Green function of correlated genes and the mechanical evolution of protein

Abstract

There has been growing evidence that cooperative interactions and configurational rearrangements underpin protein functions. But in spite of vast genetic and structural data, the information-dense, heterogeneous nature of protein has held back the progress in understanding the underlying principles. Here we outline a general theory of protein that quantitatively links sequence, dynamics and function: The protein is a strongly-coupled amino acid network whose interactions and large-scale motions are captured by the mechanical propagator, also known as the Green function. The propagator relates the gene to the connectivity of the amino acid network and the transmission of forces through the protein. How well the force pattern conforms to the collective modes of the functional protein is measured by the fitness. Mutations introduce localized perturbations to the propagator which scatter the force field. The emergence of function is manifested by a topological transition when a band of such perturbations divides the protein into subdomains. Epistasis quantifies how much the combined effect of multiple mutations departs from additivity. We find that epistasis is the nonlinearity of the Green function, which corresponds to a sum over multiple scattering paths passing through the localized perturbations. We apply this mechanical framework to the simulations of protein evolution, and observe long-range epistasis which facilitates collective functional modes. Our model lays the foundation for understanding the protein as an evolved state of matter and may be a prototype for other strongly-correlated living systems.Protein evolution | Mechanical epistasis | Genotype-to-phenotype map | Green function | Dimensional reduction

Physical Model of the Genotype-to-Phenotype Map of Proteins

“…We find that the collective mechanical interactions among the amino acids are mirrored in corresponding modes of sequence correlation in the genes. These main results do not depend on details of the model and have been reproduced in versions with (i) a different number of AA species (16 instead of 32), (ii) bonds that depend on pairwise interactions, and (iii) harmonic spring network [24]. All these suggest that the results are generic and apply to a wide range of realizations.…”

“…This graph is embedded in 2D where x r,c are the positions of the AAs, which are connected by straight bonds. We now extend the scope of our study somewhat, by assuming that the bonds are not totally rigid, but given by harmonic springs (see also [24]). This allows us to study mechanical properties which would be too stiff if we only worked with bonds which are rigid sticks.…”

“…the sought-after mechanical shear motion in Sections II E and B 3. (In[24] we take the mechanical modes themselves as the target function. )…”

Higher order analysis of gene correlations by tensor decomposition

“…Higher-order correlations among genes conditioned by the biological and experimental settings in which they are observed are not yet very well understood because of the difficulties to detect them genetic mapping studies, and not too many examples of such correlations have been described in the literature. However, evidences from model organisms suggest that these higher-order correlations among genes contribute frequently to genetic studies [9,12,15,22,26]. Informally, the main goal of this paper is to use powerful tensor analysis methods to provide the foundations of systematic and computationally efficient approaches to distinguish significant higher order correlations among the elements of biological systems.…”