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Parikh vectors
7ARX_1 1EMJ_1 7QHK_1 Letter Amino acid
7 0 11 Y Tyrosine
9 0 20 V Valine
6 0 10 Q Glutamine
13 0 10 L Leucine
10 0 10 P Proline
12 5 7 T Threonine
2 0 3 H Histidine
7 0 5 I Isoleucine
5 0 4 M Methionine
2 0 5 W Tryptophan
5 0 5 R Arginine
9 0 11 E Glutamic acid
10 1 24 G Glycine
10 0 20 S Serine
14 0 19 K Lycine
5 0 9 F Phenylalanine
5 1 19 A Alanine
9 1 14 N Asparagine
6 0 9 D Aspartic acid
9 1 6 C Cysteine 

7ARX_1|Chain A|Mannan-binding lectin serine protease 1|Homo sapiens (9606)
>1EMJ_1|Chain A[auth B]|DNA (5'-D(*TP*GP*TP*(ASU)P*AP*TP*CP*TP*T)-3')|
>7QHK_1|Chains A[auth AA], B[auth BA]|Cathepsin L2|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 \(w=f(c)\)
7ARX , Knot 80 155 0.86 40 121 150 
ASMTGNECPELQPPVHGKIEPSQAKYFFKDQVLVSCDTGYKVLKDNVEMDTFQIECLKDGTWSNKIPTCKIVDCRAPGELEHGLITFSTRNNLTTYKSEIKYSCQEPYYKMLNNNTGIYTCSAQGVWMNKVLGRSLPTCLPVCGLPKFSRKLMAR
1EMJ , Knot 6 9 0.48 10 8 7 
TGTNATCTT
7QHK , Knot 105 221 0.85 40 148 212 
LPKSVDWRKKGYVTPVKNQKQCGSAWAFSATGALEGQMFRKTGKLVSLSEQNLVDCSRPQGNQGCNGGFMARAFQYVKENGGLDSEESYPYVAVDEICKYRPENSVAQDTGFTVVAPGKEKALMKAVATVGPISVAMDAGHSSFQFYKSGIYFEPDCSSKNLDHGVLVVGYGFEGANSQNSKYWLVKNSWGPEWGSNGYVKIAKDKNNHCGIATAASYPNV

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 \(c\).

\(|P_{f(7ARX_1)}(2) \setminus P_{f(1EMJ_1)}(2)|=117\), \(|P_{f(1EMJ_1)}(2) \setminus P_{f(7ARX_1)}(2)|=4\). 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 1:10101000101011101010100100110001110000100110001010010100100101000110001100011101001110100000100000010000001000110000110000101111001110011001110111010001110
Pair \(Z_2\) Length of longest common subsequence
7ARX_1,1EMJ_1 121 2
7ARX_1,7QHK_1 157 3
1EMJ_1,7QHK_1 152 2

Newick tree

 
[
	7QHK_1:82.08,
	[
		7ARX_1:60.5,1EMJ_1:60.5
	]:21.58
]

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 \((0.85)(0.8)(\frac{164 }{\log_{20} 164}-\frac{9}{\log_{20}9})=57.1\)
Status Protein1 Protein2 d d1/2
Query variables 7ARX_1 1EMJ_1 76 40
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]