Random Spatial Networks: A Biological Solution to the Structure/Transport/Connection Problem
by
Donald A. Drew
Rensselaer Polytechnic Institute
Coauthors: Yanthe Pearson

Spatial networks are collections of fibers occupying a spatial region. The network is called random if we are interested in an ensemble of equivalent such networks, where the positions of the individual fibers are not identical but have some statistical “sameness.” Examples of random statistical networks include microtubule structures, capillaries, neurons, and trees. Each of these networks has a function that it can fulfill by producing a realization out of an ensemble of such networks having certain properties. For example, a microtubule network that is responsible for cell integrity must support the forces that maintain the cell shape; capillaries must deliver and/or absorb chemical species to the surrounding matrix. We discuss microscale (individual fiber) and structural (probability density function) models to describe random spatial networks. so as to relate network statistical structure to assembly dynamics of individual network fibers, and to determine the biological functionality of the network from network statistical properties. In addition, network assembly, disassembly, and interactions with the surroundings during network formation and structure can add to the understanding of the biochemistry and biophysics of fiber formation and guidance.

We shall focus on axonogenesis. Axons are the propagation elements in neurostructures in all higher species. During formation of the brain and nervous system, axons are generated by extension of processes from neural cells with dynamics determined at the growth cone. The progress of the growth cone is determined by the response of surface receptors to gradients of signaling molecules. Surface structures bind the signaling molecules, leading to changes in the assembly of the actin/microtubule structure that drives axonal growth cone motion. In this paper we introduce a two-dimensional stochastic model which captures the random behavior of axon growth to simulate axonal trajectories for cells in a homogeneous medium. We use data to evaluate the standard deviation of the angle changes on the axonal trajectories and to verify the validity of the structure of the stochastic differential equations for the axonal trajectories. For the model of growth of axons, we analyze trajectory data consisting of measurements of the position of the axon tip at different frames in the time sequence of micrographs. This data shows that the axon tip changes direction randomly, but the data is noisy due to the data collection procedures. We develop algorithms to filter out noise while maintaining the underlying dynamics of the axon growth process. We perform statistical analyses on relevant variables generated from our filtered data. We present Monte Carlo simulations of stochastic differential equation systems.

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Date received: May 1, 2008