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RE: Parametric plot


How about this...

data = {Km -> 10, Vmax -> 15};

eqn = (s0/Km)*x - Log[1 - x] == (Vmax/Km)*t;
Solve[eqn, t][[1,1]];
t[s0_][x_] = t /. % /. data;
(1/15)*(s0*x - 10*Log[1 - x])


ParametricPlot[Evaluate[MapThread[{t[#1][x], x} & ,
     {{1, 20, 100}}]], {x, 0, 1},
   Epilog -> {Text[Vmax == (Vmax /. data), {12, 0.8},
      {-1, 0}], Text[Km == (Km /. data), {12, 0.7}, {-1, 0}],
     Text[StyleForm[1, Background -> Linen],
      {t[1][0.8], 0.8}], Text[StyleForm[20,
       Background -> Linen], {t[20][0.7], 0.7}],
     Text[StyleForm[100, Background -> Linen],
      {t[100][0.5], 0.5}]}, Frame -> True,
   FrameLabel -> {t, x}, PlotLabel ->
   SequenceForm[ "Batch Kinetics for Various ", Subscript[S, 0]],
   Background -> Linen, ImageSize -> 450];

I used MapThread to build a table of parametrizations for the various S0's.
I used Epilog to add text labeling to the plot and to the three curves.

Many years ago when I was working as a programmer for a biochemist my wife
turned to me one evening and asked: "Was Einstein a biochemist?" This left
me completely confused but it was all cleared up several days later when she
received the following communication from The Relativity Institute.

Dear Ms. Park,

Never have we been so smitten
to be confused with Michaelis-Menten.
This grevious error as tried us sorely.

		Signed, Yours Truly,

But now it seems that Bianchi is also a biochemist!

David Park
djmp at

From: Danilo Bianchi [mailto:kaiowaa at]
To: mathgroup at

A str batch reactor model following Michaelis-Menten kinetics gives
the function:
(S0\Km)x - Log[1 - x] = (Vmax\Km) t
where x is the proportion of substrate converted, S0 is, say, 1, 20
and 100 g/L, Km is 10 and Vmax is 15.
How to plot the curves (in the same graphic) for x varying from zero
to 1 against t (from zero to 15, with the given values?

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