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Parikh vectors
8ALZ_1 6FAF_1 5JXM_1 Letter Amino acid
32 7 38 R Arginine
34 14 27 D Aspartic acid
15 1 4 C Cysteine
14 11 18 H Histidine
54 18 3 K Lycine
23 11 13 F Phenylalanine
42 30 30 S Serine
27 19 49 A Alanine
50 13 23 E Glutamic acid
39 16 40 G Glycine
60 20 49 L Leucine
25 17 34 P Proline
24 14 23 T Threonine
9 6 3 W Tryptophan
9 8 8 Y Tyrosine
21 17 3 N Asparagine
42 9 5 Q Glutamine
30 10 3 I Isoleucine
5 3 7 M Methionine
30 17 25 V Valine 

8ALZ_1|Chain A|Activating signal cointegrator 1|Homo sapiens (9606)
>6FAF_1|Chains A, B|Carbonic anhydrase 1|Homo sapiens (9606)
>5JXM_1|Chain A|PriB|Streptomyces sp. RM-5-8 (1429103)
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 \(w=f(c)\)
8ALZ , Knot 233 585 0.84 40 262 538 
GAEFMAVAGAVSGEPLVHWCTQQLRKTFGLDVSEEIIQYVLSIESAEEIREYVTDLLQGNEGKKGQFIEELITKWQKNDQELISDPLQQCFKKDEILDGQKSGDHLKRGRKKGRNRQEVPAFTEPDTTAEVKTPFDLAKAQENSNSVKKKTKFVNLYTREGQDRLAVLLPGRHPCDCLGQKHKLINNCLICGRIVCEQEGSGPCLFCGTLVCTHEEQDILQRDSNKSQKLLKKLMSGVENSGKVDISTKDLLPHQELRIKSGLEKAIKHKDKLLEFDRTSIRRTQVIDDESDYFASDSNQWLSKLERETLQKREEELRELRHASRLSKKVTIDFAGRKILEEENSLAEYHSRLDETIQAIANGTLNQPLTKLDRSSEEPLGVLVNPNMYQSPPQWVDHTGAASQKKAFRSSGFGLEFNSFQHQLRIQDQEFQEGFDGGWCLSVHQPWASLLVRGIKRVEGRSWYTPHRGRLWIAATAKKPSPQEVSELQATYRLLRGKDVEFPNDYPSGCLLGCVDLIDCLSQKQFKEQFPDISQESDSPFVFICKNPQEMVVKFPIKGNPKIWKLDSKIHQGAKKGLMKQNKAV
6FAF , Knot 121 261 0.86 40 179 253 
MASPDWGYDDKNGPEQWSKLYPIANGNNQSPVDIKTSETKHDTSLKPISVSYNPATAKEIINVGHSFHVNFEDNDNRSVLKGGPFSDSYRLFQFHFHWGSTNEHGSEHTVDGVKYSAELHVAHWNSAKYSSLAEAASKADGLAVIGVLMKVGEANPKLQKVLDALQAIKTKGKRAPFTNFDPSTLLPSSLDFWTYPGSLTHPPLYESVTWIICKESISVSSEQLAQFRSLLSNVEGDNAVPMQHNNRPTQPLKGRTVRASF
5JXM , Knot 157 405 0.77 40 170 362 
MGSSHHHHHHSSGLVPRGSHMGGPMSGFHSGEALLGDLATGQLTRLCEVAGLTEADTAAYTGVLIESLGTSAGRPLSLPPPSRTFLSDDHTPVEFSLAFLPGRAPHLRVLVEPGCSSGDDLAENGRAGLRAVHTMADRWGFSTEQLDRLEDLFFPSSPEGPLALWCALELRSGGVPGVKVYLNPAANGADRAAETVREALARLGHLQAFDALPRADGFPFLALDLGDWDAPRVKIYLKHLGMSAADAGSLPRMSPAPSREQLEEFFRTAGDLPAPGDPGPTEDTGRLAGRPALTCHSFTETATGRPSGYTLHVPVRDYVRHDGEARDRAVAVLREHDMDSAALDRALAAVSPRPLSDGVGLIAYLALVHQRGRPTRVTVYVSSEAYEVRPPRETVPTRDRARARL

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 \(c\).

\(|P_{f(8ALZ_1)}(2) \setminus P_{f(6FAF_1)}(2)|=130\), \(|P_{f(6FAF_1)}(2) \setminus P_{f(8ALZ_1)}(2)|=47\). 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 1:110111111110101110100001000111010001100110100100100010011010010010110011001000000110011000100001101000100100100010000011110010001010011011010000001000001101000010001111111001000110000110001101011000010110110101100000001100000000011001101100010101000011100010100110011000001101000010000110000001100000110010000100000010010010010001010111001100000110000010001011101010011001000000111111010100011011000111000011000111101001000101000010011011101010011101110110010100100100101111101001010010010100011010010110001010111010110010000100011010000001111100010011101110101011010001001100111000011
Pair \(Z_2\) Length of longest common subsequence
8ALZ_1,6FAF_1 177 4
8ALZ_1,5JXM_1 190 4
6FAF_1,5JXM_1 153 4

Newick tree

 
[
	8ALZ_1:96.37,
	[
		6FAF_1:76.5,5JXM_1:76.5
	]:19.87
]

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 \((0.85)(0.8)(\frac{846 }{\log_{20} 846}-\frac{261}{\log_{20}261})=160.\)
Status Protein1 Protein2 d d1/2
Query variables 8ALZ_1 6FAF_1 201 144
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]