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Parikh vectors
4WLM_1 2OMK_1 6SCL_1 Letter Amino acid
22 15 6 E Glutamic acid
16 12 2 H Histidine
10 4 4 M Methionine
19 8 7 P Proline
10 14 17 T Threonine
5 3 2 C Cysteine
12 11 7 Q Glutamine
22 12 12 V Valine
15 7 22 R Arginine
21 10 12 F Phenylalanine
5 2 1 W Tryptophan
11 12 4 Y Tyrosine
12 19 12 I Isoleucine
14 8 12 K Lycine
15 11 5 D Aspartic acid
17 19 18 G Glycine
34 20 8 L Leucine
15 16 20 S Serine
21 17 17 A Alanine
10 11 12 N Asparagine 

4WLM_1|Chains A, B|Xyloside xylosyltransferase 1|Mus musculus (10090)
>2OMK_1|Chains A, B|Hypothetical protein|Bacteroides thetaiotaomicron (226186)
>6SCL_1|Chains A, B, C|Coat protein|Barley yellow dwarf virus (12037)
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 \(w=f(c)\)
4WLM , Knot 138 306 0.86 40 204 293 
SLEGGVVVPVDYHLLMMFTKAEHNAPLQAKARVALSSLLRLAKFEAHEVLNLHFVSEEASREVAKALLRELLPPAAGFKCKVIFHDVAVLTDKLFPVVEAMQKYFSAGSGTYYSDSIFFLSVAMHQIMPKEIPRIIQLDLDLKYKTNIRELFEEFDNFLPGAVIGIAREMQPVYRHTFWQFRHENPKTRVGDPPPEGLPGFNSGVMLLNLEAMRQSPLYSHLLEPSWVQQLADKYHFRGHLGDQDFFTMIGMEHPELFHVLDCTWNRQLCTWWRDHGYSDVFQAYFRCEGHVKIYHGNCNTPIPED
2OMK , Knot 104 231 0.81 40 166 222 
MHHHHHHSSGVDLGTENLYFQSNAMINEHYIPQAIILANGEYPAHELPLRLLAEAQFVVCCNGAANEYISRGHTPDVIIGDGDSLLPEYKKRFSSIILQISDQETNDQTKAVHYLQSKGIRKIAIVGATGKREDHTLGNISLLVEYMRSGMEVRTVTDYGTFIPVSDTQSFASYPGQQVSIINFGAKGLKAEGLFYPLSDFTNWWQGTLNEAIADEFTIHCTGEYLVFLAY
6SCL , Knot 88 200 0.77 40 132 183 
MNSVGRRGPRRANQNGTRRRRRRTVRPVVVVQPNRAGPRRRNGRRKGRGGANFVFRPTGGTEVFVFSVDNLKANSSGAIKFGPSLSQCPALSDGILKSYHRYKITSIRVEFKSHASANTAGAIFIELDTACKQSALGSYINSFTISKTASKTFRSEAINGKEFQESTIDQFWMLYKANGTTTDTAGQFIITMSVSLMTAK

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 \(c\).

\(|P_{f(4WLM_1)}(2) \setminus P_{f(2OMK_1)}(2)|=101\), \(|P_{f(2OMK_1)}(2) \setminus P_{f(4WLM_1)}(2)|=63\). 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 1:010111111100011111001000111010101110011011010100110101100010001101110011111111000111001111000111110110001011010000001111011100111001101101010100000100110010011111111110010110000110100001000110111011111001111101011000110001101011001100001010110001101111001011011000100010011000100011010100010101001000011100
Pair \(Z_2\) Length of longest common subsequence
4WLM_1,2OMK_1 164 3
4WLM_1,6SCL_1 188 3
2OMK_1,6SCL_1 160 3

Newick tree

 
[
	4WLM_1:90.77,
	[
		2OMK_1:80,6SCL_1:80
	]:10.77
]

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 \((0.85)(0.8)(\frac{537 }{\log_{20} 537}-\frac{231}{\log_{20}231})=87.5\)
Status Protein1 Protein2 d d1/2
Query variables 4WLM_1 2OMK_1 110 96
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]