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Parikh vectors
4GYI_1 5AKV_1 2ISG_1 Letter Amino acid
14 5 31 F Phenylalanine
14 8 19 P Proline
26 11 22 S Serine
20 12 30 T Threonine
2 2 2 W Tryptophan
9 3 38 N Asparagine
9 0 10 Q Glutamine
12 4 6 H Histidine
12 5 23 Y Tyrosine
1 1 2 C Cysteine
26 10 26 G Glycine
26 5 29 I Isoleucine
34 7 34 L Leucine
31 8 35 K Lycine
35 12 19 A Alanine
28 4 13 R Arginine
35 5 24 D Aspartic acid
13 1 6 M Methionine
28 12 26 E Glutamic acid
22 12 26 V Valine 

4GYI_1|Chain A|Rio2 kinase|Chaetomium thermophilum (759272)
>5AKV_1|Chains A, B|TRANSTHYRETIN|Homo sapiens (9606)
>2ISG_1|Chains A, B|Neurotoxin BoNT/A|Clostridium botulinum (1491)
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 \(w=f(c)\)
4GYI , Knot 171 397 0.86 40 224 378 
SMKLDTRAMRHLTAEDWRVLTAVEMGSKNHEIVPTPLIEKIARLRGGSSGVHKSIATLAKAGLIARMKEAKYDGYRLTYGGLDYLALHTHAARKDVYSVGSRIGVGKESDIMIVADEKGKQKVLKIHRLGRISFRTVKANRDYLRNRSTGSWMYLSRLAAIKEFAFMKALYEEGFPVPEPIAQSRHTIVMSLVDALPMRQVSSVPDPASLYADLIALILRLAKHGLIHGDFNEFNILIREEKDAEDPSSITLTPIIIDFPQMVSMDHPNAEMYFDRDVQCIKRFFERRFHFVSTTPGPFYKDAKKTVGKDGAKRLDAALEASGFTKKMAKDLEAAIREQQESRNDEEDSDDYEEDSDKEKASNEDTDVDDGRLQVIGDHVQLGTEGDLKKLTINDGT
5AKV , Knot 65 127 0.82 38 101 125 
GPTGTGESKCPLMVKVLDAVRGSPAINVAVHVFRKAADDTWEPFASGKTSESGELHGLTTEEEFVEGIYKVEIDTKSYWKALGISPFHEHAEVVFTANDSGPRRYTIAALLSPYSYSTTAVVTNPKE
2ISG , Knot 170 421 0.81 40 210 390 
PFVNKQFNYKDPVNGVDIAYIKIPNAGQMQPVKAFKIHNKIWVIPERDTFTNPEEGDLNPPPEAKQVPVSYYDSTYLSTDNEKDNYLKGVTKLFERIYSTDLGRMLLTSIVRGIPFWGGSTIDTELKVIDTNCINVIQPDGSYRSEELNLVIIGPSADIIQFECKSFGHEVLNLTRNGYGSTQYIRFSPDFTFGFEESLEVDTNPLLGAGKFATDPAVTLAHELIHAGHRLYGIAINPNRVFKVNTNAYYEMSGLEVSFEELRTFGGHDAKFIDSLQENEFRLYYYNKFKDIASTLNKAKSIVGTTASLQYMKNVFKEKYLLSEDTSGKFSVDKLKFDKLYKMLTEIYTEDNFVKFFKVLNRKTYLNFDKAVFKINIVPKVNYTIYDGFNLRNTNLAANFNGQNTEINNMNFTKLKNFTPG

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 \(c\).

\(|P_{f(4GYI_1)}(2) \setminus P_{f(5AKV_1)}(2)|=153\), \(|P_{f(5AKV_1)}(2) \setminus P_{f(4GYI_1)}(2)|=30\). 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 1:0101000110010100101101101100000111011100110101100110001101101111101001000100100111001110001100010011001111000011111000100011010011010100101000010000010110100111100111101100011111011100000111011011110010011011010101111110110011101010010111000001001001010111101101101001010101000100100110001011000111100010001100110010111010110001100101110000000000000000000000001000000100101011100101100101001010010
Pair \(Z_2\) Length of longest common subsequence
4GYI_1,5AKV_1 183 3
4GYI_1,2ISG_1 144 4
5AKV_1,2ISG_1 173 3

Newick tree

 
[
	5AKV_1:94.03,
	[
		4GYI_1:72,2ISG_1:72
	]:22.03
]

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 \((0.85)(0.8)(\frac{524 }{\log_{20} 524}-\frac{127}{\log_{20}127})=117.\)
Status Protein1 Protein2 d d1/2
Query variables 4GYI_1 5AKV_1 150 97
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]