I am trying to construct a simple ODE system model for experimental data describing yeast's growth and its glucose consumption. So far my attempt is the following:
$\frac{dG}{dt} =-v_1\\ \frac{dY}{dt} =v_1-v_2\\ v_1= \frac{k_1\,.\,Y\,.\,G}{1+k_2\,.\,G}\\ v_2= k_3\,.\,Y\\$
I have no knowledge about my three kinetic constants $k$, so I am trying at the same time to estimate them through fitting my system to experimental data.
So far, my fit looks like this (in my graph the M stands for G)
My question is, if someone could advise me what kind of function should I try out in order to avoid that the decrease of yeast reach values so near to zero?
I was thinking about a piecewise function; however, I have no idea if that could be a good solution, or there is a way to do it through a continuous one.
In case that could be worth to know, I am estimating the unknown constants with Pyomo for Python.
* EDIT 08.08.2016: * My data is time(hours), glucose(mg/L) and yeast (mg/L)
time glucose yeast
0 94.4 3.43
4 83.5 4.83
8 72.2 6.13
12 61.2 7.06
16 50.6 7.84
20 41.4 8.19
24 33.5 8.05
28 26.8 6.94
32 20.9 5.44
36 15.6 4.43
40 11.6 3.96
44 8.63 3.75
48 6.171 3.61
52 4.473 3.55
56 3.069 3.49
60 2.309 3.4
64 1.608 3.33
68 1.199 3.28
72 1.025 3.21
76 0.9097 3.14
80 0.8527 3.09
84 0.7957 3.02
88 0.6802 2.95
92 0.5647 2.9
96 0.4491 2.86
100 0.3336 2.79
104 0 2.74
108 0 2.69
112 0 2.64
116 0 2.6
120 0 2.55
124 0 2.5
128 0 2.45
132 0 2.41
136 0 2.36
140 0 2.31
144 0 2.27
From my tests it seems that the experimental data is inconsistent with the system of differential equations : $$\begin{cases} \frac{dG}{dt} =-v_1\\ \frac{dY}{dt} =v_1-v_2\\ v_1= \frac{k_1\,.\,Y\,.\,G}{1+k_2\,.\,G}\\ v_2= k_3\,.\,Y\\ \end{cases}$$ Is there not a typo ? Are you sure that these differential equations are a correct model for the physical experiment ?
For example, among other tests : $$\frac{v_2}{Y}=-\frac{\frac{dG}{dt}+\frac{dY}{dt}}{Y}$$ The drawing should show approximately an horizontal line, since $\frac{v_2}{Y}=k_3$
It is far to be true (figure below). I cannot say where is the mistake.
I think that this inconsistency is the main cause of your difficulty to make the curve issued from the system of equations to fit with the experimental data.
In preliminary tests, one can use only simple approximates, such as : $$\left(\frac{dG}{dt}\right)_k\simeq \frac{G_{k+1}-G_{k-1}}{t_{k+1}-t_{k-1}}$$ $$\left(\frac{dY}{dt}\right)_k\simeq \frac{Y_{k+1}-Y_{k-1}}{t_{k+1}-t_{k-1}}$$ For each point, the sum gives an approximate of $v_2$, then of $-\frac{v_2}{Y}$
.
PROCEDURE for computing approximates of $k_1$ , $k_2$ , $k_3$ :
For information, the integral equations used are : $$\begin{cases} G(t)=-\frac{k_2}{2}\left(G(t)\right)^2-k_1\int G(t)Y(t)dt \\ G(t)+Y(t)=k_3\int Y(t)dt \end{cases}$$ This system of integral equations comes from the integration of the above system of differential equations. The general principle of the method is presented in the paper : https://fr.scribd.com/doc/14674814/Regressions-et-equations-integrales .
NUMERICAL EXAMPLE :
The goal is ( for an example of exact values $k_1$ , $k_2$ , $k_3$ ) to compare to the results $k_1$ , $k_2$ , $k_3$ obtained with the above procedure of computation.
So, it is necessary to chose an arbitrary example of values $k_1$ , $k_2$ , $k_3$ , together with initial values $G(t_1)$ and $Y(t_1)$, for example $G(0)=80$ , $Y(0)=4$.
Then, the system of ODEs is numerically solved and the values $G(t_k)$ and $Y(t_k)$ are recorded along the computation. These values are rounded. They are shown on the above figure.
From this data, the procedure of regression leads to the computed values of $G(t_k)$ and $Y(t_k)$ shown on the figure.