Looking at structure, stability, and evolution of proteins through the principal eigenvector of contact matrices and hydrophobicity profiles

Abstract

We review and further develop an analytical model that describes how thermodynamic constraints on the stability of the native state influence protein evolution in a site-specific manner. To this end, we represent both protein sequences and protein structures as vectors: Structures are represented by the principal eigenvector (PE) of the protein contact matrix, a quantity that resembles closely the effective connectivity of each site; Sequences are represented through the "interactivity" of each amino acid type, using novel parameters that are correlated with hydropathy scales. These interactivity parameters are more strongly correlated than the other hydropathy scales that we examine with: (1) The change upon mutations of the unfolding free energy of proteins with two-states thermodynamics; (2) Genomic properties as the genome-size and the genome-wide GC content; (3) The main eigenvectors of the substitution matrices. The evolutionary average of the interactivity vector correlates very strongly with the PE of a protein structure. Using this result, we derive an analytic expression for site-specific distributions of amino acids across protein families in the form of Boltzmann distributions whose "inverse temperature" is a function of the PE component. We show that our predictions are in agreement with site-specific amino acid distributions obtained from the Protein Data Bank, and we determine the mutational model that best fits the observed site-specific amino acid distributions. Interestingly, the optimal model almost minimizes the rate at which deleterious mutations are eliminated by natural selection.

A protein evolution model with independent sites that reproduces site-specific amino acid distributions from the Protein Data Bank

“…Here we simulate the SCN model with a more realistic mutation process that takes into account the genetic code and represent mutations nucleotide level. The stochastic process that corresponds to this modified SCN model is the combination of two processes [ 53 ]: (1) A mutation process identical to the one simulated in the SCN model; (2) The selection process described by Eq. (7), which imposes that the site-specific average HP is perfectly correlated with the PE.…”

“…We have modified the mutation process in order to take into account the genetic code and the mutation bias at the DNA level (see also Ref. [ 53 ]). We represent each amino acid site by 3 nucleotides, and consider two mutation schemes: (1) Independent and identical mutation processes at each nucleotide site, each one satisfying detailed balance.…”

Positive and Negative Design in Stability and Thermal Adaptation of Natural Proteins

“…In particular, our recent analysis of complete bacterial proteomes [8] revealed that proteomes of thermophilic bacteria are enriched in both hydrophobic residues (IVYLW) and charged ones (ER), while all polar residues are suppressed. Discrepancies between different hydrophobicity scales [42], the statistical nature of knowledge-based Miyazawa–Jernigan potential [22], and limitations of the lattice model make it impossible to quantitatively compare the content of individual amino acids in lattice and natural proteomes or exactly predict the amino acid composition of thermophilic proteomes with very high accuracy from lattice model calculations. Nevertheless our lattice calculations are in semiquantitative agreement with data on natural proteomes, (see Figures 1 and 3) and exhibit the same “from both ends of hydrophobicity scale” trend in amino acid composition adaptation in response to elevated habitat temperature.…”

“…To relate the trends in amino acid composition with the physical properties and interaction energies of individual amino acids, we use hydrophobicity as a generic parameter characterizing an amino acid [42]. To characterize the hydrophobicity of amino acids in the simulations, we make use of the fact that the Miyazawa–Jernigan interaction energy matrix is very well approximated by its spectral decomposition [43].…”

Efficient Parameter Estimation of Generalizable Coarse-Grained Protein Force Fields Using Contrastive Divergence: A Maximum Likelihood Approach

“…83−85 It has been argued that the Boltzmann hypothesis represents an evolutionary equilibrium where these structural features are maintained around a narrow set of values, 83 for example it has been proposed that protein sequences have evolved maintaining an optimal mean hydrophobicity profile. 84 According to the maximum entropy principle, these may be considered as evolutionary constraints on the evolution of protein sequences (see the Discussion section in the work of Podtelezhnikov et al 27 ). This argument suggests the existence of a generalizable protein force field that captures these evolutionary constraints, which we infer using a training set of protein conformations that is representative of the proteins to be modeled (that is, proteins with a globular structure).…”