% File src/library/stats/man/SSmicmen.Rd
% Part of the R package, https://www.R-project.org
% Copyright 1995-2017 R Core Team
% Distributed under GPL 2 or later

\name{SSmicmen}
\title{Self-Starting \code{nls} \I{Michaelis}-\I{Menten} Model}
\usage{
SSmicmen(input, Vm, K)
}
\alias{SSmicmen}
\arguments{
 \item{input}{a numeric vector of values at which to evaluate the model.}
 \item{Vm}{a numeric parameter representing the maximum value of the response.}
 \item{K}{a numeric parameter representing the \code{input} value at
   which half the maximum response is attained.  In the field of enzyme
   kinetics this is called the \I{Michaelis} parameter.}
}
\description{
  This \code{selfStart} model evaluates the \I{Michaelis}-\I{Menten} model and
  its gradient.  It has an \code{initial} attribute that
  will evaluate initial estimates of the parameters \code{Vm} and \code{K}
}
\value{
  a numeric vector of the same length as \code{input}.  It is the value of
  the expression \code{Vm*input/(K+input)}.  If both
  the arguments \code{Vm} and \code{K} are
  names of objects, the gradient matrix with respect to these names is
  attached as an attribute named \code{gradient}.
}
\author{\enc{José}{Jose} Pinheiro and Douglas Bates}
\seealso{\code{\link{nls}}, \code{\link{selfStart}}
}
\examples{
PurTrt <- Puromycin[ Puromycin$state == "treated", ]
SSmicmen(PurTrt$conc, 200, 0.05)  # response only
local({  Vm <- 200; K <- 0.05
  SSmicmen(PurTrt$conc, Vm, K)    # response _and_ gradient
})
print(getInitial(rate ~ SSmicmen(conc, Vm, K), data = PurTrt), digits = 3)
## Initial values are in fact the converged values
fm1 <- nls(rate ~ SSmicmen(conc, Vm, K), data = PurTrt)
summary(fm1)
## Alternative call using the subset argument
fm2 <- nls(rate ~ SSmicmen(conc, Vm, K), data = Puromycin,
           subset = state == "treated")
summary(fm2) # The same indeed:
stopifnot(all.equal(coef(summary(fm1)), coef(summary(fm2))))

## Visualize the SSmicmen()  Michaelis-Menton model parametrization :

  xx <- seq(0, 5, length.out = 101)
  yy <- 5 * xx/(1+xx)
  stopifnot(all.equal(yy, SSmicmen(xx, Vm = 5, K = 1)))
  require(graphics)
  op <- par(mar = c(0, 0, 3.5, 0))
  plot(xx, yy, type = "l", lwd = 2, ylim = c(-1/4,6), xlim = c(-1, 5),
       ann = FALSE, axes = FALSE, main = "Parameters in the SSmicmen model")
  mtext(quote(list(phi[1] == "Vm", phi[2] == "K")))
  usr <- par("usr")
  arrows(usr[1], 0, usr[2], 0, length = 0.1, angle = 25)
  arrows(0, usr[3], 0, usr[4], length = 0.1, angle = 25)
  text(usr[2] - 0.2, 0.1, "x", adj = c(1, 0))
  text(     -0.1, usr[4], "y", adj = c(1, 1))
  abline(h = 5, lty = 3)
  arrows(-0.8, c(2.1, 2.9),
         -0.8, c(0,   5  ),  length = 0.1, angle = 25)
  text(  -0.8,     2.5, quote(phi[1]))
  segments(1, 0, 1, 2.7, lty = 2, lwd = 0.75)
  text(1, 2.7, quote(phi[2]))
  par(op)
}
\keyword{models}
