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Parikh vectors
3EUA_1 6XQJ_1 2JCD_1 Letter Amino acid
9 12 10 H Histidine
10 14 20 P Proline
3 3 8 W Tryptophan
20 7 26 V Valine
11 10 33 R Arginine
3 2 2 C Cysteine
29 27 38 L Leucine
7 4 9 M Methionine
23 8 13 S Serine
36 18 38 A Alanine
13 12 5 N Asparagine
26 29 32 E Glutamic acid
19 16 18 G Glycine
20 15 19 I Isoleucine
21 4 4 K Lycine
14 5 18 T Threonine
19 5 6 Y Tyrosine
15 7 23 D Aspartic acid
16 26 13 Q Glutamine
15 10 10 F Phenylalanine 

3EUA_1|Chains A, B, C, D, E, F, G, H|PUTATIVE FRUCTOSE-AMINOACID-6-PHOSPHATE DEGLYCASE|Bacillus subtilis (1423)
>6XQJ_1|Chain A|Protein Vpr,UV excision repair protein RAD23 homolog A|Human immunodeficiency virus type 1 group M subtype B (isolate NY5) (11698)
>2JCD_1|Chains A, B|N-OXIDASE|STREPTOMYCES THIOLUTEUS (66431)
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 \(w=f(c)\)
3EUA , Knot 147 329 0.86 40 206 320 
GMSQATAKVNREVQAFLQDLKGKTIDHVFFVACGGSSAIMYPSKYVFDRESKSINSDLYSANEFIQRNPVQLGEKSLVILCSHSGNTPETVKAAAFARGKGALTIAMTFKPESPLAQEAQYVAQYDWGDEALAINTNYGVLYQIVFGTLQVLENNTKFEQAIEGLDQLQAVYEKALKQEADNAKQFAKAHEKESIIYTMASGANYGVAYSYSICILMEMQWIHSHAIHAGEYFHGPFEIIDESVPFIILLGLDETRPLEERALTFSKKYGKKLTVLDAASYDFTAIDDSVKGYLAPLVLNRVLRSYADELAEERNHPLSHRRYMWKVEY
6XQJ , Knot 104 234 0.80 40 147 216 
MEQAPEDQGPQREPYNEWTLELLEELKSEAVRHFPRIWLHNLGQHIYETYGDTWAGVEAIIRILQQLLFIHFRIGCRHSGGSGGSATEAAGENPLEFLRDQPQFQNMRQVIQQNPALLPALLQQLGQENPQLLQQISRHQEQFIQMLNEPPGELADISDVEGEVGAIGEEAPQMNYIQVTPQEKEAIERLKALGFPESLVIQAYFACEKNENLAANFLLSQNFDDELEHHHHHH
2JCD , Knot 148 345 0.83 40 189 328 
ISEFDSGETMREEQPHLATTWAARGWVEEEGIGSATLGRLVRAWPRRAAVVNKADILDEWADYDTLVPDYPLEIVPFAEHPLFLAAEPHQRQRVLTGMWIGYNERVIATEQLIAEPAFDLVMHGVFPGSDDPLIRKSVQQAIVDESFHTYMHMLAIDRTRELRKISERPPQPELVTYRRLRRVLADMPEQWERDIAVLVWGAVAETCINALLALLARDATIQPMHSLITTLHLRDETAHGSIVVEVVRELYARMNEQQRRALVRCLPIALEAFAEQDLSALLLELNAAGIRGAEEIVGDLRSTAGGTRLVRDFSGARKMVEQLGLDDAVDFDFPERPDWSPHTPR

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 \(c\).

\(|P_{f(3EUA_1)}(2) \setminus P_{f(6XQJ_1)}(2)|=123\), \(|P_{f(6XQJ_1)}(2) \setminus P_{f(3EUA_1)}(2)|=64\). 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 1:11001010100010111001010010011111011001110100011000000100010010011000110110001111000010010010111110101110111010100111001001100011001111000011100111101011000001001101100101100011000100100110100000110011011001110000101110101100011011001011101100011111111100001100011010000100101101100010110001010111111001100010011000001100000110100
Pair \(Z_2\) Length of longest common subsequence
3EUA_1,6XQJ_1 187 4
3EUA_1,2JCD_1 169 4
6XQJ_1,2JCD_1 154 4

Newick tree

 
[
	3EUA_1:92.80,
	[
		2JCD_1:77,6XQJ_1:77
	]:15.80
]

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 \((0.85)(0.8)(\frac{563 }{\log_{20} 563}-\frac{234}{\log_{20}234})=93.7\)
Status Protein1 Protein2 d d1/2
Query variables 3EUA_1 6XQJ_1 123 103
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]