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Parikh vectors
7ZCG_1 3CNO_1 6YIR_1 Letter Amino acid
14 17 27 A Alanine
10 21 33 I Isoleucine
6 14 15 T Threonine
1 3 12 Y Tyrosine
6 21 27 V Valine
2 10 13 P Proline
3 15 14 S Serine
0 3 0 W Tryptophan
7 6 11 N Asparagine
9 12 25 D Aspartic acid
12 20 29 E Glutamic acid
4 20 26 G Glycine
16 2 14 M Methionine
13 9 13 Q Glutamine
2 17 18 F Phenylalanine
15 20 21 R Arginine
0 0 1 C Cysteine
0 2 14 H Histidine
11 23 29 L Leucine
16 27 31 K Lycine 

7ZCG_1|Chain A[auth B]|Charged multivesicular body protein 2a|Homo sapiens (9606)
>3CNO_1|Chain A|Putative uncharacterized protein|Thermotoga maritima (2336)
>6YIR_1|Chain A|Oligosaccharides import ATP-binding protein MsmX|Bacillus subtilis (strain 168) (224308)
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 \(w=f(c)\)
7ZCG , Knot 69 147 0.78 34 105 142 
RKTPEELLRQNQRALNRAMRELDRERQKLETQEKKIIADIKKMAKQGQMDAVRIMAKDLVRTRRYVRKFVLMRANIQAVSLKIQTLKSNNSMAQAMKGVTKAMGTMNRQLKLPQIQKIMMEFERQAEIMDMKEEMMNDAIDDAMGDE
3CNO , Knot 116 262 0.82 38 159 248 
MSWYPGHIEKAKRQIKDLLRLVNTVVEVRDARAPFATSAYGVDFSRKETIILLNKVDIADEKTTKKWVEFFKKQGKRVITTHKGEPRKVLLKKLSFDRLARVLIVGVPNTGKSTIINKLKGKRASSVGAQPGITKGIQWFSLENGVKILDTPGILYKNIFSEDLAAKLLLVGSLPVERIEDQRIFERAFEIFARSIGIESSFSEFFEDFARKRGLLKKGGVPDIERALMLFFTEVAQGKAGRVSFERPEDITPVQQEQTRGV
6YIR , Knot 158 373 0.83 38 207 351 
MAELRMEHIYKFYDQKEPAVDDFNLHIADKEFIVFVGPSGCGASTTLRMVAGLEEISKGDFYIEGKRVNDVAPKDRDIAMVFQNYALYPHMTVYDNIAFGLKLRKMPKPEIKKRVEEAAKILGLEEYLHRKPKALSGGQRQRVALGRAIVRDAKVFLMDEPLSNLDAKLRVQMRAEIIKLHQRLQTTTIYVTHDQTEALTMATRIVVMKDGKIQQIGTPKDVYEFPENVFVGGFIGSPAMNFFKGKLTDGLIKIGSAALTVPEGKMKVLREKGYIGKEVIFGIRPEDIHDELIVVESYKNSSIKAKINVAELLGSEIMIYSQIDNQDFIARIDARLDIQSGDELTVAFDMNKGHFFDSETEVRIRLEHHHHHH

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 \(c\).

\(|P_{f(7ZCG_1)}(2) \setminus P_{f(3CNO_1)}(2)|=48\), \(|P_{f(3CNO_1)}(2) \setminus P_{f(7ZCG_1)}(2)|=102\). 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 1:000100110000011001100100000010000001110100110010101101110011000001001111010101101010010000011011011001110100010110100111010001011010001100110011100
Pair \(Z_2\) Length of longest common subsequence
7ZCG_1,3CNO_1 150 3
7ZCG_1,6YIR_1 162 3
3CNO_1,6YIR_1 154 5

Newick tree

 
[
	6YIR_1:80.32,
	[
		7ZCG_1:75,3CNO_1:75
	]:5.32
]

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 \((0.85)(0.8)(\frac{409 }{\log_{20} 409}-\frac{147}{\log_{20}147})=78.5\)
Status Protein1 Protein2 d d1/2
Query variables 7ZCG_1 3CNO_1 98 76
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]