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Parikh vectors
7OGU_1 4BUU_1 2VDV_1 Letter Amino acid
112 15 20 L Leucine
36 8 13 P Proline
77 19 9 S Serine
22 11 23 K Lycine
7 8 8 M Methionine
23 12 14 V Valine
33 7 12 D Aspartic acid
10 5 4 C Cysteine
55 25 13 G Glycine
22 11 14 F Phenylalanine
21 9 11 T Threonine
6 1 3 W Tryptophan
13 13 10 Y Tyrosine
22 11 12 A Alanine
14 14 13 R Arginine
56 12 13 N Asparagine
15 9 8 Q Glutamine
35 17 19 E Glutamic acid
7 19 14 H Histidine
31 14 13 I Isoleucine 

7OGU_1|Chains A[auth AAA], D[auth DDD], G[auth GGG], J[auth JJJ]|Receptor-like protein kinase HSL1|Arabidopsis thaliana (3702)
>4BUU_1|Chains A, B|TANKYRASE-2|HOMO SAPIENS (9606)
>2VDV_1|Chains A[auth E], B[auth F]|TRNA (GUANINE-N(7)-)-METHYLTRANSFERASE|SACCHAROMYCES CEREVISIAE (4932)
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 \(w=f(c)\)
7OGU , Knot 220 617 0.76 40 211 480 
GSSMDNQDGFILQQVKLSLDDPDSYLSSWNSNDASPCRWSGVSCAGDFSSVTSVDLSSANLAGPFPSVICRLSNLAHLSLYNNSINSTLPLNIAACKSLQTLDLSQNLLTGELPQTLADIPTLVHLDLTGNNFSGDIPASFGKFENLEVLSLVYNLLDGTIPPFLGNISTLKMLNLSYNPFSPSRIPPEFGNLTNLEVMWLTECHLVGQIPDSLGQLSKLVDLDLALNDLVGHIPPSLGGLTNVVQIELYNNSLTGEIPPELGNLKSLRLLDASMNQLTGKIPDELCRVPLESLNLYENNLEGELPASIALSPNLYEIRIFGNRLTGGLPKDLGLNSPLRWLDVSENEFSGDLPADLCAKGELEELLIIHNSFSGVIPESLADCRSLTRIRLAYNRFSGSVPTGFWGLPHVNLLELVNNSFSGEISKSIGGASNLSLLILSNNEFTGSLPEEIGSLDNLNQLSASGNKFSGSLPDSLMSLGELGTLDLHGNQFSGELTSGIKSWKKLNELNLADNEFTGKIPDEIGSLSVLNYLDLSGNMFSGKIPVSLQSLKLNQLNLSYNRLSGDLPPSLAKDMYKNSFIGNPGLCGDIKGLCGSENEAKKRGYVLEGSENLYFQ
4BUU , Knot 112 240 0.85 40 176 232 
MHHHHHHSSGVDLGTENLYFQSMLNTSGSGTILIDLSPDDKEFQSVEEEMQSTVREHRDGGHAGGIFNRYNILKIQKVCNKKLWERYTHRRKEVSEENHNHANERMLFHGSPFVNAIIHKGFDERHAYIGGMFGAGIYFAENSSKSNQYVYGIGGGTGCPVHKDRSCYICHRQLLFCRVTLGKSFLQFSAMKMAHSPPGHHSVTGRPSVNGLALAEYVIYRGEQAYPEYLITYQIMRPEG
2VDV , Knot 113 246 0.84 40 179 238 
LPKKRYYRQRAHSNPFSDHQLEYPVSPQDMDWSKLYPYYKNAENGQMTKKVTIADIGCGFGGLMIDLSPAFPEDLILGMEIRVQVTNYVEDRIIALRNNTASKHGFQNINVLRGNAMKFLPNFFEKGQLSKMFFCFPDPHFKQRKHKARIITNTLLSEYAYVLKEGGVVYTITDVKDLHEWMVKHLEEHPLFERLSKEWEENDECVKIMRNATEEGKKVERKKGDKFVACFTRLPTPAILHHHHHH

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 \(c\).

\(|P_{f(7OGU_1)}(2) \setminus P_{f(4BUU_1)}(2)|=105\), \(|P_{f(4BUU_1)}(2) \setminus P_{f(7OGU_1)}(2)|=70\). 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 1:10010000111100101010010001001000010100101100110100100101001011111101100100110101000010001110111000100101000110101100110110110101010010101110110100101101100110101111110100101101000110100111011010010111100001110110011010011010111001110111011110011010100001010111011010010110101001010110010011100101000010101110111010100101110010111100111001101101000010101110101010100111100010111100110000100101100010101101111110101101100010101000111100101111000010101100110100100101010010101100110110110101010010101001100100100101100010101100110101100101010110101110100101001010000101011101100100001110111010101101000010001011010001010
Pair \(Z_2\) Length of longest common subsequence
7OGU_1,4BUU_1 175 6
7OGU_1,2VDV_1 174 5
4BUU_1,2VDV_1 181 6

Newick tree

 
[
	4BUU_1:89.67,
	[
		7OGU_1:87,2VDV_1:87
	]:2.67
]

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 \((0.85)(0.8)(\frac{857 }{\log_{20} 857}-\frac{240}{\log_{20}240})=169.\)
Status Protein1 Protein2 d d1/2
Query variables 7OGU_1 4BUU_1 196 139
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]