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Parikh vectors
4XRA_1 7TWZ_1 5FHS_1 Letter Amino acid
5 18 3 H Histidine
10 17 23 S Serine
19 29 18 A Alanine
6 23 9 R Arginine
14 17 14 D Aspartic acid
12 17 15 I Isoleucine
14 7 18 K Lycine
2 4 0 C Cysteine
15 18 17 E Glutamic acid
13 17 23 G Glycine
11 12 9 F Phenylalanine
8 15 12 P Proline
13 12 19 T Threonine
5 6 6 N Asparagine
5 18 6 Q Glutamine
2 10 4 M Methionine
13 19 16 V Valine
11 36 26 L Leucine
3 5 2 W Tryptophan
5 6 10 Y Tyrosine 

4XRA_1|Chains A, B, C, D, E|Alkyl hydroperoxide reductase subunit C|Salmonella enterica subsp. enterica serovar Typhimurium (90371)
>7TWZ_1|Chains A, B, C, D, E, F|Putative oxidoreductase, NAD(P)-linked|Klebsiella pneumoniae subsp. pneumoniae (strain HS11286) (1125630)
>5FHS_1|Chains A, O|Proteasome subunit alpha type-2|Saccharomyces cerevisiae (strain ATCC 204508 / S288c) (559292)
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 \(w=f(c)\)
4XRA , Knot 88 186 0.82 40 137 178 
SLINTKIKPFKNQAFKNGEFIEVTEKDTEGRWSVFFFYPADFSFVCPTELGDVADHYEELQKLGVDVYSVSTDTHFTHKAWHSSSETIAKIKYAMIGDPTGALTRNFDNMREDEGLADRATFVVDPQGIIQAIEVTAEGIGRDASDLLRKIKAAQYVAAHPGEVCPAKWKEGEATLAPSLDLVGKI
7TWZ , Knot 135 306 0.84 40 199 289 
MAHHHHHHMVQRIAMAPQGPEFSRFVMGYWRLMDWKMSPGELVSFIEQHLDLGVTTVDHADIYGDYQCEAAFGEALKRAPHLRSRMEIVSKCGIATRARAENTIGHYITDRDHIVLSAEQSLRNLATDHLDLLLIHRPDPLMDADEVAEAFLALHHSGKVRHFGVSNFTPAQFTLLQSRLPFTLATNQVEISPVHQPLLLDGTLDQLQQLRIRPMAWSCLGGGRLFNEECFQALRDELAQVAHELNADSIEQVVYAWVLRLPSQPLPIIGSGKIERVRSAIVAEKLSMTRQQWFRIRKAALGYDVP
5FHS , 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 \(c\).

\(|P_{f(4XRA_1)}(2) \setminus P_{f(7TWZ_1)}(2)|=55\), \(|P_{f(7TWZ_1)}(2) \setminus P_{f(4XRA_1)}(2)|=117\). 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 1:011000101100011001011010000001010111101101011010011011000001001110100100000100011000000110100111101011100010010000111001011101011101101010111001001100101100111011010110100101011101011101
Pair \(Z_2\) Length of longest common subsequence
4XRA_1,7TWZ_1 172 3
4XRA_1,5FHS_1 148 3
7TWZ_1,5FHS_1 166 4

Newick tree

 
[
	7TWZ_1:87.73,
	[
		4XRA_1:74,5FHS_1:74
	]:13.73
]

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 \((0.85)(0.8)(\frac{492 }{\log_{20} 492}-\frac{186}{\log_{20}186})=89.1\)
Status Protein1 Protein2 d d1/2
Query variables 4XRA_1 7TWZ_1 114 90
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]