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Parikh vectors
9EJU_1 6CVF_1 5PGV_1 Letter Amino acid
0 6 16 I Isoleucine
0 2 9 F Phenylalanine
0 11 5 P Proline
0 14 15 T Threonine
0 1 10 Y Tyrosine
0 13 12 R Arginine
1 3 10 N Asparagine
0 3 4 C Cysteine
0 5 16 M Methionine
3 50 27 A Alanine
3 27 26 L Leucine
0 3 17 K Lycine
0 2 1 W Tryptophan
0 26 25 V Valine
0 14 8 D Aspartic acid
0 27 22 G Glycine
1 8 10 H Histidine
0 7 9 Q Glutamine
1 6 18 E Glutamic acid
0 7 26 S Serine 

9EJU_1|Chains A, B, C, D|Co-MAHF-9 A8N|synthetic construct (32630)
>6CVF_1|Chains A, B, C, D|ATP-dependent dethiobiotin synthetase BioD|Mycobacterium tuberculosis (strain ATCC 25618 / H37Rv) (83332)
>5PGV_1|Chains A, B, C[auth D], D[auth E]|Corticosteroid 11-beta-dehydrogenase isozyme 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 \(w=f(c)\)
9EJU , Knot 8 11 0.58 12 9 9 
XLAEALHNALX
6CVF , Knot 91 235 0.70 40 122 197 
MGHHHHHHGGTILVVTGTGTGVGKTVVCAALASAARQAGIDVAVCKPVQTGTARGDDDLAEVGRLAGVTQLAGLARYPQPMAPAAAAEHAGMALPARDQIVRLIADLDRPGRLTLVEGAGGLLVELAEPGVTLRDVAVDVAAAALVVVTADLGTLNHTKLTLEALAAQQVSCAGLVIGSWPDPPGLVAASNRSALARIAMVRAALPAGAASLDAGDFAAMSAAAFDRNWVAGLVG
5PGV , Knot 129 286 0.85 40 183 279 
GSHMASMTGGQQMGRGSNEEFRPEMLQGKKVIVTGASKGIGREMAYHLAKMGAHVVVTARSKETLQKVVSHCLELGAASAHYIAGTMEDMTFAEQFVAQAGKLMGGLDMLILNHITNTSLNLFHDDIHHVRKSMEVNFLSYVVLTVAALPMLKQSNGSIVVVSSLAGKVAYPMVAAYSASKFALDGFFSSIRKEYSVSRVNVSITLCVLGLIDTETAMKAVSGIVHMQAAPKEECALEIIKGGALRQEEVYYDSSRWTTLLIRNPCRKILEELYSTSYNMDRFINK

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 \(c\).

\(|P_{f(9EJU_1)}(2) \setminus P_{f(6CVF_1)}(2)|=5\), \(|P_{f(6CVF_1)}(2) \setminus P_{f(9EJU_1)}(2)|=118\). 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 1:01101100110
Pair \(Z_2\) Length of longest common subsequence
9EJU_1,6CVF_1 123 3
9EJU_1,5PGV_1 186 2
6CVF_1,5PGV_1 171 5

Newick tree

 
[
	5PGV_1:96.84,
	[
		9EJU_1:61.5,6CVF_1:61.5
	]:35.34
]

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 \((0.85)(0.8)(\frac{246 }{\log_{20} 246}-\frac{11}{\log_{20}11})=81.6\)
Status Protein1 Protein2 d d1/2
Query variables 9EJU_1 6CVF_1 89 47
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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