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Parikh vectors
8RLC_1 7AUQ_1 5KSS_1 Letter Amino acid
18 10 23 V Valine
8 13 26 A Alanine
18 11 23 D Aspartic acid
7 5 10 Q Glutamine
22 12 20 G Glycine
20 20 4 K Lycine
9 5 8 Y Tyrosine
13 9 13 N Asparagine
11 10 15 I Isoleucine
20 12 23 L Leucine
5 3 6 M Methionine
1 3 4 W Tryptophan
2 4 2 C Cysteine
18 11 25 T Threonine
8 13 16 R Arginine
16 20 7 E Glutamic acid
10 6 14 H Histidine
12 4 3 F Phenylalanine
10 11 17 P Proline
10 9 11 S Serine 

8RLC_1|Chain A[auth B]|Green fluorescent protein|Aequorea victoria (6100)
>7AUQ_1|Chain A|Diphosphoinositol polyphosphate phosphohydrolase DDP1|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
>5KSS_1|Chains A, B|5'-nucleotidase SurE|Xylella fastidiosa (strain 9a5c) (160492)
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 \(w=f(c)\)
8RLC , Knot 108 238 0.82 40 160 227 
MSKGEELFTGVVPILVELDGDVNGHKFSVRGEGEGDATNGKLTLKFICTTGKLPVPWPTLVTTLTYGVQCFSRYPDHMKRHDFFKSAMPEGYVQERTISFKDDGTYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNFNSHNVYITADKQKNGIKANFKIRHNVEDGSVQLADHYQQNTPIGDGPVLLPDNHYLSTQSVLSKDPNEKRDHMVLLEFVTAAGITHGMDELYK
7AUQ , Knot 91 191 0.83 40 137 187 
GGSMGKTADNHGPVRSETAREGRENQVYSPVTGARLVAGCICLTPDKKQVLMITSSAHKKRWIVPKGGVEKDEPNYETTAQRETWEEAGCIGKIVANLGTVEDMRPPKDWNKDIKQFENSRKDSEVAKHPPRTEFHFYELEIENLLDKFPECHKRHRKLYSYTEAKQNLIDAKRPELLEALNRSAIIKDDK
5KSS , Knot 117 270 0.80 40 168 260 
MRVLVSNDDGVDAPGIKILADALRNAGHEVMVVAPDRDRSGASNSLTLDTPIRAKQIDMHTYSVAGTPTDCVHLALTGLLNYDPDIVVSGINNTGNLGDDVIYSGTVSAAMEGRFLGLPAVAVSLVTLYREGQQAPQYETAAHAAINIVAQLKTDPLPADTILNVNVPDVTWQQMRGFKVTRLGNRHRSAPCLTQTDPRGHTIYWIGPAGPEQDAGPGTDFDAVRNTYISITPIHVDLTRYQALENVTRWTDRLTAHMDWPTLEHHHHHH

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 \(c\).

\(|P_{f(8RLC_1)}(2) \setminus P_{f(7AUQ_1)}(2)|=94\), \(|P_{f(7AUQ_1)}(2) \setminus P_{f(8RLC_1)}(2)|=71\). 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 1:1001001101111111010101010010101010101001010101100010111111011001001100100010010000110011101010000101000100000101010100110010101101000101110010001000010101000001101010100010010101100000001110111111000010000110001000000111101101111001100100
Pair \(Z_2\) Length of longest common subsequence
8RLC_1,7AUQ_1 165 3
8RLC_1,5KSS_1 170 4
7AUQ_1,5KSS_1 173 3

Newick tree

 
[
	5KSS_1:86.81,
	[
		8RLC_1:82.5,7AUQ_1:82.5
	]:4.31
]

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 \((0.85)(0.8)(\frac{429 }{\log_{20} 429}-\frac{191}{\log_{20}191})=70.0\)
Status Protein1 Protein2 d d1/2
Query variables 8RLC_1 7AUQ_1 92 82.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]