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Parikh vectors
3UDL_1 5JIW_1 1RWD_1 Letter Amino acid
38 41 0 R Arginine
14 31 2 F Phenylalanine
9 21 1 W Tryptophan
38 29 3 V Valine
52 55 3 A Alanine
26 17 7 D Aspartic acid
18 10 0 Q Glutamine
33 45 5 G Glycine
12 10 0 M Methionine
30 38 5 P Proline
23 18 3 Y Tyrosine
18 10 1 N Asparagine
19 1 4 C Cysteine
19 13 0 H Histidine
58 54 1 L Leucine
38 15 1 T Threonine
26 50 6 E Glutamic acid
23 11 4 I Isoleucine
32 17 5 K Lycine
50 14 2 S Serine 

3UDL_1|Chains A, B, C, D|HCV NS5B polymerase|Hepatitis C virus subtype 1b (31647)
>5JIW_1|Chain A|4-alpha-glucanotransferase|Thermus aquaticus (271)
>1RWD_1|Chain A|Rubredoxin|Pyrococcus furiosus (2261)
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 \(w=f(c)\)
3UDL , Knot 231 576 0.85 40 277 535 
HHHHHHSMSYTWTGALITPCAAEESKLPINALSNSLLRHHNMVYATTSRSAGLRQKKVTFDRLQVLDDHYRDVLKEMKAKASTVKAKLLSVEEACKLTPPHSAKSKFGYGAKDVRNLSSKAVNHIHSVWKDLLEDTVTPIDTTIMAKNEVFCVQPEKGGRKPARLIVFPDLGVRVCEKMALYDVVSTLPQVVMGSSYGFQYSPGQRVEFLVNTWKSKKNPMGFSYDTRCFDSTVTENDIRVEESIYQCCDLAPEARQAIKSLTERLYIGGPLTNSKGQNCGYRRCRASGVLTTSCGNTLTCYLKASAACRAAKLQDCTMLVNGDDLVVICESAGTQEDAASLRVFTEAMTRYSAPPGDPPQPEYDLELITSCSSNVSVAHDASGKRVYYLTRDPTTPLARAAWETARHTPVNSWLGNIIMYAPTLWARMILMTHFFSILLAQEQLEKALDCQIYGACYSIEPLDLPQIIERLHGLSAFSLHSYSPGEINRVASCLRKLGVPPLRVWRHRARSVRARLLSQGGRAATCGKYLFNWAVKTKLKLTPIPAASQLDLSGWFVAGYSGGDIYHSLSRARPR
5JIW , Knot 198 500 0.82 40 228 444 
MELPRAFGLLLHPTSLPGPYGVGVLGREARDFLRFLKEAGGRYWQVLPLGPTGYGDSPYQSFSAFAGNPYLIDLRPLAERGYVRLEDPGFPQGRVDYGLLYAWKWPALKEAFRGFKEKASPEEREAFAAFREREAWWLEDYALFMALKGAHGGLPWNRWPLPLRKREEKALREAKSALAEEVAFHAFTQWLFFRQWGALKAEAEALGIRIIGDMPIFVAEDSAEVWAHPEWFHLDEEGRPTVVAGVPPDYFSETGQRWGNPLYRWDVLEREGFSFWIRRLEKALELFHLVRIAHFRGFEAYWEIPASCPTAVEGRWVKAPGEKLFQKIQEVFGEVPVLAEDLGVITPEVEALRDRFGLPGMKVLQFAFDNGMENPFLPHNYPAHGRVVVYTGTHNNDTTLGWYRTATPHEKAFMARYLADWGITFREEEEVPWALMHLGMKSVARLAVYPVQDVLALGSEARMNYPGRPSGNWAWRLLPGELSPEHGARLRAMAEATERL
1RWD , Knot 30 53 0.75 32 46 50 
AKYVCKICGYIYDEDAGDPDNGVSPGTKFEEIPDDWVCPICGAPKSEFEKLED

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 \(c\).

\(|P_{f(3UDL_1)}(2) \setminus P_{f(5JIW_1)}(2)|=115\), \(|P_{f(5JIW_1)}(2) \setminus P_{f(3UDL_1)}(2)|=66\). 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 1:000000010001011110101100001110110001100001101000001110000101001011000000110010101001010110100100101100100011011001001000110010011001100010110001110001101010011001101111101110100011100110011011110001100011001011100100000111100000010001000010100010000011101001100100010111110000100010000010111000010010001010110011010000111010011110001100001101011001100001111011010001011000000101100101001001000100111011100100011001110111011011101111001101111000100110001011000101101101100101101101000011010011001001111110110001001010110011011001001101110001010111110010101111110011010001001010
Pair \(Z_2\) Length of longest common subsequence
3UDL_1,5JIW_1 181 5
3UDL_1,1RWD_1 251 3
5JIW_1,1RWD_1 212 3

Newick tree

 
[
	1RWD_1:12.53,
	[
		3UDL_1:90.5,5JIW_1:90.5
	]:33.03
]

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 \((0.85)(0.8)(\frac{1076 }{\log_{20} 1076}-\frac{500}{\log_{20}500})=150.\)
Status Protein1 Protein2 d d1/2
Query variables 3UDL_1 5JIW_1 194 180.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]