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Parikh vectors
5JOC_1 5KNS_1 4UXQ_1 Letter Amino acid
22 16 10 I Isoleucine
29 17 38 L Leucine
8 8 9 M Methionine
22 7 7 T Threonine
13 0 5 W Tryptophan
40 7 24 A Alanine
13 16 19 D Aspartic acid
24 3 8 Q Glutamine
9 6 11 F Phenylalanine
15 6 8 Y Tyrosine
20 10 14 K Lycine
27 6 25 P Proline
17 14 18 S Serine
12 14 22 R Arginine
14 15 21 E Glutamic acid
30 12 24 G Glycine
22 18 25 V Valine
17 2 7 N Asparagine
0 2 6 C Cysteine
12 3 8 H Histidine 

5JOC_1|Chains A, B|Beta-lactamase|Escherichia coli (562)
>5KNS_1|Chains A, B|Hypoxanthine-guanine phosphoribosyltransferase|Escherichia coli (562)
>4UXQ_1|Chain A|FIBROBLAST GROWTH FACTOR RECEPTOR 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 \(w=f(c)\)
5JOC , Knot 155 366 0.83 38 211 350 
MHHHHHHPQQINDIVHRTITPLIEQQKIPGMAVAVIYQGKPYYFTWGYADIAKKQPVTQQTLFELGAVSKTFTGVLGGDAIARGEIKLSDPATKYWPELTAKQWNGITLLHLATYTAGGLPLQVPDEVKSSSDLLRFYQNWQPAWAPGTQRLYANSSIGLFGALAVKPSGLSFEQAMQTRVFQPLKLNHTWINVPPPEEKNYAWGYREGKAVHVSPGALDAEAYGVKSTIEDMARWVRSNMNPRDINDKTLQQGIQLAQSRYWQTGDMYQGLGWEMLDWPVNPDSIINGSGNKIALAAAAHPVKAITPPTPAVRASWVHKTGATGGFGSYVAFIPEKELGIVMLANKNYPNPARVAAAWQILNALQ
5KNS , Knot 84 182 0.80 38 131 175 
MVRDMKHTVEVMIPEAEIKARIAELGRQITERYKDSGSDMVLVGLLRGSFMFMADLCREVQVSHEVDFMTASSYGSGMSTTRDVKILKDLDEDIRGKDVLIVEDIIDSGNTLSKVREILSLREPKSLAICTLLDKPSRREVNVPVEFIGFSIPDEFVVGYGIDYAQRYRHLPYIGKVILLDE
4UXQ , Knot 135 309 0.83 40 193 293 
GSAGLVSLDLPLDPLWEFPRDRLVLGKPLGEGAFGQVVRAEAFGMDPARPDQASTVAVKMLKDNASDKDLADLVSEMEVMKLIGRHKNIINLLGVCTQEGPLYVIVECAAKGNLREFLRARRPPGPDLSPDGPRSSEGPLSFPVLVSCAYQVARGMQYLESRKCIHRDLAARNVLVTEDNVMKIADFGLARGVHHIDYYKKTSNGRLPVKWMAPEALFDRVYTHQSDVWSFGILLWEIFTLGGSPYPGIPVEELFSLLREGHRMDRPPHCPPELYGLMRECWHAAPSQRPTFKQLVEALDKVLLAVSEE

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 \(c\).

\(|P_{f(5JOC_1)}(2) \setminus P_{f(5KNS_1)}(2)|=125\), \(|P_{f(5KNS_1)}(2) \setminus P_{f(5JOC_1)}(2)|=45\). 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 1:100000010010011000101110000111111111001010010110101100011000011011110001011111011101010100110001101010010110110110001111110110010000011010001011111100010100011111111101011010011000110110100011011110000011100010110101111010101100010011011000101001000010011011000010010100111101101110100110101001111111011011011011101011000110111100111110001111111000010110111110110110
Pair \(Z_2\) Length of longest common subsequence
5JOC_1,5KNS_1 170 3
5JOC_1,4UXQ_1 168 3
5KNS_1,4UXQ_1 166 3

Newick tree

 
[
	5JOC_1:84.99,
	[
		4UXQ_1:83,5KNS_1:83
	]:1.99
]

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 \((0.85)(0.8)(\frac{548 }{\log_{20} 548}-\frac{182}{\log_{20}182})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 5JOC_1 5KNS_1 136 99.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]