gene regulation: example of delay equations

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.