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Parikh vectors
5ZHY_1 5PTY_1 5XMO_1 Letter Amino acid
1 7 4 M Methionine
6 13 7 D Aspartic acid
0 1 1 C Cysteine
15 8 2 Q Glutamine
7 10 3 H Histidine
8 5 4 I Isoleucine
6 6 14 K Lycine
4 6 6 F Phenylalanine
9 11 16 A Alanine
2 6 8 P Proline
11 7 2 S Serine
13 5 6 T Threonine
2 5 3 Y Tyrosine
6 13 7 E Glutamic acid
15 4 7 N Asparagine
10 7 14 G Glycine
19 18 6 L Leucine
1 0 0 W Tryptophan
11 10 13 V Valine
2 14 1 R Arginine 

5ZHY_1|Chains A, B, C, D, E, F|Spike glycoprotein, Spike glycoprotein|Human coronavirus 229E (11137)
>5PTY_1|Chains A, B|Bromodomain-containing protein 1|Homo sapiens (9606)
>5XMO_1|Chain A|Pseudoazurin|Achromobacter cycloclastes (223)
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 \(w=f(c)\)
5ZHY , Knot 67 148 0.75 38 100 134 
ENQKILAASFNKAMTNIVDAFTGVNDAITQTSQALQTVATALNKIQDVVNQQGNSLNHLTSQLRQNFQLVPRGSGGSGGSGGLEVLFQGPDLVVEQYNQTILNLTSEISTLENKSAELNYTVQKLQTLIDNINSTLVDLKWLHHHHHH
5PTY , Knot 74 156 0.79 38 120 151 
MHHHHHHSSGVDLGTENLYFQSMEQVAMELRLTELTRLLRSVLDQLQDKDPARIFAQPVSLKEVPDYLDHIKHPMDFATMRKRLEAQGYKNLHEFEEDFDLIIDNCMKYNARDTVFYRAAVRLRDQGGVVLRQARREVDSIGLEEASGMHLPERPA
5XMO , Knot 62 124 0.80 38 93 118 
ADFEVHMLNKGKDGAFVFEPASLKVAPGDTVTFIPKDKGHNVETIKGMIPDGAEAFKSKINENYKVTFTAPGVYGVKCTPHYGMGMVGVVQVGDAPANLEAVKGAKNPKKAQERLDAALAALGN

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 \(c\).

\(|P_{f(5ZHY_1)}(2) \setminus P_{f(5PTY_1)}(2)|=60\), \(|P_{f(5PTY_1)}(2) \setminus P_{f(5ZHY_1)}(2)|=80\). 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 1:0000111101001100110110110011000001100110110010011000100100100010001011101011011011101110110111000000110100010010000101000100100110010001101011000000
Pair \(Z_2\) Length of longest common subsequence
5ZHY_1,5PTY_1 140 6
5ZHY_1,5XMO_1 131 3
5PTY_1,5XMO_1 149 3

Newick tree

 
[
	5PTY_1:74.40,
	[
		5ZHY_1:65.5,5XMO_1:65.5
	]:8.90
]

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 \((0.85)(0.8)(\frac{304 }{\log_{20} 304}-\frac{148}{\log_{20}148})=47.9\)
Status Protein1 Protein2 d d1/2
Query variables 5ZHY_1 5PTY_1 60 56.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]