Adaptive walks in a gene network model of morphogenesis: insights into the Cambrian explosion

The emergence of complex patterns of organization close to the Cambrian boundary is known to have happened over a (geologically) short period of time. It involved the rapid diversification of body plans and stands as one of the major transitions in evolution. How it took place is a controversial issue. Here we explore this problem by considering a simple model of pattern formation in multicellular organisms. By modeling gene network-based morphogenesis and its evolution through adaptive walks, we explore the question of how combinatorial explosions might have been actually involved in the Cambrian event. Here we show that a small amount of genetic complexity including both gene regulation and cell-cell signaling allows one to generate an extraordinary repertoire of stable spatial patterns of gene expression compatible with observed anteroposterior patterns in early development of metazoans. The consequences for the understanding of the tempo and mode of the Cambrian event are outlined.Comment: to appear in International Journal of Developmental Biology, special issue on Evo-Devo (2003

Hide and seek on complex networks

Signaling pathways and networks determine the ability to communicate in systems ranging from living cells to human society. We investigate how the network structure constrains communication in social-, man-made and biological networks. We find that human networks of governance and collaboration are predictable on teat-a-teat level, reflecting well defined pathways, but globally inefficient. In contrast, the Internet tends to have better overall communication abilities, more alternative pathways, and is therefore more robust. Between these extremes the molecular network of Saccharomyces cerevisea is more similar to the simpler social systems, whereas the pattern of interactions in the more complex Drosophilia melanogaster, resembles the robust Internet.Comment: 5 pages, 5 figure

A Lax type operator for quantum finite W-algebras

For a reductive Lie algebra g, its nilpotent element f and its faithful finite dimensional representation, we construct a Lax operator L(z) with coefficients in the quantum finite W-algebra W(g, f). We show that for the classical linear Lie algebras glN, slN, soN and spN, the operator L(z) satisfies a generalized Yangian identity. The operator L(z) is a quantum finite analogue of the operator of generalized Adler type which we recently introduced in the classical affine setup. As in the latter case, L(z) is obtained as a generalized quasideterminant

Computation of cohomology of vertex algebras

We review cohomology theories corresponding to the chiral and classical operads. The first one is the cohomology theory of vertex algebras, while the second one is the classical cohomology of Poisson vertex algebras (PVA), and we construct a spectral sequence relating them. Since in “good” cases the classical PVA cohomology coincides with the variational PVA cohomology and there are well-developed methods to compute the latter, this enables us to compute the cohomology of vertex algebras in many interesting cases. Finally, we describe a unified approach to integrability through vanishing of the first cohomology, which is applicable to both classical and quantum systems of Hamiltonian PDEs

Vertex Operator Superalgebras and Odd Trace Functions

We begin by reviewing Zhu's theorem on modular invariance of trace functions associated to a vertex operator algebra, as well as a generalisation by the author to vertex operator superalgebras. This generalisation involves objects that we call `odd trace functions'. We examine the case of the N=1 superconformal algebra. In particular we compute an odd trace function in two different ways, and thereby obtain a new representation theoretic interpretation of a well known classical identity due to Jacobi concerning the Dedekind eta function.Comment: 13 pages, 0 figures. To appear in Conference Proceedings `Advances in Lie Superalgebras

Sustainable Triazine-Based Dehydro-Condensation Agents for Amide Synthesis

Conventional methods employed today for the synthesis of amides often lack of economic and environmental sustainability. Triazine-derived quaternary ammonium salts, e.g., 4-(4,6-dimethoxy-1,3,5-triazin-2-yl)-4-methylmorpholinium chloride (DMTMM(Cl)), emerged as promising dehydro-condensation agents for amide synthesis, although suffering of limited stability and high costs. In the present work, a simple protocol for the synthesis of amides mediated by 2-chloro-4,6-dimethoxy-1,3,5-triazine (CDMT) and a tert-amine has been described and data are compared to DMTMM(Cl) and other CDMT-derived quaternary ammonium salts (DMT-Ams(X), X: Cl- or ClO4-). Different tert-amines (Ams) were tested for the synthesis of various DMT-Ams(Cl), but only DMTMM(Cl) could be isolated and employed for dehydro-condensation reactions, while all CDMT/tert-amine systems tested were efficient as dehydro-condensation agents. Interestingly, in best reaction conditions, CDMT and 1,4-dimethylpiperazine gave N-phenethyl benzamide in 93% yield in 15 min, with up to half the amount of tert-amine consumption. The efficiency of CDMT/tert-amine was further compared to more stable triazine quaternary ammonium salts having a perchlorate counter anion (DMT-Ams(ClO4)). Overall CDMT/tert-amine systems appear to be a viable and more economical alternative to most dehydro-condensation agents employed today

Local and Non-local Multiplicative Poisson Vertex Algebras and Differential-Difference Equations

We develop the notions of multiplicative Lie conformal and Poisson vertex algebras, local and non-local, and their connections to the theory of integrable differential-difference Hamiltonian equations. We establish relations of these notions to q-deformed W-algebras and lattice Poisson algebras. We introduce the notion of Adler type pseudodifference operators and apply them to integrability of differential-difference Hamiltonian equations

On classical finite and affine W-algebras

This paper is meant to be a short review and summary of recent results on the structure of finite and affine classical W-algebras, and the application of the latter to the theory of generalized Drinfeld-Sokolov hierarchies.Comment: 12 page

Lagrangian phase transitions in nonequilibrium thermodynamic systems

In previous papers we have introduced a natural nonequilibrium free energy by considering the functional describing the large fluctuations of stationary nonequilibrium states. While in equilibrium this functional is always convex, in nonequilibrium this is not necessarily the case. We show that in nonequilibrium a new type of singularities can appear that are interpreted as phase transitions. In particular, this phenomenon occurs for the one-dimensional boundary driven weakly asymmetric exclusion process when the drift due to the external field is opposite to the one due to the external reservoirs, and strong enough.Comment: 10 pages, 2 figure

Preferential attachment in the protein network evolution

The Saccharomyces cerevisiae protein-protein interaction map, as well as many natural and man-made networks, shares the scale-free topology. The preferential attachment model was suggested as a generic network evolution model that yields this universal topology. However, it is not clear that the model assumptions hold for the protein interaction network. Using a cross genome comparison we show that (a) the older a protein, the better connected it is, and (b) The number of interactions a protein gains during its evolution is proportional to its connectivity. Therefore, preferential attachment governs the protein network evolution. The evolutionary mechanism leading to such preference and some implications are discussed.Comment: Minor changes per referees requests; to appear in PR