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Parikh vectors
8QNH_1 1YDG_1 6MCR_1 Letter Amino acid
20 9 6 P Proline
21 14 0 S Serine
22 18 8 T Threonine
19 29 3 A Alanine
14 7 4 N Asparagine
30 8 6 K Lycine
22 6 2 F Phenylalanine
20 9 4 R Arginine
22 10 6 Q Glutamine
23 13 4 E Glutamic acid
10 8 1 H Histidine
27 7 13 I Isoleucine
38 16 12 L Leucine
9 7 2 M Methionine
7 4 2 W Tryptophan
14 0 2 C Cysteine
13 13 6 V Valine
21 20 13 G Glycine
17 5 1 Y Tyrosine
25 8 4 D Aspartic acid 

8QNH_1|Chain A|E3 ubiquitin-protein ligase CBL-B|Homo sapiens (9606)
>1YDG_1|Chains A, B, C, D, E, F, G, H|trp repressor binding protein WrbA|Deinococcus radiodurans (1299)
>6MCR_1|Chain A|Protease|Human immunodeficiency virus 1 (11676)
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 \(w=f(c)\)
8QNH , Knot 174 394 0.88 40 248 386 
HMPKQAAADRRTVEKTWKLMDKVVRLCQNPKLQLKNSPPYILDILPDTYQHLRLILSKYDDNQKLAQLSENEYFKIYIDSLMKKSKRAIRLFKEGKERMYEEQSQDRRNLTKLSLIFSHMLAEIKAIFPNGQFQGDNFRITKADAAEFWRKFFGDKTIVPWKVFRQCLHEVHQISSGLEAMALKSTIDLTCNDYISVFEFDIFTRLFQPWGSILRNWNFLAVTHPGYMAFLTYDEVKARLQKYSTKPGSYIFRLSCTRLGQWAIGYVTGDGNILQTIPHNKPLFQALIDGSREGFYLYPDGRSYNPDLTGLCEPTPHDHIKVTQEQYELYCEMGSTFQLCKICAENDKDVKIEPCGHLMCTSCLTAWQESDGQGCPFCRCEIKGTEPIIVDPFD
1YDG , Knot 94 211 0.79 38 139 199 
MSLTAPVKLAIVFYSSTGTGYAMAQEAAEAGRAAGAEVRLLKVRETAPQDVIDGQDAWKANIEAMKDVPEATPADLEWAEAIVFSSPTRFGGATSQMRAFIDTLGGLWSSGKLANKTFSAMTSAQNVNGGQETTLQTLYMTAMHWGAVLTPPGYTDEVIFKSGGNPYGASVTANGQPLLENDRASIRHQVRRQVELTAKLLEGGSHHHHHH
6MCR , Knot 52 99 0.80 38 80 94 
PQITLWQRPLVTIKIGGQLKEALLDTGADDTVLEEMNLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF

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 \(c\).

\(|P_{f(8QNH_1)}(2) \setminus P_{f(1YDG_1)}(2)|=148\), \(|P_{f(1YDG_1)}(2) \setminus P_{f(8QNH_1)}(2)|=39\). 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 1:0110011100001000101100110100010101000110110111000001011100000000110100000101010011000001101100100010000000000100101110011101011110101010010100101101100111000111101100010010010011011110001010000010110101100110111011001011110011011110000101010000001100110100001101111010101011001100011101110100011010101000010101100101000101000000100011001010010100000101010101100001011000010101100001010011110110
Pair \(Z_2\) Length of longest common subsequence
8QNH_1,1YDG_1 187 3
8QNH_1,6MCR_1 214 3
1YDG_1,6MCR_1 153 3

Newick tree

 
[
	8QNH_1:10.28,
	[
		1YDG_1:76.5,6MCR_1:76.5
	]:30.78
]

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 \((0.85)(0.8)(\frac{605 }{\log_{20} 605}-\frac{211}{\log_{20}211})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 8QNH_1 1YDG_1 145 109
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]