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Parikh vectors
1OAI_1 9HBF_1 3DXA_1 Letter Amino acid
5 3 24 E Glutamic acid
7 1 16 Q Glutamine
2 11 18 G Glycine
5 8 21 A Alanine
1 3 26 R Arginine
3 6 5 N Asparagine
2 3 19 D Aspartic acid
1 2 4 C Cysteine
1 1 8 H Histidine
4 4 8 K Lycine
3 1 3 M Methionine
3 1 6 F Phenylalanine
4 15 24 T Threonine
1 4 16 Y Tyrosine
1 4 9 I Isoleucine
5 1 19 L Leucine
3 2 16 P Proline
5 7 13 S Serine
2 7 8 W Tryptophan
1 7 13 V Valine 

1OAI_1|Chain A|NUCLEAR RNA EXPORT FACTOR|HOMO SAPIENS (9606)
>9HBF_1|Chain A|Fucose-binding lectin protein|Ralstonia solanacearum (305)
>3DXA_1|Chains A, F, K|HLA class I histocompatibility complex HLA-B*4402|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 \(w=f(c)\)
1OAI , Knot 35 59 0.80 40 53 56 
PTLSPEQQEMLQAFSTQSGMNLEWSQKCLQDNNWDYTRSAQAFTHLKAKGEIPEVAFMK
9HBF , Knot 44 91 0.72 40 66 82 
MKSVQTAATSWGTVPSIRVYTANNGKITERCWDGKGWYTGAFNEPGDNVSVTSWLVGSAIHIRVYASTGTTTTEWCWDGNGWTKGAYTATN
3DXA , Knot 122 276 0.82 40 180 268 
GSHSMRYFYTAMSRPGRGEPRFITVGYVDDTLFVRFDSDATSPRKEPRAPWIEQEGPEYWDRETQISKTNTQTYRENLRTALRYYNQSEAGSHIIQRMYGCDVGPDGRLLRGYDQYAYDGKDYIALNEDLSSWTAADTAAQITQRKWEAARVAEQDRAYLEGLCVESLRRYLENGKETLQRADPPKTHVTHHPISDHEVTLRCWALGFYPAEITLTWQRDGEDQTQDTELVETRPAGDRTFQKWAAVVVPSGEEQRYTCHVQHEGLPKPLTLRWEP

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 \(c\).

\(|P_{f(1OAI_1)}(2) \setminus P_{f(9HBF_1)}(2)|=43\), \(|P_{f(9HBF_1)}(2) \setminus P_{f(1OAI_1)}(2)|=56\). 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 1:10101000011011000011010100001000010000010110010101011011110
Pair \(Z_2\) Length of longest common subsequence
1OAI_1,9HBF_1 99 2
1OAI_1,3DXA_1 185 2
9HBF_1,3DXA_1 180 3

Newick tree

 
[
	3DXA_1:10.42,
	[
		1OAI_1:49.5,9HBF_1:49.5
	]:51.92
]

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 \((0.85)(0.8)(\frac{150 }{\log_{20} 150}-\frac{59}{\log_{20}59})=31.5\)
Status Protein1 Protein2 d d1/2
Query variables 1OAI_1 9HBF_1 36 31.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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