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Parikh vectors
2QMI_1 6NFW_1 9CJR_1 Letter Amino acid
19 16 17 D Aspartic acid
42 17 8 E Glutamic acid
4 4 7 H Histidine
40 8 28 L Leucine
34 10 28 V Valine
21 12 17 A Alanine
1 1 12 C Cysteine
13 5 10 M Methionine
16 9 24 T Threonine
38 13 11 I Isoleucine
40 19 11 K Lycine
21 11 17 F Phenylalanine
23 7 13 P Proline
33 5 11 Y Tyrosine
3 1 3 W Tryptophan
17 13 11 R Arginine
10 9 21 N Asparagine
1 7 13 Q Glutamine
45 13 26 G Glycine
26 8 16 S Serine 

2QMI_1|Chains A, B, C, D, E, F, G, H|Pbp related beta-lactamase|Pyrococcus abyssi (29292)
>6NFW_1|Chain A|VPg|Potato virus Y (12216)
>9CJR_1|Chains A, B|3C-like proteinase nsp5|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
2QMI , Knot 179 447 0.81 40 195 406 
MDVGKLESFIVEKMAERKVPGISISIIKDGDVVYAKGFGYRNVEARLPSTPETIYGIGSITKSFTALAIMKLVEEGGLSLDDPVEKFVNIKLRPFGEPVTVHHLLTHSSGIPSLGYAEAFIDGMVGGDNWLPVSTPEETIAFARDMEKWAVAKPGERFFYLNTGYVLLGKIIEKVSGVSYEEYIKKKILEPLGMNRSYFFKEEVEKDKDVAMGYILDKEGRLVPQPFPYGITADGGLLSSVLDLAKYLKMYIERDESIVSKEYIEKMETSYIKVPWEIFGGEGYGYGLIIYPNFLGEKLVGHSGSVGMYTGYIGYIPEKKIGVAVLENSSGYPPSYIAMYALALLLGKNPEKELPFIYRERILKKVEGRYMGYKGTIKFEVKVDGDVVYLRALGRAFTYTIPLFPEVLEEDFIKCYTLSNGRKMYAEFYIKDNKVDLIFERYRLIKS
6NFW , Knot 86 188 0.79 40 133 179 
GKNKSKRIQALKFRHARDKRAGFEIDNNDDTIEEFFGSAYRKKGKGKGTTVGMGKSSRRFINMYGFDPTEYSFIQFVDPLTGAQIEENVYADIRDIQERFSEVRKKMVENDDIEMQALGSNTTIHAYFRKDWSDKALKIDLMPHNPLKVCDKTNGIAKFPERELELRQTGPAVEVDVKDIPAQEVEHE
9CJR , Knot 133 304 0.83 40 194 288 
SGFRKMAFPSGKVEGCMVQVTCGTTTLNGLWLDDVVYCPRHVICTSEDMFNPNYEDLLIRKSNHNFLVQAGNVQLRVIGHSMQNCVLKLKVDTANPKTPKYKFVRIQPGQTFSVLACYNGSPSGVYQCAMRPNFTIKGSFLNGSCGSVGFNIDYDCVSFCYMHHMVLPTGVHAGTDLEGNFYGPFVDRQTAQAAGTDTTITVNVLAWLYAAVINGDRWFLNRFTTTLNDFNLVAMKYNYEPLTQDHVDILGPLSAQTGIAVLDMCASLKELLQNGMNGRTILGSALLEDEFTPFDVVRQCSGVT

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 \(c\).

\(|P_{f(2QMI_1)}(2) \setminus P_{f(6NFW_1)}(2)|=114\), \(|P_{f(6NFW_1)}(2) \setminus P_{f(2QMI_1)}(2)|=52\). 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 1:101101001110011000111101011001011010111000101011001001011101000101111101100111010011001101010111011010011000011101101011101111100111100100011110010011110110011010010111101100101100000100011011110000110001000001111011000101110111011010111100110110010101000001100001001000010111011110101011110101110011100101110010110110001111110000101100111011111110010001111000011001010011001010101010101101011101100011111011000110000100100101010100001011100001100
Pair \(Z_2\) Length of longest common subsequence
2QMI_1,6NFW_1 166 3
2QMI_1,9CJR_1 169 4
6NFW_1,9CJR_1 185 3

Newick tree

 
[
	9CJR_1:90.37,
	[
		2QMI_1:83,6NFW_1:83
	]:7.37
]

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 \((0.85)(0.8)(\frac{635 }{\log_{20} 635}-\frac{188}{\log_{20}188})=127.\)
Status Protein1 Protein2 d d1/2
Query variables 2QMI_1 6NFW_1 153 109
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]