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Parikh vectors
4HHX_1 4PFP_1 2TSB_1 Letter Amino acid
1 25 4 H Histidine
14 52 10 S Serine
6 48 11 G Glycine
0 2 1 W Tryptophan
4 45 10 V Valine
3 39 1 R Arginine
9 46 7 N Asparagine
9 51 11 D Aspartic acid
1 30 6 Q Glutamine
6 54 4 E Glutamic acid
1 48 6 F Phenylalanine
12 81 10 L Leucine
13 62 11 K Lycine
6 13 5 M Methionine
3 25 4 P Proline
3 33 10 T Threonine
7 44 8 A Alanine
0 16 3 C Cysteine
18 47 4 I Isoleucine
4 27 2 Y Tyrosine 

4HHX_1|Chain A|Toxin coregulated pilus biosynthesis protein E|Vibrio cholerae O1 biovar El Tor str. N16961 (243277)
>4PFP_1|Chains A, B[auth C]|Unconventional myosin-VI|Sus scrofa (9823)
>2TSB_1|Chains A, B, C, D|AZURIN AZIDE|Pseudomonas aeruginosa (287)
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 \(w=f(c)\)
4HHX , Knot 59 120 0.78 36 87 115 
GSHMMKIISKKYRLELYSMLVDLLNDNIPLYDALNKIQNEGVGIYDKNFIKSIELIKDRMKSNSSLTDALTGLIPDKEVLMINVAENSGKISSGIAAIRKNIIDADEIKSKAISSMITPS
4PFP , Knot 302 788 0.85 40 303 726 
EDGKPVWAPHPTDGFQVGNIVDIGPDSLTIEPLNQKGKTFLALINQVFPAEEDSKKDVEDNCSLMYLNEATLLHNIKVRYSKDRIYTYVANILIAVNPYFDIPKIYSSETIKSYQGKSLGTMPPHVFAIADKAFRDMKVLKLSQSIIVSGESGAGKTENTKFVLRYLTESYGTGQDIDDRIVEANPLLEAFGNAKTVRNNNSSRFGKFVEIHFNEKSSVVGGFVSHYLLEKSRICVQGKEERNYHIFYRLCAGASEDIRERLHLSSPDNFRYLNRGCTRYFANKETDKQILQNRKSPEYLKAGSLKDPLLDDHGDFIRMCTAMKKIGLDDEEKLDLFRVVAGVLHLGNIDFEEAGSTSGGCNLKNKSTQALEYCAELLGLDQDDLRVSLTTRVMLTTAGGAKGTVIKVPLKVEQANNARDALAKTVYSHLFDHVVNRVNQCFPFETSSYFIGVLDIAGFEYFEHNSFEQFCINYCNEKLQQFFNERILKEEQELYQKEGLGVNEVHYVDNQDCIDLIEARLVGILDILDEENRLPQPSDQHFTSAVHQKHKDHFRLSIPRKSKLAIHRNIRDDEGFIIRHFAGAVCYETTQFVEKNNDALHMSLESLICESRDKFIRELFESSTNNNKDTKQKAGKLSFISVGNKFKTQLNLLLDKLRSTGASFIRCIKPNLKMTSHHFEGAQILSQLQCSGMVSVLDLMQGGFPSRASFHELYNMYKKYMPDKLARLDPRLFCKALFKALGLNEIDYKFGLTKVFFRPGKFAEFDQIMKSDPDHLAELVKRVNHWLI
2TSB , Knot 68 128 0.86 40 105 126 
AECSVDIQGNDQMQFNTNAITVDKSCKQFTVNLSHPGNLPKNVMGHNWVLSTAADMQGVVTDGMASGLDKDYLKPDDSRVIAHTKLIGSGEKDSVTFDVSKLKEGEQYMFFCTFPGHSALAKGTLTLK

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 \(c\).

\(|P_{f(4HHX_1)}(2) \setminus P_{f(4PFP_1)}(2)|=10\), \(|P_{f(4PFP_1)}(2) \setminus P_{f(4HHX_1)}(2)|=226\). 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 1:100110110000010100111011000111001100100011110000110010110001000001001101111000111101100010100111110001101001000110011010
Pair \(Z_2\) Length of longest common subsequence
4HHX_1,4PFP_1 236 3
4HHX_1,2TSB_1 136 2
4PFP_1,2TSB_1 234 4

Newick tree

 
[
	4PFP_1:12.87,
	[
		4HHX_1:68,2TSB_1:68
	]:61.87
]

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 \((0.85)(0.8)(\frac{908 }{\log_{20} 908}-\frac{120}{\log_{20}120})=220.\)
Status Protein1 Protein2 d d1/2
Query variables 4HHX_1 4PFP_1 284 161.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]