biochemistry - Scale up of Bacterial growth from 250mL to 1-5L?

I am currently growing up a specific strain of E.Coli with a knockout in 40mL of growth medium (LB) in a 250mL shaker flask... My ultimate goal is to scale up this process to a 1-5L large scale setup for the lab. But for now I'd like to quantitatively characterize this 250mL scale using Monod Kinetics so that the mathematical model can be used to build the aforementioned bio-reactor.

Currently I am making two growth curves of the E.Coli in the current 250mL flask. One is OD vs. time and the other g.Dry mass vs. time. The idea of doing so is to ascertain the $\mu_g^{max}$ (maximum specific growth rate). However, I am stuck as to how to find the saturation constant (Ks). I am also stuck on how to measure the sugar concentration ([S]) at each time-point that I would measure the OD. The idea behind these two variables is to use the equation(s):

$D = \mu_g^{max} . \frac{[S]}{Ks + [S]}\\ D^{\ max} = \mu_g^{max} . \frac{[S^{\ max}]}{Ks + [S^{\ max}]}$

Am I looking for the right variables and/or using the right equations?

Thanks again for your help!

Answer

Michaelis-Menten:

$V=V_{max}.\frac{[S]}{Ks+[S]}$

Lineweaver-Burk:

$\frac{1}{V}=\frac{Ks}{V_{max}[S]}+\frac{1}{V_{max}}$

Plot $\frac{1}{V}$ vs $\frac{1}{[S]}$; find the slope and intercept of this plot.

Depending on what you consider as primary substrate (glucose??) measure the concentrations using appropriate assays. Glucose sensors (based on glucose oxidase) are available or you can do conventional biochemical assays (DNSA) for glucose and other reducing sugars. To find the growth rate, measure OD at different time-points and get the differences of OD in a time interval. A better method would be to make a scatter-plot of OD and time and fit a polynomial function. Get rate by taking the tangent of the curve at that time-point.