The following example was obtained from the page of  Prof. Nick Monk of the University of Sheffield, England. It describes the gene regulation of the cycle of the protein Hes1. ¨X¨ is the concentration of mRNA that is translated into protein Hes1, with a concentration ¨y¨.

# gene regulation with delay

# x: concentration of Hes 1-mRNA

# y: concentration of protein Hes1

# tau: delay

# average life Hes1-mRNA 24min => mu=ln(2)/24=0.028881132523

# average life prot. Hes1 22min => nu=ln(2)/22=0.031506690025

# Axel Osses

dx/dt = -mu*x+g(delay(y,tau))

dy/dt = -nu*y+x

g(y)=1/(1+(y/y0)^n)

# Initial Conditions

init x = 3

init y = 3

y0=3

aux XP=x*40

# Parameters

param mu=0.028881132523

param nu=0.031506690025

param n = 5

param tau=18.5

@ bounds = 100000

@ TOTAL = 490

@ DELAY = 20

@ DT=0.1

@ xplot = t, yplot = y

@ xlo = -tau, xhi = 800, ylo = 0, yhi = 260

done

The experimental results show a stationary solution and a phased oscillation of both concentrations. This is modelled by assuming that it exists a delay in the influence of the concentration of the protein in the inhibition of the transcription. A simultaneous plot of ¨y¨ and ¨xp=40*x¨ is given in function of time for large times in order to obtain the behavior observed experimentally. We take p = 1, q = 1.

more about this model:

https://www.cell.com/current-biology/abstract/S0960-9822%2803%2900494-9