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Parikh vectors
9MUS_1 1LZE_1 8PNF_1 Letter Amino acid
10 12 21 A Alanine
20 8 15 L Leucine
6 3 8 Q Glutamine
16 7 23 T Threonine
9 4 15 Y Tyrosine
13 3 10 F Phenylalanine
10 2 16 P Proline
17 6 23 V Valine
7 11 13 R Arginine
17 2 17 E Glutamic acid
15 1 10 H Histidine
27 12 13 G Glycine
12 6 15 I Isoleucine
22 6 19 K Lycine
6 2 5 M Methionine
10 10 23 S Serine
12 14 15 N Asparagine
19 7 15 D Aspartic acid
3 8 3 C Cysteine
2 5 4 W Tryptophan 

9MUS_1|Chains A, B|Fluorescent thiol-disulfide redox biosensor|Aequorea victoria (6100)
>1LZE_1|Chain A|HEN EGG WHITE LYSOZYME|Gallus gallus (9031)
>8PNF_1|Chain A[auth 1]|Capsid protein VP1|rhinovirus B14 (12131)
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 \(w=f(c)\)
9MUS , Knot 114 253 0.83 40 164 237 
MKHHHHHHHGGASMVSKGEELFTGVVPILVELDGDVNGHKFSVSGEGEGDATYGKLTLKFISTTGKLPVPWPTLVTTLSWGVQCFARYPDHMKRHDFFKSAMPEGYVQERTIFFKDDGNYKTRAEVKFEGDTLVNRIELKGIDFKEDGNILGHKLEYNGGCEDNVYITADKQKNGIKANFKIRHNIEDGGVQLADHYQQNTPIGDGPVLLPDNHYLSTCSKLSKDPNEKRDHMVFLEFVTAAGITLGMDELYK
1LZE , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRYWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
8PNF , Knot 130 283 0.86 40 195 280 
EEVIVEKTKQTVASISSGPKHTQKVPILTANETGATMPVLPSDSIETRTTYMHFNGSETDVECFLGRAACVHVTEIQNKDATGIDNHREAKLFNDWKINLSSLVQLRKKLELFTYVRFDSEYTILATASQPDSANYSSNLVVQAMYVPPGAPNPKEWDDYTWQSASNPSVFFKVGDTSRFSVPYVGLASAYNCFYDGYSHDDAETQYGITVLNHMGSMAFRIVNEHDEHKTLVKIRVYHRAKHVEAWIPRAPRALPYTSIGRTNYPKNTEPVIKKRKGDIKSY

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 \(c\).

\(|P_{f(9MUS_1)}(2) \setminus P_{f(1LZE_1)}(2)|=115\), \(|P_{f(1LZE_1)}(2) \setminus P_{f(9MUS_1)}(2)|=55\). 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 1:1000000001110110010011011111110101010100101010101010010101011000101111110110010111001100100100001100111010100001110001000001010101001100101011010001011100100011000010101000001101010100010011101100000001110111111000010000010001000000111101101111011100100
Pair \(Z_2\) Length of longest common subsequence
9MUS_1,1LZE_1 170 4
9MUS_1,8PNF_1 159 4
1LZE_1,8PNF_1 179 3

Newick tree

 
[
	1LZE_1:89.72,
	[
		9MUS_1:79.5,8PNF_1:79.5
	]:10.22
]

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 \((0.85)(0.8)(\frac{382 }{\log_{20} 382}-\frac{129}{\log_{20}129})=76.8\)
Status Protein1 Protein2 d d1/2
Query variables 9MUS_1 1LZE_1 95 71.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]