How Do We Measure Protein Shape? A Pattern Matching Example. A Simple Pattern Matching Algorithm. Comparing Protein Structures II

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1 How Do We Measure Protein Shape? omparing Protein Structures II Protein function is largely based on the proteins geometric shape Protein substructures with similar shapes are likely to share a common function (e.g., a binding site) Guest Lecture for S66 Spring 006 How do we measure protein shape? ompare pictures? Bond angles? Distance matrices? RMSD (Root Mean Square Distance)? Geometric Hashing?? Finding Geometric Patterns using Hashing A Pattern Matching Example Goal: determine if a particular pattern of points appears in a scene of many points Finding a particular pattern of stars in a sky-photo Finding a tumor in a catscan Finding a pattern of atoms (an active site) on the surface of a protein Basic idea: Try every possible way to place the pattern on the scene The position that has the most hits is the best match Hit pattern point is close enough to a scene point Does the pattern appear in the scene? Assume translation, but no rotation Pattern Scene Finding Geometric Patterns Efficiently Don t look at all pattern positions Just look at the ones with at least one hit For this example (translation only), we try each pattern point on each scene point There are mn possibilities where m is the size of the pattern and n is the size of the scene A Simple Pattern Matching Algorithm O(m n ) O(m n) for each pattern point p for each scene point s Determine translation so that p falls on s; count = 0; for each translated pattern point q Find the scene point r closest to q; if (r is close enough to q) count++; Remember which p,s gives max count; Total time = O(m n ) 6

2 How Do We Detect a Hit? Why heck Every Pattern Point? an t afford to check every point in the scene to see if it s close enough to a pattern point One solution is to rasterize the scene Place a grid on the scene Each grid-square is a bin Points are close enough if they are in the same bin Disadvantage: We ve lost accuracy We may miss some nearby points Advantage: Instead of searching for arbitrary points, we search for occupied bins The bins follow a regular pattern The bins can be stored as an array! We could save time (a factor of m) by choosing just one pattern point p to translate to each scene point s, but Some pattern points may be missing from the scene Due to occlusion or errors in the scene data, for example The pattern may not align well with the bins 7 8 An Improved Pattern Matching Algorithm Is the Improved Algorithm Better? O(b) O(n) O(m n) lear all the bins; for each scene point s Mark the bin for s as occupied; for each pattern point p for each scene point s Determine translation so that p falls on s; count = 0; for each translated pattern point q if (q falls in an occupied bin) count++; Remember which p,s gives max count; Total time is now O(b+m n) instead of O(m n ) b is the number of bins Note: m is small; n is big But we have lost accuracy and the array of bins can be expensive an be a very large array Most bins are empty anyway Note: could use a bit-array, but it s still large Total time = O(b+m n) 9 10 Improvement Using Hashing Both Translation and Rotation? Idea: Don t keep an array of bins Instead, place the occupied bins in a hash table Hash on the bin s array indices Result: Initialization takes time O(n) instead of O(b) Still fast to test whether q falls in an occupied bin A single test takes expected time O(1) instead of worst-case time O(1) Algorithm now runs in expected time O(m n) instead of worst-case time O(b+m n) A single pattern point p and single scene point s aren t enough to determine a transformation Instead, we need pattern points and scene point The first (p,s) pair gives the translation; the second gives the rotation This leads to an algorithm that runs in expected time O(m n ) If more-complicated transformations are allowed then more points are needed and the time bound grows Example: For D translation and rotation, three points are needed and the time bound is O(m n ) 11 1

3 Geometric Hashing Problem: A running time of O(m n ) (this is the time bound assuming both translation and rotation in D) may be impractical Idea: Use hashing to separate the runtime factors Preprocessing: [O(m ) time] for each pair (p,q) of pattern points Transform all pattern points to reference frame (p,q) [i.e., reference frame with origin at p, +X-axis through q]; for each pattern point r Determine bin containing the transformed r Place (p,q) in Hashtable, hashed by bin number of r Geometric Hashing (continued) To test for a match: [O(n ) time] for each pair (s,t) of scene points Find transformation with origin at s, +X-axis through t; for each transformed scene point u if u falls in an occupied pattern bin [tested using hashing] Increment the vote for that pattern transformation; Report a match if some transformation gets enough votes; 1 1 Geometric Hashing Advantages Applications of Geometric Hashing Advantages Depending on problem, O(n ) may be significantly better than O(m n ) If we want to search for multiple patterns, this affects the preprocessing time, but not the matching time Time is better in practice than big-o bound an often reject a pair (s,t) of scene points quickly because, for instance, they re too far apart to match any pair of pattern points 1 Applications to omputer vision Astronomy Medical imaging Fingerprint matching Molecular docking Protein structure matching References: Lamdan & Wolfson, Geometric hashing: A general and efficient model-based recognition scheme (1988); lots of papers since then Web tutorial on protein structure comparison Includes geometric hashing (among other methods) 16 Protein Structure omparison Summary URMS is Being Used Methods discussed (a subset of the many available) ompare pictures Torsion Angles Distance Matrices RMSD (Root Mean Square Distance) We talked about using it on α-carbons Has also been used to compare Torsion Angles and Distance Matrices URMS (Unit-vector Root Mean Square) Geometric Hashing Why so many methods? Proteins are complex three-dimensional structures hoice depends on Purpose of comparison Whether method is appropriate Protein backbone only Torsion Angles, URMS Any cloud of points (or atoms) Distance Matrices, RMSD, Geometric Hashing ASP (ritical Assessment of Techniques for Protein Structure Prediction) A contest for researchers trying to predict protein structure from sequence Held every years ASP7 will be held in 006 Entrants are asked to predict protein structures that are known, but not yet published Evaluation is done by experts AFASP (ritical Assessment of Fully Automated Structure Prediction) Held in conjunction with ASP Only automated web servers are eligible Evaluation is automated URMS has been used as part of evaluation process 17 18

4 Protein Families Alignment via Dynamic Programming Evolution a protein ancient ancestor evolved into a of proteins Membership in a protein is expressed by sequence similarity, but is more strongly expressed by structure similarity -0% sequence resemblance (almost always) ensures shape resemblance There are databases of protein families: SOP (Structural lassification of Proteins), also ATH, FSSP They are mainly classified by their secondary structures (e.g., all-helix, all-strand, helix-strand) 19 Alignment RHYPGDFSPA ARFPADFTAE corresponds to Alignment -RHYPGDFSPA- ARF-PADFT-AE corresponds to RHYPGDFSPA RHYPGDFSPA R R P P D D T T E E (gap penalty = -1) 0 A String-Based Tool: the Profile Members of a protein can be aligned P1 A B - A P A B A B A P A B - P B - B This multiple alignment can be used to build a profile 1 A B A B - Using the Profile Given a new protein string, one can determine its protein by finding the best-matching profile The profile summarizes a protein s string-information Goal: Summarize a protein s shape-information Our Result: the onsensus Shape Previous Related Work Goal: a shape-based profile Summarize a protein by using a concise representation of the s shape information The onsensus Shape Algorithm produces A multiple alignment of structures, and A single (core) structure that summarizes the structural information for the Gerstein/Altman 9 Multiple alignment based on sequence to derive a structural core Orengo/Taylor 96 Multiple structure alignment based on the structural environment of each residue Gerstein/Levitt 96,98 Alignment, average core structure, automatic structure alignment against a manual standard Gelfand et al 98 Geometric invariant core based on distance matrices Leibowitz et al 99 Multiple structural alignment and core detection based on geometric hashing Many papers on pairwise structural alignment

5 Protein Shape and URMS Building a onsensus Shape y Our technique: URMS (Unit-vector Root Mean Square distance) y Suppose we have a multiple alignment of proteins in a protein y Advantages y Idea: an build a consensus shape by averaging the corresponding unit vectors Insensitive to outliers Efficient to compute Equal weight for all portions y How do we get the initial multiple alignment? y How do we make the proteins have consistent orientations? y What about gaps? y A profile can be iteratively improved; can we iteratively improve the consensus shape? 6 Averaging Unit Vectors 1 onsensus Shape Algorithm y Input: a set of proteins belonging to a single protein?? 1 y Output: the consensus shape, a 1 A gap is treated here as a zero vector, but there is a better strategy pseudo-protein that summarizes shape information for the y Initialization: Arbitrarily choose one member of the to be, the initial consensus shape Use Dynamic Programming to orient the other members of the to y Optimization: Loop until quits changing For each protein P in the, use Dynamic Programming to align P with Improve 7 Initial Orientations 8 Orientation via Dynamic Programming y Want pairwise alignment y Idea: Use a block of, say, using Dynamic Programming unit-vectors as a single character y We fill in a DP table by computing the distance for each pair of characters y DP can find the least-cost path y But This also provides a correspondence between unitvectors We don t know the proper orientation If the orientation is y Instead of a 0-character arbitrary then all unitalphabet of amino acids, we vectors look alike have an infinite alphabet of unit-vector characters y Two such characters can be compared using, for instance, the URMS distance y The correspondence can then be used to find an optimal orientation Via URMS distance calculation, for instance (gap penalty = 0.) 0

6 Optimization: Align Each Protein with Optimization: Align Each Protein with There is a DP table for each protein of the There is a DP table for each protein of the P 1 P There is an optimal path through each such table P 1 P There is an optimal path through each such table P P P P Each vector of maps to set of vectors, one vector for each path In some cases (e.g, P 1, P, P ) a vector of maps to a gap vector P P 6 P P 6 1 Improving the onsensus Shape Preserved Structure Property 1. Recompute by averaging corresponding unit-vectors from each protein of the. For each protein P in the, orient P to minimize its distance from. Introduce an additional consensus vector. Delete an existing (useless) consensus vector Each of these improvement methods can be shown to decrease the total cost The total cost is the sum of the squared distances between and each protein P of the Theorem: If all member of a protein exhibit a geometric relationship between corresponding α-carbons then that relationship is preserved in the consensus shape In particular, distances and angles are preserved This holds even though vectors between adjacent α-carbons are short in regions where the proteins disagree Requires the use a special gap-direction (a th dimension) to distinguish between short vectors due to disagreement vs. short vectors due to gaps An Alpha Protein Family (Globins) A Beta Protein Family 6

7 Unrelated Proteins Discrete/ontinuous Optimization To find an optimal structure-match between proteins Need to determine correspondence Need to determine orientation If either is known, the other is easy Goal is to do both at once For matching of D pseudoproteins We have a new algorithm Based on finding convex hulls for a sequence of point-sets in the plane For real proteins (in D) The D algorithm generalizes to D But requires finding extreme points of pointsets in 10D 7 8 Summary: onsensus Structure Input A of related proteins Main ideas Use blocks of, say, unit vectors to determine initial orientations of proteins Use a special thdimensional gap-vector Use paths in DP tables to iteratively improve the consensus structure and the orientations of the proteins Result A compact summary of the significant structural information for a protein 9