Synchronization of Circadian Neurons and Protein Synthesis Control - Advanced Models of Neural Networks - page 111

5.2.2 Mathematical Model of the Circadian Oscillator

The considered model of the circadian oscillator comes from Neurospora Crassa
(fungus), which is a benchmark biological system. This is a third order oscillator
model. Oscillator models of higher order describe the functioning of human
circadian neurons.
The considered state variables are: x 1 : describing the concentration of mRNA,
x 2 : describing the concentration of the FRQ (frequency) protein outside the nucleus,
and x 3 : describing the concentration of the FRQ protein inside the nucleus. The
Neurospora oscillator model is described by a set of three differential equations:

K i

x 1
K M C x 1

x 1 D v s

K i C x 3 v m

(5.2)

x 2
K d C x 2 K 1 x 2 C K 2 x 3

x 2 D K s x 1 v d (5.3)

x 3 D K 1 x 2 K 2 x 3 (5.4)

This model describes the molecular mechanism of circadian rhythms in Neurospora.
The molecular mechanism is based on a negative feedback loop that regulates the
synthesis of FRQ protein which is extracted from the frq gene. Transcription of the
frq gene produces messenger RNA (mRNA). Transcription occurs in the nucleus.
At a next step, the translation of the mRNA produces FRQ protein in the cytoplasm.
Once translation process takes place in the cytoplasm, the generated FRQ protein is
transferred in the nucleus and this has as a result to inhibit further transcription of
the frq gene into mRNA. This completes the feedback loop.
The rest of the model's parameters are defined as follows: v s denotes the rate of
transcription of the frq gene into FRQ protein inside the nucleus. K i represents the
threshold constant beyond which nuclear FRQ protein inhibits transcription of the
frq gene (Fig. 5.2 ). n is the Hill coefficient showing the degree of cooperativity of
the inhibition process, v m is the maximum rate of frq mRNA degradation, K M is
the Michaelis constant related to the transcription process. Parameter K s defines the
rate of FRQ synthesis and stands for the control input to the model. Parameter K 1
denotes the rate of transfer of FRQ protein from the cytoplasm to the nucleus and K 2
denotes the rate of transfer of the FRQ protein from the nucleus into the cytoplasm.
Parameter v d denotes the maximum rate of degradation of the FRQ protein and K d
is the Michaelis constant associated with this process.
Typical values for the aforementioned parameters are: v s D 1:6 nMh 1 , K i D
1 nM , n D 4, v m D 0:7 nMh 1 , K M D 0:4 nM , k s D 1h 1 (when no control is
exerted to the protein synthesis), v d D 4 nMh 1 , K d D 1:4 nM , k 1 D 0:3h 1 , and
k 2 D 0:15h 1 .