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Parikh vectors
3LXO_1 5POJ_1 4NBC_1 Letter Amino acid
2 18 31 L Leucine
3 6 13 F Phenylalanine
4 6 21 P Proline
10 5 14 T Threonine
4 14 16 R Arginine
8 1 7 C Cysteine
5 13 29 E Glutamic acid
3 5 23 I Isoleucine
9 10 30 V Valine
4 10 15 H Histidine
10 6 29 K Lycine
0 0 13 W Tryptophan
6 5 15 Y Tyrosine
3 7 31 G Glycine
4 7 7 M Methionine
15 7 12 S Serine
12 11 22 A Alanine
10 4 20 N Asparagine
5 13 31 D Aspartic acid
7 8 13 Q Glutamine 

3LXO_1|Chain A|Ribonuclease pancreatic|Bos taurus (9913)
>5POJ_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
>4NBC_1|Chains A, B, C|Terminal oxygenase component of carbazole|Janthinobacterium (213804)
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 \(w=f(c)\)
3LXO , Knot 64 124 0.83 38 107 121 
KETAAAKFERQHMDSSTSAASSSNYCNQMMKSRNLTKDRCKPVNTFVHESLADVQAVCSQKNVACKNGQTNCYQSYSTMSITDCRETGSSKYPNCAYKTTQANKHIIVACEGNPYVPVHFDASV
5POJ , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA
4NBC , Knot 172 392 0.87 40 238 375 
MANVDEAILKRVKGWAPYVDAKLGFRNHWYPVMFSKEINEGEPKTLKLLGENLLVNRIDGKLYCLKDRCLHRGVQLSVKVECKTKSTITCWYHAWTYRWEDGVLCDILTNPTSAQIGRQKLKTYPVQEAKGCVFIYLGDGDPPPLARDTPPNFLDDDMEILGKNQIIKSNWRLAVENGFDPSHIYIHKDSILVKDNDLALPLGFAPGGDRKQQTRVVDDDVVGRKGVYDLIGEHGVPVFEGTIGGEVVREGAYGEKIVANDISIWLPGVLKVNPWPNPDMMQFEWYVPIDENTHYYFQTLGKPCANDEERKKYEQEFESKWKPMALEGFNNDDIWAREAMVDFYADDKGWVNEILFESDEAIVAWRKLASEHNQGIQTQAHVSGLEHHHHHH

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 \(c\).

\(|P_{f(3LXO_1)}(2) \setminus P_{f(5POJ_1)}(2)|=74\), \(|P_{f(5POJ_1)}(2) \setminus P_{f(3LXO_1)}(2)|=87\). 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 1:0001110100001000001100000000110000100000011001100011010110000011000100000000001010000001000010010000010001111001010111010101
Pair \(Z_2\) Length of longest common subsequence
3LXO_1,5POJ_1 161 2
3LXO_1,4NBC_1 217 4
5POJ_1,4NBC_1 200 6

Newick tree

 
[
	4NBC_1:11.15,
	[
		3LXO_1:80.5,5POJ_1:80.5
	]:30.65
]

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 \((0.85)(0.8)(\frac{280 }{\log_{20} 280}-\frac{124}{\log_{20}124})=48.8\)
Status Protein1 Protein2 d d1/2
Query variables 3LXO_1 5POJ_1 66 59
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]