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Parikh vectors
1YIH_1 3WYB_1 7VZS_1 Letter Amino acid
21 25 61 A Alanine
1 2 1 C Cysteine
10 15 7 H Histidine
18 22 50 L Leucine
11 18 30 K Lycine
7 16 26 P Proline
1 2 11 W Tryptophan
1 12 17 Q Glutamine
4 24 28 E Glutamic acid
7 15 16 F Phenylalanine
11 23 43 S Serine
9 15 30 T Threonine
13 28 48 V Valine
7 27 53 G Glycine
0 20 28 I Isoleucine
2 6 8 M Methionine
3 17 23 R Arginine
4 15 40 N Asparagine
8 21 30 D Aspartic acid
3 11 22 Y Tyrosine 

1YIH_1|Chains A, C|Hemoglobin alpha chain|Homo sapiens (9606)
>3WYB_1|Chains A, B|Meso-diaminopimelate D-dehydrogenase|Ureibacillus thermosphaericus (51173)
>7VZS_1|Chains A, B|GMC oxidoreductase|Aspergillus oryzae (5062)
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 \(w=f(c)\)
1YIH , Knot 69 141 0.80 38 99 137 
VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHFDLSHGSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTSKYR
3WYB , Knot 146 334 0.84 40 199 318 
MSKIRIGIVGYGNLGRGVEAAIQQNPDMELVAVFTRRDPKTVAVKSNVKVLHVDDAQSYKDEIDVMILCGGSATDLPEQGPYFAQYFNTIDSFDTHARIPDYFDAVNAAAEQSGKVAIISVGWDPGLFSLNRLLGEVVLPVGNTYTFWGKGVSQGHSDAIRRIQGVKNAVQYTIPIDEAVNRVRSGENPELSTREKHARECFVVLEEGADPAKVEHEIKTMPNYFDEYDTTVHFISEEELKQNHSGMPHGGFVIRSGKSDEGHKQIIEFSLNLESNPMFTSSALVAYARAAYRLSQNGDKGAKTVFDIPFGLLSPKSPEDLRKELLTRHHHHHH
7VZS , Knot 220 572 0.81 40 239 522 
MKNTTTYDYIVVGGGTSGLVVANRLSENPDVSVLLLEAGASVFNNPDVTNANGYGLAFGSAIDWQYQSINQSYAGGKQQVLRAGKALGGTSTINGMAYTRAEDVQIDVWQKLGNEGWTWKDLLPYYLKSENLTAPTSSQVAAGAAYNPAVNGKEGPLKVGWSGSLASGNLSVALNRTFQAAGVPWVEDVNGGKMRGFNIYPSTLDVDLNVREDAARAYYFPYDDRKNLHLLENTTANRLFWKNGSAEEAIADGVEITSADGKVTRVHAKKEVIISAGALRSPLILELSGVGNPTILKKNNITPRVDLPTVGENLQDQFNNGMAGEGYGVLAGASTVTYPSISDVFGNETDSIVASLRSQLSDYAAATVKVSNGHMKQEDLERLYQLQFDLIVKDKVPIAEILFHPGGGNAVSSEFWGLLPFARGNIHISSNDPTAPAAINPNYFMFEWDGKSQAGIAKYIRKILRSAPLNKLIAKETKPGLSEIPATAADEKWVEWLKANYRSNFHPVGTAAMMPRSIGGVVDNRLRVYGTSNVRVVDASVLPFQVCGHLVSTLYAVAERASDLIKEDAKSA

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 \(c\).

\(|P_{f(1YIH_1)}(2) \setminus P_{f(3WYB_1)}(2)|=28\), \(|P_{f(3WYB_1)}(2) \setminus P_{f(1YIH_1)}(2)|=128\). 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 1:110110000101111011101100110110011101100000110101001010101010011011001110100110110110010100101011010110001110111011101011101010011101001100000
Pair \(Z_2\) Length of longest common subsequence
1YIH_1,3WYB_1 156 4
1YIH_1,7VZS_1 184 3
3WYB_1,7VZS_1 138 4

Newick tree

 
[
	1YIH_1:90.06,
	[
		3WYB_1:69,7VZS_1:69
	]:21.06
]

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 \((0.85)(0.8)(\frac{475 }{\log_{20} 475}-\frac{141}{\log_{20}141})=98.9\)
Status Protein1 Protein2 d d1/2
Query variables 1YIH_1 3WYB_1 126 87
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]