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Parikh vectors
7WBF_1 2AUZ_1 5JBH_1 Letter Amino acid
6 4 0 W Tryptophan
3 13 0 Y Tyrosine
6 15 0 V Valine
3 9 0 Q Glutamine
12 24 556 G Glycine
1 2 0 H Histidine
6 19 0 K Lycine
10 16 0 S Serine
12 18 296 A Alanine
6 8 0 I Isoleucine
8 12 0 L Leucine
7 5 0 T Threonine
11 7 0 R Arginine
14 19 0 N Asparagine
8 8 445 C Cysteine
3 5 0 F Phenylalanine
2 8 0 P Proline
7 9 0 D Aspartic acid
2 10 0 E Glutamic acid
2 4 0 M Methionine 

7WBF_1|Chain A|Lysozyme C|Gallus gallus (9031)
>2AUZ_1|Chain A|Cathepsin K|Homo sapiens (9606)
>5JBH_1|Chain A[auth 2]|16S ribosomal RNA|Pyrococcus abyssi GE5 (272844)
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 \(w=f(c)\)
7WBF , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
2AUZ , Knot 99 215 0.82 40 148 209 
APDSVDYRKKGYVTPVKNQGQCGSCWAFSSVGALEGQLKKKTGKLLNLSPQNLVDCVSENDGCGGGYMTNAFQYVQKNRGIDSEDAYPYVGQEESCMYNPTGKAAKCRGYREIPEGNEKALKRAVARVGPVSVAIDASLTSFQFYSKGVYYDESCNSDNLNHAVLAVGYGIQKGNKHWIIKNSWGENWGNKGYILMARNKNNACGIANLASFPKM
5JBH , Knot 271 1518 0.43 8 16 64 
GUACUCCCUUUAAUUCCGGUUGAUCCUGCCGGAGGCCACUGCUAUGGGGGUCCGACUAAGCCAUGCGAGUCAAGGGGGCGUCCCUUCUGGGACGCCACCGGCGGACGGCUCAGUAACACGUCGGUAACCUACCCUCGGGAGGGGGAUAACCCCGGGAAACUGGGGCUAAUCCCCCAUAGGCCUGGGGUACUGGAAGGUCCCCAGGCCGAAAGGGAGCCGUAAGGCCCGCCCGAGGAUGCGGCGGCGGCCGAUUAGGUAGUUGGUGGGGUAACGGCCCACCAAGCCGAAGAUCGGUACGGGCCGUGAGAGCGGGAGCCCGGAGAUGGACACUGAGACACGGGUCCAGGCCCUACGGGGCGCAGCAGGCGCGAAACCUCCGCAAUGCGGGAAACCGCGACGGGGGGACCCCCAGUGCCGUGCCUCUGGCACGGCUUUUCCGGAGUGUAAAAAGCUCCGGGAAUAAGGGCUGGGCAAGGCCGGUGGCAGCCGCCGCGGUAAUACCGGCGGCCCGAGUGGUGGCCACUAUUAUUGGGCCUAAAGCGGCCGUAGCCGGGCCCGUAAGUCCCUGGCGAAAUCCCACGGCUCAACCGUGGGGCUCGCUGGGGAUACUGCGGGCCUUGGGACCGGGAGAGGCCGGGGGUACCCCCGGGGUAGGGGUGAAAUCCUAUAAUCCCGGGGGGACCGCCAGUGGCGAAGGCGCCCGGCUGGAACGGGUCCGACGGUGAGCGCGGAAGGCCAGGGGAGCGAACCGGAUUAGAUACCCGGGUAGUCCUGGCUGUAAAGGAUGCGGGCUAGGUGUCGGGCGAGCUUCGAGCUCGCCCGGUGCCGUAGGGAAGCCGUUAAGCCCGCCGCCUGGGGAGUACGGCCGCAAGGCUGAAACUUAAAGGAAUUGGCGGGGGAGCACUACAAGGGGUGGAGCGUGCGGUUUAAUUGGAUUCAACGCCGGGAACCUCACCGGGGGCGACGGCAGGAUGAAGGCCAGGCUGAAGGUCUUGCCGGACGCGCCGAGAGGAGGUGCAUGGCCGCCGUCAGCUCGUACCGUGAGGCGUCCACUUAAGUGUGGUAACGAGCGAGACCCGCGCCCCCAGUUGCCAGUCCCUCCCGCUCGGGAGGGAGGCACUCUGGGGGGACUGCCGGCGAUAAGCCGGAGGAAGGGGCGGGCGACGGUAGGUCAGUAUGCCCCGAAACCCCCGGGCUACACGCGCGCUACAAUGGGCGGGACAAUGGGACCCGACCUCGAAAGGGGAAGGGAAUCCCCUAAACCCGCCCUCAGUUCGGAUCGCGGGCUGCAACUCGCCCGCGUGAAGCUGGAAUCCCUAGUACCCGCGUGUCAUCAUCGCGCGGCGAAUACGUCCCUGCUCCUUGCACACACCGCCCGUCACUCCACCCGAGCGGAGUCCGGGUGAGGCCUGAUCUCCUUCGGGAGGUCAGGUCGAGCCCGGGCUCCGUGAGGGGGGAGAAGUCGUAACAAGGUAACCGUAGGGGAACCUACGGCUCGAUCACCUCCUAUCGCCGGAA

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 \(c\).

\(|P_{f(7WBF_1)}(2) \setminus P_{f(2AUZ_1)}(2)|=60\), \(|P_{f(2AUZ_1)}(2) \setminus P_{f(7WBF_1)}(2)|=104\). 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 1:011100011111000110000100110110110100010001000000100001110100011000100110001001100111000101010010011001011011111000001001011101001
Pair \(Z_2\) Length of longest common subsequence
7WBF_1,2AUZ_1 164 3
7WBF_1,5JBH_1 114 3
2AUZ_1,5JBH_1 152 5

Newick tree

 
[
	2AUZ_1:85.14,
	[
		7WBF_1:57,5JBH_1:57
	]:28.14
]

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 \((0.85)(0.8)(\frac{344 }{\log_{20} 344}-\frac{129}{\log_{20}129})=65.9\)
Status Protein1 Protein2 d d1/2
Query variables 7WBF_1 2AUZ_1 83 68
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]