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Parikh vectors
6DUJ_1 5WXN_1 5CTX_1 Letter Amino acid
1 2 2 W Tryptophan
5 14 10 Y Tyrosine
2 3 0 C Cysteine
8 31 21 E Glutamic acid
13 12 18 G Glycine
3 7 5 H Histidine
18 22 11 K Lycine
3 6 4 F Phenylalanine
5 10 12 N Asparagine
3 15 14 D Aspartic acid
2 14 10 Q Glutamine
5 26 11 A Alanine
8 10 17 I Isoleucine
2 22 9 S Serine
4 12 21 V Valine
2 11 12 R Arginine
6 27 14 L Leucine
3 8 2 M Methionine
4 4 4 P Proline
7 11 15 T Threonine 

6DUJ_1|Chains A, B[auth C]|Cytochrome c|Homo sapiens (9606)
>5WXN_1|Chains A, B|14-3-3 protein zeta/delta|Homo sapiens (9606)
>5CTX_1|Chains A, B|DNA gyrase subunit B|Staphylococcus aureus (1280)
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 \(w=f(c)\)
6DUJ , Knot 54 104 0.80 40 87 99 
GDVEKGKKIFIMKCSQCHTVEKGGKHKTGPNLHGLFGRKTGQAPGYSYTAVNKNKGIIWGEDTLMEYLENPKKYIPGTKMIFVGIKKKEERADLIAYLKKATNE
5WXN , Knot 117 267 0.81 40 162 255 
MSYYHHHHHHLESTSLYKKAGLMDKNELVQKAKLAEQAERYDDMAACMKSVTEQGAELSNEERNLLSVAYKNVVGARRSSWRVVSSIEQKTEGAEKKQQMAREYREKIETELRDICNDVLSLLEKFLIPNASQAESKVFYLKMKGDYYRYLAEVAAGDDKKGIVDQSQQAYQEAFEISKKEMQPTHPIRLGLALNFSVFYYEILNSPEKACSLAKTAFDEAIAELDTLSEESYKDSTLIMQLLRDNLTLWTSDTQGDEAEAGEGGEN
5CTX , Knot 98 212 0.82 38 151 207 
GSVTALSDVNNTDNYGAGQIQVLEGLEAVRKRPGMYIGSTSERGLHHLVWEIVDNSIDEALAGYANQIEVVIEKDNWIKVTDNGRGIPVDIQEKMGRPAVEVILTSSVVNALSQDLEVYVHRNETIYHQAYKKGVPQFDLKEVGTTDKTGTVIRFKADGEIFTETTVYNYETLQQRIRELAFLNKGIQITLRDERDEENVREDSYHYEGGIK

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 \(c\).

\(|P_{f(6DUJ_1)}(2) \setminus P_{f(5WXN_1)}(2)|=47\), \(|P_{f(5WXN_1)}(2) \setminus P_{f(6DUJ_1)}(2)|=122\). 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 1:10100100111100000001001100001101011110001011100001100001111100011001001000111001111110000001011101001000
Pair \(Z_2\) Length of longest common subsequence
6DUJ_1,5WXN_1 169 3
6DUJ_1,5CTX_1 148 4
5WXN_1,5CTX_1 153 3

Newick tree

 
[
	5WXN_1:82.68,
	[
		6DUJ_1:74,5CTX_1:74
	]:8.68
]

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 \((0.85)(0.8)(\frac{371 }{\log_{20} 371}-\frac{104}{\log_{20}104})=82.1\)
Status Protein1 Protein2 d d1/2
Query variables 6DUJ_1 5WXN_1 102 71
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]