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Parikh vectors
4YAT_1 6ZMN_1 6BGF_1 Letter Amino acid
12 5 10 D Aspartic acid
4 5 8 Q Glutamine
18 11 16 E Glutamic acid
16 14 12 K Lycine
10 2 9 F Phenylalanine
11 4 14 S Serine
2 4 3 W Tryptophan
5 8 9 R Arginine
3 2 6 M Methionine
7 6 13 T Threonine
7 5 11 A Alanine
12 5 4 C Cysteine
5 6 11 G Glycine
4 6 11 H Histidine
20 14 17 L Leucine
8 3 6 Y Tyrosine
10 4 14 N Asparagine
14 7 7 P Proline
9 10 11 V Valine
7 4 8 I Isoleucine 

4YAT_1|Chains A, B|Transcription intermediary factor 1-alpha|Homo sapiens (9606)
>6ZMN_1|Chains A, B|Mothers against decapentaplegic homolog 3|Homo sapiens (9606)
>6BGF_1|Chain A|Cysteine dioxygenase type 1|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 \(w=f(c)\)
4YAT , Knot 92 184 0.87 40 140 178 
SPNEDWCAVCQNGGELLCCEKCPKVFHLSCHVPTLTNFPSGEWICTFCRDLSKPEVEYDCDAPSHNSEKKKTEGLVKLTPIDKRKCERLLLFLYCHEMSLAFQDPVPLTVPDYYKIIKNPMDLSTIKKRLQEDYSMYSKPEDFVADFRLIFQNCAEFNEPDSEVANAGIKLENYFEELLKNLYP
6ZMN , Knot 63 125 0.81 40 102 121 
GPAVKRLLGWKQGDEEEKWCEKAVKSLVKKLKKTGQLDELEKAITTQNVNTKCITIPRSLDGRLQVSHRKGLPHVIYCRLWRWPDLHSHHELRAMELCEFAFNMKKDEVCVNPYHYQRVETPVLP
6BGF , Knot 98 200 0.86 40 160 194 
SEQTEVLKPRTLADLIRILHQLFAGDEVNVEEVQAIMEAYESDPTEWAMYAKFDQYRYTRNLVDQGNGKFNLMILCWGEGHGSSIHDHTNSHCFLKMLQGNLKETLFAWPDKKSNEMVKKSERVLRENQCAYINDSVGLHRVENISHTEPAVSLHLYSPPFDTCHAFDQRTGHKNKVTMTFHSKFGIRTPNATSGSLENN

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 \(c\).

\(|P_{f(4YAT_1)}(2) \setminus P_{f(6ZMN_1)}(2)|=100\), \(|P_{f(6ZMN_1)}(2) \setminus P_{f(4YAT_1)}(2)|=62\). 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 1:0100010110001101100000101101000110100110101100100010010100000110000000000111010110000000111110000101110011110110000110011010010001000001000100111010111000101001000110111010001001100101
Pair \(Z_2\) Length of longest common subsequence
4YAT_1,6ZMN_1 162 3
4YAT_1,6BGF_1 178 3
6ZMN_1,6BGF_1 176 3

Newick tree

 
[
	6BGF_1:90.86,
	[
		4YAT_1:81,6ZMN_1:81
	]:9.86
]

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 \((0.85)(0.8)(\frac{309 }{\log_{20} 309}-\frac{125}{\log_{20}125})=57.0\)
Status Protein1 Protein2 d d1/2
Query variables 4YAT_1 6ZMN_1 73 60.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]