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Parikh vectors
8HCC_1 2XKP_1 4POF_1 Letter Amino acid
21 14 10 A Alanine
16 16 16 R Arginine
16 10 7 Q Glutamine
6 13 12 F Phenylalanine
3 0 1 W Tryptophan
11 7 18 D Aspartic acid
23 14 14 G Glycine
14 4 6 H Histidine
7 7 5 M Methionine
23 9 12 P Proline
20 12 9 T Threonine
12 16 26 V Valine
4 5 10 N Asparagine
7 1 4 C Cysteine
11 16 29 E Glutamic acid
29 18 12 S Serine
5 3 4 Y Tyrosine
8 16 26 I Isoleucine
33 32 18 L Leucine
7 9 18 K Lycine 

8HCC_1|Chains A, B, C, D|Three-prime repair exonuclease 1|Mus musculus (10090)
>2XKP_1|Chains A, B, C, D, E, F|GLOBAL NITROGEN REGULATOR|SYNECHOCOCCUS ELONGATUS (1140)
>4POF_1|Chains A, B, C, D, E, F|Cell division control protein 21|Pyrococcus furiosus (186497)
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 \(w=f(c)\)
8HCC , Knot 121 276 0.82 40 168 263 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSMGSQTLPHGHMQTLIFLDLEATGLPSSRPEVTELCLLAVHRRALENTSISQGHPPPVPRPPRVVDKLSLCIAPGKACSPGASEITGLSKAELEVQGRQRFDDNLAILLRAFLQRQPQPCCLVAHNGDRYDFPLLQTELARLSTPSPLDGTFCVDSIAALKALEQASSPSGNGSRKSYSLGSIYTRLYWQAPTDSHTAEGDVLTLLSICQWKPQALLQWVDEHARPFSTVKPMYGTPATTGTT
2XKP , Knot 100 222 0.81 38 146 211 
MLANENSLLTMFRELGSGKLPLQIEQFERGKTIFFPGDPAERVYLLVKGAVKLSRVYESGEEITVALLRENSVFGVLSLLTGQRSDRFYHAVAFTPVQLFSVPIEFMQKALIERPELANVMLQGLSSRILQTEMMIETLAHRDMGSRLVSFLLILCRDFGIPSPDGITIDLKLSHQAIAEAIGSTRVTVTRLLGDLRESKLIAIHKKRITVFNPVALSQQFS
4POF , Knot 110 257 0.79 40 164 244 
SVDREEMIERFANFLREYTDEDGNPVYRGKITDLLTITPKRSVAIDWMHLNSFDSELAHEVIENPEEGISAAEDAIQIVLREDFQREDVGKIHARFYNLPETLMVKDIGAEHINKLIQVEGIVTRVGEIKPFVSVAVFVCKDCGHEMIVPQKPYESLEKVKKCEQCGSKNIELDVNKSSFVNFQSFRIQDRPETLKGGEMPRFIDGILLDDIVDVALPGDRVIVTGILRVVLEKREKTPIFRKILEVNHIEPVSKEI

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 \(c\).

\(|P_{f(8HCC_1)}(2) \setminus P_{f(2XKP_1)}(2)|=85\), \(|P_{f(2XKP_1)}(2) \setminus P_{f(8HCC_1)}(2)|=63\). 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 1:110000000000111101001101011001101011000110101001111010101110001010010111100011000010010111110110110010101111010011100101100101010100010001111101110001010011100100001111000110100101101010100111101100100101010000001101000101011000001010110110100101011101100010110010110101100100
Pair \(Z_2\) Length of longest common subsequence
8HCC_1,2XKP_1 148 4
8HCC_1,4POF_1 176 3
2XKP_1,4POF_1 142 3

Newick tree

 
[
	8HCC_1:84.45,
	[
		2XKP_1:71,4POF_1:71
	]:13.45
]

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 \((0.85)(0.8)(\frac{498 }{\log_{20} 498}-\frac{222}{\log_{20}222})=79.6\)
Status Protein1 Protein2 d d1/2
Query variables 8HCC_1 2XKP_1 99 88
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]