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Parikh vectors
4BML_1 7XUL_1 8UBU_1 Letter Amino acid
18 68 33 I Isoleucine
12 42 68 K Lycine
15 49 29 F Phenylalanine
32 51 48 S Serine
18 34 40 T Threonine
0 5 6 W Tryptophan
2 10 12 C Cysteine
14 22 62 E Glutamic acid
13 20 35 Y Tyrosine
4 11 14 H Histidine
5 10 17 M Methionine
40 53 46 A Alanine
19 26 37 N Asparagine
14 28 38 R Arginine
27 82 92 L Leucine
35 46 31 G Glycine
8 32 16 P Proline
22 68 47 V Valine
18 38 41 D Aspartic acid
16 23 52 Q Glutamine 

4BML_1|Chains A, B, C, D, E, F, G|MAJOR CAPSID PROTEIN|SYNECHOCOCCUS PHAGE SYN5 (438482)
>7XUL_1|Chains A, B|Chloride anion exchanger|Homo sapiens (9606)
>8UBU_1|Chains A, G[auth H]|Cullin-1|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 \(w=f(c)\)
4BML , Knot 142 332 0.82 38 179 310 
MTTLSNFSLPNQANGGARNADYDVRYATALKLFSGEVFTAFNNASIFKGLVRSYDLRGGKSKQFMFTGKLSAGYHTPGTPIVGDAGIKANEKTLVMDDLLVSSQFVYSLDEIFSQYSTRAEVSKQIGEALATHYDERIARVLAKASAEASPVTGEPGGFHVNIGAGNTNDAQAIVDGFFEAAAVLDERSAPQEGRVAVLSPRQYYSLISSVDTNILNREIGNSQGDMNSGKGLYSIAGIRILKSNNLAGLYGQDLSSAAVTGENNDYQVDASALAGLIFHREAAGCIQSVAPTIQTTSGDFNVQYQGDLIVGKLAMGCGSLRTSVAGSFQAA
7XUL , Knot 272 718 0.83 40 273 636 
QYIVARPVYSTNAFEENHKKTGRHHKTFLDHLKVCCSCSPQKAKRIVLSLFPIASWLPAYRLKEWLLSDIVSGISTGIVAVLQGLAFALLVDIPPVYGLYASFFPAIIYLFFGTSRHISVGPFPILSMMVGLAVSGAVSKAVPDRNATTLGLPNNSNNSSLLDDERVRVAAAASVTVLSGIIQLAFGILRIGFVVIYLSESLISGFTTAAAVHVLVSQLKFIFQLTVPSHTDPVSIFKVLYSVFSQIEKTNIADLVTALIVLLVVSIVKEINQRFKDKLPVPIPIEFIMTVIAAGVSYGCDFKNRFKVAVVGDMNPGFQPPITPDVETFQNTVGDCFGIAMVAFAVAFSVASVYSLKYDYPLDGNQELIALGLGNIVCGVFRGFAGSTALSRSAVQESTGGKTQIAGLIGAIIVLIVVLAIGFLLAPLQKSVLAALALGNLKGMLMQFAEIGRLWRKDKYDCLIWIMTFIFTIVLGLGLGLAASVAFQLLTIVFRTQFPKCSTLANIGRTNIYKNKKDYYDMYEPEGVKIFRCPSPIYFANIGFFRRKLIDAVGFSPLRILRKRNKALRKIRKLQKQGLLQVTPKGFICTVDTIKDSDEELDNNQIEVLDQPINTTDLPFHIDWNDDLPLNIEVPKISLHSLILDFSAVSFLDVSSVRGLKSILQEFIRIKVDVYIVGTDDDFIEKLNRYEFFDGEVKSSIFFLTIHDAVLHILMKKD
8UBU , Knot 294 764 0.85 40 287 694 
KQIGLDQIWDDLRAGIQQVYTRQSMAKSRYMELYTHVYNYCTSVHQSNQARGAGVPPSKSKKGQTPGGAQFVGLELYKRLKEFLKNYLTNLLKDGEDLMDESVLKFYTQQWEDYRFSSKVLNGICAYLNRHWVRRECDEGRKGIYEIYSLALVTWRDCLFRPLNKQVTNAVLKLIEKERNGETINTRLISGVVQSYVELGLNEDDAFAKGPTLTVYKESFESQFLADTERFYTRESTEFLQQNPVTEYMKKAEARLLEEQRRVQVYLHESTQDELARKCEQVLIEKHLEIFHTEFQNLLDADKNEDLGRMYNLVSRIQDGLGELKKLLETHIHNQGLAAIEKCGEAALNDPKMYVQTVLDVHKKYNALVMSAFNNDAGFVAALDKACGRFINNNAVTKMAQSSSKSPELLARYCDSLLKKSSKNPEEAELEDTLNQVMVVFKYIEDKDVFQKFYAKMLAKRLVHQNSASDDAEASMISKLKQACGFEYTSKLQRMFQDIGVSKDLNEQFKKHLTNSEPLDLDFSIQVLSSGSWPFQQSCTFALPSELERSYQRFTAFYASRHSGRKLTWLYQLSKGELVTNCFKNRYTLQASTFQMAILLQYNTEDAYTVQQLTDSTQIKMDILAQVLQILLKSKLLVLEDENANVDEVELKPDTLIKLYLGYKNKKLRVNINVPMKTEQKQEQETTHKNIEEDRKLLIQAAIVRIMKMRKVLKHQQLLGEVLTQLSSRFKPRVPVIKKCIDILIEKEYLERVDGEKDTYSYLA

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 \(c\).

\(|P_{f(4BML_1)}(2) \setminus P_{f(7XUL_1)}(2)|=31\), \(|P_{f(7XUL_1)}(2) \setminus P_{f(4BML_1)}(2)|=125\). 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 1:10010010110010111001000100101101101011011001011011100001011000011101010110001101111011101000011100111000110010011000000101000110111000000110111010101011010111101011110000101110111011111000011001011110100000110010001100011000101001011001111011000011110100100111010000001010111111100011101001110100001010100010111101111010100011101011
Pair \(Z_2\) Length of longest common subsequence
4BML_1,7XUL_1 156 4
4BML_1,8UBU_1 164 4
7XUL_1,8UBU_1 120 5

Newick tree

 
[
	4BML_1:85.66,
	[
		7XUL_1:60,8UBU_1:60
	]:25.66
]

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 \((0.85)(0.8)(\frac{1050 }{\log_{20} 1050}-\frac{332}{\log_{20}332})=190.\)
Status Protein1 Protein2 d d1/2
Query variables 4BML_1 7XUL_1 236 170
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]