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Parikh vectors
6WZZ_1 2ALX_1 6RHL_1 Letter Amino acid
15 21 6 S Serine
4 7 1 W Tryptophan
7 21 6 A Alanine
5 17 9 R Arginine
12 20 7 D Aspartic acid
7 27 10 I Isoleucine
5 15 14 N Asparagine
11 34 13 L Leucine
2 5 1 C Cysteine
6 19 4 Q Glutamine
15 16 8 K Lycine
8 19 5 T Threonine
13 16 8 F Phenylalanine
3 12 10 P Proline
14 15 3 Y Tyrosine
8 18 13 V Valine
11 29 6 E Glutamic acid
13 12 8 G Glycine
7 8 4 H Histidine
1 9 2 M Methionine 

6WZZ_1|Chain A|Glucose-induced degradation protein 4 homolog|Homo sapiens (9606)
>2ALX_1|Chain A|Ribonucleoside-diphosphate reductase 1|Escherichia coli (562)
>6RHL_1|Chain A|Galectin-3|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 \(w=f(c)\)
6WZZ , Knot 82 167 0.83 40 129 162 
GSGSKFRGHQKSKGNSYDVEVVLQHVDTGNSYLCGYLKIKGLTEEYPTLTTFFEGEIISKKHPFLTRKWDADEDVDRKHWGKFLAFYQYAKSFNSDDFDYEELKNGDYVFMRWKEQFLVPDHTIKDISGASFAGFYYICFQKSAASIEGYYYHRSSEWYQSLNLTHV
2ALX , Knot 147 340 0.84 40 213 332 
MAYTTFSQTKNDQLKEPMFFGQPVNVARYDQQKYDIFEKLIEKQLSFFWRPEEVDVSRDRIDYQALPEHEKHIFISNLKYQTLLDSIQGRSPNVALLPLISIPELETWVETWAFSETIHSRSYTHIIRNIVNDPSVVFDDIVTNEQIQKRAEGISSYYDELIEMTSYWHLLGEGTHTVNGKTVTVSLRELKKKLYLCLMSVNALEAIRFYVSFACSFAFAERELMEGNAKIIRLIARDEALHLTGTQHMLNLLRSGADDPEMAEIAEECKQECYDLFVQAAQQEKDWADYLFRDGSMIGLNKDILCQYVEYITNIRMQAVGLDLPFQTRSNPIPWINTWL
6RHL , Knot 68 138 0.81 40 112 136 
PLIVPYNLPLPGGVVPRMLITILGTVKPNANRIALDFQRGNDVAFHFNPRFNENNRRVIVCNTKLDNNWGREERQSVFPFESGKPFKIQVLVEPDHFKVAVNDAHLLQYNHRVKKLNEISKLGISGDIDLTSASYTMI

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 \(c\).

\(|P_{f(6WZZ_1)}(2) \setminus P_{f(2ALX_1)}(2)|=54\), \(|P_{f(2ALX_1)}(2) \setminus P_{f(6WZZ_1)}(2)|=138\). 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 1:10100101000001000010111001001000101010101100001010011010110000111000101000100001101111000100100001000010010011101000111100010010110111100101000110101000000001000101001
Pair \(Z_2\) Length of longest common subsequence
6WZZ_1,2ALX_1 192 3
6WZZ_1,6RHL_1 161 3
2ALX_1,6RHL_1 197 3

Newick tree

 
[
	2ALX_1:10.23,
	[
		6WZZ_1:80.5,6RHL_1:80.5
	]:21.73
]

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 \((0.85)(0.8)(\frac{507 }{\log_{20} 507}-\frac{167}{\log_{20}167})=99.3\)
Status Protein1 Protein2 d d1/2
Query variables 6WZZ_1 2ALX_1 128 95
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]