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Parikh vectors
4WMJ_1 3FLM_1 5NSF_1 Letter Amino acid
30 35 18 V Valine
54 64 20 A Alanine
37 16 15 N Asparagine
30 26 15 D Aspartic acid
29 31 33 E Glutamic acid
41 13 33 K Lycine
9 17 1 W Tryptophan
18 9 13 Y Tyrosine
15 35 16 R Arginine
17 19 13 H Histidine
11 8 12 M Methionine
24 30 14 S Serine
36 27 7 T Threonine
2 7 2 C Cysteine
21 36 5 Q Glutamine
52 69 31 L Leucine
31 14 8 F Phenylalanine
40 40 23 G Glycine
34 28 27 I Isoleucine
25 32 12 P Proline 

4WMJ_1|Chains A, B, C, D|Glucose-6-phosphate isomerase|Colias eurytheme (42296)
>3FLM_1|Chains A, B|2-succinyl-5-enolpyruvyl-6-hydroxy-3-cyclohexene-1-carboxylate synthase|Escherichia coli K-12 (83333)
>5NSF_1|Chains A, B, C, D|Tyrosine--tRNA ligase|Methanocaldococcus jannaschii (strain ATCC 43067 / DSM 2661 / JAL-1 / JCM 10045 / NBRC 100440) (243232)
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 \(w=f(c)\)
4WMJ , Knot 226 556 0.85 40 260 522 
MEPKVNLKQDPAYQKLQEYYDNNADKINILQLFQQDADRFNKYSLRIPTPNDGEILLDYSKNRIDDTTFSLLLNLAKSRNVEKARDAMFAGEKINFTEDRAVLHVALRNRQNRPIMVNGKDVTPDVNAVLAHMKEFSTQVISGAWKGYTGKPITDVINIGIGGSDLGPLMVTEALKPYANHLKVHFVSNIDGTHLAEVLKRLNPETALFIIASKTFTTQETITNATSAKTWFLEAAKDPAAVSKHFVALSTNGEKVTAFGIDPKNMFGFWDWVGGRYSLWSAIGLSISLYIGFENFEKLLDGANFMDNHFCTAPLEKNAPVILALLGVWYGNFYGAETHALLPYDQYLHRFAAYFQQGDMESNGKYVTRSGATVDYSTGPVVWGEPGTNGQHAFYQLVHQGTRLIPCDFLAPAQTHNPIANGAHHKILLANFLAQTEALMKGKTDAEAKAELEKSGMAPEAIAKILPHKVFKGNRPTNSIVVKKFTPFTLGALIAMYEHKIFTQGVIWDINSFDQWGVELGKQLAKAIEPELQDGKKITSHDASTNGLINFLKENF
3FLM , Knot 227 556 0.86 40 250 531 
MSVSAFNRRWAAVILEALTRHGVRHICIAPGSRSTLLTLAAAENSAFIHHTHFDERGLGHLALGLAKVSKQPVAVIVTSGTAVANLYPALIEAGLTGEKLILLTADRPPELIDCGANQAIRQPGMFASHPTHSISLPRPTQDIPARWLVSTIDHALGTLHAGGVHINCPFAEPLYGEMDDTGLSWQQRLGDWWQDDKPWLREAPRLESEKQRDWFFWRQKRGVVVAGRMSAEEGKKVALWAQTLGWPLIGDVLSQTGQPLPCADLWLGNAKATSELQQAQIVVQLGSSLTGKRLLQWQASCEPEEYWIVDDIEGRLDPAHHRGRRLIANIADWLELHPAEKRQPWCVEIPRLAEQAMQAVIARRDAFGEAQLAHRICDYLPEQGQLFVGNSLVVRLIDALSQLPAGYPVYSNRGASGIDGLLSTAAGVQRASGKPTLAIVGDLSALYDLNALALLRQVSAPLVLIVVNNNGGQIFSLLPTPQSERERFYLMPQNVHFEHAAAMFELKYHRPQNWQELETAFADAWRTPTTTVIEMVVNDTDGAQTLQQLLAQVSHL
5NSF , Knot 136 318 0.82 40 181 299 
MRGSHHHHHHGSMDEFEMIKRNTSEIISEEELREVLKKDEKSAGIGFEPSGKIHLGHYLQIKKMIDLQNAGFDIIIWLADLGAYLNQKGELDEIRKIGDYNKKVFEAMGLKAKYVYGSEHNLDKDYTLNVYRLALKTTLKRARRSMELIAREDENPKVAEVIYPIMQVNAIHYNGVDVAVGGMEQRKIHMLARELLPKKVVCIHNPVLTGLDGEGKMSSSKGNFIAVDDSPEEIRAKIKKAYCPAGVVEGNPIMEIAKYFLEYPLTIKRPEKFGGDLTVNSYEELESLFKNKELHPMRLKNAVAEELIKILEPIRKRL

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 \(c\).

\(|P_{f(4WMJ_1)}(2) \setminus P_{f(3FLM_1)}(2)|=78\), \(|P_{f(3FLM_1)}(2) \setminus P_{f(4WMJ_1)}(2)|=68\). 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 1:1010101000110001000000010010110110001001000010110100101110000001000010111011000010010011111001010000111011100000011110100101010111101001000110111010010110011011111001111110011010100101011001010011011001010011111100010000010010010011101100111100011110001001011110100111110111100011011110101011100100110110110001001110001111111111101010110001111000010011101001010001001000110100001111110110010011001100100111001111100001110110001111011100011101000101010100011110111011100110100100011100101101111111000011001111010010011101100110110101001001000010001110110001
Pair \(Z_2\) Length of longest common subsequence
4WMJ_1,3FLM_1 146 4
4WMJ_1,5NSF_1 171 4
3FLM_1,5NSF_1 173 4

Newick tree

 
[
	5NSF_1:89.91,
	[
		4WMJ_1:73,3FLM_1:73
	]:16.91
]

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 \((0.85)(0.8)(\frac{1112 }{\log_{20} 1112}-\frac{556}{\log_{20}556})=143.\)
Status Protein1 Protein2 d d1/2
Query variables 4WMJ_1 3FLM_1 186 183.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]