Exponential Time Complexity of Weighted Counting of Independent Sets

We consider weighted counting of independent sets using a rational weight x: Given a graph with n vertices, count its independent sets such that each set of size k contributes x^k. This is equivalent to computation of the partition function of the lattice gas with hard-core self-repulsion and hard-core pair interaction. We show the following conditional lower bounds: If counting the satisfying assignments of a 3-CNF formula in n variables (#3SAT) needs time 2^{\Omega(n)} (i.e. there is a c>0 such that no algorithm can solve #3SAT in time 2^{cn}), counting the independent sets of size n/3 of an n-vertex graph needs time 2^{\Omega(n)} and weighted counting of independent sets needs time 2^{\Omega(n/log^3 n)} for all rational weights x\neq 0. We have two technical ingredients: The first is a reduction from 3SAT to independent sets that preserves the number of solutions and increases the instance size only by a constant factor. Second, we devise a combination of vertex cloning and path addition. This graph transformation allows us to adapt a recent technique by Dell, Husfeldt, and Wahlen which enables interpolation by a family of reductions, each of which increases the instance size only polylogarithmically.Comment: Introduction revised, differences between versions of counting independent sets stated more precisely, minor improvements. 14 page

An Evolutionary Reduction Principle for Mutation Rates at Multiple Loci

A model of mutation rate evolution for multiple loci under arbitrary selection is analyzed. Results are obtained using techniques from Karlin (1982) that overcome the weak selection constraints needed for tractability in prior studies of multilocus event models. A multivariate form of the reduction principle is found: reduction results at individual loci combine topologically to produce a surface of mutation rate alterations that are neutral for a new modifier allele. New mutation rates survive if and only if they fall below this surface - a generalization of the hyperplane found by Zhivotovsky et al. (1994) for a multilocus recombination modifier. Increases in mutation rates at some loci may evolve if compensated for by decreases at other loci. The strength of selection on the modifier scales in proportion to the number of germline cell divisions, and increases with the number of loci affected. Loci that do not make a difference to marginal fitnesses at equilibrium are not subject to the reduction principle, and under fine tuning of mutation rates would be expected to have higher mutation rates than loci in mutation-selection balance. Other results include the nonexistence of 'viability analogous, Hardy-Weinberg' modifier polymorphisms under multiplicative mutation, and the sufficiency of average transmission rates to encapsulate the effect of modifier polymorphisms on the transmission of loci under selection. A conjecture is offered regarding situations, like recombination in the presence of mutation, that exhibit departures from the reduction principle. Constraints for tractability are: tight linkage of all loci, initial fixation at the modifier locus, and mutation distributions comprising transition probabilities of reversible Markov chains.Comment: v3: Final corrections. v2: Revised title, reworked and expanded introductory and discussion sections, added corollaries, new results on modifier polymorphisms, minor corrections. 49 pages, 64 reference

The repulsive lattice gas, the independent-set polynomial, and the Lov\'asz local lemma

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovasz local lemma in probabilistic combinatorics. We show that the conclusion of the Lovasz local lemma holds for dependency graph G and probabilities {p_x} if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p_x}. Furthermore, we show that the usual proof of the Lovasz local lemma -- which provides a sufficient condition for this to occur -- corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer and explicitly by Dobrushin. We also present some refinements and extensions of both arguments, including a generalization of the Lovasz local lemma that allows for "soft" dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.Comment: LaTex2e, 97 pages. Version 2 makes slight changes to improve clarity. To be published in J. Stat. Phy

Synonymous Genes Explore Different Evolutionary Landscapes

The evolutionary potential of a gene is constrained not only by the amino acid sequence of its product, but by its DNA sequence as well. The topology of the genetic code is such that half of the amino acids exhibit synonymous codons that can reach different subsets of amino acids from each other through single mutation. Thus, synonymous DNA sequences should access different regions of the protein sequence space through a limited number of mutations, and this may deeply influence the evolution of natural proteins. Here, we demonstrate that this feature can be of value for manipulating protein evolvability. We designed an algorithm that, starting from an input gene, constructs a synonymous sequence that systematically includes the codons with the most different evolutionary perspectives; i.e., codons that maximize accessibility to amino acids previously unreachable from the template by point mutation. A synonymous version of a bacterial antibiotic resistance gene was computed and synthesized. When concurrently submitted to identical directed evolution protocols, both the wild type and the recoded sequence led to the isolation of specific, advantageous phenotypic variants. Simulations based on a mutation isolated only from the synthetic gene libraries were conducted to assess the impact of sub-functional selective constraints, such as codon usage, on natural adaptation. Our data demonstrate that rational design of synonymous synthetic genes stands as an affordable improvement to any directed evolution protocol. We show that using two synonymous DNA sequences improves the overall yield of the procedure by increasing the diversity of mutants generated. These results provide conclusive evidence that synonymous coding sequences do experience different areas of the corresponding protein adaptive landscape, and that a sequence's codon usage effectively constrains the evolution of the encoded protein

A Universal Trend of Reduced mRNA Stability near the Translation-Initiation Site in Prokaryotes and Eukaryotes

Recent studies have suggested that the thermodynamic stability of mRNA secondary structure near the start codon can regulate translation efficiency in Escherichia coli, and that translation is more efficient the less stable the secondary structure. We survey the complete genomes of 340 species for signals of reduced mRNA secondary structure near the start codon. Our analysis includes bacteria, archaea, fungi, plants, insects, fishes, birds, and mammals. We find that nearly all species show evidence for reduced mRNA stability near the start codon. The reduction in stability generally increases with increasing genomic GC content. In prokaryotes, the reduction also increases with decreasing optimal growth temperature. Within genomes, there is variation in the stability among genes, and this variation correlates with gene GC content, codon bias, and gene expression level. For birds and mammals, however, we do not find a genome-wide trend of reduced mRNA stability near the start codon. Yet the most GC rich genes in these organisms do show such a signal. We conclude that reduced stability of the mRNA secondary structure near the start codon is a universal feature of all cellular life. We suggest that the origin of this reduction is selection for efficient recognition of the start codon by initiator-tRNA

The genome of the seagrass Zostera marina reveals angiosperm adaptation to the sea

Seagrasses colonized the sea(1) on at least three independent occasions to form the basis of one of the most productive and widespread coastal ecosystems on the planet(2). Here we report the genome of Zostera marina (L.), the first, to our knowledge, marine angiosperm to be fully sequenced. This reveals unique insights into the genomic losses and gains involved in achieving the structural and physiological adaptations required for its marine lifestyle, arguably the most severe habitat shift ever accomplished by flowering plants. Key angiosperm innovations that were lost include the entire repertoire of stomatal genes(3), genes involved in the synthesis of terpenoids and ethylene signalling, and genes for ultraviolet protection and phytochromes for far-red sensing. Seagrasses have also regained functions enabling them to adjust to full salinity. Their cell walls contain all of the polysaccharides typical of land plants, but also contain polyanionic, low-methylated pectins and sulfated galactans, a feature shared with the cell walls of all macroalgae(4) and that is important for ion homoeostasis, nutrient uptake and O-2/CO2 exchange through leaf epidermal cells. The Z. marina genome resource will markedly advance a wide range of functional ecological studies from adaptation of marine ecosystems under climate warming(5,6), to unravelling the mechanisms of osmoregulation under high salinities that may further inform our understanding of the evolution of salt tolerance in crop plants(7)

Epithelial markers and differentiation in adnexal neoplasms of the skin: An immunohistochemical study including individual cytokeratins

Applying immunohistochemical procedures for the detection of eight different cytokeratin (CK) polypeptides and other differentiation markers, we compared the staining patterns of normal cutaneous structures with those of benign adnexal tumors (n=65), Syringomas exhibited a marker pattern highly reminiscent of that seen in normal dermal eccrine ducts (EMA in peripheral cells, CK 10 in intermediate cells, and CK 6, CK 19, and CEA in luminal cells). Nodular hidradenomas exhibited complex patterns suggesting relationship between tumor cells, including clear cells, and normal secretory coil cells (CK 7, CK 8, CK 19, and EMA); however, dermal-duct and epidermoid differentiation were also detectable. In both cylindromas and spiradenomas, zonal staining patterns were apparent: modified myoepithelial cells were positive for smooth-muscle-type actin, while the luminal cells mainly expressed ductal markers (CK 6 and CK 19) and, less prominently, secretory-coil markers including CK 7. Eccrine poromas exhibited a widespread reaction for CK 5/6 and EMA, analogous to peripheral dermal duct cells, but focal maturation toward inner-ductal and secretory-coil cells was also demonstrable. The staining pattern observed in trichoepitheliomas resembled that of the outer but not the inner root sheath. In conclusion, the detailed marker profiles obtained in the present study have broadened our understanding of the differentiation and nature of these highly singular tumor types