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Parikh vectors
2KHA_1 2UZI_1 2DRS_1 Letter Amino acid
4 5 8 Q Glutamine
16 12 33 G Glycine
5 3 13 I Isoleucine
10 10 26 L Leucine
4 3 13 W Tryptophan
9 7 30 A Alanine
8 8 17 R Arginine
1 2 4 C Cysteine
5 7 25 Y Tyrosine
9 4 42 E Glutamic acid
2 2 12 M Methionine
5 14 19 S Serine
4 4 20 N Asparagine
9 4 19 K Lycine
7 2 19 P Proline
7 9 16 T Threonine
6 8 13 V Valine
10 4 25 D Aspartic acid
2 0 17 H Histidine
7 6 25 F Phenylalanine 

2KHA_1|Chain A|Beta-1,3-glucan-binding protein|Plodia interpunctella (58824)
>2UZI_1|Chain A[auth H]|ANTI-RAS FV HEAVY CHAIN|HOMO SAPIENS (9606)
>2DRS_1|Chain A|Xylanase Y|Bacillus halodurans (272558)
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 \(w=f(c)\)
2KHA , Knot 68 130 0.84 40 105 124 
GAMGSQPRAQQYVVPSAKLEAIYPRGLRVSIPDDGFSLFAFHGKLNEEMDGLEAGHWARDITKPKEGRWTFRDRNAKLKLGDKIYFWTYVIKDGLGYRQDNGEWTVTEFVNEDGTPADTSLEPKLGCFGG
2UZI , Knot 57 114 0.79 38 86 108 
EVQLLESGGGLVQPGGSLRLSCAASGFTFSTFSMNWVRQAPGKGLEWVSYISRTSKTIYYADSVKGRFTISRDNSKNTLYLQMNSLRAEDTAVYYCARGRFFDYWGQGTLVTVS
2DRS , Knot 166 396 0.83 40 223 376 
MEKTTEGAFYTREYRNLFKEFGYSEAEIQERVKDTWEQLFGDNPETKIYYEVGDDLGYLLDTGNLDVRTEGMSYGMMMAVQMDRKDIFDRIWNWTMKNMYMTEGVHAGYFAWSCQPDGTKNSWGPAPDGEEYFALALFFASHRWGDGDEQPFNYSEQARKLLHTCVHNGEGGPGHPMWNRDNKLIKFIPEVEFSDPSYHLPHFYELFSLWANEEDRVFWKEAAEASREYLKIACHPETGLAPEYAYYDGTPNDEKGYGHFFSSSYRVAANIGLDAEWFGGSEWSAEEINKIQAFFADKEPEDYRRYKIDGEPFEEKSLHPVGLIATNAMGSLASVDGPYAKANVDLFWNTPVRTGNRRYYDNCLYLFAMLALSGNFKIWFPEGQEEEHLEHHHHHH

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 \(c\).

\(|P_{f(2KHA_1)}(2) \setminus P_{f(2UZI_1)}(2)|=77\), \(|P_{f(2UZI_1)}(2) \setminus P_{f(2KHA_1)}(2)|=58\). 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 1:1111001010001110101011010110101100110111101010001011011011001001001010100001010110010110011001110000010101001100010110001010110111
Pair \(Z_2\) Length of longest common subsequence
2KHA_1,2UZI_1 135 3
2KHA_1,2DRS_1 196 4
2UZI_1,2DRS_1 205 3

Newick tree

 
[
	2DRS_1:10.03,
	[
		2KHA_1:67.5,2UZI_1:67.5
	]:41.53
]

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 \((0.85)(0.8)(\frac{244 }{\log_{20} 244}-\frac{114}{\log_{20}114})=41.3\)
Status Protein1 Protein2 d d1/2
Query variables 2KHA_1 2UZI_1 54 49
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]