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Parikh vectors
1UHH_1 2GCB_1 7BDV_1 Letter Amino acid
7 19 18 S Serine
10 32 19 T Threonine
7 13 16 Y Tyrosine
17 52 25 A Alanine
9 14 26 N Asparagine
5 24 29 Q Glutamine
5 5 4 M Methionine
6 22 19 R Arginine
5 13 12 H Histidine
6 22 17 P Proline
19 24 19 D Aspartic acid
8 13 15 F Phenylalanine
6 6 5 W Tryptophan
12 40 36 L Leucine
15 16 21 K Lycine
9 35 19 V Valine
3 0 5 C Cysteine
15 16 17 E Glutamic acid
15 31 13 G Glycine
12 31 31 I Isoleucine 

1UHH_1|Chains A, B|Aequorin 2|Aequorea victoria (6100)
>2GCB_1|Chain A|Folylpolyglutamate synthase|Lactobacillus casei (1582)
>7BDV_1|Chains A, B, C, D|Can2|Sulfobacillus thermosulfidooxidans (28034)
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 \(w=f(c)\)
1UHH , Knot 94 191 0.86 40 145 183 
ANSKLTSDFDNPRWIGRHKHMFNFLDVNHNGKISLDEMVYKASDIVINNLGATPEQAKRHKDAVEAFFGGAGMKYGVETDWPAYIEGWKKLATDELEKYAKNEPTLIRIWGDALFDIVDKDQNGAITLDEWKAYTKAAGIIQSSEDCEETFRVCDIDESGQLDVDEMTRQHLGFWYTMDPACEKLYGGAVP
2GCB , Knot 177 428 0.83 38 209 396 
MNYTETVAYIHSFPRLAKTGDHRRILTLLHALGNPQQQGRYIHVTGTNGKSTAANAIAHVLEASGLTVGLYTSPFIMRFNERIMIDHEPIPDAALVNAVAFVRAALERLQQQQADFNVTEFEFITALGYWYFRQRQVDVAVIEVGIGGDTDSTNVITPVVSVLTSVALDHQKLLGHTITAIAKHKAGIIKRGIPVVTGNLVPDAAAVVAAKVATTGSQWLRFDRDFSVPKAKLHGWGQRFTYEDQDGRISDLEVPLVGDYQQRNMAIAIQTAKVYAKQTEWPLTPQNIRQGLAASHWPARLEKISDTPLIVIDGAHNPDGINGLITALKQLFSQPITVIAGILADKDYAAMADRLTAAFSTVYLVPVPGTPRALPEAGYEALHEGRLKDSWQEALAASLNDVPDQPIVITGSLYLASAVRQTLLGGKS
7BDV , Knot 162 366 0.87 40 221 356 
MARLDDLFIIHDTYVCLLSDHLLPNVIPVIQAPPQRVILLYTPNNKERVQRFRQATESVPTEIIEKQVHPYQYAQTQRICDEILEQFPNAILNVTGGTKIMALAAFDRFRHNHRPIIYVDSDSQRILYLHNGESERLGDPLTVKQYLACYGFKADNINRQDNLPKTWREVEDLFAQNSTKWQNQLGRLNWIAAQQQPIFTLQTGELQDLLLKANLIKPAEAKNAGFQFTSDQARQFINGGWFEHYVYSLLRQISAQYPIKNLTKNIEISNDSVSNELDVVFLYHNKLHVIECKTRHFTADGKINPMETIYKIDSVTNRVAGIKGKSMFASYYPLTQAAKKRCLNNSIYVSDQPSQLHHQLIKWINA

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 \(c\).

\(|P_{f(1UHH_1)}(2) \setminus P_{f(2GCB_1)}(2)|=55\), \(|P_{f(2GCB_1)}(2) \setminus P_{f(1UHH_1)}(2)|=119\). 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 1:10001000100101110000110110100010101001100100111001110100100000110111111110011000111010110011000100010001011011101110110000011101001010001111100000000010100100010101001000011110010110001011111
Pair \(Z_2\) Length of longest common subsequence
1UHH_1,2GCB_1 174 4
1UHH_1,7BDV_1 184 3
2GCB_1,7BDV_1 164 4

Newick tree

 
[
	1UHH_1:91.90,
	[
		2GCB_1:82,7BDV_1:82
	]:9.90
]

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 \((0.85)(0.8)(\frac{619 }{\log_{20} 619}-\frac{191}{\log_{20}191})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 1UHH_1 2GCB_1 149 108.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]