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Parikh vectors
4EOR_1 2ZNB_1 1KZD_1 Letter Amino acid
10 8 0 H Histidine
16 17 2 I Isoleucine
38 19 8 L Leucine
19 11 4 P Proline
17 18 7 T Threonine
16 16 5 D Aspartic acid
21 15 5 K Lycine
16 4 5 F Phenylalanine
10 9 2 Y Tyrosine
23 18 13 V Valine
17 9 9 E Glutamic acid
15 15 7 S Serine
4 5 2 W Tryptophan
16 21 8 G Glycine
16 1 8 R Arginine
8 12 14 N Asparagine
3 3 4 C Cysteine
9 11 3 Q Glutamine
5 5 2 M Methionine
19 15 7 A Alanine

4EOR_1|Chains A, C|Cyclin-dependent kinase 2|Homo sapiens (9606)
>2ZNB_1|Chains A, B|METALLO-BETA-LACTAMASE|Bacteroides fragilis (817)
>1KZD_1|Chains A[auth 1], B[auth 2]|MANNOSE-BINDING PROTEIN C|Rattus norvegicus (10116)
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 \(w=f(c)\)
4EOR , Knot 135 298 0.86 40 191 284
SMENFQKVEKIGEGTYGVVYKARNKLTGEVVALKKIRLDTETEGVPSTAIREISLLKELNHPNIVKLLDVIHTENKLYLVFEFLHQDLKKFMDASALTGIPLPLIKSYLFQLLQGLAFCHSHRVLHRDLKPQNLLINTEGAIKLADFGLARAFGVPVRTYTHEVVTLWYRAPEILLGCKYYSTAVDIWSLGCIFAEMVTRRALFPGDSEIDQLFRIFRTLGTPDEVVWPGVTSMPDYKPSFPKWARQDFSKVVPPLDEDGRSLLSQMLHYDPNKRISAKAALAHPFFQDVTKPVPHLR
2ZNB , Knot 107 232 0.83 40 164 225
AQKSVKISDDISITQLSDKVYTYVSLAEIEGWGMVPSNGMIVINNHQAALLDTPINDAQTETLVNWVADSLHAKVTTFIPNHWHGDCIGGLGYLQKKGVQSYANQMTIDLAKEKGLPVPEHGFTDSLTVSLDGMPLQCYYLGGGHATDNIVVWLPTENILFGGCMLKDNQATSIGNISDADVTAWPKTLDKVKAKFPSARYVVPGHGDYGGTELIEHTKQIVNQYIESTSKP
1KZD , Knot 59 115 0.81 38 92 111
NVGKKYFMSSVRRMPLNRAKALCSELQGTVATPRNAEENRAIQNVAKDVAFLGITDQRTENVFEDLTGNRVRYTNWNEGEPNNVGSGENCVVLLTNGKWNDVPCSDSFLVVCEFS

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 \(c\).

\(|P_{f(4EOR_1)}(2) \setminus P_{f(2ZNB_1)}(2)|=93\), \(|P_{f(2ZNB_1)}(2) \setminus P_{f(4EOR_1)}(2)|=66\). 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 1:0100100100110100111001000101011110010100000111001100101100100101101101100000101110110001001101011011111110001101101111000001100010100111000111011011110111111000000110110011011110000001101101101110110001111100010011011001101001111110011000101101100010011111000100110011000100010101111011100100111010
Pair \(Z_2\) Length of longest common subsequence
4EOR_1,2ZNB_1 159 3
4EOR_1,1KZD_1 177 3
2ZNB_1,1KZD_1 178 3

Newick tree

 
[
	1KZD_1:91.62,
	[
		4EOR_1:79.5,2ZNB_1:79.5
	]:12.12
]

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 \((0.85)(0.8)(\frac{530 }{\log_{20} 530}-\frac{232}{\log_{20}232})=85.3\)
Status Protein1 Protein2 d d1/2
Query variables 4EOR_1 2ZNB_1 105 93.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]