# [R] Michaelis-menten equation

Peter Dalgaard p.dalgaard at biostat.ku.dk
Tue Jul 19 11:34:43 CEST 2005

```"Chun-Ying Lee" <u9370004 at cc.kmu.edu.tw> writes:

> Dear R users:
>    I encountered difficulties in michaelis-menten equation. I found
> that when I use right model definiens, I got wrong Km vlaue,
> and I got right Km value when i use wrong model definiens.
> The value of Vd and Vmax are correct in these two models.

How do you know what the correct value is? Are you sure that the other
values are right?

I'm a bit rusty on MM, but are you sure your "right" model is right?
Try doing a dimensional analysis on the ODE. I kind of suspect that
Vd is entering in the wrong way. Since you're dealing in
concentrations, should it enter at all (except via the conc. at time
0, of course)?

Not knowing the context, I can't be quite sure, but generally, I'd
expect Vm*Km/(Km+y) to be the reaction rate, so that Vm is the maximum
rate, attained when y is zero and Km is the conc. at half-maximum
rate. This doesn't look quit like what you have.

> #-----right model definiens--------
> PKindex<-data.frame(time=c(0,1,2,4,6,8,10,12,16,20,24),
>        conc=c(8.57,8.30,8.01,7.44,6.88,6.32,5.76,5.20,4.08,2.98,1.89))
> mm.model <- function(time, y, parms) {
>        dCpdt <- -(parms["Vm"]/parms["Vd"])*y/(parms["Km"]+y)
>        list(dCpdt)}
> Dose<-300
> modfun <- function(time,Vm,Km,Vd) {
>        out <- lsoda(Dose/Vd,time,mm.model,parms=c(Vm=Vm,Km=Km,Vd=Vd),
>               rtol=1e-8,atol=1e-8)
>           out[,2] }
> objfun <- function(par) {
>    out <- modfun(PKindex\$time,par,par,par)
>    sum((PKindex\$conc-out)^2) }
> print(fit\$par)
>  10.0390733  0.1341544 34.9891829  #--Km=0.1341544,wrong value--
>
>
> #-----wrong model definiens--------
> #-----Km should not divided by Vd--
> PKindex<-data.frame(time=c(0,1,2,4,6,8,10,12,16,20,24),
>        conc=c(8.57,8.30,8.01,7.44,6.88,6.32,5.76,5.20,4.08,2.98,1.89))
> mm.model <- function(time, y, parms) {
>    dCpdt <- -(parms["Vm"]/parms["Vd"])*y/(parms["Km"]/parms["Vd"]+y)
>    list(dCpdt)}
> Dose<-300
> modfun <- function(time,Vm,Km,Vd) {
> out <- lsoda(Dose/Vd,time,mm.model,parms=c(Vm=Vm,Km=Km,Vd=Vd),
>             rtol=1e-8,atol=1e-8)
>        out[,2]
> }
> objfun <- function(par) {
>     out <- modfun(PKindex\$time,par,par,par)
>     sum((PKindex\$conc-out)^2)}
> print(fit\$par)
>  10.038821  4.690267 34.989239  #--Km=4.690267,right value--
>
> What did I do wrong, and how to fix it?
> Any suggestions would be greatly appreciated.
>
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