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DCDS-S

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.

DCDS-B

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.

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