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Parikh vectors
1MYF_1 3VSL_1 4KSC_1 Letter Amino acid
5 36 60 Q Glutamine
6 41 89 S Serine
17 33 117 A Alanine
7 50 55 D Aspartic acid
11 54 102 G Glycine
3 29 36 Y Tyrosine
2 19 34 M Methionine
6 14 67 F Phenylalanine
2 3 11 W Tryptophan
0 0 9 C Cysteine
14 31 74 E Glutamic acid
12 8 24 H Histidine
18 54 117 L Leucine
19 72 85 K Lycine
8 37 93 V Valine
4 22 58 R Arginine
1 36 50 N Asparagine
9 37 104 I Isoleucine
4 27 31 P Proline
5 43 68 T Threonine 

1MYF_1|Chain A|MYOGLOBIN|Physeter catodon (9755)
>3VSL_1|Chains A, B|Penicillin-binding protein 3|Staphylococcus aureus (196620)
>4KSC_1|Chain A|Multidrug resistance protein 1A|Mus musculus (10090)
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 \(w=f(c)\)
1MYF , Knot 73 153 0.80 38 111 148 
VLSEGEWQLVLHVWAKVEADVAGHGQDILIRLFKSHPETLEKFDRFKHLKTEAEMKASEDLKKHGVTVLTALGAILKKKGHHEAELKPLAQSHATKHKIPIKYLEFISEAIIHVLHSRHPGDFGADAQGAMNKALELFRKDIAAKYKELGYQG
3VSL , Knot 249 646 0.83 38 257 590 
AQGSHYKQIIKNDENITVNESVPRGRILDRNGKVLVDNASKMAITYTRGRKTTQSEMLDTAEKLSKLIKMDTKKITERDKKDFWIQLHPKKAKAMMTKEQAMLADGSIKQDQYDKQLLSKIRKSQLDELSSKDLQVLAIFREMNAGTVLDPQMIKNEDVSEKEYAAVSQQLSKLPGVNTSMDWDRKYPYGDTLRGIFGDVSTPAEGIPKELTEHYLSKGYSRNDRVGKSYLEYQYEDVLRGKKKEMKYTTDKSGKVTSSEVLNPGARGQDLKLTIDIDLQKEVEALLDKQIKKLRSQGAKDMDNAMMVVQNPKNGDILALAGKQINKSGKMTDYDIGTFTSQFAVGSSVKGGTLLAGYQNKAIKVGETMVDEPLHFQGGLTKRSYFNKNGHVSINDKQALMHSSNVYMFKTALKLAGDPYYSGMALPSDISSPAQKLRRGLNQVGLGVKTGIDLPNETRGQIEPLTNNPGNYLDLSIGQYDTYTPLQLSQYVSTIANDGYRIQPHIGLTIHESTNKDEVGPLKKKINGTVLNKVNNTEKEIKQIQEGFKMAFNDKDGTGYVSFKDTVVPTAGKTGTAEVFQNGEPRVNSTYIGYAPIDDPKLAFSIVYTNQPVPPPWLTGGDLGRDVINYYFKQLGKDDKNKDKDK
4KSC , Knot 436 1284 0.81 40 317 1041 
MELEEDLKGRADKNFSKMGKKSKKEKKEKKPAVSVLTMFRYAGWLDRLYMLVGTLAAIIHGVALPLMMLIFGDMTDSFASVGNVSKNSTNMSEADKRAMFAKLEEEMTTYAYYYTGIGAGVLIVAYIQVSFWCLAAGRQIHKIRQKFFHAIMNQEIGWFDVHDVGELNTRLTDDVSKINEGIGDKIGMFFQAMATFFGGFIIGFTRGWKLTLVILAISPVLGLSAGIWAKILSSFTDKELHAYAKAGAVAEEVLAAIRTVIAFGGQKKELERYNNNLEEAKRLGIKKAITANISMGAAFLLIYASYALAFWYGTSLVISKEYSIGQVLTVFFSVLIGAFSVGQASPNIEAFANARGAAYEVFKIIDNKPSIDSFSKSGHKPDNIQGNLEFKNIHFSYPSRKEVQILKGLNLKVKSGQTVALVGNSGCGKSTTVQLMQRLYDPLDGMVSIDGQDIRTINVRYLREIIGVVSQEPVLFATTIAENIRYGREDVTMDEIEKAVKEANAYDFIMKLPHQFDTLVGERGAQLSGGQKQRIAIARALVRNPKILLLDEATSALDTESEAVVQAALDKAREGRTTIVIAHRLSTVRNADVIAGFDGGVIVEQGNHDELMREKGIYFKLVMTQTAGNEIELGNEACKSKDEIDNLDMSSKDSGSSLIRRRSTRKSICGPHDQDRKLSTKEALDEDVPPASFWRILKLNSTEWPYFVVGIFCAIINGGLQPAFSVIFSKVVGVFTNGGPPETQRQNSNLFSLLFLILGIISFITFFLQGFTFGKAGEILTKRLRYMVFKSMLRQDVSWFDDPKNTTGALTTRLANDAAQVKGATGSRLAVIFQNIANLGTGIIISLIYGWQLTLLLLAIVPIIAIAGVVEMKMLSGQALKDKKELEGSGKIATEAIENFRTVVSLTREQKFETMYAQSLQIPYRNAMKKAHVFGITFSFTQAMMYFSYAACFRFGAYLVTQQLMTFENVLLVFSAIVFGAMAVGQVSSFAPDYAKATVSASHIIRIIEKTPEIDSYSTQGLKPNMLEGNVQFSGVVFNYPTRPSIPVLQGLSLEVKKGQTLALVGSSGCGKSTVVQLLERFYDPMAGSVFLDGKEIKQLNVQWLRAQLGIVSQEPILFDCSIAENIAYGDNSRVVSYEEIVRAAKEANIHQFIDSLPDKYNTRVGDKGTQLSGGQKQRIAIARALVRQPHILLLDEATSALDTESEKVVQEALDKAREGRTCIVIAHRLSTIQNADLIVVIQNGKVKEHGTHQQLLAQKGIYFSMVSVQAGAKRSLEHHHHHH

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 \(c\).

\(|P_{f(1MYF_1)}(2) \setminus P_{f(3VSL_1)}(2)|=23\), \(|P_{f(3VSL_1)}(2) \setminus P_{f(1MYF_1)}(2)|=169\). 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 1:110010101110111010101110100111011000100100100100100010101000100011011011111100010001010111000100001110010110011101100001101110101110011011000111000011001
Pair \(Z_2\) Length of longest common subsequence
1MYF_1,3VSL_1 192 4
1MYF_1,4KSC_1 228 4
3VSL_1,4KSC_1 108 5

Newick tree

 
[
	1MYF_1:11.62,
	[
		3VSL_1:54,4KSC_1:54
	]:63.62
]

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 \((0.85)(0.8)(\frac{799 }{\log_{20} 799}-\frac{153}{\log_{20}153})=181.\)
Status Protein1 Protein2 d d1/2
Query variables 1MYF_1 3VSL_1 228 139.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]