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Parikh vectors
5CSJ_1 5NBC_1 2LMV_1 Letter Amino acid
0 0 1 W Tryptophan
1 5 1 Y Tyrosine
8 15 7 K Lycine
3 9 8 N Asparagine
3 8 7 Q Glutamine
6 4 0 H Histidine
7 7 11 F Phenylalanine
3 4 10 T Threonine
7 9 4 V Valine
5 5 10 A Alanine
2 4 1 C Cysteine
8 12 9 L Leucine
1 4 7 R Arginine
16 12 24 E Glutamic acid
5 7 11 G Glycine
3 12 11 I Isoleucine
5 4 9 M Methionine
0 3 2 P Proline
5 9 3 S Serine
7 7 12 D Aspartic acid 

5CSJ_1|Chains A, B|Protein S100-B|Homo sapiens (9606)
>5NBC_1|Chains A, B, C, D|Ferric uptake regulation protein|Francisella tularensis (263)
>2LMV_1|Chain A|Calmodulin-related protein 97A|Drosophila melanogaster (7227)
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 \(w=f(c)\)
5CSJ , Knot 50 95 0.80 36 76 91 
GSHMSELEKAMVALIDVFHQYSGREGDKHKLKKSELKELINNELSHFLEEIKEQEVVDKVMETLDNDGDGECDFQEFMAFVAMVTTACHEFFEHE
5NBC , Knot 72 140 0.84 38 112 135 
MNSKNLDLKEFGFKVTQPRVEILKLFEKNKDKHLSPDDVFSKLKAQGSTTGIATVYRVLNQFESAGIINRLKLDNEQVMYELNQGEHHDHIICVKCNMIQEFYSPGIEALQKQIVESFGAEMIDYSLNIYVKCKSCREKI
2LMV , Knot 67 148 0.75 38 95 136 
MSELTEEQIAEFKDAFVQFDKEGTGKIATRELGTLMRTLGQNPTEAELQDLIAEAENNNNGQLNFTEFCGIMAKQMRETDTEEEMREAFKIFDRDGDGFISPAELRFVMINLGEKVTDEEIDEMIREADFDGDGMINYEEFVWMISQK

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 \(c\).

\(|P_{f(5CSJ_1)}(2) \setminus P_{f(5NBC_1)}(2)|=44\), \(|P_{f(5NBC_1)}(2) \setminus P_{f(5CSJ_1)}(2)|=80\). 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 1:10010010011111101100001001000010000100110001001100100001100110010001010001001111111100100011000
Pair \(Z_2\) Length of longest common subsequence
5CSJ_1,5NBC_1 124 3
5CSJ_1,2LMV_1 115 4
5NBC_1,2LMV_1 135 3

Newick tree

 
[
	5NBC_1:67.06,
	[
		5CSJ_1:57.5,2LMV_1:57.5
	]:9.56
]

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 \((0.85)(0.8)(\frac{235 }{\log_{20} 235}-\frac{95}{\log_{20}95})=45.1\)
Status Protein1 Protein2 d d1/2
Query variables 5CSJ_1 5NBC_1 57 46.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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