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Parikh vectors
3ZUP_1 3KLL_1 2YDM_1 Letter Amino acid
0 49 33 Q Glutamine
7 56 30 K Lycine
5 61 33 S Serine
34 74 44 A Alanine
34 75 62 L Leucine
9 84 32 T Threonine
23 57 28 V Valine
9 67 26 Y Tyrosine
12 89 30 D Aspartic acid
1 0 7 C Cysteine
25 44 40 E Glutamic acid
2 20 20 H Histidine
3 45 25 I Isoleucine
21 33 32 P Proline
4 19 19 W Tryptophan
31 35 26 R Arginine
2 97 30 N Asparagine
30 82 30 G Glycine
1 22 17 M Methionine
6 30 25 F Phenylalanine 

3ZUP_1|Chains A, B|MANNOSYL-3-PHOSPHOGLYCERATE PHOSPHATASE|THERMUS THERMOPHILUS (262724)
>3KLL_1|Chain A|Glucansucrase|Lactobacillus reuteri (1598)
>2YDM_1|Chain A|ANGIOTENSIN CONVERTING ENZYME|HOMO SAPIENS (9606)
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 \(w=f(c)\)
3ZUP , Knot 104 259 0.74 38 128 223 
MIVFTDLDGTLLDERGELGPAREALERLRALGVPVVPVTAKTRKEVEALGLEPPFIVENGGGLYLPRDWPVRAGRPKGGYRVVSLAWPYRKVRARLREAEALAGRPILGYGDLTAEAVARLTGLSREAARRAKAREYDETLVLCPEEVEAVLEALEAVGLEWTHGGRFYHAAKGADKGRAVARLRALWPDPEEARFAVGLGDSLNDLPLFRAVDLAVYVGRGDPPEGVLATPAPGPEGFRYAVERYLLPRLSRRGGSGP
3KLL , Knot 378 1039 0.84 38 300 922 
MGINGQQYYIDPTTGQPRKNFLLQNGNDWIYFDKDTGAGTNALKLQFDKGTISADEQYRRGNEAYSYDDKSIENVNGYLTADTWYRPKQILKDGTTWTDSKETDMRPILMVWWPNTVTQAYYLNYMKQYGNLLPASLPSFSTDADSAELNHYSELVQQNIEKRISETGSTDWLRTLMHEFVTKNSMWNKDSENVDYGGLQLQGGFLKYVNSDLTKYANSDWRLMNRTATNIDGKNYGGAEFLLANDIDNSNPVVQAEELNWLYYLMNFGTITGNNPEANFDGIRVDAVDNVDVDLLSIARDYFNAAYNMEQSDASANKHINILEDWGWDDPAYVNKIGNPQLTMDDRLRNAIMDTLSGAPDKNQALNKLITQSLVNRANDNTENAVIPSYNFVRAHDSNAQDQIRQAIQAATGKPYGEFNLDDEKKGMEAYINDQNSTNKKWNLYNMPSAYTILLTNKDSVPRVYYGDLYQDGGQYMEHKTRYFDTITNLLKTRVKYVAGGQTMSVDKNGILTNVRFGKGAMNATDTGTDETRTEGIGVVISNNTNLKLNDGESVVLHMGAAHKNQKYRAVILTTEDGVKNYTNDTDAPVAYTDANGDLHFTNTNLDGQQYTAVRGYANPDVTGYLAVWVPAGAADDQDARTAPSDEAHTTKTAYRSNAALDSNVIYEGFSNFIYWPTTESERTNVRIAQNADLFKSWGITTFELAPQYNSSKDGTFLDSIIDNGYAFTDRYDLGMSTPNKYGSDEDLRNALQALHKAGLQAIADWVPDQIYNLPGKEAVTVTRSDDHGTTWEVSPIKNVVYITNTIGGGEYQKKYGGEFLDTLQKEYPQLFSQVYPVTQTTIDPSVKIKEWSAKYFNGTNILHRGAGYVLRSNDGKYYNLGTSTQQFLPSQLSVQDNEGYGFVKEGNNYHYYDENKQMVKDAFIQDSVGNWYYLDKNGNMVANQSPVEISSNGASGTYLFLNNGTSFRSGLVKTDAGTYYYDGDGRMVRNQTVSDGAMTYVLDENGKLVSESFDSSATEAHPLKPGDLNGQKHHHHHH
2YDM , Knot 240 589 0.86 40 296 558 
LVTDEAEASKFVEEYDRTSQVVWNEYAEANWNYNTNITTETSKILLQKNMQIANHTLKYGTQARKFDVNQLQNTTIKRIIKKVQDLERAALPAQELEEYNKILLDMETTYSVATVCHPNGSCLQLEPDLTNVMATSRKYEDLLWAWEGWRDKAGRAILQFYPKYVELINQAARLNGYVDAGDSWRSMYETPSLEQDLERLFQELQPLYLNLHAYVRRALHRHYGAQHINLEGPIPAHLLGNMWAQTWSNIYDLVVPFPSAPSMDTTEAMLKQGWTPRRMFKEADDFFTSLGLLPVPPEFWNKSMLEKPTDGREVVCHASAWDFYNGKDFRIKQCTTVNLEDLVVAHHEMGHIQYFMQYKDLPVALREGANPGFHEAIGDVLALSVSTPKHLHSLNLLSSEGGSDEHDINFLMKMALDKIAFIPFSYLVDQWRWRVFDGSITKENYNQEWWSLRLKYQGLCPPVPRTQGDFDPGAKFHIPSSVPYIRYFVSFIIQFQFHEALCQAAGHTGPLHKCDIYQSKEAGQRLATAMKLGFSRPWPEAMQLITGQPNMSASAMLSYFKPLLDWLRTENELHGEKLGWQQYNWTPNS

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 \(c\).

\(|P_{f(3ZUP_1)}(2) \setminus P_{f(3KLL_1)}(2)|=18\), \(|P_{f(3KLL_1)}(2) \setminus P_{f(3ZUP_1)}(2)|=190\). 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 1:1111001010110001011110011001011111111101000001011110111110011110110011101101011001101111000101010010111101111010101011101011000110010100000011101001011101101111010011010011011001011101011110100101111110010011110110111011010110111101111101100110001110100011011
Pair \(Z_2\) Length of longest common subsequence
3ZUP_1,3KLL_1 208 4
3ZUP_1,2YDM_1 202 4
3KLL_1,2YDM_1 108 5

Newick tree

 
[
	3ZUP_1:11.18,
	[
		2YDM_1:54,3KLL_1:54
	]:60.18
]

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 \((0.85)(0.8)(\frac{1298 }{\log_{20} 1298}-\frac{259}{\log_{20}259})=273.\)
Status Protein1 Protein2 d d1/2
Query variables 3ZUP_1 3KLL_1 349 214.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]