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Parikh vectors
1JBD_1 4FNX_1 6BSX_1 Letter Amino acid
5 17 15 V Valine
3 18 5 R Arginine
2 12 6 N Asparagine
4 24 14 E Glutamic acid
1 14 7 F Phenylalanine
8 26 4 P Proline
6 22 16 S Serine
2 16 8 D Aspartic acid
6 13 13 K Lycine
1 4 1 W Tryptophan
10 10 0 C Cysteine
1 13 9 Q Glutamine
2 6 9 H Histidine
2 22 17 I Isoleucine
2 30 14 L Leucine
1 12 4 M Methionine
5 17 9 A Alanine
4 24 10 G Glycine
7 15 9 T Threonine
2 12 8 Y Tyrosine 

1JBD_1|Chain A|LONG NEUROTOXIN 1|Bungarus multicinctus (8616)
>4FNX_1|Chain A|ALK tyrosine kinase receptor|Homo sapiens (9606)
>6BSX_1|Chains A, B, C, D|GTP-binding protein Rheb|Homo sapiens (9606)
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 \(w=f(c)\)
1JBD , Knot 43 74 0.83 40 66 70 
IVCHTTATSPISAVTCPPGENLCYRKMWCDAFCSSRGKVVELGCAATCPSKKPYEEVTCCSTDKCNPHPKQRPG
4FNX , Knot 139 327 0.82 40 206 316 
RTSTIMTDYNPNYSFAGKTSSISDLKEVPRKNITLIRGLGHGAFGEVYEGQVSGMPNDPSPLQVAVKTLPEVCSEQDELDFLMEALIISKFNHQNIVRCIGVSLQSLPRFILLELMAGGDLKSFLRETRPRPSQPSSLAMLDLLHVARDIACGCQYLEENHFIHRDIAARNCLLTCPGPGRVAKIGDFGMAQDIYRASYYRKGGCAMLPVKWMPPEAFMEGIFTSKTDTWSFGVLLWEIFSLGYMPYPSKSNQEVLEFVTSGGRMDPPKNCPGPVYRIMTQCWQHQPEDRPNFAIILERIEYCTQDPDVINTALPIEYGPLVEEEEK
6BSX , Knot 82 178 0.79 38 130 170 
SMPQSKSRKIAILGYRSVGKSSLTIQFVEGQFVDSYDPTIENTFTKLITVNGQEYHLQLVDTAGQDEYSIFPQTYSIDINGYILVYSVTSIKSFEVIKVIHGKLLDMVGKVQIPIMLVGNKKDLHMERVISYEEGKALAESWNAAFLESSAKENQTAVDVFRRIILEAEKLEHHHHHH

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 \(c\).

\(|P_{f(1JBD_1)}(2) \setminus P_{f(4FNX_1)}(2)|=27\), \(|P_{f(4FNX_1)}(2) \setminus P_{f(1JBD_1)}(2)|=167\). 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 1:11000010011011001110010000110011000010110110110010001000100000000010100011
Pair \(Z_2\) Length of longest common subsequence
1JBD_1,4FNX_1 194 3
1JBD_1,6BSX_1 150 3
4FNX_1,6BSX_1 178 4

Newick tree

 
[
	4FNX_1:98.37,
	[
		1JBD_1:75,6BSX_1:75
	]:23.37
]

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 \((0.85)(0.8)(\frac{401 }{\log_{20} 401}-\frac{74}{\log_{20}74})=101.\)
Status Protein1 Protein2 d d1/2
Query variables 1JBD_1 4FNX_1 128 77.5
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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