Exponential growth step by step

A cell that splits in two

In the chapter How do Organisms Reproduce? you met binary fission. A bacterium copies its DNA, grows, and splits into two cells. Each new cell does the same. If a culture has $500$ cells and they all split every $20$ minutes, the counts go

• $500$, $1000$, $2000$, $4000$, $8000$, ...

This is exponential growth: the count is multiplied by the same number in each equal step of time. In this lesson you compute it three ways.

Way 1: a loop

A loop does exactly what the cells do. One pass of the loop is one doubling:

It prints $1000$, $2000$ and $4000$ cells at $20$, $40$ and $60$ minutes. The line N = 2 * N replaces the old count with a new one, twice as large.

Way 2: the formula

After $n$ doublings the count is $N_0 \times 2^n$. If the doubling time is $T$, then after a time $t$ there have been $\dfrac{t}{T}$ doublings:

• $N = N_0 \times 2^{t/T}$

In Python, ** is the power: 500 * 2 ** (180 / 20) gives $256\,000.0$ cells after $3$ hours. The formula also works between the steps. After $30$ minutes, 2 ** (30 / 20) is about $2.83$, so the culture holds about $1414$ cells.

Way 3: a numpy array

np.arange(0, 181, 20) makes the times $0, 20, 40, \ldots, 180$ minutes. The formula then works on the whole array at once:

There is no loop in your code, but numpy loops over the array for you. This is the form you need for a plot.

Linear and log axes

On an ordinary, linear axis, exponential growth hides its start. The first hour looks flat, and the last hour fills the plot. A log axis spaces $1000$, $10\,000$ and $100\,000$ equally. On it, every doubling climbs the same height, so the curve becomes a straight line.

In matplotlib, one line changes the axis: plt.yscale("log"), or ax.set_yscale("log") for one panel of a figure. A straight line on a log axis is the quickest test of exponential growth in real data.

The growth rate and the doubling time

Biologists often give a growth rate $r$ instead of a doubling time. Then the growth is written with the number $e$:

• $N = N_0 e^{rt}$

The two forms agree when $e^{rT} = 2$, that is

• $T = \dfrac{\ln 2}{r}$ and $r = \dfrac{\ln 2}{T}$.

For a doubling time of $20$ minutes, $r = \dfrac{0.693}{20} = 0.0347$ per minute, or $2.08$ per hour. In numpy, np.log is the natural logarithm, $\ln$, and np.exp is the power of $e$.

Practise with new numbers

Try it in Python

The program grows the culture with a loop, checks the formula, and plots the counts on both kinds of axis.

Common mistakes

• Adding instead of multiplying. N = N + 2 adds two cells each step. Doubling is N = 2 * N.
• Mixing units of time. With $T$ in minutes, $t$ must be in minutes too. Three hours is $180$ min, not $3$.
• Writing ^ for a power. In Python, 2 ^ 3 is a bitwise operation and gives 1. The power is 2 ** 3.
• Using np.log10 for $\ln$. The doubling time needs the natural log: np.log(2) is $0.693$, but np.log10(2) is $0.301$.
• Trusting a linear plot. Real counts that look flat at the start may still be growing fast. Look at them on a log axis.

Remember these

• Exponential growth: the count is multiplied by the same number in each equal step of time.
• $N = N_0 \times 2^{t/T}$, or $N = N_0 e^{rt}$, with $T = \dfrac{\ln 2}{r}$.
• A loop with N = 2 * N follows the doublings; N0 * 2 ** (t / T) jumps to any time, for a number or a numpy array.
• On a log axis, plt.yscale("log"), exponential growth is a straight line.

Check yourself

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

1. A culture starts with $50$ cells and doubles every hour. How many cells are there after $6$ hours?

2. What does 2 ** (90 / 30) give, and what does it mean for a culture?

3. Why does exponential growth look like a straight line on a log axis?

4. A growth rate is $r = 0.1$ per minute. What is the doubling time?

5. A student writes N = 500 * 2 ^ (t / 20). What goes wrong?