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Parikh vectors
2OQP_1 3ZZY_1 8GSF_1 Letter Amino acid
10 4 16 R Arginine
7 7 12 I Isoleucine
5 9 18 V Valine
5 2 7 H Histidine
13 7 6 K Lycine
4 2 6 M Methionine
5 14 14 A Alanine
8 9 14 N Asparagine
7 9 7 D Aspartic acid
9 8 9 Q Glutamine
4 6 14 G Glycine
8 5 17 P Proline
12 12 21 S Serine
2 5 13 Y Tyrosine
9 1 9 E Glutamic acid
11 15 11 L Leucine
4 2 3 C Cysteine
5 6 13 F Phenylalanine
5 7 22 T Threonine
1 0 5 W Tryptophan 

2OQP_1|Chain A|Interleukin-21|Homo sapiens (9606)
>3ZZY_1|Chains A, B|POLYPYRIMIDINE TRACT-BINDING PROTEIN 1|HOMO SAPIENS (9606)
>8GSF_1|Chains A[auth 1], D[auth A]|VP1|Echovirus E3 (47516)
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 \(w=f(c)\)
2OQP , Knot 67 134 0.81 40 111 129 
MQGQDRHMIRMRQLIDIVDQLKNYVNDLVPEFLPAPEDVETNCEWSAFSCFQKAQLKSANTGNNERIINVSIKKLKRKPPSTNAGRRQKHRLTCPSCDSYEKKPPKEFLERFKSLLQKMIHQHLSSRTHGSEDS
3ZZY , Knot 65 130 0.81 38 103 126 
SVQSGNLALAASAAAVDAGMAMAGQSPVLRIIVENLFYPVTLDVLHQIFSKFGTVLKIITFTKNNQFQALLQYADPVSAQHAKLSLDGQNIYNACCTLRIDFSKLTSLNVKYNNDKSRDYTRPDLPSGDS
8GSF , Knot 112 237 0.86 40 173 230 
YHSRTESSIENFLCRAACVYIATYASAGGTPTERYASWRINTRQMVQLRRKFELFTYLRFDMEITFVITSTQDPGTQLAQDMPVLTHQIMYIPPGGPVPNSATDFAWQSSTNPSIFWTEGCAPARMSVPFISIGNAYSNFYDGWSHFTQEGVYGFNSLNNMGHIYVRHVNEQSLGVSTSTLRVYFKPKHVRAWVPRPPRLSPYVKSSNVNFKPTAVTTERKDINDVGTLRPVGYTNH

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 \(c\).

\(|P_{f(2OQP_1)}(2) \setminus P_{f(3ZZY_1)}(2)|=76\), \(|P_{f(3ZZY_1)}(2) \setminus P_{f(2OQP_1)}(2)|=68\). 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 1:10100001101001101100100010011101111100100000101100100101001001000011010100100011000110000001001000000001100110010011001100010000010000
Pair \(Z_2\) Length of longest common subsequence
2OQP_1,3ZZY_1 144 3
2OQP_1,8GSF_1 184 3
3ZZY_1,8GSF_1 170 3

Newick tree

 
[
	8GSF_1:93.44,
	[
		2OQP_1:72,3ZZY_1:72
	]:21.44
]

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 \((0.85)(0.8)(\frac{264 }{\log_{20} 264}-\frac{130}{\log_{20}130})=42.0\)
Status Protein1 Protein2 d d1/2
Query variables 2OQP_1 3ZZY_1 54 53.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]