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Parikh vectors
8EVW_1 2PHY_1 6YKF_1 Letter Amino acid
4 1 3 C Cysteine
16 5 18 Q Glutamine
13 11 20 K Lycine
11 9 15 F Phenylalanine
2 6 21 N Asparagine
17 12 15 D Aspartic acid
28 7 23 E Glutamic acid
24 5 23 S Serine
2 1 7 W Tryptophan
12 4 10 P Proline
11 7 11 T Threonine
5 5 14 Y Tyrosine
48 9 18 A Alanine
21 2 11 R Arginine
38 13 22 G Glycine
11 5 18 I Isoleucine
44 7 23 L Leucine
28 9 20 V Valine
6 2 12 H Histidine
6 5 5 M Methionine 

8EVW_1|Chains A, B|D-alanine--D-alanine ligase A|Pseudomonas aeruginosa (208964)
>2PHY_1|Chain A|PHOTOACTIVE YELLOW PROTEIN|Halorhodospira halophila (1053)
>6YKF_1|Chain A|HNH endonuclease|Vibrio campbellii (680)
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 \(w=f(c)\)
8EVW , Knot 143 347 0.80 40 183 316 
GMGQDKLKVAVLFGGSSEERDVSIASGAQVIQALRSAGHQVLAVDTASGLLGAEEERRLLASKVKEVPPDSDSLAIIRSGKQSLLSAGELAGVDVFFLALHGGTGEDGTLQALLDAGGFAYTGSGHLASAMAMDKDVAKRLFLAAGVETASWLMAPASEEEVREQLGFPLVVKPNSQGSTVGLSIVHSQAELQPAIELAGRYGDEVMLERFVAGREVTVGVLDDQALPVGEILLGGQEVFDYEHKYQAGAVREVFPADLPPAIAAEAQRLALKVHRALKLSGYSRTDFRLDEQGRLWCLEVNTLPGMTATSLLPQAAAAAGIGFAELCERICRLGIERCKGARKARS
2PHY , Knot 65 125 0.83 40 102 120 
MEHVAFGSEDIENTLAKMDDGQLDGLAFGAIQLDGDGNILQYNAAEGDITGRDPKQVIGKNFFKDVAPCTDSPEFYGKFKEGVASGNLNTMFEYTFDYQMTPTKVKVHMKKALSGDSYWVFVKRV
6YKF , Knot 136 309 0.84 40 199 302 
MNYWWVSQKQTFKQEFEGGYMWSPKENKNGTQSHYYNNMTLVQPGDVVFSFANGLILSVGIARSHAYSYNKPTEFGVAGADWANDGWKIDLEYHLVENKIRPKAHIDFIRPYLPQKYSPLQDNGNGNQAYLFSVPHELASKVVELIGSEAEEVIFGFADTTEITTTADAIECQISNDASIDETEKHQLVKSRRGQGIFRSRLEQVESRCRVTGVQLKNHLIASHIKPWAVSNNQERLDGHNGLLLAPHVDHLFDKGFISFEDNGEMIVSEKLNLDVLKAWSISQGNYGYFSKQQQEYMCYHRENVFKKL

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 \(c\).

\(|P_{f(8EVW_1)}(2) \setminus P_{f(2PHY_1)}(2)|=124\), \(|P_{f(2PHY_1)}(2) \setminus P_{f(8EVW_1)}(2)|=43\). 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 1:11100010111111100000010110110110110011001111001011111000001110010011100001111001000110110111101111110110100101011101111100101011011110001100111111100101111110000100011111110100010011101100010101110111001001110011110010111100011111011111001100000001111001111011111110100111010011010100000101000101101010011110100111011111111110100010011100001100100
Pair \(Z_2\) Length of longest common subsequence
8EVW_1,2PHY_1 167 4
8EVW_1,6YKF_1 162 5
2PHY_1,6YKF_1 183 3

Newick tree

 
[
	2PHY_1:89.68,
	[
		8EVW_1:81,6YKF_1:81
	]:8.68
]

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 \((0.85)(0.8)(\frac{472 }{\log_{20} 472}-\frac{125}{\log_{20}125})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 8EVW_1 2PHY_1 124 85
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]