hes1modelODE {magi}

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)

Arguments

theta

vector of parameters.

x

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

tvec

vector of time points

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.

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)