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Parikh vectors
1TBP_1 3WLD_1 5QUP_1 Letter Amino acid
11 17 15 R Arginine
2 16 9 H Histidine
5 18 5 M Methionine
10 19 11 S Serine
10 31 13 T Threonine
17 22 21 A Alanine
5 41 11 D Aspartic acid
10 38 17 E Glutamic acid
15 35 27 L Leucine
4 18 9 N Asparagine
5 16 11 Q Glutamine
8 13 7 P Proline
0 2 1 W Tryptophan
14 26 13 V Valine
2 2 0 C Cysteine
11 47 21 G Glycine
17 22 17 I Isoleucine
16 32 12 K Lycine
12 20 7 F Phenylalanine
6 15 4 Y Tyrosine 

1TBP_1|Chains A, B|TATA-BINDING PROTEIN|Saccharomyces cerevisiae (4932)
>3WLD_1|Chain A|Myosin light chain kinase, Green fluorescent protein, Calmodulin|Gallus gallus (9031)
>5QUP_1|Chain A|RadA|Pyrococcus furiosus (2261)
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 \(w=f(c)\)
1TBP , Knot 82 180 0.78 38 127 169 
MGIVPTLQNIVATVTLGCRLDLKTVALHARNAEYNPKRFAAVIMRIREPKTTALIFASGKMVVTGAKSEDDSKLASRKYARIIQKIGFAAKFTDFKIQNIVGSCDVKFPIRLEGLAFSHGTFSSYEPELFPGLIYRMVKPKIVLLIFVSGKIVLTGAKQREEIYQAFEAIYPVLSEFRKM
3WLD , Knot 183 450 0.82 40 224 405 
MGSHHHHHHGMASMTGGQQMGRDLYDDDDKDLATMVDSSRRKWNKTGHAVRAIGRLSSLENVYIKADKQKNGIKANFKIRHNIEDGGVQLAYHYQQNTPIGDGPVLLPDNHYLSVQSKLSKDPNEKRDHMVLLEFVTAAGITLGMDELYKGGTGGSMVSKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFICTTGKLPVPWPTLVTTLTYGVQCFSRYPDHMKQHDFFKSAMPEGYIQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNLPDQLTEEQIAEFKEAFSLFDKDGDGTITTKELGTVMRSLGQNPTEAELQDMINEVDADGDGTIDFPEFLTMMARKGSYRDTEEEIREAFGVFDKDGNGYISAAELRHVMTNLGEKLTDEEVDEMIREADIDGDGQVNYEEFVQMMTAK
5QUP , Knot 102 231 0.80 38 154 223 
MATIGRISTGSKSLDKLLGGGIETQAITEVFGEFGSGKTQLAHTLAVMVQLPPEEGGLNGSVMWIDTENTFRPERIREIAQNRGLDPDEVLKHIAYARAFNSNHQMLLVQQAEDMIKELLNTDRPVKLLIVDSLTSHFRSEYIGRGALAERQQKLAKHLADLHRLANLYDIAVFVTNQVQANGGHILAHSATLRVYLRKGKGGKRIARLIDAPHLPEGEAVFSITEKGIED

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 \(c\).

\(|P_{f(1TBP_1)}(2) \setminus P_{f(3WLD_1)}(2)|=34\), \(|P_{f(3WLD_1)}(2) \setminus P_{f(1TBP_1)}(2)|=131\). 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 1:111110100111010110010100111010010001001111110100100011111010111011000000011000010110011111010010100111000101110101111001010000101111110011010111111101011101100000100110110111001001
Pair \(Z_2\) Length of longest common subsequence
1TBP_1,3WLD_1 165 3
1TBP_1,5QUP_1 155 4
3WLD_1,5QUP_1 154 3

Newick tree

 
[
	1TBP_1:81.02,
	[
		5QUP_1:77,3WLD_1:77
	]:4.02
]

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 \((0.85)(0.8)(\frac{630 }{\log_{20} 630}-\frac{180}{\log_{20}180})=128.\)
Status Protein1 Protein2 d d1/2
Query variables 1TBP_1 3WLD_1 159 109.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]