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Parikh vectors
7ZLF_1 8QJN_1 5SXG_1 Letter Amino acid
3 22 16 R Arginine
5 3 10 F Phenylalanine
9 10 9 V Valine
5 12 6 N Asparagine
10 16 13 D Aspartic acid
9 15 11 Q Glutamine
4 20 9 G Glycine
9 7 7 I Isoleucine
4 3 5 M Methionine
2 5 5 W Tryptophan
16 32 11 A Alanine
2 15 4 H Histidine
15 8 3 K Lycine
8 12 8 P Proline
8 10 11 S Serine
8 10 8 T Threonine
5 4 6 C Cysteine
16 13 12 E Glutamic acid
19 30 21 L Leucine
5 7 10 Y Tyrosine 

7ZLF_1|Chain A[auth AAA]|Beta-lactoglobulin|Bos taurus (9913)
>8QJN_1|Chains A, B|S1/P1 Nuclease|Stenotrophomonas maltophilia (40324)
>5SXG_1|Chains A, B|DNA dC->dU-editing enzyme APOBEC-3B|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 \(w=f(c)\)
7ZLF , Knot 80 162 0.83 40 120 157 
ASVTQTMKGLDIQKVAGTWYSLAMAASDISLLDAQSAPYRVYVEELKPTPEGDLEILFQKWENGECAQKKIIAEKTKIPAVFKIDALNENKVLVLDTDYKKYLLFCMENSAEPEQSLACQCLVRTPEVDDEALEKFDKALKALPMHIRLSFNPTQLEEQCHI
8QJN , Knot 113 254 0.82 40 164 240 
WGAQGHRLVAEVADARLNPTARAEVDRLLATEPDATLASIAPWADQLRAKDPGLGRRSAGWHYVNIAEDNCHYEAPKHCRNGNCIVEALKAQSTILGDRSLTDGERLQALKFVVHLVGDIHQPMHAGYAHDKGGNDFQLQFGNRGTNLHSLWDSGMLNTRKLDDAGYLPLLQSQRAPKLARQSNPQRDPQTWAEASCRISMQAGVYPATRKIGDEYTERYRPLAEAQLRLAGENLAQLLNRVLGARLEHHHHHH
5SXG , Knot 90 185 0.84 40 139 179 
SEILRYLMDPDTFTSNFNNDPLVLRRRQTYLCYEVERLDNGTSVKMDQHMGFLCNESGRHAELRFLDLVPSLQLDPAQIYRVTWFISWSPCFSWGCAGEVRAFLQENTHVRLRIKAARIYDYDPLYKEALQMLRDAGAQVSIMTYDEFEYCWDTFVYRQGCPFQPWDGLEEHSQALSGRLRAILQ

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 \(c\).

\(|P_{f(7ZLF_1)}(2) \setminus P_{f(8QJN_1)}(2)|=61\), \(|P_{f(8QJN_1)}(2) \setminus P_{f(7ZLF_1)}(2)|=105\). 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 1:101000101101001110100111110010110100110010100101010101011100100100100011100001111101011000011110000000111010001010001100011001010001100100110111101010101001000001
Pair \(Z_2\) Length of longest common subsequence
7ZLF_1,8QJN_1 166 4
7ZLF_1,5SXG_1 165 3
8QJN_1,5SXG_1 165 3

Newick tree

 
[
	8QJN_1:82.83,
	[
		7ZLF_1:82.5,5SXG_1:82.5
	]:0.33
]

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 \((0.85)(0.8)(\frac{416 }{\log_{20} 416}-\frac{162}{\log_{20}162})=75.6\)
Status Protein1 Protein2 d d1/2
Query variables 7ZLF_1 8QJN_1 95 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]