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Parikh vectors
8BZM_1 4WEU_1 1JVM_1 Letter Amino acid
0 9 20 L Leucine
0 2 3 M Methionine
0 3 3 F Phenylalanine
0 4 3 I Isoleucine
0 4 4 P Proline
0 11 5 S Serine
9 9 9 T Threonine
0 5 9 R Arginine
0 6 1 D Aspartic acid
4 2 1 C Cysteine
2 12 14 G Glycine
0 4 5 W Tryptophan
0 7 15 V Valine
0 9 2 Q Glutamine
0 3 4 E Glutamic acid
0 6 3 H Histidine
0 6 1 K Lycine
2 14 19 A Alanine
0 3 0 N Asparagine
0 8 4 Y Tyrosine 

8BZM_1|Chains A[auth D], F[auth G]|DNA|Homo sapiens (9606)
>4WEU_1|Chains A[auth D], D[auth E]|Anti-F4+ETEC bacteria VHH variable region|Lama glama (9844)
>1JVM_1|Chains A, B, C, D|Voltage-gated potassium channel|Streptomyces lividans (1916)
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 \(w=f(c)\)
8BZM , Knot 8 17 0.44 8 9 13 
TTACTTCCTGTTTACGT
4WEU , Knot 61 127 0.77 40 101 121 
QVQLQESGGGLVQAGGSLRLSCAASGLTFDTYAMGWFRQAPGKKREYVAAISWTGISTYYADIAKGRFTISRDNAKNTLYLQMDSLKPEDTAVYYCAAQKSLNVPAPWDYWGQGTQVTVSSHHHHHH
1JVM , Knot 59 125 0.76 38 84 121 
MAPMLSGLLARLVKLLLGRHGSALHWRAAGAATVLLVIVLLAGSYLAVLAERGAPGAQLITYPRALWWSVETATTVGYGDLYPVTLWGRCVAVVVMVAGITSFGLVTAALATWFVGREQERRGHF

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 \(c\).

\(|P_{f(8BZM_1)}(2) \setminus P_{f(4WEU_1)}(2)|=6\), \(|P_{f(4WEU_1)}(2) \setminus P_{f(8BZM_1)}(2)|=98\). 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 1:00100000010001010
Pair \(Z_2\) Length of longest common subsequence
8BZM_1,4WEU_1 104 2
8BZM_1,1JVM_1 89 2
4WEU_1,1JVM_1 131 3

Newick tree

 
[
	4WEU_1:63.26,
	[
		8BZM_1:44.5,1JVM_1:44.5
	]:18.76
]

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 \((0.85)(0.8)(\frac{144 }{\log_{20} 144}-\frac{17}{\log_{20}17})=46.8\)
Status Protein1 Protein2 d d1/2
Query variables 8BZM_1 4WEU_1 58 32
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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