Parameter transformation functions in dMod are generated with P(). The method argument of P() allows to specify the transformation explicitly or implicitly. Here, we explain the difference.

library(dMod)
library(dplyr)

Explicit transformations

Explicit parameter transformations express parameters (e.g. model parameters) in terms of other or new parameters. Let us assume we have model parameters X and Y. Then we could write X = a + b and Y = a - b to parameterize the original parameters in terms of new parameters a and b. The implementation reads:

p <- eqnvec(
  X = "a + b",
  Y = "a - b"
) %>% P(method = "explicit")

Here, the names of the vector denote the original parameters and the vector entries are expressions of the new parameters. We test the result:

pars <- c(a = 3, b = 2)
p(pars)

## [[1]]
## X:  5
## Y:  1

In dMod, we call X and Y the inner parameters and a and b the outer parameters.

Important: inner and outer parameters can have the same names!

Implicit transformations

Implicit parameter transformations express the current parameters in terms of constraints which need to be satisfied. The constraint is that a certain function of the parameters vanishes. We could for example require that X + Y - 2*a = 0 and X - Y - 2*b = 0 (the transformation from above reformulated). X and Y are the inner parameters, a and b are the outer parameters which determine the inner ones. To implement this, we write:

p <- eqnvec(
  X = "X + Y - 2*a",
  Y = "X - Y - 2*b"
) %>% P(method = "implicit")

Here, the names of the vector denote the parameters which should be expressed in terms of other parameters and the vector entries denote the constraints which must cancel out.

We test the result:

pars <- c(a = 3, b = 2)
p(pars)

## [[1]]
## a:  3
## b:  2
## X:  5.00000002430988
## Y:  1

Compared to the result above, the parameter transformation not only returns the implicitly defined inner parameters X and Y but also returns the outer parameters a and b which were used to determine X and Y. The reason for the different default behavior is the following application of implicit transformations.

Application of implicit transformations

A frequent application of implicit parameter transformations are steady state constraints. Let’s assume the following reactions:

reactions <- eqnlist() %>% 
  addReaction("A", "B", "k_on*A") %>% 
  addReaction("B", "A", "k_off*B")

reactions

##   Conserved quantities: 1
## 1                 1*A+1*B
## 
##   Check Educt -> Product    Rate Description
## 1           A ->       B  k_on*A            
## 2           B ->       A k_off*B

The reactions correspond to the following differential equations:

equations <- as.eqnvec(reactions)
paste("d/dt", names(equations), "=", equations)

## [1] "d/dt A = -1*(k_on*A) +1*(k_off*B)" "d/dt B = 1*(k_on*A) -1*(k_off*B)"

\((A^*, B^*)\) is a steady state if the derivatives vanish for \(A = A^*\) and \(B = B^*\). Therefore, the idea is that equations itself takes the role of the constraints. The names of equations are already A and B, i.e. the initial value parameters we aim to implicitly define.

However, frequently with steady state constraints, the equations are not independent. In our case \(\frac{dA}{dt} = - \frac{dB}{dt}\). Consequently, whenever \(\frac{dA}{dt} = 0\) is satisfied, \(\frac{dB}{dt} = 0\) is automatically satisfied, too, for all \(k_{\rm on}\) and \(k_{\rm off}\).

Keeping states

Because effectively we have only one constraint equation, we could keep one initial value, say A, and determine B such that we are in steady state. We would do this writing:

constraint <- eqnvec(
  B = equations[["B"]]
)

p <- P(constraint, method = "implicit")


pars <- c(k_on = 1, k_off = 3, A = 2)
p(pars)

## [[1]]
##     A:  2
##     B:  0.666666664640843
## k_off:  3
##  k_on:  1

Using additional constraints

Another way to deal with the situation would be to introduce additional constraints. In our case we could introduce total = A + B as a new parameter. To implement this, we write:

constraint <- eqnvec(
  A = equations[["A"]],
  B = "A + B - total"
)

p <- P(constraint, method = "implicit")

pars <- c(k_on = 1, k_off = 3, total = 3)
p(pars)

## [[1]]
##     A:  2.25000000759684
##     B:  0.749999998480632
## k_off:  3
##  k_on:  1
## total:  3

In the spirit of dMod, the parameter transformations can be concatenated. This means that parameter estimation with steady state constraints can be implemented using a mixture of explicit and implicit parameter transformations: prd <- x*pSS*p with the implicit steady state transformation pSS and the conventional transformation p, e.g. log-transformation of parameters, etc.

Some additional notes

dMod uses the package rootSolve to find the root of the constraint function. The method used is a Newton method. This means, the algorithm needs to start with some initial guess of the parameters to be determined. The user can provide initial guesses with pars, i.e., we could call p() with pars = c(k_on = 1, k_off = 3, total = 3, A = 0.1, B = 0.1). If no guess is provided, the algorithm starts with 1 for all unknown parameters.

Happy parameter transforming!!