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Parikh vectors
5UPD_1 5EQX_1 4XQQ_1 Letter Amino acid
7 17 5 Y Tyrosine
5 9 5 H Histidine
13 40 8 I Isoleucine
16 39 6 L Leucine
12 24 0 P Proline
11 34 2 A Alanine
16 10 2 C Cysteine
8 24 3 Q Glutamine
7 21 7 F Phenylalanine
10 43 2 T Threonine
2 4 0 W Tryptophan
17 26 8 R Arginine
11 35 6 D Aspartic acid
4 8 1 M Methionine
9 44 11 S Serine
11 46 8 V Valine
18 38 8 N Asparagine
21 35 8 E Glutamic acid
18 27 2 G Glycine
17 31 8 K Lycine 

5UPD_1|Chain A|Histone-lysine N-methyltransferase NSD3|Homo sapiens (9606)
>5EQX_1|Chain A|Desmoglein-3|Homo sapiens (9606)
>4XQQ_1|Chains A, B, C, D|Accessory gene regulator A|Staphylococcus aureus (strain COL) (93062)
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 \(w=f(c)\)
5UPD , Knot 106 233 0.82 40 167 228 
GQRESKEALEIEKNSRKPPPYKHIKANKVIGKVQIQVADLSEIPRCNCKPADENPCGLESECLNRMLQYECHPQVCPAGDRCQNQCFTKRLYPDAEIIKTERRGWGLRTKRSIKKGEFVNEYVGELIDEEECRLRIKRAHENSVTNFYMLTVTKDRIIDAGPKGNYSRFMNHSCNPNCETQKWTVNGDVRVGLFALCDIPAGMELTFNYNLDCLGNGRTECHCGADNCSGFLG
5EQX , Knot 222 555 0.84 40 253 509 
EWVKFAKPCREGEDNSKRNPIAKITSDYQATQKITYRISGVGIDQPPFGIFVVDKNTGDINITAIVDREETPSFLITCRALNAQGLDVEKPLILTVKILDINDNPPVFSQQIFMGEIEENSASNSLVMILNATDADEPNHLNSKIAFKIVSQEPAGTPMFLLSRNTGEVRTLTNSLDREQASSYRLVVSGADKDGEGLSTQCECNIKVKDVNDNFPMFRDSQYSARIEENILSSELLRFQVTDLDEEYTDNWLAVYFFTSGNEGNWFEIQTDPRTNEGILKVVKALDYEQLQSVKLSIAVKNKAEFHQSVISRYRVQSTPVTIQVINVREGIAFRPASKTFTVQKGISSKKLVDYILGTYQAIDEDTNKAASNVKYVMGRNDGGYLMIDSKTAEIKFVKNMNRDSTFIVNKTITAEVLAIDEYTGKTSTGTVYVRVPDFNDNCPTAVLEKDAVCSSSPSVVVSARTLNNRYTGPYTFALEDQPVKLPAVWSITTLNATSALLRAQEQIPPGVYHISLVLTDSQNNRCEMPRSLTLEVCQCDNRGICGTSHHHHHH
4XQQ , Knot 52 100 0.79 36 83 98 
SVETIELKRGSNSVYVQYDDIMFFESSTKSHRLIAHLDNRQIEFYGNLKELSQLDDRFFRCHNSFVVNRHNIESIDSKERIVYFKNKEHCYASVRNVKKI

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 \(c\).

\(|P_{f(5UPD_1)}(2) \setminus P_{f(5EQX_1)}(2)|=49\), \(|P_{f(5EQX_1)}(2) \setminus P_{f(5UPD_1)}(2)|=135\). 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 1:10000001101000000111000101001110101011010011000001100010110000100110000010101110000000100010101011000001111000001001011000110110000001010010000100101101000011011101000011000001000000101010101111110011111010100010011010000001100001111
Pair \(Z_2\) Length of longest common subsequence
5UPD_1,5EQX_1 184 3
5UPD_1,4XQQ_1 166 3
5EQX_1,4XQQ_1 214 4

Newick tree

 
[
	5EQX_1:10.78,
	[
		5UPD_1:83,4XQQ_1:83
	]:21.78
]

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 \((0.85)(0.8)(\frac{788 }{\log_{20} 788}-\frac{233}{\log_{20}233})=153.\)
Status Protein1 Protein2 d d1/2
Query variables 5UPD_1 5EQX_1 194 137.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]