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Parikh vectors
6ZUJ_1 8YLA_1 1ODV_1 Letter Amino acid
13 13 5 N Asparagine
12 10 1 H Histidine
25 23 4 I Isoleucine
29 27 5 L Leucine
24 18 4 S Serine
6 1 1 C Cysteine
15 11 4 Q Glutamine
20 26 4 E Glutamic acid
22 18 10 G Glycine
21 17 10 K Lycine
20 14 6 T Threonine
36 19 8 V Valine
10 28 2 R Arginine
16 24 8 D Aspartic acid
7 11 8 F Phenylalanine
20 16 4 P Proline
3 9 1 W Tryptophan
33 30 7 A Alanine
5 11 3 M Methionine
9 13 5 Y Tyrosine 

6ZUJ_1|Chains A[auth AAA], B[auth BBB], C[auth CCC], D[auth DDD]|Serine racemase|Homo sapiens (9606)
>8YLA_1|Chains A, B|Sesterfisherol synthase|Neosartorya fischeri (strain ATCC 1020 / DSM 3700 / CBS 544.65 / FGSC A1164 / JCM 1740 / NRRL 181 / WB 181) (331117)
>1ODV_1|Chains A, B|PHOTOACTIVE YELLOW PROTEIN|ECTOTHIORHODOSPIRA HALOPHILA (1053)
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 \(w=f(c)\)
6ZUJ , Knot 146 346 0.82 40 198 323 
MDAQYDISFADVEKAHINIRDSIHLTPVLTSSILNQLTGRNLFFKCELFQKTGSFKIRGALNAVRSLVPDALERKPKAVVTHSSGNHGQALTYAAKLEGIPAYIVVPQTAPDCKKLAIQAYGASIVYCEPSDESRENVAKRVTEETEGIMVHPNQEPAVIAGQGTIALEVLNQVPLVDALVVPVGGGGMLAGIAITVKALKPSVKVYAAEPSNADDCYQSKLKGKLMPNLYPPETIADGVKSSIGLNTWPIIRDLVDDIFTVTEDEIKCATQLVWERMKLLIEPTAGVGVAAVLSQHFQTVSPEVKNICIVLSGGNVDLTSSITWVKQAERPASYQSVSVHHHHHH
8YLA , Knot 149 339 0.85 40 208 329 
MEVWEHSRPIADDTIKKTPSFTTLPIRINKQNDVADAATRRALRDWDYYLHDGLAERALISISELGNLGAFAYPEVPPERLAIVTYLTDLGILHDDGYEAMDMDQARTEHREFGALFDPHEQLPSRRGTRAAKLKKLVSQILLEAIRIDRDMGMYMFDMYNKGWLSVAGGEGKVPQFKSVEEYQAYRRDDFGIRAFWPMVEFGMAMRLSDEDKKLIEPVMEPIDKAIIWTNDYWSFDREYHESITNGSRLTNVVEVVRQIENKSIDEAKAAVRQLLVNLEQQYLERKRAIYAQNPSIPSHLRKWIEVVGITVAGTHFWASCSPRHHAWRNNSRNGLKPA
1ODV , Knot 52 100 0.79 40 87 98 
LAFGAIQLDGDGNILQYNAAEGDITGRDPKQVIGKNFFKDVAPCTDSPEFYGKFKEGVASGNLNTMFEYTFDYQMTPTKVKVHMKKALSGDSYWVFVKRV

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 \(c\).

\(|P_{f(6ZUJ_1)}(2) \setminus P_{f(8YLA_1)}(2)|=78\), \(|P_{f(8YLA_1)}(2) \setminus P_{f(6ZUJ_1)}(2)|=88\). 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 1:1010001011010010101000101011100011001010011100011000101010111011001110110001011100001001011001101011110111100110000111010110110001000000011001000001111010001111110101110110011110111111111111111110101101010101101001000000010101110101100110110001110011110011001101000010010011100101110101111111110001001010100101110110101000101100100110000101000000
Pair \(Z_2\) Length of longest common subsequence
6ZUJ_1,8YLA_1 166 4
6ZUJ_1,1ODV_1 173 3
8YLA_1,1ODV_1 213 3

Newick tree

 
[
	1ODV_1:10.25,
	[
		6ZUJ_1:83,8YLA_1:83
	]:18.25
]

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 \((0.85)(0.8)(\frac{685 }{\log_{20} 685}-\frac{339}{\log_{20}339})=95.1\)
Status Protein1 Protein2 d d1/2
Query variables 6ZUJ_1 8YLA_1 120 119.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]