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Parikh vectors
4OAK_1 6PZU_1 2MYT_1 Letter Amino acid
11 17 8 D Aspartic acid
12 8 7 K Lycine
8 14 5 F Phenylalanine
8 16 6 T Threonine
10 17 13 V Valine
12 31 8 A Alanine
7 16 5 N Asparagine
9 21 9 I Isoleucine
11 32 14 S Serine
16 21 12 E Glutamic acid
5 11 5 M Methionine
10 18 5 P Proline
11 22 6 R Arginine
2 7 2 C Cysteine
16 8 6 Q Glutamine
12 25 11 G Glycine
14 21 2 H Histidine
22 37 7 L Leucine
4 4 2 W Tryptophan
11 11 1 Y Tyrosine 

4OAK_1|Chains A, B|D,D-dipeptidase/D,D-carboxypeptidase|Enterococcus gallinarum (1353)
>6PZU_1|Chains A, B|Hdac6 protein|Danio rerio (7955)
>2MYT_1|Chain A|Glutaredoxin arsenate reductase|Synechocystis sp. (1111708)
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 \(w=f(c)\)
4OAK , Knot 94 211 0.79 40 151 200 
MGSSHHHHHHSSGRENLYFQGMNTLQLINKNHPLKKNQEPPHLVLAPFSDHDVYLQPEVAKQWERLVRATGLEKDIRLVSGYRTEKEQRRLWEYSLKENGLAYTKQFVALPGCSEHQIGLAIDVGLKKQEDDDLICPHFRDSAAADLFMQQMMNYGFILRYPEDKQEITGISYEPWHFRYVGLPHSQVITAQKWTLEEYHDYLAQTVRQFA
6PZU , Knot 154 357 0.84 40 210 344 
PITGLVYDQRMMLHHNMWDSHHPELPQRISRIFSRHEELRLLSRCHRIPARLATEEELALCHSSKHISIIKSSEHMKPRDLNRLGDEYNSIFISNESYTCALLAAGSCFNSAQAILTGQVRNAVAIVRPPGHHAEKDTACGFCFFNTAALTARYAQSITRESLRVLIVDWDVHHGNGTQHIFEEDDSVLYISLHRYEDGAFFPNSEDANYDKVGLGKGRGYNVNIPWNGGKMGDPEYMAAFHHLVMPIAREFAPELVLVSAGFDAARGDPLGGFQVTPEGYAHLTHQLMSLAAGRVLIILEGGYNLTSISESMSMCTSMLLGDSPPSLDHLTPLKTSATVSINNVLRAHAPFWSSLR
2MYT , Knot 68 134 0.82 40 108 131 
GSHMKKVMFVCKRNSSRSQMAEGFAKTLGAGKIAVTSSGLESSRVHPTAIAMMEEVGIDISGQTSDPIENFNADDYDVVISLCGSGVNLPPEWVTQEIFEDWQLEDPDGQSLEVFRTVRGQVKERVENLIAKIS

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 \(c\).

\(|P_{f(4OAK_1)}(2) \setminus P_{f(6PZU_1)}(2)|=69\), \(|P_{f(6PZU_1)}(2) \setminus P_{f(4OAK_1)}(2)|=128\). 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 1:1100000000001000101011001011000011000001101111110000101010110010011010110001011010000000001100010001110000111111000001111101110000000110101000111011100110011110010000010110001101001111000110100101000000110010011
Pair \(Z_2\) Length of longest common subsequence
4OAK_1,6PZU_1 197 3
4OAK_1,2MYT_1 159 3
6PZU_1,2MYT_1 188 3

Newick tree

 
[
	6PZU_1:10.25,
	[
		4OAK_1:79.5,2MYT_1:79.5
	]:21.75
]

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 \((0.85)(0.8)(\frac{568 }{\log_{20} 568}-\frac{211}{\log_{20}211})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 4OAK_1 6PZU_1 133 104
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]