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Parikh vectors
1XDD_1 9GQJ_1 4EAJ_1 Letter Amino acid
3 3 8 R Arginine
16 7 6 D Aspartic acid
0 0 2 C Cysteine
7 2 5 Q Glutamine
17 9 9 L Leucine
21 16 6 K Lycine
5 2 4 M Methionine
15 6 4 F Phenylalanine
4 6 4 P Proline
13 8 5 A Alanine
12 10 4 E Glutamic acid
9 15 10 G Glycine
4 3 3 H Histidine
14 9 7 I Isoleucine
6 4 5 Y Tyrosine
11 9 6 V Valine
5 4 4 N Asparagine
12 8 8 S Serine
13 6 4 T Threonine
1 1 2 W Tryptophan 

1XDD_1|Chains A, B|Integrin alpha-L|Homo sapiens (9606)
>9GQJ_1|Chains A, B|Peptidyl-prolyl cis-trans isomerase FKBP5|Homo sapiens (9606)
>4EAJ_1|Chain A|5'-AMP-activated protein kinase catalytic subunit alpha-1|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 \(w=f(c)\)
1XDD , Knot 86 188 0.79 38 129 181 
MASKGNVDLVFLFDGSMSLQPDEFQKILDFMKDVMKKLSNTSYQFAAVQFSTSYKTEFDFSDYVKWKDPDALLKHVKHMLLLTNTFGAINYVATEVFREELGARPDATKVLIIITDGEATDSGNIDAAKDIIRYIIGIGKHFQTKESQETLHKFASKPASEFVKILDTFEKLKDLFTELQKKIYVIEG
9GQJ , Knot 64 128 0.80 38 103 125 
GAPATVTEQGEDITSKKDRGVLKIVKRVGNGEETPMIGDKVYVHYKGKLSNGKKFDSSHDRNEPFVFSLGKGQVIKAWDIGVATMKKGEIAHLLIKPEYAYGSAGSLPKIPSNATLFFEIELLDFKGE
4EAJ , Knot 57 106 0.83 40 93 100 
GPHMGAKWHLGIRSQSRPNDIMAEVCRAIKQLDYEWKVVNPYYLRVRRKNPVTSTFSKMSLQLYQVDSRTYLLDFRSIDDGGGGGGGSHTIEFFEMCANLIKILAQ

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 \(c\).

\(|P_{f(1XDD_1)}(2) \setminus P_{f(9GQJ_1)}(2)|=80\), \(|P_{f(9GQJ_1)}(2) \setminus P_{f(1XDD_1)}(2)|=54\). 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 1:11001010111110101010100100110110011001000000111101000000010100010100101110010011110001111001100110001110101001111100101000101011001100111110010000000010011001100110110010010011001000101101
Pair \(Z_2\) Length of longest common subsequence
1XDD_1,9GQJ_1 134 4
1XDD_1,4EAJ_1 142 4
9GQJ_1,4EAJ_1 148 4

Newick tree

 
[
	4EAJ_1:74.26,
	[
		1XDD_1:67,9GQJ_1:67
	]:7.26
]

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 \((0.85)(0.8)(\frac{316 }{\log_{20} 316}-\frac{128}{\log_{20}128})=58.1\)
Status Protein1 Protein2 d d1/2
Query variables 1XDD_1 9GQJ_1 70 59
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]