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Parikh vectors
5IAB_1 2PJU_1 2UUF_1 Letter Amino acid
26 13 3 S Serine
10 7 1 Y Tyrosine
14 15 3 R Arginine
8 1 1 C Cysteine
12 14 0 V Valine
19 16 1 I Isoleucine
15 9 2 F Phenylalanine
8 10 0 H Histidine
21 24 5 L Leucine
22 10 3 K Lycine
7 7 1 P Proline
16 17 2 T Threonine
2 1 0 W Tryptophan
12 24 1 A Alanine
4 7 0 Q Glutamine
20 12 6 E Glutamic acid
16 13 4 G Glycine
10 4 0 M Methionine
15 9 0 N Asparagine
21 12 3 D Aspartic acid 

5IAB_1|Chains A, B[auth C]|Caspase-3|Homo sapiens (9606)
>2PJU_1|Chains A, B, C, D|Propionate catabolism operon regulatory protein|Escherichia coli (562)
>2UUF_1|Chain A|HUMAN ALPHA THROMBIN|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 \(w=f(c)\)
5IAB , Knot 123 278 0.83 40 183 273 
MENTENSVDSKSIKNLEPKIIHGSESMDSGISLDNSYKMDYPEMGLCIIINNKNFHKSTGMTSRSGTDVDAANLRETFRNLKYEVRNKNDLTREEIVELMRDVSKEDHSKRSSFVCVLLSHGEEGIIFGTNGPVDLKKITNFFRGDRCRSLTGKPKLFIIQACRGTELDCGIETDSGVDDDMACHKIPVEADFLYAYSTAPGYYSWRNSKDGSWFIQSLCAMLKQYADKLEFMHILTRVNRKVATEFESFSFDATFHAKKQIPCIDSMLTKELYFYHL
2PJU , Knot 104 225 0.83 40 161 216 
MSLAHPPRLNDDKPVIWTVSVTRLFELFRDISLEFDHLANITPIQLGFEKAVTYIRKKLANERCDAIIAAGSNGAYLKSRLSVPVILIKPSGYDVLQFLAKAGKLTSSIGVVTYQETIPALVAFQKTFNLRLDQRSYITEEDARGQINELKANGTEAVVGAGLITDLAEEAGMTGIFIYSAATVRQAFSDALDMTRMSLRHNTHDATRNALRTRYVLEGHHHHHH
2UUF , Knot 24 36 0.79 28 34 34 
TFGSGEADCGLRPLFEKKSLEDKTERELLESYIDGR

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 \(c\).

\(|P_{f(5IAB_1)}(2) \setminus P_{f(2PJU_1)}(2)|=97\), \(|P_{f(2PJU_1)}(2) \setminus P_{f(5IAB_1)}(2)|=75\). 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 1:10000001000010010101101000100110100000100101110111000010000110000100101101000100100010000010000110110010000000000110111001001111100111010010011010000010101011110100100100110000110001100011101011010001110001000001011100101110001001011011001000110010010101010100011010011000101001
Pair \(Z_2\) Length of longest common subsequence
5IAB_1,2PJU_1 172 3
5IAB_1,2UUF_1 163 3
2PJU_1,2UUF_1 153 3

Newick tree

 
[
	5IAB_1:86.07,
	[
		2UUF_1:76.5,2PJU_1:76.5
	]:9.57
]

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 \((0.85)(0.8)(\frac{503 }{\log_{20} 503}-\frac{225}{\log_{20}225})=80.0\)
Status Protein1 Protein2 d d1/2
Query variables 5IAB_1 2PJU_1 105 94.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]