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Parikh vectors
2RLW_1 6PSD_1 3OQB_1 Letter Amino acid
3 3 25 R Arginine
0 9 31 L Leucine
1 2 13 P Proline
5 5 32 A Alanine
0 6 17 Q Glutamine
2 2 8 H Histidine
3 3 14 S Serine
0 4 19 T Threonine
1 0 11 W Tryptophan
2 1 11 Y Tyrosine
1 6 29 D Aspartic acid
1 4 23 K Lycine
2 2 16 N Asparagine
0 7 20 E Glutamic acid
4 7 34 G Glycine
2 1 21 I Isoleucine
0 2 8 M Methionine
2 9 18 F Phenylalanine
5 2 28 V Valine
0 1 5 C Cysteine 

2RLW_1|Chain A|PlnF|Lactobacillus plantarum
>6PSD_1|Chains A, B[auth C], C[auth E], D[auth G], E[auth I], F[auth K], G[auth M], H[auth O]|EF-hand calcium-binding domain-containing protein 4B|Homo sapiens (9606)
>3OQB_1|Chains A, B, C, D, E, F, G, H|Oxidoreductase|Bradyrhizobium japonicum (224911)
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 \(w=f(c)\)
2RLW , Knot 21 34 0.72 28 28 31 
VFHAYSARGVRNNYKSAVGPADWVISAVRGFIHG
6PSD , Knot 42 76 0.79 38 64 73 
GGQLVMLRKAQEFFQTCDAEGKGFIARKDMQRLHKELPLSLEELEDVFDALDADGNGYLTPQEFTTGFSHFFFSQN
3OQB , Knot 165 383 0.85 40 231 367 
SLTTQRLGLIMNGVTGRMGLNQHLIRSIVAIRDQGGVRLKNGDRIMPDPILVGRSAEKVEALAKRFNIARWTTDLDAALADKNDTMFFDAATTQARPGLLTQAINAGKHVYCEKPIATNFEEALEVVKLANSKGVKHGTVQDKLFLPGLKKIAFLRDSGFFGRILSVRGEFGYWVFEGGWQEAQRPSWNYRDEDGGGIILDMVCHWRYVLDNLFGNVQSVVCIGNTDIPERFDEQGKKYKATADDSAYATFQLEGGVIAHINMSWVTRVYRDDLVTFQVDGTHGSAVAGLSDCMIQARQATPRPVWNPDEKRLHDFYGDWQKLPDNVSYDNGFKEQWEMFIRHVYEDAPYKFTLLEGAKGVQLAECALKSWKERRWIDVAPIK

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 \(c\).

\(|P_{f(2RLW_1)}(2) \setminus P_{f(6PSD_1)}(2)|=23\), \(|P_{f(6PSD_1)}(2) \setminus P_{f(2RLW_1)}(2)|=59\). 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 1:1101001011000000111110111011011101
Pair \(Z_2\) Length of longest common subsequence
2RLW_1,6PSD_1 82 3
2RLW_1,3OQB_1 221 3
6PSD_1,3OQB_1 205 3

Newick tree

 
[
	3OQB_1:12.76,
	[
		2RLW_1:41,6PSD_1:41
	]:79.76
]

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 \((0.85)(0.8)(\frac{110 }{\log_{20} 110}-\frac{34}{\log_{20}34})=28.0\)
Status Protein1 Protein2 d d1/2
Query variables 2RLW_1 6PSD_1 35 24.5
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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