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Parikh vectors
2BDT_1 2YKG_1 1WGN_1 Letter Amino acid
11 33 2 N Asparagine
14 37 1 D Aspartic acid
3 20 2 C Cysteine
8 17 1 H Histidine
11 66 3 K Lycine
5 40 11 S Serine
3 5 0 W Tryptophan
13 40 3 A Alanine
16 51 6 E Glutamic acid
9 32 7 G Glycine
13 54 2 I Isoleucine
6 22 4 Y Tyrosine
21 59 5 L Leucine
4 14 3 M Methionine
11 36 1 T Threonine
10 28 2 R Arginine
6 34 3 Q Glutamine
7 37 1 F Phenylalanine
7 29 2 P Proline
11 42 4 V Valine 

2BDT_1|Chain A|BH3686|Bacillus halodurans (272558)
>2YKG_1|Chain A|PROBABLE ATP-DEPENDENT RNA HELICASE DDX58|HOMO SAPIENS (9606)
>1WGN_1|Chain A|ubiquitin associated protein|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 \(w=f(c)\)
2BDT , Knot 86 189 0.79 40 141 180 
MKKLYIITGPAGVGKSTTCKRLAAQLDNSAYIEGDIINHMVVGGYRPPWESDELLALTWKNITDLTVNFLLAQNDVVLDYIAFPDEAEALAQTVQAKVDDVEIRFIILWTNREELLRRDALRKKDEQMGERCLELVEEFESKGIDERYFYNTSHLQPTNLNDIVKNLKTNPRFIFCMAGDPLEHHHHHH
2YKG , Knot 274 696 0.86 40 296 645 
SEVSDTNLYSPFKPRNYQLELALPAMKGKNTIICAPTGCGKTFVSLLICEHHLKKFPQGQKGKVVFFANQIPVYEQNKSVFSKYFERHGYRVTGISGATAENVPVEQIVENNDIIILTPQILVNNLKKGTIPSLSIFTLMIFDECHNTSKQHPYNMIMFNYLDQKLGGSSGPLPQVIGLTASVGVGDAKTTDEALDYICKLCASLDASVIATVKHNLEELEQVVYKPQKFFRKVESRISDKFKYIIAQLMRDTESLAKRICKDLENLSQIQNREFGTQKYEQWIVTVQKACMVFQMPDKDEESRICKALFLYTSHLRKYNDALIISEHARMKDALDYLKDFFSNVRAAGFDEIEQDLTQRFEEKLQELESVSRDPSNENPKLEDLCFILQEEYHLNPETITILFVKTRALVDALKNWIEGNPKLSFLKPGILTGRGKTNQNTGMTLPAQKCILDAFKASGDHNILIATSVADEGIDIAQCNLVILYEYVGNVIKMIQTRGRGRARGSKCFLLTSNAGVIEKEQINMYKEKMMNDSILRLQTWDEAVFREKILHIQTHEKFIRDSQEKPKPVPDKENKKLLCRKCKALACYTADVRVIEDCHYTVLGDAFKECFVSRPHPKPKQFSSFEKRAKIFCARQNCSHDWGIHVKYKTFEIPVIKIESFVVEDIATGVQTLYSKWKDFHFEKIPFDPAEMSK
1WGN , Knot 34 63 0.74 38 51 58 
GSSGSSGAYSELQMLSPSERQCVETVVNMGYSYECVLRAMKKKGENIEQILDYLFAHSGPSSG

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 \(c\).

\(|P_{f(2BDT_1)}(2) \setminus P_{f(2YKG_1)}(2)|=25\), \(|P_{f(2YKG_1)}(2) \setminus P_{f(2BDT_1)}(2)|=180\). 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 1:100101101111110000000111010001010101100111110011100001111010010010101111000111001111001011100101010010101111100000110001100000011000101100100011000010000010100100110010001011101110110000000
Pair \(Z_2\) Length of longest common subsequence
2BDT_1,2YKG_1 205 4
2BDT_1,1WGN_1 144 3
2YKG_1,1WGN_1 261 4

Newick tree

 
[
	2YKG_1:12.95,
	[
		2BDT_1:72,1WGN_1:72
	]:56.95
]

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 \((0.85)(0.8)(\frac{885 }{\log_{20} 885}-\frac{189}{\log_{20}189})=192.\)
Status Protein1 Protein2 d d1/2
Query variables 2BDT_1 2YKG_1 246 153
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]