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Parikh vectors
1FWC_1 9CTH_1 4JLL_1 Letter Amino acid
4 47 5 D Aspartic acid
2 25 3 Q Glutamine
6 49 9 I Isoleucine
9 36 16 V Valine
0 10 1 C Cysteine
5 20 2 M Methionine
5 59 6 S Serine
9 34 24 A Alanine
2 33 4 N Asparagine
2 23 10 H Histidine
5 46 10 K Lycine
5 49 9 T Threonine
1 35 3 Y Tyrosine
7 32 10 R Arginine
11 48 15 E Glutamic acid
6 38 19 G Glycine
12 47 12 L Leucine
3 30 2 F Phenylalanine
6 37 5 P Proline
0 11 2 W Tryptophan 

1FWC_1|Chain A|UREASE|Klebsiella aerogenes (548)
>9CTH_1|Chain A|Activated Factor V (FVa) heavy chain|Homo sapiens (9606)
>4JLL_1|Chain A|Evolved variant of computationally designed serine hydrolase OSH55.4_H1|synthetic construct (32630)
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 \(w=f(c)\)
1FWC , Knot 51 100 0.78 36 79 96 
MELTPREKDKLLLFTAALVAERRLARGLKLNYPESVALISAFIMEGARDGKSVASLMEEGRHVLTREQVMEGVPEMIPDIQVEATFPDGSKLVTVHNPII
9CTH , Knot 278 709 0.85 40 302 644 
AQLRQFYVAAQGISWSYRPEPTNSSLNLSVTSFKKIVYREYEPYFKKEKPQSTISGLLGPTLYAEVGDIIKVHFKNKADKPLSIHPQGIRYSKLSEGASYLDHTFPAEKMDDAVAPGREYTYEWSISEDSGPTHDDPPCLTHIYYSHENLIEDFNSGLIGPLLICKKGTLTEGGTQKTFDKQIVLLFAVFDESKSWSQSSSLMYTVNGYVNGTMPDITVCAHDHISWHLLGMSSGPELFSIHFNGQVLEQNHHKVSAITLVSATSTTANMTVGPEGKWIISSLTPKHLQAGMQAYIDIKNCPKKTRNLKKITREQRRHMKRWEYFIAAEEVIWDYAPVIPANMDKKYRSQHLDNFSNQIGKHYKKVMYTQYEDESFTKHTVNPNMKEDGILGPIIRAQVRDTLKIVFKNMASRPYSIYPHGVTFSPYEDEVNSSFTSGRNNTMIRAVQPGETYTYKWNILEFDEPTENDAQCLTRPYYSDVDIMRDIASGLIGLLLICKSRSLDRRGIQRAADIEQQAVFAVFDENKSWYLEDNINKFCENPDEVKRDDPKFYESNIMSTINGYVPESITTLGFCFDDTVQWHFCSVGTQNEILTIHFTGHSFIYGKRHEDTLTLFPMRGESVTVTMDNVGTWMLTSMNSSPRSKKLRLKFRDVKCIPDDDEDSYEIFEPPESTVMATRKMHDRLEPEDEESDADYDYQNRLAAALGIR
4JLL , Knot 78 167 0.79 40 113 154 
MARIRHVQGDITEFQGDAIVNAANNYLKLGAGVAGAILRKGGPSIQEECDRIGKIRVGEAAVTGAGNLPVRYVIHAAVLGDEPASLETVRKATKSALEKAVELGLKTVAFTSWGAWVGGLPAEAVHRVMGEEIKKAPDTLEVTGVHGTEKSAEAYRRASLEHHHHHH

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 \(c\).

\(|P_{f(1FWC_1)}(2) \setminus P_{f(9CTH_1)}(2)|=12\), \(|P_{f(9CTH_1)}(2) \setminus P_{f(1FWC_1)}(2)|=235\). 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 1:1010100000111101111100011011010010011110111101100100110110010011000011011101110101010110100110100111
Pair \(Z_2\) Length of longest common subsequence
1FWC_1,9CTH_1 247 4
1FWC_1,4JLL_1 122 3
9CTH_1,4JLL_1 235 3

Newick tree

 
[
	9CTH_1:13.65,
	[
		1FWC_1:61,4JLL_1:61
	]:73.65
]

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 \((0.85)(0.8)(\frac{809 }{\log_{20} 809}-\frac{100}{\log_{20}100})=201.\)
Status Protein1 Protein2 d d1/2
Query variables 1FWC_1 9CTH_1 257 144.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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