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19 October 2001
From the Forum: Science Fair Projects
We would like to learn a bit more about the latest developments in trangenic plants for a school project we're working on. We are also planning to obtain a tomato's DNA so any kind of information would be highly apreciatted. thanx.
Sigmoid Growth Curve of Microbes
Hello!
I am a senior in high school. I am performing a follow-up to a previous experiment I performed last year. I am studying the growth rate of Rhizopus stolonifer, or common bread mold. It is actually quite interesting. I should have predicted it before starting, but the growth is turning out to plot on an Area vs. Time graph as a sigmoid curve. The carrying capacity being the maximum area the mold can obtain in the petri dish. My problem is that I am having trouble formulating a growth analysis equation. I hope to take into account error as well as the carrying capacity. Does anyone have information on the logistics of growth curves for molds or fungi? Does anyone know of references that I could look for? Please help.
I have a question along these lines myself. I would guess that actual growth would almost never be an actual sigmoid curve y = 1/(1+e**-ax) but only take roughly that shape. My reasoning is that the sigmoid curve is symmetrical and I would guess it unlikely that the factors involved in the rapid growth would exactly but inversely mirror the factors which are involved in the slowdown of growth. Perhaps a better way to model the process is to consider the two parts separately. But if Iım wrong, it will be interesting to learn the details.
Note: you can show that the curve is symmetrical by noting that
- 1/(1+e**-ax) = 1 - (1/(1+e**ax))
- all the curves go through (0,0.5) regardless of the value of a
- all the curves approach y=1 toward + infinity
- all the curves approach and y=0 toward - infinity
There are lots of sigmoid curves, but the one most often encountered in population biology is the logistic (the 1/(1+e**-ax) form that Peter suggests). The logistic curve is the solution of the differential equation
dN/dt = rN(1 - N/K)
where N is the number of critters you're talking about (perhaps in a unit like area if the organism is a mold or fungus), and K is the carrying capacity.
It is true that the logistic curve is symmetrical. It's because of an important assumption that is made when deriving the expression for dN/dt. The assumption states that the per capita birth rate decreases linearly with the population, and the per capita mortality rate increases linearly with the population. The population where the per capita mortality rate equals the per capita birth rate is the carrying capacity. The equations look like this:
b = -bo*N + b1
m = mo*N + m1where b is the per capita birth rate, and m is the per capita mortality rate. Subtracting m from b and multiplying it by N gives the rate of change of the population:
dN/dt = (b-m)N
= [(-bo-mo)N + (b1 - m1)]N
= (b1-m1)N[1 - ((bo+mo)/(b1-m1))N]
= rN[1-(N/K)]where r = (b1-m1) and K = (b1-m1)/(bo+mo).
One of the cool things about the logistic is that it shows up in lots of different contexts. I remember stumbling on it (at least the logistic curve fit the data very well) a few years ago by accident in a high school lab involving reversible chemical reactions in sulfur compounds.
I agree with you, Peter. We can't in general expect the per capita birth and death rate (or the forward and reverse reaction rates) to vary linearly with the population (or the concentration of one reactant and one product). Perhaps in some instances the per capita mortality rate rises quadratically with population due to overcrowding. We could try using polynomial fits as a second approximation, and then derive a new expression for dN/dt.
About two years ago I started messing around with population models that didn't assume linear changes of the per capita birth and death rates with population. If people are interested in talking about it I'll dig 'em up and post them.
Best Regards, Joseph Geddes
Logistic Growth Model
The exponential growth model
A(t)=A0ekt, k>0,
assumes uninhibited growth, meaning that the value of the function grows without limit, assuming that no cells die and no by-products are produced. In reality, cell division would eventually be limited by factors such as living space and food supply. The logistic growth model is an expotential function that can model situations where the growth of the dependent variable is limited.
Logistic Growth Model
P(t)=c/1+ae-bt
where a, b, and c are constants with c>0 and b>0, t is time and P is population, e is the natural logarithm. The number c is the carrying capacity because the value P(t) approaches c as t approaches infinity. Thus, c represents the maximum value that the function can attain.\
Logistic Growth Model is non-linear...
Oops, I forgot to mention the fact that the above equation is not linear...for that matter nothing in nature is, whether its astronomical, biological, chemical, or physical...even quantum mechanics is not linear. Thus the relationship between a population's growth and mortality is not linear, which is why science makes great use of logarithmic functions of one sort or another.
The logistic growth model can be applied to any natural population, the differences of course is the values of the constants.
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