Architecture and coevolution of allosteric materials

Abstract

We introduce a numerical scheme to evolve functional elastic materials that can accomplish a specified mechanical task. In this scheme, the number of solutions, their spatial architectures, and the correlations among them can be computed. As an example, we consider an "allosteric" task, which requires the material to respond specifically to a stimulus at a distant active site. We find that functioning materials evolve a less-constrained trumpetshaped region connecting the stimulus and active sites, and that the amplitude of the elastic response varies nonmonotonically along the trumpet. As previously shown for some proteins, we find that correlations appearing during evolution alone are sufficient to identify key aspects of this design. Finally, we show that the success of this architecture stems from the emergence of soft edge modes recently found to appear near the surface of marginally connected materials. Overall, our in silico evolution experiment offers a window to study the relationship between structure, function, and correlations emerging during evolution. disordered materials | proteins | evolution P roteins are long polymers that can fold in a reproducible way and achieve a specific function. Often, the activity of the main functional site depends on the binding of an effector on a distant site (1, 2). Such an allosteric behavior can occur over large distances, such as 20 residues or more (3), and often involves only a sparse subset of residues in the protein (3, 4). Allosteric regulation offers an appealing target for drug design (5), and there is considerable interest in predicting allosteric pathways (6, 7). One central difficulty is that the physical mechanisms allowing such an "action at a distance" remain elusive. In some cases, allostery can be understood as the modulation of a hinge connecting two extended rigid parts of the protein (8, 9), but, often, the displacement field induced by the binding of the effector cannot be described in these terms (4, 10, 11). Another route, statistical coupling analysis (12), considers correlations within sequences of proteins of the same family to infer allosteric pathways (4, 7). The generality of this elegant approach is, however, debated (13).From a physical viewpoint, specific response at a distance is surprising. The structure of proteins is similar to randomly packed spheres (14). Generically, the response of such systems is nonspecific and decays rapidly in space (in a manner similar to a continuum medium) at distances larger than the particle size; this is true, except close to a critical point where the number of constraints coming from strongly interacting particles is just sufficient to match the number of degrees of freedom of the particles (15). There, the elastic response becomes heterogeneous on all scales (16,17). This point is illustrated in Fig. 1A, showing the rapidly decaying response of a random spring network to a stimulus. However, as shown in Fig. 1B (and independently found in ref. 18 using a different algorithm), springs can be move...

Inferring interaction partners from protein sequences using mutual information

“…Global statistical 15 models allow direct and indirect correlations to be disentangled [2][3][4]. Such models, built using 16 the maximum entropy principle [5], and assuming pairwise interactions, known in the field of 17 proteins as Direct Coupling Analysis (DCA), have been used with success to determine three- 18 dimensional protein structures from sequences [6,7], to predict mutational effects [8][9][10], to find 19 residue contacts between known interaction partners [4,[11][12][13][14][15], and most recently to predict 20 interaction partners from sequence data [16,17]. DCA models lay the emphasis on interactions 21 between residues that are in direct contact in the three-dimensional protein structure, and have 22 been optimized for contact prediction.…”

“…DCA models lay the emphasis on interactions 21 between residues that are in direct contact in the three-dimensional protein structure, and have 22 been optimized for contact prediction. However, correlations in protein sequences also have 23 important collective modes [18,19], which can arise from functional selection [20,21], and addi- 24 tional correlations are due to phylogeny [18,22,23]. These contributions are deleterious to the 25 prediction of contacts [23] but not necessarily to the prediction of interacting partners, since a 26 pair of interacting partners may be subject to common functional selection, and may also have 27 a more strongly shared phylogenetic history than non-interacting proteins.…”

“…In particular, global 364 functional selection could potentially affect both interacting partners together. For instance, a 365 functionally important mechanical deformation of the complex formed by these partners could 366 be subject to selection, yielding collective correlations that extend in the sequences of both 367 partners [20,21]. Besides, interacting partners may also share more phylogenetic history than 368 non-interacting proteins, e.g.…”

Mechanics of allostery: contrasting the induced fit and population shift scenarios

“…This result supports that in at least some proteins elasticity -possibly non-linear -is an appropriate language to describe allostery (in contrast to intrinsically disordered proteins that may be considered more as liquids than solids, for which the analysis proposed here would not hold). Very recently, there has been a considerable effort to use insilico evolution [16,17] to study how linear elastic materials can evolve to accomplish an allosteric task [18][19][20][21][22][23][24][25][26]. In general, binding a ligand locally distorts the protein, which is modelled by imposing local displacements at some site, generating an extended elastic response that in turn determines fitness (chosen specifically to accomplish a given task).…”

Green function of correlated genes and the mechanical evolution of protein

“…For example, many enzymes are known to deform upon binding to their targets (15)(16)(17) by hinge-like rotations, twists or shear-like sliding (18,19). Recent works link such large-scale motions with the emergence of a marginally stable 'shear band' or 'channel' in the protein (18)(19)(20)(21)(22)(23), and other works highlighted the role of such collective interactions in allostery, the transduction of signal between distant sites in the protein (24)(25)(26)(27)(28)(29)(30). Previously, we examined the shear band as a mechanism for allostery, and our simple model distilled certain basic features of the genotype-to-phenotype map (22,23).…”