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Parikh vectors
7SRR_1 1YUX_1 7BLP_1 Letter Amino acid
6 11 35 Y Tyrosine
16 33 71 A Alanine
20 10 48 D Aspartic acid
17 11 36 G Glycine
4 6 24 H Histidine
3 0 5 W Tryptophan
4 5 13 C Cysteine
6 9 40 P Proline
20 7 58 R Arginine
7 4 52 Q Glutamine
16 18 65 E Glutamic acid
17 11 41 I Isoleucine
17 14 31 K Lycine
14 12 51 T Threonine
14 10 54 V Valine
13 8 31 N Asparagine
18 11 109 L Leucine
5 4 19 M Methionine
14 9 26 F Phenylalanine
15 9 60 S Serine 

7SRR_1|Chain A[auth B]|G protein subunit q (Gi2-mini-Gq chimera)|Homo sapiens (9606)
>1YUX_1|Chains A, B|Nigerythrin|Desulfovibrio vulgaris subsp. vulgaris (882)
>7BLP_1|Chains A, C|Vacuolar protein sorting-associated protein 35|Chaetomium thermophilum (strain DSM 1495 / CBS 144.50 / IMI 039719) (759272)
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 \(w=f(c)\)
7SRR , Knot 109 246 0.81 40 167 234 
MGSTVSAEDKAAAERSKMIDKNLREDGEKARRTLRLLLLGADNSGKSTIVKQMRILHGGSGGSGGTSGIFETKFQVDKVNFHMFDVGGQRDERRKWIQCFNDVTAIIFVVDSSDYNRLQEALNDFKSIWNNRWLRTISVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRKEFVDISTASGDGRHICYPHFTCAVDTENARRIFNDCKDIILQMNLREYNLV
1YUX , Knot 96 202 0.84 38 147 195 
MKVRAQVPTVKNATNFNMVADSKTSVGSTLENLKAAIAGETGAHAKYTAFAKAAREQGYEQIARLFEATAAAELIHIGLEYALVAEMEPGYEKPTVAAPSAYSCDLNLISGANGEIYETSDMYPAFIRKAQEEGNSKAVHVFTRAKLAESVHAERYLAAYNDIDAPDDDKFHLCPICGYIHKGEDFEKCPICFRPKDTFTAY
7BLP , Knot 322 869 0.83 40 290 777 
MSTPAPPEEQARLLEDALIAVRQQTAMMRKFLDTPGKLMDALKCCSTLVSELRTSSLSPKQYYELYMAVFDALRYLSAHLRENHPVNHLADLYELVQYAGNIIPRLYLMITVGTAYMSIDGAPVKELMKDMMDMSRGVQHPVRGLFLRYYLSGQARDYLPTGDSDGPEGNLQDSINFILTNFVEMNKLWVRLQHQGHSRERDLRTQERRELQLLVGSNIVRLSQLVDLPTYRDSILGPLLEQIVQCRDILAQEYLLEVITQVFPDEYHLHTLDQFLGAVSRLNPHVNVKAIVIGMMNRLSDYAERESQNEPEEDRAKLEEEALAKLLEKTKLGQNSELEPQNGDHPDTEVSSTTDSAQAPSTADSDTTAVNGEEEPVRKRRGIPVNVPLYDIFFDQVQHLVQAQHLPIQDTIALCCSLANLSLNIYPERLDYVDGILAYALAKVKEHANSADLHSQPAQQSLLSLLQSPLRRYVSIFTALSLPTYVSLFQAQTYPTRRAIAGEIVRTLLKNQTLISTPAHLENVLEILKVLIKEGSQPPAGYPGVVQPRARPLETDETMEEQGWLARLVHLIHSDDNDTQFRLLQMTRKAYAEGNERIRTTTPPLITAGLKLARRFKAREHYDDNWSSQSSSLFKFLHSAISTLYTRVNGPGVADLCLRLFCSCGQVADMTEFEEVAYEFFAQAFTVYEESISDSKAQFQAVCVIASALHRTRNFGRENYDTLITKCAQHASKLLRKPDQCRAVYLASHLWWATPIAARGETEDTELYRDGKRVLECLQRALRVADSCMETATSIELFVEILDRYVYYFDQRNESVTTKYLNGLIELIHSNLAGNQQDSASVEASRKHFIQTLEMIQSKEFEGIVVAPK

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 \(c\).

\(|P_{f(7SRR_1)}(2) \setminus P_{f(1YUX_1)}(2)|=88\), \(|P_{f(1YUX_1)}(2) \setminus P_{f(7SRR_1)}(2)|=68\). 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 1:110010100011100001100010001001000101111110001000110010110110110110011100010100101011011100000001100100101111110000000100110010011000110010111110000111001111000100011011000010010101100101001001100011010010101001001010011000010011000001110101000011
Pair \(Z_2\) Length of longest common subsequence
7SRR_1,1YUX_1 156 4
7SRR_1,7BLP_1 185 3
1YUX_1,7BLP_1 181 5

Newick tree

 
[
	7BLP_1:95.58,
	[
		7SRR_1:78,1YUX_1:78
	]:17.58
]

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 \((0.85)(0.8)(\frac{448 }{\log_{20} 448}-\frac{202}{\log_{20}202})=71.9\)
Status Protein1 Protein2 d d1/2
Query variables 7SRR_1 1YUX_1 88 80.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]