dMod -- Dynamic Modeling and Parameter Estimation in R

The dMod package is a framework that provides functions to generate ODEs of reaction networks, parameter transformations, observation functions, residual functions, etc. The framework follows the paradigm that derivative information should be used for optimization whenever possible. Therefore, all major functions produce and can handle expressions for symbolic derivatives.

System requirements

dMod uses the package cOde to set up ODE models as compiled C code (deSolve) or C++ code (Sundials). This means that C and C++ compilers are required on the system. On Linux, the compilers are installed by default. Windows users need to install RTools.

For parallelization, dMod uses mclapply() which is available on Linux and Mac but not on Windows. Windows users may use the foreach package to put dMod functions in a %dopar% loop.

To install dMod from the git repository, it is convenient to use RStudio. Create a "New Project" -> "Version Control" -> "Git". Use the address https://github.com/dkaschek/dMod and create project. Next, go to menu "Build" -> "Build and Reload". Once theses steps are completed, it should be possible to run the following example.

Simple example: enzyme kinetics

Load required packages

library(dMod)
library(ggplot2)

Generate an ODE model of enzyme kinetics with enzyme degradation

# Reactions
f <- NULL
f <- addReaction(f, 
                 from = "Enz + Sub", 
                 to = "Compl", 
                 rate = "k1*Enz*Sub",
                 description = "production of complex")
f <- addReaction(f, 
                 from = "Compl", 
                 to = "Enz + Sub", 
                 rate = "k2*Compl",
                 description = "decay of complex")
f <- addReaction(f, 
                 from = "Compl", 
                 to = "Enz + Prod", 
                 rate = "k3*Compl",
                 description = "production of product")
f <- addReaction(f, 
                 from = "Enz", 
                 to = ""     , 
                 rate = "k4*Enz",
                 description = "enzyme degradation")

# ODE model
model <- odemodel(f, modelname = "enzymeKinetics")

# Prediction function
x <- Xs(model)

Define observables and generate observation function g

observables <- eqnvec(
  product = "Prod", 
  substrate = "(Sub + Compl)", 
  enzyme = "(Enz + Compl)"
)

# Generate observation functions
g <- Y(observables, x, compile = TRUE, modelname = "obsfn", attach.input = FALSE)

Define parameter transformation for two experimental conditions

# Get all parameters
innerpars <- getParameters(g*x)
# Identity transformation
trafo <- repar("x~x", x = innerpars)
# Set some initial value parameters
trafo <- repar("x~0", x = c("Compl", "Prod"), trafo)
# Explicit log-transform of all parameters
trafo <- repar("x~exp(x)", x = innerpars, trafo)

## Split transformation into two
trafo1 <- trafo2<- trafo

# Set the degradation rate in the first condition to 0
trafo1["k4"] <- "0"

# Generate parameter transformation functions
p <- NULL
p <- p + P(trafo1, condition = "noDegradation")
p <- p + P(trafo2, condition = "withDegradation")

Initialize parameters and make prediction

# Initialize with randomly chosen parameters
set.seed(1)
outerpars <- getParameters(p)
pouter <- structure(rnorm(length(outerpars), -2, .5), names = outerpars)
times <- 0:100


plot((g*x*p)(times, pouter))

Define data to be fitted by the model

data <- datalist(
  noDegradation = data.frame(
    name = c("product", "product", "product", "substrate", "substrate", "substrate"),
    time = c(0, 25, 100, 0, 25, 100),
    value = c(0.0025, 0.2012, 0.3080, 0.3372, 0.1662, 0.0166),
    sigma = 0.02),
  withDegradation = data.frame(
    name = c("product", "product", "product", "substrate", "substrate", "substrate", "enzyme", "enzyme", "enzyme"),
    time = c(0, 25, 100, 0, 25, 100, 0, 25, 100),
    value = c(-0.0301,  0.1512, 0.2403, 0.3013, 0.1635, 0.0411, 0.4701, 0.2001, 0.0383),
    sigma = 0.02)
)

timesD <- sort(unique(unlist(lapply(data, function(d) d$time))))

# Compare data to prediction
plot(data) + geom_line()

plot((g*x*p)(times, pouter), data)

Define an objective function to be minimized and run minimization by trust()

# Define prior values for parameters
prior <- structure(rep(0, length(pouter)), names = names(pouter))

# Set up objective function
obj <- normL2(data, g*x*p) + constraintL2(mu = prior, sigma = 10)

# Optimize the objective function
myfit <- trust(obj, pouter, rinit = 1, rmax = 10)

plot((g*x*p)(times, myfit$argument), data)

Compute the profile likelihood to analyze parameter identifiability

# Compute the profile likelihood around the optimum
bestfit <- myfit$argument
profiles <- profile(obj, bestfit, names(bestfit), limits = c(-10, 10), cores = 4)

# Take a look at each parameter
plotProfile(profiles)