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:

enter image description here

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:

enter image description here

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?

  • $\begingroup$ Welcome to Biology.SE. I’m voting to close this question because this isn't really a biology question — it seems closer to statistics to me. Some questions for you: How much uncertainty is associated with each of the estimates? Based on that are they actually different answers? When measuring things you typically aren't finding the "correct value" you are estimating (with some amount of error). For details see the tour and help center pages starting with How to Ask questions effectively on this site. $\endgroup$
    – tyersome
    Oct 24 '21 at 19:34
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    $\begingroup$ @user68022, this seems to be a continuation to the/your question on MSE. I also think that the question is not about biology or directly related to biology (even after your edit), but falls under data analysis/statistics. You have clearly defined your model, which is a nonlinear model. You have also clearly defined $k$. Therefore, there is no question anymore about what $k$ is because you have defined it in your model. What remains is how to properly estimate $k$. $\endgroup$
    – Domen
    Oct 24 '21 at 20:11
  • $\begingroup$ If given a chance, I would prefer the first approach. Reasons being: It looks similar to a normal Bacterial growth curve ( I can see different phases lag, log etc.), Easy to track the growth pattern, easy to understand. $\endgroup$ Oct 25 '21 at 3:29
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    $\begingroup$ @jakebeal, I don't understand: Which part of my comment is it that you strongly disagree with? I don't see how your answer (below) contradicts with anything I said in the comment. In your answer, you advise to use more datapoints and obtain error estimations. But both of these are, once again, in the scope of how to properly estimate model parameter $k$. You see, everything you said in the answer will naturally follow from what I commented: When you start to think about how to properly estimate parameters, things like replicates, error estimations etc. will naturally emerge. $\endgroup$
    – Domen
    Oct 25 '21 at 14:06
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    $\begingroup$ @Domen I disagree with your statement that "I also think that the question is not about biology or directly related to biology". Please note that critical portions of my answer have to do not just with data per se, but about thinking through the relationship between data and the underlying biological system (e.g., media effects, how OD measurements actually work). $\endgroup$
    – jakebeal
    Oct 25 '21 at 14:14

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.

  • $\begingroup$ This answer suggests fitting the same equation to the same data on different scales to get better results. This is a very poor scientific practice. Note also that logistic is an ad-hoc choice - one could propose any number of curves with any number of parameters that rise exponentially in the beginning and flatten in the end, and have extremely good fits - mainly because one downweights the results in non-exponential region and "fits the noise" with extra parameters. This is no different from trancation, but uncontrolled and subject to arbitrary errors due to the curve choice. $\endgroup$ Oct 28 '21 at 7:48
  • $\begingroup$ @RogerVadim Every scientific data fit is done with respect to a model, and part of that model is the expected distribution of errors and outliers. Consider, for example, the difference between mean and median household income which are two different interpretations of the same dataset aimed at extracting different knowledge. For OP's data, different goals (e.g., focus on growth rate vs. focus on level) and knowledge about instrument range entail different choices. $\endgroup$
    – jakebeal
    Oct 28 '21 at 10:08
  • $\begingroup$ Likewise, you are incorrect about the logistic function being an ad-hoc choice. The logistic function was derived specifically as the solution for constrained population growth ODEs, and remains the standard starting point for modeling for that reason. Since you are not familiar with the derivation of growth models, here is a Khan Academy article that can serve as a starting point for you. $\endgroup$
    – jakebeal
    Oct 28 '21 at 10:21
  • $\begingroup$ @jakebeal I know where the logistic function comes from - and this is precisely why I call it ad hoc - the basis for this ODE is that it is easy to solve, i.e., it is mathematical convenience rather than biology. Biology only tells us that the equation is $dN/dt=f(N)$, where $f(N)$ has two zeros and is positive between them: $f(N_{1,2})=0, f'(N_1)>0, f'(N_2)<0$. The early exponential growth than can be shown to be a general feature when expending near $N_1$, just like the saturation. The rest is completely arbitrary. $\endgroup$ Oct 28 '21 at 10:28
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    $\begingroup$ @RogerVadim Sounds like you've got a more general argument with population biology then, which supports the use of this model as a first approximation on the basis of far more than mathematical convenience. I will prefer to follow the scientific consensus until given reason to believe that it is incorrect. $\endgroup$
    – jakebeal
    Oct 28 '21 at 10:41

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 $k(t_0-t)\gg 1$.

  • $\begingroup$ @user68022 Indeed, this makes sense, but you use only the part of the data points, where the grow can be viewed as exponential. I added a discussion to my answer. $\endgroup$ Oct 25 '21 at 8:38
  • $\begingroup$ As an alternative to simple logistic (with more parameters obviously): pnas.org/content/116/29/14698 They seem to be doing non-linear regression though. $\endgroup$ Oct 25 '21 at 8:55
  • $\begingroup$ Fitting log-logistic vs. log-OD is perfectly reasonable as a data-handling choice: it's just an effective rebalancing of error weighting and doesn't require restriction to "far from inflection point" $\endgroup$
    – jakebeal
    Oct 25 '21 at 11:01
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    $\begingroup$ @RogerVadim Mathematically, it simply depends on your expectation of distributions. If you are expecting variation to be dominated by +/- errors in the measurement, then you want to stick with an arithmetic scale. If you are expecting variation to be dominated by fluctuations in the exponential process, however, then the fit error magnitude will scale with the value, and you want to switch to a geometric (log) scale in order to avoid overfitting on the highest few data points. $\endgroup$
    – jakebeal
    Oct 25 '21 at 11:43
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    $\begingroup$ Let us continue this discussion in chat. $\endgroup$
    – jakebeal
    Oct 25 '21 at 13:53

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