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Parikh vectors
6MXB_1 2FOI_1 7OOI_1 Letter Amino acid
19 8 8 R Arginine
16 16 6 D Aspartic acid
22 16 13 G Glycine
14 6 1 H Histidine
20 19 8 L Leucine
17 22 17 S Serine
24 10 9 V Valine
12 24 4 I Isoleucine
0 2 6 W Tryptophan
12 11 5 E Glutamic acid
16 27 3 K Lycine
9 7 2 F Phenylalanine
14 14 10 Y Tyrosine
14 18 12 A Alanine
11 34 2 N Asparagine
3 2 2 C Cysteine
5 7 4 Q Glutamine
10 9 2 M Methionine
13 6 4 P Proline
21 11 7 T Threonine 

6MXB_1|Chains A, B, C, D, E, F|Hypoxanthine-guanine phosphoribosyltransferase, putative|Trypanosoma brucei brucei (strain 927/4 GUTat10.1) (185431)
>2FOI_1|Chains A, B|enoyl-acyl carrier reductase|Plasmodium falciparum (36329)
>7OOI_1|Chains A, B|JD1 VH domain|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 \(w=f(c)\)
6MXB , Knot 120 272 0.82 38 177 262 
MGSSHHHHHHDYDIPTTENLYFQGHMASMTGGQQMGRGSHSGHPLKPNFVGRDADGNVTVDGRSYPMAESVVATESTIHRSMKEMAQTLANAYKTLKHRDTHNKGNSALAPITDENPLIIISVLKGSYIFTADMVRYLGDCGLPNVVDFIRITSYRGTTKSSGTVQVLDNLRFTELTGKHVLIMEDIADTGRTMKLLVEKIRREYRPASLKVCVLVDKPGGRVVDFKPEFVCLTAPTRYVVGYGFEVNDRYRNYRHVFVLKPEYAKRYPSKL
2FOI , Knot 118 269 0.81 40 165 258 
EDICFIAGIGDTNGYGWGIAKELSKRNVKIIFGIWPPVYNIFMKNYKNGKFDNDMIIDKDKKMNILDMLPFDASFDTANDIDEETKNNKRYNMLQNYTIEDVANLIHQKYGKINMLVHSLANAKEVQKDLLNTSRKGYLDALSKSSYSLISLCKYFVNIMKPQSSIISLTYHASQKVVPGYGGGMSSAKAALESDTRVLAYHLGRNYNIRINTISAGPLKSRAATAINKLNNTYENNTNQNKNRNSHDVHNIMNNSGEKEEKKNSASQN
7OOI , Knot 62 125 0.79 40 96 121 
EVQLVESGGGLVQPGRSLRLSCAASGFDIDYYSIGWVRRAPGKGEELVARIYPSSSSTSYADSVKGRFTISADTSKNTAYLQMNSLRAEDTAVYYCARWHWAYSAWPGWYGMDYWGQGTLVTVSS

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 \(c\).

\(|P_{f(6MXB_1)}(2) \setminus P_{f(2FOI_1)}(2)|=89\), \(|P_{f(2FOI_1)}(2) \setminus P_{f(6MXB_1)}(2)|=77\). 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 1:11000000000001100001010101101011001101000101101011100101010101000111001110000100010011001101000100000000100111110000111110110100110101100110011101101101000010000010101100101001010011110011001001011100100000110101011100111011010101101011000111011010000000001111010010001001
Pair \(Z_2\) Length of longest common subsequence
6MXB_1,2FOI_1 166 4
6MXB_1,7OOI_1 163 3
2FOI_1,7OOI_1 161 4

Newick tree

 
[
	6MXB_1:82.82,
	[
		7OOI_1:80.5,2FOI_1:80.5
	]:2.32
]

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 \((0.85)(0.8)(\frac{541 }{\log_{20} 541}-\frac{269}{\log_{20}269})=77.1\)
Status Protein1 Protein2 d d1/2
Query variables 6MXB_1 2FOI_1 100 98
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]