1. Sequence order independent structural alignment Joe Dundas, Andrew Binkowski, Bhaskar DasGupta, Jie Liang Department of Bioengineering/Bioinformatics, University of Illinois at Chicago

2. Background oExtended Central Dogma of molecular biology DNA  RNA  primary structure  3D structure  function oEvolution conserves the 3D structure more than amino acid sequence. oStructural similarity often reflects a common function or origin of proteins. [1] oIt is useful to classify proteins based on their structures. (SCOP, CATH, FSSP). oMany methods for structure alignment have been reported. (CE, DALI, FAST, Matchprot)

3. Circular Permutation oLigation of the N and C termini, and subsequent cleavage elsewhere. oIn 1979, first natural circular permutation was observed in favin vs. concanavalin A. [2] oIn 1983, the first engineered circular permutation was performed on bovine pancreatic trypsin inhibitor. [3] oSince, studies have shown that artificially permuted proteins are able to fold into a stable structures that are similar to the native protein. [4] oCircular permutations have been discovered in lectins, β-glucanases, swaposin… [5] Uliel S., Fliess A., Amir A., Unger R. (1999) [6] Uliel S., Fliess A., Unger R. (2001) [7]

4. Alignment Problem oMost structural alignment methods rely on the structural units of each protein to align sequentially i.e. CE, FAST. oSome newer methods will perform non-sequential alignments i.e. Dali, Matchprot. After explaining our method, will we compare the results against Dali and Matchprot.

5. Our Method We exhaustively fragment protein A and protein B into lengths ranging from 4 to 7 residues. Notation: fragment λ a = (a 1, a 2 ), where a 1 and a 2 are the beginning and ending positions relative to the N termini of protein A. Π a = {λ a,1, λ a,2,… λ a,n } is the set of all fragments from protein A. L a,i is the length of fragment Π a,I Each fragment from protein A is aligned to all fragments of protein B if L a,I = L b,j, forming a set of Aligned Fragment Pairs ( Λ Π a x Π b ). A similarity function σ maps Λ 

6. Similarity Function All Λ i with σ(Λ i ) > Threshold are used to create a conflict graph.

7. Conflict Graph Two fragment pairs Λ i and Λ j are in conflict if any residue in λ i,A is also in λ j,A or any residue in λ i,B is also in λ j,B. Conflicts can be found by a vertex sweep. Query Protein Residues Reference Protein Residues δ3δ3 δ2δ2 δ4δ4 δ1δ1 Simplified Example

8. LP Formulation No conflicting residues in query or reference protein. Consistency between variables All variables are between 0 and 1 x is a relaxed integer between 0 and 1 0 = don’t use fragment 1 = use fragment Solve using linear programming package Subject to:

9. Local Conflict Number LP will assign a number between 0 and 1 for each x δ. For each Λ compute a local conflict number Θ Define δ min as the vertex with the smallest local conflict number. Assign a new σ Remove all vertices with σ ≤ 0 from Λ and push them onto a stack Ω in descending order of σ δ4δ4 δ2δ2 δ1δ1 δ3δ3 δ4δ4 δ2δ2 δ1δ1 δ3δ3 σ(Λ 1 ) = 50 x Λ1 =.85 Θ Λ1 = 1.10 σ(Λ 2 ) = 20 x Λ2 =.25 Θ Λ2 = 1.46 σ(Λ 3 ) = 20 x Λ3 = 0.6 Θ Λ3 = 0.85 σ(Λ 4 ) = 15 x Λ4 = 0.01 Θ Λ4 = 0.26 δ min σ(Λ1) = 50 σ(Λ2) = 15 σ(Λ4) = 0 σ(Λ3) = 20

10. Repeat Repeat LP formulation until all vertices have been pushed onto the stack Ω. Begin with 5 empty alignments. While the stack is not empty, retrieve a aligned pair by popping the stack. Insert it into each non-empty alignment if and only if: 1.No residue conflicts occur. 2.The global RMSD does not change by some threshold. If it can not be inserted into any alignment, insert it into an available empty alignment. Determine which alignment with highest similarity score.

11. Results – Circular Permutation? 1jqsC 70s ribosome functional complex Fold: Ribosome & Ribosomal fragments 2pii PII (Product of glnB) Fold: Ferredoxin-like RMSD: 2.3194

12. Results – Circular Permutation 1iudA Aspartate Racemase Fold: ATC-like 1h0rA Type II 3-dehydrogenate dehydralase Fold: Flavodoxin

13. Results 1fe0 ATX1 Metallochaperone Fold: ferredoxin-like 1vet Mitogen activated protein kinase

14. Results 1e50 Core binding factor Fold: Core binding factor beta 1pkv Riboflavin Synthase