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Parikh vectors
2JHY_1 3ATT_1 2NPG_1 Letter Amino acid
9 24 14 E Glutamic acid
10 25 58 G Glycine
7 10 34 I Isoleucine
11 16 15 S Serine
9 23 20 T Threonine
7 9 11 Y Tyrosine
7 25 9 R Arginine
4 11 13 H Histidine
13 3 9 K Lycine
6 22 10 P Proline
3 13 9 W Tryptophan
2 14 10 Q Glutamine
9 24 10 D Aspartic acid
11 35 45 L Leucine
5 18 24 F Phenylalanine
5 32 54 A Alanine
1 1 8 C Cysteine
4 9 11 M Methionine
12 38 45 V Valine
3 5 15 N Asparagine 

2JHY_1|Chain A|RHO GDP-DISSOCIATION INHIBITOR 1|HOMO SAPIENS (9606)
>3ATT_1|Chain A|Putative uncharacterized protein|Mycobacterium tuberculosis (1773)
>2NPG_1|Chain A|Ammonia channel|Escherichia coli (562)
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 \(w=f(c)\)
2JHY , Knot 67 138 0.79 40 109 133 
AMVPNVVVTGLTLVCSSAPGPLELDLTGDLESFKKQSFVLKEGVEYRIKISFRVNREIVSGMKYIQHTYRKGVKIDKTDYMVGSYGPRAEHYHFLTPVEEAPKGMLARGSYSIKSRFTDDDKTDHLSWEWNLTIKKDW
3ATT , Knot 148 357 0.81 40 201 338 
TLPAVISRWLSSVLPGGAAPEVTVESGVDSTGMSSETIILTARWQQDGRSIQQKLVARVAPAAEDVPVFPTYRLDHQFEVIRLVGELTDVPVPRVRWIETTGDVLGTPFFLMDYVEGVVPPDVMPYTFGDNWFADAPAERQRQLQDATVAALATLHSIPNAQNTFSFLTQGRTSDTTLHRHFNWVRSWYDFAVEGIGRSPLLERTFEWLQSHWPDDAAAREPVLLWGDARVGNVLYRDFQPVAVLDWEMVALGPRELDVAWMIFAHRVFQELAGLATLPGLPEVMREDDVRATYQALTGVELGDLHWFYVYSGVMWACVFMRTGARRVHFGEIEKPDDVESLFYHAGLMKHLLGEEH
2NPG , Knot 165 424 0.78 40 196 378 
APAVADKADNAFMMICTALVLFMTIPGIALFYGGLIRGKNVLSMLTQVTVTFALVCILWVVYGYSLAFGEGNNFFGNINWLMLKNIELTAVMGSIYQYIHVAFQGSFACITVGLIVGALAERIRFSAVLIFVVVWLTLSYIPIAHMVWGGGLLASHGALDFAGGTVVHINAAIAGLVGAYLIGKRVGFGKEAFKPHNLPMVFTGTAILYIGWFGFNAGSAGTANEIAALAFVNTVVATAAAILGWIFGEWALRGKPSLLGACSGAIAGLVGVTPACGYIGVGGALIIGVVAGLAGLWGVTMLKRLLRVDDPCDVFGVFGVCGIVGCIMTGIFAASSLGGVGFAEGVTMGHQLLVQLESIAITIVWSGVVAFIGYKLADLTVGLRVPEEQEREGLDVNSHGENAYNADQAQQPAQADLEHHHHHH

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 \(c\).

\(|P_{f(2JHY_1)}(2) \setminus P_{f(3ATT_1)}(2)|=44\), \(|P_{f(3ATT_1)}(2) \setminus P_{f(2JHY_1)}(2)|=136\). 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 1:111101110110110001111101010101001000011100110001010101000110110010000001101000001110011010000110110011011110100010001000000001010101010001
Pair \(Z_2\) Length of longest common subsequence
2JHY_1,3ATT_1 180 3
2JHY_1,2NPG_1 187 3
3ATT_1,2NPG_1 169 4

Newick tree

 
[
	2JHY_1:94.06,
	[
		3ATT_1:84.5,2NPG_1:84.5
	]:9.56
]

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 \((0.85)(0.8)(\frac{495 }{\log_{20} 495}-\frac{138}{\log_{20}138})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 2JHY_1 3ATT_1 129 89
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]