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Parikh vectors
1HWL_1 6KRB_1 7WUN_1 Letter Amino acid
33 6 1 S Serine
22 4 1 R Arginine
23 3 0 N Asparagine
12 2 0 D Aspartic acid
18 6 0 Q Glutamine
31 2 0 I Isoleucine
24 5 0 K Lycine
12 4 0 Y Tyrosine
33 9 0 E Glutamic acid
42 10 1 G Glycine
42 11 0 L Leucine
20 2 0 P Proline
24 4 0 T Threonine
2 1 0 W Tryptophan
50 3 0 A Alanine
16 3 0 C Cysteine
8 4 0 H Histidine
17 1 0 M Methionine
7 3 0 F Phenylalanine
31 8 0 V Valine 

1HWL_1|Chains A, B, C, D|HMG-COA REDUCTASE|Homo sapiens (9606)
>6KRB_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|Acylphosphatase|Vibrio cholerae serotype O1 (strain ATCC 39541 / Classical Ogawa 395 / O395) (345073)
>7WUN_1|Chain A[auth X]|3-mer peptide|Oryza sativa (4530)
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 \(w=f(c)\)
1HWL , Knot 194 467 0.85 40 238 444 
GAMASSVLVTQEPEIELPREPRPNEECLQILGNAEKGAKFLSDAEIIQLVNAKHIPAYKLETLIETHERGVSIRRQLLSKKLSEPSSLQYLPYRDYNYSLVMGACCENVIGYMPIPVGVAGPLCLDEKEFQVPMATTEGCLVASTNRGCRAIGLGGGASSRVLADGMTRGPVVRLPRACDSAEVKAWLETSEGFAVIKEAFDSTSRFARLQKLHTSIAGRNLYIRFQSRSGDAMGMNMISKGTEKALSKLHEYFPEMQILAVSGNYCTDKKPAAINWIEGRGKSVVCEAVIPAKVVREVLKTTTEAMIEVNINKNLVGSAMAGSIGGYNAHAANIVTAIYIACGQDAAQNVGSSNCITLMEASGPTNEDLYISCTMPSIEIGTVGGGTNLLPQQACLQMLGVQGACKDNPGENARQLARIVCGTVMAGELSLMAALAAGHLVKSHMIHNRSKINLQDLQGACTKKTA
6KRB , Knot 50 91 0.82 40 82 89 
MEKQCSKFIVSGHVQGVGFCYHTSHQGLKLGLTGYAKNLNNGDVEVVACGTPERLEELYLWLQEGPKTASVRQVRRLSSELEHDYQGFEIL
7WUN , Knot 3 3 0.36 6 2 1 
RSG

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 \(c\).

\(|P_{f(1HWL_1)}(2) \setminus P_{f(6KRB_1)}(2)|=180\), \(|P_{f(6KRB_1)}(2) \setminus P_{f(1HWL_1)}(2)|=24\). 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 1:11110011100010101100101000010111010011011001011011010011100100110000011010001100010010010011000000011111000011101111111111101000010111100010111000010011111111000111011001111011010001010111000011111001100000110100100011100101010000101111011001000110010001101011110100000001111011010100110011111011001100000111010100011101111011100101101101101101001100110000101101011000010100011010110111100111001010111101100001100100110110101111010111111110110001100000101001011000001
Pair \(Z_2\) Length of longest common subsequence
1HWL_1,6KRB_1 204 3
1HWL_1,7WUN_1 236 3
6KRB_1,7WUN_1 82 2

Newick tree

 
[
	1HWL_1:12.13,
	[
		6KRB_1:41,7WUN_1:41
	]:84.13
]

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 \((0.85)(0.8)(\frac{558 }{\log_{20} 558}-\frac{91}{\log_{20}91})=138.\)
Status Protein1 Protein2 d d1/2
Query variables 1HWL_1 6KRB_1 178 104.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]