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Parikh vectors
7XHP_1 7DGR_1 5EAG_1 Letter Amino acid
2 8 4 C Cysteine
16 2 24 H Histidine
9 2 7 W Tryptophan
37 16 30 A Alanine
32 10 33 I Isoleucine
10 4 18 M Methionine
19 6 27 F Phenylalanine
28 12 28 T Threonine
13 10 29 Y Tyrosine
16 4 17 Q Glutamine
39 9 36 G Glycine
31 7 32 S Serine
27 19 31 E Glutamic acid
14 7 19 N Asparagine
38 8 25 D Aspartic acid
50 15 52 L Leucine
25 8 36 K Lycine
20 10 30 P Proline
37 5 35 V Valine
30 14 26 R Arginine 

7XHP_1|Chains A, B, C, D, E, F, G, H|Glucose 6-Phosphate Dehydrogenase|Zymomonas mobilis (542)
>7DGR_14|Chain N[auth E]|NADH dehydrogenase [ubiquinone] iron-sulfur protein 8, mitochondrial|Bos taurus (9913)
>5EAG_1|Chain A|Lanosterol 14-alpha demethylase|Saccharomyces cerevisiae (307796)
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 \(w=f(c)\)
7XHP , Knot 197 493 0.82 40 242 458 
MTNTVSTMILFGSTGDLSQRMLLPSLYGLDADGLLADDLRIVCTSRSEYDTDGFRDFAEKALDRFVASDRLNDDAKAKFLNKLFYATVDITDPTQFGKIADLCGPVEKGIAIYLSTSPSLFEGAIAGLKQAGLAGPTSRLALEKPLGQDLASSDHINDAVLKVFSEKQVYRIDHYLGKETVQNLLTLRFGNALFEPLWNSKGIDHVQISVAETVGLEGRIGYFDSSGSLRDMVQSHILQLVALVAMEPPAHMEANAVRDEKVKVFRALRPINNDTVITHTVTGQYGAGVSGGKEVAGYIDELGQPSDTETFVAIKAHVDNWRWHGVPFYIRTGKRLPARRSEIVVQFKPVPHSIFSSSGGILQPNKLRIVLQPDETIQISIMVKEPGLDRNGAHMREVWLDLSLTDVFKDRKRRIAYERLMLDLIEGDATLFVRRDEVEAQWIWIDGIREGWKANSMKPKTYVSGTWGPITAIALVERDGVTWYDLEHHHHHH
7DGR , Knot 84 176 0.82 40 132 172 
TYKYVNLREPSMDMKSVTDRAAQTLLWTELIRGLGMTLSYLFREPATINYPFEKGPLSPRFRGEHALRRYPSGEERCIACKLCEAVCPAQAITIEAEPRADGSRRTTRYDIDMTKCIYCGFCQEACPVDAIVEGPNFEFSTETHEELLYNKEKLLNNGDKWEAEIAANIQADYLYR
5EAG , Knot 220 539 0.85 40 268 510 
MSATKSIVGEALEYVNIGLSHFLALPLAQRISLIIIIPFIYNIVWQLLYSLRKDRPPLVFYWIPWVGSAVVYGMKPYEFFEECQKKYGDIFSFVLLGRVMTVYLGPKGHEFVFNAKLADVSAEAAYAHLTTPVFGKGVIYDCPNSRLMEQKKFVKGALTKEAFKSYVPLIAEEVYKYFRDSKNFRLNERTTGTIDVMVTQPEMTIFTASRSLLGKEMRAKLDTDFAYLYSDLDKGFTPINFVFPNLPLEHYRKRDHAQKAISGTYMSLIKERRKNNDIQDRDLIDSLMKNSTYKDGVKMTDQEIANLLIGVLMGGQHTSAATSAWILLHLAERPDVQQELYEEQMRVLDGGKKELTYDLLQEMPLLNQTIKETLRMHHPLHSLFRKVMKDMHVPNTSYVIPAGYHVLVSPGYTHLRDEYFPNAHQFNIHRWNNDSASSYSVGEEVDYGFGAISKGVSSPYLPFGGGRHRCIGEHFAYCQLGVLMSIFIRTLKWHYPEGKTVPPPDFTSMVTLPTGPAKIIWEKRNPEQKIGGRHHHHHH

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 \(c\).

\(|P_{f(7XHP_1)}(2) \setminus P_{f(7DGR_1)}(2)|=151\), \(|P_{f(7DGR_1)}(2) \setminus P_{f(7XHP_1)}(2)|=41\). 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 1:1000100111110010100011110101101011110010110000000000110011001100111000100010101100110101010010011011010111001111010001011011111100111111000111001110011000010011101100001001000110001001101011011101110001100101011001110101101000101001100011011111110111010101100001011011011000011000101001111011001110100110100000111101010010101111010010011100001110101110011000111101001011101000101011100111000110100111010100110000001100011101101010111000010101111011001101001010001010111101111100011010010000000
Pair \(Z_2\) Length of longest common subsequence
7XHP_1,7DGR_1 192 3
7XHP_1,5EAG_1 136 6
7DGR_1,5EAG_1 192 3

Newick tree

 
[
	7DGR_1:10.66,
	[
		7XHP_1:68,5EAG_1:68
	]:35.66
]

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 \((0.85)(0.8)(\frac{669 }{\log_{20} 669}-\frac{176}{\log_{20}176})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 7XHP_1 7DGR_1 178 119.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]