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Parikh vectors
6OGD_1 7HRA_1 2UUX_1 Letter Amino acid
87 11 4 A Alanine
67 15 0 Q Glutamine
10 13 0 H Histidine
75 15 1 I Isoleucine
15 7 1 W Tryptophan
41 19 4 R Arginine
63 19 0 D Aspartic acid
7 4 8 C Cysteine
147 23 1 L Leucine
57 14 3 K Lycine
36 8 5 Y Tyrosine
72 12 2 N Asparagine
55 21 8 G Glycine
18 7 2 M Methionine
47 9 3 P Proline
108 21 2 S Serine
64 22 3 V Valine
76 21 4 E Glutamic acid
50 6 1 F Phenylalanine
69 28 3 T Threonine 

6OGD_1|Chains A, D, G, J, M|Toxin subunit YenA1|Yersinia entomophaga (935293)
>7HRA_1|Chains A, B|Oleoyl-acyl carrier protein thioesterase 1, chloroplastic|Arabidopsis thaliana (3702)
>2UUX_1|Chain A|TRYPTASE INHIBITOR|RHIPICEPHALUS APPENDICULATUS (34631)
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 \(w=f(c)\)
6OGD , Knot 412 1164 0.83 40 311 997 
MDKYNNYSNVIKNKSSISPLLAAAAKIEPEITVLSSASKSNRSQYSQSLADTLLGLGYRSIFDIAKVSRQRFIKRHDESLLGNGAVIFDKAVSMANQVLQKYRKNRLEKSNSPLVPQTSSSTDASSESQTNKLPEYNQLFPEPWDNFCRPGAIEALDSPASYLLDLYKFIQSVELDGSNQARKLETRRADIPKLSLDNDALYKEVTALSIVNDVLSGSAREYIDQSGQADKAVNQILGDTHFPFTLPYSLPTQQINKGLGASNIELGTVIQRVDPQFSWNTTQEKYNQVLLAYTQLSSEQIALLSLPDVFTQNFLTQTELSAGYLSASTTEILAEKDLSRHGYIVKAADNIKGPTQLVEHSDASYDVIELTCTNQAKETITVKLRGENIITYQRTKARMVPFDNSSPFSRQLKLTFVAEDNPSLGNLDKGPYFANMDIYAAEWVRENVSSETMVSRPFLTMTYRIAIAKAGASLEELQPEADAFFINNFGLSAEDSSQLVKLVAFGDQTGSKAEEIESLLSCGENLPIVSPNVIFANPIFGSYFNDEPFPAPYHFGGVYINAHQRNAMTIIRAEGGREIQSLSNFRLERLNRFIRLQRWLDLPSHQLDLLLTSVMQADADNSQQEITEPVLKSLGLFRHLNLQYKITPEIFSSWLYQLTPFAVSGEIAFFDRIFNREQLFDQPFILDGGSFTYLDAKGSDAKSVKQLCAGLNISAVTFQFIAPLVQSALGLEAGTLVRSFEVVSSLYRLVSIPQTFGLSTEDGLILMNILTDEMGYLAKQPAFDDKQTQDKDFLSIILKMEALSAWLTKNNLTPASLALLLGVTRLAVVPTNNMVTFFKGIANGLSENVCLTTDDFQRQELEGADWWTLLSTNQVIDDMGLVLDIHPVWGKSDEEMLMEKIQSIGVSNDNNTLSIIVQILIQAKNAQENLLSQTISAEYGVERSVVPLQLRWLGSNVYSVLNQVLNNTPTDISSIVPKLSELTYSLLIYTQLINSLKLNKEFIFLRLTQPNWLGLTQPKLSTQLSLPEIYLITCYQDWVVNANKNEDSIHEYLEFANIKKTEAEKTLVDNSEKCAELLAEILAWDAGEILKAASLLGLNPPQATNVFEIDWIRRLQTLSEKTMISTEYLWQMGDLTENSEFSLKEGVGEAVMAALKAQGDSDNV
7HRA , Knot 131 295 0.84 40 189 276 
MGSLTEDGLSYKEKFVVRSYEVGSNKTATVETIANLLQEVGCNHAQSVGFSTDGFATTTTMRKLHLIWVTARMHIEIYKYPAWGDVVEIETWCQSEGRIGTRRDWILKDSVTGEVTGRATSKWVMMNQDTRRLQKVSDDVRDEYLVFCPQEPRLAFPEENNRSLKKIPKLEDPAQYSMIGLKPRRADLDMNQHVNNVTYIGWVLESIPQEIVDTHELQVITLDYRRECQQDDVVDSLTTTTSEIGGTNGSATSGTQGHNDSQFLHLLRLSGDGQEINRGTTLWRKKPSSHHHHHH
2UUX , Knot 34 55 0.82 34 49 53 
AAECTVPIGWSEPVKGLCKARFTRYYCMGNCCKVYEGCYTGGYSRMGECARNCPG

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 \(c\).

\(|P_{f(6OGD_1)}(2) \setminus P_{f(7HRA_1)}(2)|=152\), \(|P_{f(7HRA_1)}(2) \setminus P_{f(6OGD_1)}(2)|=30\). 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 1:100000000110000010111111101010101100100000000000110011111000110110100001100000011101111100110110011000000010000011110000000100000000110000111011001001111011001100110100110010101000100100001011010100011000101101100110101000100010100110011100011101100110001001111001011011001010101000000000111100010000111101101100011000010110101000011100010001011011001011001100001000110100000100010101010011000000101111000011000101011100010110100110110101011011000100001100111010001111011101001010101111001110100000110111110001001001001100100111101011110111100100011111001111010100001101101011001001001010010011010011011000101110011010100000010011100111100101000101011001100101111010111100110000110011110110100101010010010010111010110101111110011110110110010110010011011001110000111110110001101100111000000000110111010110111000010110111111100111110001101101110110001010000100001011011011000011001111101011110000011100100111000000101110111010010001100010100110001111010111001001100110001001001110100100011100011001010001111010010111100101000101101011000001110100000010001011010000100011000000101110111101101101101111011010011010110010010000110000110110100000101001110111111010100001
Pair \(Z_2\) Length of longest common subsequence
6OGD_1,7HRA_1 182 5
6OGD_1,2UUX_1 288 3
7HRA_1,2UUX_1 198 3

Newick tree

 
[
	2UUX_1:13.65,
	[
		6OGD_1:91,7HRA_1:91
	]:41.65
]

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 \((0.85)(0.8)(\frac{1459 }{\log_{20} 1459}-\frac{295}{\log_{20}295})=302.\)
Status Protein1 Protein2 d d1/2
Query variables 6OGD_1 7HRA_1 380 236
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]