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Parikh vectors
5TAS_1 5YUP_1 2NXC_1 Letter Amino acid
5 9 18 R Arginine
2 5 0 C Cysteine
2 2 5 H Histidine
6 6 6 I Isoleucine
7 15 38 L Leucine
9 9 8 K Lycine
6 15 36 A Alanine
5 10 12 D Aspartic acid
13 18 29 G Glycine
7 14 21 P Proline
4 6 3 N Asparagine
4 30 2 S Serine
0 5 9 W Tryptophan
7 8 24 E Glutamic acid
3 3 4 M Methionine
6 7 9 F Phenylalanine
8 21 7 T Threonine
2 11 5 Y Tyrosine
7 20 17 V Valine
5 5 1 Q Glutamine 

5TAS_1|Chains A, B[auth F], C[auth H], D[auth J]|Peptidyl-prolyl cis-trans isomerase FKBP1B|Homo sapiens (9606)
>5YUP_1|Chain A[auth H]|the heavy chain of the Fab fragment of FVIIa antibody mAb4F5|Mus musculus (10090)
>2NXC_1|Chain A|Ribosomal protein L11 methyltransferase|Thermus thermophilus (300852)
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 \(w=f(c)\)
5TAS , Knot 57 108 0.82 38 92 106 
MGVEIETISPGDGRTFPKKGQTCVVHYTGMLQNGKKFDSSRDRNKPFKFRIGKQEVIKGFEEGAAQMSLGQRAKLTCTPDVAYGATGHPGVIPPNATLIFDVELLNLE
5YUP , Knot 101 219 0.82 40 152 210 
EVKLVESGGGLVQPGGSRKLSCAASGFTFSDYGMAWVRQAPGKGPEWIAFISNLSRIYYADTVTGRFTISRENDKNTLFLEMSSLRSEDSAIYYCTRDDGYYRVDYWGQGATLTVSSAKTTPPSVYPLAPGSAAQTNSMVTLGCLVKGYFPEPVTVTWNSGSLSSGVHTFPAVLQSDLYTLSSSVTVPSSTWPSETVTCNVAHPASSTKVDKKIVPRDC
2NXC , Knot 112 254 0.81 38 143 236 
MWVYRLKGTLEALDPILPGLFDGGARGLWEREGEVWAFFPAPVDLPYEGVWEEVGDEDWLEAWRRDLKPALAPPFVVLAPWHTWEGAEIPLVIEPGMAFGTGHHETTRLALKALARHLRPGDKVLDLGTGSGVLAIAAEKLGGKALGVDIDPMVLPQAEANAKRNGVRPRFLEGSLEAALPFGPFDLLVANLYAELHAALAPRYREALVPGGRALLTGILKDRAPLVREAMAGAGFRPLEEAAEGEWVLLAYGR

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 \(c\).

\(|P_{f(5TAS_1)}(2) \setminus P_{f(5YUP_1)}(2)|=45\), \(|P_{f(5YUP_1)}(2) \setminus P_{f(5TAS_1)}(2)|=105\). 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 1:111010010110100110010001100011100100100000000110101100011011001110101100101000101101101011111101011101011010
Pair \(Z_2\) Length of longest common subsequence
5TAS_1,5YUP_1 150 3
5TAS_1,2NXC_1 141 3
5YUP_1,2NXC_1 157 3

Newick tree

 
[
	5YUP_1:78.74,
	[
		5TAS_1:70.5,2NXC_1:70.5
	]:8.24
]

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 \((0.85)(0.8)(\frac{327 }{\log_{20} 327}-\frac{108}{\log_{20}108})=68.0\)
Status Protein1 Protein2 d d1/2
Query variables 5TAS_1 5YUP_1 88 64.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]