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Parikh vectors
8RBJ_1 2ZKF_1 7WQV_1 Letter Amino acid
5 3 8 C Cysteine
17 6 1 H Histidine
25 12 13 T Threonine
6 5 2 W Tryptophan
25 19 9 R Arginine
9 6 7 Q Glutamine
33 27 17 G Glycine
16 4 17 F Phenylalanine
19 16 22 V Valine
40 12 15 A Alanine
32 12 7 E Glutamic acid
21 5 10 I Isoleucine
39 11 11 K Lycine
4 4 2 M Methionine
27 12 18 S Serine
19 10 18 Y Tyrosine
13 12 19 N Asparagine
32 11 9 D Aspartic acid
35 11 17 L Leucine
17 12 12 P Proline 

8RBJ_1|Chain A|tRNA ligase|Thermochaetoides thermophila (209285)
>2ZKF_1|Chain A|E3 ubiquitin-protein ligase UHRF1|Mus musculus (10090)
>7WQV_1|Chains A, C, E, G, I, K, M, O|Spike protein S1|Severe acute respiratory syndrome coronavirus 2 (2697049)
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 \(w=f(c)\)
8RBJ , Knot 180 434 0.84 40 231 412 
MGSSHHHHHHSSGLVPRGSHMDGTAEKQRDSLASRYKASTDYAKDAEGLTAPYVSQDPQETAALVRALDDAAKKGKKSGGFSVKKTRYAVASSPTGAEVDSWRFNDWDYKKPDLPTYARGLFTTRTQHGVPEIAVRGYDKFFNIDETRDTAWSAIRERTKGPYELTLKENGCIIFISGLEDGTLLVCSKHSTGDRSDVALSHSSAGEKHLEAQLERIGKTKEELARELRKRNATAVAELCDDSFEEHILAYGPDKAGLYLHGINLNIPEFITYPSPLVQKFAEDWGFRKTGLIIIDNIDDVKAFLEEVAETGAHDGRDVEGFVIRCKKSTNPGVGPYHDWFFKYKFEEPYLMYRQWRECTKALISGKQPKIKKHVKITEEYLLYARKRLAADPKLAKLYNQNHGIIKLRNDFLEYKNMKGTDAANLEDDGAASV
2ZKF , Knot 99 210 0.84 40 155 208 
SGMACVGRTTECTIVPANHFGPIPGVPVGTMWRFRVQVSESGVHRPHVAGIHGRSNDGAYSLVLAGGYEDDVDNGNYFTYTGSGGRDLSGNKRTAGQSSDQKLTNNNRALALNCHSPINEKGAEAEDWRQGKPVRVVRNMKGGKHSKYAPAEGNRYDGIYKVVKYWPERGKSGFLVWRYLLRRDDTEPEPWTREGKDRTRQLGLTMQYPE
7WQV , Knot 106 234 0.82 40 156 226 
MKFLVNVALVFMVVYISYIYAAGSPNITNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKKSTGTLEVLFQ

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 \(c\).

\(|P_{f(8RBJ_1)}(2) \setminus P_{f(2ZKF_1)}(2)|=127\), \(|P_{f(2ZKF_1)}(2) \setminus P_{f(8RBJ_1)}(2)|=51\). 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 1:11000000000011110100101010000001100001000010010110110100010001111011001100100011101000001110010110100101001000010110010111000000111011101000110100000011011000001100101000101111011001011100000010000111000011000101010011000001100100001011101000010001110110011101011010110110010111001100111000111110010010111001100110010010111100000001111100011100010010110001000001110100101000101000011010001110101101000001110100011000010100110100011101
Pair \(Z_2\) Length of longest common subsequence
8RBJ_1,2ZKF_1 178 4
8RBJ_1,7WQV_1 173 4
2ZKF_1,7WQV_1 181 3

Newick tree

 
[
	2ZKF_1:90.81,
	[
		8RBJ_1:86.5,7WQV_1:86.5
	]:4.31
]

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 \((0.85)(0.8)(\frac{644 }{\log_{20} 644}-\frac{210}{\log_{20}210})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 8RBJ_1 2ZKF_1 154 113.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]