Graph-based Methods for the Design of DNA Computations

Abstract

DNA computing is a rapidly evolving field utilizing DNA molecules instead of silicon-based electronic units to perform calculations. The reliability of such computations strongly depends on the DNA sequences that represent units of information. Recently, the thermodynamic constraints, based on the free energy of hybridization between pairs of DNA single strands, are considered as the most reliable criterion to compose such sequences. The purpose of this thesis is to provide a contribution to the field of finding reliable DNA sequences for the encoding of entities in mathematical problems. The developed methods make use of the nearest neighbor DNA thermodynamic model as a biological fundament. The modelling method uses graph theory. The work addresses the following issues. The first one is a thermodynamic evaluation of a DNA encoding. The performance of predeveloped published sets in vitro differs for particular DNA computations that follow distinct models, since the intended reactions are not the same. The models using an encoding principle proposed by Adleman [2] imply interactions, that are not taken into account by modern strand design applications. Therefore, an evaluation method comprising additional restrictions is proposed, in order to more accurately assess the performance of a candidate DNA encoding for these models. The evaluation is performed with respect to thermodynamic restrictions and allows to find weak spots in the encoding set of DNA words. The second issue is the prediction of the hybridization energy for a pair of DNA single strands. This comprises finding the secondary structure of the DNA/DNA complex with minimal free energy (MFE), which is usually referred to as MFE problem. The effective solution methods for this problem are the main prerequisites for the thermodynamically based DNA sequence design. Contemporary methods are based on the nearest neighbor model to assess the thermal stability of biomolecular DNA complexes and utilize the paradigm of dynamic programming to find an energetically optimal complex. In this work, a novel graph model for DNA/DNA hybridization complexes is developed. Based on this, two methods for the solution of the MFE problem using the paradigm of dynamic programming are proposed. The performance of the methods is compared with that of currently used methods for this task. An additional issue concerns one of the basic problems in algorithmic graph theory. Namely, the all-pairs shortest path problem that comprises finding of the paths with minimal weight between each pair of nodes in a graph. There was developed a memory saving technique to perform the calculations for the particular case of bipartite graphs. It is based on transformation of the weight matrix by rules of tropical (min-plus) algebra.