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Parikh vectors
5OOL_1 2ZEJ_1 6AHO_1 Letter Amino acid
5 1 17 C Cysteine
8 7 18 M Methionine
9 11 54 T Threonine
23 8 82 E Glutamic acid
4 4 12 H Histidine
22 18 131 L Leucine
21 19 59 K Lycine
6 3 6 W Tryptophan
15 12 88 S Serine
6 4 27 Y Tyrosine
19 11 75 V Valine
25 12 28 R Arginine
16 8 47 Q Glutamine
16 9 22 G Glycine
13 7 49 F Phenylalanine
26 5 29 P Proline
20 13 56 A Alanine
8 8 59 N Asparagine
11 13 70 D Aspartic acid
19 11 80 I Isoleucine 

5OOL_15|Chain O[auth Q]|39S ribosomal protein L19, mitochondrial|Homo sapiens (9606)
>2ZEJ_1|Chains A, B|Leucine-rich repeat kinase 2|Homo sapiens (9606)
>6AHO_1|Chain A|Importin subunit beta-5|Saccharomyces cerevisiae (559292)
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 \(w=f(c)\)
5OOL , Knot 129 292 0.83 40 184 279 
MAACIAAGHWAAMGLGRSFQAARTLLPPPASIACRVHAGPVRQQSTGPSEPGAFQPPPKPVIVDKHRPVEPERRFLSPEFIPRRGRTDPLKFQIERKDMLERRKVLHIPEFYVGSILRVTTADPYASGKISQFLGICIQRSGRGLGATFILRNVIEGQGVEICFELYNPRVQEIQVVKLEKRLDDSLLYLRDALPEYSTFDVNMKPVVQEPNQKVPVNELKVKMKPKPWSKRWERPNFNIKGIRFDLCLTEQQMKEAQKWNQPWLEFDMMREYDTSKIEAAIWKEIEASKRS
2ZEJ , Knot 90 184 0.85 40 142 180 
NRMKLMIVGNTGSGKTTLLQQLMKTKKSDLGMQSATVGIDVKDWPIQIRDKRKRDLVLNVWDFAGREEFYSTHPHFMTQRALYLAVYDLSKGQAEVDAMKPWLFNIKARASSSPVILVGTHLDVSDEKQRKACMSKITKELLNKRGFPAIRDYHFVNATEESDALAKLRKTIINESLNFKIRDQ
6AHO , Knot 371 1009 0.84 40 289 858 
GPLGSMDINELIIGAQSADKHTREVAETQLLQWCDSDASQVFKALANVALQHEASLESRQFALLSLRKLITMYWSPGFESYRSTSNVEIDVKDFIREVLLKLCLNDNENTKIKNGASYCIVQISAVDFPDQWPQLLTVIYDAISHQHSLNAMSLLNEIYDDVVSEEMFFEGGIGLATMEIVFKVLNTETSTLIAKIAALKLLKACLLQMSSHNEYDEASRKSFVSQCLATSLQILGQLLTLNFGNVDVISQLKFKSIIYENLVFIKNDFSRKHFSSELQKQFKIMAIQDLENVTHINANVETTESEPLLETVHDCSIYIVEFLTSVCTLQFSVEEMNKIITSLTILCQLSSETREIWTSDFNTFVSKETGLAASYNVRDQANEFFTSLPNPQLSLIFKVVSNDIEHSTCNYSTLESLLYLLQCILLNDDEITGENIDQSLQILIKTLENILVSQEIPELILARAILTIPRVLDKFIDALPDIKPLTSAFLAKSLNLALKSDKELIKSATLIAFTYYCYFAELDSVLGPEVCSETQEKVIRIINQVSSDAEEDTNGALMEVLSQVISYNPKEPHSRKEILQAEFHLVFTISSEDPANVQVVVQSQECLEKLLDNINMDNYKNYIELCLPSFINVLDSNNANNYRYSPLLSLVLEFITVFLKKKPNDGFLPDEINQYLFEPLAKVLAFSTEDETLQLATEAFSYLIFNTDTRAMEPRLMDIMKVLERLLSLEVSDSAAMNVGPLVVAIFTRFSKEIQPLIGRILEAVVVRLIKTQNISTEQNLLSVLCFLTCNDPKQTVDFLSSFQIDNTDALTLVMRKWIEAFEVIRGEKRIKENIVALSNLFFLNDKRLQKVVVNGNLIPYEGDLIITRSMAKKMPDRYVQVPLYTKIIKLFVSELSFQSKQPNPEQLITSDIKQEVVNANKDDDNDDWEDVDDVLDYDKLKEYIDDDVDEEADDDSDDITGLMDVKESVVQLLVRFFKEVASKDVSGFHCIYETLSDSERKVLSEALL

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 \(c\).

\(|P_{f(5OOL_1)}(2) \setminus P_{f(2ZEJ_1)}(2)|=108\), \(|P_{f(2ZEJ_1)}(2) \setminus P_{f(5OOL_1)}(2)|=66\). 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 1:1110111101111111001011001111110110010111100000110011110111011110000110100011010111001000110101000011000011011010110110100101010101001111010001011110111001101011010101001010010110100010001101001110000101010111001000111001010101011000100101010110101010000100100100111010110000000101111001010000
Pair \(Z_2\) Length of longest common subsequence
5OOL_1,2ZEJ_1 174 4
5OOL_1,6AHO_1 173 5
2ZEJ_1,6AHO_1 195 5

Newick tree

 
[
	2ZEJ_1:94.28,
	[
		5OOL_1:86.5,6AHO_1:86.5
	]:7.78
]

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 \((0.85)(0.8)(\frac{476 }{\log_{20} 476}-\frac{184}{\log_{20}184})=85.3\)
Status Protein1 Protein2 d d1/2
Query variables 5OOL_1 2ZEJ_1 108 88
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]