Equations for Biology
103
Update:
April 29, 2008
The following are the mathematical relations that have been discussed
in class or are needed for the Biomorph Challenge assignments.
Population Growth and Decay
(1) Population based on generation number:
N_{n} = N_{0}Q^{n}
This equation shows the population N_{n}
as a function of n generations, when N_{0}
is the initial population and Q is the number
of offspring per parent in one generation. Does not include time.
(2) Population based on time
and rate of doubling:
N_{t} =
N_{0}2^{t/d}
This equation shows the population
N_{n} after a given time t,
depending on the doubling time d. The units
of t and d must be the same.
Does NOT include the actual number of offspring per parent. The "generation"
is defined arbitrarily as 2. This is the way populations are usually measured.
It avoids complications arising from variable family size.
Hardy Weinberg Law
p + q = 1
p^{2} + 2pq + q^{2} = 1
Description: This law predicts how gene frequencies will
be transmitted from generation to generation.
Assumptions necessary: an infinitely large, random mating population that
is free from outside evolutionary forces (i.e. mutation, migration and
natural selection), individuals survive equally.
Definitions: p = frequency of allele 'A'; q = frequency
of allele 'a'; p^{2} = AA genotype frequency; 2pq = Aa genotype
frequency; and q^{2} = aa genotype frequency.
Consequences: genetic variability can be maintained in
a population; gene frequencies will remain unchanged from one generation
to the next unless selection pressure is exerted; frequencies of heterozygous
carriers can be calculated.
Basic Wave Equation
lambda = c/f
Description: This equation shows the relation between wavelength
and frequency. Quantities: c is the speed of light (3.00 x 10^{8}
meter/sec); lambda is the wavelength; and f is the frequency of the light
wave (sometimes written as Greek nu). Units: Wavelength should be in meters
and frequency in Hertz (1 Hz = 1 cycle/second).
Photon Energy
E = hc/lambda
Description: This equation relates the photon energy to the wavelength
of the light wave. Quantities: E is the photon energy; lambda is the wavelength;
h is Planck's constant (6.626 x 10^{-34} Joules sec); c is the
speed of light.
Energy relationships
Watt = Joule per second
W = J/sec
Measuring Evolution Time with a Molecular Clock
A "molecular clock" is a gene that evolves
at a steady rate and is present in many related species.
The percent similarity of this gene between any pair of species is given
by the number of base positions in the gene that are the same between
two species.
The time that has passed since the point when two species
diverged varies approximately with the percent difference between
the two; that is:
Time since divergence of two species
is given by
(100 - X% sequence similarity) / (% change
/ years).
[Note: This simple equation only approximates real
biology. Actual animals and plants show different molecular clock rates
for different genes and species; thus it takes a supercomputer with complex
analysis to work it out.]
pH
pH = -log[H^{+}] or pH = -log[H_{3}O^{+}]
Description: pH is a measure of the acidity (also basicity)
of a solution
Quantities: [H^{+}] is the concentration of
the hydronium ion. Units: concentrations should be in the units moles/liter.
Ionization of water
K_{w}
= 1.0 x 10^{-14} for the reaction: 2H_{2}O
<=> H_{3}O^{+} + OH^{-}
K_{w}
= 1.0 x 10^{-14} = [H_{3}O^{+}] [OH^{-}]
Taking
the negative log of both sides of the equation gives you: pH + pOH =
14
Description: Water ionizes to form the
hydronium ion (H_{3}O^{+}) and the hydroxide ion (OH^{-}).
Quantities: K_{w} is the equilibrium
constant for this reaction. [H_{3}O^{+}] is the hydronium
ion concentration and [OH^{-}] is the hydroxide ion concentration.
Units: Ion concentrations are in moles
per liter (moles/liter = M).
Weak acid dissociation
K_{a} for the reaction:
HA + H_{2}O <=> H_{3}O^{+} + A^{-}
K_{a} = [H_{3}O^{+}]
[A^{-}] / [HA]
% dissociation =
{[H_{3}O^{+}] / original [HA]} x 100 or
{[A^{-}] / original [HA]} x 100
Description: Weak acids dissociate only
partially in water to form the hydronium ion (H_{3}O^{+})
and the conjugate base of the weak acid (A^{-}). K_{a}
is the equilibrium constant that mathematically describes the concentration
of all chemical species at equilibrium.
Quantities: K_{a} is the dissociation
constant for a weak acid. Values for individual acids are either supplied
in the question or located in reference tables. [H_{3}O^{+}]
is the hydronium ion concentration and [A^{-}] is the concentration
of the conjugate base. [HA] is the concentration of the acid that remains
undissociated.
Units: Concentrations are in moles per
liter (moles/liter = M). |