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Parikh vectors
1QFQ_1 6IRE_1 6GOU_1 Letter Amino acid
0 2 8 W Tryptophan
0 3 21 Y Tyrosine
5 27 58 A Alanine
0 10 26 N Asparagine
0 18 33 D Aspartic acid
0 5 17 H Histidine
0 3 30 P Proline
0 15 37 R Arginine
0 30 39 K Lycine
0 9 12 M Methionine
0 5 29 F Phenylalanine
0 10 35 V Valine
4 0 14 C Cysteine
0 21 25 Q Glutamine
0 21 44 E Glutamic acid
5 6 55 G Glycine
0 12 40 I Isoleucine
0 16 54 L Leucine
0 8 43 S Serine
0 13 38 T Threonine 

1QFQ_1|Chain A|15-mer nutRboxB RNA hairpin|null
>6IRE_1|Chain A|1-phosphatidylinositol 4,5-bisphosphate phosphodiesterase|Drosophila melanogaster (7227)
>6GOU_1|Chains A, B, C, D|Alkyldihydroxyacetonephosphate synthase, peroxisomal|Cavia porcellus (10141)
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 \(w=f(c)\)
1QFQ , Knot 6 15 0.36 8 8 11 
GCCCUGAAAAAGGGC
6IRE , Knot 106 234 0.82 38 143 225 
AEPPLVFEPVTLESLRQEAGFQAVGAAQIAELDTLRAAHAAERTSVQKTQNAAIDKLIKGKSKDDIRNDANIKNSINDQTKQWTDMIARHRKEEWDMLRQHVQDSQDAMKALMLTVQAAQIKQLEDRHARDIKDLNAKQAKMSADTAKEVQNDKTLKTKNEKDRRLREKRQNNVKRFMEEKKQIGVKQGRAMEKLKLAHSKQIEEFSTDVQKLMDMYKIEEEAYKTQGKTEFYA
6GOU , Knot 256 658 0.84 40 289 616 
MAEAAAAAAAAAAAGETSASSGSAAERDPDQDRAGRRLRVLSGHLLGRPQEALSTNECKARRAASAATAAPTATPAAPESGIIPKKRQELMKWNGWGYNDSKFFLNKKGQLELTGKRYPLSGVALPTFKDWIQNTFGINLDHKTTSKASLNPSDTPPSIVNEDFLHELKKTNISYSQEADDRVFRAHGHCLHEIFLLREGMFERIPDIVLWPTCHDDVVKIVNLACKYNLCIIPIGGGTSVSYGLMCPADETRTIISLDTSQMNRILWVDENNLTAHVEAGITGQELERQLKESGYCTGHEPDSLEFSTVGGWISTRASGMKKNIYGNIEDLVVHMKVVTPRGVIEKSCQGPRMSTGPDIHHFIMGSEGTLGVITEATIKIRPTPEYQKYGSVAFPNFEQGVACLREIAKQRCAPASIRLMDNQQFQFGHALKPQVSSIFTSFLDGLKKFYITKFKGFDPNQLSVATLLFEGDREKVLQHEKQVYDIAAKFGGLAAGEDNGQRGYLLTYVIAYMRDLGLEYYIIGESFETSAPWDRVVDLCRNVKERIRRECKEKGVQFPPLSTCRVTQTYDAGACIYFYFAFNYRGISDPLAVFEQTEAAAREEILANGGSLSHHHGVGKLRKQWLKESISDVGFGMLKSVKDYVDPTNIFGNRNLL

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 \(c\).

\(|P_{f(1QFQ_1)}(2) \setminus P_{f(6IRE_1)}(2)|=5\), \(|P_{f(6IRE_1)}(2) \setminus P_{f(1QFQ_1)}(2)|=140\). 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 1:100001111111110
Pair \(Z_2\) Length of longest common subsequence
1QFQ_1,6IRE_1 145 3
1QFQ_1,6GOU_1 289 6
6IRE_1,6GOU_1 200 4

Newick tree

 
[
	6GOU_1:13.23,
	[
		1QFQ_1:72.5,6IRE_1:72.5
	]:64.73
]

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 \((0.85)(0.8)(\frac{249 }{\log_{20} 249}-\frac{15}{\log_{20}15})=80.6\)
Status Protein1 Protein2 d d1/2
Query variables 1QFQ_1 6IRE_1 104 54
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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