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Parikh vectors
1FSB_1 4EOF_1 4MEJ_1 Letter Amino acid
1 12 13 A Alanine
2 3 4 Q Glutamine
4 12 13 G Glycine
1 6 11 K Lycine
4 3 7 Y Tyrosine
1 14 9 N Asparagine
1 7 13 D Aspartic acid
1 8 9 L Leucine
2 2 11 P Proline
0 6 1 W Tryptophan
1 6 9 V Valine
6 8 2 C Cysteine
0 1 3 H Histidine
1 6 11 I Isoleucine
1 2 7 M Methionine
1 3 11 F Phenylalanine
1 11 6 R Arginine
5 2 11 E Glutamic acid
4 10 7 S Serine
3 7 9 T Threonine 

1FSB_1|Chain A|P-SELECTIN|Homo sapiens (9606)
>4EOF_1|Chain A|Lysozyme|Gallus gallus (9031)
>4MEJ_1|Chains A, B, C|Nucleoside deoxyribosyltransferase|Lactobacillus helveticus (767462)
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 \(w=f(c)\)
1FSB , Knot 26 40 0.80 36 35 38 
TASCQDMSCSKQGECLETIGNYTCSCYPGFYGPECEYVRE
4EOF , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
4MEJ , Knot 81 167 0.82 40 127 162 
MKAVVPTGKIYLGSPFYSDAQRERAAKAKELLAKNPSIAHVFFPFDDGFTDPDEKNPEIGGIRSMVWRDATYQNDLTGISNATCGVFLYDMDQLDDGSAFEIGFMRAMHKPVILVPFTEHPEKEKKMNLMIAQGVTTIIDGNTEFEKLADYNFNECPSNPVRGYGIY

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 \(c\).

\(|P_{f(1FSB_1)}(2) \setminus P_{f(4EOF_1)}(2)|=27\), \(|P_{f(4EOF_1)}(2) \setminus P_{f(1FSB_1)}(2)|=96\). 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 1:0100001000001001001100000001110110000100
Pair \(Z_2\) Length of longest common subsequence
1FSB_1,4EOF_1 123 3
1FSB_1,4MEJ_1 136 2
4EOF_1,4MEJ_1 157 3

Newick tree

 
[
	4MEJ_1:77.00,
	[
		1FSB_1:61.5,4EOF_1:61.5
	]:15.50
]

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 \((0.85)(0.8)(\frac{169 }{\log_{20} 169}-\frac{40}{\log_{20}40})=45.0\)
Status Protein1 Protein2 d d1/2
Query variables 1FSB_1 4EOF_1 57 36.5
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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