Using three-dimensional microfluidic networks for solving computationally hard problems

This paper describes the design of a parallel algorithm that uses moving fluids in a three-dimensional microfluidic system to solve a nondeterministically polynomial complete problem (the maximal clique problem) in polynomial time. This algorithm relies on (i) parallel fabrication of the microfluidic system, (ii) parallel searching of all potential solutions by using fluid flow, and (iii) parallel optical readout of all solutions. This algorithm was implemented to solve the maximal clique problem for a simple graph with six vertices. The successful implementation of this algorithm to compute solutions for small-size graphs with fluids in microchannels is not useful, per se, but does suggest broader application for microfluidics in computation and control.

A positivity result in the theory of Macdonald polynomials

We outline here a proof that a certain rational function Cn(q, t), which has come to be known as the “q, t-Catalan,” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because Cn(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the collection

Gene recognition via spliced sequence alignment.

Gene recognition is one of the most important problems in computational molecular biology. Previous attempts to solve this problem were based on statistics, and applications of combinatorial methods for gene recognition were almost unexplored. Recent advances in large-scale cDNA sequencing open a way toward a new approach to gene recognition that uses previously sequenced genes as a clue for recognition of newly sequenced genes. This paper describes a spliced alignment algorithm and software tool that explores all possible exon assemblies in polynomial time and finds the multiexon structure with the best fit to a related protein. Unlike other existing methods, the algorithm successfully recognizes genes even in the case of short exons or exons with unusual codon usage; we also report correct assemblies for genes with more than 10 exons. On a test sample of human genes with known mammalian relatives, the average correlation between the predicted and actual proteins was 99%. The algorithm correctly reconstructed 87% of genes and the rare discrepancies between the predicted and real exon-intron structures were caused either by short (less than 5 amino acids) initial/terminal exons or by alternative splicing. Moreover, the algorithm predicts human genes reasonably well when the homologous protein is nonvertebrate or even prokaryotic. The surprisingly good performance of the method was confirmed by extensive simulations: in particular, with target proteins at 160 accepted point mutations (PAM) (25% similarity), the correlation between the predicted and actual genes was still as high as 95%.

The cutoff phenomenon in finite Markov chains.

Natural mixing processes modeled by Markov chains often show a sharp cutoff in their convergence to long-time behavior. This paper presents problems where the cutoff can be proved (card shuffling, the Ehrenfests' urn). It shows that chains with polynomial growth (drunkard's walk) do not show cutoffs. The best general understanding of such cutoffs (high multiplicity of second eigenvalues due to symmetry) is explored. Examples are given where the symmetry is broken but the cutoff phenomenon persists.