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Parikh vectors
3QGA_1 6PYW_1 6WGV_1 Letter Amino acid
1 5 2 C Cysteine
4 16 4 Q Glutamine
21 18 15 G Glycine
21 26 5 E Glutamic acid
15 7 8 I Isoleucine
10 16 12 P Proline
6 12 13 S Serine
1 8 3 W Tryptophan
18 24 27 A Alanine
10 4 6 N Asparagine
20 21 23 L Leucine
12 13 14 V Valine
9 22 11 T Threonine
4 13 5 Y Tyrosine
8 25 15 R Arginine
12 20 18 D Aspartic acid
9 10 6 H Histidine
28 8 4 K Lycine
6 2 2 M Methionine
10 6 3 F Phenylalanine 

3QGA_1|Chains A, C[auth D], E[auth G], G[auth J], I[auth M], K[auth P]|Fusion of urease beta and gamma subunits|Helicobacter mustelae (679897)
>6PYW_1|Chain A|HLA class I histocompatibility antigen, B-27 alpha chain|Homo sapiens (9606)
>6WGV_1|Chain A|Corrinoid adenosyltransferase|Mycobacterium tuberculosis (1773)
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 \(w=f(c)\)
3QGA , Knot 101 225 0.81 40 154 217 
MKLTPKEQEKFLLYYAGEVARKRKEEGLKLNQPEAIAYISAHIMDEARRGKKTVAQLMEECVHFLKKDEVMPGVGNMVPDLGVEANFPDGTKLVTVNWPIEPDDFKAGEIKFASDKDIELNAGKEITELKVTNKGPKSLHVGSHFHFFEANRALEFDREKAYGKRLDIPSGNTLRIGAGETKTVHLIPIGGSKKIIGMNGLLNGIADDLHKQKALEKAKHHGFIK
6PYW , Knot 123 276 0.83 40 179 265 
GSHSMRYFHTSVSRPGRGEPRFITVGYVDDTLFVRFDSDAASPREEPRAPWIEQEGPEYADRETQICKAKAQTDREDLRTLLRYYNQSEAGSHTLQNMYGCDVGPDGRLLRGYHQDAYDGKDYIALNEDLSSWTAADTAAQITQRKWEAARVAEQLRAYLEGECVEWLRRYLENGKETLQRADPPKTHVTHHPISDHEATLRCWALGFYPAEITLTWQRDGEDQTQDTELVETRPAGDRTFQKWAAVVVPSGEEQRYTCHVQHEGLPKPLTLRWEP
6WGV , Knot 91 196 0.81 40 131 187 
GSHMAVHLTRIYTRTGDDGTTGLSDMSRVAKTDARLVAYADCDEANAAIGAALALGHPDTQITDVLRQIQNDLFDAGADLSTPIVENPKHPPLRIAQSYIDRLEGWCDAYNAGLPALKSFVLPGGSPLSALLHVARTVVRRAERSAWAAVDAHPEGVSVLPAKYLNRLSDLLFILSRVANPDGDVLWRPGGDRTAS

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 \(c\).

\(|P_{f(3QGA_1)}(2) \setminus P_{f(6PYW_1)}(2)|=73\), \(|P_{f(6PYW_1)}(2) \setminus P_{f(3QGA_1)}(2)|=98\). 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 1:101010000011100110110000001101001011101010110010010001101100010110000111111011101110101101001101011101001011010110000101011001001010001100101100101101001101000010100101101001011110000101111110001111011101110010000110010001110
Pair \(Z_2\) Length of longest common subsequence
3QGA_1,6PYW_1 171 3
3QGA_1,6WGV_1 157 3
6PYW_1,6WGV_1 146 4

Newick tree

 
[
	3QGA_1:84.88,
	[
		6WGV_1:73,6PYW_1:73
	]:11.88
]

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 \((0.85)(0.8)(\frac{501 }{\log_{20} 501}-\frac{225}{\log_{20}225})=79.5\)
Status Protein1 Protein2 d d1/2
Query variables 3QGA_1 6PYW_1 105 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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]