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Parikh vectors
1WLX_1 4JME_1 8YYP_1 Letter Amino acid
5 1 11 N Asparagine
9 2 6 K Lycine
5 30 34 P Proline
9 24 19 S Serine
6 16 18 T Threonine
2 5 7 Y Tyrosine
7 18 27 D Aspartic acid
6 10 9 H Histidine
11 22 56 L Leucine
12 25 49 A Alanine
5 21 35 R Arginine
15 15 29 E Glutamic acid
3 38 28 G Glycine
11 4 17 I Isoleucine
3 7 6 M Methionine
4 29 26 V Valine
0 4 2 C Cysteine
11 10 14 Q Glutamine
3 14 13 F Phenylalanine
2 7 6 W Tryptophan 

1WLX_1|Chain A|Alpha-actinin 4|Homo sapiens (9606)
>4JME_1|Chains A, B|Putative uncharacterized protein mppR|Streptomyces hygroscopicus (1912)
>8YYP_1|Chains A, B|PtmB|Kitasatospora mediocidica KCTC 9733 (1449347)
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 \(w=f(c)\)
1WLX , Knot 68 129 0.85 38 105 126 
GSTEKQLEAIDQLHLEYAKRAAPFNNWMESAMEDLQDMFIVHTIEEIEGLISAHDQFKSTLPDADREREAILAIHKEAQRIAESNHIKLSGSNPYTTVTPQIINSKWEKVQQLVPKRDHALLEEQSKQQ
4JME , Knot 129 302 0.81 40 170 284 
MVRQHAGPRDRSEIVTTSTGTNGRHTVAGPGSAGPVGYSLPLSPTGESAMLTPPPWHFSGEVVMVDYRVDPDAARRFLPPGLEPGADPGAAAAVFATWQWCSQDGAELTDPGRCQFGEFLILLSCEFEGRPMARCPYAWVDQAVPMMRGWVQGMPKQFGVIHQSRPVTVGKAGSRLAPGGRFDGALSVHGRRVVEASVTVDRSTDQPPALHDVPLAHTLVFPEWVPSGGGPRPRLVASEVSDVEFSPIWTGSGDLTFFDGLGDDFGALAPLEVGSGHVFSYGETLHGGRLLSDYSVSERHQP
8YYP , Knot 169 412 0.82 40 204 382 
MTTTAQRDPVLPGTAHHPPLPDFPLSRRGDILPAEAERLRAEQPVARVRTMTGDEAWLVSSYELAKQVLEDDRFSLKDTANPGVPRQYALTIPPEVVNNMGNINSAGLRNAVMKTLSPQADRELGGWLEAQAHQLLDRLIEQGPPADLRDGFTEPYSAALHCRLLGIPTDDWRRLMSGIDVAFITSPRTFEGSAVNWYKDLGYMVDRLNADPEPTEGLLGRFAELRRSPDVSDQVSDELLATVALSLFGAGAVSTSAFLQHAIIALAQQPELADRLRAEPAVIGRAVDELLRYNLSIGDALPRIALADVRLGEVEIRAGELVLVLIEGANYDPAVFPHPERIDFDRESNPHLAFGGGQHFCPASALGRTHAEIALTALVEKLPALRLALPVEQLAWRPGFIKRLPERLPVLW

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 \(c\).

\(|P_{f(1WLX_1)}(2) \setminus P_{f(4JME_1)}(2)|=53\), \(|P_{f(4JME_1)}(2) \setminus P_{f(1WLX_1)}(2)|=118\). 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 1:100000101100101001001111001100110010011110010010111010001000110100000111110001001100001010100100010101100010010011100001110000000
Pair \(Z_2\) Length of longest common subsequence
1WLX_1,4JME_1 171 4
1WLX_1,8YYP_1 171 4
4JME_1,8YYP_1 146 3

Newick tree

 
[
	1WLX_1:89.27,
	[
		4JME_1:73,8YYP_1:73
	]:16.27
]

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 \((0.85)(0.8)(\frac{431 }{\log_{20} 431}-\frac{129}{\log_{20}129})=90.6\)
Status Protein1 Protein2 d d1/2
Query variables 1WLX_1 4JME_1 114 82
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]