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Parikh vectors
6RHW_1 7GPW_1 6IJK_1 Letter Amino acid
13 13 14 I Isoleucine
17 13 19 L Leucine
5 6 8 M Methionine
8 7 15 P Proline
7 12 46 A Alanine
11 10 23 R Arginine
29 10 4 N Asparagine
23 12 18 D Aspartic acid
8 0 4 W Tryptophan
18 16 21 V Valine
0 3 2 C Cysteine
18 6 15 E Glutamic acid
32 7 6 K Lycine
13 10 17 F Phenylalanine
27 4 25 S Serine
26 17 16 T Threonine
13 7 3 Y Tyrosine
11 5 8 Q Glutamine
21 18 21 G Glycine
9 6 14 H Histidine

6RHW_1|Chain A[auth G]|Beta-channel forming cytolysin|Staphylococcus aureus (1280)
>7GPW_1|Chains A, B|Protease 3C|Human Enterovirus D68 (42789)
>6IJK_1|Chains A, B|Enoyl-CoA hydratase|Cupriavidus necator (strain ATCC 17699 / H16 / DSM 428 / Stanier 337) (381666)
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 \(w=f(c)\)
6RHW , Knot 135 309 0.83 38 196 301
KINSEIKQVSEKNLDGDTKMYTRTATTSDSQKNITQSLQFNFLTEPNYDKETVFIKAKGTIGSGLRILDPNGYWNSTLRWPGSYSVSIQNVDDNNNTNVTDFAPKNQDESREVKYTYGYKTGGDFSINRGGLTGNITKESNYSETISYQQPSYRTLLDQSTSHKGVGWKVEAHLINNMGHDHTRQLTNDSDNRTKSEIFSLTRNGNLWAKDNFTPKDKMPVTVSEGFNPEFLAVMSHDKKDKGKSQFVVHYKRSMDEFKIDWNRHGFWGYWSGENHVDKKEEKLSALYEVDWKTHNVKFVKVLNDNEKK
7GPW , Knot 86 182 0.82 38 139 179
MGPGFDFAQAIMKKNTVIARTEKGEFTMLGVYDRVAVIPTHASVGEIIYINDVETRVLDACALRDLTDTNLEITIVKLDRNQKFRDIRHFLPRCEDDYNDAVLSVHTSKFPNMYIPVGQVTNYGFLNLGGTPTHRILMYNFPTRAGQCGGVVTTTGKVIGIHVGGNGAQGFAAMLLHSYFTD
6IJK , Knot 127 299 0.80 40 171 281
MTQATPSFETLRYAVADGVATITLHRPDQLNAFTAQMMHELIAAFDATDADDNVRAVIVTGSGRAFCAGADLSGGSSTFDFEKRYGASPDSAHRDGGGRVSLRIFRSLKPVIAAVNGAAVGVGVTMQLPMDIRLASTDAKFGFVFARRGITPEAASSWFLSRVVGISTALEWCYTGRVFSAQEAHERGLVRSLHAPEDLLPAAQAIAREIAANAAPVSVAISRQLIWRMAGASHPMEAHKLDSRAIQSRGRSADVKEGVSAFLEKRPAAFPETVSHDMPDFFDWTSEPPFILEHHHHHH

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 \(c\).

\(|P_{f(6RHW_1)}(2) \setminus P_{f(7GPW_1)}(2)|=118\), \(|P_{f(7GPW_1)}(2) \setminus P_{f(6RHW_1)}(2)|=61\). 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 1:010001001000010100010000100000000100010101100100000011101010110110110101010001011100010100100000001001110000000010000100011010100111010100000000010000100001100000001111010101100110000001000000000001101000101110001010001110100110101111100000001000111000001001010100011110101000100000010110010100001011011000000
Pair \(Z_2\) Length of longest common subsequence
6RHW_1,7GPW_1 179 3
6RHW_1,6IJK_1 193 3
7GPW_1,6IJK_1 172 4

Newick tree

 
[
	6RHW_1:95.30,
	[
		7GPW_1:86,6IJK_1:86
	]:9.30
]

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 \((0.85)(0.8)(\frac{491 }{\log_{20} 491}-\frac{182}{\log_{20}182})=90.1\)
Status Protein1 Protein2 d d1/2
Query variables 6RHW_1 7GPW_1 112 90.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]