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Parikh vectors
1VQQ_1 3ZKM_1 6IVO_1 Letter Amino acid
42 21 17 E Glutamic acid
43 36 14 G Glycine
9 1 9 H Histidine
41 36 19 S Serine
31 26 18 T Threonine
33 24 15 V Valine
12 12 20 R Arginine
55 17 12 N Asparagine
27 30 24 A Alanine
44 32 49 L Leucine
13 23 15 F Phenylalanine
0 6 3 C Cysteine
27 13 8 Q Glutamine
55 27 22 I Isoleucine
86 13 5 K Lycine
17 7 7 M Methionine
16 20 13 P Proline
7 7 6 W Tryptophan
35 18 9 Y Tyrosine
53 17 12 D Aspartic acid 

1VQQ_1|Chains A, B|penicillin-binding protein mecA, low-affinity|Staphylococcus aureus (1280)
>3ZKM_1|Chains A, B|BETA-SECRETASE 2|HOMO SAPIENS (9606)
>6IVO_1|Chains A, B, C, D, E|bestrophin|Klebsiella pneumoniae (573)
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 \(w=f(c)\)
1VQQ , Knot 248 646 0.82 38 246 582 
MASKDKEINNTIDAIEDKNFKQVYKDSSYISKSDNGEVEMTERPIKIYNSLGVKDINIQDRKIKKVSKNKKRVDAQYKIKTNYGNIDRNVQFNFVKEDGMWKLDWDHSVIIPGMQKDQSIHIENLKSERGKILDRNNVELANTGTAYEIGIVPKNVSKKDYKAIAKELSISEDYIKQQMDQNWVQDDTFVPLKTVKKMDEYLSDFAKKFHLTTNETESRNYPLEKATSHLLGYVGPINSEELKQKEYKGYKDDAVIGKKGLEKLYDKKLQHEDGYRVTIVDDNSNTIAHTLIEKKKKDGKDIQLTIDAKVQKSIYNNMKNDYGSGTAIHPQTGELLALVSTPSYDVYPFMYGMSNEEYNKLTEDKKEPLLNKFQITTSPGSTQKILTAMIGLNNKTLDDKTSYKIDGKGWQKDKSWGGYNVTRYEVVNGNIDLKQAIESSDNIFFARVALELGSKKFEKGMKKLGVGEDIPSDYPFYNAQISNKNLDNEILLADSGYGQGEILINPVQILSIYSALENNGNINAPHLLKDTKNKVWKKNIISKENINLLTDGMQQVVNKTHKEDIYRSYANLIGKSGTAELKMKQGETGRQIGWFISYDKDNPNMMMAINVKDVQDKGMASYNAKISGKVYDELYENGNKKYDIDE
3ZKM , Knot 164 386 0.84 40 220 370 
ANFLAMVDNLQGDSGRGYYLEMLIGTPPQKLQILVDTGSSNFAVAGTPHSYIDTYFDTERSSTYRSKGFDVTVKYTQGSWTGFVGEDLVTIPKGFNTSFLVNIATIFESENFFLPGIKWNGILGLAYATLAKPSSSLETFFDSLVTQANIPNVFSMQMCGAGLPVAGSGTNGGSLVLGGIEPSLYKGDIWYTPIKEEWYYQIEILKLEIGGQSLNLDCREYNADKAIVDSGTTLLRLPQKVFDAVVEAVARASLIPEFSDGFWTGSQLACWTNSETPWSYFPKISIYLRDENSSRSFRITILPQLYIQPMMGAGLNYECYRFGISPSTNALVIGATVMEGFYVIFDRAQKRVGFAASPCAEIAGAAVSEISGPFSTEDVASNCVPA
6IVO , Knot 130 297 0.83 40 180 282 
SNAMIIRPEQHWFLRLFDWHGSVLSKIIFRLLLNVLMSIIAIISYQWYEQLGIHLTVAPFSLLGIAIAIFLGFRNSASYSRFVEARNLWGTVLIAERTLVRQLRNILPAEHDAHRRIVSYLVAFSWSLKHQLRKTDPTADLRRLLPEERVTEILASSMPTNRILLLAGNEIGQLREAGKLSDITYGLMDNKLDELAHVLGGCERLATTPVAFAYTLILQRTVYLFCTLLPFALVGDLHYMTPFVSVFISYTFLSWDSLAEELEDPFGTAANDLPLNAMCNTIERNLLDMTGQHPLPE

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 \(c\).

\(|P_{f(1VQQ_1)}(2) \setminus P_{f(3ZKM_1)}(2)|=90\), \(|P_{f(3ZKM_1)}(2) \setminus P_{f(1VQQ_1)}(2)|=64\). 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 1:1100000100010110000100100000010000010101000110100011100101000010010000001010001000010100010101100011101010001111110000010100100001011000010110010100111110010000001110010100001000100011000011110010010001001100101000000000011001000111011110000100000010000111100110010000100001001011000000110011000000100101010101000100010000101011010010111110010001011101100000001000000111001010001100001101111100001000000010101100000111001000011010101001100000111101110110001001100111100110001100101000010001111001010101110110110100110001010110110000001100011000010110011001100000001000010111001010101001001001111100000010111110100100011100010101010001000100000100
Pair \(Z_2\) Length of longest common subsequence
1VQQ_1,3ZKM_1 154 4
1VQQ_1,6IVO_1 186 4
3ZKM_1,6IVO_1 180 4

Newick tree

 
[
	6IVO_1:95.86,
	[
		1VQQ_1:77,3ZKM_1:77
	]:18.86
]

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 \((0.85)(0.8)(\frac{1032 }{\log_{20} 1032}-\frac{386}{\log_{20}386})=170.\)
Status Protein1 Protein2 d d1/2
Query variables 1VQQ_1 3ZKM_1 215 171.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]