The Simplified Partial Digest Problem:Hardness and a Probabilistic Analysis

Zoë Abrams zoea@stanford.edu

Ho-Lin Chen holin@stanford.edu

An enzyme cuts a target DNA strand to into DNA fragments, and these DNA fragments are used to reconstruct the restriction site locations of the enzyme.

Two common Approaches

Double Digest Problem (NP-complete) [Goldstein, Waterman ’87]

Partial Digest Problem

Restriction Site Analysis

Reconstruct the locations using the length of all fragments that can possibly be produced.

The hardness of the problem is unknown. [Skiena, Sundaram ’93][Lemke, Skiena, Smith ’02]

Adding the primary fragments to the information used, we can find a unique reconstruction in polynomial time. [Pandurangan, Ramesh ’01]

Information is susceptible to experimental error caused by missing fragments.

Partial Digest Problem

Proposed by Blazewicz et. Al. ’01 Uses primary fragments and base fr

agments to reconstruct restriction sites Primary fragments: One of the endpoin

ts is the endpoint of the original DNA strand

Base fragments: two endpoints are consecutive sites on the DNA strand

Simplified Partial Digest Problem

Problem Definition

Given

X0 = 0, Xn+1 = D

A set of base fragments

{Xi - Xi-1}1 i n+1

A set of primary fragments

{(Xn+1 - Xi) (Xi – X0)}1 i n

Reconstruct the original series X1,...,Xn,

Theoretical and Algorithmic Issues

The algorithm that finds the exact solution may take 2n time in the worst case. [Blazewicz, Jaroszewski ’03]

The Simplified Partial Digest Problem may have exponential number of solutions.

The problem is APX-hard.

Simple algorithms can give correct solution with high probability.

Proof of APX-Hardness

We proved Simplified Partial Digest Problem is APX-hard by reducing the Tripartite-Matching problem to it.

Tripartite-Matching Problem:

Given a set S of triples in {1,2,3..n}3 , |S|=T.

Find whether there exists a subset M of S such that |M| = n, and no two triples in M are the same in some coordinates.

Tripartite Matching Problem

Tripartite Matching Problem

Proof of APX-Hardness Use symmetric restriction sites to cut the segment into

2T equal-length segments

…….1 2 2T

Proof of APX-Hardness Use symmetric restriction sites to cut the segment into

2T equal-length segments

…….

Pairs of symmetric restriction sites

Proof of APX-Hardness Use symmetric restriction sites to cut the segment into

2T equal-length segments

…….

Pairs of symmetric restriction sites

Proof of APX-Hardness Use symmetric restriction sites to cut the segment into

2T equal-length segments

…….

Pairs of symmetric restriction sites

Proof of APX-Hardness Use symmetric restriction sites to cut the segment into

2T equal-length segments. In each pair of equal-length segments, there are seven

restriction sites that can be put on either side.

…….1 2 2T

Sites “x" can be on either side

Proof of APX-Hardness Use symmetric restriction sites to cut the segment into

2T equal-length segments. In each pair of equal-length segments, there are seven

restriction sites that can be put on either side.

…….1 2 2T

Sites “x" can be on either side

Proof of APX-Hardness Those seven restriction sites can be divided into two

groups, denoted by “o” and “x” respectively.

Proof of APX-Hardness Those seven restriction sites can be divided into two

groups, denoted by “o” and “x” respectively. In each segment, restriction sites in the same group

must be put on the same side.

Proof of APX-Hardness Those seven restriction sites can be divided into two

groups, denoted by “o” and “x” respectively. In each segment, restriction sites in the same group

must be put on the same side. Each placement of restriction sites corresponds to a set

of triples chosen in the Tripartite Matching Problem.

not chosen

chosen

Proof of APX-Hardness Those seven restriction sites can be divided in

to two groups, denoted by “o” and “x” respectively.

In each segment, restriction sites in the same group must be put on the same side.

Each placement of restriction sites corresponds to a set of triples chosen in the Tripartite Matching Problem.

The current placement of restriction sites is a solution iff the corresponding set of triples is a solution to the Tripartite Matching Problem.

A Simple Algorithm Put all symmetric points at correct locations Put all asymmetric points on the left side

A Simple Algorithm Put all symmetric points at correct locations Put all asymmetric points on the left side From each site, do (from endpoints to the middle)

If the base segment is matched, fix its location

A Simple Algorithm Put all symmetric points at correct locations Put all asymmetric points on the left side From each site, do (from endpoints to the middle)

If the base segment is matched, fix its location If the base segment isn’t matched, move it and all points

toward middle to the other side.

A Simple Algorithm Put all symmetric points at correct locations Put all asymmetric points on the left side From each site, do (from endpoints to the middle)

If the base segment is matched, fix its location If the base segment isn’t matched, move it and all points

toward middle to the other side.

Analysis of the Algorithm Assuming a uniform distribution for restriction

sites, for many practical parameters*, with probability at least 0.4 the algorithm outputs correct locations.

All the primary fragments are matched, and at least ¼ of all base fragments will be matched in the worst case.

Runs in time linear to the number of sites

*Ex: Length of the DNA strand around 20,000, 10-20 restriction sites

Future Work

Construct better heuristics to solve SPDP

Analyze the hardness of Partial Digest Problem

Find other characterizations of restriction sites that are both easy to measure and can be used to reconstruct the sites