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Parikh vectors
5URZ_1 3OKU_1 6OEE_1 Letter Amino acid
13 17 8 P Proline
8 13 14 A Alanine
9 7 11 R Arginine
5 10 29 N Asparagine
9 22 11 G Glycine
12 26 30 L Leucine
6 2 6 M Methionine
6 12 6 F Phenylalanine
8 18 15 S Serine
2 7 0 W Tryptophan
7 17 18 V Valine
7 19 11 D Aspartic acid
10 1 2 C Cysteine
4 9 16 I Isoleucine
4 12 12 T Threonine
6 8 16 Y Tyrosine
7 11 21 Q Glutamine
10 12 4 H Histidine
10 13 20 E Glutamic acid
3 24 30 K Lycine 

5URZ_1|Chains A, B|Frizzled-5|Homo sapiens (9606)
>3OKU_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>6OEE_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N|Type IV secretion system apparatus protein CagT|Helicobacter pylori (210)
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 \(w=f(c)\)
5URZ , Knot 73 146 0.83 40 115 139 
AHHHHHHGENLYFQGSKAPVCQEITVPMCRGIGYNLTHMPNQFNHDTQDEAGLEVHQFWPLVEIQCSPDLRFFLCSMYTPICLPDYHKPLPPCRSVCERAKAGCSPLMRQYGFAWPERMSCDRLPVLGRDAEVLCMDYERSEAGNS
3OKU , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
6OEE , Knot 121 280 0.81 38 169 271 
MKLRASVLIGATILCLILSACSNYAKKVVKQKNHVYTPVYNELIEKYSEIPLNDKLKDTPFMVQVKLPNYKDYLLDNKQVVLTFKLVHHSKKITLIGDANKILQYKNYFQANGARSDIDFYLQPTLNQKGVVMIASNYNDNPNSKEKPQTFDVLQGSQPMLGANTKNLHGYDVSGANNKQVINEVAREKAQLEKINQYYKTLLQDKEQEYTTRKNNQREILETLSNRAGYQMRQNVISSEIFKNGNLNMQAKEEEVREKLQEERENEYLRNQIRSLLSGK

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 \(c\).

\(|P_{f(5URZ_1)}(2) \setminus P_{f(3OKU_1)}(2)|=68\), \(|P_{f(3OKU_1)}(2) \setminus P_{f(5URZ_1)}(2)|=129\). 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 1:10000001001010100111000101110011100100110010000000111010011111010001010111001001101100001111000100010110011100011111001000011111001011010000001100
Pair \(Z_2\) Length of longest common subsequence
5URZ_1,3OKU_1 197 3
5URZ_1,6OEE_1 178 3
3OKU_1,6OEE_1 175 3

Newick tree

 
[
	5URZ_1:95.89,
	[
		6OEE_1:87.5,3OKU_1:87.5
	]:8.39
]

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 \((0.85)(0.8)(\frac{406 }{\log_{20} 406}-\frac{146}{\log_{20}146})=78.0\)
Status Protein1 Protein2 d d1/2
Query variables 5URZ_1 3OKU_1 101 79.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]