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Parikh vectors
8XHQ_1 4GJP_1 6GCG_1 Letter Amino acid
15 0 13 F Phenylalanine
19 53 35 V Valine
13 15 6 S Serine
23 28 32 T Threonine
24 66 46 A Alanine
5 12 1 C Cysteine
10 56 10 Q Glutamine
17 23 17 H Histidine
10 14 12 I Isoleucine
29 75 29 L Leucine
11 0 13 Y Tyrosine
22 17 14 R Arginine
26 21 22 E Glutamic acid
5 15 17 K Lycine
5 1 12 M Methionine
20 30 24 P Proline
4 2 4 W Tryptophan
5 13 14 N Asparagine
16 8 20 D Aspartic acid
32 50 37 G Glycine 

8XHQ_1|Chains A, B, C, D|Fe/2OG dependent dioxigenase|Streptomyces ossamyceticus (249581)
>4GJP_1|Chains A, D[auth B]|Hax3|Xanthomonas campestris pv. armoraciae (329463)
>6GCG_1|Chain A|Copper-containing nitrite reductase|Achromobacter cycloclastes (223)
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 \(w=f(c)\)
8XHQ , Knot 133 311 0.81 40 185 293 
HMTTKEMTLQRARTASGELVFETGGGLSQALQDGCFYLAIPEDIDLEPGKLLCRQFYRPAHPGSPELRPYRGFRRNDGIYFDREYYQTEHILADGPAREKYLPPDVVALCERMTSLALLVLTSTLTGLGIDEAVWEKVTGGAVGGGGTQWFAASHYRPERHQLGCAPHKDTGFVTVLYIEQDGLESSVGGEWIPIAPLPGYFLVNFGGATELLTARMGRPVQAILHRVRSCVTEPAREDRFSFAVFANPPATGDLYQMSESGEPVAVRGVEEFLRDFNNETWSDRHTDFGITTTAPGEVHDLEHHHHHHHH
4GJP , Knot 78 499 0.32 36 105 142 
MQWSGARALEALLTVAGELRGPPLQLDTGQLLKIAKRGGVTAVEAVHAWRNALTGAPLNLTPEQVVAIASHDGGKQALETVQRLLPVLCQAHGLTPQQVVAIASHDGGKQALETVQRLLPVLCQAHGLTPEQVVAIASHDGGKQALETVQALLPVLCQAHGLTPEQVVAIASNGGGKQALETVQRLLPVLCQAHGLTPQQVVAIASNGGGKQALETVQRLLPVLCQAHGLTPQQVVAIASNGGGKQALETVQRLLPVLCQAHGLTPQQVVAIASNNGGKQALETVQRLLPVLCQAHGLTPQQVVAIASNGGGKQALETVQRLLPVLCQAHGLTPQQVVAIASNNGGKQALETVQRLLPVLCQAHGLTPEQVVAIASNGGGKQALETVQRLLPVLCQAHGLTPEQVVAIASHDGGKQALETVQRLLPVLCQAHGLTPQQVVAIASNGGGRPALESIVAQLSRPDPALAALTNDHLVALACLGGRPALDAVKKLEHHHHHH
6GCG , Knot 151 378 0.79 40 203 343 
MTEQLQMTRRTMLAGAALAGAVAPLLHTAQAHAAGAAAAAGAAPVDISTLPRVKVDLVKPPFVHAHDQVAKTGPRVVEFTMTIEEKKLVIDREGTEIHAMTFNGSVPGPLMVVHENDYVELRLINPDTNTLLHNIDFHAATGALGGGALTQVNPGEETTLRFKATKPGVFVYHCAPEGMVPWHVTSGMNGAIMVLPRDGLKDEKGQPLTYDKIYYVGEQDFYVPKDEAGNYKKYETPGEAYEDAVKAMRTLTPTHIVFNGAVGALTGDHALTAAVGERVLVVHSQANRDTRPHLIGGHGDYVWATGKFRNPPDLDQETWLIPGGTAGAAFYTFRQPGVYAYVNHNLIEAFELGAAGHFKVTGEWNDDLMTSVVKPASM

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 \(c\).

\(|P_{f(8XHQ_1)}(2) \setminus P_{f(4GJP_1)}(2)|=117\), \(|P_{f(4GJP_1)}(2) \setminus P_{f(8XHQ_1)}(2)|=37\). 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 1:01000010100100101011100111100110010101111001010110110001001101101010100110000110100000000011101110000111011110001001111110001011110011100101111111100111100001000011011000011101101000110001110111111111011101111001101011011011100100010011000010111110111010100100010111101100110010000100000011100011101001000000000
Pair \(Z_2\) Length of longest common subsequence
8XHQ_1,4GJP_1 154 8
8XHQ_1,6GCG_1 168 4
4GJP_1,6GCG_1 166 4

Newick tree

 
[
	6GCG_1:85.55,
	[
		8XHQ_1:77,4GJP_1:77
	]:8.55
]

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 \((0.85)(0.8)(\frac{810 }{\log_{20} 810}-\frac{311}{\log_{20}311})=136.\)
Status Protein1 Protein2 d d1/2
Query variables 8XHQ_1 4GJP_1 113 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]