Otdfbt

I want to draw a simplified Michaelis-Menten kinetic (monod-function) to compare it with a linear function.

Minimum Working Example (MWE):

documentclassstandalone
usepackagepgfplots
usepackageamsmath

pgfplotssetcompat=1.14, /pgf/declare function=f1(x)=ln(x);% <- This is the exponential function which needs to be optimized

begindocument
 begintikzpicture
 beginaxis[
 ymin = 0,
 xmin = 0,
 xmax = 1,
 ymax = 0.9,
 axis x line = bottom,
 axis y line = left,
 ]

% addplot[no marks, samples=100, draw=blue] f1(x);% This is the exponential graph based on the function
 addplot[no marks, samples=100, draw=black, thick] coordinates(0,0) (0.2020725942,0.35);%
 addplot[no marks, samples=100, draw=black, thick] (0.2020725942,0.35) to [out=60,in=180] (0.8,0.7) to [out=0,in=0] (1,0.7);%
 draw[draw=black, dashed] (0,0.7) -- node[above] (y_texttot) ++(0.8,0.0);%
 draw[draw=black, dashed] (0,0.35) -- node[above] (fracy_texttot2) ++(0.2020725942,0) -- (0.2020725942,-0.35);%
 endaxis
 endtikzpicture
enddocument

Screenshot of the result:

Description of the issue:

How can I replace the current graph with an exponential graph?

Start point of the exponential graph:

• Start point: x = 0.2020725942,
  y = 0.35,
  angle = 60°,


• End point: y = ~ 0.7 (of course, wherever the e-function would end)

As soon as I activate the graph with the exponential function, my whole diagram will be distorted. How to implement an exponential graph based on the upper values correctly?

One way is via this (note this uses a differnt function than yours). Your MWE is not wrong IMO. However, due to varying domains, your final axis is getting mixed-up.

Nevertheless, you can obtain your desired solution with a summation of two-exponents.

documentclassamsart
usepackagepgfplots
pgfplotssetcompat=newest
usepackagetikz
begindocument 
 begintikzpicture
 beginaxis[
 scaled ticks=false,
 xmin=0,
 xmax=1,
 ymin=0,
 ymax=1.2,
 xlabel=x axis label,
 ylabel=y axis label,
 axis x line = bottom,
 axis y line = left,
 ]
 addplot[domain=0.2:1.2, samples=1000, red, ultra thick,smooth] (1-e^(-5*x)-exp(-10*x))*0.7;
addplot[no marks, samples=100, draw=black, thick] coordinates(0,0) (0.2020725942,0.35);%
draw[draw=black, dashed] (0,0.7) -- node[above] (y_texttot) ++(1,0.0);%
draw[draw=black, dashed] (0,0.35) -- node[above] (fracy_texttot2) ++(0.2020725942,0) -- (0.2020725942,-0.35);%
 endaxis
 endtikzpicture
enddocument

to get:

I would not find a function for that. A curve with exact starting angle (60°) and ending angle (180°) is enough here.

And also, why don't you simply use tan function in TikZ? 0.2020725942 ≈ 0.35 × tan(30°), but certainly if you type .35*tan(30) it is more accurate than 0.2020725942.

documentclass[tikz]standalone
begindocument
begintikzpicture[scale=8,>=stealth]
draw[<->] (1,0) -- (0,0) -- (0,.9);
draw[thick] (0,0) -- (.35*tan(30),0.35) coordinate (a);
draw[thick] (a) to[out=60,in=180] (0.8,0.7) -- (1,0.7);
foreach i in 0,0.2,0.4,0.6,0.8 
 draw (i,.01) -- (i,-.01) node[below] $i$;
 draw (.01,i) -- (-.01,i) node[left] $i$;

draw (1,.01) -- (1,-.01) node[below] $1$;
draw[dashed] (0.8,0.7) -- (0,0.7) node[midway,above] $y_mathrmtot$;
draw[dashed] (.35*tan(30),0) -- (.35*tan(30),0.35);
draw[dashed] (.35*tan(30),0.35) -- (0,0.35) node[midway,above] $y_mathrmtot/2$;
endtikzpicture
enddocument