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Parikh vectors
8HYN_1 2DLB_1 1MTC_1 Letter Amino acid
14 3 15 D Aspartic acid
6 2 12 P Proline
9 2 13 Y Tyrosine
5 1 13 R Arginine
14 7 9 N Asparagine
1 1 3 C Cysteine
6 12 15 E Glutamic acid
10 2 9 G Glycine
1 6 4 H Histidine
12 5 19 K Lycine
4 4 11 A Alanine
1 2 8 M Methionine
17 4 13 I Isoleucine
17 12 25 L Leucine
9 2 13 F Phenylalanine
10 6 12 S Serine
13 4 7 T Threonine
0 0 4 W Tryptophan
7 5 5 V Valine
7 0 7 Q Glutamine 

8HYN_1|Chains A, B|CD-NTase-associated protein 12|Riemerella anatipestifer Yb2 (1455062)
>2DLB_1|Chains A, B|yopT|Bacillus subtilis (1423)
>1MTC_1|Chains A, B|Glutathione S-transferase YB1|Rattus norvegicus (10116)
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 \(w=f(c)\)
8HYN , Knot 77 163 0.80 38 116 157 
SGGGFPSSTLAAVYYENFVKPTCLHIIQNGGIQDDDGTKYENSTIKIIIPQKLTTDVNSQFQTLKKSFQTKKLTFDYLGRPRNIDVETLIQDGKLYVIDFPTVLSGINYAISNLLPNDFNSMSDDYELILNREFDRFIYTLNKLALRDGYNNLITVINEKDIK
2DLB , Knot 42 80 0.76 36 61 73 
MAGYLNNIALNLEIVLKNKADSPEVSETLVTRICENLLLSKEVSFLKADGSVENFKLSDMEYEITNTEELPELEHHHHHH
1MTC , Knot 104 217 0.86 40 158 208 
PMILGYWNVRGLTHPIRLLLEYTDSSYEEKRYAMGDAPDYDRSQWLNEKFKLGLDFPNLPYLIDGSRKITQSNAIMRYLARKHHLCGETEEERIRADIVENQVMDNRMQLIMLCFNPDFEKQKPEFLKTIPEKMKLYSEFLGKRPWFAGDKVTYVDFLAYDILDQYHIFEPKCLDAFPNLKDFLARFEGLKKISAYMKSSRYLSTPIFSKLAQWSNK

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 \(c\).

\(|P_{f(8HYN_1)}(2) \setminus P_{f(2DLB_1)}(2)|=91\), \(|P_{f(2DLB_1)}(2) \setminus P_{f(8HYN_1)}(2)|=36\). 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 1:0111110001111000011010010110011100001000000010111100100010001001000100001010011010010100110010101101101101100110011100100100000111000100110010011100100011011000010
Pair \(Z_2\) Length of longest common subsequence
8HYN_1,2DLB_1 127 3
8HYN_1,1MTC_1 160 3
2DLB_1,1MTC_1 153 3

Newick tree

 
[
	1MTC_1:82.60,
	[
		8HYN_1:63.5,2DLB_1:63.5
	]:19.10
]

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 \((0.85)(0.8)(\frac{243 }{\log_{20} 243}-\frac{80}{\log_{20}80})=52.9\)
Status Protein1 Protein2 d d1/2
Query variables 8HYN_1 2DLB_1 69 49.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]