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Parikh vectors
6GUC_1 4DDB_1 5BUX_1 Letter Amino acid
10 1 10 H Histidine
16 6 8 I Isoleucine
21 6 12 K Lycine
16 3 14 F Phenylalanine
16 11 11 R Arginine
3 8 2 C Cysteine
9 3 3 Q Glutamine
17 2 15 E Glutamic acid
10 3 2 Y Tyrosine
16 7 9 D Aspartic acid
15 10 8 S Serine
4 6 0 W Tryptophan
19 12 13 A Alanine
18 12 14 G Glycine
39 8 15 L Leucine
23 6 22 V Valine
8 14 3 N Asparagine
5 2 6 M Methionine
20 2 15 P Proline
17 7 9 T Threonine 

6GUC_1|Chains A, C|Cyclin-dependent kinase 2|Homo sapiens (9606)
>4DDB_1|Chain A|Lysozyme C|Gallus gallus (9031)
>5BUX_1|Chains A, B|3-hydroxyacyl-[acyl-carrier-protein] dehydratase FabZ|Yersinia pestis KIM10+ (187410)
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 \(w=f(c)\)
6GUC , Knot 136 302 0.85 40 193 288 
GPGSMENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHLRL
4DDB , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
5BUX , Knot 88 191 0.80 38 134 177 
HHHHHHSSGLVPRGSMTTDTHTLHIEEILDLLPHRFPFLLVDRVLDFEEGKFLRAVKNVSFNEPFFQGHFPGKPIFPGVLILEAMAQATGILAFKSRGKLEPGELYYFAGIDEARFKRPVVPGDQMIMEVEFVKERRGLTRFTGVAKVDGEIVCTATMMCARSKPAAPAESVVVKPDVVKPDVVNPVVKES

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 \(c\).

\(|P_{f(6GUC_1)}(2) \setminus P_{f(4DDB_1)}(2)|=141\), \(|P_{f(4DDB_1)}(2) \setminus P_{f(6GUC_1)}(2)|=52\). 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 1:11101001001001101001110010001010111100101000001110011001011001001011011011000001011101100010011010110111111100011011011110000011000101001110001110110111101111110000001101100110111100000011011011011101100011111000100110110011010011111100110001011011000100111110001001100110001000101011110111001001110101
Pair \(Z_2\) Length of longest common subsequence
6GUC_1,4DDB_1 193 3
6GUC_1,5BUX_1 161 4
4DDB_1,5BUX_1 170 3

Newick tree

 
[
	4DDB_1:94.15,
	[
		6GUC_1:80.5,5BUX_1:80.5
	]:13.65
]

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 \((0.85)(0.8)(\frac{431 }{\log_{20} 431}-\frac{129}{\log_{20}129})=90.6\)
Status Protein1 Protein2 d d1/2
Query variables 6GUC_1 4DDB_1 116 83
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]