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Parikh vectors
6RYX_1 6XQW_1 8HUZ_1 Letter Amino acid
38 9 7 N Asparagine
62 20 14 G Glycine
12 0 6 M Methionine
14 6 1 W Tryptophan
30 3 5 R Arginine
17 5 2 C Cysteine
25 10 5 Q Glutamine
23 10 2 Y Tyrosine
51 5 11 D Aspartic acid
39 6 7 I Isoleucine
41 16 6 P Proline
53 20 11 V Valine
21 14 13 K Lycine
28 6 10 F Phenylalanine
51 33 6 S Serine
56 20 8 T Threonine
63 12 12 A Alanine
35 6 13 E Glutamic acid
17 4 8 H Histidine
38 18 22 L Leucine 

6RYX_1|Chain A|Kelch domain-containing protein|Colletotrichum graminicola (strain M1.001 / M2 / FGSC 10212) (645133)
>6XQW_1|Chain A[auth H]|MaliM03 Fab Heavy Chain|Homo sapiens (9606)
>8HUZ_1|Chains A, B, C, D, E, F|Prolyl-tRNA synthetase associated domain-containing protein 1|Mus musculus (10090)
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 \(w=f(c)\)
6RYX , Knot 276 714 0.84 40 285 641 
EFAITSCPNNETVWETPIGVKYTLCPGSDYQNGGASLQTVRDIQSSLECAKICDSDARCNRAVYDNVNKACDVKNSTNPMQWAADDRFETIRLTNDLPEGAFISTCSFNETSYRVPETNAEYRICPDTDYTGVNAKVVEGVTTIQACAELCSNTQDCRKSVFDHINNACAIKAAEPATSIFWVQDKQFSTIRLPENIDPAVKGKWGDLIRLPVIPVAAYIVPSYPEPSRLLFFSSWSNDAFSGASGMTQFGDYDFATGAISQRTVTNTHHDMFCPGISQLEDGRILIQGGSDADTVSIYDPATNEFTRGPNMTLARGYQTSCTLSNGKVFTIGGAWSGERVGKNGEVYDPVANAWTYLPGADFRPMLTNDHEGIWREDNHAWLFGWKNGSIFQAGPSKDQHWYGIQGNGTVAKAATRDDDDAMCGVWVMYDAVAGKIFSAGGSPDYTDSPATQRAHITTIGEPNTPAEVERVADMGFPRGFANAVVLPDGQVLVTGGQRMSLVFTNTDGILVAELFNPETREWKQMAPMAVPRNYHSVSILLPDATVFSGGGGMCWVQNVGDSTAGCDKTVDHSDGEIFEPPYLFNEDGSRAARPVISAISADPIKAGATLTFTVEGVEGQGTAALIRLGSVTHSVNSDQRRVPLNVTVSGNEYSATLPDDYGILLPGYYYLFVSTPQGTPSIAKTVHVILGLEQKLISEEDLNSAVDHHHHHH
6XQW , Knot 102 223 0.82 38 141 211 
QVQLQQWGAGLLKPSETLSLTCAVYGGSFSGYYWTWIRQPPGKGLEWIGEINNSGKTNYNPSLKSRVSISIDTSKNQFSLKVTSVTAADTAVYYCARGPQQHLEPPFDYWGHGTLVTVSSASTKGPSVFPLAPSSKSTSGGTAALGCLVKDYFPEPVTVSWNSGALTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYICNVNHKPSNTKVDKKVEPKSC
8HUZ , Knot 85 169 0.86 40 135 166 
MAGSELRAELEQRLGALAIRTEVVEHPEVFTIEEMMPHIQHLKGAHSKNLFLKDKKKKNYWLVTVLHDRQINLNDLGKQLGVGSGNLRFADETAMLEKLKVGQGCATPLSLFCDDGDVKFVLDSAFLEGGHEKVYFHPMTNAATMGLSPEDFLIFVKATGHDPIILNFD

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 \(c\).

\(|P_{f(6RYX_1)}(2) \setminus P_{f(6XQW_1)}(2)|=168\), \(|P_{f(6XQW_1)}(2) \setminus P_{f(6RYX_1)}(2)|=24\). 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 1:011100010000110011110001011000001110100100100010010100001000011000100100100000110111000100101000110111100001000000110001000101000001101011011001010101000000000011001001011011011001111000010010110010111010110110111111110111001010011110010001101101100110001101110000100000011011100100101110110010010100110001001101011010000001001011011111010011001010011101100111101011100000111000001111110010110111000001011010101101100000011011111001111011011101000001100010100110100110100110111101110111110101110110010111000011111011010000100111111100000101111010110111110110011000110000100001011011011000100110111011010110111010101011010101111011010001000000111010101000010110001111110001110010101011001011111000110000100110000000
Pair \(Z_2\) Length of longest common subsequence
6RYX_1,6XQW_1 192 5
6RYX_1,8HUZ_1 206 4
6XQW_1,8HUZ_1 170 4

Newick tree

 
[
	6RYX_1:10.96,
	[
		6XQW_1:85,8HUZ_1:85
	]:18.96
]

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 \((0.85)(0.8)(\frac{937 }{\log_{20} 937}-\frac{223}{\log_{20}223})=194.\)
Status Protein1 Protein2 d d1/2
Query variables 6RYX_1 6XQW_1 248 160
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]