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Parikh vectors
4LPJ_1 4XVA_1 5AZP_1 Letter Amino acid
4 9 14 I Isoleucine
19 25 53 L Leucine
14 23 9 K Lycine
10 15 37 S Serine
5 15 17 T Threonine
9 10 48 R Arginine
13 25 34 D Aspartic acid
8 9 28 Q Glutamine
6 5 0 M Methionine
8 14 9 Y Tyrosine
12 17 72 A Alanine
9 20 7 N Asparagine
2 1 1 C Cysteine
7 24 30 G Glycine
8 18 11 F Phenylalanine
1 7 6 W Tryptophan
19 20 31 E Glutamic acid
8 6 7 H Histidine
3 16 16 P Proline
11 14 25 V Valine

4LPJ_1|Chain A|Ferritin, middle subunit|Rana catesbeiana (8400)
>4XVA_1|Chains A, B[auth C], C[auth E], D[auth G]|Cytochrome c peroxidase, mitochondrial|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
>5AZP_1|Chains A, B, C|Multidrug efflux outer membrane protein OprN|Pseudomonas aeruginosa (strain ATCC 15692 / PAO1 / 1C / PRS 101 / LMG 12228) (208964)
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 \(w=f(c)\)
4LPJ , Knot 85 176 0.83 40 135 172
MVSQVRQNYHSDCEAAVNRMLNLELYASYTYSSMYAFFDRDDVALHNVAEFFKEHSHEEREHAEKFMKYQNKRGGRVVLQDIKKPERDEWGNTLEAMQAALQLEKTVNQALLDLHKLATDKVDPHLCDFLESEYLEEQVKDIKRIGDFITNLKRLGLPENGMGEYLFDKHSVKESS
4XVA , Knot 134 293 0.86 40 195 286
TPLVHVASVEKGRSYEDFQKVYNAIALKLREDDEYDNYIGYGPVLVRLAWHTSGTWDKHDNTGGSYGGTYRFKKEFNDPSNAGLQNGFKFLEPIHKEFPWISSGDLFSLGGVTAVQEMQGPKIPWRCGRVDTPEDTTPDNGRLPDADKDADYVRTFFQRLNMNDREVVALMGAHALGKTHLKNSGYEGPWGAANNVFTNEFYLNLLNEDWKLEKNDANNEQWDSKSGYMMLPTDYSLIQDPKYLSIVKEYANDQDKFFKDFSKAFEKLLENGITFPKDAPSPFIFKTLEEQGL
5AZP , Knot 180 455 0.80 38 185 402
CTVGPDYRTPDTAAAKIDATASKPYDRSRFESLWWKQFDDPTLNQLVEQSLSGNRDLRVAFARLRAARALRDDVANDRFPVVTSRASADIGKGQQPGVTEDRVNSERYDLGLDSAWELDLFGRIRRQLESSDALSEAAEADLQQLQVSLIAELVDAYGQLRGAQLREKIALSNLENQKESRQLTEQLRDAGVGAELDVLRADARLAATAASVPQLQAEAERARHRIATLLGQRPEELTVDLSPRDLPAITKALPIGDPGELLRRRPDIRAAERRLAASTADVGVATADLFPRVSLSGFLGFTAGRGSQIGSSAARAWSVGPSISWAAFDLGSVRARLRGAKADADAALASYEQQVLLALEESANAFSDYGKRQERLVSLVRQSEASRAAAQQAAIRYREGTTDFLVLLDAEREQLSAEDAQAQAEVELYRGIVAIYRSLGGGWQPSAGSHHHHHH

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 \(c\).

\(|P_{f(4LPJ_1)}(2) \setminus P_{f(4XVA_1)}(2)|=55\), \(|P_{f(4XVA_1)}(2) \setminus P_{f(4LPJ_1)}(2)|=115\). 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 1:11001000000000111001101010100000010111000011100110110000000000100110000001101110010010000110010110111010001001110100110001010100110000100010010011011001001111001110011000010000
Pair \(Z_2\) Length of longest common subsequence
4LPJ_1,4XVA_1 170 4
4LPJ_1,5AZP_1 170 5
4XVA_1,5AZP_1 162 4

Newick tree

 
[
	4LPJ_1:86.29,
	[
		4XVA_1:81,5AZP_1:81
	]:5.29
]

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 \((0.85)(0.8)(\frac{469 }{\log_{20} 469}-\frac{176}{\log_{20}176})=85.9\)
Status Protein1 Protein2 d d1/2
Query variables 4LPJ_1 4XVA_1 110 87
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]