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Parikh vectors
3VIB_1 1DUO_1 2ITY_1 Letter Amino acid
4 0 6 C Cysteine
9 11 8 H Histidine
17 9 25 I Isoleucine
21 18 33 L Leucine
2 4 18 P Proline
14 5 13 T Threonine
9 1 8 N Asparagine
15 7 21 D Aspartic acid
7 5 12 Q Glutamine
15 19 23 K Lycine
16 14 26 E Glutamic acid
4 2 12 M Methionine
12 6 8 F Phenylalanine
7 6 16 S Serine
23 17 20 A Alanine
13 4 16 R Arginine
8 12 20 G Glycine
5 2 6 W Tryptophan
3 3 13 Y Tyrosine
6 8 23 V Valine 

3VIB_1|Chains A, B, C, D|MtrR|Neisseria gonorrhoeae (485)
>1DUO_1|Chain A|SPERM WHALE METAQUOMYOGLOBIN VARIANT H93G|Physeter catodon (9755)
>2ITY_1|Chain A|EPIDERMAL GROWTH FACTOR RECEPTOR|HOMO SAPIENS (9606)
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 \(w=f(c)\)
3VIB , Knot 94 210 0.79 40 149 200 
MRKTKTEALKTKEHLMLAALETFYRKGIARTSLNEIAQAAGVTRDALYWHFKNKEDLFDALFQRICDDIENCIAQDAADAEGGSWTVFRHTLLHFFERLQSNDIHYKFHNILFLKCEHTEQNAAVIAIARKHQAIWREKITAVLTEAVENQDLADDLDKETAVIFIKSTLDGLIWRWFSSGESFDLGKTAPRIIGIMMDNLENHPCLRRK
1DUO , Knot 72 153 0.79 38 111 148 
VLSEGEWQLVLHVWAKVEADVAGHGQDILIRLFKSHPETLEKFDRFKHLKTEAEMKASEDLKKHGVTVLTALGAILKKKGHHEAELKPLAQSGATKHKIPIKYLEFISEAIIHVLHSRHPGDFGADAQGAMNKALELFRKDIAAKYKELGYQG
2ITY , Knot 145 327 0.85 40 210 321 
GEAPNQALLRILKETEFKKIKVLGSGAFGTVYKGLWIPEGEKVKIPVAIKELREATSPKANKEILDEAYVMASVDNPHVCRLLGICLTSTVQLITQLMPFGCLLDYVREHKDNIGSQYLLNWCVQIAKGMNYLEDRRLVHRDLAARNVLVKTPQHVKITDFGLAKLLGAEEKEYHAEGGKVPIKWMALESILHRIYTHQSDVWSYGVTVWELMTFGSKPYDGIPASEISSILEKGERLPQPPICTIDVYMIMVKCWMIDADSRPKFRELIIEFSKMARDPQRYLVIQGDERMHLPSPTDSNFYRALMDEEDMDDVVDADEYLIPQQG

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 \(c\).

\(|P_{f(3VIB_1)}(2) \setminus P_{f(1DUO_1)}(2)|=90\), \(|P_{f(1DUO_1)}(2) \setminus P_{f(3VIB_1)}(2)|=52\). 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 1:100000011000001111110010001110001001101111000110101000001101110010001000110011010110101100011011001000010001001111000000001111111000011100010111001100001100100001111100010111101100100101100110111111001000101000
Pair \(Z_2\) Length of longest common subsequence
3VIB_1,1DUO_1 142 4
3VIB_1,2ITY_1 185 4
1DUO_1,2ITY_1 175 4

Newick tree

 
[
	2ITY_1:95.54,
	[
		3VIB_1:71,1DUO_1:71
	]:24.54
]

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 \((0.85)(0.8)(\frac{363 }{\log_{20} 363}-\frac{153}{\log_{20}153})=63.4\)
Status Protein1 Protein2 d d1/2
Query variables 3VIB_1 1DUO_1 80 67.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]