An investigation into the occurrence of nonlinear attractor in cellular dynamics

Abstract

High resolution observation of a cellular system of which the components are on the scale of micro-meters or smaller is typically thwarted by several aspects: the difficulty to obtain the signal from the subject itself, the high irregularity of the biological environment, and the measurement noise. Quantitative research over this subject, because of this, could not be easy. However, when we view the system from the perspective of attractors and its properties, it is possible to achieve better understanding of the dynamics of cellular processes even with the observations of limited resolution. In this thesis, we present our research on biochemical network and neuronal network from the perspective of the occurrence of nonlinear attractors in them.

In the chapter on biochemical network, we discuss the different statistical property between two attractors that induces rhythmic behavior while interacting with intrinsic noise: limit cycle and weakly attracting stable spiral. The first type of attractor gives rise to rhythmic behavior from its deterministic dynamics alone, noise only perturbs the already existing oscillation, giving it variation; the second type of attractor gives rise to rhythmic behavior only through its interaction with noise, hence both the oscillation amplitude and its variation are closely related to the strength of the noise. Because of this difference, the statistics of the rhythmic behavior induced by these two attractors behave differently when system size changes.

Based on our theoretical result, a new method is proposed to determine which of these two mechanisms act behind an observed rhythmic behavior in biochemical system. In particular, we have applied our method in a subsequent chapter to in vitro hes1 data. We have uncovered that the observed hes1 rhythmic behavior is driven by Hopf bifurcation induced limit cycle in its deterministic dynamics.

Finally in the chapter on neuronal network, we explored into the possible presence of a chaotic attractor. Our results established that if a chaotic attractor do exist, it cannot possess an embedding dimension of 10 and less. Our analysis instead found that the neuronal network potentially exhibits a critical attractor that arises from the phenomenon of self-organized criticality. Ongoing research is actively being pursued to affirm this latter result definitely.