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Parikh vectors
8GKV_1 7HNH_1 5UUV_1 Letter Amino acid
7 10 18 D Aspartic acid
10 7 23 E Glutamic acid
1 11 8 F Phenylalanine
8 8 42 A Alanine
3 10 10 N Asparagine
8 7 23 K Lycine
3 6 14 M Methionine
5 14 41 V Valine
7 10 15 Q Glutamine
13 12 29 L Leucine
1 10 15 H Histidine
2 7 10 Y Tyrosine
4 9 14 R Arginine
3 14 44 G Glycine
3 9 13 P Proline
3 15 23 S Serine
2 10 17 T Threonine
0 6 1 W Tryptophan
0 4 3 C Cysteine
5 9 21 I Isoleucine 

8GKV_1|Chains A, B|Anti-adapter protein IraP|Escherichia coli K-12 (83333)
>7HNH_1|Chain A[auth B]|E3 ubiquitin-protein ligase TRIM21|Mus musculus (10090)
>5UUV_1|Chains A, B, C, D|Inosine-5'-monophosphate dehydrogenase|Bacillus anthracis (1392)
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 \(w=f(c)\)
8GKV , Knot 44 88 0.74 36 69 84 
GGMKNLIAELLFKLAQKEEESKELSAQVEALEIIVTAMLRNMAQNDQQRLIDQVEGALYEVKPDASIPDDDTELLRDYVKKLLKHPRQ
7HNH , Knot 91 188 0.84 40 148 179 
MHHHHHHMVHITLDRNTANSWLIISKDRRQVRMGDTHQNVSDNKERFSNYPMVLGAQRFSSGKMYWEVDVTQKEAWDLGVCRDSVQRKGQFSLSPENGFWTIWLWQDSYEAGTSPQTTLHIQVPPCQIGIFVDYEAGVVSFYNITDHGSLIYTFSECVFAGPLRPFFNVGFNYSGGNAAPLKLCPLKM
5UUV , Knot 158 384 0.81 40 204 357 
MHHHHHHSSGVDLGTENLYFQSNAMWESKFVKEGLTFDDVLLVPAKSDVLPREVSVKTVLSESLQLNIPLISAGMDTVTEADMAIAMARQGGLGIIHKNMSIEQQAEQVDKVKRSGGLLVGAAVGVTADAMTRIDALVKASVDAIVLDTAHGHSQGVIDKVKEVRAKYPSLNIIAGNVATAEATKALIEAGANVVKVGIGPGSICTTRVVAGVGVPQLTAVYDCATEARKHGIPVIADGGIKYSGDMVKALAAGAHVVMLGSMFAGVAESPGETEIYQGRQFKVYRGMGSVGAMEKGSKDRYFQEGNKKLVPEGIEGRVPYKGPLADTVHQLVGGLRAGMGYCGAQDLEFLRENAQFIRMSGAGLLESHPHHVQITKEAPNYSL

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 \(c\).

\(|P_{f(8GKV_1)}(2) \setminus P_{f(7HNH_1)}(2)|=44\), \(|P_{f(7HNH_1)}(2) \setminus P_{f(8GKV_1)}(2)|=123\). 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 1:1110011101110110000000010101011011101110011000000110010111001010101100000110001001100100
Pair \(Z_2\) Length of longest common subsequence
8GKV_1,7HNH_1 167 3
8GKV_1,5UUV_1 173 3
7HNH_1,5UUV_1 186 7

Newick tree

 
[
	5UUV_1:91.81,
	[
		8GKV_1:83.5,7HNH_1:83.5
	]:8.31
]

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 \((0.85)(0.8)(\frac{276 }{\log_{20} 276}-\frac{88}{\log_{20}88})=59.9\)
Status Protein1 Protein2 d d1/2
Query variables 8GKV_1 7HNH_1 79 57
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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