Local Parallel Biomolecular Computation

Abstract

Biomolecular Computation(BMC) is computation at the molecular scale, using biotechnology engineering techniques. Most proposed methods for BMC used distributed (molecular) parallelism (DP); where operations are executed in parallel on large numbers of distinct molecules. BMC done exclusively by DP requires that the computation execute sequentially within any given molecule (though done in parallel for multiple molecules). In contrast, local parallelism (LP) allows operations to be executed in parallel on each given molecule. Winfree, et al [W96, WYS96]) proposed an innovative method for LPBMC, that of computation by unmediated self-assembly of 2D arrays of DNA molecules, applying known domino tiling techniques (see Buchi [B62], Berger [B66], Robinson [R71], and Lewis and Papadimitriou [LP81]) in combination with the DNA self-assembly techniques of Seeman et al [SZC94]. We develop improved techniques to more fully exploit the potential power of LP-BMC. we propose a refined step-wise assembly method, which provides control of the assembly in distinct steps. Step-wise assembly may increase the likelihood of success of assembly, decrese the number of tiles required, and provide additional control of the assembly process. The assembly depth is the number of stages of assembly required and the assembly size is the number of tiles required. We also introduce the assembly frame, a rigid nanostructure which binds the input DNA strands in place on its boundaries and constrains the shape of the assembly. Our main results are LP-BMC algorithms for some fundamental problems that form the basis of many parallel computations. For these problems we decrease the assembly size to linear in the input size and and significantly decrease the assembly depth. We give LP-BMC algorithms with linear assembly size and logarithmic assembly depth, for the parallel prefix computation problems, which include integer addition, subtraction, multiplication by a constant number, finite state automata simulation, and ∗A preliminary version of this paper appeared in Proc. DNA-Based Computers, III: University of Pennsylvania, June 23-26, 1997. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, H. Rubin and D. H. Wood, editors. American Mathematical Society, Providence, RI, vol. 48, 1999, pp. 217-254. †Department of Computer Science, Duke University, Durham, NC , USA and Adjunct, King Abdulaziz University (KAU), Jeddah, Saudi Arabia