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 = N₀Qⁿ

This equation shows the population N_n as a function of n generations, when N₀ 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 = N₀2^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
p² + 2pq + q² = 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'; p² = AA genotype frequency; 2pq = Aa genotype frequency; and q² = 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 10⁸ 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[H₃O⁺]

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: 2H₂O <=> H₃O⁺ + OH^-

                        K_w = 1.0 x 10^-14 =  [H₃O⁺] [OH^-]          

                        Taking the negative log of both sides of the equation gives you:  pH + pOH = 14

Description: Water ionizes to form the hydronium ion (H₃O⁺) and the hydroxide ion (OH^-).

Quantities: K_w is the equilibrium constant for this reaction.  [H₃O⁺] 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 + H₂O  <=> H₃O⁺ + A^-

                       K_a = [H₃O⁺] [A^-] / [HA]

                       % dissociation = {[H₃O⁺] / original [HA]} x 100           or         {[A^-] / original [HA]} x 100

Description: Weak acids dissociate only partially in water to form the hydronium ion (H₃O⁺) 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.  [H₃O⁺] 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).