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Parikh vectors
1BDY_1 2FQZ_1 6IJH_1 Letter Amino acid
5 0 14 R Arginine
3 0 14 N Asparagine
10 0 24 E Glutamic acid
6 0 14 P Proline
6 0 22 T Threonine
4 0 12 Y Tyrosine
10 0 16 V Valine
11 1 21 A Alanine
1 0 14 H Histidine
4 0 21 I Isoleucine
7 0 23 S Serine
2 0 7 W Tryptophan
6 0 15 D Aspartic acid
7 0 17 Q Glutamine
10 0 18 K Lycine
6 0 13 M Methionine
6 0 15 F Phenylalanine
3 4 12 C Cysteine
11 0 36 L Leucine
5 4 15 G Glycine 

1BDY_1|Chains A, B|PROTEIN KINASE C|Rattus norvegicus (10116)
>2FQZ_1|Chains A[auth E], C[auth G]|DNA STRAND 1|
>6IJH_1|Chains A, B|cAMP and cAMP-inhibited cGMP 3',5'-cyclic phosphodiesterase 10A|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 \(w=f(c)\)
1BDY , Knot 63 123 0.82 40 105 119 
MAPFLRISFNSYELGSLQAEDDASQPFCAVKMKEALTTDRGKTLVQKKPTMYPEWKSTFDAHIYEGRVIQIVLMRAAEDPMSEVTVGVSVLAERCKKNNGKAEFWLDLQPQAKVLMCVQYFLE
2FQZ , Knot 6 9 0.48 6 6 7 
CGCCAGGGC
6IJH , Knot 154 343 0.87 40 222 334 
HMSICTSEEWQGLMQFTLPVRLCKEIELFHFDIGPFENMWPGIFVYMVHRSCGTSCFELEKLCRFIMSVKKNYRRVPYHNWKHAVTVAHCMYAILQNNHTLFTDLERKGLLIACLCHDLDHRGFSNSYLQKFDHPLAALYSTSTMEQHHFSQTVSILQLEGHNIFSTLSSSEYEQVLEIIRKAIIATDLALYFGNRKQLEEMYQTGSLNLNNQSHRDRVIGLMMTACDLCSVTKLWPVTKLTANDIYAEFWAEGDEMKKLGIQPIPMMDRDKKDEVPQGQLGFYNAVAIPCYTTLTQILPPTEPLLKACRDNLSQWEKVIRGEETATWISSPSVAQKAAASED

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 \(c\).

\(|P_{f(1BDY_1)}(2) \setminus P_{f(2FQZ_1)}(2)|=104\), \(|P_{f(2FQZ_1)}(2) \setminus P_{f(1BDY_1)}(2)|=5\). 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 1:111110101000011010100010011011010011000010011000101010100010101001011011110110011001011101110000000101011101010101110100110
Pair \(Z_2\) Length of longest common subsequence
1BDY_1,2FQZ_1 109 2
1BDY_1,6IJH_1 193 4
2FQZ_1,6IJH_1 226 2

Newick tree

 
[
	6IJH_1:11.17,
	[
		1BDY_1:54.5,2FQZ_1:54.5
	]:62.67
]

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 \((0.85)(0.8)(\frac{132 }{\log_{20} 132}-\frac{9}{\log_{20}9})=46.7\)
Status Protein1 Protein2 d d1/2
Query variables 1BDY_1 2FQZ_1 61 32.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]