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Parikh vectors
8JUB_1 9JRI_1 3VPY_1 Letter Amino acid
30 67 4 A Alanine
47 45 6 G Glycine
17 12 4 M Methionine
43 57 9 S Serine
3 13 1 W Tryptophan
15 33 8 Y Tyrosine
29 30 6 D Aspartic acid
19 28 5 Q Glutamine
32 28 17 E Glutamic acid
19 19 12 H Histidine
25 36 5 T Threonine
38 55 4 V Valine
17 44 10 R Arginine
22 17 6 N Asparagine
13 8 3 C Cysteine
19 26 7 I Isoleucine
50 84 12 L Leucine
37 29 10 K Lycine
28 37 5 F Phenylalanine
30 22 11 P Proline 

8JUB_1|Chains A, B, C, D|Glutaminase kidney isoform, mitochondrial|Homo sapiens (9606)
>9JRI_1|Chain A|Soluble cytochrome b562,Reduced folate transporter,fusion protein|Escherichia coli (562)
>3VPY_1|Chain A|FHA domain-containing protein DDL|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
8JUB , Knot 215 533 0.84 40 265 496 
MGHHHHHHGLSSSPSEILQELGKGSTHPQPGVSPPAAPAAPGPKDGPGETDAFGNSEGKELVASGENKIKQGLLPSLEDLLFYTIAEGQEKIPVHKFITALKSTGLRTSDPRLKECMDMLRLTLQTTSDGVMLDKDLFKKCVQSNIVLLTQAFRRKFVIPDFMSFTSHIDELYESAKKQSGGKVADYIPQLAKFSPDLWGVSVCTVDGQRHSTGDTKVPFCLQSCVKPLKYAIAVNDLGTEYVHRYVGKEPSGLRFNKLFLNEDDKPHNPMVNAGAIVVTSLIKQGVNNAEKFDYVMQFLNKMAGNEYVGFSNATFQSERESGDRNFAIGYYLKEKKCFPEGTDMVGILDFYFQLCSIEVTCESASVMAATLANGGFCPITGERVLSPEAVRNTLSLMHSCGMYDFSGQFAFHVGLPAKSGVAGGILLVVPNVMGMMCWSPPLDKMGNSVKGIHFCHDLVSLCNFHNYDNLRHFAKKLDPRREGGDQRHSFGPLDYESLQQELALKETVWKKVSPESNEDISTTVVYRMESLG
9JRI , Knot 265 690 0.83 40 272 629 
MADLEDNWETLNDNLKVIEKADNAAQVKDALTKMRAAALDAQKATPPKLEDKSPDSPEMKDFRHGFDILVGQIDDALKLANEGKVKEAQAAAEQLKTTRNAYIQKYLSWRHLVCYLCFYGFMAQIRPGESFITPYLLGPDKQFTREQVTNEITPVLSYSYLAVLVPVFLLTDYLRYTPVLLLQGLSFVSVWLLLLLGHSVAHMQLMELFYSVTMAARIAYSSYIFSLVRPARYQRVAGYSRAAVLLGVFTSSVLGQLLVTVGRVSFSTLNYISLAFLTFSVVLALFLKRPKRSLFFNRDDRGRCETSASELERMNPGPGGKLGHALRVACGDSVLARMLRELGDSLRRPQLRLWSLWWVFNSAGYYLVVYYVHILWNEVDPTTNSARVYNWAADAASTLLGAITSFAAGFVKIRWARWSKLLIAGVTATQAGLVFLLAHTRHPSSIWLCYAAFVLFRGSYQFLVPIATFQIASSLSKELCALVFGVNTFFATIVKTIITFIVSDVRGLGLPVRKQFQLYSVYFLILSIIYFLGAMLDGLRHAEEEKRKAEEEKRKGSGSQVQLQESGGGLVQPGGSLRLSCAASGRTISRYAMSWFRQAPGKEREFVAVARRSGDGAFYADSVQGRFTVSRDDAKNTVYLQMNSLKPEDTAVYYCAIDSDTFYSGSYDYWGQGTQVTVSSHHHHHHHHHH
3VPY , Knot 69 145 0.79 40 111 136 
TLLFNEPPEARKPSERWRLYVFKDGEPLNEPLCLHRQSCYLFGRERRIADIPTDHPSCSKQHAVIQYREMEKEKPDGMMGKQVKPYIMDLGSTNKTYINESPIEPQRYYELFEKDTIKFGNSSREYVLLHENSAELEHHHHHHHH

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 \(c\).

\(|P_{f(8JUB_1)}(2) \setminus P_{f(9JRI_1)}(2)|=67\), \(|P_{f(9JRI_1)}(2) \setminus P_{f(8JUB_1)}(2)|=74\). 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 1:11000000110001001100110100010111011111111110011100011100010011101000100111101001110011010001110011011000110000101000101101010000011110001100010001111001100011110110100010010001000011011001101101010111101001010000010001110100010110011110011000100011001011010011100000100111011111100110011001001001101100111000111001010000001000111100100000110100111110101010010100001011110110111011010011010110001011000110010101110111110011111111111011111010111001100101101000110100100000100110010100011000001111000010001110001100101000001000110010011
Pair \(Z_2\) Length of longest common subsequence
8JUB_1,9JRI_1 141 6
8JUB_1,3VPY_1 210 6
9JRI_1,3VPY_1 221 8

Newick tree

 
[
	3VPY_1:11.61,
	[
		8JUB_1:70.5,9JRI_1:70.5
	]:47.11
]

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 \((0.85)(0.8)(\frac{1223 }{\log_{20} 1223}-\frac{533}{\log_{20}533})=177.\)
Status Protein1 Protein2 d d1/2
Query variables 8JUB_1 9JRI_1 224 200.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]