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Finding Compact Structural Motifs Presented By: Xin Gao Authors: Jianbo Qian, Shuai Cheng Li, Dongbo...
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Transcript of Finding Compact Structural Motifs Presented By: Xin Gao Authors: Jianbo Qian, Shuai Cheng Li, Dongbo...
Finding Compact Structural Motifs
Presented By: Xin GaoAuthors: Jianbo Qian, Shuai Cheng Li, Dongbo Bu,
Ming Li, and Jinbo XuUniversity of Waterloo, Ontario, Canada [email protected]
Outline
Introduction to Structural Motif
Related Work
Compact Motif-finding Problem Formulation
NP-Hard of the Compact Motif-finding Problem
A Polynomial Time Approximate Scheme
Outline
Introduction to Structural Motif
Related Work
Compact Motif-finding Problem Formulation
NP-Hard of the Compact Motif-finding Problem
A Polynomial Time Approximate Scheme
Introduction
Protein is a sequence of amino acids. A protein always folds into a specific
3-D shape. Structures are important to proteins:
The functional properties of proteins depend on their 3-D structures.
Structures are more conserved than sequence during the evolution of proteins.
Structural Motif
Structural motif is a frequently occurring substructure of proteins.
Motifs are thought to be tightly related to protein functions.
Identifying motifs from a set of proteins can help us to know their evolutionary history and functions.
Structural Motif Finding Problem Given a set of protein structures, to find the
frequently occurring substructure. Informally, to find one substructure from each
protein, that exhibit the highest degree of similarity.
How to measure the similarity of two substructures?
Two popular measurements: dRMSD: measure the root mean square Euclidean
distance between the corresponding residues from different protein
structures.
cRMSD: calculate the internal distance matrix for each protein, and compare the distance matrices for input structures.
Outline
Introduction to Structural Motif
Related Work
Compact Motif-finding Problem Formulation
NP-Hard of the Compact Motif-finding Problem
A Polynomial Time Approximate Scheme
Related Work L.P.Chew proposed an iterative algorithm to
compute the conserved shape and proved its convergence. (2002)
D. Bandyopadhyay applied graph-based data-mining tools to find the family-specific fingerprints. (2006)
M. Shatsky presented an algorithm to uncover the binding pattern. (2006)
DALI and CE attempt to identify structural alignment with minimal dRMSD.
STRUCTRAL and TM-Align employ heuristics to detect the alignment with minimal cRMSD.
Related Work (continued)
However, these methods are all heuristic; the solutions are not guaranteed to be optimal or near optimal.
The first PTAS for pairwise structural alignment: R. Kolodny explored the Lipschitz property of the scoring
function. (2004) Though this algorithm can be extended to the case
of multiple structure alignment, the simple extension has a time complexity exponential in the number of proteins.
Is there a PTAS to multiple structure motif finding?
Outline
Introduction to Structural Motif
Related Work
Compact Motif-finding Problem Formulation
NP-Hard of the Compact Motif-finding Problem
A Polynomial Time Approximate Scheme
We focus on (R, C)-Compact Motif.
What is (R, C)-compact motif? A motif is bounded in a minimum ball with radius R. In this ball, at most C residues do not belong to this motif.
(R,C)-compact motif is biologically meaningful since We focus on globular proteins. We allows at most C exceptions.
(R, C)-Compact Motif Finding Problem Input: protein structures S1 …, Sn, and length l
Output: a consensus consists of l 3D points q=(q1, …, ql ) a substructure ui from each protein Si
Objective: min (1 in d2(q, ui))1/2
Here, we adopt the dRMSD distance function, i.e., d(q, ui)=min||q- (ui)||2
consists of a rotation and a translation ||*||2 is the Euclidean metric.
n
Outline
Introduction to Structural Motif
Related Work
Compact Motif-finding Problem Formulation
NP-Hard of the Compact Motif-finding Problem
A Polynomial Time Approximate Scheme
(R,C)-compact motif finding is still NP-Hard. Reduction from the Sequence Consensus Problem
Input: n binary strings S1, …, Sn, each is of length m
Output: A substring ti of length l from each string Si, 1i n,
Objective: minimize 1 i <i’ n dH(ti, ti’), where dH is Hamming distance.
Basic Idea: Try to find a way of reduction to make:
dRMSD=Hamming Distance
(R,C)-compact motif finding is still NP-Hard. Each l-mer is
transformed into 6l 3D points.
110 110 001 000000 111111
0(0, 2i, 0), 1(1, 2i, 0)
(R,C)-compact motif finding is still NP-Hard. Each l-mer is transformed into 6l 3D points.
110 110 001 000000 111111 0(0, 2i, 0), 1(1, 2i, 0)
The centroid will be (1/2, 2i, 0) (Easy translation) Large “tail” no rotation RMSD = Hamming Distance
Small distortion to each point to make it protein-like.
Sequence Consensus Problem (1,0)-Compact Motif Finding Problem
Outline
Introduction to Structural Motif
Related Work
Compact Motif-finding Problem Formulation
NP-Hard of the Compact Motif-finding Problem
A Polynomial Time Approximate Scheme
The Basic Idea of Our PTAS
There are always a few “important” sub-structures, whose consensus holds most of the “secrets” of the true optimal motif.
Therefore, if we can simply do exhaustive search to find these few sub-structures, then the trivial optimal solution for these sub-structures is a good approximation to the real optimal solution.
Technique 1: Sampling
We sample only r proteins, consider each motif in a sampled protein, we can say we almost know the optimal
solution.
Sampling will introduce only a bit of error. There is at least one selection schema,
whose consensus has a cost value less than (1+1/r)OPT.
So, we can find this schema by simply enumerating operation.
Technique 2: Discretize the Rotation Space
Each rotation is parameterized by three angles 1, 2, 3[0, 2)
Discretize the angles with step size ’ we get an ’-rotation net.
Discretized rotation will not introduce a large error, either.
A parameterized algorithm for protein structure alignment. J. Xu, F. Jiao, and B. Berger. RECOMB2006.
Running Time
Each protein contains M motifs M is a polynomial of protein length
Each motif can adopt W rotations
W depends on the constant
So the number of consensus is less than O(nr(MW)r)= O((nMW)r)
)()( llm mOCOM
Conclusion and Future Work
We prove the (R,C)-compact motif finding problem is NP-hard
We obtain a PTAS for this problem. Future Work:
Further reduce the time complexity Design some practical algorithms. Solve a more general case.