Ph.D Thesis

Ph.D StudentSchreier Hallel Ilan
SubjectA Random Network Model of Cellular Adaptation
DepartmentDepartment of Applied Mathematics
Supervisor PROF. Naama Brenner


The capacity of cells and organisms to respond to challenging conditions in a repeatable manner is limited by a finite repertoire of pre-evolved adaptive responses. Beyond this capacity, cells can use exploratory dynamics to cope with a much broader array of conditions. However, the process of adaptation by exploratory dynamics within the lifetime of a cell is not well understood. In this thesis we demonstrate the feasibility of exploratory adaptation in a high-dimensional network model of gene regulation. Exploration is initiated by failure to comply with a constraint and is implemented by random sampling of network configurations. It ceases if and when the network reaches a stable state satisfying the constraint. We find that successful convergence (adaptation) in high dimensions requires that the network has a heavy tail out-going degree distribution (HTO), and is enhanced by the auto-regulation of the networks's hubs. The ability of these empirically-validated features of gene regulatory networks to support exploratory adaptation without fine-tuning, makes it plausible for biological implementation.

In order to explain the adaptation advantage of HTO networks we develop a novel theory which enables us to better understand and  analyze such networks. We focus on the probability of random network ensembles to converge to attractors, and how this probability is modulated by ensemble topology. Here, we propose to take advantage of the finite size of real networks in order to develop a novel approximation for network ensembles with a broad connectivity distribution. Specifically, we approximate heterogeneous networks using "star networks" (STR) which are homogeneous networks with a single central outgoing hub. This approximation is based on the dominant role of a handful of hubs in a finite networks. We use this approximation in order to map the problem of dynamics of heterogeneous networks to the dynamics of a homogeneous network driven by an external input.  

Finally, we examine further dynamical properties beyond the probability of convergence to fixed-points such as node switching rates, node clustering, and the effective dimesionality of the trajectories of the networks. We demonstrate that the STR approximation and the HTO networks also show similar behavior for these dynamical properties.