Equivalent Distance Geometry Error for Molecular Conformation Comparison

1 INTRODUCTION . Small organic molecules , conventionally a chemical or biomedical topic , are now being widely studied by artificial intelligence researchers . Based on prior natural science knowledge , machine learning models aid in multiple tasks on molecules including 3-D Quantitative Structure-Activity Relationship ( QSAR ) analysis , virtual screening for drug discovery , and protein-ligand binding analysis . Before any real industrial applications , molecular conformation generation draws particular concern . The task is also challenging as ( 1 ) conformation contains the 3-D geometry which can be translated or rotated , making it complicated to compare the generated one with ground-truth ; ( 2 ) the number of molecules found in nature presents combination explosive growth , so the procedure of generating conformations is supposed to be fast enough for applicability . In the past few years , different machine learning models have been proposed to handle the molecular conformation generation task . Generally , the molecules are featurized into token sequences ( e.g . SMILES ( Weininger , 1988 ) ) or 2-D molecular graphs 1 ( Gilmer et al. , 2017 ) to feed the neural networks , while the way to obtain conformations is optional : ( 1 ) Straight-forward methods directly generate the 3-D coordinates of atoms from the inputs ; ( 2 ) Step-wise methods first predict distances between atom pairs and then rebuild the conformations from the distances either via discrete algorithms ( e.g . EDG ( Havel , 1998 ) ) or bi-level ( Xu et al. , 2021 ) . As the expense of avoiding the troublesome comparison between conformations , step-wise methods usually adopt more complex architectures and endure relatively expensive time and space costs . Therefore , based on its innate advantage of high efficiency , Straight-forward methods require powerful assistance to perform equally great as step-wise ones . And a well-designed loss function can be helpful . In recent studies , the simplest way to compare the conformations is to represent the conformations with atom position matrices R ( G ) ∈ Rn×3 and calculate their Root-Mean-Squared Deviations ( RMSD ) . Since conformation is invariant to spatial permutations ( e.g . translation and rotation ) , a universal approach is to first find an optimal alignment . As an instance , Kabsch algorithm ( Kabsch , 1976 ) provides a differentiable method to align two given conformations . We referred to the socalculated RMSD as Kabsch-RMSD . However , using Kabsch-RMSD as the loss of neural networks 1The atoms and bonds inside the molecule form the vertexes and edges respectively . is neither efficient nor effective : it needs to calculate the rotation matrix for every single conformation , which stifles batch processing and increases time cost ; meanwhile , the rotation matrix is vulnerable to misplaced atoms and poorly-predicted regions ( Mariani et al. , 2013 ) , so the RMSD between aligned and reference conformation can not correctly reflect molecular similarity when input conformations are less likely . To further improve the applicability of loss function on molecular conformation comparing , Distance Geometry ( DG ) is proposed to encode a conformation into a vector of distances among the atoms , where the comparison is applied . The main idea of DG is to indirectly optimize bond lengths D , bond angles Φ and dihedral angles2 Ψ by optimizing the distances among 1-hop , 2-hop and more distant neighbors respectively in the molecular graph . However , when balancing time and effects , traditional distance geometry methods are faced with a dilemma : ( 1 ) If too many distant neighbors in the graph are considered , the loss calculation will be time-consuming . ( 2 ) If only close atompairs ( e.g . 3-hop at most ) are considered , the factors in the geometry will be treated unfairly . Based on the theoretical and experimental demonstration in Appendix B.1 , the gradient allocated on bond angles Φ and dihedral angles Ψ will become much smaller than that on bond lengths D when optimizing atom-pair distances , resulting in contorted angles and poor local geometry . Therefore , the equivalence in optimizing different factors should be stressed in the comparison . To push the boundary of efficient and effective loss function for molecular conformation generation , we propose Equivalent Distance Geometry Error ( EDGE ) . Firstly , EDGE exploits the essential factors inside the conformation geometry , including a series of bond lengths D , bond angles Φ and dihedral angles Ψ . Then the factors are jointly optimized with certain weights to guarantee that they are fairly treated . To avoid the high complexity of directly calculating the angles , we introduce Taylor Expansion to our method to approximate the angles with linear transformations of atom-pair distances within 3-hop . Extensive experiments prove that the improved version of EDGE performs as fast as optimizing 1/2/3-hop atom-pair distances and achieves better performance than existing popular loss functions on conformation comparison when applied to straight-forward backbones . To conclude , our main contributions are threefold : • We propose a novel loss function , EDGE , for straight-forward models on molecular conformation generation , which optimizes the essential factors in the conformation geometry equivalently , realizing high consistency of generation . • To avoid the high complexity and gradient explosion caused by direct calculation of angles , we introduce Taylor Expansion and Multiplier Truncation which achieves efficient model optimization with improved EDGE and guarantees its applicability . • We conduct comprehensive experiments on several recently proposed benchmarks , including QM9 ( Wu et al. , 2018 ) , GEOM-QM9 and GEOM-Drugs ( Axelrod & GómezBombarelli , 2020 ) . The outstanding performance on different metrics shows that EDGE is powerful on both definite conformation prediction and multiple conformation generation tasks , compared with existing loss functions . Besides , EDGE ’ s efficiency is guaranteed via theoretical analysis and empirical test . 2 PROPOSED METHOD . 2.1 PRELIMINARIES . Notations In the statements and equations below , we use italic letters for scalars and indices , bold lower-case letters for ( column ) vectors , bold upper-case letters for matrices , calligraphic letters for sets , and normal letters for annotations . Given a certain molecule , M = ( V , E , n , m , Xv , Xe ) is the corresponding molecular graph . Here , V is the set of all n atoms , E ⊂ V × V is the set of all m chemical bonds , Xv = ( xv1 , ... , x v n ) > ∈ Rn×dv is the matrix of atomic features , and Xe = ( xe1 , ... , x e m ) > ∈ Rm×de that of bond features . G is the geometry of its conformation and R ( G ) is the matrix of atom positions . d ( u , v ; M ) is the graph distance between atom u , v inM and d ( u , v ; G ) is their Euclidean distance in G. A conformation geometry G has three types of factors D ( G ) , Φ ( G ) , Ψ ( G ) , which refer to the set of bond length d , bond angle φ and dihedral angle ψ 2Dihedral angle is referred as torsion angle in some works . respectively . For operators , ⊕ is the operator of concatenations and ‖ · ‖ denotes the l2 norm of the input vector . Problem Definition We consider following straight-forward molecular conformation generation process : the models input a molecular graphM and output a molecular conformation R ( G ) . Our task is to find an efficient and effective loss function to calculate the differential discrepancy between generated and reference conformation G , Ĝ , which can be utilized to optimize the models backwardly . 2.2 EQUIVALENCE IN MOLECULAR GEOMETRY . When applied to straight-forward backbones on molecular conformation generation , existing DGbased loss functions such as Conn-k ( Yang et al. , 2021 ) can ’ t fairly distribute the gradient to different factors D ( G ) , Φ ( G ) and Ψ ( G ) in the conformation during optimization , resulting in contorted geometry . Simply calculating the average error between the generated D ( G ) , Φ ( G ) , Ψ ( G ) and the reference ones has the same deficiency , because the three types of factors are in different dimensions and equivalence can hardly be guaranteed while training the model . To fairly treat all the factors , it is necessary to jointly optimize them with certain weights which are determined by their corresponding degrees of freedom and value distributions in the dataset . Degree of Freedom Consider a 3-D molecular geometry G of a molecule with n atoms , it has fG = 3n−6 degree of freedom due to the existence of rotation and translation ( Simm & HernandezLobato , 2020 ) . It means that a geometry can be just recovered from a certain combination of bond lengths , bond angles and dihedral angles with a total number 3n−6 . Note that three types of factors occupy fD , fΦ , fΨ degrees of freedom out of fG respectively , and we have3 : fG = fD + fΦ + fΨ = 3n− 6 ( 1 ) To completely perceive the local structure of molecular geometries , the number of bond lengths , bond angles and dihedral angles sampled in the geometry ( i.e . f̂D , f̂Φ , f̂Ψ ) may exceed their exact degrees of freedom fD , fΦ , fΨ during optimization . So we need to balance them through a simple proportion ( e.g . fD/f̂D for bond lengths D ) . Value Distribution When jointly optimizing bond lengths D , bond angles Φ and torsion angles Ψ , which are in different dimensions , it can hardly achieve equivalence to manually give hyperparameters to balance the gradients . Therefore , we rescale D , Φ , Ψ with the reciprocals of their deviations σD , σΦ , σΨ respectively to uniformly approximate a Gaussian distribution with σ = 1 . 3The calculation of degrees of freedom for each kind of factors can be found in Appendix C.1 . 2.3 OUR METHOD : EQUIVALENT DISTANCE GEOMETRY ERROR . We propose Equivalent Distance Geometry Error ( EDGE ) as a loss function for straight-forward methods of molecular conformation generation . As illustrated in Fig.1 , EDGE calculates the discrepancy between generated conformation G and reference conformation Ĝ following4 : L ( G , Ĝ ) = λ2D ∑ d∈D ( G ) d̂∈D ( Ĝ ) ( d− d̂ ) 2 + λ2Φ ∑ φ∈Φ ( G ) φ̂∈Φ ( Ĝ ) ( φ− φ̂ ) 2 + λ2Ψ ∑ ψ∈Ψ ( G ) ψ̂∈Ψ ( Ĝ ) ( ψ − ψ̂ ) 2 where λD = fD f̂D 1 σD λΦ = fΦ f̂Φ 1 σΦ λΨ = fΨ f̂Ψ 1 σΨ ( 2 ) To be more efficient , we can encode a molecular conformation into a vector z : zD = [ λDd|d ∈ D ( G ) ] zΦ = [ λΦφ|φ ∈ Φ ( G ) ] zΨ = [ λΨψ|ψ ∈ Ψ ( G ) ] z = zD ⊕ zΦ ⊕ zΨ ( 3 ) Then Eq.2 can be represented as the discrepancy between two encodings of conformation , which achieves fast comparison among multiple conformations : L ( G , Ĝ ) = ‖z − ẑ‖2 ( 4 ) ( a ) Calculation of D For a conformation G , we first calculate the distance matrix D : Di , j = ‖R ( G ) i −R ( G ) j‖2 ( 5 ) where R ( G ) i ∈ R3 is the 3-D position of i-th atom in G. Then the value of sampled bond lengths D can be found in D. Calculation of Φ For a certain bond angle φ determined by three atoms in G , a , b are the lengths of its two sides and c is the length of the side opposite it , which is shown in Fig.2 ( a ) . As a , b , c can be found in D , the value of φ can be calculated through the cosine theorem as φ = arccos a2 + b2 − c2 2ab ( 6 ) Calculation of Ψ For a certain dihedral angle ψ determined by four sequentially-connected atoms in G , a , b , c are the lengths of the sides in the atom chain and d , e , f are 2/3-hop distances among the atoms , which is shown in Fig.2 ( b ) . Following Euler ’ s formula in the tetrahedron , we have5 : ψ = arcsin √ 4a2b2e2 − b2t22 − a2r22 − e2t21 + r2t1t2 4a2b2c2 − a2r21 ( 7 )

Summary: The work  introduces a custom loss function for the molecular conformation similarity. The authors combine this loss function with the PhysNet and GraphDG models to either predict a single conformation or to generate an ensemble of conformations. Other loss functions, including Naive RMSD, Kabsch algorithm, Conn-k and lDDT-k are compared under the same settings.  The main contributions of the paper are as follow: authors proposed a novel loss function for conformation generation, that can be used in *straight-forward* models, also provide Taylor Expansion and Multiplier Truncation techniques to speed up this metric.