Ján Manuch

Two lower bounds for self-assemblies at temperature 1.

Using the Tile Assembly Model proposed by Rothemund and Winfree, we give two lower bounds on the minimum number of tile types needed to uniquely assemble a shape at temperature 1 under a natural assumption that there are no binding domain mismatches (any two adjacent tiles either form a bond or else both touching sides of the tiles are without glues). Rothemund and Winfree showed that uniquely assembling a full N x N square (a square where there is a bond between any two adjacent tiles) at temperature 1 requires N(2) distinct tile types, and conjectured that the minimum number of tile types needed to self-assemble an N x N square (not a full square) is 2N - 1. Our lower bounds imply that a tile system that uniquely assembles an N x N square without binding domains mismatches, requires at least 2N - 1 tile types.

Characterization of the existence of galled-tree networks.

In this paper, we give a complete characterization of the existence of a galled-tree network in the form of simple sufficient and necessary conditions for both root-known and root-unknown cases. As a by-product we obtain a simple algorithm for constructing galled-tree networks. We also introduce a new necessary condition for the existence of a galled-tree network similar to bi-convexity.

On f-wise Arc Forwarding Index and Wavelength Allocations in Faulty All-optical Hypercubes

Motivated by the wavelength division multiplexing in all-optical networks, we consider the problem of finding an optimal (with respect to the least possible number of wavelengths) set of ƒ+1 internally node disjoint dipaths connecting all pairs of distinct nodes in the binary r -dimensional hypercube, where 0 ≤ ƒ . This system of dipaths constitutes a routing protocol that remains functional in the presence of up to ƒ faults (of nodes and/or links). The problem of constructing such protocols for general networks was mentioned in [1]. We compute precise values of ƒ -wise arc forwarding indexes and give (describe dipaths and color them) nearly optimal all-to-all ƒ -fault tolerant protocols for the hypercube network. Our results generalize corresponding results from [1, 4, 14].

Inverse protein folding in 2D HP model

The inverse protein folding problem is that of designing an amino acid sequence which has a particular native protein fold. This problem arises in drug design where a particular structure is necessary to ensure proper protein-protein interactions. In this paper we show that in the 2D HP model of Dill it is possible to solve this problem for a broad class of structures. These structures can be used to closely approximate any given structure. One of the most important properties of a good protein is its stability - the aptitude not to fold simultaneously into other structures. We show that for a number of basic structures, our sequences have a unique fold.

Haplotype inferring via galled-tree networks is NP-complete.

The problem of determining haplotypes from genotypes has gained considerable prominence in the research community since the beginning of the HapMap project. Here the focus is on determining the sets of SNP values of individual chromosomes (haplotypes), since such information better captures the genetic causes of diseases. One of the main algorithmic tools for haplotyping is based on the assumption that the evolutionary history for the original haplotypes satisfies perfect phylogeny. This tool can be applied only on individual blocks of chromosomes, in which it is assumed that recombinations do not happen. However, exact determination of blocks is usually not possible. It would be desirable to develop a method for haplotyping which can account for recombinations, and thus can be applied on multiblock sections of chromosomes. A natural candidate for such a method is haplotyping via phylogenetic networks (which model recombinations) or their simplified version: galled-tree networks. However, even haplotyping via galled-tree networks appears hard, as the efficient algorithms exist only for very special cases: the galled-tree network has either a single gall or only small galls with two mutations each. Building on our previous results, we show that, in general, haplotyping via galled-tree networks is NP-complete, and it remains NP-complete when galls are allowed to have at most k mutations, for any k ≥ 3.

Structure-approximating design of stable proteins in 2D HP model fortified by cysteine monomers

divides amino acids to two groups: hydrophobic (H) and polar (P), and considers only hydrophobic interactions between neighboring H amino in the energy formula. Another significant force acting during the protein folding are sulfide (SS) bridges between two cysteine amino acids. In this paper, we will enrich the HP model by adding cysteines as the third group of amino acids. A cysteine monomer acts as an H amino acid, but in addition two neighboring cysteines can form a bridge to further reduce the energy of the fold. We call our model the HPC model. We consider a subclass of linear structures designed in Gupta et al. 1 which is rich enough to approximate (although more coarsely) any given structure. We refine the structures for the HPC model by setting approximately a half of H amino acids to cysteine ones. We conjecture that these structures are stable under the HPC model and prove it under an additional assumption that non-cysteine amino acids act as cysteine ones, i.e., they tend to form their own bridges to reduce the energy. In the proof we will make an efficient use of a computational tool 2DHPSolver which significantly speeds up the progress in the technical part of the proof. This is a preliminary work, and we believe that the same techniques can be used to prove this result without the artificial assumption about non-cysteine H monomers.

Inverse Protein Folding in 2D HP Mode (Extended Abstract)

The inverse protein folding problem is that of designing an amino acid sequence which has a particular native protein fold. This problem arises in drug design where a particular structure is necessary to ensure proper protein-protein interactions. In this paper we show that in the 2D HP model of Dill it is possible to solve this problem for a broad class of structures. These structures can be used to closely approximate any given structure. One of the most important properties of a good protein is its stability -- the aptitude not to fold simultanously into other structures. We show that for a number of basic structures, our sequences have a unique fold.

Algorithms and Complexity Results for Genome Mapping Problems

Genome mapping algorithms aim at computing an ordering of a set of genomic markers based on local ordering information such as adjacencies and intervals of markers. In most genome mapping models, markers are assumed to occur uniquely in the resulting map. We introduce algorithmic questions that consider repeats, i.e., markers that can have several occurrences in the resulting map. We show that, provided with an upper bound on the copy number of repeated markers and with intervals that span full repeat copies, called repeat spanning intervals, the problem of deciding if a set of adjacencies and repeat spanning intervals admits a genome representation is tractable if the target genome can contain linear and/or circular chromosomal fragments. We also show that extracting a maximum cardinality or weight subset of repeat spanning intervals given a set of adjacencies that admits a genome realization is NP-hard but fixed-parameter tractable in the maximum copy number and the number of adjacent repeats, and tractable if intervals contain a single repeated marker.

Two Lower Bounds for Self-Assemblies at Temperature 1

Using the Tile Assembly Model proposed by Rothemund and Winfree, we give two lower bounds on the minimum number of tile types needed to uniquely assemble a shape at temperature 1 under a natural assumption that there are no binding domain mismatches (any two adjacent tiles either form a bond or else both touching sides of the tiles are without glues). Rothemund and Winfree showed that uniquely assembling a full N times N square (a square where there is a bond between any two adjacent tiles) at temperature 1 requires N 2 distinct tile types, and conjectured that the minimum number of tile types needed to self-assemble an N times N square (not a full square) is 2N - 1. Our lower bounds imply that a tile system that uniquely assembles an N times N square without binding domains mismatches, requires at least 2N - 1 tile types.