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Parikh vectors
4PTA_1 7KDQ_1 7LZK_1 Letter Amino acid
13 2 2 I Isoleucine
6 3 10 F Phenylalanine
6 0 7 T Threonine
1 0 1 W Tryptophan
8 0 11 D Aspartic acid
11 0 6 E Glutamic acid
3 2 9 G Glycine
0 0 2 H Histidine
4 1 2 P Proline
3 1 12 S Serine
4 0 6 R Arginine
5 0 7 Q Glutamine
16 3 12 K Lycine
6 1 8 V Valine
6 1 12 A Alanine
14 0 6 N Asparagine
1 0 1 C Cysteine
15 3 13 L Leucine
1 0 6 M Methionine
8 0 4 Y Tyrosine 

4PTA_1|Chain A[auth D]|Replication initiator protein|Staphylococcus aureus subsp. aureus (367830)
>7KDQ_1|Chain A|Stigmurin analog StigA15|Tityus stigmurus (50344)
>7LZK_1|Chains A, B|Dehaloperoxidase B|Amphitrite ornata (129555)
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 \(w=f(c)\)
4PTA , Knot 63 131 0.78 38 95 126 
MSKQFFTVEENYKERFYQLPKVFFTNPNYKDLSNDAKIAYAILRDRLQLSIKNNWIDTEGNIYFIYTVADLEVILNCGNKKITKIKKELENVDLLIQKRQGLNKPNLLYLLKPAITKNDIYEIDKAENEVE
7KDQ , Knot 13 18 0.69 20 16 16 
FFSLIPKLVGGLIKAFKX
7LZK , Knot 68 137 0.81 40 105 134 
GFKQDIATLRGDLRTYAQDIFLAFLNKYPDEKRNFKNYVGKSDQELKSMAKFGDHTEKVFNLMMEVADRATDCVPLASDASTLVQMKQHSGLTTGNFEKLFVALVEYMRASGQSFDSQSWDRFGKNLVSALSSAGMK

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 \(c\).

\(|P_{f(4PTA_1)}(2) \setminus P_{f(7KDQ_1)}(2)|=89\), \(|P_{f(7KDQ_1)}(2) \setminus P_{f(4PTA_1)}(2)|=10\). 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 1:10001101000000010011011100100001000101101110001010100011000101011001101011100100010010001001011100001100101101101110000100100100010
Pair \(Z_2\) Length of longest common subsequence
4PTA_1,7KDQ_1 99 2
4PTA_1,7LZK_1 144 3
7KDQ_1,7LZK_1 109 2

Newick tree

 
[
	7LZK_1:67.96,
	[
		4PTA_1:49.5,7KDQ_1:49.5
	]:18.46
]

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 \((0.85)(0.8)(\frac{149 }{\log_{20} 149}-\frac{18}{\log_{20}18})=47.9\)
Status Protein1 Protein2 d d1/2
Query variables 4PTA_1 7KDQ_1 58 32.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]