in a structure on the eigenvalues of the structure’s contact matrix, and then use a Monte Carlo simulation to test the applicability of this analytical result to cubic lattice proteins. We find that the lattice structures with the most low-energy sequences are the same as those predicted by the theory. We argue that, given sufficiently strict requirements for foldability, these structures are the most designable, and we propose a simple means to test whether the results in this paper hold true for real proteins.

Figure 1

The difference in sequence space entropy between an energy near the peak of all structural sequence spectra ($E=-2$) and one in the lower tail of all spectra ($E=-8$) as a function of the contact trace, measured here by the largest eigenvalue of the structure’s contact matrix (which follows from $\mathrm{T}\mathrm{r}{C}^n=\sum _{}^{}{\lambda }_i^n$). Each point was generated from data collected while slowly annealing a Monte Carlo sequence design simulation from high temperature ($T=2$) to low ($T=0.2$), with ${10}^7$ Monte Carlo steps taken at each temperature. The boxed points correspond to structures which were chosen by hand so as to ensure that the extrema of the range of possible eigenvalues were represented. All other structures were chosen randomly.

Figure 2

The change in sequence space entropy from energy $⟨E⟩(T=\infty )=\overline{E}$ to energy $E$ for three structures with largest contact matrix eigenvalues of 3.02 (high trace), 2.78 (intermediate trace), and 2.60 (low trace).

Figure 3

We designed sequences on two different target structures in Monte Carlo annealing simulations of $5\times {10}^8$ steps sampled at every sequence in ${10}^4$. Here, we plot the number of sequences whose unique compact ground states were the structure on which they were designed, as a function of the difference in energy between the sequence’s ground state and the lowest-lying excited state which shares fewer than $25\%$ of its contacts with the ground state.