# Fun with Rhizobia

By now you know how to find the doubling time from a variety of different situations, and how to convert doubling time into a "growth rate" to use with the exponential growth equation, so you can compare 2 or more populations.

So let's try some practice problems.

### You order two bacterial strains from Amazon: L. acidophilous to make yogurt, and R. japonicum to help your vegetable garden. According to the website, L acidophilous has a doubling time of 66 minutes, while R. japonicum has a doubling time of 344 minutes. But when you receive the package, you find that only one strain has been included, and its label was mangled so that it only reads "N_{t} = N_{0} * e^{(0.002 t)}". Should you put this one in the yogurt, or dig it into your garden?

(To make this problem interactive, turn on javascript!)

- I need a hint ... : what is the growth rate, m?

- ...another hint ... : you need to solve the equation m = ln(2)/doubling time to find out what the doubling time is.

#### I think I have the answer: in the garden.

### After putting the Rhizobium in your garden, you don't notice much difference for several days. After a few irate phone calls, you discover that you applied the Rhizobium at 1/100 of the necessary strength. Given the growth rate of Rhizobium above, how long will it take for Rhizobium to get up to a useful level

(To make this problem interactive, turn on javascript!)

- Try an estimate first: how many doubling times? between 6 and 7. So, 6.5 * 344 minutes = 2236 minutes, or 37 hours

- Now try an exact answer : Increasing 100 fold is the same as setting N
_{0}to 1 and N_{t}to 100.

- Plug and chug... : 100 = 1 * e
^{(0.002t)}

- take the ln of both sides : ln 100 = 0.002 t

#### I think I have the answer: t = 2302 minutes, or 38.4 hours.

### But wait, the conditions in your garden are hardly optimal. In fact, the doubling time for R. japonicum is about 10 times longer than in the lab. How does this affect the time you need in your garden? And what is the new growth rate m?

(To make this problem interactive, turn on javascript!)

- I need a hint ... : You still need the same number of doublings to get to a 100-fold increase

- ...another hint ... : But each doubling takes 10 times as long

- ...another hint ... : So you need 10 times as much time, that is, 384 hours, or 16 days.

- ...the new growth rate... : Use m = ln(2)/doubling time

- ...another hint ... : if doubling time got 10 times bigger, m must have gotten 10 times smaller

#### I think I have the answer: time needed = 384 hours, m = 0.0002.

### So after 8 hours in the garden vs. 8 hours on the labbench, starting with 2.5 million Rhizo each, how many Rhizo do you expect?

(To make this problem interactive, turn on javascript!)

- on the lab bench... : N
_{t}= 2.5 million * e^{(0.002*8*60)}

- in the garden... : N
_{t}= 2.5 million * e^{(0.0002*8*60)}

#### I think I have the answer: 2.7x10^{6} in the garden, 6.5x10^{6} on the labbench.

Note that you did NOT get 10 times more Rhizo with a 10-times increase in growth rate. You need to be careful with exponential growth problems -- always use the equation to double-check your intuition.

### You decide to measure the growth rate in your garden. You culture the Rhizo at three different times, with the following results:

time | number cells (x10^{6}) |

day 0 | 1.2 |

day 7 | 4.0 |

day 14 | 13.5 |

What is the actual rate of growth (m) in your garden?

(To make this problem interactive, turn on javascript!)

- I need a hint ... : Find the doubling time first
- ...another hint...: m = ln(2)/doubling time

#### I think I have the answer: m = 0.17

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