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Parikh vectors
4LNA_1 3ZTE_1 2BTS_1 Letter Amino acid
2 1 3 C Cysteine
8 7 21 K Lycine
18 0 19 P Proline
21 7 23 V Valine
11 3 16 R Arginine
15 3 9 Q Glutamine
23 6 16 G Glycine
23 5 16 I Isoleucine
33 3 39 L Leucine
23 7 17 T Threonine
1 0 4 W Tryptophan
17 4 19 A Alanine
11 3 10 H Histidine
13 4 5 M Methionine
9 0 10 Y Tyrosine
7 6 8 N Asparagine
16 7 17 E Glutamic acid
12 3 16 F Phenylalanine
16 5 14 S Serine
14 4 16 D Aspartic acid 

4LNA_1|Chain A|Purine nucleoside phosphorylase|Spirosoma linguale (504472)
>3ZTE_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V|TRYPTOPHAN OPERON RNA-BINDING ATTENUATION PROTEIN (TRAP)|BACILLUS LICHENIFORMIS (1402)
>2BTS_1|Chain A|CELL DIVISION PROTEIN KINASE 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 \(w=f(c)\)
4LNA , Knot 127 293 0.82 40 186 281 
MHHHHHHSSGVDLGTENLYFQSMFEQIQETTQFIQSKITLRPAIGIILGTGLGALTNELDIDTTIPYETIPHFPLSTVEFHSGKLLIGTLGGKSVVVMQGRFHYYEGYTMQQVTYPVRVMHALGIQTLLVSNAAGGMNPTFQTSDLMVIDDHISLLLPQNPLICPNPPIFGDRFPDMSEPYRKSLIDLAFSVAAELDIPLKRGVYVSVTGPQLETRAEYRMLRQWGADAVGMSTVPEVIVANQLGMDVFGISVITDLCFPDTLEKAELVKILATAAQAEPKLTMLIREMIGRL
3ZTE , Knot 43 78 0.80 34 70 76 
MNMTNNQSNDFVVIKAVEDGVNVIGLTRGTDTRFHHSEKLDKGEVMICQFTEHTSAIKVRGEALIQTANGEMKSESKK
2BTS , Knot 134 298 0.85 40 191 284 
MENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHLRL

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 \(c\).

\(|P_{f(4LNA_1)}(2) \setminus P_{f(3ZTE_1)}(2)|=141\), \(|P_{f(3ZTE_1)}(2) \setminus P_{f(4LNA_1)}(2)|=25\). 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 1:10000000011011000101001100100000110001010111111110111110001010001100011011100101001011110111001111010100001001001001101101111001110011111010100001111000101111001110101111100110100100001101110111010111001101010110100010001100111011110011011110011101111011001011001001011011101101010101110011101
Pair \(Z_2\) Length of longest common subsequence
4LNA_1,3ZTE_1 166 3
4LNA_1,2BTS_1 165 4
3ZTE_1,2BTS_1 187 3

Newick tree

 
[
	3ZTE_1:90.28,
	[
		4LNA_1:82.5,2BTS_1:82.5
	]:7.78
]

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 \((0.85)(0.8)(\frac{371 }{\log_{20} 371}-\frac{78}{\log_{20}78})=91.2\)
Status Protein1 Protein2 d d1/2
Query variables 4LNA_1 3ZTE_1 113 70.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]