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Parikh vectors
8DUN_1 4AGG_1 4ESM_1 Letter Amino acid
9 4 11 N Asparagine
6 7 12 I Isoleucine
1 1 6 M Methionine
5 2 3 C Cysteine
7 9 13 E Glutamic acid
3 0 7 H Histidine
7 7 17 F Phenylalanine
13 11 20 P Proline
36 9 10 S Serine
6 2 7 W Tryptophan
6 7 16 R Arginine
10 9 30 D Aspartic acid
7 6 10 Q Glutamine
18 17 21 G Glycine
10 6 11 K Lycine
8 9 11 Y Tyrosine
25 16 18 V Valine
11 7 30 A Alanine
21 11 35 L Leucine
22 6 14 T Threonine 

8DUN_1|Chains A[auth H], I, K[auth N]|Antibody SKW11 heavy chain|Macaca fascicularis (9541)
>4AGG_1|Chains A, B|GALECTIN|CINACHYRELLA SP. (76781)
>4ESM_1|Chain A|Phenylalanine-4-hydroxylase|Chromobacterium violaceum (243365)
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 \(w=f(c)\)
8DUN , Knot 103 231 0.81 40 147 216 
QVQLQESGPGLVKPSETLSLTCAVSGDSISNNYWTWIRHFPGKGLEWIGRIYGDAGSTDYNPSLKSRVTISMDLSKNQFSLDLTSVTVADTALYYCASRTTVADNWFDVWGPGVLVTVSSASTKGPSVFPLAPSSRSTSESTAALGCLVKDYFPEPVTVSWNSGSLTSGVHTFPAVLQSSGLYSLSSVVTVPSSSLGTQTYVCNVNHKPSNTKVDKRVEIKTCGGLEVLFQ
4AGG , Knot 72 146 0.82 38 117 143 
VGPIQSIKVDPMKSGGLGVVYRSPDKGRVSLYLYNDGEDILLVVDARFDWRGEQNVLVLNSKFAGGEWGPEVRPEGFPFPCCGYVTTITVRVEIGADGFTLSANGIEIVKYPYRDGLPPPVTKFQYVFQDQGASETAQLESLSAYY
4ESM , Knot 134 302 0.84 40 188 287 
GLPGSMNDRADFVVPDITTRKNVGLSHDANDFTLPQPLDRYSAEDHATWATLYQRQCKLLPGRACDEFLEGLERLEVDADRVPDFNKLNEKLMAATGWKIVAVPGLIPDDVFFEHLANRRFPVTWWLREPHQLDYLQEPDVFHDLFGHVPLLINPVFADALEAYGKGGVKAKALGALPMLARLYWYTVEFGLINTPAGMRIYGAGILSSKSESIYCLDSASPNRVGFDLMRIMNTRYRIDTFQKTYFVIDSFKQLFDATAPDFAPLYLQLADAQPWGAGDIAPDDLVLNAGDHQGWADTEDV

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 \(c\).

\(|P_{f(8DUN_1)}(2) \setminus P_{f(4AGG_1)}(2)|=86\), \(|P_{f(4AGG_1)}(2) \setminus P_{f(8DUN_1)}(2)|=56\). 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 1:010100011111010001010011010010000101100111011011101010110000010100010101010000101010010110011000100001100110111111110100100011011111100000000011110110001101101010010100110011111000110010011011000110000100100010000100010100011101110
Pair \(Z_2\) Length of longest common subsequence
8DUN_1,4AGG_1 142 4
8DUN_1,4ESM_1 175 4
4AGG_1,4ESM_1 171 4

Newick tree

 
[
	4ESM_1:91.08,
	[
		8DUN_1:71,4AGG_1:71
	]:20.08
]

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 \((0.85)(0.8)(\frac{377 }{\log_{20} 377}-\frac{146}{\log_{20}146})=69.7\)
Status Protein1 Protein2 d d1/2
Query variables 8DUN_1 4AGG_1 86 70.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]