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Parikh vectors
2KAV_1 3HNZ_1 7PXT_1 Letter Amino acid
5 18 7 R Arginine
6 54 18 G Glycine
12 34 20 S Serine
1 10 10 Y Tyrosine
1 4 2 C Cysteine
12 29 11 L Leucine
5 17 3 K Lycine
7 15 20 P Proline
1 2 6 W Tryptophan
3 23 16 T Threonine
8 16 8 H Histidine
6 28 16 I Isoleucine
7 14 1 M Methionine
11 54 22 A Alanine
2 16 17 N Asparagine
11 20 16 D Aspartic acid
3 10 8 Q Glutamine
13 22 5 E Glutamic acid
9 14 6 F Phenylalanine
6 27 23 V Valine 

2KAV_1|Chain A|Sodium channel protein type 2 subunit alpha|Homo sapiens (9606)
>3HNZ_1|Chain A|3-oxoacyl-[acyl-carrier-protein] synthase 2|Escherichia coli (83333)
>7PXT_1|Chain A|Auxiliary activity 9|Lentinus similis (292560)
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 \(w=f(c)\)
2KAV , Knot 62 129 0.77 40 98 122 
MGSSHHHHHHSSGLVPRGSHMASENFSVATEESAEPLSEDDFEMFYEVWEKFDPDATQFIEFAKLSDFADALDPPLLIAKPNKVQLIAMDLPMVSGDRIHCLDILFAFTKRVLGESGEMDALRIQMEER
3HNZ , Knot 174 427 0.82 40 218 396 
MRGSHHHHHHGSACVSKRRVVVTGLGMLSPVGNTVESTWKALLAGQSGISLIDHFDTSAYATKFAGLVKDFNCEDIISRKEQRKMDAFIQYGIVAGVQAMQDSGLEITEENATRIGAAIGSGIGGLGLIEENHTSLMNGGPRKISPFFVPSTIVNMVAGHLTIMYGLRGPSISIATAATSGVHNIGHAARIIAYGDADVMVAGGAEKASTPLGVGGFGAARALSTRNDNPQAASRPWDKERDGFVLGDGAGMLVLEEYEHAKKRGAKIYAELVGFGMSSDAYHMTSPPENGAGAALAMANALRDAGIEASQIGYVNAHGTSTPAGDKAEAQAVKTIFGEAASRVLVSSTKSMTGHLLGAAGAVESIYSILALRDQAVPPTINLDNPDEGCDLDFVPHEARQVSGMEYTLCNSFGFGGTNGSLIFKKI
7PXT , Knot 105 235 0.81 40 154 226 
HTLVWGVWVNGVDQGDGRNIYIRSPPNNNPVKNLTSPDMTCNVDNRVVPKSVPVNAGDTLTFEWYHNTRDDDIIASSHHGPIAVYIAPAASNGQGNVWVKLFEDAYNVTNSTWAVDRLITAHGQHSVVVPHVAPGDYLFRAEIIALHEADSLYSQNPIRGAQFYISCAQITINSSDDSTPLPAGVPFPGAYTDSTPGIQFNIYTTPATSYVAPPPSVWSGALGGSIAQVGDASLE

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 \(c\).

\(|P_{f(2KAV_1)}(2) \setminus P_{f(3HNZ_1)}(2)|=34\), \(|P_{f(3HNZ_1)}(2) \setminus P_{f(2KAV_1)}(2)|=154\). 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 1:110000000000111101001100010110000101100001011001100101010011011010011011011111101001011110111101001001011111000111001010110101000
Pair \(Z_2\) Length of longest common subsequence
2KAV_1,3HNZ_1 188 7
2KAV_1,7PXT_1 174 4
3HNZ_1,7PXT_1 170 4

Newick tree

 
[
	2KAV_1:92.34,
	[
		7PXT_1:85,3HNZ_1:85
	]:7.34
]

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 \((0.85)(0.8)(\frac{556 }{\log_{20} 556}-\frac{129}{\log_{20}129})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 2KAV_1 3HNZ_1 159 102.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]