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Parikh vectors
7GUX_1 8YFN_1 1RHQ_1 Letter Amino acid
8 7 3 A Alanine
4 11 11 K Lycine
7 5 7 F Phenylalanine
0 2 0 W Tryptophan
15 11 12 L Leucine
6 1 4 M Methionine
3 4 3 Y Tyrosine
9 7 11 R Arginine
8 5 11 D Aspartic acid
5 6 12 G Glycine
9 4 11 I Isoleucine
7 3 10 N Asparagine
4 3 1 Q Glutamine
3 3 3 H Histidine
10 10 7 V Valine
7 3 10 T Threonine
5 1 5 C Cysteine
6 6 11 E Glutamic acid
3 5 3 P Proline
9 8 12 S Serine 

7GUX_1|Chain A|B-cell lymphoma 6 protein|Homo sapiens (9606)
>8YFN_1|Chains A, C|RB1-inducible coiled-coil protein 1|Homo sapiens (9606)
>1RHQ_1|Chains A, C[auth D]|Caspase-3|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 \(w=f(c)\)
7GUX , Knot 65 128 0.82 38 98 125 
GPGADSCIQFTRHASDVLLNLNRLRSRDILTDVVIVVSREQFRAHKTVLMACSGLFYSIFTDQLKCNLSVINLDPEINPEGFCILLDFMYTSRLNLREGNIMAVMATAMYLQMEHVVDTCRKFIKASE
8YFN , Knot 57 105 0.84 40 87 101 
SRHSEKIAIRDFQVGDLVLIILDERHDNYVLFTVSPTLYFLHSESLPALDLKPGEGASGASRRPWVLGKVMEKEYCQAKKAQNRFKVPLGTKFYRVKAVSWNKKV
1RHQ , Knot 72 147 0.81 38 113 144 
SGISLDNSYKMDYPEMGLCIIINNKNFHKSTGMTSRSGTDVDAANLRETFRNLKYEVRNKNDLTREEIVELMRDVSKEDHSKRSSFVCVLLSHGEEGIIFGTNGPVDLKKITNFFRGDRCRSLTGKPKLFIIQACRGTELDCGIETD

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 \(c\).

\(|P_{f(7GUX_1)}(2) \setminus P_{f(8YFN_1)}(2)|=73\), \(|P_{f(8YFN_1)}(2) \setminus P_{f(7GUX_1)}(2)|=62\). 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 1:11110001010001001110100100001100111110000101000111100111001100010001011010101010110111011000010100101111110110101001100000110100
Pair \(Z_2\) Length of longest common subsequence
7GUX_1,8YFN_1 135 2
7GUX_1,1RHQ_1 139 4
8YFN_1,1RHQ_1 152 3

Newick tree

 
[
	1RHQ_1:74.51,
	[
		7GUX_1:67.5,8YFN_1:67.5
	]:7.01
]

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 \((0.85)(0.8)(\frac{233 }{\log_{20} 233}-\frac{105}{\log_{20}105})=41.1\)
Status Protein1 Protein2 d d1/2
Query variables 7GUX_1 8YFN_1 53 47.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]