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Parikh vectors
8DXU_1 3TYQ_1 1EFQ_1 Letter Amino acid
6 28 4 E Glutamic acid
16 32 8 G Glycine
7 26 5 I Isoleucine
11 31 6 P Proline
15 24 9 Y Tyrosine
8 20 2 C Cysteine
9 26 6 D Aspartic acid
9 29 6 K Lycine
11 40 8 T Threonine
12 54 6 A Alanine
0 12 1 M Methionine
14 15 3 F Phenylalanine
16 50 18 S Serine
16 32 6 V Valine
6 19 8 Q Glutamine
17 16 3 N Asparagine
7 12 0 H Histidine
13 59 9 L Leucine
2 9 2 W Tryptophan
9 42 4 R Arginine 

8DXU_1|Chain A[auth R]|Spike protein S1|Severe acute respiratory syndrome coronavirus 2 (2697049)
>3TYQ_1|Chains A, B|RNA-directed RNA polymerase|Hepatitis C virus (420174)
>1EFQ_1|Chain A|KAPPA-4 IMMUNOGLOBULIN (LIGHT CHAIN)|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 \(w=f(c)\)
8DXU , Knot 93 204 0.80 38 142 194 
TNLCPFGEVFNATRFASVYAWNRKRISNCVADYSVLYNSASFSTFKCYGVSPTKLNDLCFTNVYADSFVIRGDEVRQIAPGQTGKIADYNYKLPDDFTGCVIAWNSNNLDSKVGGNYNYLYRLFRKSNLKPFERDISTEIYQAGSTPCNGVEGFNCYFPLQSYGFQPTNGVGYQPYRVVVLSFELLHAPATVCGPKGSHHHHHH
3TYQ , Knot 232 576 0.85 40 277 537 
SMSYTWTGALITPCAAEESKLPINPLSNSLLRHHNMVYATTSRSASLRQKKVTFDRLQVLDDHYRDVLKEMKAKASTVKAKLLSIEEACKLTPPHSAKSKFGYGAKDVRNLSSRAVNHIRSVWEDLLEDTETPIDTTIMAKSEVFCVQPEKGGRKPARLIVFPDLGVRVCEKMALYDVVSTLPQAVMGSSYGFQYSPKQRVEFLVNTWKSKKCPMGFSYDTRCFDSTVTESDIRVEESIYQCCDLAPEARQAIRSLTERLYIGGPLTNSKGQNCGYRRCRASGVLTTSCGNTLTCYLKATAACRAAKLQDCTMLVNGDDLVVICESAGTQEDAAALRAFTEAMTRYSAPPGDPPQPEYDLELITSCSSNVSVAHDASGKRVYYLTRDPTTPLARAAWETARHTPINSWLGNIIMYAPTLWARMILMTHFFSILLAQEQLGKALDCQIYGACYSIEPLDLPQIIERLHGLSAFTLHSYSPGEINRVASCLRKLGVPPLRTWRHRARSVRAKLLSQGGRAAICGRYLFNWAVRTKLKLTPIPAASQLDLSGWFVAGYSGGDIYHSLSRARPRENLYFQ
1EFQ , Knot 59 114 0.81 38 93 110 
DIVMTQSPDSLAVSLGERATINCKSSQSVLYSSNSKNYLAWYQDKPGQPPKLLIYWASTRESGVPDRFSGSGSGTDFTLTISSLQAEDVAVYYCQQYYSTPYSFGQGTKLEIKR

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 \(c\).

\(|P_{f(8DXU_1)}(2) \setminus P_{f(3TYQ_1)}(2)|=34\), \(|P_{f(3TYQ_1)}(2) \setminus P_{f(8DXU_1)}(2)|=169\). 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 1:001011101101001101011000010001100011000101001000110100100101001010011101001001111001011000001100101011110000100011100001001100001011000100010011001001101100011100011010011100100111101011011101011010000000
Pair \(Z_2\) Length of longest common subsequence
8DXU_1,3TYQ_1 203 5
8DXU_1,1EFQ_1 133 4
3TYQ_1,1EFQ_1 214 3

Newick tree

 
[
	3TYQ_1:11.13,
	[
		8DXU_1:66.5,1EFQ_1:66.5
	]:47.63
]

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 \((0.85)(0.8)(\frac{780 }{\log_{20} 780}-\frac{204}{\log_{20}204})=160.\)
Status Protein1 Protein2 d d1/2
Query variables 8DXU_1 3TYQ_1 207 137.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]