QSD – Quadratic Shape Descriptors Surface Matching and Molecular Docking Using Quadratic Shape Descriptors Goldman BB, Wipke WT. Quadratic Shape Descriptors. 1. Rapid Superposition of Dissimilar Molecules Using Geometrically Invariant Surface Descriptors. J.Chem. Inf. Comput. Sci., 40 (3), , 2000

QSD idea Define a geometrical invariant representation of small surface sections (if two molecules have a similar surface region then its small parts are also similar). In case a geometrical invariant allows to define a reference frame then the number of all superpositions is n*m. n (m) - number of invariants in the first (second) molecule Principle curvature and principle directions provide an elegant formalism that captures these notions.

Reminder: curvature properties |k 1 | > |k 2 | > |k 3 | =0

k normal curvature - curvature of normal section at p Principal Curvatures: k max, k min - normal curvatures with maximal-minimal values Principal Directions: λ max, λ min - tangent vectors associated with principal curvatures. k max ≠ k min → λ max ┴ λ min p (a surface curve)

Molecular Surface Calculation The preprocessing stage of the algorithm computes the molecular surface of a molecule by using the original Connolly MS program. Critical Points Calculation The critical points of the surface as defined by Lin et al.40 are calculated. These critical points are the center of gravity of each face of the Connolly surface projected back onto the surface.

Critical Points To reduce the number of the critical points used to describe a molecule, the critical points associated with the toroidal sections (light purple) of the surface are not used.

S = {p 1,..., p n }, where p = (v, n) is composed of the surface point location v in three-dimensional space and n is the unit vector normal to the surface at p.v C = {c 1,..., c m } - set of critical points, where c i in S Surface neighborhood around c:

N is transformed s.t. : c.v = (0,0,0) c.n = (0,0,1) Hessian matrix (second fundamental form): Redefine points N: Local principal curvatures and directions are eigenvalues and eigenvectors, respectively, of the II matrix.

Calculate matrix II by fitting the points of N to the second order part of the Taylor expansion of w: Notice: w(0,0)=0 and so the first derivatives. w(u,v) ~

The least-squares estimator of is given by Finally, two right-handed orthogonal coordinate systems can be constructed from the local principal curvature directions:

Principal curvature directions are in cyan.

Shape Index (κ min, λ min ) and (κ max, λ max ) represent the local principal curvatures and directions of the surface patch.The shape index represents the degree of concavity of a local surface section and is defined by :

Shape Index Similarity The shape index provides a convenient mechanism for determining the similarity between two section of surface. The Similarity measure for two surface patches with shape indexes S 1 and S 2 is : 1.0 – shapes are identical 0.0 – shapes are exactly opposite

Total Shape Similarity Score Y The score is simply a summation of the individual similarity scores for each pair of matching descriptors. ML = {ml 0,…,ml n }, where ml = (r i,l j ) indicates that i th QSD on the receptor matchs the j th QSD on the ligand. S(ml.x) represent the value of the shape index S for the match list QSD ml.x.

QSD Preprocessing Algorithm. Input: MCoordinates of Molecule ρDistance parameter Variables: AAlignment Matrix SShape Index Algorithm: Create molecular surface for molecule M the Connolly algorithm. Calculate critical points C = {c 1,…,c m } of surface using Lin’s method. for each c  C (c,S,A)  Create QSD at point c with distance range ρ store (c,S,A) end

Surface matching phase This phase of the algorithm commences with the input of the ligand and proteins atomic coordinates along with the set of quadratic shape descriptors approximating threir molecular surface. The surface of the active site has been inverted, and shape complementary between the ligand and receptor surfaces is referred to as shape similarity. An additional input parameter, the shape filter ΔS, is used as a filter to determine the extant of similarity between two surface sections.

Surface matching phase Input: M L,M R Coordinates of Ligand and receptor Q L,Q R QSD set describing Ligand and receptor ΔSShape Filter Algorithm: for each q l  Q L for each q r  Q R if (|q l.S – q r.S|)  ΔS) Dock Q L to Q R as dictated by alignment of q l to q r if (sufficient QSDs from Q R superimpose on QSD from Q L ) Dock M L onto M R as dictated by alignment of q l onto q r if (acceptable steric clash* between M R and transformed M L ) store docking end if end if end if end for end for *Steric collisions are quickly evaluated using a three-dimensional grid-based procedure.

Scoring The scoring module uses three types of scoring routines to prioritize the computed dockings: Empirical estimate of Δg binding (using Bohm’s algorithm). Measure of shape similarity Υ. Clustering algorithm.

Matching & Scoring Phase Complexity Let n,m represent the number of QSDs used to describe the shape of the target molecule and the moving molecule. The total number of the dockings calculated O(mn). For each docking calculated, all of the QSDs in the moving set are transformed, matched with QSDs in the target set and then the surface similarity score assessed. The total complexity of the matching phase is thus O(nm 2 ).

Create Molecular Surface for Ligand and Receptor High level flow chart for QSD docking algorithm

Create Molecular Surface for Ligand and Receptor High level flow chart for QSD docking algorithm Calculate Molecular Surface Critical Points

Create Molecular Surface for Ligand and Receptor High level flow chart for QSD docking algorithm Calculate Molecular Surface Critical Points Calculate Quadratic Shape Descriptors Preprocessing

Create Molecular Surface for Ligand and Receptor High level flow chart for QSD docking algorithm Calculate Molecular Surface Critical Points Calculate Quadratic Shape Descriptors Dock Ligands To Receptor Using QSD Preprocessing

Create Molecular Surface for Ligand and Receptor High level flow chart for QSD docking algorithm Calculate Molecular Surface Critical Points Calculate Quadratic Shape Descriptors Dock Ligands To Receptor Using QSD Score Successful Dockings Preprocessing Object Recognition

Preprocessing Times

Crystallographic Scores

QSD Matching Results

QSD Docking Results on Ligand Into Protein and Comparison With Cocrystalized Structure Position

Comparison of QSDock a Times to DOCK2 and Geometric Hashing (GH)

Conclusion QSDock is capable of reproducing the crystallographically determined orientations using only shape. QSD for shape-based docking dretically reduces the computational complexity of the docking problem.

Preprocessing The preprocessing algorithm accepts as input the three-dimensional coordinates of a molecule and calculate the set of QSDs describing its surface shape. The preprocessing is done only once for each molecule.

Shape Descriptors Calculation A QSD is a macroscopic interpretation of the classical differential geometric surface properties of principal curvatures and principal directions. A QSD is calculated by least-squares fitting of a quadratic surface to a 2.0 Å circular patch of molecular surface surrounding a critical point. After the least-squares fitting procedure, the principal curvatures and directions of the surface at p are calculated.