Which approach is the common one in the literature for determining the bacterial growth rate?

Question

I have the following data, which is OD600 (the second component) vs. time (the first component):

data = {{0, 0.046}, {40, 0.111}, {80, 0.291}, {120, 0.808}, {160, 1.742}, {200, 3.319}, {240, 5.017}, {280, 5.503}, {320, 5.897}}

I want to obtain the growth rate of bacteria from the above data.

If I fit a logistic function, that is, $$f(t) = \frac{L}{1 + e^{-k(t - t_0)}},$$ where $k$ is the growth rate, I obtain the following curve:

In this approach, the growth rate is: $k = 0.028$.

Now, if I fit the natural logarithm of the logistic function to the natural logarithm of OD600, I obtain:

In this case, the growth rate is: $k = 0.025$.

Which of these values (or, approaches) is the correct and common approach for determining the growth rate of bacteria?

Answer 1

The simple logistic function that you are using here is a fine first-order approximation for modeling microbial growth. Both logarithmic and linear scales are reasonable, with different tradeoffs as explained below. It is unsurprising that these two scales will give slightly different numbers, however, as the exponential approach gives more relative weight to fit errors on the lower values. I wouldn't be too concerned about the difference in numbers, though, since they are only 1.1-fold different, and that's within the likely limits of your assay precision.

Thinking about this from a biological perspective, however, there are a number of key limitations to fit accuracy that you need to take into account when you consider and present your data:

• You show only a single replicate. I would not trust this data without at least three replicates, which will give some understanding of the degree of variability encountered with your protocol.
• At the low end of the scale, precision will likely be limited by instrument precision and media density. What are your control blanks? Are the OD values that you present with or without media subtraction? Since you are showing a high final OD, I would guess that you are working with rich media, which typically has a significant OD even without any cells present.
• At the high end of the scale, remember that OD itself is a logarithmic measure. Once the level of light penetrating is very low (below ~1%, i.e., above OD 2.0), most instruments will start having difficulty quantifying the penetrating light. Are you actually measuring an OD of 6, or is this a compensated measure from a much shorter path length? If it's a compensated measure, that will again imply decreased precision on the low end.

These questions can be addressed by using controls that determine the effective linear range of your instrument with respect to your media and path length, such as the method that we validated in this interlaboratory study.

Once you know the range in which you can trust your data, then you can determine your answer on how to fit:

• Since your primary interest is growth rate, you want to focus on the exponential portion of the behavior. For this, the log-scale fit is better, since it places more weight on the range where the exponential growth dominates the behavior. Based on what you've presented, this is likely the approach that will work for you.
• If your low data is mostly outside of the effective linear range, however, you can fall back to the linear-scale fit, which will de-emphasize the low data.
• If your primary interest were the saturated level, then you would want to prioritize the linear-scale fit instead, for the converse reason.

Bottom line: you need to figure out your data limitations, based on the interaction between biology and instrument, then see if that supports the log-scale fit or if you need to fall back to linear.

Answer 2

The approach depends on what is known besides the data points. Non-linear regression for parameters $L, k, t_0$ is certainly a thing to try, as the OP author has already done in MSE - judging by the shape of the curve logistic is a good candidate and it is hard to think, how one could improve upon it without introducing additional parameters (and therefore weakening the results).

Fitting the log would make sense only if $L$ (the saturated value of the population) is known. In this case one could rearrange the equation as $$ e^{-k(t-t_0)}=\frac{L-f(t)}{f(t)},$$ the logarithm of which is $$ -k(t-t_0)=\log\left[\frac{L-f(t)}{f(t)}\right]. $$ That is, plotting the logarithm of the expression on the right against $t$ and performing linear regression one could determine $k$ as the slope of the line. Taking log of the logistic function itself does not make much sense to me...

Comments follow-up
Far from the inflection point ($t=t_0$) the curve approximately behaves as an exponential one, and for $t\ll t_0$ it indeed makes sense to make a linear fit to $\log f(t)$: $$ \log f(t) = \frac{L}{1+e^{-k(t-t_0)}}\approx \log\left[Le^{k(t-t_0)}\right]= \log{L} + k(t-t_0), t\ll t_0 $$ That is, one do first rough fitting using non-linear regression to estimate the inflection point $t_0$ and $k$, and choose to do linear regression of $\log f(t)$ against $t$ only for those data points that satisfy $k(t_0-t)\gg 1$.