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Parikh vectors
4DGU_1 1WUQ_1 1USC_1 Letter Amino acid
9 17 11 R Arginine
19 16 14 A Alanine
14 6 6 D Aspartic acid
0 2 1 C Cysteine
7 10 11 F Phenylalanine
6 3 9 Y Tyrosine
1 2 3 W Tryptophan
16 24 16 V Valine
25 23 22 G Glycine
17 7 2 I Isoleucine
2 10 1 M Methionine
9 8 15 P Proline
19 9 8 S Serine
23 9 5 T Threonine
15 1 2 N Asparagine
7 12 3 Q Glutamine
21 25 13 E Glutamic acid
2 6 9 H Histidine
20 21 20 L Leucine
15 9 7 K Lycine 

4DGU_1|Chains A, B|putative cell adhesion protein|Bacteroides thetaiotaomicron (226186)
>1WUQ_1|Chains A, B, C, D, E|GTP cyclohydrolase I|Thermus thermophilus (274)
>1USC_1|Chains A, B|PUTATIVE STYRENE MONOOXYGENASE SMALL COMPONENT|THERMUS THERMOPHILUS (300852)
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 \(w=f(c)\)
4DGU , Knot 108 247 0.80 38 150 236 
GDELQETSKEATNFSISIDDALSDPLTRTSNDLFPARNSITTGEVISMAASGQDYTPFIVGKDSRAWNEIGTATGTVTFYAHYPALTDEAATRSGGNKRYLKGGQEHLFGTAEAAPGSQNVSLKFKRMTVPVIILDENDRPYEGEAKVELSLKNEGTQDLLNGTIEINENALSENIEVKKVSEGVTTNVLPQKINAGEEIGTITVGGVTQKISAVEDLDLKAGSTLSVRLSKKFGGGIIDGNVPLYR
1WUQ , Knot 101 220 0.82 40 147 211 
MSPGPQSGGQERGSMERKMVELEDTGLTFATEVDLERLQALAAEWLQVIGEDPGREGLLKTPERVAKAWAFLTRGYRQRLEEVVGGAVFPAEGSEMVVVKGVEFYSMCEHHLLPFFGKVHIGYIPDGKILGLSKFARIVDMFARRLQVQERLAVQIAEAIQEVLEPQGVGVVVEGVHLCMMMRGVEKQHSRTVTSAMLGVFRENQKTREEFLSHLRDGTA
1USC , Knot 84 178 0.81 40 121 171 
MRSYRAQGPLPGFYHYYPGVPAVVGVRVEERVNFCPAVWNTGLSADPPLFGVSISPKRFTHGLLLKARRFSASFHPFGQKDLVHWLGSHSGREVDKGQAPHFLGHTGVPILEGAYAAYELELLEVHTFGDHDLFVGRVVAVWEEEGLLDEKGRPKPGLALLYYGKGLYGRPAEETFAP

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 \(c\).

\(|P_{f(4DGU_1)}(2) \setminus P_{f(1WUQ_1)}(2)|=78\), \(|P_{f(1WUQ_1)}(2) \setminus P_{f(4DGU_1)}(2)|=75\). 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 1:1001000000100101010011001100000011110001001011011101000011111000011001101010101010011100011000110000101100011101011110001010100101111110000010010101010100010001101010100011000101001001100011100101100110101111000101100101011001010100011111101011100
Pair \(Z_2\) Length of longest common subsequence
4DGU_1,1WUQ_1 153 4
4DGU_1,1USC_1 149 3
1WUQ_1,1USC_1 148 4

Newick tree

 
[
	4DGU_1:76.00,
	[
		1USC_1:74,1WUQ_1:74
	]:2.00
]

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 \((0.85)(0.8)(\frac{467 }{\log_{20} 467}-\frac{220}{\log_{20}220})=71.6\)
Status Protein1 Protein2 d d1/2
Query variables 4DGU_1 1WUQ_1 90 85
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]