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Parikh vectors
1NQT_1 3DZA_1 4MBO_1 Letter Amino acid
10 12 14 Q Glutamine
34 11 19 E Glutamic acid
47 11 18 G Glycine
31 17 27 L Leucine
32 13 24 V Valine
36 40 12 A Alanine
21 7 27 N Asparagine
27 14 23 D Aspartic acid
13 4 7 M Methionine
23 3 6 F Phenylalanine
32 9 39 S Serine
29 5 7 R Arginine
6 0 1 C Cysteine
38 10 23 I Isoleucine
14 1 9 H Histidine
33 15 21 K Lycine
22 4 9 P Proline
26 9 36 T Threonine
4 1 3 W Tryptophan
18 5 15 Y Tyrosine 

1NQT_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Glutamate dehydrogenase 1|Bos taurus (9913)
>3DZA_1|Chains A, B, C, D|uncharacterized putative membrane protein|Klebsiella pneumoniae subsp. pneumoniae MGH 78578 (272620)
>4MBO_1|Chain A|Serine-rich Repeat Adhesion Glycoprotein (Srr1)|Streptococcus agalactiae (211110)
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 \(w=f(c)\)
1NQT , Knot 203 496 0.84 40 255 464 
DPNFFKMVEGFFDRGASIVEDKLVEDLRTRESEEQKRNRVRGILRIIKPCNHVLSLSFPIRRDDGSWEVIEGYRAQHSHQRTPCKGGIRYSTDVSVDEVKALASLMTYKCAVVDVPFGGAKAGVKINPKNYTDNELEKITRRFTMELAKKGFIGPGIDVPAPDMSTGEREMSWIADTYASTIGHYDINAHACVTGKPISQGGIHGRISATGRGVFHGIENFINEASYMSILGMTPGFGDKTFVVQGFGNVGLHSMRYLHRFGAKCIAVGESDGSIWNPDGIDPKELEDFKLQHGSILGFPKAKPYEGSILEADCDILIPAASEKQLTKSNAPRVKAKIIAEGANGPTTPEADKIFLERNIMVIPDLYLNAGGVTVSYFEWLKNLNHVSYGRLTFKYERDSNYHLLMSVQESLERKFGKHGGTIPIVPTAEFQDRISGASEKDIVHSGLAYTMERSARQIMRTAMKYNLGLDLRTAAYVNAIEKVFKVYNEAGVTFT
3DZA , Knot 86 191 0.78 38 121 182 
GATDSATAAPAAAATTQVQKEAADVLQVAVQGANAMRDIQFARLALFHGQPDSAKKLTDDAAALLAADDASWAKFVKTDAKAKMIADRYVIINASIALSEDYVATPEKESAIQSANEKLAKGDQKGAIDTLRLAGIGVIENQYLMPLNQTRKAVAQSQELLKAGKYYEANLVLKGAEEGIVVDSEMLVAGN
4MBO , Knot 142 340 0.81 40 179 317 
MEVSDTEMLGKDVSSELQKVNIALKDNTLSEPGTVKLDSSENLVLNFAFSIASVNEGDVFTVKLSDNLDTQGIGTILKVQDIMDETGQLLATGSYSPLTHNITYTWTRYASTLNNIKARVNMPVWPDQRIISKTTSDKQCFTATLNNQVASIEERVQYNSPSVTEHTNVKTNVRSRIMKLDDERQTETYITQINPEGKEMYFASGLGNLYTIIGSDGTSGSPVNLLNAEVKILKTNSKNLTDSMDQNYDSPEFEDVTSQYSYTNDGSKITIDWKTNSISSTTSYVVLVKIPKQSGVLYSTVSDINQTYGSKYSYGHTNISGDSDANAEIKLLLEHHHHHH

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 \(c\).

\(|P_{f(1NQT_1)}(2) \setminus P_{f(3DZA_1)}(2)|=157\), \(|P_{f(3DZA_1)}(2) \setminus P_{f(1NQT_1)}(2)|=23\). 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 1:0101101101110011011000110010000000000001011101101000110101110000101011010010000000100111000001010010111011000011101111110111010100000001001000101011001111111011110100100010111000100110001010101010110011101010101011101100110010010111101111000111011101110010010011100111100010110101101001001010010111110101001011010001111110000100001101010111011011001010011100011111010101111010010110010010010101000000000111010001000110011011111010100010110000110011100100010011001100011101001101011001101000111010
Pair \(Z_2\) Length of longest common subsequence
1NQT_1,3DZA_1 180 3
1NQT_1,4MBO_1 178 4
3DZA_1,4MBO_1 148 4

Newick tree

 
[
	1NQT_1:94.10,
	[
		4MBO_1:74,3DZA_1:74
	]:20.10
]

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 \((0.85)(0.8)(\frac{687 }{\log_{20} 687}-\frac{191}{\log_{20}191})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 1NQT_1 3DZA_1 180 121
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]