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Parikh vectors
1SZO_1 1HVY_1 3OSI_1 Letter Amino acid
24 13 16 A Alanine
15 19 8 R Arginine
14 20 20 D Aspartic acid
4 8 10 M Methionine
6 11 7 Y Tyrosine
8 9 7 H Histidine
6 15 24 K Lycine
14 13 13 T Threonine
20 13 14 V Valine
22 16 20 E Glutamic acid
19 24 12 G Glycine
13 19 23 I Isoleucine
31 31 39 L Leucine
4 5 0 W Tryptophan
10 8 9 N Asparagine
3 5 1 C Cysteine
9 14 17 Q Glutamine
9 14 15 F Phenylalanine
13 17 11 P Proline
13 14 19 S Serine 

1SZO_1|Chains A, B, C, D, E, F, G, H, I, J, K, L|6-oxocamphor hydrolase|Rhodococcus sp. NCIMB 9784 (157732)
>1HVY_1|Chains A, B, C, D|THYMIDYLATE SYNTHASE|Homo sapiens (9606)
>3OSI_1|Chains A, B|Peroxisome proliferator-activated receptor gamma|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 \(w=f(c)\)
1SZO , Knot 118 257 0.85 40 160 248 
MKQLATPFQEYSQKYENIRLERDGGVLLVTVHTEGKSLVWTSTAHDELAYCFHDIACDRENKVVILTGTGPSFCNEIDFTSFNLGTPHDWDEIIFEGQRLLNNLLSIEVPVIAAVNGPVTNAPEIPVMSDIVLAAESATFQDGPHFPSGIVPGDGAHVVWPHVLGSNRGRYFLLTGQELDARTALDYGAVNEVLSEQELLPRAWELARGIAEKPLLARRYARKVLTRQLRRVMEADLSLGLAHEALAAIDLGMESEQ
1HVY , Knot 129 288 0.84 40 188 276 
PPHGELQYLGQIQHILRCGVRKDDRTGTGTLSVFGMQARYSLRDEFPLLTTKRVFWKGVLEELLWFIKGSTNAKELSSKGVKIWDANGSRDFLDSLGFSTREEGDLGPVYGFQWRHFGAEYRDMESDYSGQGVDQLQRVIDTIKTNPDDRRIIMCAWNPRDLPLMALPPCHALCQFYVVNSELSCQLYQRSGDMGLGVPFNIASYALLTYMIAHITGLKPGDFIHTLGDAHIYLNHIEPLKIQLQREPRPFPKLRILRKVEKIDDFKAEDFQIEGYNPHPTIKMEMAV
3OSI , Knot 126 285 0.83 38 177 273 
GSHMEISSDIDQLNPESADLRALAKHLYDSYIKSFPLTKAKARAILTGKTTDKSPFVIYDMNSLMMGEDKIKFKHITPLQEQSKEVAIRIFQGCQFRSVEAVQEITEYAKSIPGFVNLDLNDQVTLLKYGVHEIIYTMLASLMNKDGVLISEGQGFMTREFLKSLRKPFGDFMEPKFEFAVKFNALELDDSDLAIFIAVIILSGDRPGLLNVKPIEDIQDNLLQALELQLKLNHPESSQLFAKLLQKMTDLRQIVTEHVQLLQVIKKTETDMSLHPLLQEIYKDL

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 \(c\).

\(|P_{f(1SZO_1)}(2) \setminus P_{f(1HVY_1)}(2)|=76\), \(|P_{f(1HVY_1)}(2) \setminus P_{f(1SZO_1)}(2)|=104\). 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 1:10011011000000000101000111111010001001110001000110010011000000111101011010001010010110100100111010011001101011111110111001101111001111100101001101101111101101111011100010011101001010011001110011000011101101101110011110001001100010011010101111001111101110000
Pair \(Z_2\) Length of longest common subsequence
1SZO_1,1HVY_1 180 4
1SZO_1,3OSI_1 163 4
1HVY_1,3OSI_1 157 3

Newick tree

 
[
	1SZO_1:88.17,
	[
		3OSI_1:78.5,1HVY_1:78.5
	]:9.67
]

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 \((0.85)(0.8)(\frac{545 }{\log_{20} 545}-\frac{257}{\log_{20}257})=81.8\)
Status Protein1 Protein2 d d1/2
Query variables 1SZO_1 1HVY_1 103 99.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]