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

How DNA is mapped to functional proteins is a basic question of living matter. We introduce and study a physical model of protein evolution which suggests a mechanical basis for this map. Many proteins rely on large-scale motion to function. We therefore treat protein as learning amorphous matter that evolves towards such a mechanical function: Genes are binary sequences that encode the connectivity of the amino acid network that makes a protein. The gene is evolved until the network forms a shear band across the protein, which allows for long-range, soft modes required for protein function. The evolution reduces the high-dimensional sequence space to a low-dimensional space of mechanical modes, in accord with the observed dimensional reduction between genotype and phenotype of proteins. Spectral analysis of the space of $10^6$ solutions shows a strong correspondence between localization around the shear band of both mechanical modes and the sequence structure. Specifically, our model shows how mutations are correlated among amino acids whose interactions determine the functional mode.

Popular Summary

Proteins are complex molecules that perform a range of essential tasks in a living cell. Each protein is a chain of a few hundred basic building blocks known as amino acids that are assembled according to a blueprint written in a cell’s DNA. While the structure of individual proteins is well understood, the secret of how a simple DNA gene—a long “word” that lists the amino acids—encodes the intricate structure and biochemistry of a protein remains elusive. Our paper offers a simple answer: A protein’s function relies on how it moves, and this large-scale motion is exactly what is coded for by the DNA.

We develop a simple mathematical model of a protein, where the gene is a binary sequence that determines the connectivity of the amino acid network that makes the protein. In our simulations, the gene evolves such that the protein network moves in a specific fashion, which facilitates the function of the protein. Our model is easy to calculate, and this allows us to repeat the evolutionary search millions of times. This massive statistic allows us to explain the fundamental properties of the relation between DNA and proteins. In particular, it explains why biochemical function is described by only a few parameters, a number that is much smaller than the length of the gene or the number of amino acids in the protein.

Our model shows how the parameters for a protein’s function are encoded in specific parts of the gene sequence that are surrounded by large chunks that record mostly random evolution. Understanding proteins as evolvable soft machines may open possibilities to tweak and engineer nanomachines with new biological functions.

Figure 1

The main features of the physical model. Left: The mapping from the binary gene to the connectivity of the amino acid (AA) network that makes a functional protein. AAs are beads and links are bonds. The color of the AAs represents their rigidity state as determined by the connectivity according to the algorithm of Appendix pp1-s3. Each AA can be in one of three states: rigid (gray) or fluid (i.e., nonrigid), which are divided between shearable (blue) and nonshearable (red). Right: The AAs in the model protein are arranged in the shape of a cylinder, in this case with a fluid channel (blue region). Such a configuration can transduce a mechanical signal of shear or hinge motion along the fluid channel.

Figure 2

Evolution of mechanical function. Top: An initial configuration with a given input (black ellipse at bottom) and a random sequence is required to evolve into a straight fluid channel ($S$) or a tilted one ($T$). Bottom: Following the success of evolution. In each generation, a randomly drawn bit (a letter in the five-bit codon) is flipped, and this point mutation is changing one bond (similar to point mutations that change one base in a codon). A typical run is a sequence of mostly neutral steps, a fraction of deleterious ones, and rare beneficial steps. Note that the fitness of the configuration is measured only at the top, not in the interior of the cylinder.

Figure 3

Dimensional reduction of the genotype-to-phenotype map. Dimension measurement for the straight ($S$, top) and tilted ($T$, bottom) cases. $10^6$ independent functional configurations are found for the input-output problem. An estimate for the dimension of the solutions is the correlation length, the slope of the cumulative fraction of solution pairs as a function of distance. In configuration space (red), the distance is the number of AAs (out of 540) with a different rigidity state. The estimated dimension from $10^{12}/2$ distances is about 9 (black line) for problem $S$ and 8.5 in problem $T$. The sequence space is a 2550-dimensional hypercube with $32^{510}$ sequences. Most distances are close to the typical distance between two random sequences ($2550/2=1275$), indicating a high-dimensional solution space. An estimate for the dimension is $\sim 150$ (black line) for both $S$ and $T$ problems. The similarity of the dimensions in both cases suggests that these numbers are not specific to the problem.

Figure 4

Distribution of solutions in the sequence universe. A measure for the expansion in the functional sequence universe is the backward/forward ratio, the fraction of point mutations that make two sequences closer versus the ones that increase the distance [6]. The Hamming distances $D$ (normalized by the universe diameter $d_{\max }=2550$) show that most sequences reach the edge of the universe, where no further expansion is possible. The black curve, $D/(1-D)$, is the backward/forward ratio from purely random mutations.

Figure 5

Long-range genetic correlations. (a) The sequence correlation matrix across the $10^6$ examples shows long-range correlations among the bits (codons) at the rigid or fluid boundary, and short-range correlations in the rigid domains. (b) A cross section perpendicular to the diagonal axis.

Figure 6

Correspondence of modes in sequence and configuration spaces. We produce the spectra by singular value decomposition of the ${10}^6$ solutions of problem $S$. The corresponding spectra for the $T$ case are shown in Fig. 7. (a) Top: The spectrum in configuration space exhibits about 8–10 eigenvalues outside the continuum (large first eigenvalue not shown). Bottom: The corresponding eigenvectors describe the basic modes of the fluid channel, such as side-to-side shift (second) or expansion (third). (B) Top: The spectrum of the solutions in sequence space is similar to that of random sequences (black line), except for about 8–9 high eigenvalues that are outside the continuous spectrum. Bottom: The first eight eigenvectors exhibit patterns of alternating $+/-$ stripes—which we term correlation ripples—around the fluid channel region. Seeing these ripples through the random evolutionary noise requires at least $10^5$ independent solutions [39].

Figure 7

Spectra and eigenfunctions for the tilted example ($T$). Note the similarity with Fig. 6, and also how the tilt is manifested not only in the protein modes, but also in the gene modes. This demonstrates that the gene and the protein share common features. (a) The configuration spectrum and eigenfunctions. (b) The sequence spectrum and eigenfunctions.

Figure 8

Stability of the mechanical phenotype to mutations. Mutations at sensitive positions of the sequence move the output away from the prescribed solution. (a) Fraction of runs (among $10^6$) destroyed by the $m$th mutation. A single mutation destroys about 9% of solutions. The proportion decays exponentially like $\exp (-0.09m)$. (b) The density map of such mutations for problems $S$ and $T$ (Fig. 2) shows accumulation around the fluid channel and at the top layer (dark regions). (c) The double mutations are evenly distributed in the rigid regions.

Figure 9

Mechanical shear modes. Displacement and strain fields for the tilted solution $T$ for two low eigenvalues. The vectors show the direction of the displacement and the color code denotes the strain (i.e., the local change in the vector field as a function of position; maximal stress is red).

Figure 10

Adaptation of thermal stability. Extreme configurations, with low (50%, left) and high (95%, right) bond density, solve problem $T$.

Figure 11

The AA interaction model. (a) Dimensions of the genotype and phenotype spaces are similar to the standard model (Fig. 3). (b) Left: The first few eigenfunctions for the configuration. Right: The same for the bond patterns.

Figure 12

Illustration of the percolation rules for shearability and fluid or solid states. Note that site $(r,c)$ is turned solid because it is attached to two solid sites below it. Also note that the red site above it is fluid, because it is attached to less than two solid sites below it. But it is not shearable because it does not connect to a shearable site below it. On the other hand, the top right-hand site is shearable and fluid, since it is attached to only one solid site [namely, $(r,c)$] and no others on the invisible part of the structure (as seen by its blue connections), and it is also connected to the blue site at $(r,c+2)$.

Figure 13

The distributions of the bond densities for the $10^6$ solutions. Note that these densities are just like random Gaussian variables, except for the outliers.