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Parikh vectors
2MPS_1 1ESQ_1 6MGU_1 Letter Amino acid
3 43 42 A Alanine
5 20 21 I Isoleucine
9 11 23 K Lycine
4 5 13 M Methionine
9 21 16 T Threonine
4 10 14 R Arginine
1 2 3 C Cysteine
5 25 42 G Glycine
2 13 9 H Histidine
7 18 21 S Serine
6 10 14 Q Glutamine
5 16 22 E Glutamic acid
13 23 27 L Leucine
3 5 7 F Phenylalanine
4 11 9 N Asparagine
5 11 17 D Aspartic acid
5 7 13 P Proline
0 2 1 W Tryptophan
7 3 9 Y Tyrosine
10 28 40 V Valine 

2MPS_1|Chain A|E3 ubiquitin-protein ligase Mdm2|Homo sapiens (9606)
>1ESQ_1|Chains A, B, C|HYDROXYETHYLTHIAZOLE KINASE|Bacillus subtilis (1423)
>6MGU_1|Chains A, B|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 \(w=f(c)\)
2MPS , Knot 55 107 0.80 38 90 103 
NTNMSVPTDGAVTTSQIPASEQETLVRPKPLLLKLLKSVGAQKDTYTMKEVLFYLGQYIMTKRLYDEKQQHIVYCSNDLLGDLFGVPSFSVKEHRKIYTMIYRNLVV
1ESQ , Knot 121 284 0.80 40 168 264 
MRGSHHHHHHGSMDAQSAAKCLTAVRRHSPLVHSITNNVVTNFTANGLLALGASPVMAYAKEEVADMAKIAGALVLNIGTLSKESVEAMIIAGKSANEHGVPVILDPVGAGATPFRTESARDIIREVRLAAIRGNAAEIAHTVGVTDWLIKGVDAGEGGGDIIRLAQQAAQKLNTVIAITGEVDVIADTSHVYTLHNGHKLLTKVTGAGSLLTSVVGAFCAVEENPLFAAIAAISSYGVAAQLAAQQTADKGPGSFQIELLNKLSTVTEQDVQEWATIERVTVS
6MGU , Knot 152 363 0.82 40 198 340 
SNAMWESKFVKEGLTFDDVLLVPAKSDVLPREVSVKTVLSESLQLNIPLISAGMDTVTEADMAIAMARQGGLGIIHKNMSIEQQAEQVDKVKRSGGLLVGAAVGVTADAMTRIDALVKASVDAIVLDTAHGHSQGVIDKVKEVRAKYPSLNIIAGNVATAEATKALIEAGANVVKVGIGPGSICTTRVVAGVGVPQLTAVYDCATEARKHGIPVIADGGIKYSGDMVKALAAGAHVVMLGSMFAGVAESPGETEIYQGRQFKVYRGMGSVGAMEKGSKDRYFQEGNKKLVPEGIEGRVPYKGPLADTVHQLVGGLRAGMGYCGAQDLEFLRENAQFIRMSGAGLLESHPHHVQITKEAPNYSL

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 \(c\).

\(|P_{f(2MPS_1)}(2) \setminus P_{f(1ESQ_1)}(2)|=49\), \(|P_{f(1ESQ_1)}(2) \setminus P_{f(2MPS_1)}(2)|=127\). 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 1:00010110011100001110000011010111101100111000000100111011001100010000000110000011101111101010000010011000111
Pair \(Z_2\) Length of longest common subsequence
2MPS_1,1ESQ_1 176 3
2MPS_1,6MGU_1 186 4
1ESQ_1,6MGU_1 144 4

Newick tree

 
[
	2MPS_1:95.92,
	[
		1ESQ_1:72,6MGU_1:72
	]:23.92
]

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 \((0.85)(0.8)(\frac{391 }{\log_{20} 391}-\frac{107}{\log_{20}107})=86.8\)
Status Protein1 Protein2 d d1/2
Query variables 2MPS_1 1ESQ_1 110 75.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]