On oriented graphs with minimal skew energy

Let $S(G^\sigma)$ be the skew-adjacency matrix of an oriented graph $G^\sigma$. The skew energy of $G^\sigma$ is defined as the sum of all singular values of its skew-adjacency matrix $S(G^\sigma)$. In this paper, we first deduce an integral formula for the skew energy of an oriented graph. Then we determine all oriented graphs with minimal skew energy among all connected oriented graphs on $n$ vertices with $m \ (n\le m < 2(n-2))$ arcs, which is an analogy to the conjecture for the energy of undirected graphs proposed by Caporossi {\it et al.} [G. Caporossi, D. Cvetkovi$\acute{c}$, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996.]Comment: 15 pages. Actually, this paper was finished in June 2011. This is an updated versio

Nonequilibrium Thermodynamics in Biology: From Molecular Motors to Metabolic Pathways

Biological systems need to exchange energy and matter with their environment in order to stay functional or “alive”. This exchange has to obey the laws of thermodynamics: energy cannot be created and exchange comes at the cost of dissipation, which limits the efficiency of biological function. Additionally, subcellular processes that involve only few molecules are stochastic in their dynamics and a consistent theoretical modeling has to account for that. This dissertation connects recent development in nonequilibrium thermodynamics with approaches taken in biochemical modeling. I start by a short introduction to thermodynamics and statistical mechanics, with a special emphasis on large deviation theory and stochastic thermodynamics. Building on that, I present a general theory for the thermodynamic analysis of networks of chemical reactions that are open to the exchange of matter. As a particularly insightful concrete example I discuss the mechanochemical energy conversion in stochastic models of a molecular motor protein, and show how a similar analysis can be performed for more general models. Furthermore, I compare the dissipation in stochastically and deterministically modeled open chemical networks, and present a class of chemical networks that displays exact agreement for arbitrary abundance of chemical species and arbitrary distance from thermodynamic equilibrium. My major achievement is a thermodynamically consistent coarse-graining procedure for biocatalysts, which are ubiquitous in molecular cell biology. Finally, I discuss the thermodynamics of unbranched enzymatic chains