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Parikh vectors
4OEO_1 6WHJ_1 8SJB_1 Letter Amino acid
5 16 3 T Threonine
1 1 1 W Tryptophan
5 12 6 N Asparagine
0 7 1 C Cysteine
16 33 4 G Glycine
7 34 6 A Alanine
4 18 6 D Aspartic acid
8 37 4 S Serine
4 16 4 F Phenylalanine
1 8 4 Y Tyrosine
9 26 6 V Valine
6 18 7 E Glutamic acid
6 18 11 K Lycine
3 9 2 M Methionine
7 26 7 I Isoleucine
5 25 10 L Leucine
3 13 1 P Proline
6 17 2 R Arginine
6 8 2 Q Glutamine
5 9 1 H Histidine 

4OEO_1|Chains A, B, C|Tight junction protein ZO-1|Homo sapiens (9606)
>6WHJ_1|Chains A, B, C, D|Ribokinase|Giardia lamblia (184922)
>8SJB_1|Chains A, B|Protein S100-A8|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 \(w=f(c)\)
4OEO , Knot 58 107 0.84 38 93 103 
GSHMIWEQHTVTLHRAPGFGFGIAISGGRDNPHFQSGETSIVISDVLKGGPAEGQLQENDRVAMVNGVSMDNVEHAFAVQQLRKSGKNAKITIRRKKGGGESLTGYV
6WHJ , Knot 151 351 0.84 40 200 330 
MAAASTAKIVIIGSYNRDLVWYVKDFPIGGQTINGSFASSHGGKGSNQAIACGKVLRDPSRAFFVGAVGKDTFGDEILAHYRELGIPNCIKQVSGAPTGNAGIYVAESGENMIVISEGANGMLKPSLVPDLMAVLVKATLVVMQCEISPDNTLLFVEVIKQAKAQNSSLRFVFNPAPYRADYDFSKILSITDIFCPNELEALEISGTGRLEGICDDSMMKALVEKMSSLSPSLKFVLFTLGSRGSRIVQTKSYESRTVGIYSHGRAIDTSGAGDCFIGSFCVRLMELAEESTRGPSALNDIDTIAEAARFASVAAGISVTRKGTSASVPRRQEVDDALSKFSGKGHHHHHH
8SJB , Knot 47 88 0.79 40 74 86 
MLTELEKALNSIIDVYNKYSLIKGNFNAVYRDDLKKLLETESPQYIRKKGADVWFKELDINTDGAVNFQEFLILVIKMGVAAHKKSCE

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 \(c\).

\(|P_{f(4OEO_1)}(2) \setminus P_{f(6WHJ_1)}(2)|=35\), \(|P_{f(6WHJ_1)}(2) \setminus P_{f(4OEO_1)}(2)|=142\). 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 1:10011100001010011111111110110001010010001110011011110101000001111011010010011110010001001010100001110010101
Pair \(Z_2\) Length of longest common subsequence
4OEO_1,6WHJ_1 177 4
4OEO_1,8SJB_1 125 4
6WHJ_1,8SJB_1 182 3

Newick tree

 
[
	6WHJ_1:97.16,
	[
		4OEO_1:62.5,8SJB_1:62.5
	]:34.66
]

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 \((0.85)(0.8)(\frac{458 }{\log_{20} 458}-\frac{107}{\log_{20}107})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 4OEO_1 6WHJ_1 132 85.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]