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Parikh vectors
1MVF_1 9GLB_1 3TSR_1 Letter Amino acid
6 8 10 N Asparagine
9 22 8 Q Glutamine
5 10 10 K Lycine
15 18 11 S Serine
5 22 7 D Aspartic acid
2 9 8 C Cysteine
6 18 6 H Histidine
6 38 3 L Leucine
4 11 5 M Methionine
14 43 7 A Alanine
8 21 5 R Arginine
4 21 4 E Glutamic acid
14 40 4 G Glycine
8 19 9 V Valine
4 19 4 I Isoleucine
4 15 3 F Phenylalanine
2 22 7 P Proline
8 12 8 T Threonine
3 4 0 W Tryptophan
8 10 6 Y Tyrosine 

1MVF_1|Chains A, B|immunoglobulin heavy chain variable region|Camelus dromedarius (9838)
>9GLB_1|Chain A[auth C]|Deacetylase|Klebsiella pneumoniae subsp. ozaenae (574)
>3TSR_1|Chains A, B, C, D|Ribonuclease pancreatic|Mus musculus (10090)
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 \(w=f(c)\)
1MVF , Knot 64 135 0.77 40 102 125 
QVQLVESGGGSVQAGGSLRLSCAASGFTYSRKYMGWFRQAPGKEREGVAAIFIDNGNTIYADSVQGRFTISQDNAKNTVYLQMNSLKPEDTAMYYCAASSRWMDYSALTAKAYNSWGQGTQVTVSSRGRHHHHHH
9GLB , Knot 159 382 0.82 40 213 361 
MAHHHHHHVGTMKRKTGFFFDERCFWHSTGLHAVTLPVGGWVQPPAGGGHAESPETKRRMKNLMDVSGLTPQLALRSAAPASLEDLRRIHPDSYLERFKAISDNGGGMLGKEAPLGPGSYEIACLSAGLACAAVEAVLKGELDNAYSLSRPPGHHCLPDQSMGFCFLANIPIAVERAKAQLGLGKVAIIDWDVHHGNGTQHIYLQRDDVLTISLHQDGCFPPGYAGEDDRGVGAGEGYNINIPLLAGAGDDSWRYALETIVIPALARFEPELIIIACGYDANAMDPLARMQLHSDSFRAMTEQVQQAADRLCGGKLVMVHEGGYAESYVPFCGLAVMEALSGIRTEVQDPLLEFIQQQQPRATFAQFQRQAIDRLAQQFGLQ
3TSR , Knot 67 125 0.86 38 110 123 
MRESAAQKFQRQHMDPDGSSINSPTYCNQMMKRRDMTNGSCKPVNTFVHEPLADVQAVCSQENVTCKNRKSNCYKSSSALHITDCHLKGNSKYPNCDYKTTQYQKHIIVACEGNPYVPVHFDATV

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 \(c\).

\(|P_{f(1MVF_1)}(2) \setminus P_{f(9GLB_1)}(2)|=31\), \(|P_{f(9GLB_1)}(2) \setminus P_{f(1MVF_1)}(2)|=142\). 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 1:010110011101011101010011011000000111100111000011111110010010100101010100001000101010010100011000110001100011010100011010010100010000000
Pair \(Z_2\) Length of longest common subsequence
1MVF_1,9GLB_1 173 6
1MVF_1,3TSR_1 148 3
9GLB_1,3TSR_1 215 4

Newick tree

 
[
	9GLB_1:10.24,
	[
		1MVF_1:74,3TSR_1:74
	]:30.24
]

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 \((0.85)(0.8)(\frac{517 }{\log_{20} 517}-\frac{135}{\log_{20}135})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 1MVF_1 9GLB_1 139 91
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]