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Parikh vectors
6GCJ_1 8GIF_1 1HVD_1 Letter Amino acid
7 18 16 V Valine
11 23 26 A Alanine
11 30 6 N Asparagine
10 20 25 D Aspartic acid
19 36 23 G Glycine
1 13 3 H Histidine
15 20 38 L Leucine
0 8 7 M Methionine
1 8 19 R Arginine
8 24 22 K Lycine
6 19 23 T Threonine
10 11 12 Q Glutamine
3 20 28 E Glutamic acid
13 18 18 I Isoleucine
9 20 21 S Serine
1 18 12 Y Tyrosine
8 12 1 C Cysteine
2 14 13 F Phenylalanine
7 21 5 P Proline
0 4 1 W Tryptophan 

6GCJ_1|Chain A|Hydrophobin|Neosartorya fumigata (strain CEA10 / CBS 144.89 / FGSC A1163) (451804)
>8GIF_1|Chain A|Apical membrane antigen 1, rhoptry neck protein 2 chimera|Plasmodium falciparum 3D7 (36329)
>1HVD_1|Chain A|ANNEXIN V|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 \(w=f(c)\)
6GCJ , Knot 68 142 0.79 36 101 136 
SLPQHDVNAAGNGVGNKGNANVRFPVPDDITVKQATEKCGDQAQLSCCNKATYAGDVTDIDEGILAGTLKNLIGGGSGTEGLGLFNQCSKLDLQIPIIGIPIQDLVNQKCKQNIACCQNSPSDASGSLIGLGLPCIALGSIL
8GIF , Knot 153 357 0.84 40 211 328 
ETGNYMGNPWTEYMAKYDIEEVHGSGIRVDLGEDAEVAGTQYRLPSGKCPVFGKGIIIENSNTAFLTPVATGNQYLKDGGFAFPPTEPLMSPMTLDEMRHFYKDNKYVKNLDELTLCSRHAGNMIPDNDKNSNYKYPAVYDDKDKKCHILYIAAQENNGPRYCNGGGGSGGGGSDITQQAKDIGAGPVASCFTTRMSPPQQICLNSVVNTALSGGSGMFCFRPAKDISFQNYAYLSKNVVDNWEKVCPRKNLQNAKFGLWVDGNCEDIPHVNEFPAIDLFECNKLVFELSASDQPGGSGYKSHGKGYNWGNYNTETQKCEIFNVKPTCLINNAAYIATTALSHPIEVEGTKHHHHHH
1HVD , Knot 136 319 0.82 40 183 295 
AQVLRGTVTDFPGFDGRADAETLRKAMKGLGTDEESILTLLTSRSNAQRQEISAAFKTLFGRDLLDDLKSELTGKFEKLIVALMKPSRLYDAYELKHALKGAGTNEKVLTEIIASRTPEELRAIKQVYEEEYGSSLEDDVVGDTSGYYQRMLVVLLQANRDPDAGIDEAQVEQDAQALFQAGELKWGTDEEKFITIFGTRSVSHLRKVFDKYMTISGFQIEETIDRETSGNLEQLLLAVVKSIRSIPAYLAETLYYAMKGAGTDDHTLIRVMVSRSEIDLFNIRKEFRKNFATSLYSMIKGDTSGDYKKALLLLCGEDD

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 \(c\).

\(|P_{f(6GCJ_1)}(2) \setminus P_{f(8GIF_1)}(2)|=31\), \(|P_{f(8GIF_1)}(2) \setminus P_{f(6GCJ_1)}(2)|=141\). 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 1:0110001011101110010101011110010100100001001010000010011010010011111010011111010011111000001010111111110011000000011000001001010111111101111011
Pair \(Z_2\) Length of longest common subsequence
6GCJ_1,8GIF_1 172 5
6GCJ_1,1HVD_1 162 3
8GIF_1,1HVD_1 168 4

Newick tree

 
[
	8GIF_1:86.29,
	[
		6GCJ_1:81,1HVD_1:81
	]:5.29
]

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 \((0.85)(0.8)(\frac{499 }{\log_{20} 499}-\frac{142}{\log_{20}142})=105.\)
Status Protein1 Protein2 d d1/2
Query variables 6GCJ_1 8GIF_1 134 92
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]