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Parikh vectors
1IQZ_1 3FNJ_1 4QHQ_1 Letter Amino acid
4 0 0 C Cysteine
1 4 15 Q Glutamine
4 10 27 K Lycine
6 7 10 E Glutamic acid
1 2 16 S Serine
4 11 8 T Threonine
4 6 8 Y Tyrosine
8 15 27 A Alanine
0 2 4 R Arginine
2 3 14 N Asparagine
8 5 12 I Isoleucine
2 4 1 M Methionine
0 2 2 W Tryptophan
5 9 24 V Valine
15 9 15 D Aspartic acid
6 10 14 G Glycine
0 8 1 H Histidine
2 18 24 L Leucine
3 1 6 F Phenylalanine
6 4 13 P Proline 

1IQZ_1|Chain A|Ferredoxin|Bacillus thermoproteolyticus (1427)
>3FNJ_1|Chains A, B, C, D, E, F|protein lp_1913|Lactobacillus plantarum (1590)
>4QHQ_1|Chain A|Lipoprotein|Burkholderia cenocepacia (216591)
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 \(w=f(c)\)
1IQZ , Knot 42 81 0.76 34 66 76 
PKYTIVDKETCIACGACGAAAPDIYDYDEDGIAYVTLDDNQGIVEVPDILIDDMMDAFEGCPTDSIKVADEPFDGDPNKFE
3FNJ , Knot 60 130 0.74 38 90 122 
MNDKKIELLTTYLSLYIDHHTVLADMQNATGKYVVLDVRNAPAQVKKDQIKGAIAMPAKDLATRIGELDPAKTYVVYDWTGGTTLGKTALLVLLSAGFEAYELAGALEGWKGMQLPVETLADLEHHHHHH
4QHQ , Knot 108 241 0.82 38 149 231 
VIKVGTVAGPDSEVWQVVQKVAKEKEGLNVKVIEFNDYVQPNAALDSGDLDANSFQHQPYLDSQVKQRGYKIVSAGLTYISPIGVYSKKFKSLKDLPQGAKLAVPNDPSNENRALLLLQTQGVIKLKAGAGTGGNNATVLDIAENPKKLKISELDAAQLPRVLSDVDAAVINTNYALAANLQPTKDAIALESLTSPYANLIAVRAKDKDQPWVKKLVKAYQSPEVKEFIKKQFKGSMVASF

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 \(c\).

\(|P_{f(1IQZ_1)}(2) \setminus P_{f(3FNJ_1)}(2)|=47\), \(|P_{f(3FNJ_1)}(2) \setminus P_{f(1IQZ_1)}(2)|=71\). 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 1:100011000001101101111101000000111010100001110110111001101101010001011001101010010
Pair \(Z_2\) Length of longest common subsequence
1IQZ_1,3FNJ_1 118 2
1IQZ_1,4QHQ_1 157 3
3FNJ_1,4QHQ_1 139 3

Newick tree

 
[
	4QHQ_1:78.53,
	[
		1IQZ_1:59,3FNJ_1:59
	]:19.53
]

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 \((0.85)(0.8)(\frac{211 }{\log_{20} 211}-\frac{81}{\log_{20}81})=42.7\)
Status Protein1 Protein2 d d1/2
Query variables 1IQZ_1 3FNJ_1 53 43
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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