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Parikh vectors
8ZNC_1 6OUM_1 4ZHB_1 Letter Amino acid
40 13 7 A Alanine
24 19 8 D Aspartic acid
5 11 6 H Histidine
34 23 11 K Lycine
13 1 5 M Methionine
20 7 3 R Arginine
10 12 4 Q Glutamine
10 11 2 F Phenylalanine
16 8 7 Y Tyrosine
15 9 6 N Asparagine
37 22 2 G Glycine
29 8 8 I Isoleucine
20 17 7 P Proline
1 1 1 C Cysteine
34 13 9 E Glutamic acid
24 26 12 L Leucine
12 18 5 S Serine
23 13 4 T Threonine
11 7 0 W Tryptophan
38 18 7 V Valine 

8ZNC_1|Chain A|Glutamate dehydrogenase|Thermococcus profundus (49899)
>6OUM_1|Chain A|Carbonic anhydrase 2|Homo sapiens (9606)
>4ZHB_1|Chain A|Ankyrin repeat-containing protein|Legionella pneumophila subsp. pneumophila (strain Philadelphia 1 / ATCC 33152 / DSM 7513) (272624)
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 \(w=f(c)\)
8ZNC , Knot 173 416 0.83 40 213 398 
IDPFEMAVKQLERAAQYMDISEEALEWLKKPMRIVEVSVPIEMDDGSVKVFTGFRVQHNWARGPTKGGIRWHPAETLSTVKALATFMTWKVAVVDLPYGGGKGGIIVNPKELSEREQERLARAYIRAVYDVIGPWTDIPAPDVYTNPKIMGWMMDEYETIMRRKGPAFGVITGKPLSIGGSLGRGTATAQGAIFTIREAAKALGIDLKGKKIAVQGYGNAGYYTAKLAKEQLGMTVVAVSDSRGGIYNPDGLDPDEVLKWKREHGSVKDFPGATNITNEELLELEVDVLAPAAIEEVITEKNADNIKAKIVAEVANGPVTPEADDILREKGILQIPDFLCNAGGVTVSYFEWVQNINGYYWTEEEVREKLDKKMTKAFWEVYNTHKDKNIHMRDAAYVVAVSRVYQAMKDRGWVKK
6OUM , Knot 110 257 0.79 40 168 246 
HWGYGKHNGPEHWHKDFPIAKGERQSPVDIDTHTAKYDPSLKPLSVSYDQATSLRILNNGHSFQVTFDDSQDKAVLKGGPLDGTYRLLQFHFHWGSLDGQGSEHTVDKKKYAAELHLVHWNTKYGDVGKAVQQPDGLAVLGIFLKVGSAKPGLQKVVDVLDSIKTEGKSADFTNFDPRGLLPESLDYWTYPGSLTTPPLAECVTWIVLKEPISVSSEQVLKFRKLNFNGEGEPEELMVDNWRPAQPLKNRQIKASFK
4ZHB , Knot 59 114 0.81 38 93 112 
MIKMGRSEMKIASAELRELMKAVSEGHYETVNTILDKDPELVNQYAPPTYDSPLARVLNKKHIDYKMLDILVKHHVDFDYPINYHKETPIELACKNQDLQLFKYLVQHNAPISE

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 \(c\).

\(|P_{f(8ZNC_1)}(2) \setminus P_{f(6OUM_1)}(2)|=111\), \(|P_{f(6OUM_1)}(2) \setminus P_{f(8ZNC_1)}(2)|=66\). 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 1:10110111001001100101000110110011011010111010010101101101000110110011101011001001011101101011110110111011111010010000000110101011001111100111101000101111110000011000111111101011011101101010101111010011011110101001110101011000101100011101111000011100101101001101000010100111100100001101010111111100110000100101011101101110101001100011101101100111101001011001010010000100010001001110100000000101001101111001001100011100
Pair \(Z_2\) Length of longest common subsequence
8ZNC_1,6OUM_1 177 4
8ZNC_1,4ZHB_1 188 4
6OUM_1,4ZHB_1 153 3

Newick tree

 
[
	8ZNC_1:95.71,
	[
		6OUM_1:76.5,4ZHB_1:76.5
	]:19.21
]

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 \((0.85)(0.8)(\frac{673 }{\log_{20} 673}-\frac{257}{\log_{20}257})=116.\)
Status Protein1 Protein2 d d1/2
Query variables 8ZNC_1 6OUM_1 145 118
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]