In operations research, the Big M method is a method of solving linear programming problems using the simplex algorithm. The Big M method extends the simplex algorithm to problems that contain "greater-than" constraints. It does so by associating the constraints with large negative constants which would not be part of any optimal solution, if it exists.
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The simplex algorithm is the original and still one of the most widely used methods for solving linear maximization problems. However, to apply it, the origin (all variables equal to 0) must be a feasible point. This condition is satisfied only when all the constraints (except non-negativity) are less-than constraints and with positive constant on the right-hand side. The Big M method introduces surplus and artificial variables to convert all inequalities into that form. The "Big M" refers to a large number associated with the artificial variables, represented by the letter M.
When used in the objective function, the Big M method sometimes refers to formulations of linear optimization problems in which violations of a constraint or set of constraints are associated with a large positive penalty constant, M.
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The development of computational methods to predict three-dimensional (3D) protein structures from the protein sequence has proceeded along two complementary paths that focus on either the physical interactions or the evolutionary history. The physical interaction programme heavily integrates our understanding of molecular driving forces into either thermodynamic or kinetic simulation of protein physics16 or statistical approximations thereof17. Although theoretically very appealing, this approach has proved highly challenging for even moderate-sized proteins due to the computational intractability of molecular simulation, the context dependence of protein stability and the difficulty of producing sufficiently accurate models of protein physics. The evolutionary programme has provided an alternative in recent years, in which the constraints on protein structure are derived from bioinformatics analysis of the evolutionary history of proteins, homology to solved structures18,19 and pairwise evolutionary correlations20,21,22,23,24. This bioinformatics approach has benefited greatly from the steady growth of experimental protein structures deposited in the Protein Data Bank (PDB)5, the explosion of genomic sequencing and the rapid development of deep learning techniques to interpret these correlations. Despite these advances, contemporary physical and evolutionary-history-based approaches produce predictions that are far short of experimental accuracy in the majority of cases in which a close homologue has not been solved experimentally and this has limited their utility for many biological applications.
The methodology that we have taken in designing AlphaFold is a combination of the bioinformatics and physical approaches: we use a physical and geometric inductive bias to build components that learn from PDB data with minimal imposition of handcrafted features (for example, AlphaFold builds hydrogen bonds effectively without a hydrogen bond score function). This results in a network that learns far more efficiently from the limited data in the PDB but is able to cope with the complexity and variety of structural data.
The following versions of public datasets were used in this study. Our models were trained on a copy of the PDB5 downloaded on 28 August 2019. For finding template structures at prediction time, we used a copy of the PDB downloaded on 14 May 2020, and the PDB7066 clustering database downloaded on 13 May 2020. For MSA lookup at both training and prediction time, we used Uniref9067 v.2020_01, BFD, Uniclust3036 v.2018_08 and MGnify6 v.2018_12. For sequence distillation, we used Uniclust3036 v.2018_08 to construct a distillation structure dataset. Full details are provided in Supplementary Methods 1.2.
The procedure for filtering the recent PDB dataset based on prior template identity was as follows. Hmmsearch was run with default parameters against a copy of the PDB SEQRES fasta downloaded 15 February 2021. Template hits were accepted if the associated structure had a release date earlier than 30 April 2018. Each residue position in a query sequence was assigned the maximum identity of any template hit covering that position. Filtering then proceeded as described in the individual figure legends, based on a combination of maximum identity and sequence coverage.
Structures from the PDB were used for training and as templates ( -ftp-sites; for the associated sequence data and 40% sequence clustering see also _data/ and -40.out). Training used a version of the PDB downloaded 28 August 2019, while the CASP14 template search used a version downloaded 14 May 2020. The template search also used the PDB70 database, downloaded 13 May 2020 ( compbiol/data/hhsuite/databases/hhsuite_dbs/).
Description of the method details of the AlphaFold system, model, and analysis, including data pipeline, datasets, model blocks, loss functions, training and inference details, and ablations. Includes Supplementary Methods, Supplementary Figures, Supplementary Tables and Supplementary Algorithms.
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