Shape Modeling and Matching in Protein Structure Identification Sasakthi Abeysinghe, Tao Ju Washington University, St. Louis, USA Matthew Baker, Wah Chiu Baylor College of Medicine, Houston, USA

Shape Matching Shape comparison – How similar are shape A and shape B? – Application: 3D model retrieval Shape alignment – What is the best alignment of A onto B? – Application: object recognition and registration

Shape Matching Shape comparison – How similar are shape A and shape B? – Application: 3D model retrieval Shape alignment – What is the best alignment of A onto B? – Application: object recognition and registration 3D Protein Image 1D Protein Sequence

Structural Biology Protein: a sequence of amino acids – Folds into a 3D structure in order to interact with other molecules – Protein function derived from its 3D structure Identifying protein structure – Imaging methods: X-ray, NMR – Drawback: can not resolve large assemblies, like viruses. …

Domain Problem Cryo-electron microscopy (Cryo-EM) – Produces 3D density volumes – Drawback: insufficient resolution to resolve atom locations How to determine protein structure in a cryo-EM volume? ?

Shape Matching Formulation Matching 1D protein sequence with 3D density volume Intermediate goal: Matching alpha-helices – One of the basic building blocks in a protein – Identified as cylindrical densities in the volume [Baker 07] How to align the protein sequence with the cryo-EM volume to match the two sets of helices? + ?

Method Overview Compatible shape representation – 1D sequence and 3D volume as attributed relational graphs Graph-based shape matching – A new constrained graph matching problem and an optimal solution – Error-tolerant (inexact) matching

Shape Representation Protein sequence as attributed relation graph – An edge: a helix segment or a non-helix segment Attribute: number of amino acids in the segment – A node: end of a helix of end of the sequence – Add additional edges that skip at most m helix segments To allow matching with a cryo-EM volume that has missing helices

Shape Representation Graph representation of Cryo-EM volume via skeletons – 3D Skeleton [Ju 06] builds connectivity among detected helices – An edge: a detected helix or a skeleton path between two helices Attribute: length of the helix or skeleton path – A node: end of a helix of end of the protein – Add additional edges between helix-ends less than d apart To account for missing helix connectivity in the skeleton

Shape Matching - Problem Finding two matching chains of helices – Same number of edges – Alternating types between non-helix and helix – Minimal attribute matching error Uniqueness of this problem: – Inexact: not all edges/nodes in the two graphs are used in the matched sequence – Constrained: the match must have a linear topology

Shape Matching - Review Previous work on graph matching – Exact matching Graph mono-morphism [Wong 90] Sub-graph isomorphism [Ullmann 76, Cordella 99] – Inexact matching A* search [Nilsson 80], simulated annealing [Herault 90], neural networks [Feng 94], probabilistic relaxation [Christmas 95], genetic algorithms [Wang 97], graph decomposition [Messmer 98] All designed for un-constrained problems where there is no restriction on the topology of the matched sub-graphs.

Shape Matching - Method Key idea: utilize the linearity of chains. Performing depth-first tree-search – Append matching nodes to the incomplete chain with minimal matching error A*-search – Reduce node expansion by estimating future matching error – Optimal if future error estimation is smaller than the actual error. – 3 future error functions are designed {3,3} 63 {2,2} 42 {2,3} 85 {2,4} 92 {2,5} 40 {3,2} 61 {3,4} 72 {3,4} 48 {3,5} 91 {4,3} 99 {4,5} 51 {6,6} 58 {1,1} Sequence GraphVolume Graph

Experimental Setup Test data – Simulated data: 8 proteins (taken from Protein Data Bank) – Authentic data: 3 proteins (produced at Baylor) Test modes – Automatic – With a few user-specified helix correspondences Validation with the actual helix correspondence – Produce a list of candidates sorted by their matching errors – Find out where the actual correspondence ranks in the list

Results - 1 SequenceCryo-EM volume and its skeleton + Top Matching Bluetongue Virus (simulated, 10 helices, 0 missing) – Actual correspondence ranks #1

Results - 2 SequenceCryo-EM volume and its skeleton + Human Insulin Receptor (simulated, 9 helices, 1 missing) – Actual correspondence ranks #1 + Top Matching

Results - 3 SequenceCryo-EM skeleton Top Matching Bacteriophage P22 (authentic, 11 helices, 6 missing) – Actual correspondence ranks #4 + Actual Correspondence

Results - 4 Sequence Cryo-EM skeleton with 2 use-specified helix pairs Top Matching Without user- specification Triose Phosphate Isomerase (simulated, 12 helices, 3 missing) – Before user-specification: actual correspondence not in the candidate list – Given 2 specified helix pairs: actual correspondence ranks #9 + Actual Correspondence

Result - Summary Among the 11 proteins, the correct correspondence ranks among the candidate list computed by our method: – Top 1: 4 proteins – Within top 10: 2 proteins (1 simulated) – Top 1 after user-interaction: 2 proteins (both simulated) 4 specified helix pairs in a 14/20-helix protein. – Within top 10 after user-interaction: 3 proteins 2 specified helix pairs in a 6/9/12-helix protein Performance – Under 4 seconds for proteins with 20 helices – Compare: [Wu 05] uses exhaustive search and takes 16 hours for finding correspondences in proteins with 8 helices

Conclusion Formulate protein structure identification as shape matching – 1D protein sequence vs. 3D cryo-EM density volume – Compatible representation of disparate biological data as graphs Formulate a constrained inexact matching problem and propose an optimal solution – Based on A*-search Validation on simulated and authentic data

Future Work (Bio) Incorporating beta-sheets for improved accuracy – Challenge: the match is no longer a linear chain Integrating homology and ab initio modeling – Utilizing known 3D structure of segments – Refining the alignment by molecular energy minimization

Future Work (CS) Faster graph matching algorithm – Explore variants of A*-search to reduce running time for larger proteins (>20 helices) Better skeleton generation – Generate skeletons directly from gray-scale density volume for iso-value-independent representation – Utilize cell-complex-based skeleton for better skeleton geometry Currently used for topology editing, see [Ju, Zhou and Hu. Siggraph 2007]

Pacific Graphics Hawaii 2007 Oct 29 – Nov 2, in Maui, Hawaii Conference Chair: Ron Goldman Program co-chairs: Marc Alexa, Steven Gortler, Tao Ju

Results - 1 SequenceCryo-EM volume and its skeleton + Top Matching Bluetongue Virus (simulated, 10 helices, 0 missing) – Actual correspondence ranks #1