Energy-based models for atomic-resolution protein conformations

1 INTRODUCTION . Methods for the rational design of proteins make use of complex energy functions that approximate the physical forces that determine protein conformations ( Cornell et al. , 1995 ; Jorgensen et al. , 1996 ; MacKerell Jr et al. , 1998 ) , incorporating knowledge about statistical patterns in databases of protein crystal structures ( Boas & Harbury , 2007 ) . The physical approximations and knowledge-derived features that are included in protein design energy functions have been developed over decades , building on results from a large community of researchers ( Alford et al. , 2017 ) . In this work1 , we investigate learning an energy function for protein conformations directly from protein crystal structure data . To this end , we propose an energy-based model using the Transformer architecture ( Vaswani et al. , 2017 ) , that accepts as inputs sets of atoms and computes an energy for their configuration . Our work is a logical extension of statistical potential methods ( Tanaka & Scheraga , 1976 ; Sippl , 1990 ; Lazaridis & Karplus , 2000 ) that fit energetic terms from data , which , in combination with physically motivated force ∗Work performed during an internship at Facebook 1Data and code for experiments are available at https : //github.com/facebookresearch/ protein-ebm fields , have contributed to the feasibility of de novo design of protein structures and functions ( Kuhlman et al. , 2003 ; Ambroggio & Kuhlman , 2006 ; Jiang et al. , 2008 ; King et al. , 2014 ) . To date , energy functions for protein design have incorporated extensive feature engineering , encoding knowledge of physical and biochemical principles ( Boas & Harbury , 2007 ; Alford et al. , 2017 ) . Learning from data can circumvent the process of developing knowledge-based potential functions by automatically discovering features that contribute to the protein ’ s energy , including terms that are unknown or are difficult to express with rules or simple functions . Since energy functions are additive , terms learned by neural energy-based models can be naturally composed with those derived from physical knowledge . In principle , neural networks have the ability to identify and represent non-additive higher order dependencies that might uncover features such as hydrogen bonding networks . Such features have been shown to have important roles in protein structure and function ( Guo & Salahub , 1998 ; Redzic & Bowler , 2005 ; Livesay et al. , 2008 ) , and are important in protein design ( Boyken et al. , 2016 ) . Incorporation of higher order terms has been an active research area for energy function design ( Maguire et al. , 2018 ) . Evaluations of molecular energy functions have used as a measure of fidelity , the ability to identify native side chain configurations ( rotamers ) from crystal structures where the ground-truth configuration has been masked out ( Jacobson et al. , 2002 ; Bower et al. , 1997 ) . Leaver-Fay et al . ( 2013 ) introduced a set of benchmarks for the Rosetta energy function that includes the task of rotamer recovery . In the benchmark , the ground-truth configuration of the side chain is masked and rotamers ( possible configurations of the side chain ) are sampled and evaluated within the surrounding molecular context ( the rest of the atoms in the protein structure not belonging to the side chain ) . The energy function is scored by comparing the lowest-energy rotamer ( as determined by the energy function ) against the rotamer that was observed in the empirically-determined crystal structure . This work takes an initial step toward fully learning an atomic-resolution energy function from data . Prediction of native rotamers from their context within a protein is a restricted problem setting for exploring how neural networks might be used to learn an atomic-resolution energy function for protein design . We compare the model to the Rosetta energy function , as detailed in Leaver-Fay et al . ( 2013 ) , and find that on the rotamer recovery task , deep learning-based models obtain results approaching the performance of Rosetta . We investigate the outputs and representations of the model toward understanding its representation of molecular energies and exploring relationships to physical properties of proteins . Our results open for future work the more general problem settings of combinatorial side chain optimization for a fixed backbone ( Tuffery et al. , 1991 ; Holm & Sander , 1992 ) and the inverse folding problem ( Pabo , 1983 ) – the recovery of native sequences for a fixed backbone – which has also been used in benchmarking and development of molecular energy functions for protein design ( Leaver-Fay et al. , 2013 ) . 2 BACKGROUND . Protein conformation Proteins are linear polymers composed of an alphabet of twenty canonical amino acids ( residues ) , each of which shares a common backbone moiety responsible for formation of the linear polymeric backbone chain , and a differing side chain moiety with biochemical properties that vary from amino acid to amino acid . The energetic interplay of tight packing of side chains within the core of the protein and exposure of polar residues at the surface drives folding of proteins into stable molecular conformations ( Richardson & Richardson , 1989 ; Dill , 1990 ) . The conformation of a protein can be described through two interchangeable coordinate systems . Each atom has a set of spatial coordinates , which up to an arbitrary rotation and translation of all coordinates describes a unique conformation . In the internal coordinate system , the conformation is described by a sequence of rigid-body motions from each atom to the next , structured as a kinematic tree . The major degrees of freedom in protein conformation are the dihedral rotations ( Richardson & Richardson , 1989 ) , about the backbone bonds termed phi ( φ ) and psi ( ψ ) angles , and the dihedral rotations about the side chain bonds termed chi ( χ ) angles . Within folded proteins , the side chains of amino acids preferentially adopt configurations that are determined by their molecular structure . A relatively small number of configurations separated by high energetic barriers are accessible to each side chain ( Janin et al. , 1978 ) . These configurations are called rotamers . In Rosetta and other protein design methods , rotamers are commonly represented by libraries that estimate a probability distribution over side chain configurations , conditioned on the backbone φ and ψ torsion angles . We use the Dunbrack library ( Shapovalov & Dunbrack Jr , 2011 ) for rotamer configurations . Energy-based models A variety of methods have been proposed for learning distributions of highdimensional data , e.g . generative adversarial networks ( Goodfellow et al. , 2014 ) and variational autoencoders ( Kingma & Welling , 2013 ) . In this work , we adopt energy-based models ( EBMs ) ( Dayan et al. , 1995 ; Hinton & Salakhutdinov , 2006 ; LeCun et al. , 2006 ) . This is motivated by their simplicity and scalability , as well as their compelling results in other domains , such as image generation ( Du & Mordatch , 2019 ) . In EBMs , a scalar parametric energy function Eθ ( x ) is fit to the data , with θ set through a learning procedure such that the energy is low in regions around x and high elsewhere . The energy function maps to a probability density using the Boltzmann distribution : pθ ( x ) = exp ( −Eθ ( x ) ) /Z ( θ ) , where Z = ∫ exp ( −Eθ ( x ) ) dx denotes the partition function . EBMs are typically trained using the maximum-likelihood method ( ML ) , in which θ is adjusted to minimize KL ( pD ( x ) ||pθ ( x ) ) , the Kullback-Leibler divergence between the data and the model distribution . This corresponds to maximizing the log-likelihood of the data under the model : LML ( θ ) = Ex∼pD [ log pθ ( x ) ] = Ex∼pD [ Eθ ( x ) − logZ ( θ ) ] Following Carreira-Perpinan & Hinton ( 2005 ) , the gradient of this objective can be written as : ∇θLML ≈ Ex+∼pD [ ∇θEθ ( x + ) ] − Ex−∼pθ [ ∇θEθ ( x − ) ] Intuitively , this gradient decreases the energy of samples from the data distribution x+ and increases the energy of samples drawn from the model x− . Sampling from pθ can be done in a variety of ways , such as Markov chain Monte Carlo or Gibbs sampling ( Hinton & Salakhutdinov , 2006 ) , possibly accelerated using Langevin dynamics ( Du & Mordatch , 2019 ) . Our method uses a simpler scheme to approximate ∇θLML , detailed in Section 3.4 . 3 METHOD . Our goal is to score molecular configurations of the protein side chains given a fixed target backbone structure . We define an architecture for an energy-based model and describe its training procedure . 3.1 MODEL . The model calculates scalar functions , fθ ( A ) , of size-k subsets , A , of atoms within a protein . Selection of atom subsets In our experiments , we choose A to be nearest-neighbor sets around the residues of the protein and set k = 64 . For a given residue , we construct A to be the k atoms that are nearest to the residue ’ s beta carbon . Atom input representations Each atom in A is described by its 3D Cartesian coordinates and categorical features : ( i ) the identity of the atom ( N , C , O , S ) ; ( ii ) an ordinal label of the atom in the side chain ( i.e . which specific carbon , nitrogen , etc . atom it is in the side chain ) and ( iii ) the amino acid type ( which of the 20 types of amino acids the atom belongs to ) . The coordinates are normalized to have zero mean across the k atoms . Each categorical feature is embedded into 28 dimensions , and the spatial coordinates are projected into 172 dimensions2 , which are then concatenated into a 256-dimensional atom representation . The parameters for the input embeddings and projections of spatial information are learned via training . During training , a random rotation is applied to the coordinates in order to encourage rotational invariance of the model . For visualizations , a fixed number of random rotations ( 100 ) is applied and the results are averaged . Architecture In our proposed approach , fθ ( A ) takes the form of a Transformer model ( Vaswani et al. , 2017 ) that processes a set of atom representations . The self-attention layers allow each atom to attend to the representations of other atoms in the set , modeling the energy of the molecular configuration as a non-linear interaction of single , pairwise , and higher-order interactions between the atoms . The final hidden representations of the Transformer are pooled across the atoms to produce a single vector , which is finally passed to a two-layer multilayer perceptron ( MLP ) that produces the scalar output of the model . Figure 1 illustrates the model . For all experiments , we use a 6-layer Transformer with embedding dimension of 256 ( split over 8 attention heads ) and feed-forward dimension of 1024 . The final MLP contains 256 hidden units . The models are trained without dropout . Layer normalization ( Ba et al. , 2016 ) is applied before the attention blocks . 3.2 PARAMETERIZATION OF PROTEIN CONFORMATIONS . The structure of a protein can be represented by two parameterizations : ( 1 ) absolute Cartesian coordinates of the set of atoms , and ( 2 ) internal coordinates of the atoms encoded as a set of in-plane/out-of-plane rotations and displacements relative to each atom ’ s reference frame . Out-of-plane rotations are parameterized by χ angles which are the primary degrees of freedom in the rotamer configurations . The coordinate systems are interchangeable . 2The high dimensionality of the spatial projection was important to ensure a high weighting on the spatial coordinates , which proved necessary for the model to train reliably .

Summary: The paper proposes an Energy-Based-Model (EBM) for scoring the possible configurations of amino acid side chain conformations in protein structures with known amino acid backbone structure. The energy of the side-chain conformation (the chi-angle) for a given amino acid in the structure is calculated as a function of a local neighbourhood of atoms (A), where each atom is embedded into a 256d vector using its cartesian coordinates, atom identity, atom side-chain position and amino acid identity. The model is trained using approximate likelihood where the model samples are generated using precalculated table (from literature) of possible Chi angles conformations conditioned on the back-bone amino acid identity and back-bone angles. The results seem comprehensive comparing the transformer based energy function parameterization with two sensible baselines as well as the Rosetta energy function  which is the de facto standard tool for these types of calculations. Using rotamer recovery accuracy as the benchmark measure the empirical results are close to performance as the Rosetta energy model however always slightly worse. Further visualizations of the energy levels for different Chi angles seems to support that the learned energy function captures well known characteristics of the rotamer configuration energy landscape.