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Parikh vectors
2WEJ_1 7UWL_1 5EOI_1 Letter Amino acid
8 7 8 Y Tyrosine
13 13 13 A Alanine
19 8 19 D Aspartic acid
1 10 1 C Cysteine
22 11 22 G Glycine
12 16 12 H Histidine
18 14 17 S Serine
7 4 7 W Tryptophan
17 9 17 V Valine
7 17 7 R Arginine
10 6 10 N Asparagine
17 17 17 P Proline
12 7 12 T Threonine
26 18 26 L Leucine
1 2 1 M Methionine
12 3 12 F Phenylalanine
11 6 11 Q Glutamine
13 13 13 E Glutamic acid
9 3 9 I Isoleucine
24 4 24 K Lycine 

2WEJ_1|Chain A|CARBONIC ANHYDRASE 2|HOMO SAPIENS (9606)
>7UWL_1|Chains A, B|Interleukin-25|Homo sapiens (9606)
>5EOI_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 \(w=f(c)\)
2WEJ , Knot 111 259 0.79 40 175 248 
SHHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
7UWL , Knot 88 188 0.81 40 137 180 
DASATHTYSHWPSCCPSKGQDTSEELLRWSTVPVPPLEPARPNRHPESCRASEDGPLNSRAISPWRYELDRDLNRLPQDLYHARCLCPHCVSLQTGSHMDPRGNSELLYHNQTVFYRRPCHGEKGTHKGYCLERRLYRVSLACVCVRPRVMGAPAALEVLFQGPGAAGLNDIFEAQKIEWHEHHHHHH
5EOI , Knot 110 258 0.79 40 174 247 
HHWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHAFNVEFDDSQDKAVLKGGPLDGTYRLIQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDFGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTKGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLLECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK

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 \(c\).

\(|P_{f(2WEJ_1)}(2) \setminus P_{f(7UWL_1)}(2)|=107\), \(|P_{f(7UWL_1)}(2) \setminus P_{f(2WEJ_1)}(2)|=69\). 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 1:0001101000110010001111010000110100001000101011010000100101100101101010000001110111101000110101011010101000010000011010110100001011011001011111111101101011100110110010001001010010101111001001001101001111001011110011010000110100101010101001110010110110000101010
Pair \(Z_2\) Length of longest common subsequence
2WEJ_1,7UWL_1 176 4
2WEJ_1,5EOI_1 1 258
7UWL_1,5EOI_1 177 4

Newick tree

 
[
	7UWL_1:10.90,
	[
		2WEJ_1:0.5,5EOI_1:0.5
	]:10.40
]

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 \((0.85)(0.8)(\frac{447 }{\log_{20} 447}-\frac{188}{\log_{20}188})=76.0\)
Status Protein1 Protein2 d d1/2
Query variables 2WEJ_1 7UWL_1 96 83.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]