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Parikh vectors
1PDZ_1 3FVA_1 7QTS_1 Letter Amino acid
40 0 6 A Alanine
32 0 3 L Leucine
11 0 2 M Methionine
13 0 0 P Proline
27 1 10 T Threonine
10 0 3 Y Tyrosine
9 0 0 C Cysteine
12 1 2 Q Glutamine
38 0 4 G Glycine
31 0 1 I Isoleucine
14 0 0 R Arginine
28 0 5 D Aspartic acid
27 0 5 E Glutamic acid
8 0 0 H Histidine
26 0 0 S Serine
28 0 4 V Valine
23 3 3 N Asparagine
35 0 6 K Lycine
18 1 2 F Phenylalanine
3 0 1 W Tryptophan 

1PDZ_1|Chain A|ENOLASE|Homarus gammarus (6707)
>3FVA_1|Chain A|Major prion protein|null
>7QTS_1|Chain A|Immunoglobulin G-binding protein G|Streptococcus sp. (1306)
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 \(w=f(c)\)
1PDZ , Knot 186 434 0.86 42 234 418 
XSITKVFARTIFDSRGNPTVEVDLYTSKGLFRAAVPSGASTGVHEALEMRDGDKSKYHGKSVFNAVKNVNDVIVPEIIKSGLKVTQQKECDEFMCKLDGTENKSSLGANAILGVSLAICKAGAAELGIPLYRHIANLANYDEVILPVPAFNVINGGSHAGNKLAMQEFMILPTGATSFTEAMRMGTEVYHHLKAVIKARFGLDATAVGDEGGFAPNILNNKDALDLIQEAIKKAGYTGKIEIGMDVAASEFYKQNNIYDLDFKTANNDGSQKISGDQLRDMYMEFCKDFPIVSIEDPFDQDDWETWSKMTSGTTIQIVGDDLTVTNPKRITTAVEKKACKCLLLKVNQIGSVTESIDAHLLAKKNGWGTMVSHRSGETEDCFIADLVVGLCTGQIKTGAPCRSERLAKYNQILRIEEELGSGAKFAGKNFRAPS
3FVA , Knot 4 6 0.39 8 5 4 
NNQNTF
7QTS , Knot 32 57 0.75 30 47 55 
MMQYKLILNGKTLKGETTTEAVDAATAEKVFKQYANDNGVDGEWTYDDATKTFTVTE

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 \(c\).

\(|P_{f(1PDZ_1)}(2) \setminus P_{f(3FVA_1)}(2)|=231\), \(|P_{f(3FVA_1)}(2) \setminus P_{f(1PDZ_1)}(2)|=2\). 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 1:00100111001100010101010100001110111101100110011010010000001001101100100111101100110100000000110010100000011101111101110011110111110001101100001111111101101100110011100111110110010011011001000101110101110101110011111011000011011001100110010101110111001000001001010010001000101001001010100011110100110000100100100100101110010100100100110001000111010011010001010111000111011000010000011101111100101001110000011000011010001101101110010110
Pair \(Z_2\) Length of longest common subsequence
1PDZ_1,3FVA_1 233 2
1PDZ_1,7QTS_1 205 3
3FVA_1,7QTS_1 50 2

Newick tree

 
[
	1PDZ_1:12.87,
	[
		7QTS_1:25,3FVA_1:25
	]:10.87
]

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 \((0.85)(0.8)(\frac{440 }{\log_{20} 440}-\frac{6}{\log_{20}6})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 1PDZ_1 3FVA_1 183 92.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]