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Parikh vectors
5WCI_1 5UBB_1 1GPO_1 Letter Amino acid
10 12 7 R Arginine
14 15 12 D Aspartic acid
12 12 10 Q Glutamine
7 3 4 W Tryptophan
20 6 8 Y Tyrosine
16 13 12 I Isoleucine
30 24 14 L Leucine
26 17 34 S Serine
11 12 10 F Phenylalanine
18 17 11 V Valine
8 6 12 N Asparagine
16 14 12 E Glutamic acid
14 4 3 H Histidine
29 11 11 K Lycine
4 7 1 M Methionine
17 8 10 P Proline
11 6 21 T Threonine
7 10 9 A Alanine
8 6 5 C Cysteine
17 19 13 G Glycine 

5WCI_1|Chain A|Histone acetyltransferase KAT8|Homo sapiens (9606)
>5UBB_1|Chain A|Alpha N-terminal protein methyltransferase 1B|Homo sapiens (9606)
>1GPO_1|Chains A[auth L], C[auth M]|ANTIBODY M41|Mus musculus (10090)
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 \(w=f(c)\)
5WCI , Knot 131 295 0.84 40 191 281 
MGSSHHHHHHSSGLVPRGSTKVKYVDKIHIGNYEIDAWYFSPFPEDYGKQPKLWLCEYCLKYMKYEKSYRFHLGQCQWRQPPGKEIYRKSNISVHEVDGKDHKIYCQNLCLLAKLFLDHKTLYFDVEPFVFYILTEVDRQGAHIVGYFSKEKESPDGNNVACILTLPPYQRRGYGKFLIAFSYELSKLESTVGSPEKPLSDLGKLSYRSYWSWVLLENLRDFRGTLSIKDLSQMTSITQNDIISTLQSLNMVKYWKGQHVICVTPKLVEEHLKSAQYKKPPITVDSVCLKWAPPK
5UBB , Knot 107 222 0.86 40 167 220 
GTSQVINGEMQFYARAKLFYQEVPATEEGMMGNFIELSSPDIQASQKFLRKFVGGPGRAGTDCALDCGSGIGRVSKHVLLPVFNSVELVDMMESFLLEAQNYLQVKGDKVESYHCYSLQEFTPPFRRYDVIWIQWVSGHLTDKDLLAFLSRCRDGLKENGIIILKDNVAREGCILDLSDSSVTRDMDILRSLIRKSGLVVLGQEKQDGFPEQCIPVWMFALH
1GPO , Knot 102 219 0.83 40 155 208 
DIELTQSPATLSVTPGNSVSISCRASQSLVNEDGNTYLFWYQQKSHESPRLLIKYASQSISGIPSRFSGSGSGTDFTLSINSVETEDLAVYFCQQITDWPFTFGGGTKLEIKRADAAPTVSIFPPSSEQLTSGGASVVCFLNNFYPKDINVKWKIDGSERQNGVLNSWTDQDSKDSTYSMSSTLTLTKDEYERHNSYTCEATHKTSTSPIVKSFNRNEC

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 \(c\).

\(|P_{f(5WCI_1)}(2) \setminus P_{f(5UBB_1)}(2)|=99\), \(|P_{f(5UBB_1)}(2) \setminus P_{f(5WCI_1)}(2)|=75\). 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 1:1100000000001111010001001001011000101101011100010010111000010010000000101100010011100100000101001010000100001011101110000101010111101100100011011101000000101001101101110000101011111000100100011010011001101000001011110010010101010010010010000110010010110010100110101011000100100001110100101011110
Pair \(Z_2\) Length of longest common subsequence
5WCI_1,5UBB_1 174 4
5WCI_1,1GPO_1 174 4
5UBB_1,1GPO_1 180 4

Newick tree

 
[
	1GPO_1:89.01,
	[
		5WCI_1:87,5UBB_1:87
	]:2.01
]

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 \((0.85)(0.8)(\frac{517 }{\log_{20} 517}-\frac{222}{\log_{20}222})=84.8\)
Status Protein1 Protein2 d d1/2
Query variables 5WCI_1 5UBB_1 107 93
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]