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Parikh vectors
1QFL_1 2RMQ_1 4WLS_1 Letter Amino acid
14 0 8 N Asparagine
14 0 7 Q Glutamine
49 4 7 G Glycine
26 0 15 L Leucine
20 0 7 D Aspartic acid
5 4 0 C Cysteine
20 0 13 E Glutamic acid
3 0 3 Y Tyrosine
17 2 7 T Threonine
5 0 0 W Tryptophan
64 2 9 A Alanine
24 0 7 I Isoleucine
18 0 5 K Lycine
16 0 3 M Methionine
10 0 3 F Phenylalanine
19 0 10 S Serine
19 0 10 R Arginine
6 0 3 H Histidine
13 0 5 P Proline
27 0 6 V Valine 

1QFL_1|Chains A, B, C, D|PROTEIN (ACETOACETYL-COA THIOLASE)|Zoogloea ramigera (350)
>2RMQ_1|Chains A, B|DNA (5'-D(*(C4S)P*(S4G)P*(C4S)P*(S4G)P*(S4A)P*(S4A)P*(T49)P*(T49)P*(C4S)P*(S4G)P*(C4S)P*(S4G))-3')|null
>4WLS_1|Chains A, B|HTH-type transcriptional regulator CueR|Escherichia coli DH5[alpha] (668369)
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 \(w=f(c)\)
1QFL , Knot 155 389 0.79 40 196 360 
SIVIASAARTAVGSFNGAFANTPAHELGATVISAVLERAGVAAGEVNEVILGQVLPAGEGQNPARQAAMKAGVPQEATAWGMNQLCGSGLRAVALGMQQIATGDASIIVAGGMESMSMAPHCAHLRGGVKMGDFKMIDTMIKDGLTDAFYGYHMGTTAENVAKQWQLSRDEQDAFAVASQNKAEAAQKDGRFKDEIVPFIVKGRKGDITVDADEYIRHGATLDSMAKLRPAFDKEGTVTAGNASGLNDGAAAALLMSEAEASRRGIQPLGRIVSWATVGVDPKVMGTGPIPASRKALERAGWKIGDLDLVEANEAFAAQACAVNKDLGWDPSIVNVNGGAIAIGHPIGASGARILNTLLFEMKRRGARKGLATLCIGGGMGVAMCIESL
2RMQ , Knot 6 12 0.41 8 7 8 
CGCGAATTCGCG
4WLS , Knot 65 128 0.82 36 98 125 
MNISDVAKITGLTSKAIRFYEEKGLVTPPMRSENGYRTYTQQHLNELTLLRQARQVGFNLEESGELVNLFNDPQRHSADVKRRTLEKVAEIERHIEELQSMRDQLLALANASPGDDSADSPIIENLSG

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 \(c\).

\(|P_{f(1QFL_1)}(2) \setminus P_{f(2RMQ_1)}(2)|=191\), \(|P_{f(2RMQ_1)}(2) \setminus P_{f(1QFL_1)}(2)|=2\). 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 1:01111011001110101111001100111011011100111111010011110111110100110011101111001011110010101101111110011010101111111001011100101011101101011001100110011010011001001100101000000111110000101100010100011111101001010101000100110100110101110001010110101100111111110010100011011101101101110101110111110001100111011010110100111101011000111010110101111111011110110110011101000110011101011111111101001
Pair \(Z_2\) Length of longest common subsequence
1QFL_1,2RMQ_1 193 3
1QFL_1,4WLS_1 154 3
2RMQ_1,4WLS_1 105 1

Newick tree

 
[
	1QFL_1:96.13,
	[
		4WLS_1:52.5,2RMQ_1:52.5
	]:43.63
]

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 \((0.85)(0.8)(\frac{401 }{\log_{20} 401}-\frac{12}{\log_{20}12})=126.\)
Status Protein1 Protein2 d d1/2
Query variables 1QFL_1 2RMQ_1 152 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]