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Parikh vectors
6WSY_1 2XFS_1 5JME_1 Letter Amino acid
8 1 4 C Cysteine
0 12 10 K Lycine
0 24 12 P Proline
2 29 17 T Threonine
0 31 24 D Aspartic acid
0 13 3 H Histidine
0 26 22 S Serine
0 8 12 Y Tyrosine
0 35 17 V Valine
0 37 12 R Arginine
0 3 9 N Asparagine
0 9 11 Q Glutamine
0 33 9 E Glutamic acid
4 41 10 G Glycine
0 16 9 I Isoleucine
0 10 4 W Tryptophan
7 56 12 A Alanine
0 49 17 L Leucine
0 15 5 M Methionine
0 10 11 F Phenylalanine 

6WSY_1|Chain A|DNA (5'-D(*GP*AP*GP*CP*AP*GP*AP*CP*CP*TP*GP*AP*CP*AP*CP*CP*AP*CP*TP*CP*A)-3')|synthetic construct (32630)
>2XFS_1|Chains A, B|ORF12|STREPTOMYCES CLAVULIGERUS (1901)
>5JME_1|Chains A, B, C, D, E|Soluble acetylcholine receptor|Aplysia californica (6500)
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 \(w=f(c)\)
6WSY , Knot 10 21 0.48 8 9 16 
GAGCAGACCTGACACCACTCA
2XFS , Knot 181 458 0.80 40 201 409 
MMKKADSVPTPAEAALAAQTALAADDSPMGDAARWAMGLLTSSGLPRPEDVAARFIPTFAAAGNFAETVREWRSKGPFTVRAYHPVAHKGWVVLSAPAGVRYILSLTLDSSGLIRILTLKPETVIPDMVTWNDVEETLHTPGVQHSVYAVRLTPDGHEVLHASAPERPMPTGAAYKLYLMRALVAEIEKGTVGWDEILTLTPELRSLPTGDMQDLPDGTRVTVRETAHKMIALSDNTGADLVADRLGREVVERSLAAAGHHDPSLMRPFLTSHEVFELGWGDPERRAEWVRQDEAGRRELLEKMAGVMTVRGSDLGATVHQLGIDWHMDAFDVVRVLEGLLQDSGRDTSGTVEEILTAYPGLLIDEERWRRVYFKAGSSPGVMMFCWLLQDHAGISYVLVLRQSADEQRLIGDGLFLRGIGAKIIEAEAKLLSSGERRGAGTAAAGDDRASAGEAARR
5JME , Knot 107 230 0.84 40 165 223 
DYKDDDDKLHSQANLMRLKSDLFNRSPMYPGPTKDDPLTVTLGFTLQDIVKADSSTNEVDLVYYEQQRWKLNSLMWDPNEYGNITDFRTSAADIWTPDITAYSSTRPVQVLSPQIAVVTHDGSVMFIPAQRLSFMCDPTGVDSEEGATCAVKFGSWVYSGFEIDLKTDTDQVDLSSYYASSKYEILSATQTRQVQHYSCCPEPYIDVNLVVKFRERRAGNGFFRNLFDSR

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 \(c\).

\(|P_{f(6WSY_1)}(2) \setminus P_{f(2XFS_1)}(2)|=6\), \(|P_{f(2XFS_1)}(2) \setminus P_{f(6WSY_1)}(2)|=198\). 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 1:111011100011010010001
Pair \(Z_2\) Length of longest common subsequence
6WSY_1,2XFS_1 204 3
6WSY_1,5JME_1 162 3
2XFS_1,5JME_1 174 4

Newick tree

 
[
	2XFS_1:98.96,
	[
		6WSY_1:81,5JME_1:81
	]:17.96
]

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 \((0.85)(0.8)(\frac{479 }{\log_{20} 479}-\frac{21}{\log_{20}21})=144.\)
Status Protein1 Protein2 d d1/2
Query variables 6WSY_1 2XFS_1 179 92.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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