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Parikh vectors
3ITE_1 8VIY_1 6TNQ_1 Letter Amino acid
42 49 2 A Alanine
28 36 2 R Arginine
31 28 2 D Aspartic acid
26 23 6 K Lycine
20 32 5 F Phenylalanine
13 36 6 Q Glutamine
32 32 2 I Isoleucine
25 22 3 N Asparagine
31 45 10 E Glutamic acid
47 49 8 G Glycine
14 26 2 H Histidine
7 16 4 W Tryptophan
13 18 3 Y Tyrosine
45 38 4 V Valine
7 11 0 C Cysteine
58 86 5 L Leucine
10 12 2 M Methionine
27 54 3 P Proline
51 46 5 S Serine
35 37 1 T Threonine 

3ITE_1|Chains A, B|SidN siderophore synthetase|Neotyphodium lolii (73839)
>8VIY_1|Chain A|Polyunsaturated fatty acid lipoxygenase ALOX15B|Homo sapiens (9606)
>6TNQ_1|Chains A, C, E|Activity-regulated cytoskeleton-associated protein|Homo sapiens (9606)
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 \(w=f(c)\)
3ITE , Knot 229 562 0.86 40 264 531 
GSGAVSIASSKVSAQYTSTVPPSHYIETWAKTHPEWKAVEVATGFLGSQKIVTEDWTYKKLNETANQVANLIIHASLHGRAIAVSLDRSLIAFAIIVGIMKSGNTYVPIEAGLPNDRKSFLLRDSRAAMAFVCDNNFDGVELPPETKVLDTKNQSFIENLSTQDTSDILNNYPENLDAYLLYTSGSTGTPKGVRVSRHNLSSFSDAWGKLIGNVAPKSLELGGVGKFLCLASRAFDVHIGEMFLAWRFGLCAVTGERLSMLDDLPRTFRELGVTHAGIVPSLLDQTGLVPEDAPHLVYLGVGGEKMTPRTQQIWSSSDRVALVNVYGPTEVTIGCSAGRILPDSDTRCIGHPLGDSVAHVLAPGSNEHVKKGMAGELVIEGSLVANGYLNRPDAKGFCDINGRKMYRTGDIVRMDADSSILFLGRKDEQVKVRGQRLELGEVSEVIRSLSPTDIDVVTLLLNHPGTSKQFLVSFVASSGAAVRGELRWINENYKEINNSLRQACEQTLPAYMVPDFIIPISFIPLRDTSAKTDAKALEHMFHTLSLGELFGESSSLVNKPTT
8VIY , Knot 269 696 0.84 40 296 627 
MGSSHHHHHHSSGLVPRGSHMAEFRVRVSTGEAFGAGTWDKVSVSIVGTRGESPPLPLDNLGKEFTAGAEEDFQVTLPEDVGRVLLLRVHKAPPVLPLLGPLAPDAWFCRWFQLTPPRGGHLLFPCYQWLEGAGTLVLQEGTAKVSWADHHPVLQQQRQEELQARQEMYQWKAYNPGWPHCLDEKTVEDLELNIKYSTAKNANFYLQAGSAFAEMKIKGLLDRKGLWRSLNEMKRIFNFRRTPAAEHAFEHWQEDAFFASQFLNGLNPVLIRRCHYLPKNFPVTDAMVASVLGPGTSLQAELEKGSLFLVDHGILSGIQTNVINGKPQFSAAPMTLLYQSPGCGPLLPLAIQLSQTPGPNSPIFLPTDDKWDWLLAKTWVRNAEFSFHEALTHLLHSHLLPEVFTLATLRQLPHCHPLFKLLIPHTRYTLHINTLARELLIVPGQVLDRSTGIGIEGFSELIQRNMKQLNYSLLCLPEDIRTRGVEDIPGYYYRDDGMQIWGAVERFVSEIIGIYYPSDESVQDDRELQAWVREIFSKGFLNQESSGIPSSLETREALVQYVTMVIFTCSAKHAAVSAGQFDSCAWMPNLPPSMQLPPPTSKGLATCEGFIATLPPVNATCDVILALWLLSKEPGDQRPLGTYPDEHFTEEAPRRSIATFQSRLAQISRGIQERNQGLVLPYTYLDPPLIENSVSI
6TNQ , Knot 42 75 0.80 38 61 70 
GAMGPGVDTQIFEDPREFLSHLEEYLRQVGGSEEYWLSQIQNHMNGPAKKWWEFKQGSVKNWVEFKKEFLQYSEG

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 \(c\).

\(|P_{f(3ITE_1)}(2) \setminus P_{f(8VIY_1)}(2)|=45\), \(|P_{f(8VIY_1)}(2) \setminus P_{f(3ITE_1)}(2)|=77\). 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 1:1011101100010100000111000100110001010110110111100011000100001000100110111010101011110100011111111111001000111011110000011100001111110000101101110001100000011001000000011000100101011000100101011010000100100111011101110010111110110110011010110111110111011010010110011001001110011111011000111100110110111110010100001100000111101011001011001101110000001101110011011111000010011110111010111010100101011001010010001011010100011111000001010100101101001100101001011011100110000111011100111101010110000001000100100001110111011111011110000100010110011001011011100001100100
Pair \(Z_2\) Length of longest common subsequence
3ITE_1,8VIY_1 122 4
3ITE_1,6TNQ_1 247 3
8VIY_1,6TNQ_1 259 3

Newick tree

 
[
	6TNQ_1:14.80,
	[
		3ITE_1:61,8VIY_1:61
	]:80.80
]

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 \((0.85)(0.8)(\frac{1258 }{\log_{20} 1258}-\frac{562}{\log_{20}562})=178.\)
Status Protein1 Protein2 d d1/2
Query variables 3ITE_1 8VIY_1 225 203.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]