Growth of Bacteria 

from:  http://www-micro.msb.le.ac.uk/LabWork/bact/bact17.htm

1. m - the growth rate constant

2. Mean generation time

1. Calculate m - the growth rate constant:

During the exponential (or logarathmic) growth phase, a bacterial culture mimics a first-order chemical reaction, i.e. the rate of increase of cells is proportional to the number of bacteria present at that time. The constant of proportionality, µ, is an index of the growth rate and is called the growth rate constant:

Rate of increase of cells = µ x number of cells

The value of µ can be determined from the following equation:

ln Nt - ln N0 = µ(t - t0)

in other words:

the natural log of the number of cells at time t minus the natural log of the number of cells at time zero (t0) equals the growth rate constant multiplied by the time interval.

For most purposes, it is easier to use log10 values rather than natural logs, so the above equation can be converted as follows:

log10 N - log10 N0 = (µ/2.303) (t - t0)

or alternatively:

µ = ( (log10 N - log10 N0) 2.303) / (t - t0)

By measuring the increase in the number of cells during a certain time period, the growth rate constant (µ) can be calculated. In the experiment you have just done:

t0 = 1.5h:
N = 8.4x101,
log10 N = 1.92

t = 8.5h:
N = 3.39x108,
log10 N = 8.53

Therefore in this case:

µ = ( (log10 N - log10 N0) 2.303) / (t - t0)

= ( (8.53 - 1.92) 2.303) / (8.5 - 1.5)

= (6.61 x 2.303) / 7

= 15.22 / 7

= 2.18 hour-1

Note that µ and g are related to each other: µ = ln2/g = 0.693/g

Calculate the mean generation time:

The mean generation time or "doubling time" (g) is the average time required for all the components of the culture to double. This is calculated from the following equation:

log10 Nt = log10 N0 + g log102

or alternatively:

g = (log10 Nt - log10 N0) / log102

Again, for the experiment you have just done:

t0 = 1.5h:
N = 8.4x101,
log10 N = 1.92

t = 8.5h:
N = 3.39x108,
log10 N = 8.53

g = (log10 Nt - log10 N0) / log102

= (8.53 - 1.92) / 0.301

= 21.96 generations in 7 hours

= 7 / 21.96 = 0.32 hours (x 60 = 19.2 minutes)

Note that µ and g are related to each other: µ = ln2/g = 0.693/g