13th International Conference on Membrane Computing - MTA Sztaki

T. Hinze, B. Schell, M. Schumann, C. Bodenstein
3 Exploration of Chemical Frequency Dividers Inspired
by Periodical Cicada’s Life Cycles
In a first application study inspired by periodical cicada’s life cycles, we demonstrate
the practicability of our P meta framework Π π↑↓ =(M,P) for tracing and
experimental exploration of controlled module assembly towards new (i.e. more
or less unexpected) behavioural patterns of the resulting entire system. First, we
introduce in brief a pool of non-probabilistic P modules sufficient to interact as a
chemical frequency divider 1:17. To this end, we place a selection of different core
oscillators interpreted to be employed for generation of periodical trigger signals.
A binary signal separator complements the pool of modules by its capability of
binarisation which converts gradually or smoothly altering signal courses into
a toggling manner whereas signal values ≥ 1andthosecloseto1convergeto
1 while signal values of ≈ 0.6 and smaller become forced down against 0. In
addition, we construct a logical unit whose function is a binary chemical counter
modulo 17 based on a cycle of five-bit states. Please note that the logical unit
remains unchanged during the whole study. After providing the pool of modules,
we explore the effect of different core oscillators on the behavioural pattern of the
entire frequency divider system in the presence or absence of the binary signal
separator. Although leaving intact the logical unit, we observe new frequency
division ratios of 1:3, 1:5, and 1:6 just by the effect of module assembly.
3.1 Sketching the Pool of Individual Modules
Taking into account a Brusselator, a Repressilator, and a Goodwin oscillator, we
allow for a pool of core oscillators assumed to be formerly emerged independently
from each other and based on different molecular mechanisms. Each individual
module is considered to be fixed including its previously chosen setting of kinetic
parameters. For all simulation studies carried out in this section, we utilise a
consistent time unit.
The Brusselator Module
The Brusselator derived from the Belousov-Zhabotinsky reaction is a tool approved
for the generation of spiking oscillations forming a limit cycle [1, 28]. Here,
the oscillatory persistence is exclusively reached by a positive feedback effect of
an autocatalytic loop. The non-probabilistic P module brusselator = (∅, {S},F)
is completely based on mass-action kinetics captured by five ODEs in F :
P ˙ = −k 1 PT; ˙Q = −k 3 Q; Ṡ = k 1 PT − k 2 ST 2 ; T ˙ = −k 1 PT + k 2 ST 2 +
k 3 Q−k 4 T ; Ẇ = k 4 T . Figure 1 depicts the underlying topology of the reaction
network in conjunction with the selected parameter setting. Reaction velocities,
particularly those of decay T −→ k4
W producing waste W ,mainlydeterminethe
oscillation frequency. Our parameter setting avoids a transient phase and enables
a lower frequency oscillation with distinctive spikes.
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