Understanding protein−protein interactions is central to our understanding of almost all complex biological processes. Computational tools exploiting rapidly growing genomic databases to characterize protein−protein interactions are urgently needed. Such methods should connect multiple scales from evolutionary conserved interactions between families of homologous proteins, over the identification of specifically interacting proteins in the case of multiple paralogs inside a species, down to the prediction of residues being in physical contact across interaction interfaces. Statistical inference methods detecting residue−residue coevolution have recently triggered considerable progress in using sequence data for quaternary protein structure prediction; they require, however, large joint alignments of homologous protein pairs known to interact. The generation of such alignments is a complex computational task on its own; application of coevolutionary modeling has, in turn, been restricted to proteins without paralogs, or to bacterial systems with the corresponding coding genes being colocalized in operons. Here we show that the direct coupling analysis of residue coevolution can be extended to connect the different scales, and simultaneously to match interacting paralogs, to identify interprotein residue−residue contacts and to discriminate interacting from noninteracting families in a multiprotein system. Our results extend the potential applications of coevolutionary analysis far beyond cases treatable so far.A lmost all biological processes depend on interacting proteins.Understanding protein−protein interactions is therefore key to our understanding of complex biological systems. In this context, at least two questions are of interest: First, the question "who with whom," i.e., which proteins interact; this concerns the networks connecting specific proteins inside one organism, but alsoin the context of this article-the evolutionary perspective of protein−protein interactions, which are conserved across different species. Their coevolution is at the basis of many modern computational techniques for characterizing protein−protein interactions. The second question is the question "how" proteins interact with each other, in particular, which residues are involved in the interaction interfaces, and which residues are in contact across the interfaces. Such knowledge may provide important mechanistic insight into questions related to interaction specificity or competitive interaction with partially shared interfaces.The experimental identification of protein−protein interactions is an arduous task (for reviews, cf. refs. 1 and 2): High-throughput techniques that aim to identify protein−protein interactions in vivo or in vitro are well documented and include large-scale yeast two-hybrid assays and protein affinity mass spectrometry assays. Such large-scale efforts have revealed useful information but are hampered by high false positive and false negative error rates. Structural approaches based on protein cocrystalli...

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Abstract. -We study the directed polymer with fixed endpoints near an absorbing wall, in the continuum and in presence of disorder, equivalent to the KPZ equation on the half space with droplet initial conditions. From a Bethe Ansatz solution of the equivalent attractive boson model we obtain the exact expression for the free energy distribution at all times. It converges at large time to the Tracy Widom distribution F4 of the Gaussian Symplectic Ensemble (GSE). We compare our results with numerical simulations of the lattice directed polymer, both at zero and high temperature.Much progress was achieved recently in finding exact solutions in one dimension for noisy growth models in the Kardar-Parisi-Zhang (KPZ) universality class [1,2], and for the closely related equilibrium statistical mechanics problem of the directed polymer (DP) in presence of quenched disorder [3]. The KPZ class has been explored in several recent experiments [4,5], and the DP has found applications ranging from biophysics [6] to describing the glass phase of pinned vortex lines [7] and magnetic walls [8]. The height of the growing interface, h(x, t), corresponds to the free energy of a DP of length t starting at point x, under a mapping which is exact in the continuum (Cole-Hopf), as well as for some discrete realizations. Not only the scaling exponents h ∼ t 1/3 , x ∼ t 2/3 are known [9,10], but also the one-point (and in some cases the manypoint) probability distribution (PDF) of the height have been obtained [16,17]. Their dependence in the initial condition was found to exhibit remarkable universality at large time, with only a few subclasses, most being related to Tracy Widom (TW) distributions [15] of largest eigenvalues of random matrices. Most of these subclasses were initially discovered in a discrete growth model (the PNG model) [11][12][13] which can be mapped onto the statistics of random permutations [14], and a zero temperature lattice DP model [10]. Recently, exact solutions have been obtained directly in the continuum at arbitrary time t, for the droplet [18][19][20][21], flat [22,23] and stationary [24] initial conditions. The PDF of the height h(x, t) converges at large time to F 2 , the Gaussian unitary ensemble (GUE), and to F 1 , the Gaussian orthogonal ensemble (GOE) universal TW distributions, for droplet and flat initial conditions respectively. One useful method which led to these solutions introduces n replica and maps the DP problem to the Lieb Liniger model, i.e. the quantum mechanics of n bosons with mutual delta-function attraction, a model which can be solved using the Bethe Ansatz.The KPZ equation on the half line x > 0, equivalently a DP in presence of a wall, is also of great interest. In the statistical mechanics context constrained fluctuations are important for the study of fluctuation-induced (Casimir) forces [25,26] and for extreme value statistics. In the surface growth context one can study an interface pinned at a point, or an average growth rate which jumps across a boundary. The half space probl...

Finding a good compromise between the exploitation of known resources and the exploration of unknown, but potentially more profitable choices, is a general problem, which arises in many different scientific disciplines. We propose a stylized model for these exploration-exploitation situations, including population or economic growth, portfolio optimisation, evolutionary dynamics, or the problem of optimal pinning of vortices or dislocations in disordered materials. We find the exact growth rate of this model for tree-like geometries and prove the existence of an optimal migration rate in this case. Numerical simulations in the one-dimensional case confirm the generic existence of an optimum. The exploration-exploitation tradeoff problem pervades a large number of different fields (see [1] and the many references therein). Two early examples concern the management of firms [2] (should one exploit an already known technology or explore other avenues, potentially more profitable, but risky?) and the so-called multi-arm bandit problem [3] (sticking with the seemingly most profitable arm to date, or switching in search of potentially more profitable ones?). Clearly, this is a universal paradigm that ranges from population growth and animal foraging to economic growth, investment strategies or optimal research policies. As we will show below, the same issues also arise, in a slightly disguised form, in the context of vortex or dislocation pinning by impurities, and are relevant for material design. Intuitively, neither staying at the same place (and missing interesting opportunities) nor changing places too rapidly (and failing to exploit favorable circumstances) are optimal strategies. An optimal, non zero search rate should thus exist in general. However, there are no exactly solvable cases where the exploration-exploitation tradeoff can be investigated in details. The aim of this paper is to propose a general, stylized model for these explorationexploitation situations, which encompasses all the examples given above. We obtain exact solutions of this model in two cases (a fully connected and a tree geometry), for which we explicitely prove the existence of a non-trivial optimal search rate. Euclidean geometries are also considered, as these correspond to physical situations, like the pinning problem alluded to above. In this case, perturbation theory and numerical simulations confirm the existence of an optimum as well.Our model describes the dynamics of a quantity we generically call Z i , defined on the nodes i of an arbitrary graph, that evolves according to the following equation [4]:The first two terms encode "migration" effects, with J ij the migration rate from j to i. The last term describes the growth (or decay) of the quantity Z i with a random growth rate η i (t). We will choose η i to be Gaussian, centred and uncorrelated from site to site, with a exponential time-correlator:Our qualitative conclusions are however independent of the precise form (2), provided correlations decay on a finite scale τ , whi...