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Parikh vectors
6TYH_1 4EGX_1 5PBS_1 Letter Amino acid
0 18 5 R Arginine
0 5 8 H Histidine
0 6 8 F Phenylalanine
1 11 8 V Valine
2 28 9 E Glutamic acid
1 16 6 G Glycine
2 11 4 I Isoleucine
0 5 6 M Methionine
0 7 6 P Proline
2 11 11 S Serine
1 10 8 T Threonine
2 6 6 N Asparagine
5 3 2 C Cysteine
2 3 2 Q Glutamine
0 10 14 K Lycine
0 1 2 W Tryptophan
0 9 4 A Alanine
0 7 12 D Aspartic acid
2 14 13 L Leucine
2 3 4 Y Tyrosine 

6TYH_1|Chains A, C, E, G, I, K|Insulin A chain|Homo sapiens (9606)
>4EGX_1|Chains A, B, C, D|Kinesin-like protein KIF1A|Homo sapiens (9606)
>5PBS_1|Chain A|Bromodomain adjacent to zinc finger domain protein 2B|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 \(w=f(c)\)
6TYH , Knot 15 22 0.70 22 21 20 
GIVEQCCTSICSLYQLENYCNC
4EGX , Knot 85 184 0.80 40 132 175 
GPGSEFSEEAIERLKETEKIIAELNETWEEKLRRTEAIRMEREALLAEMGVAMREDGGTLGVFSPKKTPHLVNLNEDPLMSECLLYYIKDGITRVGREDGERRQDIVLSGHFIKEEHCVFRSDSRGGSEAVVTLEPCEGADTYVNGKKVTEPSILRSGNRIIMGKSHVFRFNHPEQARQERERT
5PBS , Knot 66 138 0.78 40 109 129 
MHHHHHHSSGVDLGTENLYFQSMSVKKPKRDDSKDLALCSMILTEMETHEDAWPFLLPVNLKLVPGYKKVIKKPMDFSTIREKLSSGQYPNLETFALDVRLVFDNCETFNEDDSDIGRAGHNMRKYFEKKWTDTFKVS

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 \(c\).

\(|P_{f(6TYH_1)}(2) \setminus P_{f(4EGX_1)}(2)|=15\), \(|P_{f(4EGX_1)}(2) \setminus P_{f(6TYH_1)}(2)|=126\). 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 1:1110000001001001000000
Pair \(Z_2\) Length of longest common subsequence
6TYH_1,4EGX_1 141 2
6TYH_1,5PBS_1 120 2
4EGX_1,5PBS_1 153 3

Newick tree

 
[
	4EGX_1:77.55,
	[
		6TYH_1:60,5PBS_1:60
	]:17.55
]

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 \((0.85)(0.8)(\frac{206 }{\log_{20} 206}-\frac{22}{\log_{20}22})=64.2\)
Status Protein1 Protein2 d d1/2
Query variables 6TYH_1 4EGX_1 81 45
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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