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Parikh vectors
3LKH_1 6EUJ_1 1MEL_1 Letter Amino acid
42 16 6 R Arginine
9 13 2 W Tryptophan
15 21 2 F Phenylalanine
50 33 15 S Serine
40 29 9 T Threonine
54 23 14 A Alanine
26 36 8 D Aspartic acid
26 22 6 I Isoleucine
29 45 3 K Lycine
12 12 3 M Methionine
16 32 4 N Asparagine
32 35 24 G Glycine
59 33 7 L Leucine
31 16 5 P Proline
32 26 8 V Valine
20 7 4 C Cysteine
19 17 7 Q Glutamine
28 22 5 E Glutamic acid
12 5 1 H Histidine
24 32 15 Y Tyrosine 

3LKH_1|Chains A, B|RNA-directed RNA polymerase|Hepatitis C virus (3052230)
>6EUJ_1|Chains A, B, C, D|Beta-glucanase|Bacteroides thetaiotaomicron (strain ATCC 29148 / DSM 2079 / NCTC 10582 / E50 / VPI-5482) (226186)
>1MEL_1|Chains A, B|VH SINGLE-DOMAIN ANTIBODY|Camelus dromedarius (9838)
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 \(w=f(c)\)
3LKH , Knot 232 576 0.85 40 277 537 
SMSYTWTGALITPCAAEESKLPINPLSNSLLRHHNMVYATTSRSASLRQKKVTFDRLQVLDDHYRDVLKEMKAKASTVKAKLLSIEEACKLTPPHSAKSKFGYGAKDVRNLSSRAVNHIRSVWEDLLEDTETPIDTTIMAKSEVFCVQPEKGGRKPARLIVFPDLGVRVCEKMALYDVVSTLPQAVMGSSYGFQYSPKQRVEFLVNTWKSKKCPMGFSYDTRCFDSTVTESDIRVEESIYQCCDLAPEARQAIRSLTERLYIGGPLTNSKGQNCGYRRCRASGVLTTSCGNTLTCYLKATAACRAAKLQDCTMLVNGDDLVVICESAGTQEDAAALRAFTEAMTRYSAPPGDPPQPEYDLELITSCSSNVSVAHDASGKRVYYLTRDPTTPLARAAWETARHTPINSWLGNIIMYAPTLWARMILMTHFFSILLAQEQLGKALDCQIYGACYSIEPLDLPQIIERLHGLSAFTLHSYSPGEINRVASCLRKLGVPPLRTWRHRARSVRAKLLSQGGRAAICGRYLFNWAVRTKLKLTPIPAASQLDLSGWFVAGYSGGDIYHSLSRARPRENLYFQ
6EUJ , Knot 196 475 0.84 40 244 455 
QNTQISPGVLWNDIDGEQINAHGGCVVYEKGTYYWFGEDRTGFKSNGVSCYQSKDLYNWKRLGLSMKTTGEAREDMNDISQGRLFERPKVIYNPQTKKWVMWSHWESGDGYGAARVCVATSDKIMGPYVLYKTFRPNKNESRDQTLFVDTDGKAYHFCSTDMNTNMNIALLRDDYLEPTPTETKILKGLKYEAPAIFKVGDMYFGLFSGCTGWEPNPGRSAYSTDILGNWTTGNNFAVDKLKQVTYNSQSCYVFKVEGKEKAYIYMGDRWNSKDVGKSHHVWLPISMRSGYPVVKWYDQWDLTVFNSMYRYKRAAEIIPGNIYSLLEKTSDRLVSKPANGFSIADDDDDINLSLEFIKTNIPNVYKIKDTKTGKFLESLFGTLRLNPEKKDDAQCWVFNLQEDGYYQIQNLKDKKYVTVSGSNTFAGSNLYLTELSKKLMQDFAVYFDSNKYKYKEADIFSDAYKANNLKQMKAQ
1MEL , Knot 72 148 0.81 40 104 140 
DVQLQASGGGSVQAGGSLRLSCAASGYTIGPYCMGWFRQAPGKEREGVAAINMGGGITYYADSVKGRFTISQDNAKNTVYLLMNSLEPEDTAIYYCAADSTIYASYYECGHGLSTGGYGYDSWGQGTQVTVSSGRYPYDVPDYGSGRA

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 \(c\).

\(|P_{f(3LKH_1)}(2) \setminus P_{f(6EUJ_1)}(2)|=98\), \(|P_{f(6EUJ_1)}(2) \setminus P_{f(3LKH_1)}(2)|=65\). 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 1:010001011110101100001110110001100001101000001010000101001011000000110010101001010110100100101100100011011001001000110010011001100000110001110001101010011001101111101110100011100110011011110001100010001011100100000111100000010001000010100010000011101001100100010111110000100010000010111000010010001010110011010000111010011110001100001111011001100001111011010001011000000101100101001001000100111011100100011001110111011011101111001101111000110110001011000101101101100101101101000011010011001001111110010001001010110011011101001101110001010111110010101111110011010001001010001010
Pair \(Z_2\) Length of longest common subsequence
3LKH_1,6EUJ_1 163 4
3LKH_1,1MEL_1 213 4
6EUJ_1,1MEL_1 200 4

Newick tree

 
[
	1MEL_1:10.60,
	[
		3LKH_1:81.5,6EUJ_1:81.5
	]:28.10
]

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 \((0.85)(0.8)(\frac{1051 }{\log_{20} 1051}-\frac{475}{\log_{20}475})=150.\)
Status Protein1 Protein2 d d1/2
Query variables 3LKH_1 6EUJ_1 195 176
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]