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Parikh vectors
1XKI_1 1QTP_1 4ZKU_1 Letter Amino acid
2 5 0 C Cysteine
4 23 12 Q Glutamine
5 2 3 H Histidine
12 13 10 K Lycine
2 1 1 W Tryptophan
5 3 6 Y Tyrosine
13 26 18 L Leucine
8 12 5 P Proline
12 13 17 S Serine
7 10 12 R Arginine
4 18 13 N Asparagine
8 10 17 D Aspartic acid
7 15 14 I Isoleucine
4 19 3 F Phenylalanine
11 11 35 A Alanine
18 13 11 E Glutamic acid
14 16 16 G Glycine
4 4 2 M Methionine
11 19 24 T Threonine
11 14 18 V Valine 

1XKI_1|Chain A|Von Ebner's gland protein|Homo sapiens (9606)
>1QTP_1|Chain A|AP-2 CLATHRIN ADAPTOR ALPHA SUBUNIT (ALPHA-ADAPTIN C)|Mus musculus (10090)
>4ZKU_1|Chains A, B|Tail needle protein gp26|Enterobacteria phage P22 (10754)
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 \(w=f(c)\)
1XKI , Knot 79 162 0.82 40 128 156 
ASDEEIQDVSGTWYLKAMTVDREFPEMNLESVTPMTLTTLEGGNLEAKVTMLISGRCQEVKAVLEKTDEPGKYTADGGKHVAYIIRSHVKDHYIFYSEGELHGKPVRGVKLVGRDPKNNLEALEDFEKAAGARGLSTESILIPRQSETCSPGSAWSHPQFEK
1QTP , Knot 115 247 0.85 40 165 239 
GSPGIRLGSSEDNFARFVCKNNGVLFENQLLQIGLKSEFRQNLGRMFIFYGNKTSTQFLNFTPTLICADDLQTNLNLQTKPVDPTVDGGAQVQQVVNIECISDFTEAPVLNIQFRYGGTFQNVSVKLPITLNKFFQPTEMASQDFFQRWKQLSNPQQEVQNIFKAKHPMDTEITKAKIIGFGSALLEEVDPNPANFVGAGIIHTKTTQIGCLLRLEPNLQAQMYRLTLRTSKDTVSQRLCELLSEQF
4ZKU , Knot 105 237 0.80 38 149 221 
GPGSMADPSLNNPVVIQATRLDASILPRNVFSKSYLLYVIAQGTDVGAIAGKANEAGQGAYDAQVKNDEQDVELADHEARIKQLRIDVDDHESRITANTKAITALNVRVTTAEGEIASLQTNVSALDGRVTTAENNISALQADYVSKTATTSQSLASPLNVTTSYSVGGKKVLGARQTGWTAATGTANKGVFDADLTFAVSDTYTQSEIQAIANALITERRRTKAMEDALRAHGLID

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 \(c\).

\(|P_{f(1XKI_1)}(2) \setminus P_{f(1QTP_1)}(2)|=56\), \(|P_{f(1QTP_1)}(2) \setminus P_{f(1XKI_1)}(2)|=93\). 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 1:100001001010101011010001101010010110100101101010101110100001011100000110001011001101100010000110001010101101101110010001011001001111011000011110000000110110010100
Pair \(Z_2\) Length of longest common subsequence
1XKI_1,1QTP_1 149 3
1XKI_1,4ZKU_1 145 3
1QTP_1,4ZKU_1 158 4

Newick tree

 
[
	1QTP_1:78.15,
	[
		1XKI_1:72.5,4ZKU_1:72.5
	]:5.65
]

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 \((0.85)(0.8)(\frac{409 }{\log_{20} 409}-\frac{162}{\log_{20}162})=73.6\)
Status Protein1 Protein2 d d1/2
Query variables 1XKI_1 1QTP_1 93 76.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]