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Parikh vectors
2MGW_1 4ECE_1 4XZP_1 Letter Amino acid
2 13 7 K Lycine
1 4 0 C Cysteine
3 32 20 G Glycine
2 13 12 H Histidine
11 28 11 L Leucine
2 16 11 S Serine
2 19 6 R Arginine
2 19 6 D Aspartic acid
4 15 9 Q Glutamine
1 16 13 P Proline
2 15 12 F Phenylalanine
0 1 2 W Tryptophan
3 23 7 A Alanine
6 11 8 N Asparagine
3 18 6 E Glutamic acid
1 12 5 I Isoleucine
2 16 6 M Methionine
2 17 4 T Threonine
1 13 9 Y Tyrosine
2 23 18 V Valine 

2MGW_1|Chain A|Next to BRCA1 gene 1 protein|Homo sapiens (9606)
>4ECE_1|Chains A, B, C, D, E, F|Purine nucleoside phosphorylase|Homo sapiens (9606)
>4XZP_1|Chain A|Galectin-4|Homo sapiens (9606)
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 \(w=f(c)\)
2MGW , Knot 32 52 0.81 38 46 49 
GPLGSSEDQTAALMAHLFEMGFCDRQLNLRLLKKHNYNILQVVTELLQLNNN
4ECE , Knot 140 324 0.83 40 204 307 
MRGSHHHHHHGMASMTGGQQMGRDLYDDDDKDPTLMENGYTYEDYKNTAEYLLSHTKHRPQVAIICGSGLGGLTDKLTQAQIFDYSEIPNFPRSTVPGHAGRLVFGFLNGRACVMMQGRFHMYEGYPLYKVTFPVRVFHLLGVDTLVVTNAAGGLNPKFEVGDIMLIRDHINLPGFSGQNPLRGPNDERFGDRFPAMSDAYDRTMRQRALSTYKQMGEQRELQEGTYVMVAGPSFETVAECRVLQKLGADAVGMSTVPEVIVARHCGLRVFGFSLITNKVIMDYESLEKANWEEVLAAGKQAAQKLEQFVSILMASIPLPDKAS
4XZP , Knot 79 172 0.78 38 122 165 
MGSSHHHHHHSSGLVPRGSHMAYVPAPGYQPTYNPTLPYYQPIPGGLNVGMSVYIQGVASEHMKRFFVNFVVGQDPGSDVAFHFNPRFDGWDKVVFNTLQGGKWGSEERKRSMPFKKGAAFELVFIVLAEHYKVVVNGNPFYEYGHRLPLQMVTHLQVDGDLQLQSINFIGG

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 \(c\).

\(|P_{f(2MGW_1)}(2) \setminus P_{f(4ECE_1)}(2)|=17\), \(|P_{f(4ECE_1)}(2) \setminus P_{f(2MGW_1)}(2)|=175\). 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 1:1111000000111110110111000010101100000011011001101000
Pair \(Z_2\) Length of longest common subsequence
2MGW_1,4ECE_1 192 3
2MGW_1,4XZP_1 130 3
4ECE_1,4XZP_1 170 7

Newick tree

 
[
	4ECE_1:97.73,
	[
		2MGW_1:65,4XZP_1:65
	]:32.73
]

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 \((0.85)(0.8)(\frac{376 }{\log_{20} 376}-\frac{52}{\log_{20}52})=102.\)
Status Protein1 Protein2 d d1/2
Query variables 2MGW_1 4ECE_1 129 74
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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