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Parikh vectors
4WRU_1 2MWR_1 8QEQ_1 Letter Amino acid
25 8 139 L Leucine
2 1 105 K Lycine
8 5 149 T Threonine
8 0 81 N Asparagine
9 2 92 D Aspartic acid
16 2 226 S Serine
4 1 85 Y Tyrosine
17 2 110 V Valine
10 0 114 E Glutamic acid
7 4 98 I Isoleucine
9 2 57 F Phenylalanine
23 1 198 P Proline
6 2 14 W Tryptophan
20 1 98 R Arginine
2 0 24 C Cysteine
19 7 128 G Glycine
10 1 42 H Histidine
6 0 56 M Methionine
33 18 93 A Alanine
4 1 61 Q Glutamine 

4WRU_1|Chain A|Uracil-DNA glycosylase|Mycobacterium tuberculosis (83332)
>2MWR_1|Chain A|Acidocin B|Lactobacillus acidophilus (1579)
>8QEQ_1|Chain A|DNA-directed RNA polymerase subunit|Sus scrofa domesticus (9825)
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 \(w=f(c)\)
4WRU , Knot 104 238 0.79 40 151 225 
MHHHHHHGMASMTARPLSELVERGWAAALEPVADQVAHMGQFLRAEIAAGRRYLPAGSNVLRAFTFPFDNVRVLIVGQDPYPTPGHAVGLSFSVAPDVRPWPRSLANIFDEYTADLGYPLPSNGDLTPWAQRGVLLLNRVLTVRPSNPASHRGKGWEAVTECAIRALAARAAPLVAILWGRDASTLKPMLAAGNCVAIESPHPSPLSASRGFFGSRPFSRANELLVGMGAEPIDWRLP
2MWR , Knot 32 58 0.74 32 43 55 
IYWIADQFGIHLATGTARKLLDAVASGASLGTAFAAILGVTLPAWALAAAGALGATAA
8QEQ , Knot 597 1970 0.76 40 358 1401 
MHGGGPPSGDSACPLRTIKRVQFGVLSPDELKRMSVTEGGIKYPETTEGGRPKLGGLMDPRQGVIERTGRCQTCAGNMTECPGHFGHIELAKPVFHVGFLVKTMKVLRCVCFFCSKLLVDSNNPKIKDILAKSKGQPKKRLTHVYDLCKGKNICEGGEEMDNKFGVEQPEGDEDLTKEKGHGGCGRYQPRIRRSGLELYAEWKHVNEDSQEKKILLSPERVHEIFKRISDEECFVLGMEPRYARPEWMIVTVLPVPPLSVRPAVVMQGSARNQDDLTHKLADIVKINNQLRRNEQNGAAAHVIAEDVKLLQFHVATMVDNELPGLPRAMQKSGRPLKSLKQRLKGKEGRVRGNLMGKRVDFSARTVITPDPNLSIDQVGVPRSIAANMTFAEIVTPFNIDRLQELVRRGNSQYPGAKYIIRDNGDRIDLRFHPKPSDLHLQTGYKVERHMCDGDIVIFNRQPTLHKMSMMGHRVRILPWSTFRLNLSVTTPYNADFDGDEMNLHLPQSLETRAEIQELAMVPRMIVTPQSNRPVMGIVQDTLTAVRKFTKRDVFLERGEVMNLLMFLSTWDGKVPQPAILKPRPLWTGKQIFSLIIPGHINCIRTHSTHPDDEDSGPYKHISPGDTKVVVENGELIMGILCKKSLGTSAGSLVHISYLEMGHDITRLFYSNIQTVINNWLLIEGHTIGIGDSIADSKTYQDIQNTIKKAKQDVIEVIEKAHNNELEPTPGNTLRQTFENQVNRILNDARDKTGSSAQKSLSEYNNFKSMVVSGAKGSKINISQVIAVVGQQNVEGKRIPFGFKHRTLPHFIKDDYGPESRGFVENSYLAGLTPTEFFFHAMGGREGLIDTAVKTAETGYIQRRLIKSMESVMVKYDATVRNSINQVVQLRYGEDGLAGESVEFQNLATLKPSNKAFEKKFRFDYTNERALRRTLQEDLVKDVLSNAHIQNELEREFERMREDREVLRVIFPTGDSKVVLPCNLLRMIWNAQKIFHINPRLPSDLHPIKVVEGVKELSKKLVIVNGDDPLSRQAQENATLLFNIHLRSTLCSRRMAEEFRLSGEAFDWLLGEIESKFNQAIAHPGEMVGALAAQSLGEPATQMTLNTFHYAGVSAKNVTLGVPRLKELINISKKPKTPSLTVFLLGQSARDAERAKDILCRLEHTTLRKVTANTAIYYDPNPQSTVVAEDQEWVNVYYEMPDFDVARISPWLLRVELDRKHMTDRKLTMEQIAEKINAGFGDDLNCIFNDDNAEKLVLRIRIMNSDENKMQEEEEVVDKMDDDVFLRCIESNMLTDMTLQGIEQISKVYMHLPQTDNKKKIIITEDGEFKALQEWILETDGVSLMRVLSEKDVDPVRTTSNDIVEIFTVLGIEAVRKALERELYHVISFDGSYVNYRHLALLCDTMTCRGHLMAITRHGVNRQDTGPLMKCSFEETVDVLMEAAAHGESDPMKGVSENIMLGQLAPAGTGCFDLLLDAEKCKYGMEIPTNIPGLGAAGPTGMFFGSAPSPMGGISPAMTPWNQGATPAYGAWSPSVGSGMTPGAAGFSPSAASDASGFSPGYSPAWSPTPGSPGSPGPSSPYIPSPGGAMSPSYSPTSPAYEPRSPGGYTPQSPSYSPTSPSYSPTSPSYSPTSPNYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPSYSPTSPNYSPTSPNYTPTSPSYSPTSPSYSPTSPNYTPTSPNYSPTSPSYSPTSPSYSPTSPSYSPSSPRYTPQSPTYTPSSPSYSPSSPSYSPTSPKYTPTSPSYSPSSPEYTPTSPKYSPTSPKYSPTSPKYSPTSPTYSPTTPKYSPTSPTYSPTSPVYTPTSPKYSPTSPTYSPTSPKYSPTSPTYSPTSPKGSTYSPTSPGYSPTSPTYSLTSPAISPDDSDEEN

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 \(c\).

\(|P_{f(4WRU_1)}(2) \setminus P_{f(2MWR_1)}(2)|=123\), \(|P_{f(2MWR_1)}(2) \setminus P_{f(4WRU_1)}(2)|=15\). 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 1:1000000111010101100110011111101110011011011010111100011110011011011100101111100101011011110101110101110011011000010110111001010111001111100110101001100010110110001101111011111111110010010111111001110010101101001111001100100111111101101011
Pair \(Z_2\) Length of longest common subsequence
4WRU_1,2MWR_1 138 4
4WRU_1,8QEQ_1 223 4
2MWR_1,8QEQ_1 319 4

Newick tree

 
[
	8QEQ_1:15.82,
	[
		4WRU_1:69,2MWR_1:69
	]:84.82
]

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 \((0.85)(0.8)(\frac{296 }{\log_{20} 296}-\frac{58}{\log_{20}58})=76.8\)
Status Protein1 Protein2 d d1/2
Query variables 4WRU_1 2MWR_1 92 56
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]