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Parikh vectors
6ZZJ_1 8JQZ_1 4EUX_1 Letter Amino acid
21 25 25 T Threonine
15 20 26 D Aspartic acid
15 15 25 G Glycine
24 32 36 L Leucine
14 41 27 K Lycine
4 11 11 M Methionine
12 14 23 P Proline
13 37 26 S Serine
3 10 16 Y Tyrosine
11 14 22 R Arginine
12 18 18 N Asparagine
0 4 11 C Cysteine
8 25 19 F Phenylalanine
21 19 24 V Valine
9 11 18 Q Glutamine
0 4 13 W Tryptophan
22 33 20 A Alanine
12 39 25 E Glutamic acid
6 13 14 H Histidine
19 29 23 I Isoleucine 

6ZZJ_1|Chain A|Dihydrolipoyllysine-residue acetyltransferase component of pyruvate dehydrogenase complex|Corynebacterium glutamicum (strain ATCC 13032 / DSM 20300 / JCM 1318 / LMG 3730 / NCIMB 10025) (196627)
>8JQZ_1|Chains A, B|Immunity-related GTPase family member b10|Mus musculus molossinus (57486)
>4EUX_1|Chains A, B|Nitric oxide synthase, brain|Rattus norvegicus (10116)
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 \(w=f(c)\)
6ZZJ , Knot 106 241 0.80 36 162 230 
GSLRGTTQKVNRIREITAMKTVEALQISAQLTQLHEVDMTRVAELRKKNKPAFIEKHGVNLTYLPFFVKAVVEALVSHPNVNASFNAKTKEMTYHSSVNLSIAVDTPAGLLTPVIHDAQDLSIPEIAKAIVDLADRSRNNKLKPNDLSGGTFTITNIGSEGALSDTPILVPPQAGILGTGAIVKRPVVITEDGIDSIAIRQMVFLPLTYDHQVVDGADAGRFLTTIKDRLETANFEGDLQL
8JQZ , Knot 175 414 0.85 40 229 387 
MGQSSSKPDAKAHNKASSLTEFFKNFKMESKIISKETIDSIQSCIQEGDIQKVISIINAALTDIEKAPLNIAVTGETGAGKSTFINALRGIGHEESESAESGAVETTMDRKKYTHPKFPNVTIWDLPGVGTTNFKPEEYLKKMKFQEYDFFLIISSARFRNNEAQLAEAIKKMKKKFYFVRTKIDSDLWNEKKAKPSSYNREKILEAIRSDCVKNLQNANAASTRVFLVSSFEVAQFDFPSLESTLLEELPAHKRHIFVQCLPTITEPAIDRRRDVLKQTIWLEALKAGASATIPMMSFFNDDIEEFEKILSHYRACFGLDDESLENMAKEWSMSVEELESTIKSPHLLSSEPNESVADKLVKTMEKIFAVTGGFVATGLYFRKSYYMQNYFLDTVTEDAKVLLKKLEHHHHHH
4EUX , Knot 186 422 0.88 40 262 411 
CPRFLKVKNWETDVVLTDTLHLKSTLETGCTEHICMGSIMLPSQHTRKPEDVRTKDQLFPLAKEFLDQYYSSIKRFGSKAHMDRLEEVNKEIESTSTYQLKDTELIYGAKHAWRNASRCVGRIQWSKLQVFDARDCTTAHGMFNYICNHVKYATNKGNLRSAITIFPQRTDGKHDFRVWNSQLIRYAGYKQPDGSTLGDPANVQFTEICIQQGWKAPRGRFDVLPLLLQANGNDPELFQIPPELVLEVPIRHPKFDWFKDLGLKWYGLPAVSNMLLEIGGLEFSACPFSGWYMGTEIGVRDYCDNSRYNILEEVAKKMDLDMRKTSSLWKDQALVEINIAVLYSFQSDKVTIVDHHSATESFIKHMENEYRCRGGCPADWVWIVPPMSGSITPVFHQEMLNYRLTPSFEYQPDPWNTHVWKG

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 \(c\).

\(|P_{f(6ZZJ_1)}(2) \setminus P_{f(8JQZ_1)}(2)|=46\), \(|P_{f(8JQZ_1)}(2) \setminus P_{f(6ZZJ_1)}(2)|=113\). 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 1:1010100001001001011001011010101001001010011010000011110001101001111101110111001010101010000100000101011100111110111001001011011011101100000001010010110101001100111000111111011111011110011110001100111001111110000011011011011001000100101010101
Pair \(Z_2\) Length of longest common subsequence
6ZZJ_1,8JQZ_1 159 4
6ZZJ_1,4EUX_1 178 4
8JQZ_1,4EUX_1 169 4

Newick tree

 
[
	4EUX_1:89.07,
	[
		6ZZJ_1:79.5,8JQZ_1:79.5
	]:9.57
]

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 \((0.85)(0.8)(\frac{655 }{\log_{20} 655}-\frac{241}{\log_{20}241})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 6ZZJ_1 8JQZ_1 145 113.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]