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Parikh vectors
8GJD_1 3ZNG_1 4NVM_1 Letter Amino acid
0 14 14 P Proline
0 18 15 S Serine
0 11 20 N Asparagine
0 16 9 Q Glutamine
1 16 20 E Glutamic acid
2 12 10 I Isoleucine
2 37 25 L Leucine
2 6 18 F Phenylalanine
4 25 16 A Alanine
0 10 10 R Arginine
0 10 1 C Cysteine
1 4 5 M Methionine
0 3 14 Y Tyrosine
2 16 14 V Valine
1 11 24 D Aspartic acid
1 19 26 G Glycine
0 17 6 H Histidine
0 12 23 K Lycine
0 7 13 T Threonine
0 4 6 W Tryptophan 

8GJD_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P|Beta-hairpin peptide derived from Abeta 17-36|synthetic construct (32630)
>3ZNG_1|Chains A, D|ANKYRIN REPEAT AND SOCS BOX PROTEIN 9|HOMO SAPIENS (9606)
>4NVM_1|Chain A[auth B]|Cytochrome c peroxidase|Saccharomyces cerevisiae (285006)
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 \(w=f(c)\)
8GJD , Knot 12 16 0.69 18 15 14 
ALVFFAEDAAIIGLMV
3ZNG , Knot 114 268 0.79 40 159 249 
MSDWSPMHEAAIHGHQLSLRNLISQGWAVNIITADHVSPLHEACLGGHLSCVKILLKHGAQVNGVTADWHTPLFNACVSGSWDCVNLLLQHGASVQPESDLASPIHEAARRGHVECVNSLIAYGGNIDHKISHLGTPLYLACENQQRACVKKLLESGADVNQGKGQDSPLHAVARTASEELACLLMDFGADTQAKNAEGKRPVELVPPESPLAQLFLEREGPPSLMQLCRLRIRKCFGIQQHHKITKLVLPEDLKQFLLHLAENLYFQ
4NVM , Knot 131 289 0.85 40 191 283 
LVHVASVEKGRSYEDFQKVYNAIALKLREDDEYDNYIGYGPVLVRLAWHISGTWDKHDNTGGSYGGTYRFKKEFNDPSNAGLQNGFKFLEPIHKEFPWISSGDLFSLGGVTAVQEMQGPKIPWRCGRVDTPEDTTPDNGRLPDADKDAGYVRTFFQRLNMNDREVVALMGAHALGKTHLKNSGYEGGGANNVFTNEFYLNLLNEDWKLEKNDANNEQWDSKSGYMMLPTDYSLIQDPKYLSIVKEYANDQDKFFKDFSKAFEKLLENGITFPKDAPSPFIFKTLEEQGL

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 \(c\).

\(|P_{f(8GJD_1)}(2) \setminus P_{f(3ZNG_1)}(2)|=8\), \(|P_{f(3ZNG_1)}(2) \setminus P_{f(8GJD_1)}(2)|=152\). 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 1:1111110011111111
Pair \(Z_2\) Length of longest common subsequence
8GJD_1,3ZNG_1 160 3
8GJD_1,4NVM_1 186 2
3ZNG_1,4NVM_1 184 4

Newick tree

 
[
	4NVM_1:96.30,
	[
		8GJD_1:80,3ZNG_1:80
	]:16.30
]

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 \((0.85)(0.8)(\frac{284 }{\log_{20} 284}-\frac{16}{\log_{20}16})=90.6\)
Status Protein1 Protein2 d d1/2
Query variables 8GJD_1 3ZNG_1 110 58
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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