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Parikh vectors
4BWZ_1 3NMU_1 1ATD_1 Letter Amino acid
15 27 3 R Arginine
4 12 2 N Asparagine
6 11 3 Q Glutamine
24 13 2 F Phenylalanine
51 34 5 A Alanine
29 44 7 E Glutamic acid
3 12 0 H Histidine
8 32 7 K Lycine
15 13 5 P Proline
3 19 2 D Aspartic acid
3 0 10 C Cysteine
45 21 5 G Glycine
71 43 0 L Leucine
7 10 0 M Methionine
13 13 1 S Serine
7 14 0 Y Tyrosine
24 25 3 I Isoleucine
15 10 4 T Threonine
1 5 1 W Tryptophan
50 21 2 V Valine 

4BWZ_1|Chain A|NA(+)/H(+) ANTIPORTER|THERMUS THERMOPHILUS (274)
>3NMU_1|Chains A, B|NOP5/NOP56 related protein|Pyrococcus furiosus (2261)
>1ATD_1|Chain A|ASCARIS TRYPSIN INHIBITOR|Ascaris suum (6253)
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 \(w=f(c)\)
4BWZ , Knot 151 394 0.76 40 159 328 
MHGAEHLLEIFYLLLAAQVCAFIFKRLNQPVVIGEVLAGVLVGPALLGLVHEGEILEFLAELGAVFLLFMVGLETRLKDILAVGKEAFLVAVLGVALPFLGGYLYGLEIGFETLPALFLGTALVATSVGITARVLQELGVLSRPYSRIILGAAVIDDVLGLIVLACVNGVAETGQVEVGAITRLIVLSVVFVGLAVFLSTLIARLPLERLPVGSPLGFALALGVGMAALAASIGLAPIVGAFLGGMLLSEVREKYRLEEPIFAIESFLAPIFFAMVGVRLELSALASPVVLVAGTVVTVIAILGKVLGGFLGALTQGVRSALTVGCGMAPRGEVGLIVAALGLKAGAVNEEEYAIVLFMVVFTTLFAPFALKPLIAWTERERAAKEGSENLYFQ
3NMU , Knot 155 379 0.81 38 197 351 
MHHHHHHVMIMKAFISENVRGIYAFDENGNLIEKRYFTDKPEKVLDQLLKGEITKDLEELLNSLKEKGYDEFVFEHPELSRRAKELGFSATTEFPNIAGERLRSNPEEFLGENWFEEYYKVGVALTRMRIQEQSGARDKMVIQAIEALDDVDKVINLLVARLREWYSLHFPELDELLPKHPQYVAFVKTVGHRDNINEEVLRELGLSEEKIKKILEAKEKTMGAWMDQTDIEVVRQLAEEIDRLYQLRKKLEDYIDRAMDDVAPNLKALVGAKLAARLISLAGGLRELAMMPSSTIQVLGAEKALFRHLRTGAKPPKHGVIYQYPAINRSPWWQRGKIARALAGKLAIAARVDYFSGEYIAEELKKELEARIREIKEKY
1ATD , Knot 34 62 0.75 32 53 60 
EAEKCTKPNEQWTKCGGCEGTCAQKIVPCTRECKPPRCECIASAGFVRDAQGNCIKFEDCPK

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 \(c\).

\(|P_{f(4BWZ_1)}(2) \setminus P_{f(3NMU_1)}(2)|=57\), \(|P_{f(3NMU_1)}(2) \setminus P_{f(4BWZ_1)}(2)|=95\). 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 1:1011001101101111101011110010011111011111111111111100101101110111111111111000100111110011111111111111110101101110011111110111100111010110011110010001111111100111111110101110010101111001111011111111110011101110011110111111111111111110111111111111111100100000100111110011111111111101010111011111110110111111011111111100110011011011110101111111111011110000011111111100111111101111100000110010001010
Pair \(Z_2\) Length of longest common subsequence
4BWZ_1,3NMU_1 152 4
4BWZ_1,1ATD_1 160 2
3NMU_1,1ATD_1 200 3

Newick tree

 
[
	1ATD_1:94.91,
	[
		4BWZ_1:76,3NMU_1:76
	]:18.91
]

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 \((0.85)(0.8)(\frac{773 }{\log_{20} 773}-\frac{379}{\log_{20}379})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 4BWZ_1 3NMU_1 128 126.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]