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Parikh vectors
3LBF_1 4UCX_1 9AZW_1 Letter Amino acid
16 28 33 G Glycine
9 11 24 D Aspartic acid
18 8 25 Q Glutamine
6 17 15 K Lycine
4 4 5 M Methionine
13 15 12 I Isoleucine
7 9 18 S Serine
4 4 5 W Tryptophan
19 17 16 V Valine
17 4 25 R Arginine
4 11 9 N Asparagine
1 11 0 C Cysteine
15 18 15 E Glutamic acid
10 21 26 P Proline
10 14 14 T Threonine
3 9 18 Y Tyrosine
20 27 46 A Alanine
5 8 6 H Histidine
24 19 46 L Leucine
5 9 11 F Phenylalanine 

3LBF_1|Chains A, B, C, D|Protein-L-isoaspartate O-methyltransferase|Escherichia coli (83333)
>4UCX_1|Chains A, B, C|HYDROGENASE (NIFE) SMALL SUBUNIT HYDA|DESULFOVIBRIO FRUCTOSIVORANS (878)
>9AZW_1|Chain A|Beta-lactamase|Pseudomonas aeruginosa (287)
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 \(w=f(c)\)
3LBF , Knot 99 210 0.84 40 144 204 
SLMVSRRVQALLDQLRAQGIQDEQVLNALAAVPREKFVDEAFEQKAWDNIALPIGQGQTISQPYMVARMTELLELTPQSRVLEIGTGSGYQTAILAHLVQHVCSVERIKGLQWQARRRLKNLDLHNVSTRHGDGWQGWQARAPFDAIIVTAAPPEIPTALMTQLDEGGILVLPVGEEHQYLKRVRRRGGEFIIDTVEAVRFVPLVKGELA
4UCX , Knot 119 264 0.83 40 173 251 
LTAKHRPSVVWLHNAECGGCTEAAIRTIKPYIDALILDTISLDYQETIMAAAGEAAEAALHQALEGKDGYYLVVEGGLPTIDGGQWGMVAGHPMIETTKKAAAKAKGIICIGTCSAYGGVQKAKPNPSQAKGVSEALGVKTINIPGCPPNPINFVGAVVHVLTKGIPDLDENGRPKLFYGELVHDNCPRLPHFEASEFAPSFDSEEAKKGFCLYELGCKGPVTYNNCPKVLFNQVNWPVQAGHPCLGCSEPDFWDTMTPFYEQG
9AZW , Knot 154 369 0.82 38 200 344 
DEAPADRLKALVDAAVQPVMKANDIPGLAVAISLKGEPHYFSYGLASKEDGRRVTPETLFEIGSVSKTFTATLAGYALAQDKMRLDDRASQHWPALQGSRFDGISLLDLATYTAGGLPLQFPDSVQKDQAQIRDYYRQWQPTYAPGSQRLYSNPSIGLFGYLAARSLGQPFERLMEQQLFPALGLEQTHLDVPEAALAQYAQGYGKDDRPLRVGPGPLDAEGYGVKTSAADLLRFVDANLHPERLDRPWAQALDATHRGYYKVGDMTQGLGWEAYDWPISLKRLQAGNSMALQPHRIARLPAPQALEGQRLLNKTGSTNGFGAYVAFIPGRDLGLVILANRNYPNAERVKIAYAILSGLEQQAKVPLKR

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 \(c\).

\(|P_{f(3LBF_1)}(2) \setminus P_{f(4UCX_1)}(2)|=63\), \(|P_{f(4UCX_1)}(2) \setminus P_{f(3LBF_1)}(2)|=92\). 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 1:011100010111001010110000110111111000110011000110011111101001001011101001101010001101101010001111011001001001011010100010010100100001011011010111011110111101101110010011111111100000100100011011100101101111101011
Pair \(Z_2\) Length of longest common subsequence
3LBF_1,4UCX_1 155 3
3LBF_1,9AZW_1 146 3
4UCX_1,9AZW_1 187 5

Newick tree

 
[
	4UCX_1:89.75,
	[
		3LBF_1:73,9AZW_1:73
	]:16.75
]

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 \((0.85)(0.8)(\frac{474 }{\log_{20} 474}-\frac{210}{\log_{20}210})=76.7\)
Status Protein1 Protein2 d d1/2
Query variables 3LBF_1 4UCX_1 96 86.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]