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Parikh vectors
6RNQ_1 3VYA_1 6IWY_1 Letter Amino acid
7 2 19 Q Glutamine
3 3 10 M Methionine
9 9 30 T Threonine
4 1 0 W Tryptophan
38 13 30 A Alanine
4 6 35 N Asparagine
15 5 34 K Lycine
6 6 9 F Phenylalanine
5 5 9 P Proline
15 14 26 V Valine
21 8 25 E Glutamic acid
6 5 27 I Isoleucine
13 14 38 G Glycine
12 2 3 H Histidine
15 4 27 D Aspartic acid
5 0 0 C Cysteine
13 4 39 S Serine
9 2 6 Y Tyrosine
13 9 8 R Arginine
39 10 37 L Leucine 

6RNQ_1|Chains A, B|Gem-associated protein 5|Homo sapiens (9606)
>3VYA_1|Chain A|FMN-binding protein|Desulfovibrio vulgaris (883)
>6IWY_1|Chain A|Flagellar hook-associated protein 2|Helicobacter pylori (strain ATCC 700392 / 26695) (85962)
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 \(w=f(c)\)
6RNQ , Knot 111 252 0.81 40 157 239 
ARSLLPLSTSLDHRSKEELHQDCLVLATAKHSRELNEDVSADVEERFHLGLFTDRATLYRMIDIEGKGHLENGHPELFHQLMLWKGDLKGVLQTAAERGELTDNLVAMAPAAGYHVWLWAVEAFAKQLCFQDQYVKAASHLLSIHKVYEAVELLKSNHFYREAIAIAKARLRPEDPVLKDLYLSWGTVLERDGHYAVAAKCYLGATCAYDAAKVLAKKGDAASLRTAAELAAIVGEDELSASLALRCAQELL
3VYA , Knot 65 122 0.85 38 96 117 
MLPGTFFEVLKNEGVVAIATQGEDGPHLVNTYNSYLKVLDGNRIVVPVGGMHKTEANVARDERVLMTLGSRKVAGRNGPGTGFLIRGSAAFRTDGPEFEAIARFKWARAALVITVVSAEQTL
6IWY , Knot 162 412 0.79 36 190 375 
GSAKDPSRKSNVTGDALSASVGVGVPIQDIKVDVQNLAQGDINELGAKFSSRDDIFSQVDTTLKFYTQNKDYAVNIKAGMTLGDVAQSITDATNGEVMGIVMKTGGNDPYQLMVNTKNTGEDNRVYFGSHLQSTLTNKNALSLGVDGSGKSEVSLNLKGADGNMHEVPIMLELPESASIKQKNTAIQKAMEQALENDPNFKNLIANGDISIDTLHGGESLIINDRRGGNIEVKGSKAKELGFLQTTTQESDLLKSSRTIKEGKLEGVVSLNGQKLDLSALTKESNTSEENTDAIIQAINAKEGLSAFKNAEGKLVINSKTGMLTIKGEDALGKASLKDLGLNAGMVQSYEASQNTLFMSKNLQKASDSAFTYNGVSITRPTNEVNDVISGVNITLEQTTEPNKPAIISVSRD

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 \(c\).

\(|P_{f(6RNQ_1)}(2) \setminus P_{f(3VYA_1)}(2)|=107\), \(|P_{f(3VYA_1)}(2) \setminus P_{f(6RNQ_1)}(2)|=46\). 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 1:100111100010000000100001111010000010001010100010111100010100110101010100101011001111010101110011001010001111111110011111101110010100001011001101001001101100001000111110101010011100101011011000100111100011100100110111001011010011011111100010101110010011
Pair \(Z_2\) Length of longest common subsequence
6RNQ_1,3VYA_1 153 4
6RNQ_1,6IWY_1 159 4
3VYA_1,6IWY_1 146 4

Newick tree

 
[
	6RNQ_1:79.61,
	[
		3VYA_1:73,6IWY_1:73
	]:6.61
]

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 \((0.85)(0.8)(\frac{374 }{\log_{20} 374}-\frac{122}{\log_{20}122})=76.8\)
Status Protein1 Protein2 d d1/2
Query variables 6RNQ_1 3VYA_1 98 71
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]