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Parikh vectors
4XLQ_1 9LDT_1 1UYS_1 Letter Amino acid
19 12 39 T Threonine
26 36 52 V Valine
22 20 52 A Alanine
17 15 43 D Aspartic acid
43 36 60 L Leucine
4 8 17 M Methionine
18 12 34 P Proline
34 20 60 E Glutamic acid
22 26 57 G Glycine
6 12 9 H Histidine
19 24 51 I Isoleucine
16 23 39 S Serine
21 11 49 R Arginine
0 5 2 C Cysteine
5 8 29 Q Glutamine
5 7 25 Y Tyrosine
9 17 33 N Asparagine
17 26 40 K Lycine
10 7 32 F Phenylalanine
1 6 14 W Tryptophan 

4XLQ_1|Chains A, B, G, H|DNA-directed RNA polymerase subunit alpha|Thermus aquaticus (271)
>9LDT_1|Chains A, B|LACTATE DEHYDROGENASE|Sus scrofa (9823)
>1UYS_1|Chains A, B, C|ACETYL-COA CARBOXYLASE|SACCHAROMYCES CEREVISIAE (4932)
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 \(w=f(c)\)
4XLQ , Knot 133 314 0.81 38 169 293 
MLESKLKAPVFTATTQGDHYGEFVLEPLERGFGVTLGNPLRRILLSSIPGTAVTSVYIEDVLHEFSTIPGVKEDVVEIILNLKELVVRFLDPKMASTTLILRAEGPKEVRAGDFTPSADVEIMNPDLHIATLEEGGKLYMEVRVDRGVGYVPAERHGIKDRINAIPVDAIFSPVRRVAFQVEDTRLGQRTDLDKLTLRIWTDGSVTPLEALNQAVAILKEHLNYFANPEASLLPTPEVSKGEKRESAEEDLDLPLEELGLSTRVLHSLKEEGIESVRALLALNLKDLRNIPGIGERSLEEIRQALAKKGFTLKE
9LDT , Knot 143 332 0.83 42 198 319 
XATLKDQLIHNLLKEEHVPHNKITVVGVGAVGMACAISILMKELADEIALVDVMEDKLKGEMMDLQHGSLFLRTPKIVSGKDYNVTANSRLVVITAGARQQEGESRLNLVQRNVNIFKFIIPNIVKYSPNCKLLVVSNPVDILTYVAWKISGFPKNRVIGSGCNLDSARFRYLMGERLGVHPLSCHGWILGEHGDSSVPVWSGVNVAGVSLKNLHPELGTDADKEHWKAVHKEVVDSAYEVIKLKGYTSWAIGLSVADLAESIMKNLRRVHPISTMIKGLYGIKENVFLSVPCILGQNGISDVVKVTLTPEEEAHLKKSADTLWGIQKELQF
1UYS , Knot 289 737 0.86 40 287 676 
PIATPYPVKEWLQPKRYKAHLMGTTYVYDFPELFRQASSSQWKNFSADVKLTDDFFISNELIEDENGELTEVEREPGANAIGMVAFKITVKTPEYPRGRQFVVVANDITFKIGSFGPQEDEFFNKVTEYARKRGIPRIYLAANSGARIGMAEEIVPLFQVAWNDAANPDKGFQYLYLTSEGMETLKKFDKENSVLTERTVINGEERFVIKTIIGSEDGLGVECLRGSGLIAGATSRAYHDIFTITLVTCRSVGIGAYLVRLGQRAIQVEGQPIILTGAPAINKMLGREVYTSNLQLGGTQIMYNNGVSHLTAVDDLAGVEKIVEWMSYVPAKRNMPVPILETKDTWDRPVDFTPTNDETYDVRWMIEGRETESGFEYGLFDKGSFFETLSGWAKGVVVGRARLGGIPLGVIGVETRTVENLIPADPANPNSAETLIQEPGQVWHPNSAFKTAQAINDFNNGEQLPMMILANWRGFSGGQRDMFNEVLKYGSFIVDALVDYKQPIIIYIPPTGELRGGSWVVVDPTINADQMEMYADVNARAGVLEPQGMVGIKFRREKLLDTMNRLDDKYRELRSQLSNKSLAPEVHQQISKQLADRERELLPIYGQISLQFADLHDRSSRMVAKGVISKELEWTEARRFFFWRLRRRLNEEYLIKRLSHQVGEASRLEKIARIRSWYPASVDHEDDRQVATWIEENYKTLDDKLKGLKLESFAQDLAKKIRSDHDNAIDGLSEVIK

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 \(c\).

\(|P_{f(4XLQ_1)}(2) \setminus P_{f(9LDT_1)}(2)|=65\), \(|P_{f(9LDT_1)}(2) \setminus P_{f(4XLQ_1)}(2)|=94\). 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 1:11000101111010001000101110110011110110110011100111011001010011001001111000110111010011101101011000111010110010110101010101101010110100110101010100111011100011000101111011101100111010000110000100101011001010110110011111000100110101011101010010000010001011100111000110010001100101111101001001111100010010011100110100
Pair \(Z_2\) Length of longest common subsequence
4XLQ_1,9LDT_1 159 4
4XLQ_1,1UYS_1 162 5
9LDT_1,1UYS_1 175 5

Newick tree

 
[
	1UYS_1:85.85,
	[
		4XLQ_1:79.5,9LDT_1:79.5
	]:6.35
]

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 \((0.85)(0.8)(\frac{646 }{\log_{20} 646}-\frac{314}{\log_{20}314})=92.1\)
Status Protein1 Protein2 d d1/2
Query variables 4XLQ_1 9LDT_1 114 110
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]