Random Boolean networks that model genetic networks show transitions between ordered and disordered dynamics as a function of the number of inputs per element, K, and the probability, p, that the truth table for a given element will have a bias for being 1, in the limit as the number of elements N → ∞. We analyze transitions between ordered and disordered dynamics in randomly constructed ordinary differential equation analogues of the random Boolean networks. These networks show a transition from order to chaos for finite N. Qualitative features of the dynamics in a given network can be predicted based on the computation of the mean dimension of the subspace admitting outflows during the integration of the equations.

formulations of the regulation of gene expression as random Boolean switching networks have been studied extensively over the past three decades. These models have been developed to make statistical predictions of the types of dynamics observed in biological networks based on network topology and interaction bias, p. For values of mean connectivity chosen to correspond to real biological networks, these models predict disordered dynamics. However, chaotic dynamics seems to be absent from the functioning of a normal cell. While these models use a fixed number of inputs for each element in the network, recent experimental evidence suggests that several biological networks have distributions in connectivity. We therefore study randomly constructed Boolean networks with distributions in the number of inputs, K, to each element. We study three distributions: delta function, Poisson, and power law (scale free). We analytically show that the critical value of the interaction bias parameter, p, above which steady state behavior is observed, is independent of the distribution in the limit of the number of elements N--> infinity. We also study these networks numerically. Using three different measures (types of attractors, fraction of elements that are active, and length of period), we show that finite, scale-free networks are more ordered than either the Poisson or delta function networks below the critical point. Thus the topology of scale-free biochemical networks, characterized by a wide distribution in the number of inputs per element, may provide a source of order in living cells. (c) 2001 American Institute of Physics.

Detailed KMC-MD (kinetic Monte Carlo-molecular dynamics) simulations of hyperthermal energy (10-100 eV) copper homoepitaxy have revealed a re-entrant layer-by-layer growth mode at low temperatures (50K) and reasonable fluxes (1 ML/s). This growth mode is the result of atoms with hyperthermal kinetic energies becoming inserted into islands when the impact site is near a step edge. The yield for atomic insertion as calculated with molecular dynamics near (111) step edges reaches a maxima near 18 eV. KMC-MD simulations of growing films find a minima in the RMS roughness as a function of energy near 25 eV. We find that the RMS roughness saturates just beyond 0.5 ML of coverage in films grown with energies greater than 25 eV due to the onset of adatom-vacancy formation near 20 eV. Adatom-vacancy pairs increase the island nuclei density and the step edge density, which increases the number of sites available to insert atoms. Smoothest growth in this regime is achieved by maximizing island and step edge densities, which consequently reverses the traditional roles of temperature and flux: low temperatures and high fluxes produce the smoothest surfaces in these films. Dramatic increases in island densities are found to persist at room temperature, where island densities increase an order of magnitude from 20 to 150 eV.