Green Function of Correlated Genes in a Minimal Mechanical Model Of

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(CC 4.0 access License NonCommercial-NoDerivatives open This interest. of conflict no declare authors Diego. The Biological San for California, Center of National University M.V., Research, and Fundamental Sciences; of Institute and Tata research M.T., performed Reviewers: and designed T.T. paper. and the A.L., wrote J.-P.E., S.D., contributions: Author rti ensisfins n iet h vltoaysearch. evolutionary the directs and fitness the its across response defines mechanical protein of forces propagation of The propagation via motion. impulse and localized a pro- the to how the responds measures understand protein function to Green way The natural dynamics. a collective is tein’s 77) (76, function Green a vletwr pcfi ehnclfnto:i epnet a conforma- 1 to large-scale (Fig. a response to change undergo in tional will network protein function: the the mechanical force, allowing specific localized is acid a interactions, that amino toward the topology one evolve tweak substitute with that another network Mutations with acid gene. the amino in evolving encoded an of as interpretation tein mechanical a the give evolu- how to their aim of particular, directs epistasis. 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BIOPHYSICS AND COMPUTATIONAL BIOLOGY Fig. 1. Protein as an evolving machine and propagation of mechanical forces. (A) Formation of a softer shear band (red) separating the protein into two rigid subdomains (light blue). When a ligand binds, the biochemical function involves a low-energy hinge-like or shear motion (arrows). (B) Shear band and large-scale motion in a real protein: the arrows show the displacement of all amino acids in human glucokinase when it binds glucose (Protein Data Bank ID codes 1v4s and 1v4t). The coloring shows a high-shear region (red) separating two low-shear domains that move as rigid bodies (shear calculated as in refs. 21 and 36). (C) The mechanical model. The protein is made of two species of amino acids, polar (P; red) and hydrophobic (H; blue), with a sequence that is encoded in a gene. Each amino acid forms weak or strong bonds with its 12 near neighbors (Right) according to the interaction rule in the table (Left). (D) The protein is made of 10 × 20 = 200 amino acids with positions that are randomized from a regular triangular lattice. Strong bonds are shown as gray lines. Evolution begins from a random conﬁguration (Left) and evolves by mutating one amino acid at each step, switching between H and P. The ﬁtness is the mechanical response to a localized force probe (pinch) (2). After ∼ 103 mutations (Center; intermediate stage), the evolution reaches a solution (Right). The green arrows show the mechanical response: a hinge-like, low-energy motion with a shear band starting at the probe and traversing the protein, qualitatively similar to B. L, left; R, right.

Thus, the Green function explicitly deﬁnes the map: gene → in terms of “multiple scattering” pathways. These indirect phys- amino acid network → protein dynamics → function. We use ical interactions appear as long-range correlations in the co- this map to examine the effects of mutations and epistasis. A evolving genes. mutation perturbs the Green function and scatters the propaga- Using a Metropolis-type evolution algorithm, solutions are tion of force through the protein (Fig. 2). We quantify epistasis quickly found, typically after ∼103 steps. Mutations add localized

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BIOPHYSICS AND COMPUTATIONAL BIOLOGY rejected. Since fitness is measured by the criterion of strong will easily excite a large-scale motion with a distinct high-shear mechanical response, it induces softening of the amino acid band (Fig. 1D, Right). network. 3 The typical evolution trajectory lasts about 10 steps. Most Results are neutral mutations (δF ' 0) and deleterious ones (δF < 0); Mechanical Function Emerges at a Topological Transition. The hall- the latter are rejected. About a dozen or so beneficial muta- mark of evolution reaching a solution gene c∗ is the emer- tions (δF > 0) drive the protein toward the solution (Fig. 3A). gence of a new zero-energy mode, u∗, in addition to the three The increase in the fitness reflects the gradual formation of the Galilean symmetry modes. Near the solution, the energy of channel, while the jump in the shear signals the emergence of this mode λ∗ almost closes the spectral gap, λ∗ → 0, and G(ω) the soft mode. The first few beneficial mutations tend to form has a pole at ω ≈ 0. As a result, the emergent mode domi- weakly bonded P-enriched regions near the pinch site on the left | nates the Green function, G ' u∗u /λ∗. The response to a pinch side and close to the right boundary of the protein. The follow- ∗ will be mostly through this soft mode, and the fitness [2] will ing ones join these regions into a floppy channel (a shear band), diverge as which traverses the protein from left to right. We stop the sim- | (v|u∗)(u∗f) ulation when the fitness reaches a large positive value Fm ∼ 5. F (c∗) ' F∗ = . [4] The corresponding gene c∗ encodes the functional protein. The λ∗ ad hoc value Fm ∼ 5 signals slowing down of the fitness curve On average, we find that the fitness increases exponentially with toward saturation at F > Fm, as the channel has formed and now the number of beneficial mutations (Fig. 3A). However, ben- only continues to slightly widen. In this regime, even a tiny pinch eficial mutations are rare and are separated by long stretches

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protein -2 0 10 shear 0 0.05 0.1 0.15 pc 0 4 8 12 16 0.0 0.025 p-pc

Fig. 3. The mechanical Green function and the emergence of protein function. (A) Progression of the fitness F during the evolution run shown in Fig. 1D (black) together with the fitness trajectory averaged over ∼ 106 runs hFi (red). Shown are the last 16 beneficial mutations toward the formation of the channel. The contribution of the emergent low-energy mode hF∗i (blue) dominates the fitness [4]. (B) Landscape of the fitness change δF [3] averaged over ∼ 106 solutions for all 200 possible positions of point mutations at a solution. Underneath, the average amino acid configuration of the protein is shown in shades of red (P) and blue (H). In most sites, mutations are neutral, while mutations in the channel are deleterious on average. L, left. (C) The average magnitude of the two-codon correlation |Qij| [5] in the shear band (amino acids in rows 7–13; red) and in the whole protein (black) as a function of the number of beneficial mutations, t.(Inset) Profile of the spatial correlation g(r) within the shear band (after t = 1, 11, 16 beneficial mutations). (D) The mean shear in the protein in a single run (black) and averaged over ∼ 106 solutions (red) as a function of the fraction of P amino acids, p. The values of p are shifted by the position of the jump, pc.(Inset) Distribution of pc.

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BIOPHYSICS AND COMPUTATIONAL BIOLOGY AB

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6 Fig. 4. Mechanical epistasis. The epistasis [7], averaged over ∼ 10 solutions Eij = heiji, between a fixed amino acid 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 amino acid configuration of the protein is drawn in shades of red (P) and blue (H). Significant epistasis mostly occurs along the 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). Landscapes are plotted for specific output site at right. L, left. (D) The two-codon correlation function Qij [5] measures the coupling between mutations at positions i and j [5]. The epistasis Eij and the gene correlation Qij 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).

long ranged along the channel when both mutations are sig- out from the bulk continuous spectrum. These are the col- −1 −1 −1 nificant, hi , hj 1, and eij ' F 1 − hi − hj + (hi + hj ) ' lective dfs, which show loci in the gene and positions in the F [1 − 1/ min(hi , hj )]. We conclude that epistasis is maximal “flow” (i.e., displacement) and shear fields that tend to vary when both sites are at the start or end of the channel as illus- together. trated in Fig. 4. The nonlinearity of the fitness function gives rise We examine the correspondence among three sets of eigen- to antagonistic epistasis. vectors: {Uk } of the flow, {Ck } of the gene, and {Sk } of the shear. The first eigenvector of the flow, U1, is the hinge motion Geometry of Fitness Landscape and Gene-to-Function Map. With caused by the pinch, with two eddies rotating in opposite direc- our mechanical evolution algorithm, we can swiftly explore tions (Fig. 5A). The next two modes, shear (U2) and breathing the fitness landscape to examine its geometry. The genotype (U3), also occur in real proteins, such as glucokinase (Fig. 1B). space is a 200D hypercube with vertices that are all possi- The first eigenvectors of the shear S1 and of the gene sequence ble genes c. The phenotypes reside in a 400D space of all C1 show that the high-shear region is mirrored as a P-rich region, possible mechanical responses u. The Green function provides where a mechanical signal may cause local rearrangement of the the genotype-to-phenotype map [1]. A functional protein is amino acids by deforming weak bonds. In the rest of the pro- encoded by a gene c∗ with fitness that exceeds a large threshold, tein, the H-rich regions move as rigid bodies with insignificant F (c∗) ≥ Fm ' 5, and the functional phenotype is dominated by shear. The higher gene eigenvectors, Ck (k > 1), capture patterns the emergent zero-energy mode, u(c∗) ' u∗ (Fig. 3A). We also of correlated genetic variations. The striking similarity between characterize the phenotype by the shear field s∗ (Materials and the sequence correlation patterns Ck and the shear eigenvectors Methods). Sk shows a tight genotype-to-phenotype map, as is further shown The singular value decomposition (SVD) of the 106 solutions in the likeness of the correlation matrices of the amino acid and returns a set of eigenvectors with ordered eigenvalues that show shear flow (Fig. 5C). their significance in capturing the data (Materials and Meth- In the phenotype space, we represent the displacement field ods). The SVD spectra reveal strong correspondence between u in the SVD basis, {Uk } (Fig. 5B). Since ∼ 90% of the data are the genotype c∗ and the phenotype, u and s∗ (Fig. 5). In all explained by the first ∼ 15 Uk , we can compress the displacement three datasets, the largest eigenvalues are discrete and stand field without much loss into the first 15 coordinates. This implies

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BIOPHYSICS AND COMPUTATIONAL BIOLOGY Thus, D is a tensor (D = ∇αinij), which we store as a matrix (α is the bond deleterious mutations (a finite temperature Metropolis algorithm) gave sim- connecting vertices i and j). In each row vector of D, which we denote as ilar results. We may impose a stricter minimum condition δF ≥ ε F with mα ≡ Dα,:, there are only 2d nonzero elements. To calculate the elastic a small positive ε, say 1%. An alternative, stricter criterion would be the response of the network, we deform it by applying a force field f, which demand that each of the terms in F, vp·up and vq·uq, increases separately. leads to the displacement of each vertex by ui to a new position ri + ui The evolution is stopped when F ≥ Fm ∼ 5, which signals the formation of (39). For small displacements, the linear response of the network is given a shear band. When simulations ensue beyond Fm ∼ 5, the band slightly by Hooke’s law, f = Hu. The elastic energy is E = u|Hu/2, and the Hamilto- widens, and the fitness slows down and converges at a maximal value, 2 nian, H = D|KD, is the Hessian of the elastic energy E, Hij = δ E/(δuiδuj). By typically Fmax ∼ 8. 1/2 p rescaling, D → K D, which amounts to scaling all distances by 1/ kα, we obtain H = D|D. It follows that the Hamiltonian is a function of the gene Evolving the Green Function Using the Dyson and Woodbury Formulas. The 0 0+ + H(c), which has the structure of the Laplacian ∆ multiplied by the tensor Dyson formula follows from the identity δH ≡ H − H = G − G , which 0 product of the direction vectors. Each d × d block Hij (i 6= j) is a function of is multiplied by G on the left and G on the right to yield [6]. The for- the codons ci and cj: mula remains valid for the pseudoinverses in the nonsingular subspace. One can calculate the change in fitness by evaluating the effect of a | mutation on the Green function, G0 = G + δG, and then examining the Hij(ci, cj) = ∆ijnijnij [11] change, δF = v|δGf [3]. Using [6] to calculate the mutated Green func- = −A k + (k − k )c c n n|. 0 ij w s w i j ij ij tion G is an impractical method, as it amounts to inverting at each step a large nd × nd matrix. However, the mutation of an amino acid at i has The diagonal blocks complete the row and column sums to zero, Hii = a localized effect. It may change only up to z = 12 bonds among the − P . j =6 i Hij bonds α(i) with the neighboring amino acids. Thanks to the localized nature of the mutation, the corresponding defect Hamiltonian δHi is, The Inverse Problem: Green Function and Its Spectrum. The Green function therefore, of a small rank, r ≤ z = 12, equal to the number of switched G is defined by the inverse relation to Hooke’s law, u = Gf [1]. If H were bonds (the average r is about 9.3). δHi can be decomposed into a prod- invertible (nonsingular),G would have been just G = H−1. However, H is uct δHi = MBM|. The diagonal r × r matrix B records whether a bond α(i) always singular owing to the zero-energy (Galilean) modes of translation is switched from weak to strong (Bαα = ks − kw = +0.99) or vice versa and rotation. Therefore, one needs to define G as the Moore–Penrose pseu- (Bαα = −0.99), and M is a nd × r matrix with r columns that are the doinverse (78, 79), G = H+, on the complement of the space of Galilean bond vectors mα for the switched bonds α(i). This allows one to calculate transformations. The pseudoinverse can be understood in terms of the spec- changes in the Green function more efficiently using the Woodbury formula trum of H. There are at least n0 = d(d + 1)/2 zero modes: d translation (91, 92): modes and d(d − 1)/2 rotation modes. These modes are irrelevant and will be projected out of the calculation (note that these modes do not + δG = −GM B−1 + M|GM M|G. [12] come from missing connectivity of the graph ∆ itself but from its embed- d ding in E ). H is singular but is still diagonalizable (since it has a basis of dimension nd), and it can be written as the spectral decomposition, n = P d λ | {λ } { } H k=1 kukuk , where k is the set of eigenvalues and uk are the corresponding eigenvectors (note that k denotes the index of the eigen- value, while i and j denote amino acid positions). For a nonsingular matrix, n −1 = P d λ−1 | one may calculate the inverse simply as H k=1 k ukuk . Since H is singular, we leave out the zero modes and get the pseudoinverse H+, G = n + P d −1 | H = λ uku . It is easy to verify that, if u is orthogonal to the k=n0+1 k k zero modes, then u = GHu. The pseudoinverse obeys the four requirements (78): (i) HGH = H, (ii) GHG = G, (iii)(HG)| = HG, and (iv)(GH)| = GH. In practice, as the projection commutes with the mutations, the pseudoinverse has most virtues of a proper inverse. The reader might prefer to link G and H K = P λkt | = R ∞ K through the heat kernel, (t) k e ukuk . Then, G 0 dt (t) and d H = dt K|t=0.

Pinching the Network. A pinch is given as a localized force applied at the boundary of the “protein.” We usually apply the force on a pair of neighboring boundary vertices, p0 and q0. It seems reasonable to apply a force

dipole (i.e., two opposing forces fq0 = −fp0 ), since a net force will move the center of mass. This pinch is, therefore, speciﬁed by the force vector f (of

size nd), with the only 2d nonzero entries being fq0 = −fp0 . Hence, it has 0 the same structure as a bond vector mα of a “pseudobond” connecting p 0 and q A normal pinch f has a force dipole directed along the rp0 − rq0 line (the np0q0 direction). Such a pinch is expected to induce a hinge motion. A shear pinch will be in a perpendicular direction ⊥ np0q0 and is expected to induce a shear motion. Evolution tunes the spring network to exhibit a low-energy mode, in which the protein is divided into two subdomains moving like rigid bodies. This large-scale mode can be detected by examining the relative motion of two neighboring vertices, p and q, at another location at the boundary (usually at the opposite side). Such a desired response at the other side of the protein is specified by a response vector v, and the only nonzero entries correspond to the directions of the response at p and q. Again, we usually consider a “dipole” response vq = −vp. Fig. 6. The effect of the backbone on evolution of mechanical function. The backbone induces long-range mechanical correlations, which influence Evolution and Mutation. The quality of the response (i.e., the biological protein evolution. We examine two configurations: parallel (A and B) and fitness) is specified by how well the response follows the prescribed one perpendicular (C and D) to the channel. (A and B) Parallel. (A) The back- v. In the context of our model, we chose the (scalar) observable F as bone directs the formation of a narrow channel along the fold (compared | | F = v u = vp·up + vq·uq = v Gf [2]. In an evolution simulation, one would with Fig. 5A). (B) The first four SVD eigenvectors of the gene Ck (Top), exchange amino acids between H and P, while demanding that the fitness the flow Uk (Middle), and the shear Sk (Bottom). (C and D) Perpendic- change δF is positive or nonnegative. By this, we mean δF > 0 is thanks to ular. (C) The formation of the channel is “dispersed” by the backbone. a beneficial mutation, whereas δF = 0 corresponds to a neutral one. Delete- (D) The first four SVD eigenvectors of Ck (Top), Uk (Middle), and shear Sk rious mutations δF < 0 are generally rejected. A version that accepts mildly (Bottom).

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Broken of is and factor tensor Pathologies [ 12] a low-rank by expan- of a simulations series of advantage our a accelerates is practical This requires by (n parentheses). The it [6] in that [12]. of term (pseudo-)inversion (the in series only scattering pseudoinverse the the that the get of may sion one and alent, ut tal. et Dutta .EsnesrE,e l 20)Itiscdnmc fa nyeudriscatalysis. underlies enzyme an of dynamics kinases. Intrinsic protein (2005) of al. plasticity et conformational EZ, The Eisenmesser (2002) reaction 9. J enzymatic an Kuriyan along M, motions Huse Intrinsic 8. (2007) al. et biomolecules. Bridging KA, of proteins: simulations Henzler-Wildman dynamics of 7. Molecular dynamics (2002) Global JA (2010) McCammon E M, Karplus Eyal LW, 6. Yang TR, Lezon of landscape I, energy Bahar dynamic The (2006) 5. PE Wright HJ, in Dyson D, processes McElheny DD, Boehr Mechanical catalysis. 4. (2004) to motion protein D Relating enzyme (2006) Izhaky SJ in Benkovic NR, S, dynamics Hammes-Schiffer Forde of 3. role YR, The Chemla (2003) C, JC Smith Bustamante JL, 2. Finney RV, Dunn RM, Daniel 1. a a = h w xrsin o h uainimpact mutation the for expressions two The eeto system. detection route. common recognition. molecular of specificity the on changes Nature 109:275–282. trajectory. Biol Struct Nat function. and structure between catalysis. reductase dihydrofolate Biochem biochemistry. activity. inln n te functions. other and Signaling neato n h rbe fallostery. of problem the and interaction model. plausible 1307. /r 0 oos;( codons); 200 ) 3 ' 438:117–121. 10 nuRvBohsBoo Struct Biomol Biophys Rev Annu 75:519–541. 4 Nature s . ∗ nuRvBiochem Rev Annu n avco flength of vector (a h eeo h ucinlprotein, functional the of gene the ) :4–5,adertm(02 9:788. (2002) erratum and 9:646–652, a hmBiol Chem Nat d o Biol Mol J = o Cell Mol 450:838–844. hnesta tr n n ttetre einwith- region target the at end and start that channels ) ii 0 of 400 h o ed(displacement), field flow the ) oeaietegoer ftefins landscape fitness the of geometry the examine To 40:388–396. 12:88–118. s ∗ x urOi tutBiol Struct Opin Curr 4:474–482. edi h u fsurso h traceless the of squares of sum the is field N nuRvBiophys Rev Annu and 73:705–748. Science sol → ∼ ewr rknit ijitcompo- disjoint into broken network A y sltdndsa h onaythat boundary the at nodes isolated ) 10 n uain,(ii mutations, P 32:69–92. a Nature 313:1638–1642. eoiycmoet) n (iii and components); velocity F 6 = W m ahslto scaatrzdby characterized is solution Each . ihu rdcn functional a producing without ) 0) ecmueteseras shear the compute We 200). | W C 228:726–734. k hl h ievcosare eigenvectors the while , , 39:23–42. 20:142–147. LSOne PLoS W U δ k C [6 G, and , , sdwy”channels “sideways” ) W c u(c ∗ U and and , 2:e468. avco flength of vector (a ∗ S rti Sci Protein ) k = i V a in (as SVD via , ,aeequiv- are 12], G(c W ∗ S ecal- We . )f nuRev Annu 17:1295– avec- (a the ) Cell 1 igD alM 20)Alseyi oregandmdlo rti dynamics. protein of model functional coarse-grained enable a in to Allostery (2005) structures ME Wall protein D, Ming of Adaptability 31. (2015) I biology. Bahar structural T, in analysis Haliloglu from mode arising 30. normal Coarse-grained proteins (2005) of AJ Rader change I, Bahar Conformational 29. (2001) YH Sanejouand inhibitor, F, Trypsin dynamics: Tama normal-mode Protein 28. (1985) PS Stern C, Sander coarse- M, Levitt by 27. predicted modes soft of significance L functional 26. the On (2010) single- I Bahar a 25. from under- for proteins models network Elastic in (2005) I Bahar motions LW, Yang AJ, elastic Rader C, Chennubhotla amplitude 24. Large (1996) MM data structural Tirion protein of analysis 23. twist and strain Elastic (2018) low S and Leibler MR, structures Mitchell protein 22. of analysis Strain (2016) S Leibler T, Tlusty MR, Mitchell 21. allostery. of coupled nature ensemble form The (2014) rearrangements VJ Hilser Contact J, Li (2008) JO, Wrabl HN, JJ Motlagh Gray 18. TJ, Upadhyaya MD, Daily 17. 0 esenM ekA,CohaC(94 tutrlmcaim o oanmovements domain for mechanisms Structural (1994) C Chothia synthesis. AM, protein Lesk M, to Gerstein specificity enzyme 20. of theory a of Application (1958) D Koshland 19. rn B-00adteSmn etrfrSsesBooyo h Institute the of Biology Systems com- for Science Princeton. Basic Center Study, valuable for Advanced Simons for Institute the for by and Council supported Wyart IBS-R020 Research is T.T. Alex Grant European Matthieu and by thank Bridges, supported Olivier and Grant is also J.-P.E. Advanced Yan, and manuscript. We the Le Zocchi, on encouragement. ments Granick, Giovanni and Steve Moses, discussions Petroff, Elisha helpful for Mitchell, Stanislas thank Rivoire We R. discussions. helpful Michael for and Leibler, 1B) (Fig. glucokinase in shear ACKNOWLEDGMENTS. long-range the on and search the solutions. of among sig- convergence correlations no fast has presence the this The on but sequence effect search, model. evolutionary nificant and our the constrain field of might results backbone the the basic of of the patterns changing to without that features examination correlation certain our further from adds requires conclude backbone this we the but Overall, fold, data. the structural of of direction analysis the 6C with (Fig. “dispersion” oriented output roughly to be the leads at and signal backbone the the neces- of of configuration) deformation (perpendicular configuration) significant fold (parallel sitates the fold across the Transmission of 6A). orientation the (Fig. along easier signals be mechanical of to Transmission seems it. around correlations sequence the with eigenvectors h eiaieo h o n hrfr,hglgtmr itntythese distinctly more highlight therefore, and the flow in the differences. seen of be derivative can the as differences, eigenvectors noticeable 6C flow shows 1D (Fig. inspection oriented (Figs. the closer of less ilar, backbone progression much without evolutionary resulting is the model The configuration, channel the 6A). perpendicular the (Fig. in In fold than 5). in the narrower moving along is eddies progress constrains channel backbone to two the formation with configuration, channel case displacement parallel the the (i.e., backboneless In pattern the fashion. flow to hinge-like a a similar with is protein that the field) of mode low-energy a to a perpendicular or cases: band extreme that shear two 6). bonds the on (Fig. to strong it focused parallel We conserved either acids. of backbone amino path serpentine our all linear augmented through and a therefore, once We, acids “backbone”: passes much mutates. a amino protein are with the the bonds model among when peptide change peptide interactions The not covalent noncovalent do backbone. by the protein than linked polypep- the stronger acids, are form amino Proteins which far. of bonds, so heteropolymers described linear results tides, the affect might backbone Backbone. Protein The sfrtecrepnec ewe eeeigenvectors gene between correspondence the for As of emergence the with interfere not does backbone the of presence The neatosadeouinr implications. evolutionary and interactions Biol Struct Opin Curr calculations. mode normal lysozyme. and ribonuclease crambin, Biol Struct Opin proteins. membrane for models grained assemblies. supramolecular to Biol enzymes From machinery: biomolecular standing analysis. atomic parameter, kcsa. channel transmembrane the of allostery and E5855. couplings. allosteric mechanical of dimensionality proteins. allosteric in motions local from networks e Lett Rev proteins. in USA Sci Acad Natl Proc 508:331–339. pzBac R Chac JR, opez-Blanco ` 2:S173–S180. 95:198103. iceity(Mosc) Biochemistry S k h akoeafcstesaeo h hne nconcert in channel the of shape the affects backbone the , 37:46–53. U k 15:586–592. 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