We present a new coordinate-space model of spherically averaged exchange-hole functions in inhomogeneous systems that depends on local values of the density and its gradient and Laplacian, and also the kinetic energy density. Our model is completely nonempirical, incorporates the uniform-density electron gas and hydrogenic atom limits, and yields the proper 1lr asymptotic exchange potential in finite systems. Comparisons of model exchange energies, holes, and potentials with exact Hartree-Fock results in selected atoms are very encouraging.

A chemical mechanism is a model of a chemical reaction network consisting of a set of elementary reactions that express how molecules react with each other. In classical mass-action kinetics, a mechanism implies a set of ordinary differential equations (ODEs) which govern the time evolution of the concentrations. In this article, ODE models of chemical kinetics that have the potential for multiple positive equilibria or oscillations are studied. We begin by considering some methods of stability analysis based on the digraph of the Jacobian matrix. We then prove two theorems originally given by A. N. Ivanova which correlate the bifurcation structure of a mass-action model to the properties of a bipartite graph with nodes representing chemical species and reactions. We provide several examples of the application of these theorems.

Temporal organization of biological processes requires massively parallel processing on a synchronized time-base. We analyzed time-series data obtained from the bioenergetic oscillatory outputs of Saccharomyces cerevisiae and isolated cardiomyocytes utilizing Relative Dispersional (RDA) and Power Spectral (PSA) analyses. These analyses revealed broad frequency distributions and evidence for long-term memory in the observed dynamics. Moreover RDA and PSA showed that the bioenergetic dynamics in both systems show fractal scaling over at least 3 orders of magnitude, and that this scaling obeys an inverse power law. Therefore we conclude that in S. cerevisiae and cardiomyocytes the dynamics are scale-free in vivo. Applying RDA and PSA to data generated from an in silico model of mitochondrial function indicated that in yeast and cardiomyocytes the underlying mechanisms regulating the scale-free behavior are similar. We validated this finding in vivo using single cells, and attenuating the activity of the mitochondrial inner membrane anion channel with 4-chlorodiazepam to show that the oscillation of NAD(P)H and reactive oxygen species (ROS) can be abated in these two evolutionarily distant species. Taken together these data strongly support our hypothesis that the generation of ROS, coupled to redox cycling, driven by cytoplasmic and mitochondrial processes, are at the core of the observed rhythmicity and scale-free dynamics. We argue that the operation of scale-free bioenergetic dynamics plays a fundamental role to integrate cellular function, while providing a framework for robust, yet flexible, responses to the environment.

The time evolution of two model enzyme reactions is represented in phase space Γ. The phase flow is attracted to a unique trajectory, the slow manifold ℳ, before it reaches the point equilibrium of the system. Locating ℳ describes the slow time evolution precisely, and allows all rate constants to be obtained from steady-state data. The line set ℳ is found by solution of a functional equation derived from the flow differential equations. For planar systems, the steady-state (SSA) and equilibrium (EA) approximations bound a trapping region containing ℳ, and direct iteration and perturbation theory are formally equivalent solutions of the functional equation. The iteration’s convergence is examined by eigenvalue methods. In many dimensions, the nullcline surfaces of the flow in Γ form a prism-shaped region containing ℳ, but this prism is not a simple trap for the flow. Two of its edges are EA and SSA. Perturbation expansion and direct iteration are now no longer equivalent procedures; they are compared in a three-dimensional example. Convergence of the iterative scheme can be accelerated by a generalization of Aitken’s δ2 extrapolation, greatly reducing the global error. These operations can be carried out using an algebraic manipulative language. Formally, all these techniques can be carried out in many dimensions.