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Parikh vectors
8SWI_1 6UUX_1 7MQD_1 Letter Amino acid
32 12 6 N Asparagine
4 5 4 C Cysteine
26 9 10 Q Glutamine
27 13 12 E Glutamic acid
19 17 11 T Threonine
26 8 17 A Alanine
10 6 3 M Methionine
10 14 5 P Proline
27 13 14 S Serine
25 10 10 G Glycine
11 8 7 H Histidine
27 22 10 K Lycine
3 4 2 W Tryptophan
21 11 13 V Valine
26 12 4 Y Tyrosine
17 6 15 R Arginine
36 23 17 D Aspartic acid
32 23 14 I Isoleucine
51 34 24 L Leucine
21 9 9 F Phenylalanine 

8SWI_1|Chain A|Eukaryotic huntingtin interacting protein B|Legionella pneumophila subsp. pneumophila (91891)
>6UUX_1|Chain A|Sulfotransferase|Schistosoma mansoni (6183)
>7MQD_1|Chains A, C, E, G, I, K, M, O|NanoRNase C|Bartonella henselae (38323)
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 \(w=f(c)\)
8SWI , Knot 191 451 0.86 40 237 429 
NADEWIDTSKIMLDLHIDNMSSSDYIPSAIDRTDLVMVQSVHLLRKTGGRGLFAREDIPKGTCIGIYTGEVYSEQEFEQYLKEHVGSDKSYAMYVGGRVIDAARKGNLTRYINFSDSQDNAEFVETTLNRKKVAKVITTKNIKAGQQLLINYNTYEEQASRYYYFLNPGDGWLSAQEFYQTYQSQYRLEQMPYNLEGFDLKAGDRVLMTQIGRIILANYSLAKEQELNASDIDLPFLKVGSDEKILDFDEADTFTPLMAACYLGQVENVKWLIEHGANIDQQQSHSGHCPLSLTLKGYSLAKDTQKYIDIIQLLIKNQVNLLVHDRSDKTFLHNAALVLNNLDFQSVVKFLIGQNPIDINEYFTYIDENDFDIVMHCYNNKLFDKALVLLAFYPDYFKRNYMSDNEGHNQFNINAFRKAIKDFNSNERSILLMQLRESGLHLPEDLLEQLG
6UUX , Knot 116 259 0.83 40 167 248 
GAMIESSTTIQVISAGLPRTGTKSLKNALEIIYHKPCYHMFEIIFNKQSDIIKWQNLIHDSHMITTPPPLTTKTIAIYDKLKELLDGYIATTDLPTCGFYKDLMNIYPNAKVLLTIRDKYDWLHSLRKVVLPKSNDPWKLKIEEGDKVLGLNSDFYKLTEDSLKFAFQKDDLNFDDDQVLLECYDEYNRLVQETVPSDRLLVLRLGDGWEPLCKFLNVEIPNGIDYPCVNSHHQMTQLTEQLIKYKSLDAIIHMFPDLI
7MQD , Knot 97 207 0.83 40 149 198 
SMTEIRVHQGDLPNLDNYRIDAVAVDTETLGLQPHRDRLCVVQLSSGDGTADVIQIAKGQKSAPNLVRLLSDRDITKIFHFGRFDLAILAHTFGVMPDVVFCTKIASKLTRTYTDRHGLKEICGELLNVNISKQQQSSDWAAETLSRAQIEYAASDVLYLHRLKDIFEERLKREERESVAKACFQFLPMRANLDLLGWSEIDIFAHS

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 \(c\).

\(|P_{f(8SWI_1)}(2) \setminus P_{f(6UUX_1)}(2)|=115\), \(|P_{f(6UUX_1)}(2) \setminus P_{f(8SWI_1)}(2)|=45\). 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 1:0100110000111010100100000110110000111100101100011011110001101001110010100000100010001100000110111011011001010001010000001011000100001101100001011001110000000010000011011011101001000000000100110010110101100111001101111000110000101001011110110000110100100101111100110100101110011010000000100110101010011000000101101110001011100000001100111110010100110111100110100010010000101110000001100111111101001000010000100010101100110010000001111010001101100110011
Pair \(Z_2\) Length of longest common subsequence
8SWI_1,6UUX_1 160 4
8SWI_1,7MQD_1 178 4
6UUX_1,7MQD_1 168 3

Newick tree

 
[
	7MQD_1:88.60,
	[
		8SWI_1:80,6UUX_1:80
	]:8.60
]

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 \((0.85)(0.8)(\frac{710 }{\log_{20} 710}-\frac{259}{\log_{20}259})=125.\)
Status Protein1 Protein2 d d1/2
Query variables 8SWI_1 6UUX_1 160 122.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]