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Parikh vectors
1LFY_1 8XSK_1 6PUV_1 Letter Amino acid
21 12 8 A Alanine
1 4 8 Q Glutamine
7 7 14 G Glycine
9 5 3 T Threonine
13 11 8 V Valine
4 7 6 N Asparagine
1 2 6 C Cysteine
4 11 8 E Glutamic acid
2 5 2 M Methionine
7 8 4 F Phenylalanine
11 14 8 S Serine
3 6 6 Y Tyrosine
8 5 10 D Aspartic acid
11 12 5 K Lycine
3 5 3 R Arginine
10 7 7 H Histidine
0 13 1 I Isoleucine
18 17 6 L Leucine
7 6 6 P Proline
1 0 10 W Tryptophan 

1LFY_1|Chain A|Hemoglobin alpha chain|Homo sapiens (9606)
>8XSK_1|Chains A, B|Phosphopantetheine adenylyltransferase|Helicobacter pylori 26695 (85962)
>6PUV_1|Chain A|C-type lectin domain family 10 member A|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 \(w=f(c)\)
1LFY , Knot 69 141 0.80 38 99 137 
VLSPADKTNVKAAWGKVGAHAGEYGAEALERMFLSFPTTKTYFPHFDLSHGSAQVKGHGKKVADALTNAVAHVDDMPNALSALSDLHAHKLRVDPVNFKLLSHCLLVTLAAHLPAEFTPAVHASLDKFLASVSTVLTSKYR
8XSK , Knot 77 157 0.82 38 123 154 
MQKIGIYPGTFDPVTNGHIDIIHRSSELFEKLIVAVAHSSAKNPMFSLDERLKMIQLATKSFKNVECVAFEGLLANLAKEYHCKVLVRGLRVVSDFEYELQMGYANKSLNHELETLYFMPTLQNAFISSSIVRSIIAHKGDASHLVPKEIYPLISKA
6PUV , Knot 67 129 0.84 40 109 127 
SCPVNWVEHQDSCYWFSHSGMSWAEAEKYCQLKNAHLVVINSREEQNFVQKYLGSAYTWMGLSDPEGAWKWVDGTDYATGFQNWKPGQPDDWQGHGLGGGEDCAHFHPDGRWNDDVCQRPYHWVCEAGL

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 \(c\).

\(|P_{f(1LFY_1)}(2) \setminus P_{f(8XSK_1)}(2)|=55\), \(|P_{f(8XSK_1)}(2) \setminus P_{f(1LFY_1)}(2)|=79\). 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 1:110110000101111011101100110110011101100000110101001010101010011011001110100110110110010100101011010110001110111011101011101010011101001100000
Pair \(Z_2\) Length of longest common subsequence
1LFY_1,8XSK_1 134 4
1LFY_1,6PUV_1 158 3
8XSK_1,6PUV_1 176 3

Newick tree

 
[
	6PUV_1:88.47,
	[
		1LFY_1:67,8XSK_1:67
	]:21.47
]

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 \((0.85)(0.8)(\frac{298 }{\log_{20} 298}-\frac{141}{\log_{20}141})=48.5\)
Status Protein1 Protein2 d d1/2
Query variables 1LFY_1 8XSK_1 63 57.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]