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Parikh vectors
6LOM_1 5GNX_1 8ZDW_1 Letter Amino acid
23 45 15 A Alanine
18 15 29 N Asparagine
15 44 18 G Glycine
18 14 11 F Phenylalanine
9 19 6 W Tryptophan
24 25 21 V Valine
12 29 19 E Glutamic acid
7 22 10 H Histidine
21 21 26 I Isoleucine
15 16 25 K Lycine
13 26 18 P Proline
28 24 19 T Threonine
14 19 13 Y Tyrosine
13 32 10 R Arginine
31 40 28 L Leucine
14 33 14 D Aspartic acid
11 1 9 C Cysteine
15 12 11 Q Glutamine
4 11 7 M Methionine
24 19 28 S Serine 

6LOM_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T|Calcium homeostasis modulator protein|Caenorhabditis elegans (6239)
>5GNX_1|Chains A, B|Beta-glucosidase|metagenome (256318)
>8ZDW_1|Chains A, K[auth C], L[auth H]|Hemagglutinin|Influenza A virus (strain A/Vietnam/1203/2004 H5N1) (284218)
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 \(w=f(c)\)
6LOM , Knot 147 329 0.86 40 210 320 
MTTSINSVVTVFQNVFTNHGSTLLNGILIATTVGGQSLVRKLTFSCPCAYPLNIYHSLVFMFGPTAALLLIGITVNSTTWKLAHGFFFRVRDTRHSWKTTCVSWIEVLIQSSVAPIAWLFVVFLDGGYYRCYRSHEFCLISDAILCKNSTILNSYASTSSFNKISDNGKYCPPCICVPNPTDASYLEAESQIYAWGLLLFSGVAAFLVITCNRMCDKYTLVQRQYVETYKNVETQKFDAVAKEHASQLAEHNARAFFGQKDWTKRDWDWVSGIPEVNNPLFARLRLIAAEKTQQTMYTPLQLWNDNKGYRIPQPDLQLTQIIVDETKED
5GNX , Knot 193 467 0.84 40 244 444 
MGSMTSDTARSYRFPEGFLWGAATAAYQIEGSSMADGAGESIWDRFSHTPGNMKDGDTGDVACDHYNRWREDIELMKRLNLQAYRFSVSWSRVIPQGRGAINPKGLAFYDRLVDGLLEAGIEPLATLYHWDLPAALDDRGGWLNPDIADWFADYGQVLFEKFKGRVKTWGTINQPWVIVDGGYLHGALAPGHRSAYEAVIAGHNVLRAHGAAVRRFREVGEGQIGIVLNIEPKYPASDKPEDEAARRRAEAQMNRWFLDPLMGRGYPEELTDVYGAAWREFPKEDFELIAEPTDWMGLNWYTRAVPENAPDAWPTRSRPVRQTQHAHTETGWEVYPPALTDTLVWLSEQTGGKLPLMVTENGSAWYDPPHAIDGRIHDPMRVHYLQTHIKALHDAIGKGVDLRGYMAWSLLDNLEWSLGYSKRFGIVHVNFATQERTIKDSGLLYAEVIKTHGDVLNTLKLHHHHHH
8ZDW , Knot 147 337 0.84 40 210 323 
MEKIVLLFAIVSLVKSDQICIGYHANNSTEQVDTIMEKNVTVTHAQDILEKKHNGKLCDLDGVKPLILRDCSVAGWLLGNPMCDEFINVPEWSYIVEKANPVNDLCYPGDFNDYEELKHLLSRINHFEKIQIIPKSSWSSHEASLGVSSACPYQGKSSFFRNVVWLIKKNSTYPTIKRSYNNTNQEDLLVLWGIHHPNDAAEQTKLYQNPTTYISVGTSTLNQRLVPRIATRSKVNGQSGRMEFFWTILKPNDAINFESNGNFIAPEYAYKIVKKGDSTIMKSELEYGNCNTKCQTPMGAINSSMPFHNIHPLTIGECPKYVKSNRLVLATGLRNSP

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 \(c\).

\(|P_{f(6LOM_1)}(2) \setminus P_{f(5GNX_1)}(2)|=78\), \(|P_{f(5GNX_1)}(2) \setminus P_{f(6LOM_1)}(2)|=112\). 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 1:10001001101100110001001101111100111001100101001010110100011111110111111110100001011011110100000010000101101110001111111111110110000000001011001110000011000100001001000100011010110100100101000101111111011111111000010000011000010000010000101110001001100010111100010000101101110100111101011110000001001101100001001101010100111000000
Pair \(Z_2\) Length of longest common subsequence
6LOM_1,5GNX_1 190 4
6LOM_1,8ZDW_1 170 4
5GNX_1,8ZDW_1 170 4

Newick tree

 
[
	5GNX_1:91.78,
	[
		6LOM_1:85,8ZDW_1:85
	]:6.78
]

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 \((0.85)(0.8)(\frac{796 }{\log_{20} 796}-\frac{329}{\log_{20}329})=127.\)
Status Protein1 Protein2 d d1/2
Query variables 6LOM_1 5GNX_1 166 140.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]