Colloquium: Proteins: The physics of amorphous evolving matter

Protein is matter of dual nature. As a physical object, a protein molecule is a folded chain of amino acids with diverse biochemistry. But it is also a point along an evolutionary trajectory determined by the function performed by the protein within a hierarchy of interwoven interaction networks of the cell, the organism, and the population. A physical theory of proteins therefore needs to unify both aspects, the biophysical and the evolutionary. Specifically, it should provide a model of how the DNA gene is mapped into the functional phenotype of the protein. Several physical approaches to the protein problem are reviewed, focusing on a mechanical framework which treats proteins as evolvable condensed matter: Mutations introduce localized perturbations in the gene, which are translated to localized perturbations in the protein matter. A natural tool to examine how mutations shape the phenotype are Green’s functions. They map the evolutionary linkage among mutations in the gene (termed epistasis) to cooperative physical interactions among the amino acids in the protein. The mechanistic view can be applied to examine basic questions of protein evolution and design.

Figure 1

A representation of the genetic code, as a function of the measured polarity of each codon [values from [175] and 69]. The smoothness of the landscape shows that moving from one AA to its neighbor does not change the polarity too abruptly.

Figure 2

Schematic illustration of protein folding: Proteins are polypeptides, linear heteropolymers of AAs (colored spheres), linked by covalent peptide bonds (red sticks), which form the protein backbone. These peptide bonds are much stronger than the noncovalent interactions among the AAs (side chains) and do not change when the protein mutates.

Figure 3

The motion and deformation between two states of glucokinase, 1v4s and 1v4t. The arrows are scaled up for better visibility. They are colored green or red depending on whether they move down or up relative to a plane passing horizontally through the center of the protein, respectively. Galilean motions have been eliminated. The red coloring of the tube corresponds to concentration of shear and is the same as in the leftmost panel of Fig. 5; see Sec. 9e.

Figure 4

A schematic interpretation of Fig. 3. Emphasis is given to the two moving pieces, with a hinge between them. This kind of hinge will be called the “fluid channel” or “shear band.”

Figure 5

Evolutionary and mechanical properties of glucokinase. Left: Shear (darker is more shear), discussed in Sec. 9e. Center: Conservation (darker represents fewer mutations), discussed in Sec. 9j. Right: Correlation (dark red represents sites of correlated mutations), discussed in Sec. 9k.

Figure 6

The main features of the cylinder model. Left: The mapping from the binary gene to the connectivity of the amino acid (AA) network that makes a functional protein. The color of the AAs represents their rigidity state as determined by the connectivity according to the algorithm of Sec. 8a3. Each AA can be in one of three states: rigid (yellow) 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 hingelike motion along the fluid channel.

Figure 7

Illustration of the percolation rules for shearability and rigidity states. Note that site $(r,q)$ was 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 site is shearable and fluid, since it is attached to only one solid site [namely, $(r,q)$] 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 $(r,q+2)$.

Figure 8

Evolution of a mechanical function: A configuration (left) with a prescribed input (black ellipse at bottom) and random connectivity pattern eventually evolved to form a fluid channel (right). The initial state has six fluid points (black ellipse), and our fitness requires five fluid points at the top (black ellipse).

Figure 9

Following the progress of evolution during a typical run. It is a sequence of mostly neutral steps, a fraction of deleterious ones, and rare beneficial steps. The vertical axis is the accumulated number of steps of each type.

Figure 10

The protein is made of two species of AAs, polar ($\mathrm{P}$, red) and hydrophobic ($\mathrm{H}$, blue) whose sequence is encoded in a gene. Each AA forms weak or strong bonds with its 12 nearest neighbors on the hexagonal lattice (right) according to the interaction rule in the table (left).

Figure 11

Illustration of the deformation field $\mathbf{u}(\mathbf{c})=\mathbf{G}(\mathbf{c})\mathbf{f}$, Eq. (4) for three choices of $\mathbf{c}$. The force $\mathbf{f}=f·{\mathbf{n}}_{p^{\prime }{q}^{\prime }}$ is applied on the left of the lattice. The three panels show, from left to right, how the response $\mathbf{u}$ (shown in small arrows) evolves as the network fitness $F={\mathbf{v}}^{\mathrm{T}}\mathbf{u}(\mathbf{c})$ increases. The choice of $\mathbf{v}$ corresponds to the vertical separation of the two central points on the right.

Figure 12

Illustration of the backbone. The backbone is shown as a solid black serpentine curve. AAs in neighboring sites along the backbone tend to move together. We show two configurations: (left) parallel to the channel and (right) perpendicular to the channel. Parallel: The backbone favors the formation of a narrow channel along the fold (compared to Fig. 11). Perpendicular: The formation of the channel is “dispersed” by the backbone.

Figure 13

Dimensional reduction of the genotype-to-phenotype map: Dimension measurement from $10^6$ independent configurations (phenotypes) for the cylinder model. The dimension of the configuration space is about 9 (red curve), while in sequence space it is basically infinite (blue curve). All pairs seem to have the same distance, namely, $1275=2550/2$, which is the typical distance between two random sequences.

Figure 14

Distribution of solutions in the sequence universe: A measure for the expansion of functional genes in the sequence universe is the backward-to-forward ratio, the fraction of point mutations that make two sequences closer vs the ones that increase the distance ([133]). 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-to-forward ratio of purely random mutations. Given the overwhelming number of samples near the maximal $D$, the Gaussian distribution is well visible (in the vertical direction).

Figure 15

Correspondence of modes in sequence (genotype) and configuration (phenotype) spaces for the cylinder model: We produced the spectra by singular value decomposition of the $10^6$ solutions. (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 eight to nine high eigenvalues that are outside the continuous spectrum. (Note that the $x$ axis does not start at 0.) 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 required at least $10^5$ independent solutions; cf. [155].

Figure 16

The projections of the set of $10^6$ solutions as 2550-dimensional gene sequences (left column) and corresponding 540-dimensional phenotypes (right column) onto their SVD eigenvectors $\mathbf{c}$ and $\mathbf{u}$. The top row shows axes 2 (horizontal) and 100 (vertical), and the bottom row shows axes 3 (horizontal) and 100 (vertical). Note that the phenotypes have larger variation along their 2 and 3 components than their 100 component, unlike the genotypes which appear evenly distributed in all directions.

Figure 17

The sensitivity of solutions to a single mutation as a function of the mutation position. Mutations at the output and around the shear band tend to be more deleterious (darker blue), while in the rigid regions they are likely to be neutral.

Figure 18

The vector fields for the HP model: The first four eigendirections for the three vector fields $k=1,\dots ,4$. Top: The first four SVD eigenvectors of the gene ${\mathbf{C}}_k$. Center: The corresponding displacement flow field ${\mathbf{U}}_k$. Bottom: The corresponding shear intensity ${\mathbf{S}}_k$.

Figure 19

Progression of the fitness $F$ corresponding to the evolution of Fig. 11 (black). The fitness trajectory averaged over $\sim {10}^6$ runs $⟨F⟩$ is shown in red. Shown are the last 16 beneficial mutations toward the formation of the channel. The contribution of the emergent low-energy mode $⟨F_k⟩$ alone, shown in blue, dominates the fitness [according to Eq. (15)].

Figure 20

Landscape of the fitness change $\delta F={\mathbf{v}}^{\mathrm{T}}\delta \mathbf{G}\mathbf{f}$, averaged over $10^6$ solutions, for all 200 possible positions of point mutations at a solution. Underneath, the average AA configuration of the protein is shown in shades of red ($\mathrm{P}$) and blue ($\mathrm{H}$). In most sites, mutations are neutral, while mutations in the channel are, on average, deleterious (blue, below the flat surface).

Figure 21

The mean shear in the protein in a single run (black) and averaged over $10^6$ samples (red) as a function of the fraction $p$ of $\mathrm{P}$-amino acids. The values of $p$ are shifted by the position of the jump $p_{\mathrm{c}}$. Inset: Distribution of $p_c$.

Figure 22

The average magnitude of the two-codon correlation $|Q_{ij}|$ Eq. (16) as a function of the number of beneficial mutations $t$. The red curve shows $|Q_{ij}|$ in the shear band (AAs in rows 7–13, of Fig. 11). The black curve shows $|Q_{ij}|$ for the whole protein. The correlations in the channel are clearly larger.

Figure 23

Force propagation, mutations, and epistasis. (a) The Green’s function $\mathbf{G}$ measures the propagation of the mechanical signal across the protein (blue) from the force source $\mathbf{f}$ (pinch) to the response site $\mathbf{v}$, depicted as a “diffraction wave.” (b) A mutation $\delta {\mathbf{H}}_i$ deflects the propagation of force. The effect of the mutation on the propagator $\delta \mathbf{G}$ can be described as a series of multiple scattering paths Eq. (6). The diagram shows the first scattering path $\mathbf{G}\delta {\mathbf{H}}_i\mathbf{G}$. (c) Epistasis is the departure from additivity of the combined fitness change of two mutations. The epistasis between two mutations $\delta {\mathbf{H}}_i$ and $\delta {\mathbf{H}}_j$ is equivalent to a series of multiple scattering paths Eq. (19). The diagram shows the path $\mathbf{G}\delta {\mathbf{H}}_i\mathbf{G}\delta {\mathbf{H}}_j\mathbf{G}$.

Figure 24

Mechanical epistasis. The epistasis of Eq. (17), averaged over $10^6$ solutions $E_{ij}=⟨e_{i,j}⟩$, between a fixed AA at position $i$ (black arrow) and all other positions $j$. Here $i$ is located at (a) the binding site, (b) the center of the channel, and (c) slightly off the channel. Underneath, the average AA configuration of the protein is drawn in shades of red ($\mathrm{P}$) and blue ($\mathrm{H}$). Significant epistasis mostly occurs along the $\mathrm{P}$-rich channel, where mechanical interactions are long ranged. Although epistasis is predominantly positive, negative values also occur, mostly at the boundary of the channel (c). (d) The two-codon correlation function $Q_{ij}$ of Eq. (16) measures the coupling between mutations at positions $i$ and $j$. The epistasis $E_{ij}$ and the gene correlation $Q_{ij}$ show similar patterns. Axes are the positions of $i$ and $j$ loci. Significant correlations and epistasis occur mostly in and around the channel region (positions 70–130, rows 7–13).