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Parikh vectors
4JCI_1 4GDN_1 8VBR_1 Letter Amino acid
31 15 20 A Alanine
21 4 6 R Arginine
35 18 16 G Glycine
19 14 2 H Histidine
16 22 5 I Isoleucine
26 18 7 D Aspartic acid
6 18 10 Q Glutamine
27 32 14 L Leucine
18 27 20 T Threonine
10 0 5 C Cysteine
17 17 10 E Glutamic acid
7 7 0 M Methionine
11 11 5 F Phenylalanine
15 14 13 P Proline
22 26 39 S Serine
6 4 3 W Tryptophan
9 14 8 Y Tyrosine
9 33 8 N Asparagine
7 30 8 K Lycine
21 18 17 V Valine 

4JCI_1|Chains A, B|Proline racemase|Chromohalobacter salexigens (290398)
>4GDN_1|Chains A, B, C, D|Protein flp|Staphylococcus aureus (158879)
>8VBR_1|Chains A, C[auth L]|Bovine Fab ElsE11 light chain|Bos taurus (9913)
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 \(w=f(c)\)
4JCI , Knot 144 333 0.83 40 206 322 
MHHHHHHSSGVDLGTENLYFQSMKRVHVIDSHTAGEPTRLVMEGMPALSGRTIAEKCDDFRDNHDAWRRAIMLEPRGHDVLVGALYCAPESSDASCGVIFFNNSGYLGMCGHGTIGLVASLHHLGQLTPGCHKIDTPAGPVSATLHDDGAVTVRNVLSYRHRRRVPVEVPGYGTVHGDIAWGGNWFFLVSDHDMTLELDNVEALTDYTWAIRQALEAQSITGENGGVIDHIELFCDDREADSRNFVLCPGKAYDRSPCGTGTSAKLACLAADGKLAPGQVWTQASICGSRFEAFYEREGDGIRPSIKGRAYLSADATLLIDERDPFAWGIASP
4GDN , Knot 144 342 0.82 38 197 328 
MNTKTRLTNDSQQQIDKIIEHDLQKGHIPGASILIVKNGKVFLNKGYGYQDVDKKVKASPTTKYEIASNTKAFTGLAILKLAQEGRLNLNDDVSKHVPHFKMNYNGQNETITIKQLLAQTSGIPSDITSEDAVTNKNNRLNDVTRAIMGDELHHKPGEEFEYSNMNYDLLGLIIQNVTKQSYTKYITNSWLKPLHMTHTSFKQTNNKSKHDAIGYELQGSTPVVSKPEFNLWDTPSAYMMTSTEDLEHWIKFQLNPPDKYKSLVQQSHKNLSSTIGEPNANAYASGWFTNNDEHLVFHSGTLDNFSSFILLNPKQNYGIVVLANLNSEYVPKLVEHLNTQIV
8VBR , Knot 96 216 0.79 38 136 202 
QAVLNQPSSVSGSLGQRVSITCSGSSSNVGNGYVSWYQLIPGSAPRTLIYGDTSRASGVPDRFSGSRSGNTATLTISSLQAEDEADYFCASAEDSSSNAVFGSGTTLTVLGQPKAAPSVTLFPPSSEELQANKATLVCLISDFYPGAVTVAWKADSSPVKAGVETTTPSKQSNNKYAASSYLSLTPEQWKSHRSYSCQVTHEGSTVEKTVAPTECS

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 \(c\).

\(|P_{f(4JCI_1)}(2) \setminus P_{f(4GDN_1)}(2)|=99\), \(|P_{f(4GDN_1)}(2) \setminus P_{f(4JCI_1)}(2)|=90\). 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 1:100000000110110001010010010110000110100111011111010011000001000001100111101010011111100110000100111110001011101010111110100110101100010011111010100011101001100000001110111010101011111011111000010101001011000011100110100101001111001011000001000011101101000010101001011011101011110110010101001011000010110101010101010101110000111111101
Pair \(Z_2\) Length of longest common subsequence
4JCI_1,4GDN_1 189 4
4JCI_1,8VBR_1 170 5
4GDN_1,8VBR_1 169 3

Newick tree

 
[
	4JCI_1:91.59,
	[
		8VBR_1:84.5,4GDN_1:84.5
	]:7.09
]

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 \((0.85)(0.8)(\frac{675 }{\log_{20} 675}-\frac{333}{\log_{20}333})=94.2\)
Status Protein1 Protein2 d d1/2
Query variables 4JCI_1 4GDN_1 123 121.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]