1. Overcoming the Curse of Dimensionality in a Statistical Geometry Based Computational Protein Mutagenesis
Majid Masso
Bioinformatics and Computational Biology
George Mason University, Manassas, Virginia, USA
BioDM Workshop, IEEE ICDM 2010

2. Delaunay Tessellation of Protein Structure
Aspartic Acid
(Asp or D)
Abstract every amino acid residue to a point
Atomic coordinates – Protein Data Bank (PDB)
center of mass (CM) D3
A22
S64 L6 F7
G62
C63 K4 R5
Delaunay tessellation: 3D “tiling” of space into non-overlapping, irregular tetrahedral simplices. Each simplex objectively identifies a quadruplet of nearest-neighbor amino acids at its vertices.

3. Delaunay Tessellation of T4 Lysozyme
Ribbon diagram (left) based on PDB file 3lzm (164 residues)
Each amino acid residue represented as a CM point in 3D space
Tessellation of the 164 CM points (right) performed using a 12Å edge-length cutoff, for “true” residue quadruplet interactions

4. Four-Body Statistical Potential
Training set: 1,375 diverse high-resolution x-ray structures
PDB …
1efaB
lac repressor
1bniA
barnase
1rtjA
HIV-1 RT
Tessellate
1jli
IL-3
Pool together all simplices from the tessellations, and compute observed frequencies of simplicial quadruplets

5. Four-Body Statistical Potential

6. Computational Mutagenesis: Residual Profiles
ribbon
CM trace
10 simplices share N163 vertex, and 10 total vertices; in the structure, N163 has 9 neighbors
tessellation
nonzero components identify the mutated position 163 and its 9 neighbors
environmental
change (EC)

7. Computational Mutagenesis: Residual Profiles
Nonzero ECs identify mutated position 163 and its 9 neighbors
So, the 19 mutants (N163A, N163C, etc.) at 163 will have nonzero ECs at the same 10 positions only, but nonzero values will differ
Each position has a different number of structural neighbors (min of 6, max of 19), which can be located throughout the sequence
Number of neighbors and their locations (position numbers) are dependent on the position being mutated

8. Experimental Data: Mutant T4 Lysozyme Activity
2015 mutants synthesized by introducing the same 13 amino acids as replacements at 163 positions (all except the first)
Rennell, D., Bouvier, S.E., Hardy, L.W. & Poteete, A.R. (1991) J. Mol. Biol. 222,
Each position yields either 12 or 13 mutants, depends on whether or not native amino acid there is also one of the 13 replacements
Mutant activity is based on plaque sizes on Petri dishes, 2 classes: “unaffected” = large plaques (same as native T4 lysozyme) “affected” = medium, small, or no plaques
1377 “unaffected” and 638 “affected” T4 lysozyme mutants

9. Computational Mutagenesis: Feature Vectors
Approach 1 – represent mutants by 164D residual profile vectors; training set consists of all 2015 T4 lysozyme mutants
Approach 2 (dimensionality reduction) – select and order the 6 closest neighbors to the mutated position; create 7D vector of nonzero EC scores for mutated position and 6 closest neighbors
Approach 3 (subspace modeling) – segregate mutants by position number, consider each subset as a separate training set for classification, and combine the results; can be applied to 164D or 7D feature vectors

10. Supervised Classification
Algorithms: decision tree (DT), neural network (NN), support vector machine (SVM), and random forest (RF)
Testing: leave-one-out cross-validation (LOOCV)
Evaluation of performance:
Overall accuracy, or proportion of correct predictions: Q
Sensitivity and precision for both classes: S(U), P(U), S(A), and P(A)
Balanced error rate: BER
Matthew’s correlation coefficient: MCC

11. Results
Full training set with 164D surpasses 7D due to loss of implicit structural information (i.e., location of nonzeros in 164D vector)
Subspace modeling (SM) improves performance due to dramatic increase in S(A); 164D and 7D SM results are equal
SM with 164D vectors amounts to dimensionality reduction that uses the entire neighborhood of mutated position (unlike 7D, which uses only the 6 closest neighbors)

12. Conclusion and Future Directions
Residual profile vectors provide a natural way to introduce subspace modeling and achieve improved performance
Current work focused on inductive learning, future project could apply transductive learning to the dataset
Transduction allows us to also use vectors of all remaining mutants not classified experimentally – wet-lab collaborations can then validate our predictions
These techniques could be applied to a similarly comprehensive experimental dataset: 4041 mutants of lac repressor protein
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