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Parikh vectors
6HIE_1 8WFR_1 5AWU_1 Letter Amino acid
6 11 25 A Alanine
3 6 3 H Histidine
11 2 12 I Isoleucine
5 4 12 F Phenylalanine
6 6 20 S Serine
6 7 19 T Threonine
1 10 18 G Glycine
7 5 16 K Lycine
3 1 5 C Cysteine
4 5 15 Q Glutamine
2 1 6 M Methionine
6 5 19 P Proline
13 8 9 R Arginine
3 0 19 N Asparagine
16 8 12 D Aspartic acid
12 14 11 E Glutamic acid
13 18 28 L Leucine
0 1 4 W Tryptophan
5 4 8 Y Tyrosine
8 8 21 V Valine 

6HIE_1|Chain A|ATPase family AAA domain-containing protein 2|Homo sapiens (9606)
>8WFR_1|Chain A|Proprotein convertase subtilisin/kexin type 9|Homo sapiens (9606)
>5AWU_1|Chain A|SH3-containing GRB2-like protein 3-interacting protein 1|Homo sapiens (9606)
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 \(w=f(c)\)
6HIE , Knot 63 130 0.78 38 97 126 
SMQEEDTFRELRIFLRNVTHRLAIDKRFRVFTKPVDPDEVPDYVTVIKQPMDLSSVISKIDLHKYLTVKDYLRDIDLICSNALEYNPDRDPGDRLIRHRACALRDTAYAIIKEELDEDFEQLCEEIQESR
8WFR , Knot 59 124 0.76 38 89 117 
GGQEDEDGDYEELVLALRSEEDGLAEAPEHGTTATFHRCAKDPWRLPGTYVVVLKEETHLSQSERTARRLQAQAARRGYLTKILHVFHGLLPGFLVKMSGDLLELALKLPHVDYIEEDSSVFAQ
5AWU , Knot 125 282 0.83 40 187 273 
GPLGSLTMGAQDTLPVAAAFTETVNAYFKGADPSKCIVKITGEMVLSFPAGITRHFANNPSPAALTFRVINFSRLEHVLPNPQLLCCDNTQNDANTKEFWVNMPNLMTHLKKVSEQKPQATYYNVDMLKYQVSAQGIQSTPLNLAVNWRCEPSSTDLRIDYKYNTDAMTTAVALNNVQFLVPIDGGVTKLQAVLPPAVWNAEQQRILWKIPDISQKSENGGVGSLLARFQLSEGPSKPSPLVVQFTSEGSTLSGCDIELVGAGYRFSLIKKRFAAGKYLADN

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 \(c\).

\(|P_{f(6HIE_1)}(2) \setminus P_{f(8WFR_1)}(2)|=62\), \(|P_{f(8WFR_1)}(2) \setminus P_{f(6HIE_1)}(2)|=54\). 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 1:0100000100101110010001110001011001101001100101100110100110010100010100010010110001100010001100110001011000101110001000100100010000
Pair \(Z_2\) Length of longest common subsequence
6HIE_1,8WFR_1 116 3
6HIE_1,5AWU_1 174 3
8WFR_1,5AWU_1 168 3

Newick tree

 
[
	5AWU_1:92.89,
	[
		6HIE_1:58,8WFR_1:58
	]:34.89
]

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 \((0.85)(0.8)(\frac{254 }{\log_{20} 254}-\frac{124}{\log_{20}124})=41.0\)
Status Protein1 Protein2 d d1/2
Query variables 6HIE_1 8WFR_1 49 48
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]