{"paper_id":"1826af15-87da-4be3-a3df-01ee1baf67bd","body_text":"Structured Random Binding:\na minimal model of protein-protein interactions\nLing-Nan Zou\nEmail: zouln@psu.edu\nDepartment of Chemistry, The Pennsylvania State University\nUniversity Park, PA 16801. USA.\nMarch 26, 2025\nAbstract\nWe describe Structured Random Binding (SRB), a minimal model of protein-protein interac-\ntions rooted in the statistical physics of disordered systems. In this model, nonspecific binding is\na generic consequence of the interaction between random proteins, exhibiting a phase transition\nfrom a high temperature state where nonspecific complexes are transient and lack well-defined\ninteraction interfaces, to a low temperature state where the complex structure is frozen and\na definite interaction interface is present. Numerically, weakly-bound nonspecific complexes\ncan evolve into tightly-bound, highly specific complexes, but only if the structural correlation\nlength along the peptide backbone is short; moreover, evolved tightly-bound homodimers favor\nthe same interface structure that is predominant in real protein homodimers.\nIn cells, protein-protein interactions are vital because proteins generally execute their activities in\ncomplex with other proteins [23]. The textbook picture where proteins are depicted as rigid bodies\ninteracting via interlocking surfaces, and where the specificity of an interaction originates from\nthe specific geometry of its interface [1], collides uncomfortably with the fact pulldown experiments\ntargeting one specific protein will routinely co-precipitate hundreds of additional proteins. How can\nany protein have sufficient surface structure to mediate this many interactions? Experimentally,\ndetectable but weak interactions are often dismissed ad hoc as “nonspecific”, but what distinguishes\nnonspecific interactions from weak but specific interactions? Here, we describe Structured Random\nBinding (SRB), a minimal model of protein-protein interactions where nonspecific binding is a\ngeneric consequence of the interaction between random proteins. SRB exhibits a phase transition\nfrom a high temperature state where nonspecific complexes are transient and lack well-defined\ninteraction interfaces, to a low temperature state where the complex structure is frozen and a\ndefinite interface is present. In numerical simulations, weakly-bound nonspecific complexes can\nevolve into tightly-bound, highly specific complexes, but only if the structural correlation length\nalong the peptide backbone is short. Furthermore, evolved tightly-bound homodimers favor the\nsame interface structure that is predominant in real protein homodimers. Our model thus captures\ntwo salient features of protein-protein interactions — the ubiquity of nonspecific complexes, and\nthe structure of homodimers — that cut across the diversity of protein species, and root these\nphenomena firmly in the statistical physics of disordered systems.\n1\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 29, 2025. ; https://doi.org/10.1101/2025.03.26.645477doi: bioRxiv preprint \n\nFigure 1: Structured Random Binding model of protein-protein interactions. Sliding ξ-tuples of the\nprimary sequence p maps onto structural motifs mi ∈ M, and endows p with a secondary structure\nf(p). Paired motif-motif interactions, encoded by J(m, m′), between two secondary structures\n(shown in parallel orientation) sum to yield the peptide-peptide binding energy u(p, b).\nDefining the model\nTo begin, we first consider the interaction between a small peptide and a protein. We represent\npeptide p = p1p2 . . . pl as a sequence of length l (the primary sequence), where pi are letters\ndrawn from a primary alphabet A (the 20 proteinogenic amino acids). We represent protein B\nas a collection of nB peptide epitopes (also of length l), B = {bα, α = 1 , . . . , nB}, where bα =\nbα\n1 bα\n2 . . . bα\nl , bα\ni ∈ A; since only surface exposed residues participate in protein-protein interactions,\nnB is proportional to the protein surface area. The interaction between residues on the peptide,\nand residues on the protein, depends on their respective structural contexts. We define a map\nf : Aξ → M from ξ-tuples of A to structural motifs,\nf(pipi+1 . . . pi+ξ−1) = mi, pi ∈ A, mi ∈ M, (1)\nwhere M is the motif alphabet, andξ is the structural correlation length along the peptide backbone;\nwe assume f is bijective and ∥M ∥ = ∥A∥ξ. By acting on each sliding ξ-tuple of p, f endows p with\na secondary structure f(p) = m1m2 . . . ml−ξ+1, mi ∈ M (Fig. 1). We define the binding energy\nu(p, b) between p and b ∈ B as the sum of l′ = l − ξ + 1 paired motif-motif interactions,\nu↑ ↑(p, b) =\nl′\nX\ni=1\nJ[f(p)i, f(b)i], (2)\nu↑ ↓(p, b) =\nl′\nX\ni=1\nJ[f(p)l′−i+1, f(b)i], (3)\ndepending on whether p, b are oriented in parallel or antiparallel. Here, J(m, m′), m, m′ ∈ M, is a\n∥M ∥ × ∥M ∥ real symmetric matrix, whose elements are independently and identically distributed\nrandom variables with mean ⟨J⟩ = 0, and finite variance σ2\nJ that sets the energy scale of motif-\nmotif interactions. In the low temperature limit T → 0, the binding energy U(P, B) of the binary\ncomplex PB is given by\nU(P, B) = min{u(p, b), p ∈ P, b ∈ B}. (4)\nBinding energy of nonspecific complexes\nThese ingredients are sufficient to fix the binding energy distribution PNS(U) for nonspecific binary\ncomplexes of random proteins. Since u(p, b) between a random peptide p, and an epitope b ∈ B\n2\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 29, 2025. ; https://doi.org/10.1101/2025.03.26.645477doi: bioRxiv preprint \n\n–15 –10 –5 0 5\n0\n0.2\n0.4\n0.6(a)\n100 101 102 103\n–12\n–8\n–4\n0(b)\nFigure 2: The binding energy U for nonspecific binary complexes of random proteins. (a) The\ndistribution PNS(U) for nonspecific complexes; n = 2nPnB is twice the product of epitope counts\nfor each protein. Filled histograms are empirical distributions from numerical simulations; solid\nlines are Gumbel distributions given by Eq. 6, with µ, ϕ given by Eqs. 7, 8. (b) Mean nonspecific\nbinding energy ⟨U ⟩NS as a function of n, for epitope sizes l′ = 2, 6, 10, 14, 18.\non a random protein, is the sum of l′ = l − ξ + 1 random variables J(m, m′), for any choice of P(J)\nwith zero mean and finite variance σ2\nJ, in the limit l′ → ∞, u will be normally distributed\nP(u) = 1q\n2πl′σ2\nJ\nexp\n\u0012\n− u2\n2l′σ2\nJ\n\u0013\n. (5)\nFor a binary complex of two random proteins P, B, its binding energy U(P, B) will follow the\nextreme value distribution for sample minima of n = 2nPnB point samples from P(u). In the limit\nn → ∞, this is a Gumbel distribution [6, 18],\nPNS(U) = 1\nϕ exp\n\u0014\n− U − µ\nϕ + exp\n\u0012\n− U − µ\nϕ\n\u0013\u0015\n, (6)\nwith location µ and scale ϕ parameters given by\nµ = −\nq\n2l′σ2\nJ erf −1 (1 − 2/n) , (7)\nϕ = −µ −\nq\n2l′σ2\nJ erf −1 (1 − 2/ne) . (8)\nThe average nonspecific binding energy over all SRB realizations (“disorder averaged”) is\n⟨U ⟩NS = µ + γϕ, (9)\nwhere γ is the Euler–Mascheroni constant. Note that neither P(u) nor PNS(U) depends on the size\nof the primary alphabet ∥A∥ (assuming it is nontrivial). The weight in PNS(U) for U > 0 is negli-\ngible (Fig. 2a); the average binding energy ⟨U ⟩NS is strictly negative and decreases monotonically\nwith increasing n (Fig. 2b). Thus in the low-T limit, nonspecific binding is a generic consequence\nof the interaction between random proteins. We can understand this intuitively as follows: given\ntwo random proteins, if their surfaces are large enough, we can always find a pair of epitopes, one\non each protein, that will bind each other, even if the average epitope-epitope interaction is zero.\n3\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 29, 2025. ; https://doi.org/10.1101/2025.03.26.645477doi: bioRxiv preprint \n\n100 101 102 103\n0\n1\n2\n3\n4\n(a)\n0 1 2 3\n0\n0.2\n0.4\n0.6\n0.8\n1\n(b)\n0 1 2 3\n0\n0.2\n0.4\n0.6\n0.8\n1\n-8.5401\n-9.0920\n-9.2464\n-10.1038\n0 1 2 3\n0\n1\n2\n3\n4\n–15 0 15\n10–3\n10–2\n10–1\n100\n(c)\n(d)\nFigure 3: The condensation transition in SRB. (a) Critical temperature Tc as a function of n =\n2nPnB, for epitope sizes l′ = 2, 6, 10, 14, 18. (b) The mean participation ratio ⟨Y (T )⟩ as a function\nof T /Tc for n = 8, 72, 800, 7200, 80000. In the limit n → ∞, ⟨Y (T )⟩ → 1 − T /Tc for T < T c (dashed\nline). (c) Y (T ) for 4 specific realization of random proteins P, B, and J (l = 10, ξ = 3, and\nnP = nB = 20); the binding energy U(P, B) of the PB complex is indicated in the legend. Dashed\nline is the average ⟨Y (T )⟩ for n = 800. (d) −dY (T )/dT for the same realizations; the location of\nthe peak is usually interpreted as the melting temperature of the protein complex. Inset: energy\nlevel distributions P(u) for these realizations; dashed line is the asymptotic distribution (Eq. 5).\nThe condensation transition\nThese results describes the low- T limit of SRB. To obtain finite T behavior, we note that the\ninteraction between random proteins P, B describes a n = 2nPnB level system whose energy levels\nare independently and identically distributed per P(u), with density of states given by\n⟨ρ(u)⟩ = nP(u) = e\nlog(n)− u2\n2l′σ2\nJ\nq\n2l′σ2\nJ\n. (10)\nSo stated, the equilibrium statistical physics of SRB is analogous to that of the exactly solvable\nRandom Energy Model (REM) [7, 9]. Originally formulated as a minimal model of spin glasses,\nREM has also proven influential in informing the statistical physics of protein folding [27, 24]. In the\nthermodynamic limit, REM undergoes a phase transition, known as condensation, at temperatures\nT < T c to a frozen phase where the Boltzmann measure condenses onto a smaller-than-exponential\n4\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 29, 2025. ; https://doi.org/10.1101/2025.03.26.645477doi: bioRxiv preprint \n\nset of configurations. In SRB, the corresponding Tc (in the limit n → ∞) is\nTc = σJ\ns\nl′\n2 log(n) , (11)\nand is slowly varying in both n and l′ (Fig. 3a). For T > T c, the complex between two random\nproteins is dynamic and lacks a definite interaction interface; for T < T c, the complex freezes into\na fixed conformation and a definite interface is present. The participation ratio\nY (T ) =\nP\n{p∈P,b∈B} e−2u(p,b)/T\n\u0010P\n{p∈P,b∈B} e−u(p,b)/T\n\u00112 (12)\nis the probability that two samples of the system return the same configuration; thus 1 /Y (T ) is\neffectively the number of states that dominate the Boltzmann measure. In REM, ⟨Y (T )⟩ = 0 for\nT > T c, ⟨Y (T )⟩ = 1 − T /Tc for T < T c [8, 21]; SRB exhibits the same behavior as n → ∞ (Fig. 3b).\nFor SRB realizations with finite-sized proteinsP, B, the condensation phase transition broadens\ninto crossover representing the binding/unbinding of thePB complex. Remarkably, individual SRB\nrealizations, founded on the same underlying P(J), can have drastically different Y (T ) curves and\napparent melting temperatures (where −dY /dT is maximal); only upon averaging do we recover the\nREM-like transition (Figs. 3c, d). This chaotic behavior is a direct consequence of the condensation\ntransition, where the T < T c thermodynamics of SRB is dominated by a small number of low- u\nconfigurations. Different SRB realizations, despite their statistically similar energy spectra, have\ndistinct thermodynamics that arise entirely from subtle differences in the low-u tail (Fig. 3d, inset).\nEvolution of tightly-bound complexes\nIn SRB, nonspecific complexes have a well-defined energy scale ⟨U ⟩NS. How can we obtain tightly-\nbound complexes with U ≪ ⟨U ⟩NS? Computationally, this is an optimization problem: given J\nand peptide b, find peptide p such that u(p, b) is minimal. In the limit ξ → 1, single residue\nchanges in p results in single motif changes in f(p), and u(p, b) can be minimized term by term.\nIn the opposite limit ξ → l, single residue changes in p completely alters f(p), and there can be no\ncorrelation between binding energy and primary sequence; minimizing u(p, b) requires exhaustive\nsearch through all possible primary sequences. In the regime 1< ξ < l , fixing f(p) at motif f(p)i, by\nfixing p at residues pi, ..., pi+ξ−1, also restricts the possible values of motifsf(p)i−ξ+1, ..., f(p)i+ξ−1.\nThis means while every primary sequence p can be mapped to a secondary structure f(p), arbitrary\nmotif sequences m = m1m2 . . . ml′, mi ∈ M, are generally noninvertible, i.e. p = f −1(m) does\nnot exist. Minimizing u, which is a function of motif sequences, must be restricted to invertible\nsecondary structures, which forms a small (fraction =∥A∥l−(l−ξ+1)ξ) and sparse (single motif change\non an invertible secondary structure renders it noninvertible) subset in the space of motif sequences.\nAs is often the case with constrained optimization [22], we conjecture there is no fast algorithm for\nsolving SRB in this regime.\nBiologically, tightly-bound protein complexes are products of evolution. We simulated affinity\nevolution in a binary complex of two coevolving random proteins P, B; for tractable numerics,\nwe restrict ourselves to a small primary alphabet ∥A∥ = 4. We start from an uniform ensemble\n{PαBα, α = 1, . . . , N}, PαBα = PβBβ for all α, β. At each generation t, we mutate the binary\ncomplex (at rate ϵ = 0.01 per residue per generation), and apply selection with an affinity-dependent\nsurvival probability\nΘ(U; U0) = 1\ne(U −U0)/σJ + 1. (13)\n5\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 29, 2025. ; https://doi.org/10.1101/2025.03.26.645477doi: bioRxiv preprint \n\n100 101 102 103\n–8\n–6\n–4\n–2\n0\n–30 –20 –10 0\n0\n0.2\n0.4\n0.6(a) (b)\nFigure 4: Evolution of weakly-bound, nonspecific complexes of random proteins into tightly-bound,\nhighly specific complexes. (a) Binding energy evolution of the nonspecifc complex between random\nproteins P, B (l = 10, nP = nP = 20) over 10 3 generations, for ξ = 2, 3, 4, 5, 6, 7; ⟨⟨U ⟩⟩ is doubly\naveraged over the ensemble of N = 10 3 binary complex mutants, and over 10 2 SRB realizations.\n(b) Distribution of binding energies P(U) for coevolved proteins P, B (ξ = 3), in binary complex\nwith each other (pink fill), and in binary complex with other random proteins (blue fill). Dashed\nline is the binding energy distribution PNS(U) for nonspecific complexes of random proteins.\nHere, the selection threshold U0(t) = ⟨U[P(t), B(t)]⟩ coevolves with the binary complex ensemble\nto maintain selection pressure. As shown in Fig. 4a, the long time evolutionary dynamics is char-\nacterized by a slow log( t) behavior similar to aging dynamics in glassy systems [5]; while steady\nstate could not be reached within the simulation window, it is clear that evolution is most effective\nin enhancing binding affinity when the structural correlation length ξ is short. Where PB evolved\ninto tightly-bound complexes, the resulting interactions are highly specific: while the coevolved\nproteins bind each other with U ≪ ⟨U ⟩NS, their binding energies with other random proteins is\nindistinguishable from that between any two random proteins (Fig. 4b). This result suggests we can\nmake a physically principled distinction between specific and nonspecific protein complexes: specific\ncomplexes are those whose binding energy U is unlikely to be the result of random protein-protein\ninteractions. It also suggests the evolution of tightly-bound, highly specific protein complexes is\nnot ex nihlio, but emerges from a basis of ubiquitous nonspecific interactions.\nStructural bias in homodimeric complexes\nCompared to heterodimeric interactions, homodimeric interactions in SRB exhibit a subtle but\nsignificant distinction. For homodimers PP with dimer interface p, p′ ∈ P, if p ̸= p′ (heterologus),\nor if p = p′ (isologus) are parallel, then u(p, p′) and u↑ ↑(p, p) will be the sum of l′ random variables\nJ(m, m′). However, when p = p′ are antiparallel, u↑ ↓(p, p) is twice the sum of l′/2 random\nvariables. From the arithmetic of random variables, P[u↑ ↓(p, p)] will have twice the variance of\nP[u↑ ↑(p, p)] or P[u(p, p′)], p ̸= p′ (Fig. 5a). This means random homodimers with antiparallel\nisologus interfaces are on average more tightly-bound than homodimers in other configurations\n(Fig. 5b). As weakly-bound nonspecific homodimers evolve into tightly-bound homodimers, they\ntend to retain antiparallel isologus interfaces (if present in the ancestral complex), or adopt it de\nnovo, such that antiparallel isologus interfaces are predominant in tightly-bound homodimers. For\nl′ = 8, ξ = 3, nP = 20, random homodimers as generated were 33% antiparallel isologus, 3.3%\nparallel isologus, and 63.7% heterologus; after 1000 generations of affinity evolution, the resulting\n6\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 29, 2025. ; https://doi.org/10.1101/2025.03.26.645477doi: bioRxiv preprint \n\n–15 –10 –5 0 5 10 15\n10–3\n10–2\n10–1\n100\n–20 –15 –10 –5 0\n10–3\n10–2\n10–1\n100(a) (b)(a) (b)\nFigure 5: Structural bias in homodimeric complexes. (a) Binding energy distribution P[u(p, p)]\nfor antiparallel (pink fill) and parallel (blue fill) isologus peptide-peptide interactions, for random\npeptides (l = 10, ξ = 3). Dashed line indicates P(u) for heterlogous peptide-peptide interactions.\n(b) Empirical binding energy distributions P(U) for random homodimers PP (l = 10, ξ = 3,\nnP = 20) with antiparallel isologus (pink fill), parallel isologus (blue fill), and heterlogus (black\nline) dimer interfaces.\ntightly-bound homodimers are 98% antiparallel isologus, 1% parallel isologus, and 1% heterologus.\nReal protein homodimers predominantly have antiparallel isologus interfaces [11, 3]; our results\nsupports the view that this phenomenon results from an intrinsic bias in homodimeric protein-\nprotein interactions [19, 20, 2]. In the case of SRB, this bias is entirely statistical in origin.\nDiscussion\nSRB is a vastly simplified model of protein-protein interactions. Actual proteins are often some-\nwhat conformationally flexible, and can undergo conformational changes upon complexation [17];\nthis has entropic effects which SRB neglects. Many proteins have interaction surfaces made up of\nphysically proximal, but sequentially distal residues, and the structural context of a surface-exposed\nresidue depends on more than just nearby residues along the peptide backbone. Nevertheless, there\nis evidence the core physical insight of SRB, that the interaction between two proteins describes a\nmulti-level system with a random or weakly-correlated energy spectrum, holds for real protein com-\nplexes [15, 4]. If SRB does indeed capture the statistical physics of real protein-protein interactions\n(at least qualitatively), several implications are immediately evident:\nSuppose the energy scale σJ ∼ kB × 300K, then at physiological temperature, a significant\nfraction proteins inside a cell will be bound into nonspecific complexes that have no functional\nsignificance. If true, then we must reassess the capabilities and limitations of protein pulldown,\nproximity labelling, and two-hybrid experiments in identifying functional protein complexes [10,\n25, 29]. While nonspecific protein complexes will be weakly-bound, they are ubiquitous due to the\ngeneric nature of nonspecific binding, and are likely to dominate any experimental signal. Here,\nthe physically principled distinction between specific and nonspecific complexes offered by SRB can\ninform new experiments and analyses seeking to identify functional protein-protein interactions [13].\nProteins in the cytoplasm, and in dense protein solutions, undergo anomalous diffusion where\n⟨∆r2⟩ ∝ tβ, β < 1, and their rotational diffusion are more impeded in dense protein solutions\nthan in simple solvents of similar viscosity [30, 26, 14, 12]. SRB suggests we should not expect the\nfree diffusion of proteins in dense protein solutions, as they will be mostly bound into nonspecific\n7\n.CC-BY 4.0 International licenseavailable under a \n(which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made \nThe copyright holder for this preprintthis version posted March 29, 2025. ; https://doi.org/10.1101/2025.03.26.645477doi: bioRxiv preprint \n\ncomplexes. This will have an especially pronounced effect for rotational diffusion: while the trans-\nlational diffusion constant Dt ∝ a−1, where a is the hydrodynamics radius, the rotational diffusion\nconstant Dr ∝ a−3. Additionally, due to the condensation transition, anomalous transport in dense\nprotein solutions should be even more pronounced at low temperatures, above and beyond effects\nexpected from temperature-dependent changes in solvent viscosity.\nFinally, the sensitivity of protein complex thermodynamics to the exact disposition of low\nenergy states suggests the accurate prediction of protein complex binding free energies, which is a\nlong standing challenge in computational biology [16, 28], may be fundamentally limited. Here, it is\npossible that different choices of necessary approximations, e.g. for molecular geometry, force fields,\netc., leading to subtle differences in low lying states, can nevertheless result in large differences in\ncomputed thermodynamic properties.\nMolecular investigations of protein complexes are singularly focused on the individuality of\nthe proteins involved. Our work demonstrates that statistical physics, encapsulated in a minimal\nmodel, can illuminate features of protein-protein interactions that cut across the diversity of protein\nspecies. SRB suggests nonspecific protein-protein binding is an ineluctable physical phenomenon,\nand that the proteins inside a cell are ever participants in a plethora of specific and nonspecific\ncomplexes. While these ubiquitous interactions challenge scientists who seek to decipher specific\nfunctions of detected interactions, they may be biologically significant in a broader sense. We have\nalready suggested nonspecific interactions could be the source from which highly specific complexes\nevolved; we speculate nonspecific interactions can also be a mechanism that cells can exploit, to\nenable subtle modes of subcellular organization based on physical principles.\nAcknowledgements. We are grateful to Stephen Benkovic for his advice and support. 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It is made \nThe copyright holder for this preprintthis version posted March 29, 2025. ; https://doi.org/10.1101/2025.03.26.645477doi: bioRxiv preprint","source_license":"CC-BY-4.0","license_restricted":false}