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Parikh vectors
5CIU_1 6TQJ_1 4PGR_1 Letter Amino acid
5 2 9 M Methionine
5 2 6 R Arginine
8 4 4 E Glutamic acid
17 11 11 G Glycine
3 0 4 H Histidine
11 12 30 L Leucine
12 17 22 A Alanine
9 0 6 D Aspartic acid
3 6 23 I Isoleucine
9 3 21 F Phenylalanine
4 2 6 N Asparagine
0 0 0 C Cysteine
14 3 17 S Serine
5 3 10 T Threonine
8 7 22 V Valine
6 3 5 Q Glutamine
14 1 6 K Lycine
7 4 5 P Proline
7 0 2 W Tryptophan
3 1 8 Y Tyrosine 

5CIU_1|Chains A, B|DNA (cytosine-5)-methyltransferase 3B|Homo sapiens (9606)
>6TQJ_1|Chains A, B, C, D, E, F, G, H, I, J, K, L, M, N|ATP synthase subunit c, chloroplastic|Spinacia oleracea (3562)
>4PGR_1|Chain A|Uncharacterized protein YetJ|Bacillus subtilis (224308)
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 \(w=f(c)\)
5CIU , Knot 71 150 0.79 38 109 143 
EADSGDGDSSEYQDGKEFGIGDLVWGKIKGFSWWPAMVVSWKATSKRQAMSGMRWVQWFGDGKFSEVSADKLVALGLFSQHFNLATFNKLVSYRKAMYHALEKARVRAGKTFPSSPGDSLEDQLKPMLEWAHGGFKPTGIEGLKPNNTQP
6TQJ , Knot 38 81 0.68 32 57 76 
MNPLIAAASVIAAGLAVGLASIGPGVGQGTAAGQAVEGIARQPEAEGKIRGTLLLSLAFMEALTIYGLVVALALLFANPFV
4PGR , Knot 93 217 0.76 38 135 206 
SNAMQATVHESKQSIMQRILTVFVFTLLIATVGLFIGQFVPVALMLPLSILEVAMIILAFWMRRRKAVGYAFVYTFAFVSGITLFPIVSHYASIAGAYVVLEAFGSTFVIFAVLGTIGAKMKKDLSFLWSFLLVAVLALAVVGIFNIFSPLNSAAMMAYSVIGTIVFSLYILYDLNQIKHRHITEDLIPVMALSLYLDFINLFINLLRFFGILSSDD

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 \(c\).

\(|P_{f(5CIU_1)}(2) \setminus P_{f(6TQJ_1)}(2)|=89\), \(|P_{f(6TQJ_1)}(2) \setminus P_{f(5CIU_1)}(2)|=37\). 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 1:010010100000001001111011110101101111111010100000110110110111010100101001111111000101101001100001100110010101100110011001000101110110111010110110100001
Pair \(Z_2\) Length of longest common subsequence
5CIU_1,6TQJ_1 126 3
5CIU_1,4PGR_1 170 3
6TQJ_1,4PGR_1 126 3

Newick tree

 
[
	4PGR_1:78.35,
	[
		5CIU_1:63,6TQJ_1:63
	]:15.35
]

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 \((0.85)(0.8)(\frac{231 }{\log_{20} 231}-\frac{81}{\log_{20}81})=48.9\)
Status Protein1 Protein2 d d1/2
Query variables 5CIU_1 6TQJ_1 61 45.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]