13th International Conference on Membrane Computing - MTA Sztaki

T. Hinze, B. Schell, M. Schumann, C. Bodenstein
We parameterise the Repressilator in a way to exhibit a medium frequency limit
cycle oscillation emphasising a comparatively small amplitude in concert with
small signal values not exceeding a threshold of approximately 0.6, see Figure
2. This threshold is meant to coincide with the ambiguous “forbidden range” in
terms of a clear distinction between 0 and 1 of binarily interpreted signals.
The Goodwin Module
The Goodwin oscillator follows the scheme of a three-staged cyclic gene regulatory
network consisting of two subsequent activating transitions along with
a single inhibition completing the loop by a negative feedback [12]. According
to the internal balance of reaction velocities, the resulting oscillatory waveform
might vary from an almost sinusoidal behaviour towards an asymmetric λ-like
shape. Here, a fast growing edge is combined with a slightly sigmoidal diminishment
of the signal. This makes the Goodwin oscillator a promising candidate
for naturally plated limit cycle oscillations. The non-probabilistic P module
goodwin = (∅, {X},F) containing three ODEs
Ẋ =
H
1+Z 9 − k 4X Ẏ = k 1 X − k 5 Y Ż = k 2 Y − k 6 Z
defines the Goodwin oscillator in its original form [12]. The degradation velocities
most significantly determine its oscillatory frequency. Our configuration
shown in Figure 3 is focused on a lower frequency oscillation of a high amplitude
spanned by signal values altering between approx. 0.6 and2.5. In contrast
to the Repressilator’s parameterisation, we intend to face the binarily operating
signal postprocessing units with intense signals revealing high values for a
comparatively longer amount of time.
Fig. 3. Goodwin oscillator reaction network (left) in its original form. We assume the
concentration course of species X over time (right) to act as module output. Higherorder
Hill kinetic’s parameter setting H =1.5,k i =0.05 for i ∈{1,...,6} chosen
to maintain slightly plated oscillations whose amplitude enables signal values between
approx. 0.6 and 2.5; initial concentrations: X(0) = 1.0,Y(0) = 1.6,Z(0) = 1.7
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