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Parikh vectors
6MGC_1 8GWG_1 7ZYF_1 Letter Amino acid
15 33 48 I Isoleucine
16 75 39 D Aspartic acid
4 29 4 C Cysteine
18 28 41 Q Glutamine
40 83 102 L Leucine
21 53 75 S Serine
17 61 56 T Threonine
13 58 38 Y Tyrosine
39 64 54 A Alanine
21 43 24 R Arginine
23 45 37 G Glycine
14 27 20 H Histidine
15 51 58 K Lycine
6 25 23 M Methionine
11 57 50 F Phenylalanine
28 74 52 V Valine
10 56 47 N Asparagine
19 31 58 E Glutamic acid
23 30 24 P Proline
8 9 21 W Tryptophan 

6MGC_1|Chain A|Capsule polysaccharide export protein KpsC|Escherichia coli O1:K1 / APEC (405955)
>8GWG_1|Chain A|RNA-directed RNA polymerase|Severe acute respiratory syndrome coronavirus 2 (2697049)
>7ZYF_1|Chains A, B|Leucyl-cystinyl aminopeptidase, pregnancy serum form|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 \(w=f(c)\)
6MGC , Knot 154 361 0.83 40 213 334 
MGIGIYSPGIWRIPHLEKFLAQPCQKLSLLRPVPQEVNAIAVWGHRPSAAKPVAIAKAAGKPVIRLEDGFVRSLDLGVNGEPPLSLVVDDCGIYYDASKPSALEKLVQDKAGNTALISQAREAMHTIVTGDMSKYNLAPAFVADESERTNIVLVVDQTFNDMSVTYGNAGPHEFAAMLEAAMAENPQAEIWVKVHPDVLEGKKTGYFADLRATQRVRLIAENVSPQSLLRHVSRVYVVTSQYGFEALLAGKPVTCFGQPWYASWGLTDDRHPQSALLSARRGSATLEELFAAAYLRYCRYIDPQTGEVSDLFTVLQWLQLQRRHLQQRNGYLWAPGLTLWKSAILKPFLRTPTLEHHHHHH
8GWG , Knot 353 932 0.86 40 325 860 
SADAQSFLNRVCGVSAARLTPCGTGTSTDVVYRAFDIYNDKVAGFAKFLKTNCCRFQEKDEDDNLIDSYFVVKRHTFSNYQHEETIYNLLKDCPAVAKHDFFKFRIDGDMVPHISRQRLTKYTMADLVYALRHFDEGNCDTLKEILVTYNCCDDDYFNKKDWYDFVENPDILRVYANLGERVRQALLKTVQFCDAMRNAGIVGVLTLDNQDLNGNWYDFGDFIQTTPGSGVPVVDSYYSLLMPILTLTRALTAESHVDTDLTKPYIKWDLLKYDFTEERLKLFDRYFKYWDQTYHPNCVNCLDDRCILHCANFNVLFSTVFPPTSFGPLVRKIFVDGVPFVVSTGYHFRELGVVHNQDVNLHSSRLSFKELLVYAADPAMHAASGNLLLDKRTTCFSVAALTNNVAFQTVKPGNFNKDFYDFAVSKGFFKEGSSVELKHFFFAQDGNAAISDYDYYRYNLPTMCDIRQLLFVVEVVDKYFDCYDGGCINANQVIVNNLDKSAGFPFNKWGKARLYYDSMSYEDQDALFAYTKRNVIPTITQMNLKYAISAKNRARTVAGVSICSTMTNRQFHQKLLKSIAATRGATVVIGTSKFYGGWHNMLKTVYSDVENPHLMGWDYPKCDRAMPNMLRIMASLVLARKHTTCCSLSHRFYRLANECAQVLSEMVMCGGSLYVKPGGTSSGDATTAYANSVFNICQAVTANVNALLSTDGNKIADKYVRNLQHRLYECLYRNRDVDTDFVNEFYAYLRKHFSMMILSDDAVVCFNSTYASQGLVASIKNFKSVLYYQNNVFMSEAKCWTETDLTKGPHEFCSQHTMLVKQGDDYVYLPYPDPSRILGAGCFVDDIVKTDGTLMIERFVSLAIDAYPLTKHPNQEYADVFHLYLQYIRKLHDELTGHMLDMYSVMLTNDNTSRYWEPEFYEAMYTPHTVLQ
7ZYF , Knot 330 871 0.85 40 304 779 
ATNGKLFPWAQIRLPTAVVPLRYELSLHPNLTSMTFRGSVTISVQALQVTWNIILHSTGHNISRVTFMSAVSSQEKQAEILEYAYHGQIAIVAPEALLAGHNYTLKIEYSANISSSYYGFYGFSYTDESNEKKYFAATQFEPLAARSAFPCFDEPAFKATFIIKIIRDEQYTALSNMPKKSSVVLDDGLVQDEFSESVKMSTYLVAFIVGEMKNLSQDVNGTLVSIYAVPEKIGQVHYALETTVKLLEFFQNYFEIQYPLKKLDLVAIPDFEAGAMENWGLLTFREETLLYDSNTSSMADRKLVTKIIAHELAHQWFGNLVTMKWWNDLWLNEGFATFMEYFSLEKIFKELSSYEDFLDARFKTMKKDSLNSSHPISSSVQSSEQIEEMFDSLSYFKGSSLLLMLKTYLSEDVFQHAVVLYLHNHSYASIQSDDLWDSFNEVTNQTLDVKRMMKTWTLQKGFPLVTVQKKGKELFIQQERFFLNMKPEIQPSDTSYLWHIPLSYVTEGRNYSKYQSVSLLDKKSGVINLTEEVLWVKVNINMNGYYIVHYADDDWEALIHQLKINPYVLSDKDRANLINNIFELAGLGKVPLKRAFDLINYLGNENHTAPITEALFQTDLIYNLLEKLGYMDLASRLVTRVFKLLQNQIQQQTWTDEGTPSMRELRSALLEFACTHNLGNCSTTAMKLFDDWMASNGTQSLPTDVMTTVFKVGAKTDKGWSFLLGKYISIGSEAEKNKILEALASSEDVRKLYWLMKSSLNGDNFRTQKLSFIIRTVGRHFPGHLLAWDFVKENWNKLVQKFPLGSYTIQNIVAGSTYLFSTKTHLSEVQAFFENQSEATFRLRCVQEALEVIQLNIQWMEKNLKSLTWWL

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 \(c\).

\(|P_{f(6MGC_1)}(2) \setminus P_{f(8GWG_1)}(2)|=25\), \(|P_{f(8GWG_1)}(2) \setminus P_{f(6MGC_1)}(2)|=137\). 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 1:1111100111101101001110100010110111001011111100101101111101110111010011100101110101110111000110001001011001100011001110010011001101010000111111100000001111100010010100101110011111011110010101110101011010001011010100010111001010011001001011000011011111011001101101011100000100111010010101001111101000001010010100110110110100001000010111111011001110111001010000000
Pair \(Z_2\) Length of longest common subsequence
6MGC_1,8GWG_1 162 5
6MGC_1,7ZYF_1 169 6
8GWG_1,7ZYF_1 99 5

Newick tree

 
[
	6MGC_1:91.19,
	[
		8GWG_1:49.5,7ZYF_1:49.5
	]:41.69
]

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 \((0.85)(0.8)(\frac{1293 }{\log_{20} 1293}-\frac{361}{\log_{20}361})=242.\)
Status Protein1 Protein2 d d1/2
Query variables 6MGC_1 8GWG_1 312 213
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]