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Parikh vectors
6TIL_1 4KCE_1 7LLP_1 Letter Amino acid
7 3 8 C Cysteine
28 13 2 E Glutamic acid
17 11 3 F Phenylalanine
30 14 10 S Serine
31 17 6 V Valine
38 15 12 A Alanine
28 13 6 I Isoleucine
24 15 6 K Lycine
20 5 2 M Methionine
34 11 7 T Threonine
13 7 3 Y Tyrosine
25 10 11 R Arginine
42 14 8 L Leucine
21 8 14 N Asparagine
33 14 7 D Aspartic acid
11 3 3 Q Glutamine
40 18 12 G Glycine
21 9 1 H Histidine
36 11 2 P Proline
9 2 6 W Tryptophan 

6TIL_1|Chain A[auth AAA]|Ferulic acid decarboxylase 1|Aspergillus niger (5061)
>4KCE_1|Chains A, B|Peroxidoxin|Leishmania braziliensis (5660)
>7LLP_1|Chain A|Lysozyme C|Gallus gallus (9031)
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 \(w=f(c)\)
6TIL , Knot 209 508 0.85 40 257 479 
MSAQPAHLCFRSFVEALKVDNDLVEINTPIDPNLEAAAITRRVCETNDKAPLFNNLIGMKNGLFRILGAPGSLRKSSADRYGRLARHLALPPTASMREILDKMLSASDMPPIPPTIVPTGPCKENSLDDSEFDLTELPVPLIHKSDGGKYIQTYGMHIVQSPDGTWTNWSIARAMVHDKNHLTGLVIPPQHIWQIHQMWKKEGRSDVPWALAFGVPPAAIMASSMPIPDGVTEAGYVGAMTGSSLELVKCDTNDLYVPATSEIVLEGTLSISETGPEGPFGEMHGYIFPGDTHLGAKYKVNRITYRNNAIMPMSSCGRLTDETHTMIGSLAAAEIRKLCQQNDLPITDAFAPFESQVTWVALRVDTEKLRAMKTTSEGFRKRVGDVVFNHKAGYTIHRLVLVGDDIDVYEGKDVLWAFSTRCRPGMDETLFEDVRGFPLIPYMGHGNGPAHRGGKVVSDALMPTEYTTGRNWEAADFNQSYPEDLKQKVLDNWTKMGFSNLEHHHHHH
4KCE , Knot 97 213 0.81 40 152 206 
MGSSHHHHHHSSGLVPRGSHMRTATVRDPAPQFSGKAVVDGAIKEINSNDYKGKYIVLFFYPMDFTFVCPTEIIAFSDRYLEFEKLNTQVIAVSCDSEYSHLAWVNTPRKKGGLGEMKIPVLADKSMEIARDYGVLIESAGIALRGLFVIDKKGTLRHSTINDLPVGRNVDEVLRVVEAFQYADENGDAIPCGWTPGKPTLDTKKAGEFFEKN
7LLP , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL

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 \(c\).

\(|P_{f(6TIL_1)}(2) \setminus P_{f(4KCE_1)}(2)|=140\), \(|P_{f(4KCE_1)}(2) \setminus P_{f(6TIL_1)}(2)|=35\). 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 1:1010110101001101101000110100110101011110001000000111100111100111011111101000010001011001111101010011001101001111110111011000001000010100111111000011001000110110010101001011011100000101111110011010011000100011111111111111110011110110011011110100101100000010111000111010101000110111101010111100011100010010000011111000101000000111011110100100000111001111100010111101000010110000011000110111000110010011111001010010011111000001110001100101111110110101110011011001111000001001011010000100100011001001110010000000
Pair \(Z_2\) Length of longest common subsequence
6TIL_1,4KCE_1 175 6
6TIL_1,7LLP_1 209 4
4KCE_1,7LLP_1 164 3

Newick tree

 
[
	6TIL_1:10.71,
	[
		4KCE_1:82,7LLP_1:82
	]:18.71
]

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 \((0.85)(0.8)(\frac{721 }{\log_{20} 721}-\frac{213}{\log_{20}213})=142.\)
Status Protein1 Protein2 d d1/2
Query variables 6TIL_1 4KCE_1 180 125
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]