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Parikh vectors
5HRL_1 1TRJ_1 3BDM_1 Letter Amino acid
3 0 14 D Aspartic acid
11 14 23 G Glycine
8 0 19 T Threonine
2 0 4 M Methionine
11 0 16 V Valine
7 5 18 A Alanine
2 14 0 C Cysteine
12 0 17 E Glutamic acid
16 0 15 I Isoleucine
27 0 26 L Leucine
10 0 9 R Arginine
5 0 6 Q Glutamine
3 0 3 H Histidine
8 0 9 F Phenylalanine
7 0 23 S Serine
8 0 6 N Asparagine
24 0 18 K Lycine
6 0 12 P Proline
1 0 2 W Tryptophan
7 0 10 Y Tyrosine 

5HRL_1|Chains A, E[auth B]|DNA polymerase beta-like protein|African swine fever virus (10497)
>1TRJ_1|Chain A[auth B]|Helix 39 of 18S rRNA|
>3BDM_1|Chains A, O|Proteasome component Y7|Saccharomyces cerevisiae (4932)
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 \(w=f(c)\)
5HRL , Knot 85 178 0.82 40 124 171 
SGGGMLTLIQGKKIVNHLRSRLAFEYNGQLIKILSKNIVAVGSLRREEKMLNDVDLLIIVPEKKLLKHVLPNIRIKGLSFSVKVCGERKCVLFIEWEKKTYQLDLFTALAEEKPYAIFHFTGPVSYLIRIRAALKKKNYKLNQYGLFKNQTLVPLKITTEKELIKELGFTYRIPKKRL
1TRJ , Knot 14 41 0.42 8 14 29 
CCGCCGUUAGUUGCCAGCGGUUCGGCCGGGCACUCUAACGG
3BDM , Knot 111 250 0.81 38 159 239 
MTDRYSFSLTTFSPSGKLGQIDYALTAVKQGVTSLGIKATNGVVIATEKKSSSPLAMSETLSKVSLLTPDIGAVYSGMGPDYRVLVDKSRKVAHTSYKRIYGEYPPTKLLVSEVAKIMQEATQSGGVRPFGVSLLIAGHDEFNGFSLYQVDPSGSYFPWKATAIGKGSVAAKTFLEKRWNDELELEDAIHIALLTLKESVEGEFNGDTIELAIIGDENPDLLGYTGIPTDKGPRFRKLTSQEINDRLEAL

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 \(c\).

\(|P_{f(5HRL_1)}(2) \setminus P_{f(1TRJ_1)}(2)|=121\), \(|P_{f(1TRJ_1)}(2) \setminus P_{f(5HRL_1)}(2)|=11\). 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 1:0111110110100110010001110001011011000111110100000110010111111000110011101010110101010100001111010000001011011100010111010111001101011100000010001110000111101000001100111000110001
Pair \(Z_2\) Length of longest common subsequence
5HRL_1,1TRJ_1 132 3
5HRL_1,3BDM_1 151 3
1TRJ_1,3BDM_1 167 2

Newick tree

 
[
	3BDM_1:83.64,
	[
		5HRL_1:66,1TRJ_1:66
	]:17.64
]

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 \((0.85)(0.8)(\frac{219 }{\log_{20} 219}-\frac{41}{\log_{20}41})=60.2\)
Status Protein1 Protein2 d d1/2
Query variables 5HRL_1 1TRJ_1 82 47.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]