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Parikh vectors
9DUO_1 5GNI_1 8VQS_1 Letter Amino acid
22 9 11 D Aspartic acid
58 14 13 L Leucine
29 5 12 A Alanine
23 8 6 R Arginine
18 10 2 H Histidine
9 5 6 M Methionine
41 19 12 S Serine
34 20 8 V Valine
11 8 6 N Asparagine
22 14 7 Q Glutamine
14 13 2 I Isoleucine
14 9 10 F Phenylalanine
22 17 7 T Threonine
9 1 1 W Tryptophan
12 4 1 C Cysteine
21 19 6 E Glutamic acid
35 7 9 G Glycine
13 17 12 K Lycine
21 8 2 P Proline
6 6 4 Y Tyrosine 

9DUO_1|Chains A, B|WD repeat-containing protein 18|Homo sapiens (9606)
>5GNI_1|Chains A, B|Platelet endothelial cell adhesion molecule|Homo sapiens (9606)
>8VQS_1|Chains A, B|Dehaloperoxidase B|Amphitrite ornata (129555)
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 \(w=f(c)\)
9DUO , Knot 178 434 0.83 40 222 402 
SNMAAPMEVAVCTDSAAPMWSCIVWELHSGANLLTYRGGQAGPRGLALLNGEYLLAAQLGKNYISAWELQRKDQLQQKIMCPGPVTCLTASPNGLYVLAGVAESIHLWEVSTGNLLVILSRHYQDVSCLQFTGDSSHFISGGKDCLVLVWSLCSVLQADPSRIPAPRHVWSHHALPITDLHCGFGGPLARVATSSLDQTVKLWEVSSGELLLSVLFDVSIMAVTMDLAEHHMFCGGSEGSIFQVDLFTWPGQRERSFHPEQDAGKVFKGHRNQVTCLSVSTDGSVLLSGSHDETVRLWDVQSKQCIRTVALKGPVTNAAILLAPVSMLSSDFRPSLPLPHFNKHLLGAEHGDEPRHGGLTLRLGLHQQGSEPSYLDRTEQLQAVLCSTMEKSVLGGQDQLRVRVTELEDEVRNLRKINRDLFDFSTRFITRPAK
5GNI , Knot 101 213 0.84 40 151 203 
QENSFTINSVDMKSLPDWTVQNGKQLTLQCFADVSTTSHVKPQHQMLFYKDDVLFYQISSMKSTESYFIPEVRIYDSGTYKCTVIVNNKEKTTAEYQLLVEGVPSPRVTLDKKEAIQGGIVRVQCSVPEEKAPIHFTIEKLELNEKMVKLKREKNSRDQNFVILEFPVEEQDRVLSFRCQARIISGIHMQTSESTKSELVTVTESTGHHHHHH
8VQS , Knot 68 137 0.81 40 105 134 
GFKQDIATLRGDLRTYAQDIFLAFLNKYPDEKRNFKNYVGKSDQELKSMAKFGDHTEKVFNLMMEVADRATDCVPLASDASTLVQMKQHSGLTTGNFEKLFVALVEYMRASGQSFDSQSWDRFGKNLVSALSSAGMK

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 \(c\).

\(|P_{f(9DUO_1)}(2) \setminus P_{f(5GNI_1)}(2)|=124\), \(|P_{f(5GNI_1)}(2) \setminus P_{f(9DUO_1)}(2)|=53\). 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 1:00111110111000011111001110100110110001101110111110100111101100010110100000100011011110010101011011111100101101001011111000000100101010000110110001111101001101010011110011000111100100111111101100010001011010010111011101011110101100011011001011010110111000001010001101101000010010100010111010000010110100000100111011100111111110110001010111101000111100100100111010111000100100100000101110001000111100010101001000100100100011010001100110
Pair \(Z_2\) Length of longest common subsequence
9DUO_1,5GNI_1 177 3
9DUO_1,8VQS_1 181 4
5GNI_1,8VQS_1 170 3

Newick tree

 
[
	9DUO_1:90.95,
	[
		5GNI_1:85,8VQS_1:85
	]:5.95
]

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 \((0.85)(0.8)(\frac{647 }{\log_{20} 647}-\frac{213}{\log_{20}213})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 9DUO_1 5GNI_1 154 114
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]