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Parikh vectors
8FYD_1 9EGW_1 5EIR_1 Letter Amino acid
0 2 13 N Asparagine
0 3 18 E Glutamic acid
0 3 8 W Tryptophan
0 4 14 D Aspartic acid
22 10 4 C Cysteine
0 7 7 Q Glutamine
0 7 11 P Proline
23 6 18 T Threonine
0 2 12 S Serine
0 0 6 Y Tyrosine
0 7 12 V Valine
19 11 11 G Glycine
0 4 11 I Isoleucine
0 2 17 L Leucine
0 3 3 M Methionine
0 0 8 F Phenylalanine
14 4 11 A Alanine
0 6 12 R Arginine
0 3 5 H Histidine
0 4 16 K Lycine 

8FYD_1|Chain A[auth J]|DNA (49-MER)|Megasphaera (906)
>9EGW_1|Chains A, B|RanBP-type and C3HC4-type zinc finger-containing protein 1|Homo sapiens (9606)
>5EIR_1|Chain A|Eukaryotic translation initiation factor 4E|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 \(w=f(c)\)
8FYD , Knot 26 78 0.48 8 16 45 
TGCGCGTGGGATCACCCCCGCTCGTGCGGGAAAGACAGTAATGGATTCCTTTATTTTCGCCCTTTTACGCTTACTGAC
9EGW , Knot 48 88 0.81 36 74 85 
GPDVAARQTTEMLKVMLQQGEAMRCPQCQIVVQKKDGADWIRCTVCHTEICWVTKGPRWGPGGPGDTSGGCRCRVNGIPCHPSCQNCH
5EIR , Knot 103 217 0.85 40 160 211 
MATVEPETTPTPNPPTTEEEKTESNQEVANPEHYIKHPLQNRWALWFFKNDKSKTWQANLRLISKFDTVEDFWALYNHIQLSSNLMPGCDYSLFKDGIEPMWEDEKNKRGGRWLITLNKQQRRSDLDRFWLETLLCLIGESFDDYSDDVCGAVVNVRAKGDKIAIWTTECENREAVTHIGRVYKERLGLPPKIVIGYQSHADTATKSGSTTKNRFVV

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 \(c\).

\(|P_{f(8FYD_1)}(2) \setminus P_{f(9EGW_1)}(2)|=10\), \(|P_{f(9EGW_1)}(2) \setminus P_{f(8FYD_1)}(2)|=68\). 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 1:010101011110010000010001010111111110110110111000000010000010000000101000100110
Pair \(Z_2\) Length of longest common subsequence
8FYD_1,9EGW_1 78 2
8FYD_1,5EIR_1 162 3
9EGW_1,5EIR_1 172 3

Newick tree

 
[
	5EIR_1:93.79,
	[
		8FYD_1:39,9EGW_1:39
	]:54.79
]

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 \((0.85)(0.8)(\frac{166 }{\log_{20} 166}-\frac{78}{\log_{20}78})=29.6\)
Status Protein1 Protein2 d d1/2
Query variables 8FYD_1 9EGW_1 42 33
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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