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A Probabilistic Programming Approach to Protein Structure Superposition

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A Probabilistic Programming Approach to Protein Structure Superposition bioRxiv preprint doi: https://doi.org/10.1101/575431; this version posted June 4, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. A Probabilistic Programming Approach to Protein Structure Superposition Lys Sanz Moreta1*, Ahmad Salim Al-Sibahi1,2*, Douglas Theobald3, William Bullock4, Basile Nicolas Rommes4, Andreas Manoukian4, and Thomas Hamelryck1,4* 1 Department of Computer Science. University of Copenhagen, Denmark 2 Skanned.com, Denmark 3 Department of Biochemistry. Brandeis University. Waltham, MA 02452, USA 4 The Bioinformatics Centre. Section for Computational and RNA Biology. University of Copenhagen, Denmark * Corresponding authors. [email protected] (Lys Sanz Moreta), [email protected] (Ahmad Salim Al-Sibahi), [email protected] (Thomas Hamelryck) Abstract—Optimal superposition of protein structures is cru- assume that all atoms have equal variance (homoscedastic- cial for understanding their structure, function, dynamics and ity) and are uncorrelated. This is problematic in the case evolution. We investigate the use of probabilistic programming of proteins with flexible loops or flexible terminal regions, to superimpose protein structures guided by a Bayesian model. Our model THESEUS-PP is based on the THESEUS model, a where the atoms can posit high variance. Here we present probabilistic model of protein superposition based on rotation, a Bayesian model that is based on the previously reported translation and perturbation of an underlying, latent mean THESEUS model [4]–[6]. THESEUS is a probabilistic model structure. The model was implemented in the deep probabilistic of protein superposition that allows for regions with low and programming language Pyro. Unlike conventional methods that high variance (heteroscedasticity), corresponding respectively minimize the sum of the squared distances, THESEUS takes into account correlated atom positions and heteroscedasticity (i.e., to conserved and variable regions [4], [5]. THESEUS assumes atom positions can feature different variances). THESEUS per- that the structures which are to be superimposed are translated, forms maximum likelihood estimation using iterative expectation- rotated and perturbed observations of an underlying latent, maximization. In contrast, THESEUS-PP allows automated max- mean structure M. imum a-posteriori (MAP) estimation using suitable priors over In contrast to the THESEUS model which features max- rotation, translation, variances and latent mean structure. The results indicate that probabilistic programming is a powerful new imum likelihood parameter estimation using iterative ex- paradigm for the formulation of Bayesian probabilistic models pectation maximization, we formulate a Bayesian model concerning biomolecular structure. Specifically, we envision the (THESEUS-PP) and perform maximum a-posteriori (MAP) use of the THESEUS-PP model as a suitable error model or parameter estimation. We provide suitable prior distributions likelihood in Bayesian protein structure prediction using deep over the rotation, the translations, the variances and the la- probabilistic programming. Index Terms—protein superposition, Bayesian modelling, deep tent, mean model. We implemented the entire model in the probabilistic programming, protein structure prediction deep probabilistic programming language Pyro [7], using its automatic inference features. The results indicate that deep probabilistic programming readily allows the implementation, I. INTRODUCTION estimation and deployment of advanced non-Euclidean models In order to compare biomolecular structures, it is necessary relevant to structural bioinformatics. Specifically, we envision to superimpose them onto each other in an optimal way. The that THESEUS-PP can be used as a likelihood function in standard method minimizes the sum of the squared distances Bayesian protein structure prediction using deep probabilistic (root mean square deviation, RMSD) between the matching programming. atom pairs. This can be easily accomplished by shifting the centre of mass of the two proteins to the origin and obtaining II. METHODS the optimal rotation using singular value decomposition [1] or A. Overall model quaternion algebra [2], [3]. These methods however typically According to the THESEUS model [4], each observed protein structure Xn is a noisy observation of a rotated and bioRxiv preprint doi: https://doi.org/10.1101/575431; this version posted June 4, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. u To ensure identifiability, one (arbitrary) protein X1 is as- sumed to be a fixed noisy observation of the structure M: X1 ∼ MN (M; U; V): (4) M q The other protein X2 is assumed to be a noisy observation of the rotated as well as translated mean structure M: X2 ∼ MN (MR − 1N T; U; V): (5) U T R Thus, the model uses the same covariance matrices U and V for the matrix-normal distributions of both X1 and X2. X1 X2 B. Bayesian posterior The graphical model of THESEUS-PP is shown in Figure Fig. 1: The THESEUS-PP model as a Bayesian graphical 1. The corresponding Bayesian posterior distribution is model. M is the latent, mean structure, which is an N- by-3 coordinate matrix, where N is the number of atoms. p(R; T; M; UjX ; X ) / T is the translation. q is a unit quaternion calculated from 1 2 three random variables u sampled from the unit interval. R p(X1; X2jM; R; T; U)p(M)p(T)p(R)p(U) = is the corresponding rotation matrix. U is the among-row p(X1jM; U)p(X2jMR − 1N T; U) variance matrix of a matrix-normal distribution. X1 and X2 p(M)p(T)p(R)p(U): (6) are N-by-3 coordinate matrices representing the proteins to be superimposed. Circles denote random variables. A lozenge Below, we specify how each of the priors and the likelihood denotes a deterministic transformation of a random variable. function is formulated and implemented. Shaded circles denote observed variables. Bold capital and bold small letters represent matrices and vectors, respectively. C. Prior for the mean structure Recall that according to the THESEUS-PP model, the atoms of the structures to be superimposed are noisy observations of translated latent, mean structure M with noise E , n a mean structure M. Typically, only Cα atoms are considered and in that case, N corresponds to the number of amino acids. Xn = (M + En)Rn − 1N Tn (1) Hence, we need to formulate a prior distribution over the where n is an index that identifies the protein, R is a rotation latent, mean structure M. matrix, T is a three-dimensional translation, E is the error We use an uninformative prior for M. Each element of and M and X are matrices with the atomic coordinate vectors M is sampled from a Student’s t-distribution with degrees along the rows, of freedom (ν = 1), mean (µ = 0) and a uniform diagonal variance (σ2 = 3). The Student’s t-distribution is chosen 2 m 3 2 x 3 over the normal distribution for reasons of numerical stability: 0 n;0 the fatter tails of the Student’s t-distribution avoid numerical M = 4 ::: 5; Xn = 4 ::: 5 : (2) problems associated with highly variable regions. mN−1 xn;N−1 Another way of representing the model is seeing Xn as D. Prior over the rotation distributed according to a matrix-normal distribution with In the general case, we have no a priori information on mean M and covariance matrices U and V - one concerning the optimal rotation. Hence, we use a uniform prior over the the rows and the other the columns. space of rotations. There are several ways to construct such The matrix-normal distribution can be considered as an a uniform prior. We have chosen a method that makes use of extension of the standard multivariate normal distribution from quaternions [8]. Quaternions are the 4-dimensional extensions vector-valued to matrix-valued random variables. Consider a of the better known 2-dimensional complex numbers. Unit random variable X distributed according to a matrix-normal quaternions form a convenient way to represent rotation matri- distribution with mean M, which in our case is an N×3 matrix ces. For our goal, the overall idea is to sample uniformly from where N is the number of atoms. In this case, the matrix- the space of unit quaternions. Subsequently, the sampled unit normal distribution is further characterized by an N × N row quaternions are transformed into the corresponding rotation covariance matrix U and a 3 × 3 column covariance V. Then, matrices, which establishes a uniform prior over rotations. X ∼ MN (M; U; V) will be equal to p p A unit quaternion q = (w; x; y; z) is sampled in the X = M + UQ V; (3) following way. First, three independent random variables are sampled from the unit interval, where Q is an N × 3 matrix with elements distributed according to the standard normal distribution. u0; u1; u2 ∼ U(0; 1): (7) bioRxiv preprint doi: https://doi.org/10.1101/575431; this version posted June 4, 2019. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC-ND 4.0 International license. Then, four auxiliary deterministic variables (θ1; θ2; r1; r2) are H. Algorithm calculated from u1; u2; u3, Algorithm 1 The Theseus-PP model. Prior over the elements of M θ = 2πu ; (8a) 1 1 mi ∼ t1(0;σM I3), where i indicates the atom position θ2 = 2πu2 (8b) . Prior over the translation p T ∼ N (0; I ) r1 = 1 − u0; (8c) 3 p .
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