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Parikh vectors
1QWK_1 2NYH_1 9CUM_1 Letter Amino acid
21 8 4 S Serine
14 5 2 Q Glutamine
5 2 1 M Methionine
12 6 2 F Phenylalanine
14 4 4 P Proline
33 14 7 A Alanine
14 4 3 I Isoleucine
21 1 2 K Lycine
11 7 3 R Arginine
34 13 2 L Leucine
5 6 2 W Tryptophan
26 5 4 E Glutamic acid
17 8 7 G Glycine
10 10 2 H Histidine
18 6 3 T Threonine
8 2 3 Y Tyrosine
13 3 3 N Asparagine
17 8 1 D Aspartic acid
2 0 1 C Cysteine
22 5 6 V Valine 

1QWK_1|Chain A|aldo-keto reductase family 1 member C1|Caenorhabditis elegans (6239)
>2NYH_1|Chains A, B|Putative dioxygenase|Burkholderia xenovorans (36873)
>9CUM_1|Chain A|Dihydrofolate reductase type 2|Escherichia coli (562)
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 \(w=f(c)\)
1QWK , Knot 139 317 0.84 40 194 304 
MSSATASIKLSNGVEMPVIGLGTWQSSPAEVITAVKTAVKAGYRLIDTASVYQNEEAIGTAIKELLEEGVVKREELFITTKAWTHELAPGKLEGGLRESLKKLQLEYVDLYLAHMPAAFNDDMSEHIASPVEDVWRQFDAVYKAGLAKAVGVSNWNNDQISRALALGLTPVHNSQVELHLYFPQHDHVDFCKKHNISVTSYATLGSPGRVNFTLPTGQKLDWAPAPSDLQDQNVLALAEKTHKTPAQVLLRYALDRGCAILPKSIQENRIKENFEVFDFSLTEEDIAKLEESKNSQRLFLQDFMTGHPEDAFAAERK
2NYH , Knot 60 117 0.81 38 94 113 
GMTFRDTSAIASWHAHVYFDASSRDAAWTLREQIEAHWSGKLQLGRFHERPVGPHPMWSYQLAFTQEQFADLVGWLTLNHGALDIFLHPNTGDALRDHRDAAVWIGHSHELVLSALN
9CUM , Knot 39 62 0.86 40 60 60 
VFPSNATFGMGDRVRKKSGAAWQGQIVGWYCTNLTPEGYAVESEAHPGSVHIYPVAALERIN

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 \(c\).

\(|P_{f(1QWK_1)}(2) \setminus P_{f(2NYH_1)}(2)|=136\), \(|P_{f(2NYH_1)}(2) \setminus P_{f(1QWK_1)}(2)|=36\). 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 1:10010101010011011111110100011011011001101100110010100000111011001100111000011100011000111101011100010010100101011011111000100011011001100101100111101111001000010011111101100001010101100001010000010100010110110101011010010111110010000111110000001101110011001011110010000100010110101000011010000000011100110101001111000
Pair \(Z_2\) Length of longest common subsequence
1QWK_1,2NYH_1 172 4
1QWK_1,9CUM_1 176 3
2NYH_1,9CUM_1 118 3

Newick tree

 
[
	1QWK_1:94.51,
	[
		2NYH_1:59,9CUM_1:59
	]:35.51
]

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 \((0.85)(0.8)(\frac{434 }{\log_{20} 434}-\frac{117}{\log_{20}117})=95.5\)
Status Protein1 Protein2 d d1/2
Query variables 1QWK_1 2NYH_1 120 81.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]