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Parikh vectors
4QJB_1 6HOW_1 8IUI_1 Letter Amino acid
13 10 17 D Aspartic acid
1 1 5 W Tryptophan
22 27 17 V Valine
28 15 15 E Glutamic acid
24 24 13 L Leucine
12 7 5 F Phenylalanine
9 6 12 Y Tyrosine
9 7 3 M Methionine
12 35 10 A Alanine
8 13 6 R Arginine
29 17 13 N Asparagine
15 22 26 G Glycine
16 13 4 H Histidine
27 14 9 I Isoleucine
24 10 11 K Lycine
11 13 10 T Threonine
7 7 7 C Cysteine
6 10 10 Q Glutamine
4 12 6 P Proline
19 25 19 S Serine 

4QJB_1|Chains A, B|Haloacid dehalogenase-like hydrolase|Plasmodium falciparum (36329)
>6HOW_1|Chains A, B, C|Pteridine reductase|Trypanosoma brucei brucei (5702)
>8IUI_1|Chains A, B|Fibroinase|Bombyx mori (7091)
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 \(w=f(c)\)
4QJB , Knot 129 296 0.82 40 184 287 
MAHHHHHHMHEIVDKNGKKVQKNNLNDEIKIIFTDLDGTLLNSENKVSEQNLESLIRAQEKGIKVVIATGRSIFSVENVIGEHVKKNRISLLPGIYMNGCVTFDEKGSRVIDRIMNNDLKMEIHEFSKQINISKYAIWFCLEKTYCFEINDCIREYMEVEALNPDVIEDNMLEGLTVYKVLFSLPENILENTLKLCREKFSHRINVANTFQSYVELFHQHTNKFEGVKEICKYYNISLNNALAMGDGENDIEMLSGLTHSVGVHNASEKVKNSAAYVGPSNNEHAISHVLKTFCDI
6HOW , Knot 127 288 0.83 40 179 275 
MGSSHHHHHHSSGLVPRGSHMEAPAAVVTGAAKRIGRAIAVKLHQTGYRVVIHYHNSAEAAVSLADELNKERSNTAVVCQADLTNSNVLPASCEEIINSCFRAFGRCDVLVNNASAFYPTPLVQGDHEDNSNGKTVETQVAELIGTNAIAPFLLTMSFAQRQKGTNPNCTSSNLSIVNLCDAMVDQPCMAFSLYNMGKHALVGLTQSAALELAPYGIRVNGVAPGVSLLPVAMGEEEKDKWRRKVPLGRREASAEQIADAVIFLVSGSAQYITGSIIKVDGGLSLVHA
8IUI , Knot 102 218 0.84 40 152 208 
LPEQVDWRKHGAVTDIKDQGKCGSCWSFSTTGALEGQHFRQSGYLVSLSEQNLIDCSEQYGNNGCNGGLMDNAFKYIKDNGGIDTEQTYPYEGVDDKCRYNPKNTGAEDVGFVDIPEGDEQKLMEAVATVGPVSVAIDASHTSFQLYSSGVYNEEECSSTDLDHGVLVVGYGTDEQGVDYWLVKNSWGRSWGELGYIKMIRNKNNRCGIASSASYPLV

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 \(c\).

\(|P_{f(4QJB_1)}(2) \setminus P_{f(6HOW_1)}(2)|=80\), \(|P_{f(6HOW_1)}(2) \setminus P_{f(4QJB_1)}(2)|=75\). 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 1:11000000100110001001000010001011100101011000001000010011010001101111010011010011100100001011111010101010001001100110001010100100010100011110100000101000100010101101011000110110100111011001100010100001000101100100010110000001011001000001010011111010001011011000111001000100011011100000110011001001
Pair \(Z_2\) Length of longest common subsequence
4QJB_1,6HOW_1 155 6
4QJB_1,8IUI_1 172 5
6HOW_1,8IUI_1 149 3

Newick tree

 
[
	4QJB_1:84.17,
	[
		6HOW_1:74.5,8IUI_1:74.5
	]:9.67
]

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 \((0.85)(0.8)(\frac{584 }{\log_{20} 584}-\frac{288}{\log_{20}288})=83.1\)
Status Protein1 Protein2 d d1/2
Query variables 4QJB_1 6HOW_1 104 102.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]