Re: [Phys-l] Bacteria problem

[Ludwik Kowalski <kowalskil@mail.montclair.edu>]

On Jan 4, 2008, at 11:55 AM, Savinainen Antti wrote:

> Hi again!
> I seem to be asking a lot of question these days...this one is not 
> physics but it may be of interest to some of you. I'm not asking for a 
> full solution; a  scenario of a possible solution would be nice.
>
> I'll give you a bit backgound info on the problem. It was stated by a 
> 13 year old student. He has been receiving informal teaching on high 
> school mathematics and beyond given by our proficient IB students, one 
> hour per week, since he was 8 years old. The student took his first 
> university course on mathematics (calculus) last year. His problem 
> goes like this:
> > "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 with a given mean and 
> > standard deviation, let's say with a mean of 20 minutes and a 
> > standard deviation of 5 minutes? What kind of model would describe 
> > this situation, if we also assume that the initial amount of 
> > bacterial cells is quite small, let's say 2?"
* * * * * * * * * * * * * * * * * * * * * *

1) To avoid another impulsive reply I waited several hours before 
posting this one. It was clear to me from the very beginning that the 
question was about the mathematical model for colonies, long before the 
departures from an exponential growth become significant.

2) My prediction is that fluctuations in the life times of individual 
bacteria, for example, between 1.5 and 2.5 days (the mean being 2 days) 
will have negligible effect for a bell-shape distribution. The reason 
is simple. For a given even number of bacteria, beginning with N=2 at 
time zero, one can calculate the mean life time. On the average, about 
one half of bacteria will live a little shorter and another  half a 
little longer. The mean time, for each generation, will fluctuate 
around 2 days but not much, especially for larger N. I composed a short 
Mone Carlo code which confirmed my intuitive conclusion. The results 
are shown below.

The first column shows the results of consecutive doubling, starting 
with N=2 while the second column shows times (in days) at which 
doublings are expected for the ideal case (standard deviation zero). 
Columns #3 and #4 show the corresponding times for the standard 
deviations of 0.25 and 0.75  days.

I also repeated the last column simulations several times. times 
fluctuate slightly but not much more. Here an illustration? How long 
does it take to reach N=256 when the standard deviation is 0.75 days? 
Here are nine consecutive results:

14.2, 14.3, 13.5, 14.2, 14.6, 13.6. 14.6, 13.7, 14.7

The mean, 14.1 does not differ much from the ideal case (mean=2 and 
sigma=0). Yes, some tendency seems to be present but it is small and 
not worth investigating (for example, using the Poisson distribution 
instead of Gaussian. Keep in mind that my initial N is 2, to keep all N 
even. That is why the table shows N=4 after 2 days. The mean is 2 days 
for each column.

 . . sigma=0 . . . sigma=0.25 . . . . . .  sigma = 0.75
N . . . days . . . . . days . . . . . . . . . . . days
4 . . . .2 . . . . . . 2.04 . . . . . . . . . . . 2.13
8 . . . .4 . . . . . . 3.98 . . . . . . . . . . . 3.94
16 . . . 6 . . . . . . 6.09 . . . . . . . . . . . 6.27
32 . . . 8 . . . . . . 8.10 . . . . . . . . . . . 8.32
64 . .  10 . . . . . . 10.1 . . . . . . . . . . . 10.3
128 . . 12 . . . . . . 12.1  . . . . . . . . . .  12.2
256 . . 14 . . . . . . 14.1 . . . . . . . . . . . 14.2

P.S. What follows is my code (in True Basic).

PROGRAM Bacteria
!************************
randomize
DIM ht(1:200)
let mean=2        ! fissions after 2 days, on the average
let stdev=0.50   ! stand. dev. in ht (bacteria's lifetime)
let t=0
let N=2               ! Initially 2 bacteria
do while N<200
  let sum=0
  for i=1 to N
       call GET_Y(mean,stdev,  ht(N))
       let sum=sum+ht(N)
   next i
       let meanht=sum/N
       let t=t+meanht
       let N=2*N
   print "t=";t;"   N=";N
loop
end

SUB GET_Y(mean,stdev,x)  ! Random gaussian
!******************************
let sum=0
for i=1 to 12
   let sum=sum+rnd       ! mean is 6 and st.dev. is 1
next i
let z=sum-6
let x=z*stdev+mean
end sub
_______________________________________________________
Ludwik Kowalski, a retired physicist
5 Horizon Road, apt. 2702, Fort Lee, NJ, 07024, USA
Also an amateur journalist at http://csam.montclair.edu/~kowalski/cf/