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Parikh vectors
1PIR_1 2EUC_1 4WTQ_1 Letter Amino acid
8 12 17 A Alanine
9 8 19 D Aspartic acid
5 11 8 F Phenylalanine
5 10 12 I Isoleucine
7 10 16 L Leucine
2 4 9 M Methionine
10 9 12 S Serine
6 4 21 T Threonine
13 4 9 N Asparagine
14 1 1 C Cysteine
1 5 5 Q Glutamine
6 9 17 E Glutamic acid
9 6 8 K Lycine
1 1 5 W Tryptophan
2 6 22 V Valine
4 4 18 R Arginine
6 5 40 G Glycine
3 13 29 H Histidine
5 3 14 P Proline
8 5 11 Y Tyrosine 

1PIR_1|Chain A|PHOSPHOLIPASE A2|Sus scrofa (9823)
>2EUC_1|Chains A, B|Hypothetical protein yfmB|Bacillus subtilis (1423)
>4WTQ_1|Chains A, B, C|Copper oxidase|Streptomyces sviceus ATCC 29083 (463191)
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 \(w=f(c)\)
1PIR , Knot 62 124 0.80 40 99 119 
ALWQFRSMIKCAIPGSHPLMDFNNYGCYCGLGGSGTPVDELDRCCETHDNCYRDAKNLDSCKFLVDNPYTESYSYSCSNTEITCNSKNNACEAFICNCDRNAAICFSKAPYNKEHKNLDTKKYC
2EUC , Knot 66 130 0.82 40 106 124 
MQYFSPEQQYNAWIVSDLVKQIFHKRAGCSPGIHELAVFAEEHFHIDIDFVFSIIMNIGDIEFALTDEIEKKLSGYLSTLLPYVTADMFETSKANAHAFLSRRHGNAAYHLFVSDDAFMRKQLEHHHHHH
4WTQ , Knot 123 293 0.79 40 173 275 
MHHHHHHAPGGEVRRIKLYAERLADGQMGYGLEKGRATIPGPLIELNEGDTLHIEFENTMDVRASLHVHGLDYEVSSDGTTLNKSDVEPGGTRTYTWRTHAPGRRSDGTWRAGSAGYWHYHDHVVGTEHGTGGIRKGLYGPVIVRRKGDVLPDATHTIVFNDMLINNRPAHSGPNFEATVGDRVEFVMITHGEYYHTFHMHGHRWADNRTGMLTGPDDPSQVVDNKIVGPADSFGFQVIAGEGVGAGAWMYHCHVQSHSDMGLVGLFLVKKTDGTIPGYEPHEHSGQRAEHHH

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 \(c\).

\(|P_{f(1PIR_1)}(2) \setminus P_{f(2EUC_1)}(2)|=75\), \(|P_{f(2EUC_1)}(2) \setminus P_{f(1PIR_1)}(2)|=82\). 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 1:1110100110011110011101000100011110101100100000000000001001000011100100000000000001000000010011100000011101001100000001000000
Pair \(Z_2\) Length of longest common subsequence
1PIR_1,2EUC_1 157 3
1PIR_1,4WTQ_1 196 3
2EUC_1,4WTQ_1 187 6

Newick tree

 
[
	4WTQ_1:10.87,
	[
		1PIR_1:78.5,2EUC_1:78.5
	]:22.37
]

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 \((0.85)(0.8)(\frac{254 }{\log_{20} 254}-\frac{124}{\log_{20}124})=41.0\)
Status Protein1 Protein2 d d1/2
Query variables 1PIR_1 2EUC_1 54 52.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]