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Parikh vectors
7GEO_1 6DPM_1 1NBE_1 Letter Amino acid
21 6 15 N Asparagine
9 16 14 E Glutamic acid
26 11 15 G Glycine
11 11 15 I Isoleucine
29 10 38 L Leucine
17 2 12 F Phenylalanine
17 8 34 A Alanine
11 6 15 R Arginine
17 7 21 D Aspartic acid
12 0 1 C Cysteine
11 14 15 K Lycine
10 2 8 M Methionine
13 5 12 P Proline
3 6 2 W Tryptophan
14 3 14 Q Glutamine
24 10 18 T Threonine
11 5 8 Y Tyrosine
27 9 22 V Valine
7 3 11 H Histidine
16 8 20 S Serine 

7GEO_1|Chain A|3C-like proteinase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>6DPM_1|Chain A|Ribonuclease H|Bacillus halodurans (86665)
>1NBE_1|Chains A, C|ASPARTATE TRANSCARBAMOYLASE|Escherichia coli (562)
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 \(w=f(c)\)
7GEO , Knot 134 306 0.83 40 196 290 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMLNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMELPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVTFQ
6DPM , Knot 70 142 0.81 38 112 140 
GSHMAKEEIIWESLSVDVGSQGNPGIVEYKGVDTKTGEVLFEREPIPIGTNNMGEFLAIVHGLRYLKERNSRKPIYSDSQTAIKWVKDKKAKSTLVRNEETALIWKLVDEAEEWLNTHTYETPILKWQTDKWGEIKADYGRA
1NBE , Knot 138 310 0.85 40 188 299 
ANPLYQKHIISINDLSRDDLNLVLATAAKLKANPQPELLKHKVIASCFFEASTRTRLSFQTSMHRLGASVVGFSDSANTSLGKKGETLADTISVISTYVDAIVMRHPQEGAARLATEFSGNVPVLNAGDGSNQHPTQTLLDLFTIQQTEGRLDNLHVAMVGDLKYGRTVHSLTQALAKFDGNRFYFIAPDALAMPEYILDMLDEKGIAWSLHSSIEEVMAEVDILYMTRVQKERLDPSEYANVKAQFVLRASDLHNAKANMKVLHPLPRVDEIATDVDKTPHAWYFQQAGNGIFARQALLALVLNRDLVL

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 \(c\).

\(|P_{f(7GEO_1)}(2) \setminus P_{f(6DPM_1)}(2)|=143\), \(|P_{f(6DPM_1)}(2) \setminus P_{f(7GEO_1)}(2)|=59\). 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 1:011001111010101011010010001011110011001001100000110100001110000001110110101011100100011010100101001000110101100101110001010110001101010101011010010111010000101001001011011011001010101111000010111000010101111101111010011100100010010111100000110000101111101001111101010100110011010011101110001011011000011010
Pair \(Z_2\) Length of longest common subsequence
7GEO_1,6DPM_1 202 3
7GEO_1,1NBE_1 162 4
6DPM_1,1NBE_1 176 3

Newick tree

 
[
	6DPM_1:98.87,
	[
		7GEO_1:81,1NBE_1:81
	]:17.87
]

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 \((0.85)(0.8)(\frac{448 }{\log_{20} 448}-\frac{142}{\log_{20}142})=91.1\)
Status Protein1 Protein2 d d1/2
Query variables 7GEO_1 6DPM_1 121 87.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]