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Parikh vectors
8DZN_1 8BCQ_1 2ESI_1 Letter Amino acid
8 6 4 A Alanine
26 19 0 L Leucine
12 25 0 K Lycine
9 31 0 N Asparagine
10 13 0 D Aspartic acid
12 28 0 I Isoleucine
11 15 0 F Phenylalanine
14 16 0 T Threonine
13 10 0 E Glutamic acid
18 7 7 G Glycine
5 12 0 H Histidine
5 4 0 P Proline
3 2 0 W Tryptophan
6 16 0 Y Tyrosine
12 12 0 V Valine
9 9 0 R Arginine
2 3 7 C Cysteine
6 6 0 Q Glutamine
6 2 0 M Methionine
11 13 0 S Serine 

8DZN_1|Chains A, B|GTP-binding protein SAR1a|Homo sapiens (9606)
>8BCQ_1|Chains A, B, C, D, E|Glutamate--tRNA ligase|Plasmodium berghei ANKA (5823)
>2ESI_1|Chains A, B|5'-R(*UP*UP*GP*CP*GP*UP*CP*AP*CP*AP*CP*CP*GP*GP*UP*GP*AP*AP*GP*UP*CP*GP*C)-3'|null
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 \(w=f(c)\)
8DZN , Knot 93 198 0.82 40 143 188 
MSFIFEWIYNGFSSVLQFLGLYKKSGKLVFLGLDNAGKTTLLHMLKDDRLGQHVPTLHPTSEELTIAGMTFTTFDLGGHEQARRVWKNYLPAINGIVFLVDCADHSRLVESKVELNALMTDETISNVPILILGNKIDRTDAISEEKLREIFGLYGQTTGKGNVTLKELNARPMEVFMCSVLKRQGYGEGFRWLSQYID
8BCQ , Knot 110 249 0.81 40 153 231 
MGIKDSKINIYYGKNYPFLCRTVFNIYQNNIKKKTANNLNDKKIKEICVNFINDKTVVEDIKVEFVRNTVSDQTRKNTNTNNSVTSSDKIFAINLDFLLKTNLYYFTSYRENINRNIITNVFFQAQYNEWIDFLRNKDIEKNIIPICEHINKHLYLNTFLSFHYLTLSDIYIYYEMHKYFSGNITTNLKYPKQYKNINRWFRLIKALLHDHVATDAELIQNLKVKEKIFIDGGSSGLVPRGSSHHHHHH
2ESI , Knot 11 23 0.50 8 13 19 
UUGCGUCACACCGGUGAAGUCGC

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 \(c\).

\(|P_{f(8DZN_1)}(2) \setminus P_{f(8BCQ_1)}(2)|=71\), \(|P_{f(8BCQ_1)}(2) \setminus P_{f(8DZN_1)}(2)|=81\). 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 1:101110110011001101111000010111111001100011011000011001101010000101111010010111000100110001111011111100100001100010101110000100111111100100001100001001111010001010101001010110111001100010101101100010
Pair \(Z_2\) Length of longest common subsequence
8DZN_1,8BCQ_1 152 3
8DZN_1,2ESI_1 150 2
8BCQ_1,2ESI_1 164 2

Newick tree

 
[
	8BCQ_1:80.36,
	[
		8DZN_1:75,2ESI_1:75
	]:5.36
]

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 \((0.85)(0.8)(\frac{447 }{\log_{20} 447}-\frac{198}{\log_{20}198})=72.9\)
Status Protein1 Protein2 d d1/2
Query variables 8DZN_1 8BCQ_1 87 81
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]