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Parikh vectors
2HFH_1 5YWY_1 1WID_1 Letter Amino acid
5 15 4 Y Tyrosine
2 12 0 C Cysteine
5 13 4 Q Glutamine
3 7 0 M Methionine
2 4 4 W Tryptophan
2 30 12 V Valine
7 5 3 H Histidine
7 24 2 I Isoleucine
9 45 10 L Leucine
7 17 6 F Phenylalanine
5 18 13 G Glycine
8 30 23 S Serine
9 18 8 R Arginine
7 10 7 N Asparagine
4 8 6 D Aspartic acid
4 11 5 E Glutamic acid
3 23 6 A Alanine
8 10 10 K Lycine
9 14 5 P Proline
3 18 2 T Threonine 

2HFH_1|Chain A|GENESIS|Rattus norvegicus (10116)
>5YWY_1|Chain A|Prostaglandin E2 receptor EP4 subtype,Prostaglandin E2 receptor EP4 subtype|Homo sapiens (9606)
>1WID_1|Chain A|DNA-binding protein RAV1|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
2HFH , Knot 57 109 0.81 40 90 104 
MVKPPYSYIALITMAILQSPQKKLTLSGICEFISNRFPYYREKFPAWQNSIRHNLSLNDCFVKIPREPGNPGKGNYWTLDPQSEDMFDNGSFLRRRKRFKRLQHHHHHH
5YWY , Knot 141 332 0.82 40 200 310 
GTSPGVQSSASLSPDRLNSPVTIPAVMFIFGVVGNLVAIVVLCKSRKEQKETTFYTLVCGLLVTDLLGTLLVSPVTIATYMKGQWPGGQPLCEYSTFILLFFSLSRLSIICAMSVERYLAINHAYFYSHYVDKRLAGLTLFAVYASNVLFCALPNMGLGSSRLQYPDTWCFIDWTTQVTAHAAYSYMYAGFSSFLILATVLCNVLVCGALLRMHRQFFRRIAGAEIQMVILLIATSLVVLICSIPLVVRVFVNQLYQPSLEREVSKNPDLQAIRIASVNPILDPWIYILLRKTVLSKAIEKIKCLFCRIGGSRRERSGQHCSDSLEENLYFQ
1WID , Knot 61 130 0.76 36 94 122 
GSSGSSGRSAEALFEKAVTPSDVGKLNRLVIPKHHAEKHFPLPSSNVSVKGVLLNFEDVNGKVWRFRYSYWNSSQSYVLTKGWSRFVKEKNLRAGDVVSFSRSNGQDQQLYIGWKSRSGSDLDASGPSSG

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 \(c\).

\(|P_{f(2HFH_1)}(2) \setminus P_{f(5YWY_1)}(2)|=39\), \(|P_{f(5YWY_1)}(2) \setminus P_{f(2HFH_1)}(2)|=149\). 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 1:1101100011110111100100010101100110001100000111100010001010001101100110110100101010000110010110000010010000000
Pair \(Z_2\) Length of longest common subsequence
2HFH_1,5YWY_1 188 3
2HFH_1,1WID_1 124 3
5YWY_1,1WID_1 176 3

Newick tree

 
[
	5YWY_1:98.85,
	[
		2HFH_1:62,1WID_1:62
	]:36.85
]

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 \((0.85)(0.8)(\frac{441 }{\log_{20} 441}-\frac{109}{\log_{20}109})=100.\)
Status Protein1 Protein2 d d1/2
Query variables 2HFH_1 5YWY_1 129 84
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]