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Parikh vectors
2XHA_1 5UIQ_1 4BGQ_1 Letter Amino acid
15 10 16 R Arginine
19 28 30 E Glutamic acid
8 17 9 T Threonine
8 21 10 A Alanine
1 15 9 H Histidine
0 1 3 W Tryptophan
9 10 11 Y Tyrosine
23 21 20 V Valine
12 21 18 G Glycine
15 19 16 I Isoleucine
12 29 42 L Leucine
1 5 5 C Cysteine
12 21 12 D Aspartic acid
4 12 10 Q Glutamine
24 23 24 K Lycine
2 11 8 M Methionine
4 13 12 F Phenylalanine
11 9 16 P Proline
8 21 14 S Serine
5 16 19 N Asparagine 

2XHA_1|Chains A, B|TRANSCRIPTION ANTITERMINATION PROTEIN NUSG|THERMOTOGA MARITIMA (2336)
>5UIQ_1|Chains A, B, C, D|Interleukin-1 receptor-associated kinase 4|Homo sapiens (9606)
>4BGQ_1|Chain A|CYCLIN-DEPENDENT KINASE-LIKE 5|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 \(w=f(c)\)
2XHA , Knot 85 193 0.77 38 126 185 
PEEVVLDATSPSERLILSPKAKLHVNNGKDVNKGDLIAEEPPIYARRSGVIVDVKNVRKIVVETIDRKYTKTYYIPESAGIEPGLRVGTKVKQGLPLSKNEEYICELDGKIVEIERMKKVVVQTPDGEQDVYYIPLDVFDRDRIKKGKEVKQGEMLAEARKFFAKVSGRVEVVDYSTRKEIRIYKTKRRKLFP
5UIQ , Knot 139 323 0.82 40 198 305 
MHHHHHHGGENLYFQGENKSLEVSDTRFHSFSFYELKNVTNNFDERPISVGGNKMGEGGFGVVYKGYVNNTTVAVKKLAAMVDITTEELKQQFDQEIKVMAKCQHENLVELLGFSSDGDDLCLVYVYMPNGSLLDRLSCLDGTPPLSWHMRCKIAQGAANGINFLHENHHIHRDIKSANILLDEAFTAKISDFGLARASEKFAQTVMTSRIVGTTAYMAPEALRGEITPKSDIYSFGVVLLEIITGLPAVDEHREPQLLLDIKEEIEDEEKTIEDYIDKKMNDADSTSVEAMYSVASQCLHEKKNKRPDIKKVQQLLQEMTAS
4BGQ , Knot 137 304 0.86 40 198 291 
SMKIPNIGNVMNKFEILGVVGEGAYGVVLKCRHKETHEIVAIKKFKDSEENEEVKETTLRELKMLRTLKQENIVELKEAFRRRGKLYLVFEYVEKNMLELLEEMPNGVPPEKVKSYIYQLIKAIHWCHKNDIVHRDIKPENLLISHNDVLKLCDFGFARNLSEGNNANYDEEVATRWYRSPELLLGAPYGKSVDMWSVGCILGELSDGQPLFPGESEIDQLFTIQKVLGPLPSEQMKLFYSNPRFHGLRFPAVNHPQSLERRYLGILNSVLLDLMKNLLKLDPADRYLTEQCLNHPTFQTQRLL

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 \(c\).

\(|P_{f(2XHA_1)}(2) \setminus P_{f(5UIQ_1)}(2)|=44\), \(|P_{f(5UIQ_1)}(2) \setminus P_{f(2XHA_1)}(2)|=116\). 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 1:1001110100100011101010101001001001011100111010001111010010011100100000000011001110111011001001111000000100101011010010011100101000100111011000010010010010111010011101010101100000001010000000111
Pair \(Z_2\) Length of longest common subsequence
2XHA_1,5UIQ_1 160 3
2XHA_1,4BGQ_1 170 3
5UIQ_1,4BGQ_1 164 5

Newick tree

 
[
	4BGQ_1:84.65,
	[
		2XHA_1:80,5UIQ_1:80
	]:4.65
]

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 \((0.85)(0.8)(\frac{516 }{\log_{20} 516}-\frac{193}{\log_{20}193})=93.5\)
Status Protein1 Protein2 d d1/2
Query variables 2XHA_1 5UIQ_1 117 92.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]