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Parikh vectors
4ATK_1 3ZCF_1 3WPB_1 Letter Amino acid
2 3 10 M Methionine
1 5 17 Y Tyrosine
16 2 45 R Arginine
7 2 38 Q Glutamine
12 18 20 K Lycine
3 4 51 P Proline
1 2 19 C Cysteine
11 8 24 E Glutamic acid
6 3 34 D Aspartic acid
3 13 45 G Glycine
3 2 79 S Serine
2 3 38 V Valine
10 6 45 A Alanine
9 5 50 N Asparagine
13 6 154 L Leucine
2 3 41 F Phenylalanine
4 7 39 T Threonine
1 1 10 W Tryptophan
4 3 25 H Histidine
8 8 18 I Isoleucine 

4ATK_1|Chains A, B|MICROPHTHALMIA-ASSOCIATED TRANSCRIPTION FACTOR|MUS MUSCULUS (10090)
>3ZCF_1|Chains A, B, C, D|CYTOCHROME C|HOMO SAPIENS (9606)
>3WPB_1|Chain A|Toll-like receptor 9|Equus caballus (9796)
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 \(w=f(c)\)
4ATK , Knot 58 118 0.78 40 90 112 
ANIKRELTACIFPTESEARALAKERQKKDNHNLIERRRRFNINDRIKELGTLIPKSNDPDMRWNKGTILKASVDYIRKLQREQQRAKDLENRQKKLEHANRHLLLRVQELEMQARAHG
3ZCF , Knot 54 104 0.80 40 87 99 
GDVEKGKKIFIMKCSQCHTVEKGGKHKTGPNLHGLFGRKTGQAPGYSYTAANKNKGIIWGEDTLMEYLENPKKYIPGTKMIFVGIKKKEERADLIAYLKKATNE
3WPB , Knot 285 802 0.79 40 274 657 
RSPWQGTLPPFLPCELQPHGLVNCNWLFLKSVPHFSAAAPRDNVTSLSLLSNRIHHLHDSDFAQLSNLQKLNLKWNCPPAGLSPMHFPCHMTIEPNTFLAVPTLEELNLSYNGITTVPALPSSLVSLILSRTNILQLDPTSLTGLHALRFLYMDGNCYYKNPCGRALEVAPGALLGLGNLTHLSLKYNNLTTVPRSLPPSLEYLLLSYNHIVTLAPEDLANLTALRVLDVGGNCRRCDHARNPCVECPHKFPQLHSDTFSHLSRLEGLVLKDSSLYQLNPRWFRGLGNLTVLDLSENFLYDCITKTKAFQGLAQLRRLNLSFNYHKKVSFAHLTLAPSFGSLLSLQELDMHGIFFRSLSQKTLQPLARLPMLQRLYLQMNFINQAQLGIFKDFPGLRYIDLSDNRISGAVEPVATTGEVDGGKKVWLTSRDLTPGPLDTPSSEDFMPSCKNLSFTLDLSRNNLVTVQPEMFAQLSRLQCLRLSHNSISQAVNGSQFVPLTSLQVLDLSHNKLDLYHGRSFTELPRLEALDLSYNSQPFSMRGVGHNLSFVAQLPTLRYLSLAHNGIHSRVSQQLCSTSLWALDFSGNSLSQMWAEGDLYLRFFQGLRSLIRLDLSQNRLHTLLPCTLGNLPKSLQLLRLRNNYLAFFNWSSLTLLPNLETLDLAGNQLKALSNGSLPSGTQLQRLDVSRNSIIFVVPGFFALATRLRELNLSANALRTVEPSWFGFLAGSLEVLDVSANPLHCACGAAFVDFLLQVQAAVPGLPSRVKCGSPGQLQGRSIFAQDLRLCLDESLSWDEFLVPR

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 \(c\).

\(|P_{f(4ATK_1)}(2) \setminus P_{f(3ZCF_1)}(2)|=65\), \(|P_{f(3ZCF_1)}(2) \setminus P_{f(4ATK_1)}(2)|=62\). 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 1:1010001010111000010111000000000011000001010001001101110000101010010110101001001000000100100000010010001110100101010101
Pair \(Z_2\) Length of longest common subsequence
4ATK_1,3ZCF_1 127 3
4ATK_1,3WPB_1 220 3
3ZCF_1,3WPB_1 235 3

Newick tree

 
[
	3WPB_1:12.20,
	[
		4ATK_1:63.5,3ZCF_1:63.5
	]:62.70
]

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 \((0.85)(0.8)(\frac{222 }{\log_{20} 222}-\frac{104}{\log_{20}104})=38.0\)
Status Protein1 Protein2 d d1/2
Query variables 4ATK_1 3ZCF_1 46 44.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]