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Parikh vectors
7QZK_1 1ONA_1 3SKG_1 Letter Amino acid
16 18 28 A Alanine
7 6 24 R Arginine
11 5 20 Q Glutamine
16 7 28 E Glutamic acid
23 12 15 K Lycine
11 11 26 P Proline
11 19 28 T Threonine
1 14 15 N Asparagine
3 0 6 C Cysteine
4 6 9 H Histidine
2 2 11 M Methionine
14 31 34 S Serine
13 17 35 V Valine
9 18 23 D Aspartic acid
25 16 46 G Glycine
5 15 25 I Isoleucine
11 11 20 F Phenylalanine
8 7 19 Y Tyrosine
28 18 36 L Leucine
3 4 7 W Tryptophan 

7QZK_1|Chain A|2',3'-cyclic-nucleotide 3'-phosphodiesterase|Mus musculus (10090)
>1ONA_1|Chains A, B, C, D|CONCANAVALIN A|Canavalia ensiformis (3823)
>3SKG_1|Chains A, B, C[auth D], D[auth E]|Beta-secretase 1|Homo sapiens (9606)
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 \(w=f(c)\)
7QZK , Knot 98 221 0.79 40 142 210 
GGLEKDFLPLYFGWFLTKKSSETLRKAGQVFLEELGNHKAFKKELRHFISGDEPKEKLELVSYFGKRPPGVLHCTTKFCDYGKAAGAEEYAQQEVVKRSYGKAFKLSISALFVTPKTAGAQVVLTDQELQLWPSDLDKPSASEGLPPGSRAHVTLGCAADIQPVQTGLDLLDILQQVKGGSQGEAVGELPRGKLYSLGKGRWMLSLTKKMEVKAIFTGYYG
1ONA , Knot 104 237 0.80 38 151 226 
ADTIVAVELDTYPNTDIGDPSYPHIGIDIKSVRSKKTAKWNMQNGKVGTAHIIYNSVDKRLSAVVSYPNADSATVSYDVDLDNVLPEWVRVGLSASTGLYKETNTILSWSFTSKLKSNSTHETNALHFMFNQFSKDQKDLILQGDATTGTDGNLELTRVSSNGSPQGSSVGRALFYAPVHIWESSAVVASFEATFTFLIKSPDSHPADGIAFFISNIDSSIPSGSTGRLLGLFPDAN
3SKG , Knot 191 455 0.85 40 238 431 
MASMTGGQQMGRGSAGVLPAHGTQHGIRLPLRSGLGGAPLGLRLPRETDEEPEEPGRRGSFVEMVDNLRGKSGQGYYVEMTVGSPPQTLNILVDTGSSNFAVGAAPHPFLHRYYQRQLSSTYRDLRKGVYVPYTQGKWEGELGTDLVSIPHGPNVTVRANIAAITESDKFFINGSNWEGILGLAYAEIARPDDSLEPFFDSLVKQTHVPNLFSLQLCGAGFPLNQSEVLASVGGSMIIGGIDHSLYTGSLWYTPIRREWYYEVIIVRVEINGQDLKMDCKEYNYDKSIVDSGTTNLRLPKKVFEAAVKSIKAASSTEKFPDGFWLGEQLVCWQAGTTPWNIFPVISLYLMGEVTNQSFRITILPQQYLRPVEDVATSQDDCYKFAISQSSTGTVMGAVIMEGFYVVFDRARKRIGFAVSACHVHDEFRTAAVEGPFVTLDMEDCGYNIPQTDEST

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 \(c\).

\(|P_{f(7QZK_1)}(2) \setminus P_{f(1ONA_1)}(2)|=80\), \(|P_{f(1ONA_1)}(2) \setminus P_{f(7QZK_1)}(2)|=89\). 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 1:11100011110111110000000100110111001100011000100110100100010110011001111100000100010111100010001100001011010101111010011101110000101110010010100111110010101101101011001101101100101100101110110101001101011101000101011101001
Pair \(Z_2\) Length of longest common subsequence
7QZK_1,1ONA_1 169 3
7QZK_1,3SKG_1 190 3
1ONA_1,3SKG_1 173 3

Newick tree

 
[
	3SKG_1:92.86,
	[
		7QZK_1:84.5,1ONA_1:84.5
	]:8.36
]

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 \((0.85)(0.8)(\frac{458 }{\log_{20} 458}-\frac{221}{\log_{20}221})=68.8\)
Status Protein1 Protein2 d d1/2
Query variables 7QZK_1 1ONA_1 88 85
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]