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Parikh vectors
4LXY_1 2POT_1 7EJX_1 Letter Amino acid
8 18 27 V Valine
5 32 24 R Arginine
31 8 8 N Asparagine
11 33 33 G Glycine
17 30 31 S Serine
9 38 61 A Alanine
6 10 10 H Histidine
9 18 18 T Threonine
1 4 10 W Tryptophan
23 20 7 F Phenylalanine
5 17 28 P Proline
16 24 4 D Aspartic acid
12 18 15 Q Glutamine
14 23 8 E Glutamic acid
47 22 7 I Isoleucine
11 8 9 Y Tyrosine
2 5 10 C Cysteine
20 27 68 L Leucine
26 14 2 K Lycine
1 17 4 M Methionine 

4LXY_1|Chains A, B|WlaRD, a sugar 3N-formyl transferase|Campylobacter jejuni subsp. jejuni (407148)
>2POT_1|Chain A|Queuine tRNA-ribosyltransferase|Zymomonas mobilis (542)
>7EJX_1|Chain A[auth R]|Probable G-protein coupled receptor 88|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 \(w=f(c)\)
4LXY , Knot 116 274 0.79 40 154 255 
GHMIKICIAGKNNIAVNSLQFILKNYFEADQIVVIPNKNDKGIDSWQKSLLKFALDNNIKIVTLDEIYNIEQIIFFSLEFDQIIKIENFKSDRLFNIHFSALPKYKGVFTSITPILNNELESGVTLHRIDNGIDTGNIIDQHCFPIDINDTARDLYFNYLKYGESIFKKNIQTIINNSYKDLKQTNINSSYFSRKDINLVHKINFKKTSFEIHNQIRAFIFQEYQLPIINNSKIIKSILANEFIGYNVFEEFENYFIISGIDGFKIIAQKLNKL
2POT , Knot 160 386 0.82 40 215 372 
MVEATAQETDRPRFSFSIAAREGKARTGTIEMKRGVIRTPAFMPVGTAATVKALKPETVRATGADIILGNTYHLMLRPGAERIAKLGGLHSFMGWDRPILTDSGGYQVMSLSSLTKQSEEGVTFKSHLDGSRHMLSPERSIEIQHLLGSDIVMAFDECTPYPATPSRAASSMERSMRWAKRSRDAFDSRKEQAENAALFGIQQGSVFENLRQQSADALAEIGFDGYAVGGLAVGQGQDEMFRVLDFSVPMLPDDKPHYLMGVGKPDDIVGAVERGIDMFDCVLPTRSGRNGQAFTWDGPINIRNARFSEDLKPLDSECHCAVCQKWSRAYIHHLIRAGEILGAMLMTEHNIAFYQQLMQKIRDSISEGRFSQFAQDFRARYFARNS
7EJX , Knot 151 384 0.78 40 168 323 
MTNSSSTSTSSTTGGSLLLLCEEEESWAGRRIPVSLLYSGLAIGGTLANGMVIYLVSSFRKLQTTSNAFIVNGCAADLSVCALWMPQEAVLGLLPTGSAEPPADWDGAGGSYRLLRGGLLGLGLTVSLLSHCLVALNRYLLITRAPATYQALYQRRHTAGMLALSWALALGLVLLLPPWAPRPGAAPPRIHYPALLAAAALLAQTALLLHCYLGIVRRVRVSVKRVSVLNFHLLHQLPGCAAAAAAFPGAQHAPGPGGAAHPAQAQPLPPALHPRRAQRRLSGLSVLLLCCVFLLATQPLVWVSLASGFSLPVPWGVQAASWLLCCALSALNPLLYTWRNEEFRRSVRSVLPGVGDAAAAAVAATAVPAVSQAQLGTRAAGQHW

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 \(c\).

\(|P_{f(4LXY_1)}(2) \setminus P_{f(2POT_1)}(2)|=64\), \(|P_{f(2POT_1)}(2) \setminus P_{f(4LXY_1)}(2)|=125\). 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 1:1011010111000111001011100010100111110000011001000110111000101101001001001111010100110100100001101010111000111001011100010011010010011001011000011101000100101001001001100010011000000100001000010000101100101000010100010111100001111000011001110011100110010001110110110111001001
Pair \(Z_2\) Length of longest common subsequence
4LXY_1,2POT_1 189 3
4LXY_1,7EJX_1 194 3
2POT_1,7EJX_1 181 4

Newick tree

 
[
	4LXY_1:97.44,
	[
		2POT_1:90.5,7EJX_1:90.5
	]:6.94
]

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 \((0.85)(0.8)(\frac{660 }{\log_{20} 660}-\frac{274}{\log_{20}274})=107.\)
Status Protein1 Protein2 d d1/2
Query variables 4LXY_1 2POT_1 139 116.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]