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Parikh vectors
3MLB_1 6BAB_1 1VIJ_1 Letter Amino acid
13 14 6 P Proline
11 15 8 T Threonine
2 5 2 W Tryptophan
19 21 4 E Glutamic acid
12 18 13 G Glycine
7 12 2 F Phenylalanine
4 6 2 M Methionine
9 17 4 R Arginine
6 13 6 Q Glutamine
12 22 6 K Lycine
2 8 2 C Cysteine
6 12 1 H Histidine
9 15 0 S Serine
12 17 6 V Valine
8 24 3 A Alanine
6 8 4 N Asparagine
5 17 4 D Aspartic acid
13 18 13 I Isoleucine
22 29 12 L Leucine
13 11 1 Y Tyrosine 

3MLB_1|Chains A, B|nicotinate-nucleotide adenylyltransferase|Bacillus anthracis (1392)
>6BAB_1|Chains A, B, C, D|Calcium/calmodulin-dependent protein kinase type II subunit delta|Mus musculus (10090)
>1VIJ_1|Chains A, B|HIV-1 PROTEASE|Human immunodeficiency virus 1 (11676)
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 \(w=f(c)\)
3MLB , Knot 93 191 0.85 40 145 185 
GAMRKIGIIGGTFDPPHYGHLLIANEVYHALNLEEVWFLPNQIPPHKQGRNITSVESRLQMLELATEAEEHFSICLEELSRKGPSYTYDTMLQLTKKYPDVQFHFIIGGDMVEYLPKWYNIEALLDLVTFVGVARPGYKLRTPYPITTVEIPEFAVSSSLLRERYKEKKTCKYLLPEKVQVYIERNGLYES
6BAB , Knot 138 302 0.87 40 202 293 
GHMSTTTCTRFTDEYQLFEELGKGAFSVVRRCMKIPTGQEYAAKIINTKKLSARDHQKLEREARICRLLKHPNIVRLHDSISEEGFHYLVFDLVTGGELFEDIVAREYYSEADASHCIQQILESVNHCHLNGIVHRDLKPENLLLASKSKGAAVKLADFGLAIEVQGDQQAWFGFAGTPGYLSPEVLRKDPYGKPVDMWACGVILYILLVGYPPFWDEDQHRLYQQIKAGAYDFPSPEWDTVTPEAKDLINKMLTINPAKRITASEALKHPWICQRSTVASMMHRQETVDCLKKFNARRKLK
1VIJ , Knot 52 99 0.80 38 80 94 
PQITLWQRPLVTIKIGGQLKEALLDTGADDTVLEEMNLPGRWKPKMIGGIGGFIKVRQYDQILIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF

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 \(c\).

\(|P_{f(3MLB_1)}(2) \setminus P_{f(6BAB_1)}(2)|=63\), \(|P_{f(6BAB_1)}(2) \setminus P_{f(3MLB_1)}(2)|=120\). 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 1:11100111111010110010111100100110100111110011100010010010001011011001000101010010001100000011010000101010111110110011010010111011011111011001001011001011011100011000000000000111001010100011000
Pair \(Z_2\) Length of longest common subsequence
3MLB_1,6BAB_1 183 4
3MLB_1,1VIJ_1 139 5
6BAB_1,1VIJ_1 178 3

Newick tree

 
[
	6BAB_1:96.18,
	[
		3MLB_1:69.5,1VIJ_1:69.5
	]:26.68
]

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 \((0.85)(0.8)(\frac{493 }{\log_{20} 493}-\frac{191}{\log_{20}191})=87.8\)
Status Protein1 Protein2 d d1/2
Query variables 3MLB_1 6BAB_1 113 90.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]