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Parikh vectors
2QQC_1 8PFU_1 4QGN_1 Letter Amino acid
0 3 6 Q Glutamine
1 2 7 M Methionine
4 10 2 S Serine
2 7 6 T Threonine
1 3 12 Y Tyrosine
6 12 13 A Alanine
1 11 15 R Arginine
0 8 1 C Cysteine
0 6 5 W Tryptophan
1 7 18 D Aspartic acid
2 1 5 H Histidine
2 3 7 F Phenylalanine
3 6 9 V Valine
7 14 5 N Asparagine
7 8 13 L Leucine
1 6 12 K Lycine
3 2 11 P Proline
3 2 15 E Glutamic acid
5 12 11 G Glycine
4 6 10 I Isoleucine 

2QQC_1|Chains A, C, E, G, I, K|Pyruvoyl-dependent arginine decarboxylase subunit beta|Methanocaldococcus jannaschii (2190)
>8PFU_1|Chain A|Lysozyme C|Gallus gallus (9031)
>4QGN_1|Chain A|1,2-dihydroxy-3-keto-5-methylthiopentene dioxygenase|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 \(w=f(c)\)
2QQC , Knot 32 53 0.80 34 47 50 
HMNAEINPLHAYFKLPNTVSLVAGSSEGETPLNAFDGALLNAGIGNVNLIRIS
8PFU , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
4QGN , Knot 87 183 0.82 40 142 179 
GAAAMVQAWYMDDAPGDPRQPHRPDPGRPVGLEQLRRLGVLYWKLDADKYENDPELEKIRRERNYSWMDIITICKDKLPNYEEKIKMFYEEHLHLDDEIRYILDGSGYFDVRDKEDQWIRIFMEKGDMVTLPAGIYHRFTVDEKNYTKAMRLFVGEPVWTAYNRPADHFEARGQYVKFLAQTA

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 \(c\).

\(|P_{f(2QQC_1)}(2) \setminus P_{f(8PFU_1)}(2)|=29\), \(|P_{f(8PFU_1)}(2) \setminus P_{f(2QQC_1)}(2)|=86\). 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 1:01010101101010110010111100010011011011110111101011010
Pair \(Z_2\) Length of longest common subsequence
2QQC_1,8PFU_1 115 3
2QQC_1,4QGN_1 155 4
8PFU_1,4QGN_1 176 4

Newick tree

 
[
	4QGN_1:89.80,
	[
		2QQC_1:57.5,8PFU_1:57.5
	]:32.30
]

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 \((0.85)(0.8)(\frac{182 }{\log_{20} 182}-\frac{53}{\log_{20}53})=44.0\)
Status Protein1 Protein2 d d1/2
Query variables 2QQC_1 8PFU_1 57 39
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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