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Parikh vectors
3NOV_1 3ABR_1 5QXJ_1 Letter Amino acid
13 22 14 R Arginine
10 31 16 D Aspartic acid
15 17 3 Q Glutamine
7 5 3 H Histidine
2 13 2 M Methionine
14 27 6 T Threonine
3 9 3 C Cysteine
9 27 13 E Glutamic acid
21 38 1 G Glycine
7 18 5 F Phenylalanine
16 13 6 P Proline
10 23 6 S Serine
9 31 11 I Isoleucine
19 34 7 V Valine
37 45 6 A Alanine
2 27 3 N Asparagine
29 38 13 L Leucine
3 18 7 K Lycine
2 2 0 W Tryptophan
3 15 5 Y Tyrosine 

3NOV_1|Chain A|ThiJ/PfpI family protein|Pseudomonas fluorescens (294)
>3ABR_1|Chains A, C|Ethanolamine ammonia-lyase heavy chain|Escherichia coli (83333)
>5QXJ_1|Chain A|ATPase family AAA domain-containing protein 2|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 \(w=f(c)\)
3NOV , Knot 102 231 0.80 40 135 217 
GSHMAVQIGFLLFPEVQQLELTGPHDVLASLPDVQVHLIWKEPGPVVASSGLVLQATTSFADCPPLDVICIPGGTGVGALMEDPQALAFIRQQAARARYVTSVCTGSLVLGAAGLLQGKRATTHWAYHELLAPLGAIPVHERVVRDGNLLTGGGITAGIDFALTLAAELFDAATAQRVQLQLEYAPAPPFNAGSPDTAPASVVQQARQRAADSLHKRREITLRAAARLAAG
3ABR , Knot 191 453 0.86 40 241 430 
MKLKTTLFGNVYQFKDVKEVLAKANELRSGDVLAGVAAASSQERVAAKQVLSEMTVADIRNNPVIAYEDDCVTRLIQDDVNETAYNQIKNWSISELREYVLSDETSVDDIAFTRKGLTSEVVAAVAKICSNADLIYGAKKMPVIKKANTTIGIPGTFSARLQPNDTRDDVQSIAAQIYEGLSFGVGDAVIGVNPVTDDVENLSRVLDTIYGVIDKFNIPTQGCVLAHVTTQIEAIRRGAPGGLIFQSICGSEKGLKEFGVELAMLDEARAVGAEFNRIAGENCLYFETGQGSALSAGANFGADQVTMEARNYGLARHYDPFIVNTVVGFIGPEYLYNDRQIIRAGLEDHFMGKLSGISMGCDCCYTNHADADQNLNENLMILLATAGCNYIMGMPLGDDIMLNYQTTAFHDTATVRQLLNLRPSPEFERWLESMGIMANGRLTKRAGDPSLFF
5QXJ , Knot 63 130 0.78 38 97 125 
SMQEEDTFRELRIFLRNVTHRLAIDKRFRVFTKPVDPDEVPDYRTVIKEPMDLSSVISKIDLHKYLTVKDYLRDIDLICSNALEYNPDRDPGDRLIRHRACALRDTAYAIIKEELDEDFEQLCEEIQESR

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 \(c\).

\(|P_{f(3NOV_1)}(2) \setminus P_{f(3ABR_1)}(2)|=39\), \(|P_{f(3ABR_1)}(2) \setminus P_{f(3NOV_1)}(2)|=145\). 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 1:100111011111110100101011001110110101011100111111001111010001100111011011110111111001011111000110100100100101111111110100100011000111111111100011001011011110111011101110110110100101010011111101101001110110010001100100000101011101111
Pair \(Z_2\) Length of longest common subsequence
3NOV_1,3ABR_1 184 4
3NOV_1,5QXJ_1 146 3
3ABR_1,5QXJ_1 202 5

Newick tree

 
[
	3ABR_1:10.28,
	[
		3NOV_1:73,5QXJ_1:73
	]:30.28
]

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 \((0.85)(0.8)(\frac{684 }{\log_{20} 684}-\frac{231}{\log_{20}231})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 3NOV_1 3ABR_1 160 118.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]