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Parikh vectors
4PIB_1 1YXD_1 5TGC_1 Letter Amino acid
22 31 31 A Alanine
11 13 30 N Asparagine
10 18 30 I Isoleucine
9 13 34 K Lycine
3 9 17 M Methionine
7 5 19 Y Tyrosine
15 17 25 D Aspartic acid
8 8 28 Q Glutamine
10 30 46 L Leucine
7 7 18 F Phenylalanine
3 1 7 W Tryptophan
2 5 4 C Cysteine
5 14 32 E Glutamic acid
5 9 10 H Histidine
18 15 37 S Serine
15 23 30 T Threonine
4 13 16 R Arginine
12 23 27 G Glycine
10 13 22 P Proline
17 25 27 V Valine 

4PIB_1|Chains A, B, C, D, E, F, G, H, I, J|inclusion body protein|Burkholderia thailandensis (271848)
>1YXD_1|Chains A, B|dihydrodipicolinate synthase|Escherichia coli (562)
>5TGC_1|Chains A, D|Actin-related protein 7|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
4PIB , Knot 92 193 0.83 40 140 190 
SNAMVTVNSRVDSARASSHITINVMTVIDVAAIIDELKAQEKKLGQTPDTATSIGHKYIYMSSDDPRGWSVGNDPGNITLNAHVGDTLSFFCASTSDNSEYAAFIYRLTGGDPHLDPSHVEVIKLQNAAQPTPSNGYPFTTAPVAFSSCDAKVAQQGQAKNFYVWAALFTLDDSGEKQVLAGYVKWDPTVNVA
1YXD , Knot 132 292 0.85 40 177 282 
MFTGSIVAIVTPMDEKGNVCRASLKKLIDYHVASGTSAIVSVGTTGESATLNHDEHADVVMMTLDLADGRIPVIAGTGANATAEAISLTQRFNDSGIVGCLTVTPYYNRPSQEGLYQHFKAIAEHTDLPQILYNVPSRTGCDLLPETVGRLAKVKNIIGIKEATGNLTRVNQIKELVSDDFVLLSGDDASALDFMQLGGHGVISVTANVAARDMAQMCKLAAEGHFAEARVINQRLMPLHNKLFVEPNPIPVKWACKELGLVATDTLRLPMTPITDSGRETVRAALKHAGLL
5TGC , Knot 203 490 0.85 40 255 465 
MMTLNRKCVVIHNGSHRTVAGFSNVELPQCIIPSSYIKRTDEGGEAEFIFGTYNMIDAAAEKRNGDEVYTLVDSQGLPYNWDALEMQWRYLYDTQLKVSPEELPLVITMPATNGKPDMAILERYYELAFDKLNVPVFQIVIEPLAIALSMGKSSAFVIDIGASGCNVTPIIDGIVVKNAVVRSKFGGDFLDFQVHERLAPLIKEENDMENMADEQKRSTDVWYEASTWIQQFKSTMLQVSEKDLFELERYYKEQADIYAKQQEQLKQMDQQLQYTALTGSPNNPLVQKKNFLFKPLNKTLTLDLKECYQFAEYLFKPQLISDKFSPEDGLGPLMAKSVKKAGASINSMKANTSTNPNGLGTSHINTNVGDNNSTASSSNISPEQVYSLLLTNVIITGSTSLIEGMEQRIIKELSIRFPQYKLTTFANQVMMDRKIQGWLGALTMANLPSWSLGKWYSKEDYETLKRDRKQSQATNATNLVPRGSHHHHHH

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 \(c\).

\(|P_{f(4PIB_1)}(2) \setminus P_{f(1YXD_1)}(2)|=61\), \(|P_{f(1YXD_1)}(2) \setminus P_{f(4PIB_1)}(2)|=98\). 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 1:0011101000100101000101011011011111001010000110010010011000101000010110110011010101011001011010000000011110010110101010010110100110101001011001111100001011001010010111111010001000111101010101011
Pair \(Z_2\) Length of longest common subsequence
4PIB_1,1YXD_1 159 3
4PIB_1,5TGC_1 199 3
1YXD_1,5TGC_1 164 4

Newick tree

 
[
	5TGC_1:94.74,
	[
		4PIB_1:79.5,1YXD_1:79.5
	]:15.24
]

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 \((0.85)(0.8)(\frac{485 }{\log_{20} 485}-\frac{193}{\log_{20}193})=85.0\)
Status Protein1 Protein2 d d1/2
Query variables 4PIB_1 1YXD_1 108 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]