Re: Correct codes of modeling Michaelis-Menten elimination using concentration-time data

From: alison@c255.ucsf.edu Subject: Re: Correct codes of modeling Michaelis-Menten elimination using concentration-time data Date: Fri, 29 Sep 2000 11:15:41 -0700 (PDT) Jianguo Li compared these two differential equations: DADT(1)=-VM*(A(1)/S1)/(KM+A(1)/S1) (Equation 1) DADT(1)=-VM*A(1)/(KM+A(1)/S1) (Equation 2) He got better results with the second. He said: I think the amount A(1) in the numerator of M-M equation was implicitly scaled by S1 if S1 had been defined in $PK block, just like the case of defining a deferential equation for a first-order elimination ( DADT(1)=-Ke*A1(1)), it is incorrect to explicitly code A1(1) as A(1)/S1 if S1 had been defined in $PK. Lewis Sheiner responded: There is no scaling in DES. The reason for the difference is that the standard definition of Vm is a maximum rate, i.e. amount/time. The Users Guide equation uses that convention. If you do not scale A(1) by S(1) then the Vm estimate you get is really Vm/Vd, with units of concentration per time. This is the parameter Dr Li uses (e.g. "Vm=0.253 mg/(ml*hr)"); it is no mistake to redefine a parameter in this way if one chooses, so long as the definition is clearly stated and the usage is consistent. However I believe it is inadvisable in this instance to do so. To minimize confusion, we should try, I think, to stick to the the conventions of the fields from which we borrow. I just want to amplify what Lewis said a bit. PREDPP uses scale S1 only in the ERROR block, after compartment amounts have been computed. The reserved value F in the ERROR block is a scaled compartment amount. (In this case, F = A(1)/S1.) The kinetics implemented in the differential equation can be written in terms of amounts A(1), or concentration A(1)/S1. It is simply a question of the units of KM and VM. WIth this change to $PK in the first control stream, he can leave Equation 1 as-is and get the same results as with Equation 2: VM=THETA(1)*THETA(3) (instead of VM=THETA(1)) Or, he can leave the $PK block (and definition of VM) as-is, and change the differential equation thus: DADT(1) =- VM*V*(A(1)/S1)/(KM+A(1)/S1)