# Complementary Sequence

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A complementary sequence is the inverse of a strictly increasing sequence of non-negative integers (the counting numbers, 1, 2, 3…). “Inverse” refers to the fact it contains numbers not in the original sequence.

More specifically, if a natural-numbered sequence (an) is a strictly increasing sequence of non-negative integers then the complementary sequence is the set of all non-negative integers which are not terms of the sequence [1].

A sequence of positive integers and its complement contain all possible positive integers.

## Example of a Complementary Sequence

The complementary sequence of even non-negative integers {2, 4, 6, 8, …} is the odd non-negative integer sequence {1, 3, 5, 7, …}.

This also works in reverse: The complementary sequence of {1, 3, 5, 7, …} is the sequence {2, 4, 6, 8, …}.

## Complementary Sequence in Biology

The definition of a complementary sequence in biology is slightly different than the mathematical definition, but the principal is the same. In a way, you can think of them as opposites.

DNA double helix. Image: Zephyris | Wikimedia Commons

Complementary sequences of nucleotides commonly arise in DNA sequencing. DNA is double-stranded; For each strand with a certain sequence, there is another strand that complements it (called the Watson-Crick pairing rule). For each strand of DNA with a given sequence, there is a complementary one where Adenine (A) complements Thymine (T) and Guanine (G) complements Cytosine (C) [4].

There is a similar complementarity with RNA: Adenine complements Uracil and Guanine complements Cytosine [5].

A pair of Golay complementary sequences is a pair (a, b) of sequences with out-of-phase (i.e. shifts) autocorrelations equal to zero. In other words, Ca(u) + Cb(u) = 0, 0 < u < n [2]. For example, (1 1 -1 1) and (1 1 1 -1) are a Golay complementary pair [3]. These sequences have many applications, including infrared multislit spectrometry and Orthogonal Frequency-Division Multiplexing.

## References

[1] Mortich, S. (2010). Remarks on Complementary Sequences. Retrieved April 10, 2021 from: https://www.fq.math.ca/Papers1/48-4/Mortici.pdf
[2] Fieldler, F. & Jedwab, J. (2006). How Do More Golay Sequences Arise? Retrieved April 10, 2021 from: https://www.sfu.ca/~jed/Papers/Fiedler%20Jedwab.%20More%20Golay%20Sequences.%202006.pdf
[3] Kalashnikov, E. (2014). An Introduction to Golay Complementary Sequences. Retrieved April 10 2021 from:https://journals.library.ualberta.ca/eureka/index.php/eureka/article/view/22829
[4] Wellcome Genome Campus Advanced Courses and Scientific Conferences. Grammatical rules for DNA sequence representation. https://www.futurelearn.com/info/courses/bacterial-genomes-bioinformatics/0/steps/47002
[5] Laventa, R. & Cesarini, G. (1981). Base pairing of RNA I with its complementary sequence in the primer precursor inhibits ColE1 replication. Nature 294. 623-626.

Image: Zephyris, CC BY-SA 3.0 , via Wikimedia Commons

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Stephanie Glen. "Complementary Sequence" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/complementary-sequence/
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