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Parikh vectors
6KEP_1 8SNQ_1 9ITN_1 Letter Amino acid
13 9 5 D Aspartic acid
3 1 0 C Cysteine
16 5 3 F Phenylalanine
19 10 5 S Serine
13 3 22 R Arginine
12 8 12 Q Glutamine
17 15 21 E Glutamic acid
5 7 2 Y Tyrosine
14 13 30 A Alanine
8 7 6 N Asparagine
8 14 3 G Glycine
9 3 1 H Histidine
9 16 13 I Isoleucine
10 12 4 T Threonine
2 0 0 W Tryptophan
32 14 21 L Leucine
10 10 5 K Lycine
11 5 3 M Methionine
11 6 2 P Proline
17 12 6 V Valine 

6KEP_1|Chains A[auth AA], B[auth BA]|Glutathione S-transferase E14|Drosophila melanogaster (7227)
>8SNQ_1|Chain A|Sortase|Streptococcus pyogenes (1314)
>9ITN_1|Chains A[auth U], B[auth V], C[auth X], D[auth Y]|ATP synthase subunit b|Chloroflexus aurantiacus J-10-fl (324602)
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 \(w=f(c)\)
6KEP , Knot 111 239 0.84 40 163 233 
MNHKVHMMSQPKPILYYDERSPPVRSCLMLIKLLDIDVELRFVNLFKGEQFQKDFLALNPQHSVPTLVHGDLVLTDSHAILIHLAEKFDEGGSLWPQEHAERMKVLNLLLFECSFLFRRDSDFMSAIVRQGFANVDVAHHERKLTEAYIIMERYLENSDFMAGPQLTLADLSIVTTLSTVNLMFPLSQFPRLRRWFTAMQQLDAYEANCSGLEKLRQTMESVGSFQFPSSSAVVTEKVE
8SNQ , Knot 81 170 0.81 38 130 168 
SSVLQAQMAAQQLPVIGGIAIPELGINLPIFKGLGNTELIYGAGTMKEEQVMGGENNYSLASHHIEGITGSSQMLFSPLERAQNGMSIYLTDKEKIYEYIIKDVFTVAPERVDVIDDTAGLKEVTLVTCTDIEATERIIVKGELKTEYDFDKAPADVLKAFNHSYNQVST
9ITN , Knot 75 164 0.77 36 97 151 
MEALGINPTLFIAQLINFLLLIFILRALLYRPVMNLLNERTRRIEESVRDAEKVREQLANARRDYEAEIARARQEAAKIVAQAQERAKQQEAEIIAQARREAERLKEEARAQAEQERIRMLSEAKSQIADLVTLTASRVLGAELQARGHDALIAESLAALDRRN

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 \(c\).

\(|P_{f(6KEP_1)}(2) \setminus P_{f(8SNQ_1)}(2)|=96\), \(|P_{f(8SNQ_1)}(2) \setminus P_{f(6KEP_1)}(2)|=63\). 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 1:10001011001011100000011100011110110101010110110100100011110100011011010111000011110110010011011100010010110111100011100000110111001110101100000100101110001000011111010110101100100101111100110100110110010100100011001000100110101100011100010
Pair \(Z_2\) Length of longest common subsequence
6KEP_1,8SNQ_1 159 3
6KEP_1,9ITN_1 154 3
8SNQ_1,9ITN_1 133 4

Newick tree

 
[
	6KEP_1:81.80,
	[
		9ITN_1:66.5,8SNQ_1:66.5
	]:15.30
]

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 \((0.85)(0.8)(\frac{409 }{\log_{20} 409}-\frac{170}{\log_{20}170})=71.1\)
Status Protein1 Protein2 d d1/2
Query variables 6KEP_1 8SNQ_1 92 77.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]