1. Protein Encoding Optimization Student: Logan Everett Mentor: Endre Boros Funded by DIMACS REU 2004

2. Project Overview Model Biological Scoring Matrices Weighted Binary Hamming Space Optimize Using Linear Programming Accurate Random Generation

3. Scoring Matrices A Q M K R H… A R M I F L… 4 1 5 –3 –3 -3…

4. Encode To Binary Strings Hamming Distances Easy to Approximate on Binary Strings Statistically Proven Methods More Efficient How Do Similarity and Distance Relate? Inverse Relationship First Create “Real” Distance Vector: D

5. Precise Problem: Distortion D ij (1–  )  h[  i,  j ]  D ij (1+  )  unique pairs i,j ( n C 2 ) s.t. 0    1 and 0 

6. Encoding Scheme as Vector C = 0110101101010101010 S = 1010110101011010101 T = 0110101010011011010 P = 1010101011010100110 A = 1011010101011011010 G = 1010101100111010101 y2y1y2y1

7. Modified Inequality D(1–  )  Ax  D(1+  ) s.t. 0    1 and 0  Let x = y

8. Linear Programming Problem Need All Linear Expressions D(1 –  )  Ax and Ax  D(1 +  ) -Ax – D   -D and Ax – D   D All x i,   0 Goal: Minimize  Solve with CPLEX

9. Problem Size Number of Constraints (Rows) 2( n C 2 ) = 380 Number of Variables (Columns) 2 n-1 = 524,288 Total Size – App. 2x10 8 CPLEX – App. 1 Minute

10. Linear Programming Solution Solution Contains: Min Value of  Scaled Weight Vector x Non-Integral Values in x Convert to p Vector X =  x i p i = x i / X

11. Random Encodings Randomly Select Cross Sections Based on Percent Weights Can Scale For Any N-Length Encoding Longer Encodings Should Approach Minimum Distortion

12. Results

13. Courtesy of DIMACS Mentor: Endre Boros – RUTCOR Logan Everett – DIMACS REU 2004