I am trying to solve a problem which may seem quite simple, although I have been unable to find any solution in the documentation or the discussion forums. I am not a mathematician, so please bare with me.
Can Mathematica group together constant parameters in a function, such that the "true" number of parameters will be more apparent? The simplest example would be to assume we have the following function: $$ f(x)=k_1 \times k_2 \times x $$ where $k_1$ and $k_2$ are constant parameters, and $x$ is the variable. The function $f(x)$ really has only one parameter, which we can see by grouping together the two parameters such that:
$$k_3=k_1 \times k_2$$
to get the function
$$g(x)=k_3 \times x$$
Both functions $f(x)$ and $g(x)$ are equivalent. This example is very simple, and can be done manually. However, I am working on deriving complex expressions of enzyme kinetics, which can be quite large and have many parameters. The aim is to simplify the expressions, and group together the constant parameters to help reduce the expressions to more "practical" forms. To give another simple but better example from a real-world application, let:
$$ v(s)=\frac{k_1 \times k_{cat} \times e \times s}{k_1 \times s + k_2 + k_{cat}} $$
where $k_1$. $k_2$ and $k_{cat}$ are constants pertaining to reaction rates in the different catalytic steps, and $e$ is the concentration of enzyme (also constant). $v(s)$ is rate of reaction where a chemical substrate $s$ is consumed.
By dividing the numerator and denominator with $k_1$, we get:
$$ v(s)=\frac{k_{cat} \times e \times s}{s + \frac{k_2 + k_{cat}}{k_1} } $$
Now we can group the parameters. Let:
$$ v_{max} = k_{cat} \times e$$
$$ k_m = \frac{k_2 + k_{cat}}{k_1} $$
The result is an example of the most common and simplest known enzyme kinetic expression known as Michaelis-Menten kinetics, which has the form:
$$ v(s)=\frac{v_{max} \times s}{k_m + s} $$
We have reduced the number of constant parameters from 4 down to 2, and created an equivalent expression for the reaction rate $v(s)$ which is more readable by a human. Can something like this be automated or semi-automated in Mathematica?
Any help would be greatly appreciated.
