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Parikh vectors
6UEP_1 8ZLP_1 2LHE_1 Letter Amino acid
10 8 7 E Glutamic acid
7 2 0 M Methionine
12 2 2 F Phenylalanine
18 3 0 S Serine
15 18 4 V Valine
7 1 2 N Asparagine
3 0 0 C Cysteine
7 3 1 Q Glutamine
13 14 2 G Glycine
17 2 4 L Leucine
20 14 9 K Lycine
18 16 7 A Alanine
11 0 0 R Arginine
9 1 0 H Histidine
17 1 5 I Isoleucine
7 1 3 Y Tyrosine
8 2 2 D Aspartic acid
12 0 0 P Proline
8 10 7 T Threonine
0 0 1 W Tryptophan 

6UEP_1|Chains A, D[auth B]|TATA-box-binding protein 1|Arabidopsis thaliana (3702)
>8ZLP_1|Chains A, B, C, D[auth F], E[auth G], F[auth H]|Alpha-synuclein|Homo sapiens (9606)
>2LHE_1|Chain A|Gb98|artificial gene (32630)
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 \(w=f(c)\)
6UEP , Knot 95 219 0.78 38 142 205 
MSSHHHHHHSSGLVPRGSHMTDQGLEGSNPVDLSKHPSGIVPTLQNIVSTVNLDCKLDLKAIALQARNAEYNPKRFAAVIMRIREPKTTALIFASGKMVCTGAKSEDFSKMAARKYARIVQKLGFPAKFKDFKIQNIVGSCDVKFPIRLEGLAYSHAAFSSYEPELFPGLIYRMKVPKIVLLIFVSGKIVITGAKMRDETYKAFENIYPVLSEFRKIQQ
8ZLP , Knot 42 99 0.65 34 56 82 
XMDVFMKGLSKAKEGVVAAAEKTKQGVAEAAGKTKEGVLYVGSKTKEGVVHGVATVAEKTKEQVTNVGGAVVTGVTAVAQKTVEGAGSIAAATGFVKKD
2LHE , Knot 29 56 0.69 28 44 52 
TTYKLILNLKQAKEEAIKEAVDAGIAEKYFKLIANAKTVEGVWTYKDEIKTFTVTE

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 \(c\).

\(|P_{f(6UEP_1)}(2) \setminus P_{f(8ZLP_1)}(2)|=115\), \(|P_{f(8ZLP_1)}(2) \setminus P_{f(6UEP_1)}(2)|=29\). 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 1:100000000001111010010001101001101000101111010011001010001010111101001000100111111010010001111101011001100001001110001011001111101001010011100010111010111000111000010111111001011011111110101110110100000011001011100100100
Pair \(Z_2\) Length of longest common subsequence
6UEP_1,8ZLP_1 144 3
6UEP_1,2LHE_1 140 3
8ZLP_1,2LHE_1 68 5

Newick tree

 
[
	6UEP_1:79.60,
	[
		2LHE_1:34,8ZLP_1:34
	]:45.60
]

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 \((0.85)(0.8)(\frac{318 }{\log_{20} 318}-\frac{99}{\log_{20}99})=68.5\)
Status Protein1 Protein2 d d1/2
Query variables 6UEP_1 8ZLP_1 87 61.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]