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Parikh vectors
1LKP_1 7SQJ_1 6FLQ_1 Letter Amino acid
14 12 23 R Arginine
11 29 16 K Lycine
14 10 4 F Phenylalanine
6 48 21 D Aspartic acid
22 11 36 E Glutamic acid
2 7 5 M Methionine
2 0 1 W Tryptophan
27 79 23 A Alanine
5 30 10 Q Glutamine
15 44 20 G Glycine
8 0 8 H Histidine
5 7 16 P Proline
7 53 17 S Serine
7 65 19 T Threonine
6 31 30 V Valine
7 45 9 N Asparagine
4 0 4 C Cysteine
12 29 24 I Isoleucine
11 36 38 L Leucine
6 12 5 Y Tyrosine 

1LKP_1|Chain A|Rubrerythrin|Desulfovibrio vulgaris (881)
>7SQJ_1|Chains A, B|Flagellin|Escherichia coli O127:H6 (168807)
>6FLQ_1|Chains A, B|DNA-directed RNA polymerase subunit alpha|Escherichia coli (strain K12) (83333)
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 \(w=f(c)\)
1LKP , Knot 87 191 0.79 40 136 184 
MKSLKGSRTEKNILTAFAGESQARNRYNYFGGQAKKDGFVQISDIFAETADQEREHAKRLFKFLEGGDLEIVAAFPAGIIADTHANLIASAAGEHHEYTEMYPSFARIAREEGYEEIARVFASIAVAEEFHEKRFLDFARNIKEGRVFLREQATKWRCRNCGYVHEGTGAPELCPACAHPKAHFELLGINW
7SQJ , Knot 205 548 0.78 34 189 477 
MAQVINTNSLSLITQNNINKNQSALSSSIERLSSGLRINSAKDDAAGQAIANRFTSNIKGLTQAARNANDGISVAQTTEGALSEINNNLQRIRELTVQASTGTNSDSDLDSIQDEIKSRLDEIDRVSGQTQFNGVNVLAKDGSMKIQVGANDGQTITIDLKKIDSDTLGLNGFNVNGKGETANTAATLKDMSGFTAAAAPGGTVGVTQYTDKSAVASSVDILNAVAGADGNKVTTSADVGFGTPAAAVTYTYNKDTNSYSAASDDISSANLAAFLNPQARDTTKATVTIGGKDQDVNIDKSGNLTAADDGAVLYMDATGNLTKNNAGGDTQATLAKVATATGAKAATIQTDKGTFTSDGTAFDGASMSIDANTFANAVKNDTYTATVGAKTYSVTTGSAAADTAYMSNGVLSDTPPTYYAQADGSITTTEDAAAGKLVYKGSDGKLTTDTTSKAESTSDPLAALDDAISQIDKFRSSLGAVQNRLDSAVTNLNNTTTNLSEAQSRIQDADYATEVSNMSKAQIIQQAGNSVLAKANQVPQQVLSLLQG
6FLQ , Knot 146 329 0.85 40 180 315 
MQGSVTEFLKPRLVDIEQVSSTHAKVTLEPLERGFGHTLGNALRRILLSSMPGCAVTEVEIDGVLHEYSTKEGVQEDILEILLNLKGLAVRVQGKDEVILTLNKSGIGPVTAADITHDGDVEIVKPQHVICHLTDENASISMRIKVQRGRGYVPASTRIHSEEDERPIGRLLVDACYSPVERIAYNVEAARVEQRTDLDKLVIEMETNGTIDPEEAIRRAATILAEQLEAFVDLRDVRQPEVKEEKPEFDPILLRPVDDLELTVRSANCLKAEAIHYIGDLVQRTEVELLKTPNLGKKSLTEIKDVLASRGLSLGMRLENWPPASIADE

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 \(c\).

\(|P_{f(1LKP_1)}(2) \setminus P_{f(7SQJ_1)}(2)|=54\), \(|P_{f(7SQJ_1)}(2) \setminus P_{f(1LKP_1)}(2)|=107\). 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 1:10010100000011011110001000000111010001110100111001000000100110110110101111111111100010111011100000001010110110001000110111011110010000110110010010111000100100000101001011101011010101010111101
Pair \(Z_2\) Length of longest common subsequence
1LKP_1,7SQJ_1 161 3
1LKP_1,6FLQ_1 162 4
7SQJ_1,6FLQ_1 137 4

Newick tree

 
[
	1LKP_1:84.43,
	[
		7SQJ_1:68.5,6FLQ_1:68.5
	]:15.93
]

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 \((0.85)(0.8)(\frac{739 }{\log_{20} 739}-\frac{191}{\log_{20}191})=153.\)
Status Protein1 Protein2 d d1/2
Query variables 1LKP_1 7SQJ_1 185 126.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]