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Parikh vectors
5XJA_1 5UAZ_1 4EGC_1 Letter Amino acid
19 5 24 F Phenylalanine
28 8 66 A Alanine
21 7 29 N Asparagine
36 10 49 L Leucine
14 3 8 M Methionine
11 4 31 P Proline
34 12 22 S Serine
24 5 26 T Threonine
32 2 24 R Arginine
30 5 43 E Glutamic acid
19 12 24 I Isoleucine
24 7 40 G Glycine
13 2 11 H Histidine
1 2 13 W Tryptophan
18 5 26 D Aspartic acid
13 1 4 C Cysteine
19 3 18 Q Glutamine
33 10 49 K Lycine
11 5 21 Y Tyrosine
26 3 31 V Valine 

5XJA_1|Chains A, B|Kinesin-like protein KIF2C|Mus musculus (10090)
>5UAZ_1|Chains A, B|Nucleoporin NUP53|Saccharomyces cerevisiae (559292)
>4EGC_1|Chain A|Maltose-binding periplasmic protein, Homeobox protein SIX1 chimera|Escherichia coli (562)
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 \(w=f(c)\)
5XJA , Knot 179 426 0.84 40 222 407 
MASMTGGQQMGRDPNSSVRRKSCIVKEMEKMKNKREEKRAQNSELRIKRAQEYDSSFPNWEFARMIKEFRVTMECSPLTVTDPIEEHRICVCVRKRPLNKQELAKKEIDVISVPSKCLLLVHEPKLKVDLTKYLENQAFCFDFAFDETASNEVVYRFTARPLVQTIFEGGKATCFAYGQTGSGKTHTMGGDLSGKSQNASKGIYAMASRDVFLLKNQPRYRNLNLEVYVTFFEIYNGKVFDLLNKKAKLRVLEDSRQQVQVVGLQEYLVTCADDVIKMINMGSACRTSGQTFANSNSSRSHACFQILLRTKGRLHGKFSLVDLAGNERGADTSSADRQTRMEGAEINKSLLALKECIRALGQNKAHTPFRESKLTQVLRDSFIGENSRTCMIAMISPGISSCEYTLNTLRYADRVKELSHHHHHHH
5UAZ , Knot 56 111 0.79 40 91 108 
SMNWDDHAIIIFGYPETIANSIILHFANFGEILEDFRVIKDFKKLNSKNMSKSPSLTAQKYPIYTGDGWVKLTYKSELSKSRALQENGIIMNGTLIGCVSYSPAALKQLAS
4EGC , Knot 218 559 0.82 40 252 509 
KIEEGKLVIWINGDKGYNGLAEVGKKFEKDTGIKVTVEHPDKLEEKFPQVAATGDGPDIIFWAHDRFGGYAQSGLLAEITPDKAFQDKLYPFTWDAVRYNGKLIAYPIAVEALSLIYNKDLLPNPPKTWEEIPALDKELKAKGKSALMFNLQEPYFTWPLIAADGGYAFKYAAGKYDIKDVGVDNAGAKAGLTFLVDLIKNKHMNADTDYSIAEAAFNKGETAMTINGPWAWSNIDTSKVNYGVTVLPTFKGQPSKPFVGVLSAGINAASPNKELAKEFLENYLLTDEGLEAVNKDKPLGAVALKSYEEELAKDPRIAATMENAQKGEIMPNIPQMSAFWYAVRTAVINAASGRQTVDAALAAAQTNAAAMSMLPSFGFTQEQVACVCEVLQQGGNLERLGRFLWSLPACDHLHKNESVLKAKAVVAFHRGNFRELYKILESHQFSPHNHPKLQQLWLKAHYVEAEKLRGRPLGAVGKYRVRRKFPLPRTIWDGEETSYCFKEKSRGVLREWYAHNPYPSPREKRELAEATGLTTTQVSNWFKNRRQRDRAAEAKEREN

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 \(c\).

\(|P_{f(5XJA_1)}(2) \setminus P_{f(5UAZ_1)}(2)|=162\), \(|P_{f(5UAZ_1)}(2) \setminus P_{f(5XJA_1)}(2)|=31\). 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 1:110101100110010001000001100100100000000100001010010000001101011011001010100011010011000010101000110000110001011011000111100101010100010001101011100010001100101011100110110100110100101000011101010000100110111000111100010000101010101101001011011000101011000000101111000110010011011011010000100110000000010101110001010101011011100011000010000010110100011110001011100010011000010011000111000000111110111000000100100100100100000000
Pair \(Z_2\) Length of longest common subsequence
5XJA_1,5UAZ_1 193 3
5XJA_1,4EGC_1 162 4
5UAZ_1,4EGC_1 207 4

Newick tree

 
[
	5UAZ_1:10.65,
	[
		5XJA_1:81,4EGC_1:81
	]:24.65
]

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 \((0.85)(0.8)(\frac{537 }{\log_{20} 537}-\frac{111}{\log_{20}111})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 5XJA_1 5UAZ_1 163 101.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]