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Parikh vectors
6XML_1 1QQV_1 6ICJ_1 Letter Amino acid
24 5 19 E Glutamic acid
9 1 6 H Histidine
19 0 21 I Isoleucine
4 1 9 M Methionine
20 3 16 S Serine
14 4 8 N Asparagine
17 2 16 Q Glutamine
15 3 8 R Arginine
3 1 0 W Tryptophan
13 0 8 Y Tyrosine
22 4 14 V Valine
14 5 18 D Aspartic acid
34 9 38 L Leucine
30 3 11 G Glycine
26 7 24 K Lycine
8 5 15 F Phenylalanine
19 5 11 P Proline
22 4 13 T Threonine
42 5 16 A Alanine
9 0 1 C Cysteine 

6XML_1|Chains A, B|Fructose-bisphosphate aldolase A|Homo sapiens (9606)
>1QQV_1|Chain A|VILLIN HEADPIECE DOMAIN|Gallus gallus (9031)
>6ICJ_1|Chain A|Peroxisome proliferator-activated receptor gamma|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 \(w=f(c)\)
6XML , Knot 156 364 0.84 40 209 346 
MPYQYPALTPEQKKELSDIAHRIVAPGKGILAADESTGSIAKRLQSIGTENTEENRRFYRQLLLTADDRVNPCIGGVILFHETLYQKADDGRPFPQVCKSKGGVVGIKVDKGVVPLAGTNGETTTQGLDGLSERCAQYKKDGADFAKWRCVLKIGEHTPSALAIMENANVLARYASICQQNGIVPIVEPEILPDGDHDLKRCQYVTEKVLAAVYKALSDHHIYLEGTLLKPNMVTPGHACTQKFSHEEIAMATVTALRRTVPPAVTGITFLSGGQSEEEASINLNAINKCPLLKPWALTFSYGRALQASALKAWGGKKENLKAAQEEYVKRALANSLACQGKYTPSGQAGAAASESLFVSNHAY
1QQV , Knot 38 67 0.79 34 61 65 
PTKLETFPLDVLVNTAAEDLPRGVDPSRKENHLSDEDFKAVFGMTRSAFANLPLWKQQNLKKEKGLF
6ICJ , Knot 120 272 0.82 38 172 260 
PESADLRALAKHLYDSYIKSFPLTKAKARAILTGKTTDKSPFVIYDMNSLMMGEDKIKFKHITPLQEQSKEVAIRIFQGCQFRSVEAVQEITEYAKSIPGFVNLDLNDQVTLLKYGVHEIIYTMLASLMNKDGVLISEGQGFMTREFLKSLRKPFGDFMEPKFEFAVKFNALELDDSDLAIFIAVIILSGDRPGLLNVKPIEDIQDNLLQALELQLKLNHPESSQLFAKLLQKMTDLRQIVTEHVQLLQVIKKTETDMSLHPLLQEIYKDLY

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 \(c\).

\(|P_{f(6XML_1)}(2) \setminus P_{f(1QQV_1)}(2)|=172\), \(|P_{f(1QQV_1)}(2) \setminus P_{f(6XML_1)}(2)|=24\). 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 1:1100011101000001001100111110111110000101100100110000000001000111010001010111111100010001001011101000011111101001111111001000001101100001000001101101001101100010111110010111001010000111111010111010001000001000111110011000010101011010110110100001000011110101100011111011011011000001010101100011101111010010110101101111000010110000100111001100100010101111100011100010
Pair \(Z_2\) Length of longest common subsequence
6XML_1,1QQV_1 196 3
6XML_1,6ICJ_1 159 4
1QQV_1,6ICJ_1 167 3

Newick tree

 
[
	1QQV_1:94.57,
	[
		6XML_1:79.5,6ICJ_1:79.5
	]:15.07
]

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 \((0.85)(0.8)(\frac{431 }{\log_{20} 431}-\frac{67}{\log_{20}67})=112.\)
Status Protein1 Protein2 d d1/2
Query variables 6XML_1 1QQV_1 142 83
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]