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Parikh vectors
4TLN_1 8BPU_1 2BAB_1 Letter Amino acid
10 11 22 R Arginine
0 8 5 C Cysteine
28 12 28 A Alanine
25 7 38 D Aspartic acid
18 6 21 I Isoleucine
26 10 14 S Serine
28 3 27 Y Tyrosine
36 12 31 G Glycine
8 1 20 H Histidine
11 6 19 K Lycine
25 7 23 T Threonine
3 6 7 W Tryptophan
10 3 19 F Phenylalanine
8 2 19 P Proline
22 6 28 V Valine
19 14 12 N Asparagine
13 3 14 Q Glutamine
8 2 32 E Glutamic acid
16 8 28 L Leucine
2 2 17 M Methionine 

4TLN_1|Chain A|THERMOLYSIN|Bacillus thermoproteolyticus (1427)
>8BPU_1|Chain A[auth AAA]|Lysozyme|Gallus gallus (9031)
>2BAB_1|Chain A|putative aminooxidase|Propionibacterium acnes (1747)
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 \(w=f(c)\)
4TLN , Knot 131 316 0.79 38 180 301 
ITGTSTVGVGRGVLGDQKNINTTYSTYYYLQDNTRGDGIFTYDAKYRTTLPGSLWADADNQFFASYDAPAVDAHYYAGVTYDYYKNVHNRLSYDGNNAAIRSSVHYSQGYNNAFWNGSEMVYGDGDGQTFIPLSGGIDVVAHELTHAVTDYTAGLIYQNESGAINEAISDIFGTLVEFYANKNPDWEIGEDVYTPGISGDSLRSMSDPAKYGDPDHYSKRYTGTQDNGGVHINSGIINKAAYLISQGGTHYGVSVVGIGRDKLGKIFYRALTQYLTPTSNFSQLRAAAVQSATDLYGSTSQEVASVKQAFDAVGVK
8BPU , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
2BAB , Knot 177 424 0.84 40 250 406 
MSISKDSRIAIIGAGPAGLAAGMYLEQAGFHDYTILERTDHVGGKCHSPNYHGRRYEMGAIMGVPSYDTIQEIMDRTGDKVDGPKLRREFLHEDGEIYVPEKDPVRGPQVMAAVQKLGQLLATKYQGYDANGHYNKVHEDLMLPFDEFLALNGCEAARDLWINPFTAFGYGHFDNVPAAYVLKYLDFVTMMSFAKGDLWTWADGTQAMFEHLNATLEHPAERNVDITRITREDGKVHIHTTDWDRESDVLVLTVPLEKFLDYSDADDDEREYFSKIIHQQYMVDACLVKEYPTISGYVPDNMRPERLGHVMVYYHRWADDPHQIITTYLLRNHPDYADKTQEECRQMVLDDMETFGHPVEKIIEEQTWYYFPHVSSEDYKAGWYEKVEGMQGRRNTFYAGEIMSFGNFDEVCHYSKDLVTRFFV

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 \(c\).

\(|P_{f(4TLN_1)}(2) \setminus P_{f(8BPU_1)}(2)|=122\), \(|P_{f(8BPU_1)}(2) \setminus P_{f(4TLN_1)}(2)|=46\). 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 1:1010001111011110000100000000010000010111000100000111011101000111000111101000111000000010001000100111000100001000111010011010101001111011101110010011000011110000011100110011101101010001010110010011101001001001100101000000001000011101001110011011001100011011111000110110011000101000100101111001001010000011010011011110
Pair \(Z_2\) Length of longest common subsequence
4TLN_1,8BPU_1 168 3
4TLN_1,2BAB_1 170 4
8BPU_1,2BAB_1 238 3

Newick tree

 
[
	2BAB_1:10.11,
	[
		4TLN_1:84,8BPU_1:84
	]:25.11
]

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 \((0.85)(0.8)(\frac{445 }{\log_{20} 445}-\frac{129}{\log_{20}129})=94.5\)
Status Protein1 Protein2 d d1/2
Query variables 4TLN_1 8BPU_1 119 82.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]