A natural class of robust networks
Original entry: Ian Burgess, Fall 2009
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This article does not discuss a specific topic in soft-matter or biophysics, but rather provides a broader mathematical analysis of a specific class of dynamical systems that exists in many biological pathways and processes, and identifies dynamics that are universal to this type of system, thus having implications on the dynamics of many biological processes.
The type of system considered is what is called a scale-free network. A network in general is a collection of elements (genes, proteins etc.) whose interactions obey a certain set of rules in both how one element affects the other and which elements can interact. The toy-model network considered by the authors is that in which each element is a switch with an "on" and "off" state and the type of interaction considered is the induced switching of the state of the target element, conditional on the state of the control element. What distinguishes so-called scale-free networks from other types of networks is the manner in which the elements are allowed to interact, specifically the degree of order in the network.At one end of the spectrum are lattice or Ising type networks where each element can be pictured as being on a lattice with only nearest-neighbor interactions allowed, and is not generally applicable to biological systems. On the other end there are completely random networks where interactions occur at random and can occur between any two elements. This type of random network has two important parameters: the average number K of elements that a given element can interact with and the probability p of an element changing its state after interaction. In a biological system, p would be related to the concentrations of each element and the inter-elemental reaction rates (i.e. chemical kinetics)
The time-evolution of random networks is found to approach a cyclic behavior. One can draw analogy between such a limit cycle and a cyclic metabolic pathway in a cell. The stability of such cycles against random perturbations (e.g. switching of a single element's state) turns out to fall into two categories: chaotic, and robust. In chaotic networks, the perturbation of a single element can cause the system to jump from one cycle to another, while in the robust case, perturbations diminish over time and the final cyclic function of the network is reproducible. Clearly, metabolic pathways in functioning organisms must be robust. However for random networks, only a very small region of the K,p space allows robust behavior.
Scale-free networks differ from random networks in that there is some degree of order, and all elements are not treated as equivalent. Specifically, there exists high connectivity between only a certain small fraction of the elements, while most (but not all) of the elements behave more like in a random network. It has been found recently that many biological processes fit this type of network topology, thus drawing interest into its dynamics[2-4]. This topology is formally applied to the two-state model, by defining the probability that one element interacts with n other elements according to the relationship:
where Z is a normalization constant and <math>\gamma</math> is called the scale-free exponent, a key bifurcation parameter for determining the stability of the dynamics. In this model, robustness is determined by <math>\gamma</math> and p. The figure below compares the robustness of random networks (top) and scale-free networks (bottom). The remarkable result is that scale-free networks display a much larger robust percentage of the parameter space, compared to the random networks.
The authors do a survey of the literature on real intracellular networks in order to fit them to their model. They find a remarkable result that the majority of these can be fit with scale-free exponents that lie in the range of <math>2<\gamma<3</math>, at the onset of robust stability. What is most significant about this result is that the ubiquity of networks of this topology in this range suggests an evolutionary advantage of this type of network, as its robustness allows adaptation to varying environmental conditions.