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Parikh vectors
9MQH_1 7VIT_1 5LDH_1 Letter Amino acid
21 12 17 D Aspartic acid
12 9 24 K Lycine
33 10 25 S Serine
37 6 14 T Threonine
27 15 19 A Alanine
23 2 11 P Proline
7 7 7 H Histidine
9 15 18 E Glutamic acid
15 11 25 G Glycine
36 4 23 I Isoleucine
38 27 36 L Leucine
20 8 5 F Phenylalanine
16 6 7 Y Tyrosine
7 11 15 Q Glutamine
24 6 19 N Asparagine
17 3 5 C Cysteine
13 3 9 M Methionine
7 1 6 W Tryptophan
29 8 40 V Valine
20 10 8 R Arginine 

9MQH_1|Chain A|Mu-type opioid receptor|Homo sapiens (9606)
>7VIT_1|Chain A|Ferritin light chain|Equus caballus (9796)
>5LDH_1|Chains A, B|LACTATE DEHYDROGENASE|Sus scrofa (9823)
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 \(w=f(c)\)
9MQH , Knot 171 411 0.83 40 230 393 
DYKDDDDASIDMDSSAAPTNASNCTDALAYSSCSPAPSPGSWVNLSHLDGNLSDPCGPNRTDLGGRDSLCPPTGSPSMITAITIMALYSIVCVVGLFGNFLVMYVIVRYTKMKTATNIYIFNLALADALATSTLPFQSVNYLMGTWPFGTILCKIVISIDYYNMFTSIFTLCTMSVDRYIAVCHPVKALDFRTPRNAKIINVCNWILSSAIGLPVMFMATTKYRQGSIDCTLTFSHPTWYWENLLKICVFIFAFIMPVLIITVCYGLMILRLKSVRLLSGSREKDRNLRRITRMVLVVVAVFIVCWTPIHIYVIIKALVTIPETTFQTVSWHFCIALGYTNSCLNPVLYAFLDENFKRCFREFCIPTSSNIEQQNSTRIRQNTRDHPSTANTVDRTNHQLENLEAETAPLP
7VIT , Knot 84 174 0.83 40 130 171 
SSQIRQNYSTEVEAAVNRLVNLYLRASYTYLSLGFYFDRDDVALEGVCHFFRELAEEKREGAERLLKMQNQRGGRALFQDLQKPSQDEWGTTLDAMKAAIVLEKSLNQALLDLHALGSAQADPHLCDFLESHFLDEEVKLIKKMGDHLTNIQRLVGSQAGLGEYLFEHCTLKHD
5LDH , Knot 142 334 0.82 42 195 316 
XATLKEKLIAPVAQQETTIPDNKITVVGVGQVGMACAISILGKSLTDELALVDVLEDKLKGEMMDLQHGSLFLQTPKIVANKDYSVTANSKIVVVTAGVRQQEGESRLNLVQRNVNVFKFIIPQIVKYSPNCIIIVVSNPVDILTYVAWKLSGLPKHRVIGSGCNLDSARFRYLMGEKLGVHPSSCHGWILGEHGDSSVAVWSGVNVAGVVLQQLNPEMGTDNDSENWKEVHKMVVESAYEVIKLKGYTNWAIGLSVADLIESMLKNLSRIHPVSTMVQGMYGIENEVFLSLPCVLNARGLTSVINQKLKDDEVAQLKNSADTLWGIQKDLKDL

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 \(c\).

\(|P_{f(9MQH_1)}(2) \setminus P_{f(7VIT_1)}(2)|=150\), \(|P_{f(7VIT_1)}(2) \setminus P_{f(9MQH_1)}(2)|=50\). 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 1:000000010101000111001000001110000011101101101001010100101100001110001011010101101101111001101111110111101110000100100101101111011100011100100111011110110011101000011001101001010001110011011010010010110100111001111111111000000101000101001010100110101111111111111010011111010010110100000001001001111111111101011010111011101100010010101011110000010111011100010001001011000010000000100000001001001000000100101001111
Pair \(Z_2\) Length of longest common subsequence
9MQH_1,7VIT_1 200 4
9MQH_1,5LDH_1 175 4
7VIT_1,5LDH_1 181 3

Newick tree

 
[
	7VIT_1:97.85,
	[
		9MQH_1:87.5,5LDH_1:87.5
	]:10.35
]

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 \((0.85)(0.8)(\frac{585 }{\log_{20} 585}-\frac{174}{\log_{20}174})=118.\)
Status Protein1 Protein2 d d1/2
Query variables 9MQH_1 7VIT_1 153 107.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]