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Parikh vectors
6NRZ_1 4TRN_1 2WKI_1 Letter Amino acid
32 20 16 I Isoleucine
63 27 18 L Leucine
27 13 11 T Threonine
20 6 6 Y Tyrosine
31 13 8 R Arginine
15 8 14 N Asparagine
34 15 6 D Aspartic acid
19 5 2 H Histidine
34 17 20 V Valine
35 36 18 A Alanine
8 1 3 C Cysteine
16 10 5 M Methionine
26 16 22 S Serine
24 7 12 F Phenylalanine
3 4 7 W Tryptophan
20 13 8 P Proline
18 10 10 Q Glutamine
47 11 23 E Glutamic acid
30 28 19 G Glycine
32 9 20 K Lycine 

6NRZ_1|Chains A, B|Lysine--tRNA ligase|Chlamydia trachomatis serovar L2b (strain UCH-1/proctitis) (471473)
>4TRN_1|Chain A|INHA|Mycobacterium tuberculosis (1773)
>2WKI_1|Chains A, B|BETA-LACTAMASE OXA-10|PSEUDOMONAS AERUGINOSA (287)
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 \(w=f(c)\)
6NRZ , Knot 210 534 0.82 40 251 490 
MAHHHHHHMSVEVEYLQHEDYLYRTSKLKEIRDLGINPYPYQYTDCLEVQEIRNQFVDNELGDSEAAFRKETPKVRFAGRLVLFRSMGKNAFGQILDNDAKIQVMFNRDFSAVAGLAADAGISPIKFIEKKLDLGDILGLEGYLFFTHSGELTVLVETVTLLCKSLISLPDKHAGLADKEIRYRKRWADLISSEDVRKTFLTRSRILKLIREYMDQQSFLEVETPILQTVYGGAEATPFVTTLQALHAEMFLRISLEIALKKLLVGGMSRVYEIGKVFRNEGIDRTHNPEFTMIEAYAAYWDYNDVMKCVENLVEYIVRALNNGETQVQYSHLKSGPQVVDFKAPWIRMTMKESISVYGGVDVDLHADHELRKILETQTSLPEKTYVHASRGELIALLFDELVCDKLIAPHHITDHPLETTPLCKTLRSGDETLVERFESFCLGKELCNAYSELNDPLQQRKLLEEQMRKKALNPDSEYHPIDEEFLEALCQGMPPAGGFGIGIDRLVMMLTDAASIRDVLFFPVMRRIEAKKD
4TRN , Knot 115 269 0.79 40 167 253 
MTGLLDGKRILVSGIITDSSIAFHIARVAQEQGAQLVLTGFDRLRLIQRITDRLPAKAPLLELDVQNEEHLASLAGRVTEAIGAGNKLDGVVHSIGFMPQTGMGINPFFDAPYADVSKGIHISAYSYASMAKALLPIMNPGGSIVGMDFDPSRAMPAYNWMTVAKSALESVNRFVAREAGKYGVRSNLVAAGPIRTLAMSAIVGGALGEEAGAQIQLLEEGWDQRAPIGWNMKDATPVAKTVCALLSDWLPATTGDIIYADGGAHTQLL
2WKI , Knot 115 248 0.85 40 170 240 
MGSITENTSWNKEFSAEAVNGVFVLCKSSSKSCATNDLARASKEYLPASTFCIPNAIIGLETGVIKNEHQVFKWDGKPRAMKQWERDLTLRGAIQVSAVPVFQQIAREVGEVRMQKYLKKFSYGNQNISGGIDKFWLEGQLRISAVNQVEFLESLYLNKLSASKENQLIVKEALVTEAAPEYLVHSKTGFSGVGTESNPGVAWWVGWVEKETEVYFFAFNMDIDNESKLPLRKSIPTKIMESEGIIGG

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 \(c\).

\(|P_{f(6NRZ_1)}(2) \setminus P_{f(4TRN_1)}(2)|=123\), \(|P_{f(4TRN_1)}(2) \setminus P_{f(6NRZ_1)}(2)|=39\). 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 1:110000001010100100000100000100100111010100000010100100011000110001110000101011101111001100111011000101011100010111111101110110110001011011110101110001010111001011000110110001111000100000110110000100011000011011000100001101001110010111010111001011010111010101110011111100100110110001100000101011010110100001100100110011011001000100001001101101011110101000101011101010100010011000001100001010010111111001100011110010001100011000100100011001001011001001000100110000110001000110100000110001101100111111111111001111100110100111111100101000
Pair \(Z_2\) Length of longest common subsequence
6NRZ_1,4TRN_1 162 4
6NRZ_1,2WKI_1 195 4
4TRN_1,2WKI_1 157 4

Newick tree

 
[
	6NRZ_1:93.04,
	[
		4TRN_1:78.5,2WKI_1:78.5
	]:14.54
]

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 \((0.85)(0.8)(\frac{803 }{\log_{20} 803}-\frac{269}{\log_{20}269})=146.\)
Status Protein1 Protein2 d d1/2
Query variables 6NRZ_1 4TRN_1 184 137
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]