The Simplified Partial Digest Problem: Hardness and a Probabilistic Analysis

Introduction We study the problem of genome mapping using restriction site analysis. In restriction site analysis, an enzyme cuts a target DNA strand into DNA fragments, and these DNA fragments are used to reconstruct the restriction site locations of the enzyme. Two common approaches are the Double Digest Problem and the Partial Digest Problem. The Double Digest Problem is known to be NP-Complete[4], but the hardness of the Partial Digest Problem is unknown, despite there being no known polynomial algorithm[5][7]. Alternative approaches to restriction site analysis use primary fragments, which are DNA fragments with one endpoint on either the left or right side of the target DNA strand. Blazewicz et. al. [1] [2] present a technique for finding primary fragment lengths called a short digestion, in which the reaction time is chosen so that all molecules are cut at most once. In [6], an alternative approach for finding primary fragment lengths is presented: before the digestion experiment, the endpoints of the molecule are labeled (one method is radio-labeling with radioactive phosphate). [6] uses the primary fragments, in addition to information used in the Partial Digest Problem approach, to find a unique reconstruction in polynomial time. However, the information from the Partial Digest Problem is susceptible to experimental errors caused by missing fragments, and therefore it is still useful to develop techniques based on other types of information that are more robust against these types of errors. The Simplified Partial Digest Problem, first proposed in [1], uses primary fragments and base fragments to locate restriction sites. Base fragments have two endpoints that were consecutive sites on the target DNA strand and can be obtained by exposing the strand to the enzyme until the digestion process is complete. We consider there are n restriction sites where the enzyme cuts along a DNA strand of length D. Simplified Partial Digest Problem (SPDP) Statement: Given X0 = 0, Xn+1 = D, and a set of base fragments {Xi Xi-1}1 ≤ i ≤ n+1 and primary fragments {(Xn+1 Xi) ∪ Xi}1 ≤ i ≤ n, reconstruct the original series X1,...,Xn, where Xi corresponds to the distance between the leftmost end of the target DNA strand and the i furthest restriction site along the strand. In this paper, we show that the SPDP is NP-Complete, an open problem in [3]. Therefore, if we desire efficient algorithms for this problem, we must relax our end criteria. We propose an efficient algorithm that in practice finds a solution for many instances of the SPDP.