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Parikh vectors
3HMC_1 1ZNX_1 2YKP_1 Letter Amino acid
15 25 13 A Alanine
6 1 5 M Methionine
7 9 13 Q Glutamine
19 19 7 G Glycine
7 19 20 L Leucine
7 15 10 P Proline
10 1 4 Y Tyrosine
13 23 10 V Valine
6 18 26 R Arginine
3 2 1 C Cysteine
15 9 18 I Isoleucine
9 5 8 F Phenylalanine
7 2 2 W Tryptophan
14 11 8 D Aspartic acid
5 12 25 E Glutamic acid
14 5 19 K Lycine
11 11 11 S Serine
7 14 10 T Threonine
11 2 12 N Asparagine
6 4 7 H Histidine 

3HMC_1|Chain A|Putative prophage LambdaBa04, glycosyl hydrolase, family 25|Bacillus anthracis (1392)
>1ZNX_1|Chain A|Guanylate kinase|Mycobacterium tuberculosis (1773)
>2YKP_1|Chains A, B, C|LINE-1 ORF1P|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 \(w=f(c)\)
3HMC , Knot 92 192 0.84 40 138 185 
GSHMGHIIDISKWNGDINWSIAKQHIDFIIARVQDGSNYVDPLYKGYVQAMKQHGIPFGNYAFCRFVSIADAKKEAQDFWNRGDKSATVWVADVEVKTMNDMRAGTQAFIDELYRLGAKKVGLYVGHHMYTPFGMANVKSDFVWIPRYGGNKPAYPCDIWQYTETGNVPGIGKCDLNSLIGNKSLSWFTESA
1ZNX , Knot 92 207 0.79 40 134 199 
SVGEGPDTKPTARGQPAAVGRVVVLSGPSAVGKSTVVRCLRERIPNLHFSVSATTRAPRPGEVDGVDYHFIDPTRFQQLIDQGELLEWAEIHGGLHRSGTLAQPVRAAAATGVPVLIEVDLAGARAIKKTMPEAVTVFLAPPSWQDLQARLIGRGTETADVIQRRLDTARIELAAQGDFDKVVVNRRLESACAELVSLLVGTAPGSP
2YKP , Knot 102 229 0.80 40 151 213 
MASLRSRCDQLEERVSAAEDEINEIKREGKFREKRIKRNEQSLQEIWDYVKRPNLRLIGVPESDVENGTKLENTLQDIIQENFPNLARQANVQIQEIQRTPQRYSSRRATPRHIIVRFTKVEMKEKMLRAAREKGRVTLKGKPIRLTVDLSAETLQARREWGPIFNILKEKNFQPRISYPAKLSFISEGEIKYFIDKQMLRDFVTTRPALKELLKEALNMERNHHHHHH

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 \(c\).

\(|P_{f(3HMC_1)}(2) \setminus P_{f(1ZNX_1)}(2)|=80\), \(|P_{f(1ZNX_1)}(2) \setminus P_{f(3HMC_1)}(2)|=76\). 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 1:100110110100101010101100010111101001000101100101011000111110011001101101000100110010001011110101001001011001110010011100111011001001111101000111110011001101001100000101111100010011100010110001
Pair \(Z_2\) Length of longest common subsequence
3HMC_1,1ZNX_1 156 3
3HMC_1,2YKP_1 165 3
1ZNX_1,2YKP_1 143 4

Newick tree

 
[
	3HMC_1:83.00,
	[
		1ZNX_1:71.5,2YKP_1:71.5
	]:11.50
]

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 \((0.85)(0.8)(\frac{399 }{\log_{20} 399}-\frac{192}{\log_{20}192})=61.3\)
Status Protein1 Protein2 d d1/2
Query variables 3HMC_1 1ZNX_1 76 73.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]