Hermine E. Smikle
The time taken for each of the bacterial cell in a population to divide is normally distributed therefore the time taken for the average cell in the population to divide is the population's the mean generation time. The time taken for the population to double in size is called doubling time. The systematic increase in bacterial numbers can be described by the exponential growth model. This model is applicable because the rate of increase in bacterial numbers is proportional to the number of bacteria at the given time. This means that the more bacteria present, the greater the increase in number.
The number e is used to describe the population growth, because the growth is a continuous process, and indicate that the population is growing without bound. The number of bacteria that will be present after n generation can be expressed as 2
n
× N
0.
The rate of increase of bacteria can also be expressed as a geometric progression. It is sometimes convenient and easier to measure the weight or density of the population of the bacteria.
If different organisms have different growth rates under unrestricted conditions, then the rate of increase in cell number can be described numerically. This is called specific growth rate and is defined by µ. µ represents the instantaneous rate of increase of a single cell, and not the increase measured at fixed intervals, and is different from doubling time. If bacterial mass is used to measure exponential growth, the increase at any point in time is represented by µ and is given as dn/dt = µn. The expression dn/dt represents the rate of increase in the number of bacteria over time. The constant of proportionality µ, represents the number of doublings per unit of time (per hr) and describes how fast the bacteria grow.
When bacteria are grown in the lab environment two assumptions called the Most Probable number are made, these are:
1.
|
The organisms are randomly distributed in the sample being studied
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2.
|
Any viable organism in the test sample will grow.
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The statistics that underline the Most Probable Number is the Poisson distribution. The Poisson distribution is a derivative of the binomial theorem. The assumption is that if a large number of samples are examined, then the likelihood of finding a positive result is small. The Poisson distribution is appropriate to describe the pattern of results that are likely to occur. The following conditions must exist to apply the Poisson distribution.
1.
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There are whole number observations
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2.
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Each number is a small fraction of the total number
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3.
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Each observation are independent of other observations
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4.
|
The probability of each outcome does not vary from one observation to the next.
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The Poisson distribution is given as P (x) = (m
x
e
-x
) / x where P is the probability, m = the average number of organisms per sample, and x = the actual number of organisms per sample
12
.
Chemical disinfectants and antibiotics are used to control the multiplication of bacteria. Two words are used to explain the decay rate or the killing of bacteria. The term bactericidal describes the act of reducing the number of bacteria, and the term bacteriostatic refers to the act of inhibiting the spread or growth of bacteria. That is, bacteriostatic antibiotics prevent cell replication, but do not kill the bacteria.
If the bacteria are exposed to a bactericidal agent over a period of time, the number will decline progressively, rather than instantaneously. The rate of killing is constant. A proportion of the total will be killed per unit of time. This can be described mathematically as N / N
0
= e
-kt
. The expression N / N
0
is the number of survivors at any given time divided by the number at the beginning, t is the time and k is the constant that describes the relationship between time and bacterial numbers.
An infectious disease that is transmitted randomly between infectious and susceptible persons is modeled by N = ΒSI. The rate of transmission N will be proportional to the product of the number of susceptible persons S, times the number of Infected persons I. Β is the constant of proportionality, and represents the probability of effective transmission. The value of Β is dependent on the micro-organism and their route of transmission.
The number of people directly infected by someone who is infected by a particular micro organism in an entirely susceptible population is labeled R
0 .
(the basic reproductive rate). R
0
= S Β D where S is the number of susceptible persons, Β the probability of effective transmission and D the period of time each person is infectious. If S, Β, and D are increased then R
0
is also increased. If the number of persons who are immune is not calculated in the equation then R
0
gives the number of new cases of the disease in the population.
Very infectious diseases have high R
0
values. Low infectious diseases have lower values close to one. R
0
values of infectious diseases have three limits
13
.
1.
|
R
0
1: The infection declines and disappears from the community
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2.
|
R
0
= 1: The infection will persist and remains endemic
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3.
|
R
0
> 1: The infection will increase in incidence and spreads as an epidemic.
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Application to Geometry
Viruses are described as having the following shapes helical, icosahedral, or complex.
Helical: The nucleic acid is contained in a cylinder made of protein arranged in a helical stack.
Icosahedral viruses: The majority of viruses are icosahedral. An icosahedron is a regular polyhedron. Viral icosahedra are constructed from 20 equilateral triangles and have 20 flat surfaces 12 corners (apexes) and 30 edges. An icosahedron has three lines of symmetry. The icosahedron houses and protects the viral nucleic acid.
Complex: Some viruses are made up of a combination of icosahedra and helices
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