Local parallel biomolecular computation

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 LP-BMC, that of computation by unmediated self-assembly of 2D arrays of DNA molecules, applying known domino tiling techniques (see Buchi [B62], Berger [B66], and [R71, LP81]) in combination with recombinant DNA nano-fabrication techniques of Seeman et al [SZC94]. The likelihood for successful unmediated self-assembly of computations has not been determined (we discuss a simple model of assembly where there may be blockages in self-assembly, but more sophisticated models may have a higher likelihood of success). We develop improved techniques to more fully exploit the potential power of LP-BMC. To increase the likelihood of success of assembly, we propose instead a re ned step-wise assembly method, which provides control of the assembly in distinct steps. It does not require large volumes or convergence time, even in the worst case assembly model that we assume for our own assembly algorithms. We also introduce the assembly frame, a rigid nano-structure 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 signi cantly decrease the number of time steps. We give LP-BMC algorithms with linear assembly size and logarithmic time, for the parallel pre x computation problems, which include integer addition, subtraction, multiplication by a constant number, nite state automata simulation, and ngerprinting (hashing) a string. We also give LP-BMC methods for perfect shu e and pair-wise exchange using a linear size assembly and constant time. This provides logarithmic time LP-BMC algorithms for the large class of normal parallel algorithms [S71, U84, L92] on shu e-exchange networks, e.g. DFT, bitonic merge, xed permutation of data, as well as evaluation of a bounded degree Boolean circuit in time bounded by a logarithmic times the circuit depth. Our LP-BMC methods require much smaller volumes than previous DPBMC algorithms [R95,R98b, OR97] for these problems. All our LP-BMC assembly techniques can be combined with DP-BMC parallelism to simultaneously solve multiple problems with distinct inputs (e.g. do parallel arithmetic on multiple inputs, or determine satisfying inputs of a circuit), so they are an enhancement of the power of DP-BMC. Surface address: Department of Computer Science, Duke University, Box 90129, Durham, NC 27708-0129. E-mail: [email protected]. This work was supported by Grants NSF/DARPA CCR-9725021, CCR-96-33567, NSF IRI9619647 and EIA-0086015, ARO contract DAAH-04-96-1-0448, and ONR contract N00014-99-1-0406. Postscript versions of this paper and its gures are at http://www.cs.duke.edu/ reif/paper/Assembly.ps and http://www.cs.duke.edu/ reif/paper/Assembly. g.ps. An invited talk on the topic of this paper was given at Third Annual DIMACS Meeting on DNA Based Computers, Univ. of Penns., June 23-26, 1997.