An a posteriori revisit of the accumulated results on the Semiring and Valued constraint satisfaction problems and related problems, showing how semi-rings and monoids (or semi-groups) are useful objects to explore the intricate interplay between expressivity and the design of efficient, correct and convergent algorithms in computer science, showing tight relationships and scientific overlaps with probabilistic reasoning over graphical structures and linear or polynomial discrete mathematical programming.

The Quadratic Assignment Problem (QAP) consists of finding a permutation that minimizes a quadratic objective function. Exact methods generally rely on a branch-and-bound procedure, the efficiency of which depends heavily on the quality of its lower bound. In integer linear programming, several bounds have been investigated, exhibiting different trade-offs between speed and quality. The Gilmore-Lawler bound appears to be the most commonly used in practice. It requires solving a linear assignment problem (LAP) for each variable-value pair. We show how to obtain this bound using Singleton Node Consistency (SNC) and LAP. In Cost Function Networks (CFNs), we propose a reformulation that transforms the result of applying LAP to a given variable-value pair into cost functions of arity 1 and 2, which can be added to the original problem. Combined with existing lower bounds for CFNs, including EDAC and a recent CFN propagator for AllDifferent, this method (SNC-LAP-GLB), used as a preprocessing, significantly increases the initial lower bound and accelerates the search, resulting in competitive results on the QAPLIB benchmark. We then propose an extension of the AllDifferent propagator for the Global Cardinality Constraint. It allows us to exploit variable symmetries on some challenging QAPLIB instances, thus improving the results.

To efficiently solve exact discrete optimization problems, branch and bound algorithms require tight bounds. In constraint programming, for optimization, soft arc consistencies typically derive much stronger bounds than those offered by domain or bound consistencies applied to a cost variable. The reason is that soft local consistencies exchange marginal cost information between variables whereas domain consistencies rely only on shrinking domains, which is less informative. However, CP solvers equipped with soft arc consistencies have so far offered limited support for efficient processing of global constraints. In this work, we show how we can efficiently enforce soft local consistency over the AllDifferent constraint, relying on algorithms for the Linear Assignment Problem (LAP). We implement this propagator in toulbar2, the state-of-the-art weighted CP solver exploiting soft local consistencies for bounding. We show that, equipped with this new propagator, toulbar2 outperforms state-of-the-art domain consistency-based CP as well as integer programming solvers for the Quadratic Assignment Problem and shows better performance for miniCOP instances of the 2024 XCSP competition with AllDifferent constraints.

Background: In the ongoing quest for hybridizing discrete reasoning with neural nets, there is an increasing interest in neural architectures that can learn how to solve discrete reasoning or optimisation problems from natural inputs, a task that Large Language Models seem to struggle with. Objectives: We introduce a differentiable neuro-symbolic architecture and a loss function dedicated to learning how to solve NP-hard reasoning problems. Methods: Our new probabilistic loss allows for learning both the constraints and the objective-possibly non-linear-of a combinatorial problem. Thus, it delivers a complete model that can be scrutinized and completed with side constraints. By pushing the combinatorial solver out of the training loop, our architecture also offers scalable training while exact inference gives access to maximum accuracy. Results: We empirically show that it can efficiently learn how to solve NP-hard reasoning problems from natural inputs. On three variants of the Sudoku benchmark-symbolic, visual, and many-solution-, our approach requires a fraction of data and training time of other hybrid methods. On a visual Min-Cut/Max-cut task, it optimizes the regret as well as a Decision-Focused-Learning regret-dedicated loss. Finally, it efficiently learns the energy optimisation formulation of the large real-world problem of designing proteins.

Deep learning has revolutionized computational protein design, enabling the generation of sequences that fold onto target backbones with unprecedented accuracy. However, state-of-the-art inverse folding tools largely rely on auto-regressive sampling. While powerful, this paradigm is increasingly recognized for its inability to "think ahead", a crucial capacity to reliably create the complex, long-range inter-residue dependencies essential for most biological functions. To overcome these fundamental limitations, we introduced EffieDes, a generative neuro-symbolic AI framework that synergizes the predictive capabilities of deep learning with the logical precision of automated reasoning. EffieDes leverages deep learning to encode the target backbone's fitness landscape into Effie-a fully decomposable probabilistic graphical model (Potts model). This landscape can then be rigorously explored by an automated reasoning prover to identify sequences that simultaneously satisfy complex design constraints and optimize backbone fitness. We validated this neuro-symbolic approach through the design of orthogonal sequence pairs that adopt identical folds but exhibit selective self-assembly, as well as the design of a de novo selective nanobody with nanomolar affinity for an immune-evasive SARS-CoV-2 variant. EffieDes provides a robust architecture for precisely dissecting learned fitness landscapes, offering a new path toward proteins with highly optimized performances and sophisticated functional objectives.

Combining molecular modelling, machine-learned models and an increasingly detailed understanding of protein chemistry and physics, computational protein design and human expertise have been able to produce new protein structures, assemblies and functions that do not exist in nature. Currently, generative deep-learning-based methods, which exploit large databases of protein sequences and structures, are revolutionizing the field, leading to new capabilities, improved reliability and democratized access in protein design. This Primer provides an introduction to the main approaches in computational protein design, covering both physics-based and machine-learning-based tools. It aims to be accessible to biological, physical and computer scientists alike. Emphasis is placed on understanding the practical challenges arising from limitations in our fundamental understanding of protein structure and function and on recent developments and new ideas that may help transcend these. Computational protein design uses information on the constraints of the biological and physical properties of proteins for protein engineering and de novo protein design. In this Primer, Albanese et al. give an overview of the guiding principles of computational protein design and its considerations, methods and applications and conclude by discussing the future of the technique in the context of rapidly advancing computational tools.

Bacterial microcompartments (BMC) are submicrometric reactors that encapsulate dedicated metabolic activities. BMC-H hexamers, the most abundant components of BMC shells, play major roles for shell plasticity and permeability. In part, chemical exchanges between the BMC lumen and the cellular cytosol will be defined by the disposition of amino acids lining the central BMC-H pores. Current models attribute to BMC-H a homo-oligomeric nature. The hexagonal symmetry of corresponding pores, however, would break down if hetero-hexamers formed, a possibility suggested by the frequent presence of multiple paralogs within BMC operons. Here, we gauged the degree of structural promiscuity between the 11 BMC-H paralogs from Klebsiella pneumoniae, a potential human pathogen endowed with the capacity to express three different BMC types. Concomitant activation of transcription of several BMC operons was first shown to be possible. By leveraging an adapted tripartite GFP technology, all possible BMC-H pair combinations were screened in E. coli. Multiple structural cross-compatibilities were pinpointed between homologs arising not only from the same BMC operon, but also from different BMC types, results supported by Alphafold and ESMFold predictions. The structural stability and assembly propensity of selected hetero-associations was established by biochemical means. In light of these results, we reinterpreted published lysine cross-linking mass spectrometry data to demonstrate that one of these hetero-hexamers, involving PduA and PduJ, was already detected to form in the shell of a recombinantly-expressed 1,2-propanediol utilization compartment from Salmonella enterica. Altogether, this study points to the need to embrace an augmented structural complexity in BMC shells.

Actual versions of hifiasm often produce unbalanced haplotype contigs for autopolyploid genome assemblies. Adding a contraint on the distribution of proteins among haplotypes improves haplotype sizes and busco scores balances. Four species were used to benchmark this approach against hifiasm raw results. The script can be downloaded from [https://github.com/chklopp/Separate\_haplotypes\_with_toulbar2][1]. ### Competing Interest Statement The authors have declared no competing interest. [1]: https://github.com/chklopp/Separate_haplotypes_with_toulbar2

Discrete Graphical Models (GMs) are widely used in Artificial Intelligence to describe complex systems through a joint function of interest. Probabilistic GMs such as Markov Random Fields (MRFs) define a joint non-normalized probability distribution while deterministic GMs such as Cost Function Networks (CFNs) define a joint cost function. A typical query on GMs consists in finding the joint state that optimizes this joint function, a problem denoted as the Maximum a Posteriori or Weighted Constraint Satisfaction Problem respectively. In practice, more than one function of interest may need to be optimized at the same time. In this paper, we develop a two-phase scalarization method for solving bi-objective discrete graphical model optimization, with the aim of computing a set of non-dominated solutions—the Pareto frontier—representing different compromises between two GM-defined objectives. For this purpose, we introduce a dedicated higher-order constraint, which bounds the value of one GM-defined objective while minimizing another GM on the same variables. Discrete GM optimization is NP-hard, and its bi-objective variants are even harder. We show how existing GM global lower and upper bounds can be exploited to provide anytime bounds of the exact Pareto frontier. We benchmark it on various instances from Operations Research and Probabilistic Graphical Models.

Understanding how proteins evolve under selective pressure is a longstanding challenge. The immensity of the search space has limited efforts to systematically evaluate the impact of multiple simultaneous mutations, so mutations have typically been assessed individually. However, epistasis, or the way in which mutations interact, prevents accurate prediction of combinatorial mutations based on measurements of individual mutations. Here, we use artificial intelligence to define the entire functional sequence landscape of a protein binding site in silico, and we call this approach Complete Combinatorial Mutational Enumeration (CCME). By leveraging CCME, we are able to construct a comprehensive map of the evolutionary connectivity within this functional sequence landscape. As a proof of concept, we applied CCME to the ACE2 binding site of the SARS-CoV-2 spike protein receptor binding domain. We selected representative variants from across the functional sequence landscape for testing in the laboratory. We identified variants that retained functionality to bind ACE2 despite changing over 40% of evaluated residue positions, and the variants now escape binding and neutralization by monoclonal antibodies. This work represents a crucial initial stride towards achieving precise predictions of pathogen evolution, opening avenues for proactive mitigation.

In the ongoing quest for hybridizing discrete reasoning with neural nets, there is an increasing interest in neural architectures that can learn how to solve discrete reasoning or optimization problems from natural inputs. In this paper, we introduce a scalable neural architecture and loss function dedicated to learning the constraints and criteria of NP-hard reasoning problems expressed as discrete Graphical Models. We empirically show our loss function is able to efficiently learn how to solve NP-hard reasoning problems from natural inputs as the symbolic, visual or many-solutions Sudoku problems as well as the energy optimization formulation of the protein design problem, providing data efficiency, interpretability, and a posteriori control over predictions.

Host-pathogen interactions drive an evolutionary game of cat-and-mouse between a pathogen’s protein virulence factors, the host’s adaptive immune system, and therapeutics targeting the pathogen. There is an urgent need for treatments and prophylactics that remain effective as a pathogen evolves, and the ability to predict pathogen evolution is a longstanding challenge. Therefore, a common strategy has been to target conserved epitopes, but strong selective pressures can drive pathogens to evolve resistance nonetheless. Here, we report a novel, generally-applicable approach called Deep Evolutionary Forecasting that predicts protein evolution using artificial intelligence and molecular modeling. The first step is to perform a complete enumeration of the functional sequence landscape in silico for a target protein. Then, we construct a graph where the edges between sequence variants are weighted by evolutionary probability. Protein evolution is forecasted by traversing this graph. We chose the SARS-CoV-2 receptor binding domain (RBD) as a model system because highly-mutated viral variants have continued to emerge that escape available therapeutics and vaccines. The RBD variants that we forecasted carry up to 11 concurrent amino acid substitutions at the host receptor binding site. Pseudoviruses harboring forecasted RBDs are active and escape binding and neutralization by FDA-approved monoclonal antibody therapeutics. We identified bottlenecks in the evolutionary landscape of SARS-CoV-2 that are promising targets for therapeutics that preempt evolution.

Although Salmonella Typhimurium (STM) and Salmonella Paratyphi A (SPA) belong to the same phylogenetic species, share large portions of their genome and express many common virulence factors, they differ vastly in their host specificity, the immune response they elicit, and the clinical manifestations they cause. In this work, we compared their intracellular transcriptomic architecture and cellular phenotypes during human epithelial cell infection. While transcription induction of many metal transport systems, purines, biotin, PhoPQ and SPI-2 regulons was similar in both intracellular SPA and STM, we identified 234 differentially expressed genes that showed distinct expression patterns in intracellular SPA vs. STM. Surprisingly, clear expression differences were found in SPI-1, motility and chemotaxis, and carbon (mainly citrate, galactonate and ethanolamine) utilization pathways, indicating that these pathways are regulated differently during their intracellular phase. Concurring, on the cellular level, we show that while the majority of STM are non-motile and reside within Salmonella-Containing Vacuoles (SCV), a significant proportion of intracellular SPA cells are motile and compartmentalized in the cytosol. Moreover, we found that the elevated expression of SPI-1 and motility genes by intracellular SPA results in increased invasiveness of SPA, following exit from host cells. These findings demonstrate unexpected flagellum-dependent intracellular motility of a typhoidal Salmonella serovar and intriguing differences in intracellular localization between typhoidal and non-typhoidal salmonellae. We propose that these differences facilitate new cycles of host cell infection by SPA and may contribute to the ability of SPA to disseminate beyond the intestinal lamina propria of the human host during enteric fever.

In this paper, we extend a Burer-Monteiro style method to compute low rank Semi-Definite Programming (SDP) bounds for the MAP problem on discrete graphical models with an arbitrary number of states and arbitrary pairwise potentials. We consider both a penalized constraint approach and a dedicated Block Coordinate Descent (BCD) approach which avoids large penalty coefficients in the cost matrix. We show our algorithm is decreas-ing. Experiments show that the BCD approach compares favorably to the penalized approach and to usual linear bounds relying on convergent message passing approaches.

With the growing need for renewable sources of energy, the interest for enzymes capable of biomass degradation has been increasing. In this paper, we consider two different xylanases from the GH-11 family: the particularly active GH-11 xylanase from Neocallimastix patriciarum, NpXyn11A, and the hyper-thermostable mutant of the environmentally isolated GH-11 xylanase, EvXyn11TS. Our aim is to identify the molecular determinants underlying the enhanced capacities of these two enzymes to ultimately graft the abilities of one on the other. Molecular dynamics simulations of the respective free-enzymes and enzyme–xylohexaose complexes were carried out at temperatures of 300, 340, and 500 K. An in-depth analysis of these MD simulations showed how differences in dynamics influence the activity and stability of these two enzymes and allowed us to study and understand in greater depth the molecular and structural basis of these two systems. In light of the results presented in this paper, the thumb region and the larger substrate binding cleft of NpXyn11A seem to play a major role on the activity of this enzyme. Its lower thermal stability may instead be caused by the higher flexibility of certain regions located further from the active site. Regions such as the N-ter, the loops located in the fingers region, the palm loop, and the helix loop seem to be less stable than in the hyper-thermostable EvXyn11TS. By identifying molecular regions that are critical for the stability of these enzymes, this study allowed us to identify promising targets for engineering GH-11 xylanases. Eventually, we identify NpXyn11A as the ideal host for grafting the thermostabilizing traits of EvXyn11TS.

The extant complex proteins must have evolved from ancient short and simple ancestors. The double-ψ β-barrel (DPBB) is one of the oldest protein folds and conserved in various fundamental enzymes, such as the core domain of RNA polymerase. Here, by reverse engineering a modern DPBB domain, we reconstructed its plausible evolutionary pathway started by "interlacing homodimerization" of a half-size peptide, followed by gene duplication and fusion. Furthermore, by simplifying the amino acid repertoire of the peptide, we successfully created the DPBB fold with only seven amino acid types (Ala, Asp, Glu, Gly, Lys, Arg, and Val), which can be coded by only GNN and ARR (R = A or G) codons in the modern translation system. Thus, the DPBB fold could have been materialized by the early translation system and genetic code.

Structure-based computational protein design (CPD) refers to the problem of finding a sequence of amino acids which folds into a specific desired protein structure, and possibly fulfills some targeted biochemical properties. Recent studies point out the particularly rugged CPD energy landscape, suggesting that local search optimization methods should be designed and tuned to easily escape local minima attraction basins. In this article, we analyze the performance and search dynamics of an iterated local search (ILS) algorithm enhanced with partition crossover. Our algorithm, PILS, quickly finds local minima and escapes their basins of attraction by solution perturbation. Additionally, the partition crossover operator exploits the structure of the residue interaction graph in order to efficiently mix solutions and find new unexplored basins. Our results on a benchmark of 30 proteins of various topology and size show that PILS consistently finds lower energy solutions compared to Rosetta fixbb and a classic ILS, and that the corresponding sequences are mostly closer to the native.

Computational protein design (CPD) is a powerful technique for engineering new proteins, with both great fundamental implications and diverse practical interests. However, the approximations usually made for computational efficiency, using a single fixed backbone and a discrete set of side chain rotamers, tend to produce rigid and hyper-stable folds that may lack functionality. These approximations contrast with the demonstrated importance of molecular flexibility and motions in a wide range of protein functions. The integration of backbone flexibility and multiple conformational states in CPD, in order to relieve the inaccuracies resulting from these simplifications and to improve design reliability, are attracting increased attention. However, the greatly increased search space that needs to be explored in these extensions defines extremely challenging computational problems. In this review, we outline the principles of CPD and discuss recent effort in algorithmic developments for incorporating molecular flexibility in the design process.