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Parikh vectors
1FXJ_1 3SOM_1 3WOG_1 Letter Amino acid
18 20 5 R Arginine
23 13 16 D Aspartic acid
3 5 0 C Cysteine
17 14 7 Q Glutamine
14 10 13 K Lycine
8 16 27 S Serine
2 6 5 W Tryptophan
37 28 26 L Leucine
8 3 0 M Methionine
9 39 10 P Proline
9 11 6 Y Tyrosine
27 17 20 V Valine
20 16 8 E Glutamic acid
31 14 16 G Glycine
4 14 15 F Phenylalanine
19 11 25 T Threonine
32 26 18 A Alanine
17 5 20 N Asparagine
10 8 3 H Histidine
23 10 14 I Isoleucine

1FXJ_1|Chains A, B|UDP-N-ACETYLGLUCOSAMINE PYROPHOSPHORYLASE|Escherichia coli (562)
>3SOM_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P|Methylmalonic aciduria and homocystinuria type C protein|Homo sapiens (9606)
>3WOG_1|Chains A, B|Erythroagglutinin|Phaseolus vulgaris (3885)
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 \(w=f(c)\)
1FXJ , Knot 141 331 0.82 40 183 320
MLNNAMSVVILAAGKGTRMYSDLPKVLHTLAGKAMVQHVIDAANELGAAHVHLVYGHGGDLLKQALKDDNLNWVLQAEQLGTGHAMQQAAPFFADDEDILMLYGDVPLISVETLQRLRDAKPQGGIGLLTVKLDDPTGYGRITRENGKVTGIVEHKDATDEQRQIQEINTGILIANGADMKRWLAKLTNNNAQGEYYITDIIALAYQEGREIVAVHPQRLSEVEGVNNRLQLSRLERVYQSEQAEKLLLAGVMLRDPARFDLRGTLTHGRDVEIDTNVIIEGNVTLGHRVKIGTGCVIKNSVIGDDCEISPYTVVEDANLAAACTIGPFAR
3SOM , Knot 125 286 0.82 40 185 272
YFQSMEPKVAELKQKIEDTLCPFGFEVYPFQVAWYNELLPPAFHLPLPGPTLAFLVLSTPAMFDRALKPFLQSCHLRMLTDPVDQCVAYHLGRVRESLPELQIEIIADYEVHPNRRPKILAQTAAHVAGAAYYYQRQDVEADPWGNQRISGVCIHPRFGGWFAIRGVVLLPGIEVPDLPPRKPHDCVPTRADRIALLEGFNFHWRDWTYRDAVTPQERYSEEQKAYFSTPPAQRLALLGLAQPSEKPSSPSPDLPFTTPAPKKPGNPSRARSWLSPRVSPPASPGP
3WOG , Knot 112 254 0.81 36 158 242
ASQTSFSFQRFNETNLILQRDATVSSKGQLRLTNVNDNGEPTLSSLGRAFYSAPIQIWDNTTGAVASFATSFTFNIDVPNNSGPADGLAFVLLPVGSQPKDKGGLLGLFNNYKYDSNAHTVAVEFDTLYNVHWDPKPRHIGIDVNSIKSIKTTTWDFVKGENAEVLITYDSSTKLLVASLVYPSLKTSFIVSDTVDLKSVLPEWVIVGFTATTGITKGNVETNDILSWSFASKLSDGTTSEALNLANFALNQIL

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 \(c\).

\(|P_{f(1FXJ_1)}(2) \setminus P_{f(3SOM_1)}(2)|=82\), \(|P_{f(3SOM_1)}(2) \setminus P_{f(1FXJ_1)}(2)|=84\). 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 1:1100110111111101001000110110011101110011011001111010110101101100110000101110100110101100111111000011110101111010010010010101111110101001010101000010101110000100000010010011111011010011101000010100010011111000100111101001001011000101001001000001001111111100110101010100100101000111010101100101101011000111000010100110010111100111110
Pair \(Z_2\) Length of longest common subsequence
1FXJ_1,3SOM_1 166 4
1FXJ_1,3WOG_1 163 3
3SOM_1,3WOG_1 177 4

Newick tree

 
[
	3SOM_1:87.17,
	[
		1FXJ_1:81.5,3WOG_1:81.5
	]:5.67
]

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 \((0.85)(0.8)(\frac{617 }{\log_{20} 617}-\frac{286}{\log_{20}286})=92.6\)
Status Protein1 Protein2 d d1/2
Query variables 1FXJ_1 3SOM_1 115 109
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]