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Parikh vectors
4QSI_1 4NFR_1 8VUZ_1 Letter Amino acid
26 28 16 L Leucine
24 23 12 K Lycine
17 19 13 P Proline
8 13 11 Y Tyrosine
1 9 4 C Cysteine
13 22 7 E Glutamic acid
22 18 15 G Glycine
18 10 30 S Serine
7 2 4 W Tryptophan
17 15 23 V Valine
11 9 9 Q Glutamine
2 7 3 M Methionine
12 14 6 F Phenylalanine
10 15 7 N Asparagine
9 27 5 I Isoleucine
12 14 5 H Histidine
12 18 22 T Threonine
13 19 14 A Alanine
7 12 4 R Arginine
19 19 8 D Aspartic acid 

4QSI_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>4NFR_1|Chains A, B|Aspartoacylase|Homo sapiens (9606)
>8VUZ_1|Chain A|28H6E11 Fab Heavy Chain|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 \(w=f(c)\)
4QSI , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
4NFR , Knot 136 313 0.83 40 196 302 
MTSCHIAEEHIQKVAIFGGTHGNELTGVFLVKHWLENGAEIQRTGLEVKPFITNPRAVKKCTRYIDCDLNRIFDLENLGKKMSEDLPYEVRRAQEINHLFGPKDSEDSYDIIFDLHNTTSNMGCTLILEDSRNNFLIQMFHYIKTSLAPLPCYVYLIEHPSLKYATTRSIAKYPVGIEVGPQPQGVLRADILDQMRKMIKHALDFIHHFNEGKEFPPCAIEVYKIIEKVDYPRDENGEIAAIIHPNLQDQDWKPLHPGDPMFLTLDGKTIPLGGDCTVYPVFVNAAAYYEKKEAFAKTTKLTLNAKSIRCCLH
8VUZ , Knot 103 218 0.84 40 151 209 
QVQLQQSGPEVVRPGVSVKISCKGSGYTFTDHAMHWVKQSHAKSLEWIGVFNTYNGDTNYNQKFEDKATMTIDKSSSTAYLELARLTSEDSAIYYCVRLGLLGFYALDYWGQGTSVTVSSAKTTPPSVYPLAPGSAAQTNSMVTLGCLVKGYFPEPVTVTWNSGSLSSGVHTFPAVLQSDLYTLSSSVTVPSSPRPSETVTCNVAHPASSTKVDKKIV

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 \(c\).

\(|P_{f(4QSI_1)}(2) \setminus P_{f(4NFR_1)}(2)|=73\), \(|P_{f(4NFR_1)}(2) \setminus P_{f(4QSI_1)}(2)|=93\). 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 1:10001101000110010001111010000110100001000101011010000100101100101101010000001110111101000110101011010101000010000011010110100001011011001011111111101101011100110110010001001010010101111001001001101001111001011110011010000110100101010101001110010110110000101010
Pair \(Z_2\) Length of longest common subsequence
4QSI_1,4NFR_1 166 4
4QSI_1,8VUZ_1 141 4
4NFR_1,8VUZ_1 203 4

Newick tree

 
[
	4NFR_1:99.01,
	[
		4QSI_1:70.5,8VUZ_1:70.5
	]:28.51
]

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 \((0.85)(0.8)(\frac{573 }{\log_{20} 573}-\frac{260}{\log_{20}260})=88.5\)
Status Protein1 Protein2 d d1/2
Query variables 4QSI_1 4NFR_1 114 103.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]