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Parikh vectors
8CAD_1 8AVR_1 5VMB_1 Letter Amino acid
24 1 16 Q Glutamine
32 4 35 G Glycine
9 5 3 W Tryptophan
34 1 16 P Proline
40 3 16 Y Tyrosine
25 5 58 A Alanine
31 0 17 H Histidine
45 2 23 I Isoleucine
31 10 25 K Lycine
29 1 16 F Phenylalanine
36 2 22 S Serine
30 0 24 D Aspartic acid
46 2 28 E Glutamic acid
76 7 26 L Leucine
14 1 10 M Methionine
74 4 37 V Valine
31 0 15 R Arginine
49 2 16 N Asparagine
4 0 4 C Cysteine
46 1 18 T Threonine 

8CAD_1|Chains A, B, C, D, E, F|acidic juvenile hormone-suppressible protein 1|Galleria mellonella (7137)
>8AVR_1|Chains A, C|Bacteriocin aureocin A53|Staphylococcus aureus (1280)
>5VMB_1|Chain A|Serine hydroxymethyltransferase|Acinetobacter baumannii (470)
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 \(w=f(c)\)
8CAD , Knot 279 706 0.86 40 277 653 
MGRLVLCVLALLVGGGISDPVKKLQRTVDQTVLDRQYKLLTLFFHPHEPIHIKEQQEIAASWDLEKNIGLYENATAVHLTIQMLHNNYQVPRGVPFTVLESVHRFEISVYYSLLYSAKTYDTFYKTAVFLRQHVNENLFVNVLSVVILHRSDTQDIRIPPIYDVFPSYFHNGEIMTTAQRITTHGQRMLEHYPSTYVWENNVVIRHNETAWPYYCNTESMPVSYFTHDVTLNALYYNIKLAYPIWLRSDACAIKEKRGELFFFWNKQLLARYYMERLSVGLGEIPELGLNEVEEGYVSGLLYHNGIPYPVRPNHLVLNHQTWHAEAIEEIEVYENRIRDMIDQGFYITNTGEHVSINSPDSIDVLGRLIEANVDSPNVQYYKDFISIWKKVLGNSLVHESVAFNGIPLVVPSVLEQYQTALRDPAYYMIMKRVLKLFNLWHEHLPHYTTKELSVPSVKIEKVEVDKLLTYFEYTNFNVTNHLHLNEIECNNVINTKSVLVQRTRLNHKVFTVRVNVKSGVAKHVTVRFFLAPKYDSVGNEIPLNVNTQNFLLIDIFNYELKEGDNLITRVSSDNLLVTDEIDSASVLFNKVDSALQGHGQYMLNMKQNILKTPRHLLLPKGRVGGMPFVLMVYISEYHAPNDVHRGTVETSTIDNTIRLTSDTLGFPVDRPLFPWMLTGVENIFLQDVQIYHKPTTEVTGVPVYVE
8AVR , Knot 27 51 0.69 32 40 47 
MSWLNFLKYIAKYGKKAVSAAWKYKGKVLEWLNVGPTLEWVWQKLKKIAGL
5VMB , Knot 170 425 0.80 40 228 394 
MAHHHHHHMFANISISEFDPELAQAIASEDERQEAHIELIASENYCSPAVMEAQGSKLTNKYAEGYPGKRYYGGCEFVDVIEQMAIDRAKELFGADYANVQPHAGSQANSAVYLALLNPGDTVLGMSLAHGGHLTHGAKVSFSGKTYNAVQYGLNAETGEIDYEEVERLALEHKPRMIVAGFSAYSRVVDWQRFRDIADKVGAYLFVDMAHVAGLVAAGVYPNPVQIADVTTTTTHKTLRGPRSGLILAKANEEIEKKLQSAVFPGNQGGPLMHAIAAKAICFKEAMSDDFKAYQQQVVKNAQAMAEVFIARGYDVVSGGTDNHLFLLSLIKQDVTGKDADAWLGAAHITVNKNSVPNDPRSPFVTSGIRIGTPAVTTRGFGEAEVRELAGWIADVIDSKGDEKVIADVKAKVEAVCAKFPVYAK

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 \(c\).

\(|P_{f(8CAD_1)}(2) \setminus P_{f(8AVR_1)}(2)|=247\), \(|P_{f(8AVR_1)}(2) \setminus P_{f(8CAD_1)}(2)|=10\). 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 1:1101110111111111100110010001000110000011011101001101000001110101000111000101101010110000011011110110010010101000110010000010001111000100011101101111000000010111100111001001011001001000100110001000110001110000011100000001110010001010110001011011110001011000010111110001110001001011110110111001001010111000111011010011100001010110010100001001100110100010010100100101110110101001010000011011001110011000111011111110110000011001100111001101101100011000000101101010010100110010000101000101001000011000011100001000110101010011100101011111000011001110100001111011000100100110010000111000100101110010011010100110100011001001111010111111111101000011001001010000100010100001111100111111101100111001010001000101111010
Pair \(Z_2\) Length of longest common subsequence
8CAD_1,8AVR_1 257 3
8CAD_1,5VMB_1 155 4
8AVR_1,5VMB_1 212 3

Newick tree

 
[
	8AVR_1:12.43,
	[
		8CAD_1:77.5,5VMB_1:77.5
	]:50.93
]

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 \((0.85)(0.8)(\frac{757 }{\log_{20} 757}-\frac{51}{\log_{20}51})=206.\)
Status Protein1 Protein2 d d1/2
Query variables 8CAD_1 8AVR_1 261 139
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]