Fitting growth curves to data

From counts to a model

In a laboratory, you do not know $r$ or $K$. You count cells at regular times and then ask: which curve fits these counts? The model you fit tells you the growth rate, the doubling time and the carrying capacity.

Here is a culture counted every $30$ minutes:

Each count is about twice the one before, so the doubling time is near $30$ minutes. A fit makes "about" precise.

A straight line through ln N

Take the natural log of $N = N_0 e^{rt}$:

• $\ln N = \ln N_0 + rt$

This is a straight line in $t$. Its slope is $r$, and its intercept is $\ln N_0$. So fit a straight line to the points $(t, \ln N)$:

np.polyfit(x, y, 1) finds the straight line of degree $1$ that passes closest to the points. For the counts above, the slope is $0.0226$ per minute, so the doubling time is $\dfrac{\ln 2}{0.0226} = 30.7$ minutes. The intercept gives $N_0 = e^{\text{intercept}}$.

What does "closest" mean?

A fit chooses the line that makes the sum of squared misses as small as possible. For each point, the miss is the count minus the value of the curve at that time. Squaring makes every miss positive and punishes large misses most. This is the method of least squares, and both np.polyfit and curve_fit use it.

curve_fit for the logistic

A logistic curve is not a straight line, even on a log axis. So we let scipy search. First write the model as a Python function. The first argument is the time, and the others are the numbers to find:

• curve_fit(model, t, N, p0=[...]) returns the best values in the order of the model's arguments, here r and then K;
• it also returns cov, which tells how uncertain those values are;
• p0 is a starting guess. Without a sensible guess, the search can wander off and fail.

Here $N_0$ is fixed at the first count, $12$ cells, so only $r$ and $K$ are searched. For the yeast counts the fit gives $r = 0.432$ per hour and $K = 484$ cells.

Good starting guesses

Read them off the data. $K$ is a little above the last counts, where they level off. For $r$, use the doubling at the start: here the count roughly doubles every $2$ hours, so $r$ is near $\dfrac{\ln 2}{2} = 0.35$ per hour.

Practise with new numbers

Try it in Python

The program fits a straight line to $\ln N$ for the culture, then a logistic curve to the yeast counts, and plots the fit.

Common mistakes

• Fitting a line to N instead of to ln N. Exponential counts lie on a straight line only after the log.
• Using np.log10 and then $\ln 2$. With np.log10, the slope is $\dfrac{r}{\ln 10}$. Use np.log throughout.
• Time first in the wrong place. In a model for curve_fit, the time must be the first argument.
• Reading the results in the wrong order. They come back in the order of the model's arguments after the time.
• No starting guess. Without p0, curve_fit starts every number at $1$, and the search may fail.

Remember these

• For exponential growth, $\ln N = \ln N_0 + rt$: np.polyfit(t, np.log(N), 1) gives the slope $r$ and the intercept $\ln N_0$.
• Doubling time $= \dfrac{\ln 2}{\text{slope}}$.
• A fit makes the sum of squared misses as small as possible: least squares.
• curve_fit(model, t, N, p0=guess) returns the best parameters, in order, and their uncertainties.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A fit of $\ln N$ against $t$ in hours has a slope of $0.35$. What is the doubling time?

2. The intercept of the same fit is $4.6$. About how many cells were there at the start?

3. Why square the misses, instead of just adding them?

4. A model is written def model(r, K, t):. Why does curve_fit give nonsense with it?

5. Counts level off near $900$ cells and double every $3$ hours at the start. Give a starting guess for curve_fit.