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Parikh vectors
9INC_1 2UXU_1 9IJZ_1 Letter Amino acid
1 5 29 N Asparagine
0 4 12 C Cysteine
3 8 30 Q Glutamine
11 29 80 L Leucine
3 4 24 P Proline
2 3 30 Y Tyrosine
13 25 49 A Alanine
0 5 11 M Methionine
0 3 17 W Tryptophan
7 14 39 V Valine
3 16 32 D Aspartic acid
5 18 45 E Glutamic acid
5 8 40 K Lycine
1 5 55 F Phenylalanine
6 10 41 T Threonine
9 21 44 R Arginine
9 10 30 G Glycine
5 7 18 H Histidine
8 9 45 I Isoleucine
7 6 41 S Serine 

9INC_1|Chains A, D|Histone H2A.Z|Homo sapiens (9606)
>2UXU_1|Chains A, B|HTH-TYPE TRANSCRIPTIONAL REGULATOR TTGR|PSEUDOMONAS PUTIDA (303)
>9IJZ_1|Chains A, B|Solute carrier family 53 member 1|Homo sapiens (9606)
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 \(w=f(c)\)
9INC , Knot 48 98 0.74 34 78 95 
AVSRSQRAGLQFPVGRIHRHLKSRTTSHGRVGATAAVYSAAILEYLTAEVLELAGNASKDLKVKRITPRHLQLAIRGDEELDSLIKATIAGGGVIPHI
2UXU , Knot 97 210 0.82 40 141 198 
MVRRTKEEAQETRAQIIEAAERAFYKRGVARTTLADIAELAGVTRGAIYWHFNNKAELVQALLDSLHETHDHLARASESEDEVDPLGCMRKLLLQVFNELVLDARTRRINEILHHKCEFTDDMCEIRQQRQSAVLDCHKGITLALANAVRRGQLPGELDAERAAVAMFAYVDGLIRRWLLLPDSVDLLGDVEKWVDTGLDMLRLSPALRK
9IJZ , Knot 278 712 0.85 40 290 649 
MKFAEHLSAHITPEWRKQYIQYEAFKDMLYSAQDQAPSVEVTDEDTVKRYFAKFEEKFFQTCEKELAKINTFYSEKLAEAQRRFATLQNELQSSLDAQKESTGVTTLRQRRKPVFHLSHEERVQHRNIKDLKLAFSEFYLSLILLQNYQNLNFTGFRKILKKHDKILETSRGADWRVAHVEVAPFYTCKKINQLISETEAVVTNELEDGDRQKAMKRLRVPPLGAAQPAPAWTTFRVGLFCGIFIVLNITLVLAAVFKLETDRSIWPLIRIYRGGFLLIEFLFLLGINTYGWRQAGVNHVLIFELNPRSNLSHQHLFEIAGFLGILWCLSLLACFFAPISVIPTYVYPLALYGFMVFFLINPTKTFYYKSRFWLLKLLFRVFTAPFHKVGFADFWLADQLNSLSVILMDLEYMICFYSLELKWDESKGLLPNNSEESGICHKYTYGVRAIVQCIPAWLRFIQCLRRYRDTKRAFPHLVNAGKYSTTFFMVTFAALYSTHKERGHSDTMVFFYLWIVFYIISSCYTLIWDLKMDWGLFDKNAGENTFLREEIVYPQKAYYYCAIIEDVILRFAWTIQISITSTTLLPHSGDIIATVFAPLEVFRRFVWNFFRLENEHLNNCGEFRAVRDISVAPLNADDQTLLEQMMDQDDGVRNRQKNRSWKYNQSISLRRPRLASQSKARDTKVLIEDTDDEANTLEGGSSGGWSHPQFEK

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 \(c\).

\(|P_{f(9INC_1)}(2) \setminus P_{f(2UXU_1)}(2)|=40\), \(|P_{f(2UXU_1)}(2) \setminus P_{f(9INC_1)}(2)|=103\). 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 1:11000001110111101000100000001011101110011110010101101110100010100101001011101000100110101111111101
Pair \(Z_2\) Length of longest common subsequence
9INC_1,2UXU_1 143 4
9INC_1,9IJZ_1 230 4
2UXU_1,9IJZ_1 207 3

Newick tree

 
[
	9IJZ_1:11.39,
	[
		9INC_1:71.5,2UXU_1:71.5
	]:47.89
]

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 \((0.85)(0.8)(\frac{308 }{\log_{20} 308}-\frac{98}{\log_{20}98})=65.9\)
Status Protein1 Protein2 d d1/2
Query variables 9INC_1 2UXU_1 83 59.5
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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