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Parikh vectors
3ROK_1 2XHF_1 9FPR_1 Letter Amino acid
13 2 11 Q Glutamine
4 6 2 M Methionine
13 7 12 F Phenylalanine
8 1 7 W Tryptophan
17 12 17 V Valine
21 12 19 D Aspartic acid
19 13 18 S Serine
12 9 17 P Proline
15 10 7 R Arginine
8 7 10 N Asparagine
10 13 22 G Glycine
7 11 12 H Histidine
13 13 9 I Isoleucine
17 12 26 L Leucine
9 9 13 A Alanine
12 2 1 C Cysteine
16 9 13 E Glutamic acid
18 11 24 K Lycine
17 7 12 T Threonine
5 5 8 Y Tyrosine 

3ROK_1|Chains A, B|ADP-ribosyl cyclase 1|Homo sapiens (9606)
>2XHF_1|Chains A, B|PEROXIREDOXIN 5|ALVINELLA POMPEJANA (6376)
>9FPR_1|Chain A|Carbonic anhydrase 2|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 \(w=f(c)\)
3ROK , Knot 116 254 0.84 40 177 247 
EFRWRQTWSGPGTTKRFPETVLARCVKYTEIHPEMRHVDCQSVWDAFKGAFISKHPCDITEEDYQPLMKLGTQTVPCNKILLWSRIKDLAHQFTQVQRDMFTLEDTLLGYLADDLTWCGEFDTSKINYQSCPDWRKDCSNNPVSVFWKTVSRRFAEAACDVVHVMLDGSRSKIFDKDSTFGSVEVHNLQPEKVQTLEAWVIHGGREDSRDLCQDPTIKELESIISKRNIQFSCKNIYRPDKFLQCVKNPEDSSC
2XHF , Knot 83 171 0.83 40 134 165 
MRGSHHHHHHGSPIKVGDIIPDVLVYEDVPSKSFPIHDVFRGRKGILFSVVGAFVPGSNNHIPEYLSLYDKFKEEGYHTIACIAVNDPFVMAAWGKTVDPEHKIRMLADMHGEFTRALGTELDSSKMLGNNRSRRYAMLIDDNKIRSVSTEPDITGLACLLSIQRQKENKT
9FPR , Knot 112 260 0.79 40 176 249 
MSHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(3ROK_1)}(2) \setminus P_{f(2XHF_1)}(2)|=103\), \(|P_{f(2XHF_1)}(2) \setminus P_{f(3ROK_1)}(2)|=60\). 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 1:01010001011100001100111001000010101001000011011011110001001000000111011000110001111001001100100100011010001110110010101010000100000101000000011011100100011011001101110100001100000110101001010010010111101100000010001010010011000010100001001001100100100000
Pair \(Z_2\) Length of longest common subsequence
3ROK_1,2XHF_1 163 3
3ROK_1,9FPR_1 179 3
2XHF_1,9FPR_1 156 3

Newick tree

 
[
	3ROK_1:87.97,
	[
		2XHF_1:78,9FPR_1:78
	]:9.97
]

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 \((0.85)(0.8)(\frac{425 }{\log_{20} 425}-\frac{171}{\log_{20}171})=75.3\)
Status Protein1 Protein2 d d1/2
Query variables 3ROK_1 2XHF_1 95 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]