The dynamics of zeroth-order ultrasensitivity: A critical phenomenon in cell biology
Qingdao Huang Hong Qian
It is well known since the pioneering work of Goldbeter and Koshland [Proc. Natl. Acad. Sci. USA, vol. 78, pp. 6840-6844 (1981)] that cellular phosphorylation- dephosphorylation cycle (PdPC), catalyzed by kinase and phosphatase under saturated condition with zeroth order enzyme kinetics, exhibits ultrasensitivity, sharp transition. We analyse the dynamics aspects of the zeroth order PdPC kinetics and show a critical slowdown akin to the phase transition in condensed matter physics. We demonstrate that an extremely simple, though somewhat mathematically "singular" model is a faithful representation of the ultrasentivity phenomenon. The simplified mathematical model will be valuable, as a component, in developing complex cellular signaling netowrk theory as well as having a pedagogic value.
keywords: Ultrasensitivity; PdPC; critical phenomenon; cooperative processes.
Stochastic dynamics: Markov chains and random transformations
Felix X.-F. Ye Yue Wang Hong Qian
This article outlines an attempt to lay the groundwork for understanding stochastic dynamical descriptions of biological processes in terms of a discrete-state space, discrete-time random dynamical system (RDS), or random transformation approach. Such mathematics is not new for continuous systems, but the discrete state space formulation significantly reduces the technical requirements for its introduction to a much broader audiences. In particular, we establish some elementary contradistinctions between Markov chain (MC) and RDS descriptions of a stochastic dynamics. It is shown that a given MC is compatible with many possible RDS, and we study in particular the corresponding RDS with maximum metric entropy. Specifically, we show an emergent behavior of an MC with a unique absorbing and aperiodic communicating class, after all the trajectories of the RDS synchronizes. In biological modeling, it is now widely acknowledged that stochastic dynamics is a more complete description of biological reality than deterministic equations; here we further suggest that the RDS description could be a more refined description of stochastic dynamics than a Markov process. Possible applications of discrete-state RDS are systems with fluctuating law of motion, or environment, rather than inherent stochastic movements of individuals.
keywords: Markov chain entropy. random dynamical system Stochastic process

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