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Re: [Phys-l] Bacteria problem



On 01/04/2008 11:55 AM, Savinainen Antti wrote:

... this one is not physics

It might as well be physics, or mathematical physics. Problems
like this show up all the time in physics, and have been attacked
by famous physicists. Various methods of solution are known.

"When modelling bacterial growth, it is often assumed that the
bacteria have a certain "division time" (the time in which a single
bacterial cell divides into two) that is constant, let's say 20
minutes. As it can be easily shown,this leads to an exponential
increase in the number of bacteria. However, the "division time" is
definitely not constant in real life. What happens if we assume that
the "division time" is normally distributed

Radioactive decay events are in fact distributed. Not normally
distributed but Poisson distributed, but it's the same idea.

I assume the division time of the bacteria is *not* inherited,
but rather that the variability is due to statistical factors
that are /independently/ distributed, independent of which
division event we are talking about.

The easiest way to attack this is to focus attention on the
probability /distribution/. That is, do not focus on the
number of bacteria, but rather on the /distribution/ of
numbers. Then ask how the /distribution/ evolves over time.

Yeah, I know a distribution is more abstract and more
complicated than a simple number, and it takes a certain
amount of sophistication to talk about distributions rather
than numbers ... but if you are going to ask complex
questions you must accept some complexity in the answers.

The /distribution/ represents a macrostate. It gives the
probability of each microstate. For more about this concept
and the associated terminology, see
http://www.av8n.com/physics/thermo-laws.htm#sec-more-s
and
http://www.av8n.com/physics/thermo-laws.htm#sec-prob-space
especially
http://www.av8n.com/physics/thermo-laws.htm#fig-cups-dispersion

Any particular distribution can be plotted as a simple graph
of probability versus N. The initial condition of two bacteria
has 100% probability at N=2 and zero elsewhere. So this
initial macrostate consists of a single microstate.

Now just timestep the governing equation, to see what happens
when we go from time t to time t+dt. You will quickly discover
that it dt is small enough, details of the distribution of
division times do not matter; all that matters is an "average"
division time. That is what controls the probability that a
state with N bacteria will transition to a state with N+1
bacteria. Since you are free to take dt as small as you like,
you should take it to be rather small (but nonzero).

If you represent each macrostate as a column in a spreadsheet,
it is super-easy to represent the next macrostate (dt later)
by the next column in the spreadsheet. Plot each column.
Plot them all on the same graph, and observe how the distribution
evolves from column to column i.e. from time to time.