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Parikh vectors
3CWA_1 5KPG_1 6HSH_1 Letter Amino acid
9 26 36 S Serine
4 7 11 C Cysteine
13 34 22 E Glutamic acid
20 23 29 G Glycine
12 23 24 I Isoleucine
20 8 26 P Proline
13 21 19 T Threonine
9 8 8 W Tryptophan
20 13 21 R Arginine
6 8 20 N Asparagine
14 21 12 Q Glutamine
4 18 17 H Histidine
33 36 49 L Leucine
8 34 16 K Lycine
9 18 19 F Phenylalanine
15 22 28 V Valine
40 15 26 A Alanine
13 27 32 D Aspartic acid
8 18 8 M Methionine
12 17 24 Y Tyrosine 

3CWA_1|Chains A, B, C, D|DNA-3-methyladenine glycosylase 2|Escherichia coli (562)
>5KPG_1|Chains A, B|Pavine N-methyltransferase|Thalictrum flavum subsp. glaucum (150095)
>6HSH_1|Chains A, B, C, D|Histone deacetylase|Schistosoma mansoni (6183)
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 \(w=f(c)\)
3CWA , Knot 125 282 0.83 40 177 273 
MYTLNWQPPYDWSWMLGFLAARAVSSVETVADSYYARSLAVGEYRGVVTAIPDIARHTLHINLSAGLEPVAAECLAKMSRLFDLQCNPQIVNGALGRLGAARPGLRLPGCVDAFEQGVRAILGQLVSVAMAAKLTARVAQLYGERLDDFPEYICFPTPQRLAAADPQALKALGMPLKRAEALIHLANAALEGTLPMTIPGDVEQAMKTLQTFPGIGRWTANYFALRGWQAKDVFLPDDYLIKQRFPGMTPAQIRRYAERWKPWRSYALLHIWYTEGWQPDEA
5KPG , Knot 169 397 0.85 40 228 378 
MRGSHHHHHHGMASMTGGQQMGRDLYDDDDKDRWIRPRDLQMETKQTKKEAVANLIKRIEHGEVSDEEIRGMMKIQVQKRLKWGYKPTHEQQLAQLVTFAQSLKGMEMAEEVDTLDAELYEIPLPFLHIMCGKTLKFSPGYFKDESTTLDESEVYMMDLYCERAQIKDGQSILDLGCGHGSLTLHVAQKYRGCKVTGITNSVSQKEFIMDQCKKLDLSNVEIILEDVTKFETEITYDRIFAVALIEHMKNYELFLKKVSTWIAQDGLLFVEHHCHKVFAYQYEPLDEDDWYTEYIFPSGTLVMSSSSILLYFQEDVSVVNHWTLSGKHPSLGFKQWLKRLDDNIDEVKEIFESFYGSKEKAMKFITYWRVFCIAHSQMYSTNNGEEWMLSQVLFKKK
6HSH , Knot 190 447 0.86 40 247 428 
HMSVGIVYGDQYRQLCCSSPKFGDRYALVMDLINAYKLIPELSRVPPLQWDSPSRMYEAVTAFHSTEYVDALKKLQMLHCEEKELTADDELLMDSFSLNYDCPGFPSVFDYSLAAVQGSLAAASALICRHCEVVINWGGGWHHAKRSEASGFCYLNDIVLAIHRLVSSTPPETSPNRQTRVLYVDLDLHHGDGVEEAFWYSPRVVTFSVHHASPGFFPGTGTWNMVDNDKLPIFLNGAGRGRFSAFNLPLEEGINDLDWSNAIGPILDSLNIVIQPSYVVVQCGADCLATDPHRIFRLTNFYPNLNLDSDCDSECSLSGYLYAIKKILSWKVPTLILGGGGYNFPDTARLWTRVTALTIEEVKGKKMTISPEIPEHSYFSRYGPDFELDIDYFPHESHNKTLDSIQKHHRRILEQLRNYADLNKLIYDYDQVYQLYNLTGMGSLVPR

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 \(c\).

\(|P_{f(3CWA_1)}(2) \setminus P_{f(5KPG_1)}(2)|=72\), \(|P_{f(5KPG_1)}(2) \setminus P_{f(3CWA_1)}(2)|=123\). 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 1:100101011001011111111011001001100001001111000111011101100010101011101111001101001101000101101111011110111011101011001101111011011111010101101010010011001011010011110101101111110010111011011101011101110100110010011111010100111011010011110001100011110110100010010110001110110001101001
Pair \(Z_2\) Length of longest common subsequence
3CWA_1,5KPG_1 195 4
3CWA_1,6HSH_1 174 4
5KPG_1,6HSH_1 181 3

Newick tree

 
[
	5KPG_1:96.30,
	[
		3CWA_1:87,6HSH_1:87
	]:9.30
]

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 \((0.85)(0.8)(\frac{679 }{\log_{20} 679}-\frac{282}{\log_{20}282})=110.\)
Status Protein1 Protein2 d d1/2
Query variables 3CWA_1 5KPG_1 142 120.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]