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Parikh vectors
7NAN_1 5WWL_1 4OUJ_1 Letter Amino acid
9 8 41 N Asparagine
13 13 18 D Aspartic acid
8 11 12 F Phenylalanine
8 13 19 T Threonine
10 18 18 V Valine
10 10 26 S Serine
0 1 9 W Tryptophan
19 10 17 A Alanine
3 0 2 C Cysteine
11 23 6 E Glutamic acid
11 16 18 K Lycine
8 8 8 P Proline
25 23 24 L Leucine
6 5 2 M Methionine
13 9 18 Y Tyrosine
12 15 9 R Arginine
7 10 17 Q Glutamine
11 8 14 G Glycine
5 4 3 H Histidine
12 10 26 I Isoleucine 

7NAN_10|Chains J, X|Proteasome subunit beta type-2|Homo sapiens (9606)
>5WWL_1|Chain A[auth M]|Centromere protein mis12|Schizosaccharomyces pombe (strain 972 / ATCC 24843) (284812)
>4OUJ_1|Chains A, B|Hemagglutinin component HA33|Clostridium botulinum B1 (498213)
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 \(w=f(c)\)
7NAN , Knot 96 201 0.84 38 146 198 
MEYLIGIQGPDYVLVASDRVAASNIVQMKDDHDKMFKMSEKILLLCVGEAGDTVQFAEYIQKNVQLYKMRNGYELSPTAAANFTRRNLADCLRSRTPYHVNLLLAGYDEHEGPALYYMDYLAALAKAPFAAHGYGAFLTLSILDRYYTPTISRERAVELLRKCLEELQKRFILNLPTFSVRIIDKNGIHDLDNISFPKQGS
5WWL , Knot 101 215 0.84 38 156 208 
MLVELLEFTPLSFIDDVINITNQLLYKGVNGVDKAFSQTRFAKKAPQEIEEGLHKFEVLFESVVDRYYDGFEVYTLRNIFSYPPELKGYMRTFGKDVDYSITTEQDAAMDQAIQEAAEKLVVKMQLRRDLRMRLSRKREKKTEIEKHLERISFLNKVPENWQVTLPETTDFLLDQLGNLQHAVKRVVEASPTVHSREVDERITYLEKGYERLSNP
4OUJ , Knot 132 307 0.82 40 180 284 
GSEHYSVIQNSLNDKIVTISCKANTDLFFYQVPGNGNVSLFQQTRNYLERWRIIYDSNKAAYKIKSMNIYNTNLVLTWNAPTHNISAQQDSNADNQYWLLLKDIGNNSFIIASYKNPNLVLYADTVARNLKLSTLNNSSYIKFIIEDYVISDFKNFTCRISPILAGGKVVQQVSMTNLAVNLYIWNNDLNQKWTIIYNEEKAAYQFFNKILSNGVLTWIFSDGNTVRVSSSAQNNDAQYWLINPVSDNYDRYTITNLRDKTKVLDLYGGQTADGTTIQVFNSNGGDNQKWNIRNPPGSAWSHPQFEK

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 \(c\).

\(|P_{f(7NAN_1)}(2) \setminus P_{f(5WWL_1)}(2)|=67\), \(|P_{f(5WWL_1)}(2) \setminus P_{f(7NAN_1)}(2)|=77\). 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 1:100111101100111100011100110100000011010001111011011001011001000101001001001010111010000110010000100101111100000111100100111110111110101111010110000010100001101100010010001110110101011000110010010110010
Pair \(Z_2\) Length of longest common subsequence
7NAN_1,5WWL_1 144 4
7NAN_1,4OUJ_1 168 3
5WWL_1,4OUJ_1 172 3

Newick tree

 
[
	4OUJ_1:88.91,
	[
		7NAN_1:72,5WWL_1:72
	]:16.91
]

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 \((0.85)(0.8)(\frac{416 }{\log_{20} 416}-\frac{201}{\log_{20}201})=63.3\)
Status Protein1 Protein2 d d1/2
Query variables 7NAN_1 5WWL_1 82 78.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]