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Parikh vectors
2VIG_1 3WVN_1 9EGQ_1 Letter Amino acid
25 24 20 I Isoleucine
16 16 11 M Methionine
26 30 24 E Glutamic acid
17 10 23 Q Glutamine
15 35 20 P Proline
15 42 28 V Valine
16 40 16 D Aspartic acid
16 39 22 R Arginine
40 54 56 L Leucine
22 16 11 K Lycine
21 33 19 T Threonine
9 13 20 Y Tyrosine
55 62 22 A Alanine
5 6 5 C Cysteine
33 40 20 G Glycine
5 19 7 H Histidine
14 14 28 F Phenylalanine
25 31 24 S Serine
5 5 12 W Tryptophan
11 12 18 N Asparagine 

2VIG_1|Chains A, B, C, D, E, F, G, H|SHORT-CHAIN SPECIFIC ACYL-COA DEHYDROGENASE,|HOMO SAPIENS (9606)
>3WVN_1|Chains A, B|Non-ribosomal peptide synthetase|Streptomyces halstedii (1944)
>9EGQ_1|Chains A, B, C, D, E|Bestrophin-1|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 \(w=f(c)\)
2VIG , Knot 164 391 0.83 40 206 372 
MQSVELPETHQMLLQTCRDFAEKELFPIAAQVDKEHLFPAAQVKKMGGLGLLAMDVPEELGGAGLDYLAYAIAMEEISRGCASTGVIMSVNNSLYLGPILKFGSKEQKQAWVTPFTSGDKIGCFALSEPGNGSDAGAASTTARAEGDSWVLNGTKAWITNAWEASAAVVFASTDRALQNKSISAFLVPMPTPGLTLGKKEDKLGIRGSSTANLIFEDCRIPKDSILGEPGMGFKIAMQTLDMGRIGIASQALGIAQTALDCAVNYAENRMAFGAPLTKLQVIQFKLADMALALESARLLTWRAAMLKDNKKPFIKEAAMAKLAASEAATAISHQAIQILGGMGYVTEMPAERHYRDARITEIYEGTSEIQRLVIAGHLLRSYRSAENLYFQ
3WVN , Knot 214 541 0.83 40 241 501 
MNHKVHHHHHHIEGRHMYSPDRRAALNSVANMVSDNADKDLRYGGLVHDLLADSGKATPNSDAMEDAFGTWTYQELLNHSQAFSAWLDGKGVARGERIVVQLPNIRQTVAVFYGACRRGVVFVPLNPGMKPFHLRSVIADADPRLVIAEDETAADRLRDVTDLPVYSIDSLWADVERLRDAGAGAEAVEVSPEDLAVLIYTSGSTAAPKAVACPHQQIVFAASSINAVLGYHAEDIVFCRMSVSWDFGLYKVLISTLTGAKLVLAGGEPDIALVKSLRESGATMMPIVPSLASMLTTLIRRDPEGAPTLRMFTNSAAALPQVTIDALRSAFPGAQVVRMYGQTECKRISIMPPHLEHERPDSVGLPLPGTTIEILDEDGTLLPPGEPGEITVTGPHVMAGYWRAPEITARAYRRDETTGAMRLHTGDYGHLDEDGFLYFGGRRDDMFKRKGTRMSTVEIEAAALDIPGVTAAVALPPTATRDLALCVASDLEPHDVLRSLAERLEPAKVPATCRIVNDFPLTLNGKSERKQLARLLDGSDK
9EGQ , Knot 173 406 0.85 40 237 391 
TITYTSQVANARLGSFSRLLLCWRGSIYKLLYGEFLIFLLCYYIIRFIYRLALTEEQQLMFEKLTLYCDSYIQLIPISFVLGFYVTLVVTRWWNQYENLPWPDRLMSLVSGFVEGKDEQGRLLRRTLIRYANLGNVLILRSVSTAVYKRFPSAQHLVQAGFMTPAEHKQLEKLSLPHNMFWVPWVWFANLSMKAWLGGRIRDPILLQSLLNEMNTLRTQCGHLYAYDWISIPLVYTQVVTVAVYSFFLTCLVGRQFLNPAKAYPGHELDLVVPVFTFLQFFFYVGWLKVAEQLINPFGEDDDDFETNWIVDRNLQVSLLAVDEMHQDLPRMEPDMYWNKPEPQPPYTAASAQFRRASFMGSTFNISLNKEEMEFQPNQEDEEDAHAGIIGRFLGLQSSNSLEVLFQ

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 \(c\).

\(|P_{f(2VIG_1)}(2) \setminus P_{f(3WVN_1)}(2)|=58\), \(|P_{f(3WVN_1)}(2) \setminus P_{f(2VIG_1)}(2)|=93\). 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 1:1001011000011100000110001111110100001111101001111111110110011111100110111100100101001111010001011111011000000111011001001101110011010011110001010100111010011100110101111110000110000101111111011101100000111010001011100001100011101111101110010110111100111110011001100100011111110010110101101111100101101011110000011100111101110011011000110111111010011100000010100100100010011111011000001001010
Pair \(Z_2\) Length of longest common subsequence
2VIG_1,3WVN_1 151 5
2VIG_1,9EGQ_1 175 3
3WVN_1,9EGQ_1 166 4

Newick tree

 
[
	9EGQ_1:88.29,
	[
		2VIG_1:75.5,3WVN_1:75.5
	]:12.79
]

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 \((0.85)(0.8)(\frac{932 }{\log_{20} 932}-\frac{391}{\log_{20}391})=144.\)
Status Protein1 Protein2 d d1/2
Query variables 2VIG_1 3WVN_1 178 152.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]