MAT385 Equivalence Classes Example
Acknowledgment
This project is modified from
Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles
Brian Hopkins, editor
323 pp., paperbound, 2009. Series: MAA Notes #74
ISBN: 978-0-88385-184-5
http://www.maa.org/news/121808resources.html
It was provided to me for my use by Brian (who created it), after I saw him present the example at the 2009 Joint Mathematics Meetings in Washington, D.C. Thanks Brian!
Codon Classes
Ribonucleic acid (RNA) is used by cells to produce proteins, among other tasks. RNA can be thought of as a string of nucleotide bases drawn from the set adenine, cytosine, guanine, and uracil. A codon is an ordered triple of bases that signals for a certain amino acid to be included in the sequence; for example, the codon AUG signals for methionine. There are possible codons, and all occur in nature. However, there are not that many amino acids. In this project, you will explore various equivalence classes on the set of all codons and then compare the correspondence that occurs in nature.
For each of the following relations on codons, determine the number of equivalence classes, the size of each equivalence class, and describe the equivalence classes (you don’t necessarily need to fully write out each equivalence class).
- The two codons have the same middle base (e.g., ACU ~ GCU).
- The two codons have the same first two bases (e.g., ACC ~ ACU).
- The two codons have the same number of A’s (e.g., AUA ~ GAA).
- The two codons have the same number of C’s or U’s (e.g., ACU ~ GUU).
- The two codons are made up of the same three bases (e.g., AUA ~ UAA).
- The two codons use the same bases (e.g., AUA ~ UUA; this is different from #5).
- In fact, the codons fall into 21 classes, indicating twenty basic amino acids and a signal to stop the genetic process called translation:
alanine: {GCA, GCC, GCG, GCU}
arginine: {AGA, AGG, CGA, CGC, CGG, CGU}
asparagine: {AAC, AAU}
aspartic acid: {GAC, GAU}
cysteine: {UGC, UGU}
glutamic acid: {GAA, GAG}
glutamine: {CAA, CAG}
glycine: {GGA, GGC, GGG, GGU}
histidine: {CAC, CAU}
isoleucine: {AUA, AUC, AUU}
leucine: {CUA, CUC, CUG, CUU, UUA, UUG}
lysine: {AAA, AAG}
methionine: {AUG}
phenylalanine: {UUC, UUU}
proline: {CCA, CCC, CCG, CCU}
serine: {AGC, AGU, UCA, UCC, UCG, UCU}
threonine: {ACA, ACC, ACG, ACU}
tryptophan: {UGG}
tyrosine: {UAC, UAU}
valine: {GUA, GUC, GUG, GUU}
stop: {UAA, UAG, UGA}
This is not quite an easily described pattern; it is “close to” one of the previous equivalence relations. Which relation above is this close to?
AAA: lysine AAC: asparagine AAG: lysine AAU: asparagine ACA: threonine ACC: threonine ACG: threonine ACU: threonine AGA: arginine AGC: serine AGG: arginine AGU: serine AUA: isoleucine AUC: isoleucine AUG: methionine AUU: isoleucine CAA: glutamine CAC: histidine CAG: glutamine CAU: histidine CCA: proline CCC: proline CCG: proline CCU: proline CGA: arginine CGC: arginine CGG: arginine CGU: arginine CUA: leucine CUC: leucine CUG: leucine CUU: leucine GAA: glutamic_acid GAC: aspartic_acid GAG: glutamic_acid GAU: aspartic_acid GCA: alanine GCC: alanine GCG: alanine GCU: alanine GGA: glycine GGC: glycine GGG: glycine GGU: glycine GUA: valine GUC: valine GUG: valine GUU: valine UAA: stop UAC: tyrosine UAG: stop UAU: tyrosine UCA: serine UCC: serine UCG: serine UCU: serine UGA: stop UGC: cysteine UGG: tryptophan UGU: cysteine UUA: leucine UUC: phenylalanine UUG: leucine UUU: phenylalanine