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Parikh vectors
2POS_1 3RFF_1 9HZK_1 Letter Amino acid
0 4 7 H Histidine
8 7 4 P Proline
1 2 3 Y Tyrosine
5 20 5 V Valine
1 10 1 R Arginine
4 5 8 N Asparagine
4 10 4 D Aspartic acid
8 9 7 E Glutamic acid
6 10 2 T Threonine
2 0 1 W Tryptophan
3 5 6 I Isoleucine
11 14 4 L Leucine
6 5 10 K Lycine
7 9 2 F Phenylalanine
4 8 11 S Serine
9 14 2 A Alanine
6 2 4 C Cysteine
3 4 3 Q Glutamine
4 12 5 G Glycine
2 7 2 M Methionine 

2POS_1|Chains A, B, C, D|SYLVATICIN|Pythium sylvaticum (82950)
>3RFF_1|Chain A|Phosphopantetheine adenylyltransferase|Mycobacterium tuberculosis (1773)
>9HZK_1|Chain A[auth B]|Circumsporozoite protein|Plasmodium falciparum NF54 (5843)
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 \(w=f(c)\)
2POS , Knot 48 94 0.77 38 78 91 
WEETKECAFTEFFKLAPLASNPALSVCQDASGWQMLPPAGYPTPEQLKLMCGTAECFTLIDAIKALNPNDCILVFGDVRLNVKKLVTEFEPSCF
3RFF , Knot 75 157 0.80 38 113 155 
MTGAVCPGSFDPVTLGHVDIFERAAAQFDEVVVAILVNPAKTGMFDLDERIAMVKESTTHLPNLRVQVGHGLVVDFVRSCGMTAIVKGLRTGTDFEYELQMAQMNKHIAGVDTFFVATAPRYSFVSSSLAKEVAMLGGDVSELLPEPVNRRLRDRLN
9HZK , Knot 50 91 0.82 40 74 86 
MEPSDKHIKEYLNKIQNSLSTEWSPCSVTCGNGIQVRIKPGSANKPKDELDYANDIEKKICKMEKCSSVFNVVNSSENLYFQSGGHHHHHH

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 \(c\).

\(|P_{f(2POS_1)}(2) \setminus P_{f(3RFF_1)}(2)|=48\), \(|P_{f(3RFF_1)}(2) \setminus P_{f(2POS_1)}(2)|=83\). 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 1:1000000110011011111001110100010110111111010100101101010010110110110100011111010101001100101001
Pair \(Z_2\) Length of longest common subsequence
2POS_1,3RFF_1 131 3
2POS_1,9HZK_1 120 3
3RFF_1,9HZK_1 133 3

Newick tree

 
[
	3RFF_1:67.88,
	[
		2POS_1:60,9HZK_1:60
	]:7.88
]

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 \((0.85)(0.8)(\frac{251 }{\log_{20} 251}-\frac{94}{\log_{20}94})=50.3\)
Status Protein1 Protein2 d d1/2
Query variables 2POS_1 3RFF_1 61 50
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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