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Parikh vectors
3OEM_1 2BML_1 1DFB_1 Letter Amino acid
10 5 6 N Asparagine
18 12 10 D Aspartic acid
17 7 9 E Glutamic acid
19 4 7 I Isoleucine
23 5 15 L Leucine
19 4 15 V Valine
7 0 5 C Cysteine
24 12 12 G Glycine
11 6 8 F Phenylalanine
3 9 3 W Tryptophan
10 12 10 Y Tyrosine
16 9 12 A Alanine
7 1 2 H Histidine
16 13 14 K Lycine
7 4 1 M Methionine
17 4 11 P Proline
17 6 34 S Serine
19 3 7 R Arginine
9 1 14 Q Glutamine
17 9 17 T Threonine 

3OEM_1|Chain A|Glutamate [NMDA] receptor subunit epsilon-4|Rattus norvegicus (10116)
>2BML_1|Chains A, B|AUTOLYSIN|STREPTOCOCCUS PNEUMONIAE (1313)
>1DFB_1|Chain A[auth L]|IGG1-KAPPA 3D6 FAB (LIGHT CHAIN)|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 \(w=f(c)\)
3OEM , Knot 128 286 0.84 40 186 277 
GDDTQHLTVATLEERPFVIVEPADPISGTCIRDSVPCRSQLNRTHSPPPDAPRPEKRCCKGFCIDILKRLAHTIGFSYDLYLVTNGKHGKKIDGVWNGMIGEVFYQRADMAIGSLTINEERSEIVDFSVPFVETGISVMVARGTTVSGLSDRKFQRPQEQYPPLKFGTVPNGSTEKNIRSNYPDMHSYMVRYNQPRVEEALTQLKAGKLDAFIYDAAVLNYMARKDEGCKLVTIGSGKVFATTGYGIALHKGSRWKRPIDLALLQFLGDDEIEMLERLWLSGICHN
2BML , Knot 59 126 0.75 38 90 110 
SYPKDKFEKINGTWYYFDSSGYMLADRWRKHTDGNWYWFDNSGEMATGWKKIADKWYYFNEEGAMKTGWVKYKDTWYYLDAKEGAMVSNAFIQSADGTGWYYLKPDGTLADRPEFTVEPDGLITVK
1DFB , Knot 99 212 0.83 40 147 200 
DIQMTQSPSTLSASVGDRVTITCRASQSISRWLAWYQQKPGKVPKLLIYKASSLESGVPSRFSGSGSGTEFTLTISSLQPDDFATYYCQQYNSYSFGPGTKVDIKRTVAAPSVFIFPPSDEQLKSGTASVVCLLNNFYPREAKVQWKVDNALQSGNSQESVTEQDSKDSTYSLSSTLTLSKADYEKHKVYACEVTHQGLSSPVTKSFNRGEC

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 \(c\).

\(|P_{f(3OEM_1)}(2) \setminus P_{f(2BML_1)}(2)|=135\), \(|P_{f(2BML_1)}(2) \setminus P_{f(3OEM_1)}(2)|=39\). 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 1:1000001011010001111101101101001000110000100000111011010000001101011001100111000101100100100101110111101100010111101010000001101011110011011110100101100001001000011101101101000001000010100011000010100110010110101110011110011000010011011010111001011110010010011011110111000101100111011000
Pair \(Z_2\) Length of longest common subsequence
3OEM_1,2BML_1 174 3
3OEM_1,1DFB_1 165 3
2BML_1,1DFB_1 157 3

Newick tree

 
[
	3OEM_1:86.77,
	[
		1DFB_1:78.5,2BML_1:78.5
	]:8.27
]

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 \((0.85)(0.8)(\frac{412 }{\log_{20} 412}-\frac{126}{\log_{20}126})=86.3\)
Status Protein1 Protein2 d d1/2
Query variables 3OEM_1 2BML_1 112 78
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]