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Parikh vectors
3IXK_1 8OEF_1 3DPH_1 Letter Amino acid
27 138 6 E Glutamic acid
15 135 6 K Lycine
20 59 2 F Phenylalanine
30 97 4 S Serine
15 65 3 N Asparagine
22 101 4 D Aspartic acid
30 133 7 L Leucine
19 40 2 Y Tyrosine
34 78 5 V Valine
21 92 5 R Arginine
16 60 5 Q Glutamine
23 79 3 I Isoleucine
8 22 3 M Methionine
22 82 6 P Proline
24 64 7 A Alanine
6 38 2 C Cysteine
35 84 7 G Glycine
7 38 1 H Histidine
24 76 7 T Threonine
7 14 1 W Tryptophan 

3IXK_1|Chains A, B, C|Beta-secretase 1|Homo sapiens (9606)
>8OEF_1|Chain A|Terminal uridylyltransferase 7|Homo sapiens (9606)
>3DPH_1|Chains A, B|HIV-1 CAPSID PROTEIN|Human immunodeficiency virus 1 (11676)
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 \(w=f(c)\)
3IXK , Knot 173 405 0.85 40 232 387 
RLPRETDEEPEEPGRRGSFVEMVDNLRGKSGQGYYVEMTVGSPPQTLNILVDTGSSNFAVGAAPHPFLHRYYQRQLSSTYRDLRKGVYVPYTQGKWEGELGTDLVSIPHGPNVTVRANIAAITESDKFFINGSNWEGILGLAYAEIARPDDSLEPFFDSLVKQTHVPNLFSLQLCGAGFPLNQSEVLASVGGSMIIGGIDHSLYTGSLWYTPIRREWYYEVIIVRVEINGQDLKMDCKEYNYDKSIVDSGTTNLRLPKKVFEAAVKSIKAASSTEKFPDGFWLGEQLVCWQAGTTPWNIFPVISLYLMGEVTNQSFRITILPQQYLRPVEDVATSQDDCYKFAISQSSTGTVMGAVIMEGFYVVFDRARKRIGFAVSACHVHDEFRTAAVEGPFVTLDMEDCGYN
8OEF , Knot 523 1495 0.85 40 353 1247 
MGDTAKPYFVKRTKDRGTMDDDDFRRGHPQQDYLIIDDHAKGHGSKMEKGLQKKKITPGNYGNTPRKGPCAVSSNPYAFKNPIYSQPAWMNDSHKDQSKRWLSDEHTGNSDNWREFKPGPRIPVINRQRKDSFQENEDGYRWQDTRGCRTVRRLFHKDLTSLETTSEMEAGSPENKKQRSRPRKPRKTRNEENEQDGDLEGPVIDESVLSTKELLGLQQAEERLKRDCIDRLKRRPRNYPTAKYTCRLCDVLIESIAFAHKHIKEKRHKKNIKEKQEEELLTTLPPPTPSQINAVGIAIDKVVQEFGLHNENLEQRLEIKRIMENVFQHKLPDCSLRLYGSSCSRLGFKNSDVNIDIQFPAIMSQPDVLLLVQECLKNSDSFIDVDADFHARVPVVVCREKQSGLLCKVSAGNENACLTTKHLTALGKLEPKLVPLVIAFRYWAKLCSIDRPEEGGLPPYVFALMAIFFLQQRKEPLLPVYLGSWIEGFSLSKLGNFNLQDIEKDVVIWEHTDSAAGDTGITKEEAPRETPIKRGQVSLILDVKHQPSVPVGQLWVELLRFYALEFNLADLVISIRVKELVSRELKDWPKKRIAIEDPYSVKRNVARTLNSQPVFEYILHCLRTTYKYFALPHKITKSSLLKPLNAITCISEHSKEVINHHPDVQTKDDKLKNSVLAQGPGATSSAANTCKVQPLTLKETAESFGSPPKEEMGNEHISVHPENSDCIQADVNSDDYKGDKVYHPETGRKNEKEKVGRKGKHLLTVDQKRGEHVVCGSTRNNESESTLDLEGFQNPTAKECEGLATLDNKADLDGESTEGTEELEDSLNHFTHSVQGQTSEMIPSDEEEEDDEEEEEEEEPRLTINQREDEDGMANEDELDNTYTGSGDEDALSEEDDELGEAAKYEDVKECGKHVERALLVELNKISLKEENVCEEKNSPVDQSDFFYEFSKLIFTKGKSPTVVCSLCKREGHLKKDCPEDFKRIQLEPLPPLTPKFLNILDQVCIQCYKDFSPTIIEDQAREHIRQNLESFIRQDFPGTKLSLFGSSKNGFGFKQSDLDVCMTINGLETAEGLDCVRTIEELARVLRKHSGLRNILPITTAKVPIVKFFHLRSGLEVDISLYNTLALHNTRLLSAYSAIDPRVKYLCYTMKVFTKMCDIGDASRGSLSSYAYTLMVLYFLQQRNPPVIPVLQEIYKGEKKPEIFVDGWNIYFFDQIDELPTYWSECGKNTESVGQLWLGLLRFYTEEFDFKEHVISIRRKSLLTTFKKQWTSKYIVIEDPFDLNHNLGAGLSRKMTNFIMKAFINGRRVFGIPVKGFPKDYPSKMEYFFDPDVLTEGELAPNDRCCRICGKIGHFMKDCPMRRKVRRRRDQEDALNQRYPENKEKRSKEDKEIHNKYTEREVSTKEDKPIQCTPQKAKPMRAAADLGREKILRPPVEKWKRQDDKDLREKRCFICGREGHIKKECPQFKGSSGSLSSKYMTQGKASAKRTQQES
3DPH , Knot 47 86 0.81 40 75 83 
SPTSILDIRQGPKEPFRDYVDRFYKTLRAEQASQEVKNWMTETLLVQNANPDCKTILKALGPGATSEEMMTACQGVGGPGHKARVL

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 \(c\).

\(|P_{f(3IXK_1)}(2) \setminus P_{f(8OEF_1)}(2)|=17\), \(|P_{f(8OEF_1)}(2) \setminus P_{f(3IXK_1)}(2)|=138\). 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 1:011000000100110010110110010100101001010110110010111001000111111101110000000100000010011011000101010110011011011010101011110000011101001011111101011010001011100110000110110101011111100001110111011111100010010110011000100011110101010010100000000001100100010110011011100101100000110111110011010110011011111010111010000101011100010110011000000001110000010111111101101110010001111101001000100111011110101000100
Pair \(Z_2\) Length of longest common subsequence
3IXK_1,8OEF_1 155 4
3IXK_1,3DPH_1 193 3
8OEF_1,3DPH_1 284 6

Newick tree

 
[
	3DPH_1:13.84,
	[
		3IXK_1:77.5,8OEF_1:77.5
	]:55.34
]

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 \((0.85)(0.8)(\frac{1900 }{\log_{20} 1900}-\frac{405}{\log_{20}405})=375.\)
Status Protein1 Protein2 d d1/2
Query variables 3IXK_1 8OEF_1 471 298
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]