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Parikh vectors
4NEC_1 8DGU_1 7DCS_1 Letter Amino acid
3 1 7 Q Glutamine
19 1 10 H Histidine
26 3 10 L Leucine
12 1 5 T Threonine
4 1 2 Y Tyrosine
14 1 9 V Valine
18 0 5 R Arginine
4 0 2 M Methionine
13 3 6 P Proline
20 2 5 S Serine
35 0 4 A Alanine
1 1 5 N Asparagine
23 3 6 D Aspartic acid
2 0 2 C Cysteine
17 3 6 E Glutamic acid
25 0 7 G Glycine
5 3 8 K Lycine
5 0 2 W Tryptophan
11 0 3 I Isoleucine
7 2 9 F Phenylalanine 

4NEC_1|Chains A, C[auth B], E[auth C], G[auth D]|Putative SAM-dependent methyltransferase|Streptomyces lasaliensis (324833)
>8DGU_1|Chain A|Spike protein S2'|Severe acute respiratory syndrome coronavirus 2 (2697049)
>7DCS_1|Chains A, B, C, D, E, F|Heat shock factor protein 1|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 \(w=f(c)\)
4NEC , Knot 115 264 0.81 40 153 240 
MGSSHHHHHHSSGLVPRGSHMTEVFDAVYRGESPFGKRPPWDIGAPQPAYVALEKAGLIQGAVLDAGCGTGEDALHLAGLGYAVTGLDLSPTAISVARDKADARGLGAVFEVADALDLTGWEERFDTVIDSGLAHTFEGDRLRAYATALHRACRPGAVAHILSISDRGSAEMQARLAEAIDEIPAPLPDDDESPTLKRSADHLRDGFAEGWTIESIDESLMRGVIPTTSELLDVHAWLGRFRRDWNSSSVDKLAAALEHHHHHH
8DGU , Knot 16 25 0.68 26 21 22 
PLQPELDSFKEELDKYFKNHTSPDV
7DCS , Knot 57 113 0.79 40 96 107 
GHHHHHHVPAFLTKLWTLVSDPDTDALICWSPSGNSFHVFDQGQFAKEVLPKYFKHNNMASFVRQLNMYGFRKVVHIEQGGLVKPERDDTEFQHPCFLRGQEQLLENIKRKVT

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 \(c\).

\(|P_{f(4NEC_1)}(2) \setminus P_{f(8DGU_1)}(2)|=141\), \(|P_{f(8DGU_1)}(2) \setminus P_{f(4NEC_1)}(2)|=9\). 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 1:110000000000111101001001101100100111001110111101101110011110111101101010011011111011011010101101100010101111110110110101100010011001110010100101010110010011111011010001010101011011001111110000010100010010011101101001000110111100001101011110100010000100111110000000
Pair \(Z_2\) Length of longest common subsequence
4NEC_1,8DGU_1 150 2
4NEC_1,7DCS_1 169 6
8DGU_1,7DCS_1 101 4

Newick tree

 
[
	4NEC_1:87.52,
	[
		8DGU_1:50.5,7DCS_1:50.5
	]:37.02
]

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 \((0.85)(0.8)(\frac{289 }{\log_{20} 289}-\frac{25}{\log_{20}25})=88.0\)
Status Protein1 Protein2 d d1/2
Query variables 4NEC_1 8DGU_1 107 58
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]