# Example 1: The JAK-STAT signaling pathway

The JAK-STAT pathway is a typical intracellular signal transduction pathway. The JAK2 protein located at the Epo receptor gets phosphorylated as soon as Epo binds to the receptor. A cascade via STAT5 phosphorylation, dimerization, internalizaton into the nucleus and export of the unphosphorylated STAT5 begins. The mathematical model and experimental data upon which this vignette is based is published in Swameye et. al, Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by data-based modeling, Proc. Natl. Acad. Sci., 2003.



### Exporing the parameter space

Sometimes, the system has many local optima. A single fit converges to a neighboring optimum and we would like to spread the initial positions of optimization throughout the parameter space. This procedure is implemented in the multi-start trust procedure mstrust():

fitlist <- mstrust(obj + constr,
center = myfit$argument, fits = 20, cores = 1, sd = 2, fixed = fixed, conditions = "Epo") pars <- as.parframe(fitlist) plotValues(pars, tol = .1) plotPars(pars, tol = .1)  In our case we find several local optima in the proximity of the original fit. Solutions for a parameter frame can be simulated by the predict() function. To see both, internal states and observables, the controls of g are changed via the controls() function. controls(g, NULL, "attach.input") <- TRUE prediction <- predict(g*x*p, times = 0:60, pars = unique(pars[pars$converged, ], tol = .1),
data = data,
fixed = c(multiple = 2))

library(ggplot2)
ggplot(prediction, aes(x = time, y = value, color = .value, group = .value)) +
facet_grid(condition~name, scales = "free") +
geom_line() +
geom_point(data = attr(prediction, "data")) +
theme_dMod()


### Identifiability analysis by the profile likelihood method

Next, we might be interested if the parameters we get out are identifiable. Therefore, we compute so-called likelihood profile for each parameter. The profile likelihood for the $i^{\rm th}$ parameter is defined as the optimimum of the following objective function [\tilde\chi^2_i (p, \lambda, c) = \chi^2(p) + \lambda(p_i - c), ] i.e. the optimum over all parameters satisfying the constraint $p_i = c$.

myprofile <- profile(obj + constr, pars = myfit$argument, whichPar = "s_EpoR", fixed = fixed) plotProfile(myprofile)  The profile indicates that the parameter s_EpoR is entirely determined by the prior. The data contribution is constant over orders of magnitude. The reason for this is revealed by the parameter paths, i.e. the trace in parameter space plotPaths(myprofile)  The parameters s_EpoR, p1 and pEpoR compensate each other. This non-identifiability is resolved by fixing two of the three paramters, e.g. p1 = pEpoR = 1. fixed <- c(p1 = 0, pEpoR = 0, multiple = 2) # log values pars <- mu[setdiff(names(mu), names(fixed))] myfit <- trust(obj + constr, pars, rinit = 1, rmax = 10, fixed = fixed) myprofile <- profile(obj + constr, pars = myfit$argument, whichPar = "s_EpoR", fixed = fixed)

plotProfile(myprofile)


Now, the situation is reversed. The parameter s_EpoR is fully determined by the data and the influence of the prior is low.

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dMod documentation built on Jan. 27, 2021, 1:07 a.m.