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Parikh vectors
1PVB_1 3LFA_1 2BYC_1 Letter Amino acid
10 13 5 F Phenylalanine
0 5 1 W Tryptophan
1 19 18 R Arginine
2 15 3 Q Glutamine
6 23 9 E Glutamic acid
1 12 3 H Histidine
4 22 10 I Isoleucine
8 20 10 S Serine
5 22 9 V Valine
1 17 7 P Proline
14 27 11 D Aspartic acid
1 4 1 C Cysteine
12 19 1 K Lycine
2 10 3 M Methionine
2 19 5 T Threonine
1 15 2 Y Tyrosine
20 25 13 A Alanine
1 14 2 N Asparagine
7 16 6 G Glycine
9 43 18 L Leucine 

1PVB_1|Chain A|PARVALBUMIN|Esox lucius (8010)
>3LFA_1|Chain A|Mitogen-activated protein kinase 14|Homo sapiens (9606)
>2BYC_1|Chains A, B|BLUE-LIGHT RECEPTOR OF THE BLUF-FAMILY|RHODOBACTER SPHAEROIDES (1063)
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 \(w=f(c)\)
1PVB , Knot 54 108 0.78 40 77 104 
XSFAGLKDADVAAALAACSAADSFKHKEFFAKVGLASKSLDDVKKAFYVIDQDKSGFIEEDELKLFLQNFSPSARALTDAETKAFLADGDKDGDGMIGVDEFAAMIKA
3LFA , Knot 158 360 0.86 40 223 346 
GSQERPTFYRQELNKTIWEVPERYQNLSPVGSGAYGSVCAAFDTKTGLRVAVKKLSRPFQSIIHAKRTYRELRLLKHMKHENVIGLLDVFTPARSLEEFNDVYLVTHLMGADLNNIVKCQKLTDDHVQFLIYQILRGLKYIHSADIIHRDLKPSNLAVNEDCELKILDFGLARHTDDEMTGYVATRWYRAPEIMLNWMHYNQTVDIWSVGCIMAELLTGRTLFPGTDHIDQLKLILRLVGTPGAELLKKISSESARNYIQSLTQMPKMNFANVFIGANPLAVDLLEKMLVLDSDKRITAAQALAHAYFAQYHDPDDEPVADPYDQSFESRDLLIDEWKSLTYDEVISFVPPPLDQEEMES
2BYC , Knot 65 137 0.77 40 97 130 
FMDELVSLTYRSRVRLADPVADIVQIMRASRVRNLRLGITGILLYNGVHFVQTIEGPRSACDELFRLISADPRHQEILAFDLEPITARRFPDWSMRIVSRKELRALAPDLERLDLSGPEDVAELHRTIAASLSRGDA

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 \(c\).

\(|P_{f(1PVB_1)}(2) \setminus P_{f(3LFA_1)}(2)|=27\), \(|P_{f(3LFA_1)}(2) \setminus P_{f(1PVB_1)}(2)|=173\). 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 1:001111001011111110011001000011101111000100100110110000011100001011100101010110010001111010001011111001111101
Pair \(Z_2\) Length of longest common subsequence
1PVB_1,3LFA_1 200 3
1PVB_1,2BYC_1 122 3
3LFA_1,2BYC_1 190 4

Newick tree

 
[
	3LFA_1:10.97,
	[
		1PVB_1:61,2BYC_1:61
	]:45.97
]

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 \((0.85)(0.8)(\frac{468 }{\log_{20} 468}-\frac{108}{\log_{20}108})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 1PVB_1 3LFA_1 140 88.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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