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Parikh vectors
7ENR_1 5PUU_1 1DGG_1 Letter Amino acid
85 4 31 N Asparagine
50 10 36 V Valine
94 13 23 E Glutamic acid
89 5 23 I Isoleucine
26 6 36 P Proline
55 7 22 S Serine
38 11 36 A Alanine
63 13 35 D Aspartic acid
3 1 4 C Cysteine
21 10 20 H Histidine
132 6 27 K Lycine
33 6 31 F Phenylalanine
60 5 21 Y Tyrosine
57 14 28 R Arginine
37 8 25 Q Glutamine
43 7 33 G Glycine
101 18 28 L Leucine
14 7 9 M Methionine
47 5 23 T Threonine
5 0 6 W Tryptophan 

7ENR_1|Chain A|CRISPR-associated endonuclease Cas9|Staphylococcus aureus (1280)
>5PUU_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
>1DGG_1|Chains A, B, C, D|CATALASE|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 \(w=f(c)\)
7ENR , Knot 381 1053 0.84 40 288 884 
MKRNYILGLDIGITSVGYGIIDYETRDVIDAGVRLFKEANVENNEGRRSKRGARRLKRRRRHRIQRVKKLLFDYNLLTDHSELSGINPYEARVKGLSQKLSEEEFSAALLHLAKRRGVHNVNEVEEDTGNELSTKEQISRNSKALEEKYVAELQLERLKKDGEVRGSINRFKTSDYVKEAKQLLKVQKAYHQLDQSFIDTYIDLLETRRTYYEGPGEGSPFGWKDIKEWYEMLMGHCTYFPEELRSVKYAYNADLYNALNDLNNLVITRDENEKLEYYEKFQIIENVFKQKKKPTLKQIAKEILVNEEDIKGYRVTSTGKPEFTNLKVYHDIKDITARKEIIENAELLDQIAKILTIYQSSEDIQEELTNLNSELTQEEIEQISNLKGYTGTHNLSLKAINLILDELWHTNDNQIAIFNRLKLVPKKVDLSQQKEIPTTLVDDFILSPVVKRSFIQSIKVINAIIKKYGLPNDIIIELAREKNSKDAQKMINEMQKRNRQTNERIEEIIRTTGKENAKYLIEKIKLHDMQEGKCLYSLEAIPLEDLLNNPFNYEVDHIIPRSVSFDNSFNNKVLVKQEENSKKGNRTPFQYLSSSDSKISYETFKKHILNLAKGKGRISKTKKEYLLEERDINRFSVQKDFINRNLVDTRYATRGLMNLLRSYFRVNNLDVKVKSINGGFTSFLRRKWKFKKERNKGYKHHAEDALIIANADFIFKEWKKLDKAKKVMENQMFEEKQAESMPEIETEQEYKEIFITPHQIKHIKDFKDYKYSHRVDKKPNRELINDTLYSTRKDDKGNTLIVNNLNGLYDKDNDKLKKLINKSPEKLLMYHHDPQTYQKLKLIMEQYGDEKNPLYKYYEETGNYLTKYSKKDNGPVIKKIKYYGNKLNAHLDITDDYPNSRNKVVKLSLKPYRFDVYLDNGVYKFVTVKNLDVIKKENYYEVNSKCYEEAKKLKKISNQAEFIASFYNNDLIKINGELYRVIGVNNDLLNRIEVNMIDITYREYLENMNDKRPPRIIKTIASKTQSIKKYSTDILGNLYEVKSKKHPQIIKKG
5PUU , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA
1DGG , Knot 208 497 0.86 40 259 470 
RDPASDQMQHWKEQRAAQKADVLTTGAGNPVGDKLNVITVGPRGPLLVQDVVFTDEMAHFDRERIPERVVHAKGAGAFGYFEVTHDITKYSKAKVFEHIGKKTPIAVRFSTVAGESGSADTVRDPRGFAVKFYTEDGNWDLVGNNTPIFFIRDPILFPSFIHSQKRNPQTHLKDPDMVWDFWSLRPESLHQVSFLFSDRGIPDGHRHMNGYGSHTFKLVNANGEAVYCKFHYKTDQGIKNLSVEDAARLSQEDPDYGIRDLFNAIATGKYPSWTFYIQVMTFNQAETFPFNPFDLTKVWPHKDYPLIPVGKLVLNRNPVNYFAEVEQIAFDPSNMPPGIEASPDKMLQGRLFAYPDTHRHRLGPNYLHIPVNCPYRARVANYQRDGPMCMQDNQGGAPNYYPNSFGAPEQQPSALEHSIQYSGEVRRFNTANDDNVTQVRAFYVNVLNEEQRKRLCENIAGHLKDAQIFIQKKAVKNFTEVHPDYGSHIQALLDKYN

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 \(c\).

\(|P_{f(7ENR_1)}(2) \setminus P_{f(5PUU_1)}(2)|=187\), \(|P_{f(5PUU_1)}(2) \setminus P_{f(7ENR_1)}(2)|=19\). 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 1:100001111011100110111000000110111011001010000100000110010000000100100111000110000010110100101011000100001011110110001100100100001001000001000001100001101010010001010101001000001001001101001000100011000101100000000111010111100100100111100001100100100100101001100100111000000010000010110011000001010011001110000101001000101010010100010010100011001011001101101000000100010010001000010010010100100010101101110011000000111100101110010100000110011001110111000110010110111000111001110110000000100110010000000000100110001000100110010100100100100101111001100110001001110010100010001110000000010001100100000010000100011011010101000000011000010010100011000110000100111011000101001010100101110011000101000000100001001111101011100100100100110001100001001101000000001110100100100100000000100010001100010000000010011100101100000001001100010011100001000001011100010000110000000100100000000111100100010010101010000100000110101010010101001100110100101100000001000000010010010001011101000011010101001111000110010101101000001001000011011001100000100000011101001000001011001
Pair \(Z_2\) Length of longest common subsequence
7ENR_1,5PUU_1 206 4
7ENR_1,1DGG_1 121 5
5PUU_1,1DGG_1 203 4

Newick tree

 
[
	5PUU_1:11.78,
	[
		7ENR_1:60.5,1DGG_1:60.5
	]:52.28
]

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 \((0.85)(0.8)(\frac{1209 }{\log_{20} 1209}-\frac{156}{\log_{20}156})=284.\)
Status Protein1 Protein2 d d1/2
Query variables 7ENR_1 5PUU_1 349 200
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]