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Parikh vectors
3HMB_1 5KGN_1 9BAT_1 Letter Amino acid
14 47 26 G Glycine
11 25 25 I Isoleucine
7 22 28 S Serine
19 55 29 A Alanine
9 30 25 D Aspartic acid
3 16 15 Q Glutamine
6 37 28 E Glutamic acid
5 19 15 Y Tyrosine
14 25 15 N Asparagine
8 25 10 H Histidine
12 30 29 K Lycine
5 4 5 W Tryptophan
9 24 31 T Threonine
4 20 26 R Arginine
2 10 9 C Cysteine
8 39 47 L Leucine
3 19 12 M Methionine
4 20 26 F Phenylalanine
7 24 26 P Proline
7 43 22 V Valine 

3HMB_1|Chains A, B, C|N-acetylmuramoyl-L-alanine amidase xlyA|Bacillus subtilis (1423)
>5KGN_1|Chains A, B|2,3-bisphosphoglycerate-independent phosphoglycerate mutase|Caenorhabditis elegans (6239)
>9BAT_1|Chains A, B|Lanosterol 14-alpha demethylase|Coryphaenoides armatus (76798)
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 \(w=f(c)\)
3HMB , Knot 79 157 0.84 40 123 152 
GSHMVNIIQKFIPVGANNRPGYAMKPKYITVHNTANTAVGADAAAHARYLKNPDTTTSWHFTVDDKEIYQHLPLNENGWHAGDGNGSGNRASIGIEICENADGDFAKATANAQWLIKTLMAEHNISLANVVPHKYWSGKECPRKLLDKWDSFKAGIG
5KGN , Knot 214 534 0.84 40 262 501 
MAMANNSSVANKVCLIVIDGWGVSEDPYGNAILNAQTPVMDKLCSGNWAQIEAHGLHVGLPEGLMGNSEVGHLNIGAGRVIYQDIVRINLAVKNNKFVTNESLVDACDRAKNGNGRLHLAGLVSDGGVHSHIDHMFALVKAIKELGVPELYLHFYGDGRDTSPNSGVGFLEQTLEFLEKTTGYGKLATVVGRYYAMDRDNRWERINVAYEAMIGGVGETSDEAGVVEVVRKRYAADETDEFLKPIILQGEKGRVQNDDTIIFFDYRADRMREISAAMGMDRYKDCNSKLAHPSNLQVYGMTQYKAEFPFKSLFPPASNKNVLAEWLAEQKVSQFHCAETEKYAHVTFFFNGGLEKQFEGEERCLVPSPKVATYDLQPEMSAAGVADKMIEQLEAGTHPFIMCNFAPPDMVGHTGVYEAAVKACEATDIAIGRIYEATQKHGYSLMVTADHGNAEKMKAPDGGKHTAHTCYRVPLTLSHPGFKFVDPADRHPALCDVAPTVLAIMGLPQPAEMTGVSIVQKIKLAAALEHHHHHH
9BAT , Knot 185 449 0.83 40 243 429 
MAKKTKCPPYIPSRIPFLGHAVAFGKNPIEFLEKAYEKYGPVVSFTMVGKTFTFLLGSDAAALMFNSKNEDLSAEDVYSKITTPVFGKGVAYDVPNPIFLEQKKMLKTGLNIAQFKKHVPIIEQETEEYIKRWGDSGETNLFVAMSELIILTASSCLHGKEIRSLLNEKVAQLYCDLDGGFSHEAWLLPSWLPLPSFRRRDRAHREIKNMFYTAIQKRRTSNEKVDDFLQTLIDATYKDGHSLSDDEIAGLLIGLLLAGQHTSSTTSSWMSFFLARDKQLQERCLAEQKAVCGEDLPPLDFDQLKELSLLDRCLKETLRLRPPIMTMMRMARSPQCVAGYTIPVGHQVCVSPTVNQRLPDSWTERDEFNPDRYLTDNAATGEKFAYVPFGAGRHRCIGEGFAYVQLKTILSTLIRKFEFDLVDGYFPTINFTTMIHTPNNPVIRYRSRT

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 \(c\).

\(|P_{f(3HMB_1)}(2) \setminus P_{f(5KGN_1)}(2)|=34\), \(|P_{f(5KGN_1)}(2) \setminus P_{f(3HMB_1)}(2)|=173\). 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 1:1001101100111111000110110100101000100111101110100100100000101010000100011100011011010101001011101000101011010101011100111000101101110001010001001100100101111
Pair \(Z_2\) Length of longest common subsequence
3HMB_1,5KGN_1 207 3
3HMB_1,9BAT_1 200 4
5KGN_1,9BAT_1 141 5

Newick tree

 
[
	3HMB_1:11.23,
	[
		9BAT_1:70.5,5KGN_1:70.5
	]:39.73
]

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 \((0.85)(0.8)(\frac{691 }{\log_{20} 691}-\frac{157}{\log_{20}157})=152.\)
Status Protein1 Protein2 d d1/2
Query variables 3HMB_1 5KGN_1 194 124
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]