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Parikh vectors
1NXO_1 6QBL_1 5BVD_1 Letter Amino acid
8 5 15 V Valine
8 10 25 A Alanine
6 12 19 R Arginine
3 6 19 N Asparagine
0 3 13 H Histidine
5 3 13 F Phenylalanine
13 5 43 L Leucine
4 0 8 M Methionine
10 4 24 D Aspartic acid
0 6 7 C Cysteine
12 2 21 E Glutamic acid
5 5 18 G Glycine
12 3 28 I Isoleucine
11 0 23 K Lycine
5 4 21 P Proline
7 8 16 S Serine
4 3 12 Q Glutamine
5 4 14 T Threonine
0 2 3 W Tryptophan
2 8 19 Y Tyrosine 

1NXO_1|Chain A|DNA-binding response regulator|Streptococcus pneumoniae (170187)
>6QBL_1|Chain A|Big defensin 1|Crassostrea gigas (29159)
>5BVD_1|Chain A|Mitogen-activated protein kinase 1|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 \(w=f(c)\)
1NXO , Knot 60 120 0.79 34 95 115 
MKKILIVDDEKPISDIIKFNMTKEGYEVVTAFNGREALEQFEAEQPDIIILDLMLPEIDGLEVAKTIRKTSSVPILMLSAKDSEFDKVIGLELGADDYVTKPFSNRELQARVKALLRRSQ
6QBL , Knot 47 93 0.76 36 77 90 
QAQALLPIASYAGLTVSAPVFAALVTVYGAYALYRYNIRRRENSYQRIRSDHDSHSCANNRGWCRPTCFSHEYTDWFNNDVCGSYRCCRPGRR
5BVD , Knot 159 361 0.86 40 234 349 
GPAAAAAAGAGPEMVRGQVFDVGPRYTNLSYIGEGAYGMVCSAYDNVNKVRVAIKKISPFEHQTYCQRTLREIKILLRFRHENIIGINDIIRAPTIEQMKDVYIVQDLMETDLYKLLKTQHLSNDHICYFLYQILRGLKYIHSANVLHRDLKPSNLLLNTTCDLKICDFGLARVADPDHDHTGFLTEYVATRWYRAPEIMLNSKGYTKSIDIWSVGCILAEMLSNRPIFPGKHYLDQLNHILGILGSPSQEDLNCIINLKARNYLLSLPHKNKVPWNRLFPNADSKALDLLDKMLTFNPHKRIEVEQALAHPYLEQYYDPSDEPIAEAPFKFDMELDDLPKEKLKELIFEETARFQPGYRS

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 \(c\).

\(|P_{f(1NXO_1)}(2) \setminus P_{f(6QBL_1)}(2)|=76\), \(|P_{f(6QBL_1)}(2) \setminus P_{f(1NXO_1)}(2)|=58\). 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 1:100111100001100110101000100110110100110010100101111011110101101100100000111111010000100111101110001001100001010101110000
Pair \(Z_2\) Length of longest common subsequence
1NXO_1,6QBL_1 134 3
1NXO_1,5BVD_1 185 3
6QBL_1,5BVD_1 209 3

Newick tree

 
[
	5BVD_1:10.18,
	[
		1NXO_1:67,6QBL_1:67
	]:40.18
]

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 \((0.85)(0.8)(\frac{213 }{\log_{20} 213}-\frac{93}{\log_{20}93})=39.1\)
Status Protein1 Protein2 d d1/2
Query variables 1NXO_1 6QBL_1 50 45
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]