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Parikh vectors
4HJY_1 7GVW_1 2ZZK_1 Letter Amino acid
24 8 28 A Alanine
9 9 27 R Arginine
10 4 28 Q Glutamine
9 6 40 E Glutamic acid
15 5 37 G Glycine
12 9 56 I Isoleucine
3 7 39 F Phenylalanine
12 7 39 N Asparagine
0 5 11 C Cysteine
7 6 16 M Methionine
13 3 30 P Proline
15 9 58 S Serine
4 0 5 W Tryptophan
9 10 40 V Valine
9 8 42 D Aspartic acid
13 3 15 H Histidine
11 4 52 K Lycine
7 7 37 T Threonine
8 3 24 Y Tyrosine
16 15 71 L Leucine 

4HJY_1|Chains A, B|Endo-type membrane-bound lytic murein transglycosylase A|Escherichia coli (83333)
>7GVW_1|Chain A|B-cell lymphoma 6 protein|Homo sapiens (9606)
>2ZZK_1|Chains A, B|Leucine carboxyl methyltransferase 2|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
4HJY , Knot 94 206 0.81 38 153 194 
MHHHHHHHHHHGENLYFQGSSKHDYTNPPWNAKVPVQRAMQWMPISQKAGAAWGVDPQLITAIIAIQSGGNPNAVSKSNAIGLMQLKASTSGRDVYRRMGWSGEPTTSELKNPERNISMGAAYLNILETGPLAGIEDPKVLQYALVVSYANGAGALLRTFSSDRKKAISKINDLDADEFLEHVARNHPAPQAPRYIYKLEQALDAM
7GVW , Knot 65 128 0.82 38 98 125 
GPGADSCIQFTRHASDVLLNLNRLRSRDILTDVVIVVSREQFRAHKTVLMACSGLFYSIFTDQLKCNLSVINLDPEINPEGFCILLDFMYTSRLNLREGNIMAVMATAMYLQMEHVVDTCRKFIKASE
2ZZK , Knot 268 695 0.84 40 286 643 
MKNLTTIKQTNKNVKQERRKKYADLAIQGTNNSSIASKRSVELLYLPKLSSANNFQMDKNNKLLEYFKFFVPKKIKRSPCINRGYWLRLFAIRSRLNSIIEQTPQDKKIVVVNLGCGYDPLPFQLLDTNNIQSQQYHDRVSFIDIDYSDLLKIKIELIKTIPELSKIIGLSEDKDYVDDSNVDFLTTPKYLARPCDLNDSKMFSTLLNECQLYDPNVVKVFVAEVSLAYMKPERSDSIIEATSKMENSHFIILEQLIPKGPFEPFSKQMLAHFKRNDSPLQSVLKYNTIESQVQRFNKLGFAYVNVGDMFQLWESADEATKKELLKVEPFDELEEFHLFCHHYVLCHATNYKEFAFTQGFLFDRSISEINLTVDEDYQLLECECPINRKFGDVDVAGNDVFYMGGSNPYRVNEILQLSIHYDKIDMKNIEVSSSEVPVARMCHTFTTISRNNQLLLIGGRKAPHQGLSDNWIFDMKTREWSMIKSLSHTRFRHSACSLPDGNVLILGGVTEGPAMLLYNVTEEIFKDVTPKDEFFQNSLVSAGLEFDPVSKQGIILGGGFMDQTTVSDKAIIFKYDAENATEPITVIKKLQHPLFQRYGSQIKYITPRKLLIVGGTSPSGLFDRTNSIISLDPLSETLTSIPISRRIWEDHSLMLAGFSLVSTSMGTIHIIGGGATCYGFGSVTNVGLKLIAIAK

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 \(c\).

\(|P_{f(4HJY_1)}(2) \setminus P_{f(7GVW_1)}(2)|=110\), \(|P_{f(7GVW_1)}(2) \setminus P_{f(4HJY_1)}(2)|=55\). 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 1:10000000000100101010000000011101011100110111100011111110101101111100110101100001111101010001001000111010100001001000101111010110011111100101100111100101111110010000001100100101001100110001110110010010011011
Pair \(Z_2\) Length of longest common subsequence
4HJY_1,7GVW_1 165 3
4HJY_1,2ZZK_1 197 3
7GVW_1,2ZZK_1 226 4

Newick tree

 
[
	2ZZK_1:11.74,
	[
		4HJY_1:82.5,7GVW_1:82.5
	]:30.24
]

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 \((0.85)(0.8)(\frac{334 }{\log_{20} 334}-\frac{128}{\log_{20}128})=63.3\)
Status Protein1 Protein2 d d1/2
Query variables 4HJY_1 7GVW_1 82 65
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]