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Parikh vectors
7RGY_1 5ILK_1 1IVE_1 Letter Amino acid
8 18 37 G Glycine
4 14 8 H Histidine
10 41 27 I Isoleucine
8 33 14 K Lycine
3 27 12 Y Tyrosine
9 24 26 R Arginine
13 57 16 L Leucine
3 12 5 M Methionine
9 17 15 P Proline
4 20 12 Q Glutamine
6 39 42 S Serine
5 33 24 T Threonine
1 8 12 W Tryptophan
6 42 13 A Alanine
14 24 25 N Asparagine
7 34 27 D Aspartic acid
1 8 18 C Cysteine
6 45 15 E Glutamic acid
8 22 12 F Phenylalanine
13 32 28 V Valine 

7RGY_1|Chain A|Galectin-3|Homo sapiens (9606)
>5ILK_1|Chain A|5-epi-aristolochene synthase|Nicotiana tabacum (4097)
>1IVE_1|Chains A, B|INFLUENZA A SUBTYPE N2 NEURAMINIDASE|Influenza A virus (strain A/Tokyo/3/1967 H2N2) (380960)
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 \(w=f(c)\)
7RGY , Knot 68 138 0.81 40 112 135 
MLIVPYNLPLPGGVVPRMLITILGTVKPNANRIALDFQRGNDVAFHFNPRFNENNRRVIVCNTKLDNNWGREERQSVFPFESGKPFKIQVLVEPDHFKVAVNDAHLLQYNHRVKKLNEISKLGISGDIDLTSASYTMI
5ILK , Knot 221 550 0.84 40 266 508 
GSMASAAVANYEEEIVRPVADFSPSLWGDQFLSFSIKNQVAEKYAKEIEALKEQTRNMLLATGMKLADTLNLIDTIERLGISYHFEKEIDDILDQIYNQNSNCNDLCTSALQFRLLRQHGFNISPEIFSKFQDENGKFKESLASDVLGLLNLYEASHVRTHADDILEDALAFSTIHLESAAPHLKSPLREQVTHALEQCLHKGVPRVETRFFISSIYDKEQSKNNVLLRFAKLDFNLLQMLHKQELAQVSRWWKDLDFVTTLPYARDRVVECYFWALGVYFEPQYSQARVMLVKTISMISIVDDTFDAYGTVKELEAYTDAIQRWDINEIDRLPDYMKISYKAILDLYKDYEKELSSAGRSHIVCHAIERMKEVVRNYNVESTWFIEGYTPPVSEYLSNALATTTYYYLATTSYLGMKSATEQDFEWLSKNPKILEASVIICRVIDDTATYEVEKSRGQIATGIECCMRDYGISTKEAMAKFQNMAETAWKDINEGLLRPTPVSTEFLTPILNLARIVEVTYIHNLDGYTHPEKVLKPHIINLLVDSIKI
1IVE , Knot 168 388 0.86 40 224 369 
VEYRNWSKPQCQITGFAPFSKDNSIRLSAGGDIWVTREPYVSCDPVKCYQFALGQGTTLDNKHSNDTVHDRIPHRTLLMNELGVPFHLGTRQVCIAWSSSSCHDGKAWLHVCITGDDKNATASFIYDGRLVDSIGSWSQNILRTQESECVCINGTCTVVMTDGSASGRADTRILFIEEGKIVHISPLAGSAQHVEECSCYPRYPGVRCICRDNWKGSNRPVVDINMEDYSIDSSYVCSGLVGDTPRNDDRSSNSNCRDPNNERGTQGVKGWAFDNGNDLWMGRTISKDLRSGYETFKVIGGWSTPNSKSQINRQVIVDSDNRSGYSGIFSVEGKSCINRCFYVELIRGRKQETRVWWTSNSIVVFCGTSGTYGTGSWPDGANINFMPI

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 \(c\).

\(|P_{f(7RGY_1)}(2) \setminus P_{f(5ILK_1)}(2)|=22\), \(|P_{f(5ILK_1)}(2) \setminus P_{f(7RGY_1)}(2)|=176\). 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 1:111110011111111101110111010101001110100100111010101000000111000010001100000011110010110101110100101110010110000010010010011101010100100011
Pair \(Z_2\) Length of longest common subsequence
7RGY_1,5ILK_1 198 4
7RGY_1,1IVE_1 194 3
5ILK_1,1IVE_1 170 5

Newick tree

 
[
	7RGY_1:10.97,
	[
		1IVE_1:85,5ILK_1:85
	]:16.97
]

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 \((0.85)(0.8)(\frac{688 }{\log_{20} 688}-\frac{138}{\log_{20}138})=157.\)
Status Protein1 Protein2 d d1/2
Query variables 7RGY_1 5ILK_1 198 122.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]