magi (version 1.2.5)

hes1modelODE: Hes1 equations: oscillation of mRNA and protein levels

Description

The Hes1 equations model the oscillatory cycles of protein and messenger ribonucleic acid (mRNA) levels in cultured cells. The system components \(X = (P, M, H)\) represent the concentrations of protein, mRNA, and the Hes1-interacting factor that provides a negative feedback loop.

\(P\), \(M\), and \(H\) are governed by the following differential equations:

$$ \frac{dP}{dt} = -aPH + bM - cP $$ $$ \frac{dM}{dt} = -d_M M + \frac{e}{1 + P^2} $$ $$ \frac{dH}{dt} = -aPH + \frac{f}{1+ P^2} - gH $$

where \(\theta = (a,b,c,d_M,e,f,g)\) are system parameters.

Usage

hes1modelODE(theta, x, tvec)
hes1modelDx(theta, x, tvec)
hes1modelDtheta(theta, x, tvec)
hes1logmodelODE(theta, x, tvec)
hes1logmodelDx(theta, x, tvec)
hes1logmodelDtheta(theta, x, tvec)

Value

hes1modelODE returns an array with the values of the derivatives \(\dot{X}\).

hes1modelDx returns a 3-D array with the values of the gradients with respect to \(X\).

hes1modelDtheta returns a 3-D array with the values of the gradients with respect to \(\theta\).

hes1logmodelODE, hes1logmodelDx, and hes1logmodelDtheta are the log-transformed versions of hes1modelODE, hes1modelDx, and hes1modelDtheta, respectively.

Arguments

theta

vector of parameters.

x

matrix of system states (one per column) at the time points in tvec.

tvec

vector of time points

References

Hirata H, Yoshiura S, Ohtsuka T, Bessho Y, Harada T, Yoshikawa K, Kageyama R (2002). Oscillatory Expression of the bHLH Factor Hes1 Regulated by a Negative Feedback Loop. Science, 298(5594), 840–843.

Examples

theta <- c(0.022, 0.3, 0.031, 0.028, 0.5, 20, 0.3)
x <- matrix(1:15, nrow = 5, ncol = 3)
tvec <- 1:5

hes1modelODE(theta, x, tvec)