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Parikh vectors
8STL_1 7ZDL_1 6QJN_1 Letter Amino acid
3 13 1 M Methionine
3 18 4 P Proline
7 36 3 T Threonine
3 7 0 W Tryptophan
16 47 16 G Glycine
5 30 11 I Isoleucine
9 99 8 L Leucine
6 10 4 K Lycine
8 14 2 Y Tyrosine
2 2 0 C Cysteine
5 27 10 E Glutamic acid
3 21 4 F Phenylalanine
5 11 4 N Asparagine
10 20 4 D Aspartic acid
15 34 6 S Serine
11 35 5 V Valine
10 60 9 A Alanine
11 40 7 R Arginine
8 38 4 Q Glutamine
1 11 2 H Histidine 

8STL_1|Chains A, B, C, D|Nanobody PIK3_Nb16|Lama glama (9844)
>7ZDL_1|Chain A[auth C]|ATP-binding/permease protein CydC|Escherichia coli K-12 (83333)
>6QJN_1|Chains A, B|Disks large homolog 4|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 \(w=f(c)\)
8STL , Knot 69 141 0.80 40 115 136 
MAQVQLQESGGGLVQAGGSLKLSCTASGLRVDTHGMGWFRQIARKKREFVASIDWEGESTNYLDSVKGRFIISRDDAKNTLYLQMNSLKPEDTAVYYCAADRSGRIRSVVSNNDYWGQGTQVTVSSGRYPYDVPDYGSGRA
7ZDL , Knot 216 573 0.79 40 228 509 
MRALLPYLALYKRHKWMLSLGIVLAIVTLLASIGLLTLSGWFLSASAVAGVAGLYSFNYMLPAAGVRGAAITRTAGRYFERLVSHDATFRVLQHLRIYTFSKLLPLSPAGLARYRQGELLNRVVADVDTLDHLYLRVISPLVGAFVVIMVVTIGLSFLDFTLAFTLGGIMLLTLFLMPPLFYRAGKSTGQNLTHLRGQYRQQLTAWLQGQAELTIFGASDRYRTQLENTEIQWLEAQRRQSELTALSQAIMLLIGALAVILMLWMASGGVGGNAQPGALIALFVFCALAAFEALAPVTGAFQHLGQVIASAVRISDLTDQKPEVTFPDTQTRVADRVSLTLRDVQFTYPEQSQQALKGISLQVNAGEHIAILGRTGCGKSTLLQQLTRAWDPQQGEILLNDSPIASLNEAALRQTISVVPQRVHLFSATLRDNLLLASPGSSDEALSEILRRVGLEKLLEDAGLNSWLGEGGRQLSGGELRRLAIARALLHDAPLVLLDEPTEGLDATTESQILELLAEMMREKTVLMVTHRLRGLSRFQQIIVMDNGQIIEQGTHAELLARQGRYYQFKQGL
6QJN , Knot 52 104 0.77 36 84 99 
MGLGEEDIPREPRRIVIHRGSTGLGFNIVGGEGGEGIFISFILAGGPADLSGELRKGDQILSVNGVDLRNASHEQAAIALKNAGQTVTIIAQYKPEEYSRFEAK

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 \(c\).

\(|P_{f(8STL_1)}(2) \setminus P_{f(7ZDL_1)}(2)|=31\), \(|P_{f(7ZDL_1)}(2) \setminus P_{f(8STL_1)}(2)|=144\). 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 1:110101000111110111010100010110100011111001100000111010101000001001010111000010001010100101000110001100010100110000011010010100100100110010101
Pair \(Z_2\) Length of longest common subsequence
8STL_1,7ZDL_1 175 3
8STL_1,6QJN_1 123 3
7ZDL_1,6QJN_1 178 4

Newick tree

 
[
	7ZDL_1:95.52,
	[
		8STL_1:61.5,6QJN_1:61.5
	]:34.02
]

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 \((0.85)(0.8)(\frac{714 }{\log_{20} 714}-\frac{141}{\log_{20}141})=163.\)
Status Protein1 Protein2 d d1/2
Query variables 8STL_1 7ZDL_1 198 123.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]