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Proteins are involved in nearly all cellular functions, encompassing roles in transport, signaling, enzymatic activity, and more. Their functionalities crucially depend on their complex three-dimensional arrangement. For this reason, being able to predict their structure from the amino acid sequence has been and still is a phenomenal computational challenge that the introduction of AlphaFold solved with unprecedented accuracy. However, the inherent complexity of AlphaFold's architectures makes it challenging to understand the rules that ultimately shape the protein's predicted structure. This study investigates a single-layer unsupervised model based on the attention mechanism. More precisely, we explore a Direct Coupling Analysis (DCA) method that mimics the attention mechanism of several popular Transformer architectures, such as AlphaFold itself. The model's parameters, notably fewer than those in standard DCA-based algorithms, can be directly used for extracting structural determinants such as the contact map of the protein family under study. Additionally, the functional form of the energy function of the model enables us to deploy a multi-family learning strategy, allowing us to effectively integrate information across multiple protein families, whereas standard DCA algorithms are typically limited to single protein families. Finally, we implemented a generative version of the model using an autoregressive architecture, capable of efficiently generating new proteins in silico.

Generative models emerge as promising candidates for novel sequence-data driven approaches to protein design, and for the extraction of structural and functional information about proteins deeply hidden in rapidly growing sequence databases. Here we propose simple autoregressive models as highly accurate but computationally efficient generative sequence models. We show that they perform similarly to existing approaches based on Boltzmann machines or deep generative models, but at a substantially lower computational cost (by a factor between 10 2 and 10 3 ). Furthermore, the simple structure of our models has distinctive mathematical advantages, which translate into an improved applicability in sequence generation and evaluation. Within these models, we can easily estimate both the probability of a given sequence, and, using the model’s entropy, the size of the functional sequence space related to a specific protein family. In the example of response regulators, we find a huge number of ca. 10 68 possible sequences, which nevertheless constitute only the astronomically small fraction 10 −80 of all amino-acid sequences of the same length. These findings illustrate the potential and the difficulty in exploring sequence space via generative sequence models. Deep learning is a powerful tool for the design of novel protein sequences, yet can be computationally very inefficient. Here the authors propose using simple forecasting models to efficiently generate a large number of novel protein structures.

Background Boltzmann machines are energy-based models that have been shown to provide an accurate statistical description of domains of evolutionary-related protein and RNA families. They are parametrized in terms of local biases accounting for residue conservation, and pairwise terms to model epistatic coevolution between residues. From the model parameters, it is possible to extract an accurate prediction of the three-dimensional contact map of the target domain. More recently, the accuracy of these models has been also assessed in terms of their ability in predicting mutational effects and generating in silico functional sequences. Results Our adaptive implementation of Boltzmann machine learning, adabmDCA , can be generally applied to both protein and RNA families and accomplishes several learning set-ups, depending on the complexity of the input data and on the user requirements. The code is fully available at https://github.com/anna-pa-m/adabmDCA . As an example, we have performed the learning of three Boltzmann machines modeling the Kunitz and Beta-lactamase2 protein domains and TPP-riboswitch RNA domain. Conclusions The models learned by adabmDCA are comparable to those obtained by state-of-the-art techniques for this task, in terms of the quality of the inferred contact map as well as of the synthetically generated sequences. In addition, the code implements both equilibrium and out-of-equilibrium learning, which allows for an accurate and lossless training when the equilibrium one is prohibitive in terms of computational time, and allows for pruning irrelevant parameters using an information-based criterion.

MicroRNAs (miRNAs) are small RNA molecules, about 22 nucleotide long, which post-transcriptionally regulate their target messenger RNAs (mRNAs). They accomplish key roles in gene regulatory networks, ranging from signaling pathways to tissue morphogenesis, and their aberrant behavior is often associated with the development of various diseases. Recently it has been experimentally shown that the way miRNAs interact with their targets can be described in terms of a titration mechanism. From a theoretical point of view titration mechanisms are characterized by threshold effect at near-equimolarity of the different chemical species, hypersensitivity of the system around the threshold, and cross-talk among targets. The latter characteristic has been lately identified as competing endogenous RNA (ceRNA) effect to mark those indirect interactions among targets of a common pool of miRNAs they are in competition for. Here we propose a stochastic model to analyze the equilibrium and out-of-equilibrium properties of a network of [Formula: see text] miRNAs interacting with [Formula: see text] mRNA targets. In particular we are able to describe in detail the peculiar equilibrium and non-equilibrium phenomena that the system displays in proximity to the threshold: (i) maximal cross-talk and correlation between targets, (ii) robustness of ceRNA effect with respect to the model's parameters and in particular to the catalyticity of the miRNA-mRNA interaction, and (iii) anomalous response-time to external perturbations.

The similarity in the three-dimensional structures of homologous proteins imposes strong constraints on their sequence variability. It has long been suggested that the resulting correlations among amino acid compositions at different sequence positions can be exploited to infer spatial contacts within the tertiary protein structure. Crucial to this inference is the ability to disentangle direct and indirect correlations, as accomplished by the recently introduced direct-coupling analysis (DCA). Here we develop a computationally efficient implementation of DCA, which allows us to evaluate the accuracy of contact prediction by DCA for a large number of protein domains, based purely on sequence information. DCA is shown to yield a large number of correctly predicted contacts, recapitulating the global structure of the contact map for the majority of the protein domains examined. Furthermore, our analysis captures clear signals beyond intradomain residue contacts, arising, e.g., from alternative protein conformations, ligand-mediated residue couplings, and interdomain interactions in protein oligomers. Our findings suggest that contacts predicted by DCA can be used as a reliable guide to facilitate computational predictions of alternative protein conformations, protein complex formation, and even the de novo prediction of protein domain structures, contingent on the existence of a large number of homologous sequences which are being rapidly made available due to advances in genome sequencing.

The evolutionary trajectory of a protein through sequence space is constrained by its function. Collections of sequence homologs record the outcomes of millions of evolutionary experiments in which the protein evolves according to these constraints. Deciphering the evolutionary record held in these sequences and exploiting it for predictive and engineering purposes presents a formidable challenge. The potential benefit of solving this challenge is amplified by the advent of inexpensive high-throughput genomic sequencing.In this paper we ask whether we can infer evolutionary constraints from a set of sequence homologs of a protein. The challenge is to distinguish true co-evolution couplings from the noisy set of observed correlations. We address this challenge using a maximum entropy model of the protein sequence, constrained by the statistics of the multiple sequence alignment, to infer residue pair couplings. Surprisingly, we find that the strength of these inferred couplings is an excellent predictor of residue-residue proximity in folded structures. Indeed, the top-scoring residue couplings are sufficiently accurate and well-distributed to define the 3D protein fold with remarkable accuracy.We quantify this observation by computing, from sequence alone, all-atom 3D structures of fifteen test proteins from different fold classes, ranging in size from 50 to 260 residues, including a G-protein coupled receptor. These blinded inferences are de novo, i.e., they do not use homology modeling or sequence-similar fragments from known structures. The co-evolution signals provide sufficient information to determine accurate 3D protein structure to 2.7-4.8 Å C(α)-RMSD error relative to the observed structure, over at least two-thirds of the protein (method called EVfold, details at http://EVfold.org). This discovery provides insight into essential interactions constraining protein evolution and will facilitate a comprehensive survey of the universe of protein structures, new strategies in protein and drug design, and the identification of functional genetic variants in normal and disease genomes.

The recent technological advances underlying the screening of large combinatorial libraries in high-throughput mutational scans deepen our understanding of adaptive protein evolution and boost its applications in protein design. Nevertheless, the large number of possible genotypes requires suitable computational methods for data analysis, the prediction of mutational effects, and the generation of optimized sequences. We describe a computational method that, trained on sequencing samples from multiple rounds of a screening experiment, provides a model of the genotype–fitness relationship. We tested the method on five large-scale mutational scans, yielding accurate predictions of the mutational effects on fitness. The inferred fitness landscape is robust to experimental and sampling noise and exhibits high generalization power in terms of broader sequence space exploration and higher fitness variant predictions. We investigate the role of epistasis and show that the inferred model provides structural information about the 3D contacts in the molecular fold.

Interaction between proteins is a fundamental mechanism that underlies virtually all biological processes. Many important interactions are conserved across a large variety of species. The need to maintain interaction leads to a high degree of co-evolution between residues in the interface between partner proteins. The inference of protein-protein interaction networks from the rapidly growing sequence databases is one of the most formidable tasks in systems biology today. We propose here a novel approach based on the Direct-Coupling Analysis of the co-evolution between inter-protein residue pairs. We use ribosomal and trp operon proteins as test cases: For the small resp. large ribosomal subunit our approach predicts protein-interaction partners at a true-positive rate of 70% resp. 90% within the first 10 predictions, with areas of 0.69 resp. 0.81 under the ROC curves for all predictions. In the trp operon, it assigns the two largest interaction scores to the only two interactions experimentally known. On the level of residue interactions we show that for both the small and the large ribosomal subunit our approach predicts interacting residues in the system with a true positive rate of 60% and 85% in the first 20 predictions. We use artificial data to show that the performance of our approach depends crucially on the size of the joint multiple sequence alignments and analyze how many sequences would be necessary for a perfect prediction if the sequences were sampled from the same model that we use for prediction. Given the performance of our approach on the test data we speculate that it can be used to detect new interactions, especially in the light of the rapid growth of available sequence data.

It is well known that, in order to preserve its structure and function, a protein cannot change its sequence at random, but only by mutations occurring preferentially at specific locations. We here investigate quantitatively the amount of variability that is allowed in protein sequence evolution, by computing the intrinsic dimension (ID) of the sequences belonging to a selection of protein families. The ID is a measure of the number of independent directions that evolution can take starting from a given sequence. We find that the ID is practically constant for sequences belonging to the same family, and moreover it is very similar in different families, with values ranging between 6 and 12. These values are significantly smaller than the raw number of amino acids, confirming the importance of correlations between mutations in different sites. However, we demonstrate that correlations are not sufficient to explain the small value of the ID we observe in protein families. Indeed, we show that the ID of a set of protein sequences generated by maximum entropy models, an approach in which correlations are accounted for, is typically significantly larger than the value observed in natural protein families. We further prove that a critical factor to reproduce the natural ID is to take into consideration the phylogeny of sequences.

Assuming a steady-state condition within a cell, metabolic fluxes satisfy an underdetermined linear system of stoichiometric equations. Characterizing the space of fluxes that satisfy such equations along with given bounds (and possibly additional relevant constraints) is considered of utmost importance for the understanding of cellular metabolism. Extreme values for each individual flux can be computed with linear programming (as flux balance analysis), and their marginal distributions can be approximately computed with Monte Carlo sampling. Here we present an approximate analytic method for the latter task based on expectation propagation equations that does not involve sampling and can achieve much better predictions than other existing analytic methods. The method is iterative, and its computation time is dominated by one matrix inversion per iteration. With respect to sampling, we show through extensive simulation that it has some advantages including computation time, and the ability to efficiently fix empirically estimated distributions of fluxes. Large-scale metabolic models of organisms from microbes to mammals can provide great insight into cellular function, but their analysis remains challenging. Here, the authors provide an approximate analytic method to estimate the feasible solution space for the flux vectors of metabolic networks, enabling more accurate analysis under a wide range of conditions of interest.