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Parikh vectors
5VTE_1 3BDF_1 3CDV_1 Letter Amino acid
4 16 10 I Isoleucine
0 3 3 W Tryptophan
0 16 1 H Histidine
0 45 11 G Glycine
5 28 13 K Lycine
4 8 5 F Phenylalanine
0 21 3 P Proline
0 39 11 T Threonine
0 14 12 R Arginine
0 4 2 C Cysteine
5 25 8 E Glutamic acid
0 8 6 M Methionine
0 22 6 S Serine
0 11 6 Y Tyrosine
0 24 9 V Valine
0 28 10 D Aspartic acid
0 20 12 N Asparagine
2 22 5 Q Glutamine
1 40 16 L Leucine
8 64 15 A Alanine 

5VTE_1|Chain A|de novo peptide 1|synthetic construct (32630)
>3BDF_1|Chains A, B|Alkaline phosphatase|Escherichia coli (562)
>3CDV_1|Chain A|Lysozyme|Bacteriophage T4
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 \(w=f(c)\)
5VTE , Knot 12 31 0.44 16 15 17 
XELEAIAQKFEAIAKKFEAIAFKFEAIAQKX
3BDF , Knot 186 458 0.83 40 219 425 
RTPEMPVLENRAAQGDITAPGGARRLTGDQTAALRDSLSDKPAKNIILLIGDGMGDSEITAARNYAEGAGGFFKGIDALPLTGQYTHYALNKKTGKPDYVTDSAASATAWSTGVKTYNGALGVDIHEKDHPTILEMAKAAGLATGNVSTAELQDAVPAALVAHVTSRKCYGPSATSEKCPGNALEKGGKGSITEQLLNARADVTLGGGAKTFAETATAGEWQGKTLREQAQARGYQLVSDAASLNSVTEANQQKPLLGLFADGNMPVRWLGPKATYHGNIDKPAVTCTPNPQRNDSVPTLAQMTDKAIELLSKNEKGFFLQVEGASIDKQDHAANPCGQIGETVDLDEAVQRALEFAKKEGNTLVIVTADHAHASQIVAPDTKAPGLTQALNTKDGAVMVMSYGNSEEDSQEHTGSQLRIAAYGPHAANVVGLTDQTDLFYTMKAALGLKLEHHHHHH
3CDV , Knot 78 164 0.80 40 120 157 
MNIFEMLRIDEGLRLKIYKDTEGYYTIGIGHLLTKSPSLNAAKSELDKAIGRNCNGVITKDEAEKLFNQDVDAAVRGILRNAKLKPVYDSLDAVRMCALINMVFQMGETGVAGFTNSLRMLQQKRWDEAAVNLAKSRWYNQTPNRAKRVITTFRTGTWDAYKNL

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 \(c\).

\(|P_{f(5VTE_1)}(2) \setminus P_{f(3BDF_1)}(2)|=5\), \(|P_{f(3BDF_1)}(2) \setminus P_{f(5VTE_1)}(2)|=209\). 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 1:0010111001011100101111010111000
Pair \(Z_2\) Length of longest common subsequence
5VTE_1,3BDF_1 214 3
5VTE_1,3CDV_1 123 2
3BDF_1,3CDV_1 171 4

Newick tree

 
[
	3BDF_1:10.04,
	[
		5VTE_1:61.5,3CDV_1:61.5
	]:44.54
]

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 \((0.85)(0.8)(\frac{489 }{\log_{20} 489}-\frac{31}{\log_{20}31})=142.\)
Status Protein1 Protein2 d d1/2
Query variables 5VTE_1 3BDF_1 178 92.5
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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