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Parikh vectors
6SQR_1 6XIE_1 2RQB_1 Letter Amino acid
4 7 5 T Threonine
5 8 9 V Valine
3 0 10 N Asparagine
0 6 7 S Serine
8 5 6 P Proline
7 1 9 C Cysteine
3 8 7 G Glycine
3 6 4 H Histidine
6 2 9 I Isoleucine
3 18 12 L Leucine
7 5 15 K Lycine
3 4 6 F Phenylalanine
1 4 5 Y Tyrosine
3 11 4 A Alanine
3 8 2 R Arginine
0 8 6 D Aspartic acid
3 5 4 Q Glutamine
5 14 9 E Glutamic acid
2 1 4 M Methionine
0 1 2 W Tryptophan 

6SQR_1|Chain A|E3 ubiquitin-protein ligase Mdm2|Felis catus (9685)
>6XIE_1|Chain A|Proprotein convertase subtilisin/kexin type 9|Homo sapiens (9606)
>2RQB_1|Chain A|Interferon-induced helicase C domain-containing protein 1|Homo sapiens (9606)
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 \(w=f(c)\)
6SQR , Knot 37 69 0.75 34 62 67 
EEIVEPEFPHNAIEPCVICQTRPKNGCIVHGKTGHLMACFTCAKKLKKRNKPCPVCRQPIQMIVLTYFP
6XIE , Knot 58 122 0.76 38 87 115 
QEDEDGDYEELVLALRSEEDGLAEAPEHGTTATFHRCAKDPWRLPGTYVVVLKEETHLSQSERTARRLQAQAARRGYLTKILHVFHGLLPGFLVKMSGDLLELALKLPHVDYIEEDSSVFAQ
2RQB , Knot 68 135 0.82 40 113 131 
GPLGSYKNNPSLITFLCKNCSVLACSGEDIHVIEKMHHVNMTPEFKELYIVRENKALQKKCADYQINGEIICKCGQAWGTMMVHKGLDLPCLKIRNFVVVFKNNSTKKQYKKWVELPITFPNLDYSECCLFSDED

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 \(c\).

\(|P_{f(6SQR_1)}(2) \setminus P_{f(6XIE_1)}(2)|=49\), \(|P_{f(6XIE_1)}(2) \setminus P_{f(6SQR_1)}(2)|=74\). 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 1:001101011001101011000010010110100101110100100100000101100011011110011
Pair \(Z_2\) Length of longest common subsequence
6SQR_1,6XIE_1 123 3
6SQR_1,2RQB_1 133 3
6XIE_1,2RQB_1 140 3

Newick tree

 
[
	2RQB_1:70.38,
	[
		6SQR_1:61.5,6XIE_1:61.5
	]:8.88
]

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 \((0.85)(0.8)(\frac{191 }{\log_{20} 191}-\frac{69}{\log_{20}69})=40.8\)
Status Protein1 Protein2 d d1/2
Query variables 6SQR_1 6XIE_1 51 40.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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