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Parikh vectors
9ARR_1 4PXF_1 6KSG_1 Letter Amino acid
5 15 12 N Asparagine
9 6 19 H Histidine
13 11 19 K Lycine
5 15 17 S Serine
4 27 18 T Threonine
1 5 2 W Tryptophan
10 7 12 R Arginine
6 23 24 G Glycine
2 16 8 M Methionine
11 31 8 F Phenylalanine
11 20 12 P Proline
4 18 8 Y Tyrosine
3 10 8 C Cysteine
13 17 23 E Glutamic acid
3 22 29 I Isoleucine
5 29 18 A Alanine
5 5 15 D Aspartic acid
3 12 9 Q Glutamine
11 28 20 L Leucine
14 31 20 V Valine 

9ARR_1|Chains A, B|Protein AF-9|Homo sapiens (9606)
>4PXF_1|Chain A|Rhodopsin|Bos taurus (9913)
>6KSG_1|Chains A, B|Methionine aminopeptidase|Vibrio cholerae serotype O1 (strain ATCC 39315 / El Tor Inaba N16961) (243277)
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 \(w=f(c)\)
9ARR , Knot 71 138 0.84 40 111 136 
MASSCAVQVKLELGHRAQVRKKPTVEGFTHDWMVFVRGPEHSNIQHFVEKVVFHLHESFPRPKRVCKDPPYKVEESGYAGFILPIEVYFKNKEEPRKVRFDYDLFLHLEGHPPVNHLRCEKLTFNNPTEDFRRKLLKA
4PXF , Knot 152 348 0.85 40 215 336 
MNGTEGPNFYVPFSNKTGVVRSPFEAPQYYLAEPWQFSMLAAYMFLLIMLGFPINFLTLYVTVQHKKLRTPLNYILLNLAVADLFMVFGGFTTTLYTSLHGYFVFGPTGCNLEGFFATLGGEIALWSLVVLAIERYVVVCKPMSNFRFGENHAIMGVAFTWVMALACAAPPLVGWSRYIPEGMQCSCGIDYYTPHEETNNESFVIYMFVVHFIIPLIVIFFCYGQLVFTVKEAAAQQQESATTQKAEKEVTRMVIIMVIAFLICWLPYAGVAFYIFTHQGSDFGPIFMTIPAFFAKTSAVYNPVIYIMMNKQFRNCMVTTLCCGKNPLGDDEASTTVSKTETSQVAPA
6KSG , Knot 134 301 0.84 40 194 288 
MGSSHHHHHHSSGLVPAGSHMMSIKIKNAVEIEKMRVAGRLAAEVLEMIEPHVKAGVTTEELDQICHKYITEVQGAIPAPLNYHGFPKSICTSINHIVCHGIPASEDTYFGQIQRPAVLRDGDILNIDITVIKDGYHGDTSKMFLIGDVSIEDKRLCHVAQECLYLALKQVKPGVQLGEIGTTIEKHIKTNNKNNPRFKFSIVRDYCGHGIGAEFHEEPQVVHYKNSDRTVLREGMIFTIEPMINAGKFGCRLDDEDSWTVYTADGKKSAQWEHTILVTATGCEILTLRSEESLPRILNNA

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 \(c\).

\(|P_{f(9ARR_1)}(2) \setminus P_{f(4PXF_1)}(2)|=39\), \(|P_{f(4PXF_1)}(2) \setminus P_{f(9ARR_1)}(2)|=143\). 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 1:110001101010110010100010101100011111011000010011001110100011010010001100100010111111101010000010010100011101010111001000010100100010001101
Pair \(Z_2\) Length of longest common subsequence
9ARR_1,4PXF_1 182 3
9ARR_1,6KSG_1 177 3
4PXF_1,6KSG_1 175 4

Newick tree

 
[
	9ARR_1:90.49,
	[
		6KSG_1:87.5,4PXF_1:87.5
	]:2.99
]

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 \((0.85)(0.8)(\frac{486 }{\log_{20} 486}-\frac{138}{\log_{20}138})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 9ARR_1 4PXF_1 133 91
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]