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Parikh vectors
1IYJ_1 3KDG_1 3NWV_1 Letter Amino acid
3 1 1 W Tryptophan
1 8 5 Y Tyrosine
4 12 18 K Lycine
3 11 3 S Serine
1 6 7 T Threonine
3 10 12 G Glycine
2 9 3 H Histidine
2 9 4 P Proline
1 10 2 R Arginine
13 8 3 D Aspartic acid
0 3 2 C Cysteine
3 7 5 N Asparagine
6 13 6 L Leucine
2 9 3 M Methionine
0 21 8 I Isoleucine
3 7 3 F Phenylalanine
3 10 3 V Valine
4 11 6 A Alanine
2 9 2 Q Glutamine
14 23 8 E Glutamic acid 

1IYJ_1|Chains A, C|Deleted in split hand/split foot protein 1|Homo sapiens (9606)
>3KDG_1|Chains A, B|DNA mismatch repair protein mutL|Bacillus subtilis (1423)
>3NWV_1|Chains A, B, C, D|Cytochrome c|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 \(w=f(c)\)
1IYJ , Knot 37 70 0.74 36 52 66 
MSEKKQPVDLGLLEEDDEFEEFPAEDWAGLDEDEDAHVWEDNWDDDNVEDDFSNQLRAELEKHGYKMETS
3KDG , Knot 95 197 0.85 40 146 192 
GAMDRVPIMYPIGQMHGTYILAQNENGLYIIDQHAAQERIKYEYFREKVGEVEPEVQEMIVPLTFHYSTNEALIIEQHKQELESVGVFLESFGSNSYIVRCHPAWFPKGEEAELIEEIIQQVLDSKNIDIKKLREEAAIMMSCKGSIKANRHLRNDEIKALLDDLRSTSDPFTCPHGRPIIIHHSTYEMEKMFKRVM
3NWV , Knot 54 104 0.80 40 87 100 
GDVEKGKKIFIMKCSQCHTVEKGGKHKTGPNLHGLFGRKTSQAPGYSYTAANKNKGIIWGEDTLMEYLENPKKYIPGTKMIFVGIKKKEERADLIAYLKKATNE

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 \(c\).

\(|P_{f(1IYJ_1)}(2) \setminus P_{f(3KDG_1)}(2)|=29\), \(|P_{f(3KDG_1)}(2) \setminus P_{f(1IYJ_1)}(2)|=123\). 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 1:1000001101111000001001110011110000010110001000010001000101010001001000
Pair \(Z_2\) Length of longest common subsequence
1IYJ_1,3KDG_1 152 3
1IYJ_1,3NWV_1 109 2
3KDG_1,3NWV_1 157 3

Newick tree

 
[
	3KDG_1:83.47,
	[
		1IYJ_1:54.5,3NWV_1:54.5
	]:28.97
]

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 \((0.85)(0.8)(\frac{267 }{\log_{20} 267}-\frac{70}{\log_{20}70})=63.7\)
Status Protein1 Protein2 d d1/2
Query variables 1IYJ_1 3KDG_1 82 54
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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