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Parikh vectors
3SFU_1 4UIK_1 1ALX_1 Letter Amino acid
7 4 0 C Cysteine
20 9 0 Q Glutamine
33 7 0 E Glutamic acid
49 15 4 L Leucine
26 11 0 K Lycine
27 20 4 V Valine
42 16 2 A Alanine
13 6 0 N Asparagine
35 17 1 G Glycine
20 4 0 M Methionine
27 26 0 T Threonine
37 5 0 R Arginine
15 3 0 H Histidine
16 5 0 F Phenylalanine
36 14 0 P Proline
30 33 0 S Serine
36 6 0 D Aspartic acid
20 3 0 I Isoleucine
10 6 3 W Tryptophan
18 13 1 Y Tyrosine 

3SFU_1|Chains A, B, C|RNA polymerase|Murine norovirus 1 (223997)
>4UIK_1|Chain A[auth H]|FAB 314.1|MUS MUSCULUS (10090)
>1ALX_1|Chain A|GRAMICIDIN A|BREVIBACILLUS BREVIS (1393)
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 \(w=f(c)\)
3SFU , Knot 215 517 0.86 40 264 480 
MLPRPSGTYAGLPIADYGDAPPLSTKTMFWRTSPEKLPPGAWEPAYLGSKDERVDGPSLQQVMRDQLKPYSEPRGLLPPQEILDAVCDAIENRLENTLEPQKPWTFKKACESLDKNTSSGYPYHKQKSKDWTGSAFIGDLGDQATHANNMYEMGKSMRPIYTAALKDELVKPDKIYGKIKKRLLWGSDLGTMIRAARAFGPFCDALKETCIFNPIRVGMSMNEDGPFIFARHANFRYHMDADYTRWDSTQQRAILKRAGDIMVRLSPEPDLARVVMDDLLAPSLLDVGDYKIVVEEGLPSGCPCTTQLNSLAHWILTLCAMVEVTRVDPDIVMQESEFSFYGDDEVVSTNLELDMVKYTMALRRYGLLPTRADKEEGPLERRQTLQGISFLRRAIVGDQFGWYGRLDRASIDRQLLWTKGPNHQNPFETLPGHAQRPSQLMALLGEAAMHGEKYYRTVASRVSKEAAQSGIEMVVPRHRSVLRWVRFGTMDAETPQERSAVFVNEDAAALEHHHHHH
4UIK , Knot 99 223 0.80 40 140 210 
AVQLQESGTVLARPGASVKMSCKASGYTFTSYWMHWVKQRPGQGLEWIGAIYPGNSDTSYNQKFKGKAKLTAVTSTSTAYMELSSLTNEDSAVYYCTRERGLYYGSSSFDYWGQGTTLTVSSAETTPPSVYPLAPGAAQTNSSMVTLGCLVKGYFPEPVTVTWNSGSLSSGVHTFPAVLQSDLYTLSSSVTVPSSTWPSQTVTCNVAHPASSTKVDKKIVPRD
1ALX , Knot 9 16 0.52 14 12 14 
VGALAVVVWLYLWLWX

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 \(c\).

\(|P_{f(3SFU_1)}(2) \setminus P_{f(4UIK_1)}(2)|=161\), \(|P_{f(4UIK_1)}(2) \setminus P_{f(3SFU_1)}(2)|=37\). 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 1:1110101001111110010111100001110001001111110110110000010110100110001010001011111001101100110001000101001101001000100000010100000000101011110110010010010011001011001110001101001010100011110011011011011111001100001101101110100011111100101000101000010000001110011011101010101101110011110110110001110011101010000100110111010111010010101110000101010001100010101100011100011110010000111000001011011001111001110101001010001110011000011001110100100111111011101000000110010001100110111100001101101101010010000111100011110000000
Pair \(Z_2\) Length of longest common subsequence
3SFU_1,4UIK_1 198 3
3SFU_1,1ALX_1 260 2
4UIK_1,1ALX_1 144 2

Newick tree

 
[
	3SFU_1:12.77,
	[
		4UIK_1:72,1ALX_1:72
	]:54.77
]

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 \((0.85)(0.8)(\frac{740 }{\log_{20} 740}-\frac{223}{\log_{20}223})=144.\)
Status Protein1 Protein2 d d1/2
Query variables 3SFU_1 4UIK_1 189 131.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]