Guaranteed Discrete Energy Optimization on Large Protein Design Problems

Hierarchical Materials from High Information Content Macromolecular Building Blocks: Construction, Dynamic Interventions, and Prediction

Hierarchical materials that exhibit order over multiple length scales are ubiquitous in nature. Because hierarchy gives rise to unique properties and functions, many have sought inspiration from nature when designing and fabricating hierarchical matter. More and more, however, nature's own high-information content building blocks, proteins, peptides, and peptidomimetics, are being coopted to build hierarchy because the information that determines structure, function, and interfacial interactions can be readily encoded in these versatile macromolecules. Here, we take stock of recent progress in the rational design and characterization of hierarchical materials produced from high-information content blocks with a focus on stimuli-responsive and "smart" architectures. We also review advances in the use of computational simulations and data-driven predictions to shed light on how the side chain chemistry and conformational flexibility of macromolecular blocks drive the emergence of order and the acquisition of hierarchy and also on how ionic, solvent, and surface effects influence the outcomes of assembly. Continued progress in the above areas will ultimately usher in an era where an understanding of designed interactions, surface effects, and solution conditions can be harnessed to achieve predictive materials synthesis across scale and drive emergent phenomena in the self-assembly and reconfiguration of high-information content building blocks.

Computational Design of Miniprotein Binders

Miniprotein binders hold a great interest as a class of drugs that bridges the gap between monoclonal antibodies and small molecule drugs. Like monoclonal antibodies, they can be designed to bind to therapeutic targets with high affinity, but they are more stable and easier to produce and to administer. In this chapter, we present a structure-based computational generic approach for miniprotein inhibitor design. Specifically, we describe step-by-step the implementation of the approach for the design of miniprotein binders against the SARS-CoV-2 coronavirus, using available structural data on the SARS-CoV-2 spike receptor binding domain (RBD) in interaction with its native target, the human receptor ACE2. Structural data being increasingly accessible around many protein-protein interaction systems, this method might be applied to the design of miniprotein binders against numerous therapeutic targets. The computational pipeline exploits provable and deterministic artificial intelligence-based protein design methods, with some recent additions in terms of binding energy estimation, multistate design and diverse library generation.

Knowledge-Based Unfolded State Model for Protein Design

The design of proteins and miniproteins is an important challenge. Designed variants should be stable, meaning the folded/unfolded free energy difference should be large enough. Thus, the unfolded state plays a central role. An extended peptide model is often used, where side chains interact with solvent and nearby backbone, but not each other. The unfolded energy is then a function of sequence composition only and can be empirically parametrized. If the space of sequences is explored with a Monte Carlo procedure, protein variants will be sampled according to a well-defined Boltzmann probability distribution. We can then choose unfolded model parameters to maximize the probability of sampling native-like sequences. This leads to a well-defined maximum likelihood framework. We present an iterative algorithm that follows the likelihood gradient. The method is presented in the context of our Proteus software, as a detailed downloadable tutorial. The unfolded model is combined with a folded model that uses molecular mechanics and a Generalized Born solvent. It was optimized for three PDZ domains and then used to redesign them. The sequences sampled are native-like and similar to a recent PDZ design study that was experimentally validated.

Protein Design with Deep Learning

Computational Protein Design (CPD) has produced impressive results for engineering new proteins, resulting in a wide variety of applications. In the past few years, various efforts have aimed at replacing or improving existing design methods using Deep Learning technology to leverage the amount of publicly available protein data. Deep Learning (DL) is a very powerful tool to extract patterns from raw data, provided that data are formatted as mathematical objects and the architecture processing them is well suited to the targeted problem. In the case of protein data, specific representations are needed for both the amino acid sequence and the protein structure in order to capture respectively 1D and 3D information. As no consensus has been reached about the most suitable representations, this review describes the representations used so far, discusses their strengths and weaknesses, and details their associated DL architecture for design and related tasks.

Iterated local search with partition crossover for computational protein design

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.

Guaranteed Diversity and Optimality in Cost Function Network Based Computational Protein Design Methods

Proteins are the main active molecules of life. Although natural proteins play many roles, as enzymes or antibodies for example, there is a need to go beyond the repertoire of natural proteins to produce engineered proteins that precisely meet application requirements, in terms of function, stability, activity or other protein capacities. Computational Protein Design aims at designing new proteins from first principles, using full-atom molecular models. However, the size and complexity of proteins require approximations to make them amenable to energetic optimization queries. These approximations make the design process less reliable, and a provable optimal solution may fail. In practice, expensive libraries of solutions are therefore generated and tested. In this paper, we explore the idea of generating libraries of provably diverse low-energy solutions by extending cost function network algorithms with dedicated automaton-based diversity constraints on a large set of realistic full protein redesign problems. We observe that it is possible to generate provably diverse libraries in reasonable time and that the produced libraries do enhance the Native Sequence Recovery, a traditional measure of design methods reliability.

Molecular flexibility in computational protein design: an algorithmic perspective

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.

The Taming of the Screw: the natural and artificial development of β-propeller proteins

Many proteins are found to possess repeated structural elements, which hint at ancient evolutionary origins and ongoing evolutionary processes. β-propeller proteins are a large family of such proteins, and a popular focus of structural analysis. This review highlights recent work to understand how they arose, and how they have developed into one of the most successful of all protein folds.

Macromolecular modeling and design in Rosetta: recent methods and frameworks

The Rosetta software for macromolecular modeling, docking and design is extensively used in laboratories worldwide. During two decades of development by a community of laboratories at more than 60 institutions, Rosetta has been continuously refactored and extended. Its advantages are its performance and interoperability between broad modeling capabilities. Here we review tools developed in the last 5 years, including over 80 methods. We discuss improvements to the score function, user interfaces and usability. Rosetta is available at http://www.rosettacommons.org.

Novel, provable algorithms for efficient ensemble-based computational protein design and their application to the redesign of the c-Raf-RBD:KRas protein-protein interface

The K* algorithm provably approximates partition functions for a set of states (e.g., protein, ligand, and protein-ligand complex) to a user-specified accuracy ε. Often, reaching an ε-approximation for a particular set of partition functions takes a prohibitive amount of time and space. To alleviate some of this cost, we introduce two new algorithms into the osprey suite for protein design: fries, a Fast Removal of Inadequately Energied Sequences, and EWAK*, an Energy Window Approximation to K*. fries pre-processes the sequence space to limit a design to only the most stable, energetically favorable sequence possibilities. EWAK* then takes this pruned sequence space as input and, using a user-specified energy window, calculates K* scores using the lowest energy conformations. We expect fries/EWAK* to be most useful in cases where there are many unstable sequences in the design sequence space and when users are satisfied with enumerating the low-energy ensemble of conformations. In combination, these algorithms provably retain calculational accuracy while limiting the input sequence space and the conformations included in each partition function calculation to only the most energetically favorable, effectively reducing runtime while still enriching for desirable sequences. This combined approach led to significant speed-ups compared to the previous state-of-the-art multi-sequence algorithm, BBK*, while maintaining its efficiency and accuracy, which we show across 40 different protein systems and a total of 2,826 protein design problems. Additionally, as a proof of concept, we used these new algorithms to redesign the protein-protein interface (PPI) of the c-Raf-RBD:KRas complex. The Ras-binding domain of the protein kinase c-Raf (c-Raf-RBD) is the tightest known binder of KRas, a protein implicated in difficult-to-treat cancers. fries/EWAK* accurately retrospectively predicted the effect of 41 different sets of mutations in the PPI of the c-Raf-RBD:KRas complex. Notably, these mutations include mutations whose effect had previously been incorrectly predicted using other computational methods. Next, we used fries/EWAK* for prospective design and discovered a novel point mutation that improves binding of c-Raf-RBD to KRas in its active, GTP-bound state (KRas^GTP). We combined this new mutation with two previously reported mutations (which were highly-ranked by osprey) to create a new variant of c-Raf-RBD, c-Raf-RBD(RKY). fries/EWAK* in osprey computationally predicted that this new variant binds even more tightly than the previous best-binding variant, c-Raf-RBD(RK). We measured the binding affinity of c-Raf-RBD(RKY) using a bio-layer interferometry (BLI) assay, and found that this new variant exhibits single-digit nanomolar affinity for KRas^GTP, confirming the computational predictions made with fries/EWAK*. This new variant binds roughly five times more tightly than the previous best known binder and roughly 36 times more tightly than the design starting point (wild-type c-Raf-RBD). This study steps through the advancement and development of computational protein design by presenting theory, new algorithms, accurate retrospective designs, new prospective designs, and biochemical validation.

Computational structure-based protein design is an innovative tool for redesigning proteins to introduce a particular or novel function. One such function is improving the binding of one protein to another, which can increase our understanding of important protein systems. Herein we introduce two novel, provable algorithms, fries and EWAK*, for more efficient computational structure-based protein design as well as their application to the redesign of the c-Raf-RBD:KRas protein-protein interface. These new algorithms speed-up computational structure-based protein design while maintaining accurate calculations, allowing for larger, previously infeasible protein designs. Additionally, using fries and EWAK* within the osprey suite, we designed the tightest known binder of KRas, a heavily studied cancer target that interacts with a number of different proteins. This previously undiscovered variant of a KRas-binding domain, c-Raf-RBD, has potential to serve as a tool to further probe the protein-protein interface of KRas with its effectors and its discovery alone emphasizes the potential for more successful applications of computational structure-based protein design.

Dynamics, a Powerful Component of Current and Future in Silico Approaches for Protein Design and Engineering

Computational prediction has become an indispensable aid in the processes of engineering and designing proteins for various biotechnological applications. With the tremendous progress in more powerful computer hardware and more efficient algorithms, some of in silico tools and methods have started to apply the more realistic description of proteins as their conformational ensembles, making protein dynamics an integral part of their prediction workflows. To help protein engineers to harness benefits of considering dynamics in their designs, we surveyed new tools developed for analyses of conformational ensembles in order to select engineering hotspots and design mutations. Next, we discussed the collective evolution towards more flexible protein design methods, including ensemble-based approaches, knowledge-assisted methods, and provable algorithms. Finally, we highlighted apparent challenges that current approaches are facing and provided our perspectives on their further development.

Minimization-Aware Recursive K*: A Novel, Provable Algorithm that Accelerates Ensemble-Based Protein Design and Provably Approximates the Energy Landscape

Protein design algorithms that model continuous sidechain flexibility and conformational ensembles better approximate the in vitro and in vivo behavior of proteins. The previous state of the art, iMinDEE-A*-K*, computes provable ɛ-approximations to partition functions of protein states (e.g., bound vs. unbound) by computing provable, admissible pairwise-minimized energy lower bounds on protein conformations, and using the A* enumeration algorithm to return a gap-free list of lowest-energy conformations. iMinDEE-A*-K* runs in time sublinear in the number of conformations, but can be trapped in loosely-bounded, low-energy conformational wells containing many conformations with highly similar energies. That is, iMinDEE-A*-K* is unable to exploit the correlation between protein conformation and energy: similar conformations often have similar energy. We introduce two new concepts that exploit this correlation: Minimization-Aware Enumeration and Recursive K*. We combine these two insights into a novel algorithm, Minimization-Aware Recursive K* (MARK*), which tightens bounds not on single conformations, but instead on distinct regions of the conformation space. We compare the performance of iMinDEE-A*-K* versus MARK* by running the Branch and Bound over K* (BBK*) algorithm, which provably returns sequences in order of decreasing K* score, using either iMinDEE-A*-K* or MARK* to approximate partition functions. We show on 200 design problems that MARK* not only enumerates and minimizes vastly fewer conformations than the previous state of the art, but also runs up to 2 orders of magnitude faster. Finally, we show that MARK* not only efficiently approximates the partition function, but also provably approximates the energy landscape. To our knowledge, MARK* is the first algorithm to do so. We use MARK* to analyze the change in energy landscape of the bound and unbound states of an HIV-1 capsid protein C-terminal domain in complex with a camelid V_HH, and measure the change in conformational entropy induced by binding. Thus, MARK* both accelerates existing designs and offers new capabilities not possible with previous algorithms.

Computational Analysis of Energy Landscapes Reveals Dynamic Features That Contribute to Binding of Inhibitors to CFTR-Associated Ligand

The CFTR-associated ligand PDZ domain (CALP) binds to the cystic fibrosis transmembrane conductance regulator (CFTR) and mediates lysosomal degradation of mature CFTR. Inhibition of this interaction has been explored as a therapeutic avenue for cystic fibrosis. Previously, we reported the ensemble-based computational design of a novel peptide inhibitor of CALP, which resulted in the most binding-efficient inhibitor to date. This inhibitor, kCAL01, was designed using osprey and evinced significant biological activity in in vitro cell-based assays. Here, we report a crystal structure of kCAL01 bound to CALP and compare structural features against iCAL36, a previously developed inhibitor of CALP. We compute side-chain energy landscapes for each structure to not only enable approximation of binding thermodynamics but also reveal ensemble features that contribute to the comparatively efficient binding of kCAL01. Finally, we compare the previously reported design ensemble for kCAL01 vs the new crystal structure and show that, despite small differences between the design model and crystal structure, significant biophysical features that enhance inhibitor binding are captured in the design ensemble. This suggests not only that ensemble-based design captured thermodynamically significant features observed in vitro, but also that a design eschewing ensembles would miss the kCAL01 sequence entirely.

Protein Design by Provable Algorithms

Protein design algorithms can leverage provable guarantees of accuracy to provide new insights and unique optimized molecules.

Positive multistate protein design

Motivation

Structure-based computational protein design (CPD) plays a critical role in advancing the field of protein engineering. Using an all-atom energy function, CPD tries to identify amino acid sequences that fold into a target structure and ultimately perform a desired function. The usual approach considers a single rigid backbone as a target, which ignores backbone flexibility. Multistate design (MSD) allows instead to consider several backbone states simultaneously, defining challenging computational problems.

Results

We introduce efficient reductions of positive MSD problems to Cost Function Networks with two different fitness definitions and implement them in the Pompd (Positive Multistate Protein design) software. Pompd is able to identify guaranteed optimal sequences of positive multistate full protein redesign problems and exhaustively enumerate suboptimal sequences close to the MSD optimum. Applied to nuclear magnetic resonance and back-rubbed X-ray structures, we observe that the average energy fitness provides the best sequence recovery. Our method outperforms state-of-the-art guaranteed computational design approaches by orders of magnitudes and can solve MSD problems with sizes previously unreachable with guaranteed algorithms.

Availability and implementation

https://forgemia.inra.fr/thomas.schiex/pompd as documented Open Source.

Supplementary information

Supplementary data are available at Bioinformatics online.

Computational design of symmetrical eight-bladed β-propeller proteins

β-Propeller proteins form one of the largest families of protein structures, with a pseudo-symmetrical fold made up of subdomains called blades. They are not only abundant but are also involved in a wide variety of cellular processes, often by acting as a platform for the assembly of protein complexes. WD40 proteins are a subfamily of propeller proteins with no intrinsic enzymatic activity, but their stable, modular architecture and versatile surface have allowed evolution to adapt them to many vital roles. By computationally reverse-engineering the duplication, fusion and diversification events in the evolutionary history of a WD40 protein, a perfectly symmetrical homologue called Tako8 was made. If two or four blades of Tako8 are expressed as single polypeptides, they do not self-assemble to complete the eight-bladed architecture, which may be owing to the closely spaced negative charges inside the ring. A different computational approach was employed to redesign Tako8 to create Ika8, a fourfold-symmetrical protein in which neighbouring blades carry compensating charges. Ika2 and Ika4, carrying two or four blades per subunit, respectively, were found to assemble spontaneously into a complete eight-bladed ring in solution. These artificial eight-bladed rings may find applications in bionanotechnology and as models to study the folding and evolution of WD40 proteins.

A structural homology approach for computational protein design with flexible backbone

Motivation

Structure-based Computational Protein design (CPD) plays a critical role in advancing the field of protein engineering. Using an all-atom energy function, CPD tries to identify amino acid sequences that fold into a target structure and ultimately perform a desired function. Energy functions remain however imperfect and injecting relevant information from known structures in the design process should lead to improved designs.

Results

We introduce Shades, a data-driven CPD method that exploits local structural environments in known protein structures together with energy to guide sequence design, while sampling side-chain and backbone conformations to accommodate mutations. Shades (Structural Homology Algorithm for protein DESign), is based on customized libraries of non-contiguous in-contact amino acid residue motifs. We have tested Shades on a public benchmark of 40 proteins selected from different protein families. When excluding homologous proteins, Shades achieved a protein sequence recovery of 30% and a protein sequence similarity of 46% on average, compared with the PFAM protein family of the target protein. When homologous structures were added, the wild-type sequence recovery rate achieved 93%.

Availability and implementation

Shades source code is available at https://bitbucket.org/satsumaimo/shades as a patch for Rosetta 3.8 with a curated protein structure database and ITEM library creation software.

Supplementary information

Supplementary data are available at Bioinformatics online.

PLUG (Pruning of Local Unrealistic Geometries) removes restrictions on biophysical modeling for protein design

Protein design algorithms must search an enormous conformational space to identify favorable conformations. As a result, those that perform this search with guarantees of accuracy generally start with a conformational pruning step, such as dead-end elimination (DEE). However, the mathematical assumptions of DEE-based pruning algorithms have up to now severely restricted the biophysical model that can feasibly be used in protein design. To lift these restrictions, I propose to prune local unrealistic geometries (PLUG) using a linear programming-based method. PLUG's biophysical model consists only of well-known lower bounds on interatomic distances. PLUG is intended as preprocessing for energy-based protein design calculations, whose biophysical model need not support DEE pruning. Based on 96 test cases, PLUG is at least as effective at pruning as DEE for larger protein designs-the type that most require pruning. When combined with the LUTE protein design algorithm, PLUG greatly facilitates designs that account for continuous entropy, large multistate designs with continuous flexibility, and designs with extensive continuous backbone flexibility and advanced nonpairwise energy functions. Many of these designs are tractable only with PLUG, either for empirical reasons (LUTE's machine learning step achieves an accurate fit only after PLUG pruning), or for theoretical reasons (many energy functions are fundamentally incompatible with DEE).

Strategies for designing non-natural enzymes and binders

The design of tailor-made enzymes is a major goal in biochemical research that can result in wide-range applications and will lead to a better understanding of how proteins fold and function. In this review we highlight recent advances in enzyme and small molecule binder design. A focus is placed on novel strategies for the design of scaffolds, developments in computational methods, and recent applications of these techniques on receptors, sensors, and enzymes. Further, the integration of computational and experimental methodologies is discussed. The outlined examples of designed enzymes and binders for various purposes highlight the importance of this topic and underline the need for tailor-made proteins.

Adaptive landscape flattening in amino acid sequence space for the computational design of protein:peptide binding

For the high throughput design of protein:peptide binding, one must explore a vast space of amino acid sequences in search of low binding free energies. This complex problem is usually addressed with either simple heuristic scoring or expensive sequence enumeration schemes. Far more efficient than enumeration is a recent Monte Carlo approach that adaptively flattens the energy landscape in sequence space of the unbound peptide and provides formally exact binding free energy differences. The method allows the binding free energy to be used directly as the design criterion. We propose several improvements that allow still more efficient sampling and can address larger design problems. They include the use of Replica Exchange Monte Carlo and landscape flattening for both the unbound and bound peptides. We used the method to design peptides that bind to the PDZ domain of the Tiam1 signaling protein and could serve as inhibitors of its activity. Four peptide positions were allowed to mutate freely. Almost 75 000 peptide variants were processed in two simulations of 10⁹ steps each that used 1 CPU hour on a desktop machine. 96% of the theoretical sequence space was sampled. The relative binding free energies agreed qualitatively with values from experiment. The sampled sequences agreed qualitatively with an experimental library of Tiam1-binding peptides. The main assumption limiting accuracy is the fixed backbone approximation, which could be alleviated in future work by using increased computational resources and multi-backbone designs.

iCFN: an efficient exact algorithm for multistate protein design

Motivation

Multistate protein design addresses real-world challenges, such as multi-specificity design and backbone flexibility, by considering both positive and negative protein states with an ensemble of substates for each. It also presents an enormous challenge to exact algorithms that guarantee the optimal solutions and enable a direct test of mechanistic hypotheses behind models. However, efficient exact algorithms are lacking for multistate protein design.

Results

We have developed an efficient exact algorithm called interconnected cost function networks (iCFN) for multistate protein design. Its generic formulation allows for a wide array of applications such as stability, affinity and specificity designs while addressing concerns such as global flexibility of protein backbones. iCFN treats each substate design as a weighted constraint satisfaction problem (WCSP) modeled through a CFN; and it solves the coupled WCSPs using novel bounds and a depth-first branch-and-bound search over a tree structure of sequences, substates, and conformations. When iCFN is applied to specificity design of a T-cell receptor, a problem of unprecedented size to exact methods, it drastically reduces search space and running time to make the problem tractable. Moreover, iCFN generates experimentally-agreeing receptor designs with improved accuracy compared with state-of-the-art methods, highlights the importance of modeling backbone flexibility in protein design, and reveals molecular mechanisms underlying binding specificity.

Availability and implementation

https://shen-lab.github.io/software/iCFN.

Supplementary information

Supplementary data are available at Bioinformatics online.

CATS (Coordinates of Atoms by Taylor Series): protein design with backbone flexibility in all locally feasible directions

Motivation

When proteins mutate or bind to ligands, their backbones often move significantly, especially in loop regions. Computational protein design algorithms must model these motions in order to accurately optimize protein stability and binding affinity. However, methods for backbone conformational search in design have been much more limited than for sidechain conformational search. This is especially true for combinatorial protein design algorithms, which aim to search a large sequence space efficiently and thus cannot rely on temporal simulation of each candidate sequence.

Results

We alleviate this difficulty with a new parameterization of backbone conformational space, which represents all degrees of freedom of a specified segment of protein chain that maintain valid bonding geometry (by maintaining the original bond lengths and angles and ω dihedrals). In order to search this space, we present an efficient algorithm, CATS, for computing atomic coordinates as a function of our new continuous backbone internal coordinates. CATS generalizes the iMinDEE and EPIC protein design algorithms, which model continuous flexibility in sidechain dihedrals, to model continuous, appropriately localized flexibility in the backbone dihedrals ϕ and ψ as well. We show using 81 test cases based on 29 different protein structures that CATS finds sequences and conformations that are significantly lower in energy than methods with less or no backbone flexibility do. In particular, we show that CATS can model the viability of an antibody mutation known experimentally to increase affinity, but that appears sterically infeasible when modeled with less or no backbone flexibility.

Availability and implementation

Our code is available as free software at https://github.com/donaldlab/OSPREY_refactor.

Supplementary information

Supplementary data are available at Bioinformatics online.

BBK* (Branch and Bound Over K*): A Provable and Efficient Ensemble-Based Protein Design Algorithm to Optimize Stability and Binding Affinity Over Large Sequence Spaces

Computational protein design (CPD) algorithms that compute binding affinity, K_a, search for sequences with an energetically favorable free energy of binding. Recent work shows that three principles improve the biological accuracy of CPD: ensemble-based design, continuous flexibility of backbone and side-chain conformations, and provable guarantees of accuracy with respect to the input. However, previous methods that use all three design principles are single-sequence (SS) algorithms, which are very costly: linear in the number of sequences and thus exponential in the number of simultaneously mutable residues. To address this computational challenge, we introduce BBK*, a new CPD algorithm whose key innovation is the multisequence (MS) bound: BBK* efficiently computes a single provable upper bound to approximate K_a for a combinatorial number of sequences, and avoids SS computation for all provably suboptimal sequences. Thus, to our knowledge, BBK* is the first provable, ensemble-based CPD algorithm to run in time sublinear in the number of sequences. Computational experiments on 204 protein design problems show that BBK* finds the tightest binding sequences while approximating K_a for up to 10⁵-fold fewer sequences than the previous state-of-the-art algorithms, which require exhaustive enumeration of sequences. Furthermore, for 51 protein-ligand design problems, BBK* provably approximates K_a up to 1982-fold faster than the previous state-of-the-art iMinDEE/[Formula: see text]/[Formula: see text] algorithm. Therefore, BBK* not only accelerates protein designs that are possible with previous provable algorithms, but also efficiently performs designs that are too large for previous methods.

LUTE (Local Unpruned Tuple Expansion): Accurate Continuously Flexible Protein Design with General Energy Functions and Rigid Rotamer-Like Efficiency

Most protein design algorithms search over discrete conformations and an energy function that is residue-pairwise, that is, a sum of terms that depend on the sequence and conformation of at most two residues. Although modeling of continuous flexibility and of non-residue-pairwise energies significantly increases the accuracy of protein design, previous methods to model these phenomena add a significant asymptotic cost to design calculations. We now remove this cost by modeling continuous flexibility and non-residue-pairwise energies in a form suitable for direct input to highly efficient, discrete combinatorial optimization algorithms such as DEE/A* or branch-width minimization. Our novel algorithm performs a local unpruned tuple expansion (LUTE), which can efficiently represent both continuous flexibility and general, possibly nonpairwise energy functions to an arbitrary level of accuracy using a discrete energy matrix. We show using 47 design calculation test cases that LUTE provides a dramatic speedup in both single-state and multistate continuously flexible designs.

Algorithms for protein design

Computational structure-based protein design programs are becoming an increasingly important tool in molecular biology. These programs compute protein sequences that are predicted to fold to a target structure and perform a desired function. The success of a program's predictions largely relies on two components: first, the input biophysical model, and second, the algorithm that computes the best sequence(s) and structure(s) according to the biophysical model. Improving both the model and the algorithm in tandem is essential to improving the success rate of current programs, and here we review recent developments in algorithms for protein design, emphasizing how novel algorithms enable the use of more accurate biophysical models. We conclude with a list of algorithmic challenges in computational protein design that we believe will be especially important for the design of therapeutic proteins and protein assemblies.

Computational protein design for given backbone: recent progresses in general method-related aspects

To achieve high success rate in protein design requires a reliable sequence design method to find amino acid sequences that stably fold into a desired backbone structure. This problem is addressed by computational protein design through the approach of energy minimization. Here we review recent method progresses related to improving the accuracy of this approach. First, the quality of the energy model is a key factor. Second, with structure sensitive energy functions, whether and how backbone flexibility is considered can have large effects on design accuracy, although usually only small adjustments of the backbone structure itself are involved. Third, the effective accuracy of design results can be boosted by post-processing a small number of designed sequences with complementary models that may not be efficient enough for full sequence optimization. Finally, computational method development will benefit greatly from increasingly efficient experimental approaches that can be applied to obtain extensive feedbacks.

Comparing three stochastic search algorithms for computational protein design: Monte Carlo, replica exchange Monte Carlo, and a multistart, steepest-descent heuristic

Computational protein design depends on an energy function and an algorithm to search the sequence/conformation space. We compare three stochastic search algorithms: a heuristic, Monte Carlo (MC), and a Replica Exchange Monte Carlo method (REMC). The heuristic performs a steepest-descent minimization starting from thousands of random starting points. The methods are applied to nine test proteins from three structural families, with a fixed backbone structure, a molecular mechanics energy function, and with 1, 5, 10, 20, 30, or all amino acids allowed to mutate. Results are compared to an exact, "Cost Function Network" method that identifies the global minimum energy conformation (GMEC) in favorable cases. The designed sequences accurately reproduce experimental sequences in the hydrophobic core. The heuristic and REMC agree closely and reproduce the GMEC when it is known, with a few exceptions. Plain MC performs well for most cases, occasionally departing from the GMEC by 3-4 kcal/mol. With REMC, the diversity of the sequences sampled agrees with exact enumeration where the latter is possible: up to 2 kcal/mol above the GMEC. Beyond, room temperature replicas sample sequences up to 10 kcal/mol above the GMEC, providing thermal averages and a solution to the inverse protein folding problem. © 2016 Wiley Periodicals, Inc.