| In this dissertation, we study three biological applications of networks. The first one is a biological coevolution model, in which a species is defined by a genome in the form of a finite bitstring and the interactions between species are given by a fixed matrix with randomly distributed elements. Here we study a version of the model, in which the matrix elements are correlated to a controllable degree by means of an averaging scheme. This method allows creation of mutants resembling their ancestors (wildtype). We compare long kinetic Monte Carlo simulations of models with uncorrelated and correlated interactions. We find that while there are quantitative differences, most qualitative features, such as 1/f behavior in power spectral densities for the diversity indices and the power-law distribution of species lifetimes, are not significantly affected by the correlations in the interaction matrix.;The second application is the growth of a directed network, in which the growth is constrained by the cost of adding links to the existing nodes. This is a new preferential-attachment scheme, in which a new node attaches to an existing node i with probability pi(k i, k'i ) ∝ ( k'i /ki)gamma, where ki and k'i are the number of outgoing and incoming links at i, respectively, and gamma is a constant. First, we calculate the degree distribution for the outgoing links for a simplified form of this function, pi( ki) ∝ k-1i , both analytically and by Monte Carlo simulations. The distribution decays like kmuk/Gamma(k) for large k, where mu is a constant. We relate this mechanism to simple food-web models by implementing it in the cascade model. We also study the generalized case, pi(ki, k'i ) ∝ ( k'i /ki)gamma, by simulations.;The third application is the evolution of robustness to mutations and noise in gene regulatory networks. It has been shown that robustness to mutations and noise can evolve through stabilizing selection for optimal phenotypes in model gene regulatory networks. The ability to evolve robust mutants is known to depend on the network architecture. We seek answers to the following questions. How do the dynamical properties and state-space structures of networks with high and low robustness differ? Does selection operate on the global dynamical behavior of the networks? What kind of state-space structures are favored by selection? First, we analytically show that the model random networks we use are intrinsically chaotic, i.e., they do not undergo an order-to-chaos phase transition with increasing connectivity, unlike their variants found in the literature. Then, we provide a damage propagation analysis and an extensive statistical analysis of state spaces of these model networks to show that the change in their dynamical properties due to stabilizing selection for optimal phenotypes is minor. Most notably, the networks that are most robust to both mutations and noise are highly chaotic. Certain properties of chaotic networks, such as being able to produce large attractor basins, can be useful for maintaining a stable gene-expression pattern. Our findings indicate that conventional measures of stability, such as the damage-propagation rate, do not provide much information about robustness to mutations or noise in model gene regulatory networks. |
