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Parikh vectors
5MNR_1 3FCI_1 1UCQ_1 Letter Amino acid
8 15 11 P Proline
36 18 25 G Glycine
18 7 15 I Isoleucine
11 18 7 K Lycine
2 3 9 M Methionine
20 8 3 N Asparagine
0 3 0 C Cysteine
7 12 9 E Glutamic acid
8 13 0 H Histidine
16 20 36 L Leucine
26 15 13 S Serine
28 10 29 A Alanine
24 9 10 D Aspartic acid
14 12 4 Q Glutamine
10 13 13 F Phenylalanine
25 8 18 T Threonine
3 7 8 W Tryptophan
28 7 11 Y Tyrosine
22 17 21 V Valine
10 8 7 R Arginine 

5MNR_1|Chain A[auth E]|Thermolysin|Bacillus thermoproteolyticus (1427)
>3FCI_1|Chain A|Uracil-DNA glycosylase|Homo sapiens (9606)
>1UCQ_1|Chain A|bacteriorhodopsin|Halobacterium salinarum (2242)
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 \(w=f(c)\)
5MNR , Knot 132 316 0.80 38 179 302 
ITGTSTVGVGRGVLGDQKNINTTYSTYYYLQDNTRGNGIFTYDAKYRTTLPGSLWADADNQFFASYDAPAVDAHYYAGVTYDYYKNVHNRLSYDGNNAAIRSSVHYSQGYNNAFWNGSQMVYGDGDGQTFIPLSGGIDVVAHELTHAVTDYTAGLIYQNESGAINEAISDIFGTLVEFYANKNPDWEIGEDVYTPGISGDSLRSMSDPAKYGDPDHYSKRYTGTQDNGGVHINSGIINKAAYLISQGGTHYGVSVVGIGRDKLGKIFYRALTQYLTPTSNFSQLRAAAVQSATDLYGSTSQEVASVKQAFDAVGVK
3FCI , Knot 103 223 0.83 40 157 216 
MEFFGESWKKHLSGEFGKPYFIKLMGFVAEERKHYTVYPPPHQVFTWTQMCDIKDVKVVILGQDPYHGPNQAHGLCFSVQRPVPPPPSLENIYKELSTDIEDFVHPGHGDLSGWAKQGVLLLNAVLTVRAHQANSHKERGWEQFTDAVVSWLNQNSNGLVFLLWGSYAQKKGSAIDRKRHHVLQTAHPSPLSVYRGFFGCRHFSKTNELLQKSGKKPIDWKEL
1UCQ , Knot 113 249 0.83 36 153 240 
QAQITGRPEWIWLALGTALMGLGTLYFLVKGMGVSDPDAKKFYAITTLVPAIAFTMYLSMLLGYGLTMVPFGGEQNPIYWARYADWLFTTPLLLLDLALLVDADQGTILALVGADGIMIGTGLVGALTKVYSYRFVWWAISTAAMLYILYVLFFGFTSKAESMRPEVASTFKVLRNVTVVLWSAYPVVWLIGSEGAGIVPLNIETLLFMVLDVSAKVGFGLILLRSRAIFGEAEAPEPSAGDGAAATSD

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 \(c\).

\(|P_{f(5MNR_1)}(2) \setminus P_{f(3FCI_1)}(2)|=96\), \(|P_{f(3FCI_1)}(2) \setminus P_{f(5MNR_1)}(2)|=74\). 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 1:1010001111011110000100000000010000010111000100000111011101000111000111101000111000000010001000100111000100001000111010011010101001111011101110010011000011110000011100110011101101010001010110010011101001001001100101000000001000011101001110011011001100011011111000110110011000101000100101111001001010000011010011011110
Pair \(Z_2\) Length of longest common subsequence
5MNR_1,3FCI_1 170 3
5MNR_1,1UCQ_1 148 4
3FCI_1,1UCQ_1 178 3

Newick tree

 
[
	3FCI_1:90.95,
	[
		5MNR_1:74,1UCQ_1:74
	]:16.95
]

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 \((0.85)(0.8)(\frac{539 }{\log_{20} 539}-\frac{223}{\log_{20}223})=90.5\)
Status Protein1 Protein2 d d1/2
Query variables 5MNR_1 3FCI_1 116 98.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]