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Parikh vectors
9MQJ_1 2YNA_1 5VPJ_1 Letter Amino acid
33 23 9 S Serine
37 26 13 T Threonine
16 9 8 Y Tyrosine
29 33 13 V Valine
20 5 14 R Arginine
21 12 10 D Aspartic acid
9 9 11 E Glutamic acid
38 25 15 L Leucine
20 13 8 F Phenylalanine
7 5 1 W Tryptophan
27 20 17 A Alanine
15 27 8 G Glycine
36 12 2 I Isoleucine
7 10 4 H Histidine
12 12 2 K Lycine
13 12 4 M Methionine
23 10 9 P Proline
24 18 4 N Asparagine
17 9 4 C Cysteine
7 16 3 Q Glutamine 

9MQJ_1|Chain A|Mu-type opioid receptor|Homo sapiens (9606)
>2YNA_1|Chains A, B|3C-LIKE PROTEINASE|TYLONYCTERIS BAT CORONAVIRUS HKU4 (694007)
>5VPJ_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Thioesterase|Actinomadura verrucosospora (46165)
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 \(w=f(c)\)
9MQJ , Knot 171 411 0.83 40 230 393 
DYKDDDDASIDMDSSAAPTNASNCTDALAYSSCSPAPSPGSWVNLSHLDGNLSDPCGPNRTDLGGRDSLCPPTGSPSMITAITIMALYSIVCVVGLFGNFLVMYVIVRYTKMKTATNIYIFNLALADALATSTLPFQSVNYLMGTWPFGTILCKIVISIDYYNMFTSIFTLCTMSVDRYIAVCHPVKALDFRTPRNAKIINVCNWILSSAIGLPVMFMATTKYRQGSIDCTLTFSHPTWYWENLLKICVFIFAFIMPVLIITVCYGLMILRLKSVRLLSGSREKDRNLRRITRMVLVVVAVFIVCWTPIHIYVIIKALVTIPETTFQTVSWHFCIALGYTNSCLNPVLYAFLDENFKRCFREFCIPTSSNIEQQNSTRIRQNTRDHPSTANTVDRTNHQLENLEAETAPLP
2YNA , Knot 134 306 0.83 40 194 291 
SGLVKMSAPSGAVENCIVQVTCGSMTLNGLWLDNTVWCPRHIMCPADQLTDPNYDALLISKTNHSFIVQKHIGAQANLRVVAHSMVGVLLKLTVDVANPSTPAYTFSTVKPGASFSVLACYNGKPTGVFTVNLRHNSTIKGSFLCGSCGSVGYTENGGVINFVYMHQMELSNGTHTGSSFDGVMYGAFEDKQTHQLQLTDKYCTINVVAWLYAAVLNGCKWFVKPTRVGIVTYNEWALSNQFTEFVGTQSIDMLAHRTGVSVEQMLAAIQSLHAGFQGKTILGQSTLEDEFTPDDVNMQVMGVVMQ
5VPJ , Knot 76 159 0.80 40 117 151 
SNAMRAYFEYRHLVTFADTNLVGNVYFTNYLSWQGACRERFLAEKAPKTVARMHDDLALVTSSCSCEFFSELYALDTVSVRMSLVGIDFHQITMGFEYYRVTDGPARLVARGEQTVACTLRAEDGLTPVEVPDELRTALDAYAPDPHGALRPRSETPPR

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 \(c\).

\(|P_{f(9MQJ_1)}(2) \setminus P_{f(2YNA_1)}(2)|=108\), \(|P_{f(2YNA_1)}(2) \setminus P_{f(9MQJ_1)}(2)|=72\). 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 1:000000010101000111001000001110000011101101101001010100101100001110001011010101101101111001101111110111101110000100100101101111011100011100100111011110110011101000011001101001010001110011011010010010110100111001111111111000000101000101001010100110101111111111111010011111010010110100000001001001111111111101011010111011101100010010101011110000010111011100010001001011000010000000100000001001001000000100101001111
Pair \(Z_2\) Length of longest common subsequence
9MQJ_1,2YNA_1 180 4
9MQJ_1,5VPJ_1 191 4
2YNA_1,5VPJ_1 181 3

Newick tree

 
[
	5VPJ_1:94.02,
	[
		9MQJ_1:90,2YNA_1:90
	]:4.02
]

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 \((0.85)(0.8)(\frac{717 }{\log_{20} 717}-\frac{306}{\log_{20}306})=113.\)
Status Protein1 Protein2 d d1/2
Query variables 9MQJ_1 2YNA_1 144 123.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]