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Parikh vectors
3EGB_1 4DOV_1 5JYK_1 Letter Amino acid
14 14 26 P Proline
3 5 4 W Tryptophan
19 8 22 R Arginine
13 7 23 N Asparagine
7 3 6 C Cysteine
18 11 41 L Leucine
9 9 20 F Phenylalanine
17 13 39 E Glutamic acid
24 7 46 G Glycine
14 12 36 I Isoleucine
22 8 51 S Serine
23 4 33 T Threonine
21 15 53 V Valine
15 7 30 D Aspartic acid
8 2 24 H Histidine
15 15 33 K Lycine
3 2 14 M Methionine
7 3 16 Y Tyrosine
19 7 30 A Alanine
14 11 19 Q Glutamine 

3EGB_1|Chains A, B|Protein pellino homolog 2|Homo sapiens (9606)
>4DOV_1|Chains A, B[auth C]|Origin recognition complex subunit 1|Mus musculus (10090)
>5JYK_1|Chains A, B|Protease Do-like 9|Arabidopsis thaliana (3702)
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 \(w=f(c)\)
3EGB , Knot 129 285 0.85 40 192 277 
GSEEHCAPNKEPVKYGELVVLGYNGALPNGDRGRRKSRFALYKRPKANGVKPSTVHVISTPQASKAISCKGQHSISYTLSRNQTVVVEYTHDKDTDMFQVGRSTESPIDFVVTDTISGSQNTDEAQITQSTISRFACRIVCDRNEPYTARIFAAGFDSSKNIFLGEKAAKWKNPDGHMDGLTTNGVLVMHPRGGFTEESQPGVWREISVCGDVYTLRETRSAQQRGKLVESETNVLQDGSLIDLCGATLLWRTADGLFHTPTQKHIEALRQEINAARPQCPVGLN
4DOV , Knot 82 163 0.85 40 126 156 
SKTRQTFSWVGRPLPNRKQFQQMYREICMKINDGSEIHIKVGQFVLIQGEDNKKPYVAKLIELFQNGAEVPPKKCARVQWFVRFLEIPVSKRHLLGRSPPAQEIFWYDCSDWDNKINVETIIGPVQVVALAPEEVIPVDQKSEETLFVKLSWNKKDFAPLPPE
5JYK , Knot 223 566 0.83 40 254 525 
MGSSHHHHHHSSGLVPRGSHMASMTGGQQMGRGSEFELLNNESEAGNQRTSSPERSRSRLHHSDTKNGDCSNGMIVSTTTESIPAAPSWETVVKVVPSMDAVVKVFCVHTEPNFSLPWQRKRQYSSGSSGFIIGGRRVLTNAHSVEHHTQVKLKKRGSDTKYLATVLAIGTECDIALLTVTDDEFWEGVSPVEFGDLPALQDAVTVVGYPIGGDTISVTSGVVSRMEILSYVHGSTELLGLQIDAAINSGNSGGPAFNDKGKCVGIAFQSLKHEDAENIGYVIPTPVIVHFIQDYEKHDKYTGFPVLGIEWQKMENPDLRKSMGMESHQKGVRIRRIEPTAPESQVLKPSDIILSFDGVNIANDGTVPFRHGERIGFSYLISQKYTGDSALVKVLRNKEILEFNIKLAIHKRLIPAHISGKPPSYFIVAGFVFTTVSVPYLRSEYGKEYEFDAPVKLLEKHLHAMAQSVDEQLVVVSQVLVSDINIGYEEIVNTQVVAFNGKPVKNLKGLAGMVENCEDEYMKFNLDYDQIVVLDTKTAKEATLDILTTHCIPSAMSDDLKTEERN

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 \(c\).

\(|P_{f(3EGB_1)}(2) \setminus P_{f(4DOV_1)}(2)|=125\), \(|P_{f(4DOV_1)}(2) \setminus P_{f(3EGB_1)}(2)|=59\). 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 1:100000110001100101111100111101001000001110001010110100101100101001100010001000100000111000000000110110000011011100010100000010100001001100110000010010111111000001111001101001010101100011111010111000001111001010101001000001000101100000110010110101101110010111001000010110001011010011110
Pair \(Z_2\) Length of longest common subsequence
3EGB_1,4DOV_1 184 3
3EGB_1,5JYK_1 172 4
4DOV_1,5JYK_1 208 4

Newick tree

 
[
	4DOV_1:10.92,
	[
		3EGB_1:86,5JYK_1:86
	]:15.92
]

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 \((0.85)(0.8)(\frac{448 }{\log_{20} 448}-\frac{163}{\log_{20}163})=84.3\)
Status Protein1 Protein2 d d1/2
Query variables 3EGB_1 4DOV_1 109 84.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]