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Parikh vectors
3NLX_1 4BMA_1 2JIY_1 Letter Amino acid
16 10 7 Y Tyrosine
11 6 2 C Cysteine
22 22 11 R Arginine
18 15 8 N Asparagine
18 21 5 Q Glutamine
25 40 25 G Glycine
23 33 13 I Isoleucine
27 44 14 K Lycine
11 8 9 M Methionine
20 38 27 A Alanine
24 37 20 V Valine
25 18 12 T Threonine
23 31 23 P Proline
26 28 12 S Serine
26 28 13 D Aspartic acid
14 12 6 H Histidine
36 37 24 L Leucine
19 29 10 F Phenylalanine
13 4 3 W Tryptophan
25 48 16 E Glutamic acid 

3NLX_1|Chains A, B|Nitric oxide synthase, brain|Rattus norvegicus (10116)
>4BMA_1|Chains A, B|UDP-N-ACETYLGLUCOSAMINE PYROPHOSPHORYLASE|ASPERGILLUS FUMIGATUS (746128)
>2JIY_1|Chain A[auth H]|REACTION CENTER PROTEIN H CHAIN|RHODOBACTER SPHAEROIDES (1063)
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 \(w=f(c)\)
3NLX , Knot 186 422 0.88 40 262 411 
CPRFLKVKNWETDVVLTDTLHLKSTLETGCTEHICMGSIMLPSQHTRKPEDVRTKDQLFPLAKEFLDQYYSSIKRFGSKAHMDRLEEVNKEIESTSTYQLKDTELIYGAKHAWRNASRCVGRIQWSKLQVFDARDCTTAHGMFNYICNHVKYATNKGNLRSAITIFPQRTDGKHDFRVWNSQLIRYAGYKQPDGSTLGDPANVQFTEICIQQGWKAPRGRFDVLPLLLQANGNDPELFQIPPELVLEVPIRHPKFDWFKDLGLKWYGLPAVSNMLLEIGGLEFSACPFSGWYMGTEIGVRDYCDNSRYNILEEVAKKMDLDMRKTSSLWKDQALVEINIAVLYSFQSDKVTIVDHHSATESFIKHMENEYRCRGGCPADWVWIVPPMSGSITPVFHQEMLNYRLTPSFEYQPDPWNTHVWKG
4BMA , Knot 205 509 0.83 40 245 476 
MAVAIKETVSNFMERFHGHPENLPREPSAEEFQQLRKKYTDAGQGHVFAFVDELQTGERSQLFHQLSSFDPVRINELADKALNPPKADDGPASLEPLPDIATASILDSDPKDLEQWYEEGLKLVAGNKVAVVLMAGGQGTRLGSSAPKGCFDIGLPSHKSLFQIQAERIAKLQLLAQRISGKEAVIPWYVMTSGPTRKPTEEFFEQHKYFGLNKSDVIIFEQGVLPCISNEGKILMESKFKVAVAPDGNGGIYQALLTSGVREDMRKRGIEHIHTYCVDNCLVKVADPVFIGFAASKQVDIATKVVRKRNATESVGLILQKNGKPDVVEYSEIDKETAEAKDPKQPDVLKFRAANIVNHYYSFKFFESIELWAHKLPHHVARKKIPCIKEGTGEFFKPEKPNGIKLEQFVFDVFPMTPLEKFACIEVRREDEFSPLKNARGTGEDDPDTSKRDIMSQGQRWIEKAGGIVITEGDVVGVEVSPLISYGGEGLEFLKGREIKAPAFIEKEE
2JIY , Knot 116 260 0.82 40 168 254 
MVGVTAFGNFDLASLAIYSFWIFLAGLIYYLQTENMREGYPLENEDGTPAANQGPFPLPKPKTFILPHGRGTLTVPGPESEDRPIALARTAVSEGFPHAPTGDPMKDGVGPASWVARRDLPELDGHGHNKIKPMKAAAGFHVSAGKNPIGLPVRGCDLEIAGKVVDIWVDIPEQMARFLEVELKDGSTRLLPMQMVKVQSNRVHVNALSSDLFAGIPTIKSPTEVTLLEEDKICGYVAGGLMYAAPKRKSVVAAMLAEYA

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 \(c\).

\(|P_{f(3NLX_1)}(2) \setminus P_{f(4BMA_1)}(2)|=80\), \(|P_{f(4BMA_1)}(2) \setminus P_{f(3NLX_1)}(2)|=63\). 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 1:01011010010001110001010001001000010110111100000010010000011111001100000010011001010010010001000000010000110110011001000110101001011010000010111001000100100010100110111000010001011000110011000101001101101010010100110110101011111101010010110111011101110010101100111010111110011101111010101101101100111000000000011001100101010000011000111010111100100001011000010001100100000001101101111111101010111000110001010100010110001101
Pair \(Z_2\) Length of longest common subsequence
3NLX_1,4BMA_1 143 4
3NLX_1,2JIY_1 186 4
4BMA_1,2JIY_1 157 4

Newick tree

 
[
	2JIY_1:90.38,
	[
		3NLX_1:71.5,4BMA_1:71.5
	]:18.88
]

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 \((0.85)(0.8)(\frac{931 }{\log_{20} 931}-\frac{422}{\log_{20}422})=135.\)
Status Protein1 Protein2 d d1/2
Query variables 3NLX_1 4BMA_1 170 157.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]