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Parikh vectors
4NFW_1 4TJY_1 8WQN_1 Letter Amino acid
4 11 6 N Asparagine
7 24 13 G Glycine
12 11 18 L Leucine
8 11 6 F Phenylalanine
10 8 5 P Proline
7 6 0 T Threonine
5 5 3 C Cysteine
15 17 12 E Glutamic acid
11 12 23 I Isoleucine
5 11 16 K Lycine
3 8 5 M Methionine
5 1 1 W Tryptophan
2 13 9 Y Tyrosine
6 19 8 H Histidine
7 18 12 S Serine
14 11 9 A Alanine
7 15 10 R Arginine
6 5 10 D Aspartic acid
9 9 0 Q Glutamine
10 12 7 V Valine 

4NFW_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Putative Nudix hydrolase ymfB|Escherichia coli (885275)
>4TJY_1|Chains A, B, C, D|Tankyrase-2|Homo sapiens (9606)
>8WQN_1|Chain A|SegC|Saccharolobus solfataricus P2 (273057)
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 \(w=f(c)\)
4NFW , Knot 77 153 0.84 40 124 150 
MFKPHVTVACVVHAEGKFLVVEETINGKALWNQPAGHLEADETLVEAAARELWEETGISAQPQHFIRMHQWIAPDKTPFLRFLFAIELEQICPTQPHDSDIDCCRWVSAEEILQASNLRSPLVAESIRCYQSGQRYPLEMIGDFNWPFTKGVI
4TJY , Knot 106 227 0.84 40 168 219 
MGSSHHHHHHSSGRENLYFQGSPDDKEFQSVEEEMQSTVREHRDGGHAGGIFNRYNILKIQKVCNKKLWERYTHRRKEVSEENHNHANERMLFHGSPFVNAIIHKGFDERHAYIGGMFGAGIYFAENSSKSNQYVYGIGGGTGCPVHKDRSCYICHRQLLFCRVTLGKSFLQFSAMKMAHSPPGHHSVTGRPSVNGLALAEYVIYRGEQAYPEYLITYQIMRPEGMV
8WQN , Knot 78 173 0.77 36 112 161 
MFEKLYSAIIYSDEFKKILLGRGVDDLEIASAYIAFLYEDLPIIGKNLCAAFLRMGLDAVYNVMPSGKVYSPRHKLYPISRYGIDGVCINCDGGKIILRISNKGYDPEDLLESKGLESRIFVSKNFKKKSMEIIEKIWDVNKIRLIARKEILERISAGGILHMIRLEHHHHHH

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 \(c\).

\(|P_{f(4NFW_1)}(2) \setminus P_{f(4TJY_1)}(2)|=69\), \(|P_{f(4TJY_1)}(2) \setminus P_{f(4NFW_1)}(2)|=113\). 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 1:110101011011010101111000101011100111010100011011100110001101010011010011110001110111110100101001000010000110100110100100111100100000100011011101011100111
Pair \(Z_2\) Length of longest common subsequence
4NFW_1,4TJY_1 182 3
4NFW_1,8WQN_1 164 3
4TJY_1,8WQN_1 164 6

Newick tree

 
[
	4TJY_1:88.10,
	[
		4NFW_1:82,8WQN_1:82
	]:6.10
]

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 \((0.85)(0.8)(\frac{380 }{\log_{20} 380}-\frac{153}{\log_{20}153})=68.3\)
Status Protein1 Protein2 d d1/2
Query variables 4NFW_1 4TJY_1 88 73.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]