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Parikh vectors
7DKU_1 5QTD_1 4MUR_1 Letter Amino acid
50 27 12 A Alanine
3 9 2 C Cysteine
65 20 12 G Glycine
80 20 12 K Lycine
55 18 11 Y Tyrosine
45 30 10 V Valine
27 15 16 Q Glutamine
44 24 16 E Glutamic acid
46 9 9 I Isoleucine
32 16 8 F Phenylalanine
30 14 10 P Proline
38 23 8 T Threonine
45 17 11 S Serine
29 5 4 W Tryptophan
12 19 11 R Arginine
57 11 7 N Asparagine
56 30 11 D Aspartic acid
13 5 14 H Histidine
46 38 22 L Leucine
26 19 5 M Methionine 

7DKU_1|Chain A|beta-glucosidase|Thermoanaerobacterium xylanolyticum LX-11 (858215)
>5QTD_1|Chain A|Farnesyl pyrophosphate synthase|Trypanosoma brucei (5691)
>4MUR_1|Chains A, B|D,D-dipeptidase/D,D-carboxypeptidase|Enterococcus gallinarum (1353)
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 \(w=f(c)\)
7DKU , Knot 300 799 0.83 40 280 717 
AMALTGCSEKININEDKISHKIDIPDSAWTIGIGEKFKNAGHPNVKYPMIDDSYVQGAPLGGFGAGTIGRTYNGGFSRWHLEIGKNKYTTVYANQFSVFQKVEGNKDGVAQVLYAGEPENGYLSSWKWDYPKESGMYYALYPNSWYTYTNKDLPVQLAVKQFSPIIPYNYKETSYPVAVFKWTAYNPTNKNVDVSIMFTWQNMIGFFGKQVNVNSGNFNKIIKDKSKDSEIVAAVMGNISNDNEEWNGEYSIGVKKVPGVDISYKAKFVTTGDGSDLWHEFSKNGILDNKDDETPTKQDGIGSAIAVNFKLQPGQTIEVPFALSWDLPIMKFGGGDKWYKMYTKYFGKNGKNSFAILKEALNNYQKWEKMIDDWQKPILSNKSKPDWYKTALFNELYYLADGGTAWENGKVGEKDKRTNNMFGLLACFDYNYYETLDVRFYGSFPLVMLWPDIEKQVMRQFADTINVQDSSEFKVGSNGAMAVKKVQGMIPHDLGSSYALPWIKINAYDWQNPNIWKDLNSKYVLLVYRDYVLTGKTDKEFLKYTWKSVKTALDKLKEMDKDNDGIPDNEGIPDQTYDTWSMKGTSAYCGSLWLAALKAAQEIGKVLKDNEAYIKYNEWYKIAQQNFEKELWNGEYYNFDTESDHKDSIMADQLAGQWYADILRLGDILPKDHVQKALKKIYEFNVMKFENGKMGAVNGMRPDGIVDESDIQAQEVWTGVTYALASFMKYRGMTEEAYNTAYGVYKMTYDKSGKGYWFRTPEAWTKDGNYRASMYMRPLSIWSMEVNYNEVLEHHHHHH
5QTD , Knot 154 369 0.82 40 216 356 
GPMPMQMFMQVYDEIQMFLLEELELKFDMDPNRVRYLRKMMDTTCLGGKYNRGLTVIDVAESLLSLSPNNNGEEDDGARRKRVLHDACVCGWMIEFLQAHYLVEDDIMDNSVTRRGKPCWYRHPDVTVQCAINDGLLLKSWTHMMAMHFFADRPFLQDLLCRFNRVDYTTAVGQLYDVTSMFDSNKLDPDVSQPTTTDFAEFTLSNYKRIVKYKTAYYTYLLPLVMGLIVSEALPTVDMGVTEELAMLMGEYFQVQDDVMDCFTPPERLGKVGTDIQDAKCSWLAVTFLAKASSAQVAEFKANYGSGDSEKVATVRRLYEEADLQGDYVAYEAAVAEQVKELIEKLRLCSPGFAASVETLWGKTYKRQK
4MUR , Knot 94 211 0.79 40 151 200 
MGSSHHHHHHSSGRENLYFQGMNTLQLINKNHPLKKNQEPPHLVLAPFSDHDVYLQPEVAKQWERLVRATGLEKDIRLVSGYRTEKEQRRLWEYSLKENGLAYTKQFVALPGCSEHQIGLAIDVGLKKQEDDDLICPHFRDSAAADLFMQQMMNYGFILRYPEDKQEITGISYEPWHFRYVGLPHSQVITAQKWTLEEYHDYLAQTVRQFA

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 \(c\).

\(|P_{f(7DKU_1)}(2) \setminus P_{f(5QTD_1)}(2)|=115\), \(|P_{f(5QTD_1)}(2) \setminus P_{f(7DKU_1)}(2)|=51\). 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 1:1111010000101000010001011001101111001001101010011100001011111111110110000111001010110000001010010110010100011101101101001010010100100011001101001000000011101110010111100000000111110101001000010101110100111111001010010100110000000011111110100000010100011100111101000101100101001100100011100000001000011101111010101100101111101011110111100100100001100100011110011000001001100100111000001010001110010011011011001011000000001111110100000001010101011111111010001100110010100000101100111110010111100110001111101010010010110010000111100001101000001100010010011001001000001110001110000001010100100101111110110011011000010100001001100010001101000010000000001110011101010110110111000100110010010110100101111011010111000010100110110011101100011000100010110010000010101100101100010001010101101101010000110000000
Pair \(Z_2\) Length of longest common subsequence
7DKU_1,5QTD_1 166 4
7DKU_1,4MUR_1 183 6
5QTD_1,4MUR_1 187 3

Newick tree

 
[
	4MUR_1:95.46,
	[
		7DKU_1:83,5QTD_1:83
	]:12.46
]

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 \((0.85)(0.8)(\frac{1168 }{\log_{20} 1168}-\frac{369}{\log_{20}369})=209.\)
Status Protein1 Protein2 d d1/2
Query variables 7DKU_1 5QTD_1 265 194
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]