Complex-linear invariants of biochemical networks

The nonlinearities found in molecular networks usually prevent mathematical analysis of network behaviour, which has largely been studied by numerical simulation. This can lead to difficult problems of parameter determination. However, molecular networks give rise, through mass-action kinetics, to polynomial dynamical systems, whose steady states are zeros of a set of polynomial equations. These equations may be analysed by algebraic methods, in which parameters are treated as symbolic expressions whose numerical values do not have to be known in advance. For instance, an "invariant" of a network is a polynomial expression on selected state variables that vanishes in any steady state. Invariants have been found that encode key network properties and that discriminate between different network structures. Although invariants may be calculated by computational algebraic methods, such as Gr\"obner bases, these become computationally infeasible for biologically realistic networks. Here, we exploit Chemical Reaction Network Theory (CRNT) to develop an efficient procedure for calculating invariants that are linear combinations of "complexes", or the monomials coming from mass action. We show how this procedure can be used in proving earlier results of Horn and Jackson and of Shinar and Feinberg for networks of deficiency at most one. We then apply our method to enzyme bifunctionality, including the bacterial EnvZ/OmpR osmolarity regulator and the mammalian 6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase glycolytic regulator, whose networks have deficiencies up to four. We show that bifunctionality leads to different forms of concentration control that are robust to changes in initial conditions or total amounts. Finally, we outline a systematic procedure for using complex-linear invariants to analyse molecular networks of any deficiency.Comment: 36 pages, 6 figure

Photon creation in a spherical oscillating cavity

We study the photon creation inside a perfectly conducting, spherical oscillating cavity. The electromagnetic field inside the cavity is described by means of two scalar fields which satisfy Dirichlet and (generalized) Neumann boundary conditions. As a preliminary step, we analyze the dynamical Casimir effect for both scalar fields. We then consider the full electromagnetic case. The conservation of angular momentum of the electromagnetic field is also discussed, showing that photons inside the cavity are created in singlet states.Comment: 14 pages, no figure

Fourth order phase-field model for local max-ent approximants applied to crack propagation

We apply a fourth order phase-field model for fracture based on local maximum entropy (LME) approximants. The higher order continuity of the meshfree LME approximants allows to directly solve the fourth order phase-field equations without splitting the fourth order differential equation into two second order differential equations. We will first show that the crack surface can be captured more accurately in the fourth order model. Furthermore, less nodes are needed for the fourth order model to resolve the crack path. Finally, we demonstrate the performance of the proposed meshfree fourth order phase-field formulation for 5 representative numerical examples. Computational results will be compared to analytical solutions within linear elastic fracture mechanics and experimental data for three-dimensional crack propagation