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Parikh vectors
8DWC_1 7HFI_1 3UGT_1 Letter Amino acid
0 3 4 C Cysteine
0 6 18 H Histidine
0 11 32 S Serine
2 7 14 Y Tyrosine
2 2 20 R Arginine
0 16 27 N Asparagine
2 13 25 G Glycine
0 19 41 L Leucine
1 13 39 K Lycine
1 2 15 M Methionine
2 0 8 W Tryptophan
1 22 20 V Valine
1 6 31 E Glutamic acid
0 15 19 A Alanine
1 9 28 D Aspartic acid
1 3 18 Q Glutamine
0 6 28 I Isoleucine
0 5 30 F Phenylalanine
1 5 24 P Proline
0 6 19 T Threonine 

8DWC_1|Chain A|Proenkephalin-A|Bos taurus (9913)
>7HFI_1|Chains A, B|Non-structural protein 3|Severe acute respiratory syndrome coronavirus 2 (2697049)
>3UGT_1|Chains A, B, C, D|Threonyl-tRNA synthetase, mitochondrial|Saccharomyces cerevisiae (559292)
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 \(w=f(c)\)
8DWC , Knot 13 15 0.78 22 14 13 
VGRPEWWMDYQKRYG
7HFI , Knot 80 169 0.81 38 119 162 
SMVNSFSGYLKLTDNVYIKNADIVEEAKKVKPTVVVNAANVYLKHGGGVAGALNKATNNAMQVESDDYIATNGPLKVGGSCVLSGHNLAKHCLHVVGPNVNKGEDIQLLKSAYENFNQHEVLLAPLLSAGIFGADPIHSLRVCVDTVRTNVYLAVFDKNLYDKLVSSFL
3UGT , Knot 187 460 0.83 40 244 429 
MGSSHHHHHHSSGLVPRGSHMASATKNASSATPATMTSMVSQRQDLFMTDPLSPGSMFFLPNGAKIFNKLIEFMKLQQKFKFGFNEVVTPLIYKKTLWEKSGHWENYADDMFKVETTDEEKEEYGLKPMNCPGHCLIFGKKDRSYNELPLRFSDFSPLHRNEASGALSGLTRLRKFHQDDGHIFCTPSQVKSEIFNSLKLIDIVYNKIFPFVKGGSGAESNYFINFSTRPDHFIGDLKVWNHAEQVLKEILEESGKPWKLNPGDGAFYGPKLDIMVTDHLRKTHQVATIQLDFQLPERFDLKFKDQDNSYKRPIMIHRATFGSIERFMALLIDSNEGRWPFWLNPYQAVIIPVNTKNVQQLDMCTALQKKLRNELEADDMEPVPLNDWHFNVDLDIRNEPVGYRIKSAILKNYSYLIIVGDEEVQLQKYNIRERDNRKSFEKLTMSQIWEKFIELEKNYK

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 \(c\).

\(|P_{f(8DWC_1)}(2) \setminus P_{f(7HFI_1)}(2)|=12\), \(|P_{f(7HFI_1)}(2) \setminus P_{f(8DWC_1)}(2)|=117\). 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 1:110101110000001
Pair \(Z_2\) Length of longest common subsequence
8DWC_1,7HFI_1 129 2
8DWC_1,3UGT_1 242 2
7HFI_1,3UGT_1 205 3

Newick tree

 
[
	3UGT_1:12.00,
	[
		8DWC_1:64.5,7HFI_1:64.5
	]:59.50
]

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 \((0.85)(0.8)(\frac{184 }{\log_{20} 184}-\frac{15}{\log_{20}15})=60.5\)
Status Protein1 Protein2 d d1/2
Query variables 8DWC_1 7HFI_1 76 41.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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