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Parikh vectors
6RPR_1 8BNY_1 1FGB_1 Letter Amino acid
6 11 5 Q Glutamine
8 17 6 L Leucine
9 8 10 T Threonine
5 10 4 D Aspartic acid
7 13 5 S Serine
9 6 4 V Valine
10 7 11 A Alanine
9 7 3 R Arginine
7 8 7 N Asparagine
7 20 9 K Lycine
2 6 3 M Methionine
4 0 1 W Tryptophan
5 2 3 P Proline
1 4 3 Y Tyrosine
1 0 2 C Cysteine
7 17 7 E Glutamic acid
10 6 3 G Glycine
2 1 4 H Histidine
3 14 10 I Isoleucine
4 4 3 F Phenylalanine 

6RPR_1|Chain A[auth B]|Prelamin-A/C|Homo sapiens (9606)
>8BNY_1|Chains A, B, C, D|Immunity repressor protein|Bacillus subtilis (1423)
>1FGB_1|Chains A[auth D], B[auth E], C[auth F], D[auth G], E[auth H]|CHOLERA TOXIN B SUBUNIT PENTAMER|Vibrio cholerae (44104)
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 \(w=f(c)\)
6RPR , Knot 60 116 0.82 40 98 112 
FSQHARTSGRVAVEEVDEEGKFVRLRNKSNEDQSMGNWQIKRQNGDDPLLTYRFPPKFTLKAGQVVTIWAAGAGATHSPPTDLVWKAQNTWGCGNSLRTALINSTGEEVAMRKLVR
8BNY , Knot 74 161 0.77 36 105 150 
MGNREQFDLSKYLLEKRKQKGLSQTQVANDTGLSSAYISMLEKGERKRPTQAVIKKLATALSIPNIEMLQIMEYIASMEEDHVKEKDKDTSKQIKLFDLTGLTEASINQIQNQIDQLKSSENNSVIRFFDVTGLSEKDIERVKEEIELLKIRNEYMKLKKD
1FGB , Knot 55 103 0.82 40 89 101 
TPQNITDLCAEYHNTQIHTLNDKIFSYTESLAGKREMAIITFKNGATFQVEVPGSQHIDSQKKAIERMKDTLRIAYLTEAKVEKLCVWNNKTPHAIAAISMAN

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 \(c\).

\(|P_{f(6RPR_1)}(2) \setminus P_{f(8BNY_1)}(2)|=63\), \(|P_{f(8BNY_1)}(2) \setminus P_{f(6RPR_1)}(2)|=70\). 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 1:10001000101110010001011010000000001101010000100111000111010101101101111111100011001110100011010010011100010011100110
Pair \(Z_2\) Length of longest common subsequence
6RPR_1,8BNY_1 133 3
6RPR_1,1FGB_1 131 3
8BNY_1,1FGB_1 122 4

Newick tree

 
[
	6RPR_1:67.58,
	[
		1FGB_1:61,8BNY_1:61
	]:6.58
]

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 \((0.85)(0.8)(\frac{277 }{\log_{20} 277}-\frac{116}{\log_{20}116})=50.6\)
Status Protein1 Protein2 d d1/2
Query variables 6RPR_1 8BNY_1 62 54.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]