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Parikh vectors
3HIT_1 2DRX_1 8VNQ_1 Letter Amino acid
1 0 0 H Histidine
13 0 0 I Isoleucine
16 0 0 K Lycine
9 0 0 S Serine
22 0 6 A Alanine
7 0 0 D Aspartic acid
0 0 4 C Cysteine
17 0 7 T Threonine
15 0 0 E Glutamic acid
18 10 4 G Glycine
11 0 0 F Phenylalanine
11 0 0 Y Tyrosine
27 0 0 V Valine
15 0 0 Q Glutamine
27 2 0 L Leucine
2 0 0 W Tryptophan
11 18 0 P Proline
11 0 0 R Arginine
24 0 0 N Asparagine
2 0 0 M Methionine 

3HIT_1|Chains A, B|Vacuolar saporin|Saponaria officinalis (3572)
>2DRX_1|Chains A, B, C|collagen like peptide|null
>8VNQ_1|Chains A[auth C], B[auth D]|DNA (5'-D(*TP*TP*GP*AP*CP*TP*CP*TP*CP*TP*TP*AP*AP*GP*AP*GP*AP*GP*TP*CP*A)-3')|synthetic construct (32630)
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 \(w=f(c)\)
3HIT , Knot 115 259 0.82 38 163 252 
VIIYELNLQGTTKAQYSTFLKQLRDDIKDPNLHYGGTNLPVIKRPVGPPKFLRVNLKASTGTVSLAVQRSNLYVAAYLAKNNNKQFRAYYFKGFQITTNQLNNLFPEATGVSNQQELGYGESYPQIQNAAGVTRQQAGLGIKKLAESMTKVNGVARVEKDEALFLLIVVQMVGEAARFKYIENLVLNNFDTAKEVEPVPDRVIILENNWGLLSRAAKTANNGVFQTPLVLTSYAVPGVEWRVTTVAEVEIGIFLNVDNN
2DRX , Knot 5 30 0.18 6 5 6 
PPGPPGPPGPPGLPGLPGPPGPPGPPGPPG
8VNQ , Knot 10 21 0.48 8 11 15 
TTGACTCTCTTAAGAGAGTCA

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 \(c\).

\(|P_{f(3HIT_1)}(2) \setminus P_{f(2DRX_1)}(2)|=158\), \(|P_{f(2DRX_1)}(2) \setminus P_{f(3HIT_1)}(2)|=0\). 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 1:1110010101000100001100100010010100110011110011111011010101001010111000010111011000000101001011010000100111010110000011010001010011110000111110011001001011101000011111111011101101001001110010010010111001111000111100110010011100111100011111010100110101111101000
Pair \(Z_2\) Length of longest common subsequence
3HIT_1,2DRX_1 158 3
3HIT_1,8VNQ_1 162 3
2DRX_1,8VNQ_1 16 1

Newick tree

 
[
	3HIT_1:92.26,
	[
		2DRX_1:8,8VNQ_1:8
	]:84.26
]

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 \((0.85)(0.8)(\frac{289 }{\log_{20} 289}-\frac{30}{\log_{20}30})=85.9\)
Status Protein1 Protein2 d d1/2
Query variables 3HIT_1 2DRX_1 108 56
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]