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Parikh vectors
2AAB_1 4GZE_1 1CEE_1 Letter Amino acid
12 21 6 Q Glutamine
12 29 12 E Glutamic acid
10 29 7 Y Tyrosine
12 39 11 A Alanine
15 39 9 G Glycine
11 29 8 I Isoleucine
13 34 17 L Leucine
12 22 15 K Lycine
31 16 9 S Serine
16 26 16 T Threonine
12 18 13 P Proline
8 22 4 R Arginine
9 24 5 N Asparagine
11 35 11 D Aspartic acid
5 7 5 C Cysteine
3 15 2 H Histidine
3 13 2 M Methionine
8 28 8 F Phenylalanine
3 11 1 W Tryptophan
12 24 18 V Valine 

2AAB_1|Chain A[auth L]|anti-idiotypic monoclonal antibody (light chain)|Mus musculus (10090)
>4GZE_1|Chains A, B, C, D, E, F|6-phospho-beta-glucosidase|Lactobacillus plantarum (220668)
>1CEE_1|Chain A|GTP-BINDING RHO-LIKE PROTEIN|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 \(w=f(c)\)
2AAB , Knot 100 218 0.82 40 155 212 
DIQLTQSPASLAVSLGQRVTISCRASESVEYYGSSLMQWYQQKPGQPPKLLIYAASNVESGVPARFSGSGSGTDFSLNIHPVEEDDIAMYFCQQSRKIPYTFGGGTKLEIKRADAAPTVSIFPPSSEQLTSGGASVVCFLNNFYPKDINVKWKIDGSERQNGVLNSWTDQDSKDSTYSMSSTLTLTKDEYERHNSYTCEATHKTSTSPIVKSFNRGEC
4GZE , Knot 200 481 0.85 40 252 459 
SNAMTIKGRAFPEGFLWGGAVAAHQLEGGYKEGGKGLSTADIMTLGTNERPREITDGVVAGKYYPNHQAIDFYHRYPEDIELFAEMGFKCFRTSIAWTRIFPNGDESEPNEAGLQFYDDLFDECLKNGIQPVVTLAHFEMPYHLVKQYGGWRNRKLIQFYLNFAKVCFERYRDKVTYWMTFNEINNQTNFESDGAMLTDSGIIHQPGENRERWMYQAAHYELVASAAAVQLGHQINPDFQIGCMIAMCPIYPLTAAPADVLFAQRAMQTRFYFADVHCNGTYPQWLRNRFESEHFNLDITAEDLKILQAGTVDYIGFSYYMSFTVKDTGKLAYNEEHDLVKNPYVKASDWGWQVDPVGLRYAMNWFTDRYHLPLFIVENGLGAIDKKTADNQIHDDYRIDYLTDHLRQIKLAVLEDGVDLIGYTPWGCIDLVAASTGQMSKRYGFIYVDENDDGSGSLKRYKKDSFTWFQHVIATNGAEIE
1CEE , Knot 86 179 0.83 40 133 174 
MQTIKCVVVGDGAVGKTCLLISYTTNKFPSEYVPTVFDNYAVTVMIGGEPYTLGLFDTAGQEDYDRLRPLSYPQTDVFLVCFSVVSPSSFENVKEKWVPEITHHCPKTPFLLVGTQIDLRDDPSTIEKLAKNKQKPITPETAEKLARDLKAVKYVECSALTQKGLKNVFDEAILAALEP

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 \(c\).

\(|P_{f(2AAB_1)}(2) \setminus P_{f(4GZE_1)}(2)|=51\), \(|P_{f(4GZE_1)}(2) \setminus P_{f(2AAB_1)}(2)|=148\). 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 1:01010001101110110010100010001000100110100001101101110110010011110101010100101010110000111010000001100111100101001011101011110000100111011011001010010101010100000111001000000000001000101000000000000001000000011100100100
Pair \(Z_2\) Length of longest common subsequence
2AAB_1,4GZE_1 199 4
2AAB_1,1CEE_1 158 3
4GZE_1,1CEE_1 189 3

Newick tree

 
[
	4GZE_1:10.33,
	[
		2AAB_1:79,1CEE_1:79
	]:23.33
]

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 \((0.85)(0.8)(\frac{699 }{\log_{20} 699}-\frac{218}{\log_{20}218})=134.\)
Status Protein1 Protein2 d d1/2
Query variables 2AAB_1 4GZE_1 174 126
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]