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Parikh vectors
1IPD_1 5AEF_1 2WZQ_1 Letter Amino acid
6 1 15 M Methionine
14 2 21 F Phenylalanine
25 0 38 P Proline
35 5 41 V Valine
5 1 17 N Asparagine
15 1 39 D Aspartic acid
6 0 11 H Histidine
11 3 41 I Isoleucine
13 0 46 T Threonine
43 3 42 A Alanine
38 2 41 L Leucine
16 2 40 K Lycine
17 1 30 S Serine
5 0 15 Y Tyrosine
0 0 6 C Cysteine
2 1 16 Q Glutamine
36 5 53 G Glycine
3 0 14 W Tryptophan
22 0 48 R Arginine
33 1 45 E Glutamic acid

1IPD_1|Chain A|3-ISOPROPYLMALATE DEHYDROGENASE|Thermus thermophilus (274)
>5AEF_1|Chains A, B|AMYLOID BETA A4 PROTEIN|HOMO SAPIENS (9606)
>2WZQ_1|Chain A|NS3 PROTEASE-HELICASE|DENGUE VIRUS (12637)
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 \(w=f(c)\)
1IPD , Knot 140 345 0.79 38 164 317
MKVAVLPGDGIGPEVTEAALKVLRALDEAEGLGLAYEVFPFGGAAIDAFGEPFPEPTRKGVEEAEAVLLGSVGGPKWDGLPRKISPETGLLSLRKSQDLFANLRPAKVFPGLERLSPLKEEIARGVDVLIVRELTGGIYFGEPRGMSEAEAWNTERYSKPEVERVARVAFEAARKRRKHVVSVDKANVLEVGEFWRKTVEEVGRGYPDVALEHQYVDAMAMHLVRSPARFDVVVTGNIFGDILSDLASVLPGSLGLLPSASLGRGTPVFEPVHGSAPDIAGKGIANPTAAILSAAMMLEHAFGLVELARKVEDAVAKALLETPPPDLGGSAGTEAFTATVLRHLA
5AEF , Knot 18 28 0.71 26 26 26
QKLVFFAEDVGSNKGAIIGLMVGGVVIA
2WZQ , Knot 247 619 0.85 40 273 568
SGALWDVPSPAATQKATLSEGVYRIMQRGLFGKTQVGVGIHMEGVFHTMWHVTRGSVICHETGRLEPSWADVRNDMISYGGGWRLGDKWDKEEDVQVLAIEPGKNPKHVQTKPGLFKTLTGEIGAVTLDFKPGTSGSPIINKKGKVIGLYGNGVVTKSGDYVSAITQAERIGEGPDYEVDEDIFRKKRLTIMDLHPGAGKTKRILPSIVREALKRRLRTLILAPTRVVAAEMEEALRGLPIRYQTPAVKSDHTGREIVDLMCHATFTTRLLSSTRVPNYNLIVMDEAHFTDPCSVAARGYISTRVEMGEAAAIFMTATPPGSTDPFPQSNSPIEDIEREIPERSWNTGFDWITDYQGKTVWFVPSIKAGNDIANCLRKSGKRVIQLSRKTFDTEYPKTKLTDWDFVVTTDISEMGANFRAGRVIDPRRCLKPVILTDGPERVILAGPIPVTPASAAQRRGRIGRNPAQEDDQYVFSGDPLKNDEDHAHWTEAKMLLDNIYTPEGIIPTLFGPEREKTQAIDGEFRLRGEQRKTFVELMRRGDLPVWLSYKVASAGISYKDREWCFTGERNNQILEENMEVEIWTREGEKKKLRPKWLDARVYADPMALKDFKEFASGRK

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 \(c\).

\(|P_{f(1IPD_1)}(2) \setminus P_{f(5AEF_1)}(2)|=147\), \(|P_{f(5AEF_1)}(2) \setminus P_{f(1IPD_1)}(2)|=9\). 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 1:101111110111101001110110110010111110011111111101110111010001100101111101111010111001010011101000001110101101111100101100011011011110010111011010110010110000000101001101110110000001101001011011011000100110101011100001011110110011010111010111011001101111011111010110101110110101101110111010111101111100111110110010011101110011101110110011010110011
Pair \(Z_2\) Length of longest common subsequence
1IPD_1,5AEF_1 156 3
1IPD_1,2WZQ_1 163 4
5AEF_1,2WZQ_1 253 3

Newick tree

 
[
	2WZQ_1:11.31,
	[
		1IPD_1:78,5AEF_1:78
	]:36.31
]

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 \((0.85)(0.8)(\frac{373 }{\log_{20} 373}-\frac{28}{\log_{20}28})=111.\)
Status Protein1 Protein2 d d1/2
Query variables 1IPD_1 5AEF_1 132 71.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]