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Parikh vectors
2ERS_1 3BXZ_1 3RQM_1 Letter Amino acid
0 13 18 Q Glutamine
2 34 25 G Glycine
2 14 14 H Histidine
1 17 11 M Methionine
4 15 23 P Proline
8 20 26 S Serine
3 13 16 Y Tyrosine
4 35 22 R Arginine
4 41 36 L Leucine
5 24 27 K Lycine
2 2 13 W Tryptophan
4 1 11 C Cysteine
3 41 25 E Glutamic acid
4 29 23 I Isoleucine
1 16 19 F Phenylalanine
6 29 25 T Threonine
4 49 20 A Alanine
3 23 18 N Asparagine
2 23 26 D Aspartic acid
4 32 24 V Valine 

2ERS_1|Chain A|Interleukin-15 receptor alpha chain|Homo sapiens (9606)
>3BXZ_1|Chains A, B|Preprotein translocase subunit secA|Escherichia coli (562)
>3RQM_1|Chains A, B|Nitric oxide synthase, brain|Rattus norvegicus (10116)
Protein code \(c\) LZ-complexity \(\mathrm{LZ}(w)\) Length \(n=|w|\) \(\frac{\mathrm{LZ}(w)}{n /\log_{20} n}\) \(p_w(1)\) \(p_w(2)\) \(p_w(3)\) Sequence \(w=f(c)\)
2ERS , Knot 37 66 0.78 38 60 64 
ITCPPPMSVEHADIWVKSYSLYSRERYICNSGFKRKAGTSSLTECVLNKATNVAHWTTPSLKCIRD
3BXZ , Knot 192 471 0.83 40 226 446 
MHHHHHHLTKVFGSRNDRTLRRMRKVVNIINAMEPEMEKLSDEELKGKTAEFRARLEKGEVLENLIPEAFAVVREASKRVFGMRHFDVQLLGGMVLNERCIAEMRTGEGKTLTATLPAYLNALTGKGVHVVTVNDYLAQRDAENNRPLFEFLGLTVGINLPGMPAPAKREAYAADITYGTNNEYGFDYLRDNMAFSPEERVQRKLHYALVDEVDSILIDEARTPLIISGANQTLASITFQNYFRLYEKLAGMTGTADTEAFEFSSIYKLDTVVVPTNRPMIRKDLPDLVYMTEAEKIQAIIEDIKERTAKGQPVLVGTISIEKSELVSNELTKAGIKHNVLNAKFHANEAAIVAQAGYPAAVTIATNMAGRGTDIVLGGSWQAEVAALENPTAEQIEKIKADWQVRHDAVLEAGGLHIIGTERHESRRIDNQLRGRSGRQGDAGSSRFYLSMEDALMRIFASDRVSGMMRK
3RQM , Knot 186 422 0.88 40 262 411 
CPRFLKVKNWETDVVLTDTLHLKSTLETGCTEHICMGSIMLPSQHTRKPEDVRTKDQLFPLAKEFLDQYYSSIKRFGSKAHMDRLEEVNKEIESTSTYQLKDTELIYGAKHAWRNASRCVGRIQWSKLQVFDARDCTTAHGMFNYICNHVKYATNKGNLRSAITIFPQRTDGKHDFRVWNSQLIRYAGYKQPDGSTLGDPANVQFTEICIQQGWKAPRGRFDVLPLLLQANGNDPELFQIPPELVLEVPIRHPKFDWFKDLGLKWYGLPAVSNMLLEIGGLEFSACPFSGWYMGTEIGVRDYCDNSRYNILEEVAKKMDLDMRKTSSLWKDQALVEINIAVLYSFQSDKVTIVDHHSATESFIKHMENEYRCRGGCPADWVWIVPPMSGSITPVFHQEMLNYRLTPSFEYQPDPWNTHVWKG

Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\). Let \(p_w(n)\) be the cardinality of \(P_w(n)\). Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).

\(|P_{f(2ERS_1)}(2) \setminus P_{f(3BXZ_1)}(2)|=29\), \(|P_{f(3BXZ_1)}(2) \setminus P_{f(2ERS_1)}(2)|=195\). Let \( Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)| \) be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:100111101001011100001000000100011000110001000110010011010010100100
Pair \(Z_2\) Length of longest common subsequence
2ERS_1,3BXZ_1 224 3
2ERS_1,3RQM_1 226 4
3BXZ_1,3RQM_1 166 3

Newick tree

 
[
	2ERS_1:12.74,
	[
		3BXZ_1:83,3RQM_1:83
	]:37.74
]

Let d be the Otu--Sayood distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{537 }{\log_{20} 537}-\frac{66}{\log_{20}66})=141.\)
Status Protein1 Protein2 d d1/2
Query variables 2ERS_1 3BXZ_1 181 103
Was not able to put for d
Was not able to put for d1

In notation analogous to [Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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