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Parikh vectors
2FLK_1 6GOB_1 4OBH_1 Letter Amino acid
16 12 3 A Alanine
9 12 13 G Glycine
0 1 1 H Histidine
15 8 11 L Leucine
11 6 7 K Lycine
5 10 0 S Serine
4 11 4 R Arginine
0 8 2 C Cysteine
11 2 4 E Glutamic acid
5 7 8 T Threonine
1 3 1 Y Tyrosine
9 6 6 V Valine
8 14 5 N Asparagine
2 3 5 Q Glutamine
7 6 13 I Isoleucine
7 2 2 M Methionine
7 3 2 F Phenylalanine
3 2 7 P Proline
8 7 3 D Aspartic acid
1 6 2 W Tryptophan 

2FLK_1|Chain A|Chemotaxis protein cheY|Salmonella typhimurium (99287)
>6GOB_1|Chain A|Lysozyme C|Gallus gallus (9031)
>4OBH_1|Chains A, B, C, D|HIV-1 Protease|Human immunodeficiency virus type 1 (11685)
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 \(w=f(c)\)
2FLK , Knot 62 129 0.77 36 96 122 
MADKELKFLVVDDFSTMRRIVRNLLKELGFNNVEEAEDGVDALNKLQAGGFGFIISDWNMPNMDGLELLKTIRADSAMSALPVLMVTAEAKKENIIAAAQAGASGYVVKPFTAATLEEKLNKIFEKLGM
6GOB , Knot 66 129 0.82 40 104 127 
KVFGRCELAAAMKRHGLDNYRGYSLGNWVCAAKFESNFNTQATNRNTDGSTDYGILQINSRWWCNDGRTPGSRNLCNIPCSALLSSDITASVNCAKKIVSDGNGMNAWVAWRNRCKGTDVQAWIRGCRL
4OBH , Knot 53 99 0.82 38 81 94 
PQITLWKRPLVTIRIGGQLKEALLNTGADDTVLEEMNLPGKWKPKMIGGIGGFIKVRQYDQIPIEICGHKAIGTVLVGPTPVNIIGRNLLTQIGCTLNF

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 \(c\).

\(|P_{f(2FLK_1)}(2) \setminus P_{f(6GOB_1)}(2)|=66\), \(|P_{f(6GOB_1)}(2) \setminus P_{f(2FLK_1)}(2)|=74\). 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 1:110001011110010010011001100111001001001101100101111111100101101011011001010011011111110101000011111011101011011011010001001100111
Pair \(Z_2\) Length of longest common subsequence
2FLK_1,6GOB_1 140 3
2FLK_1,4OBH_1 119 4
6GOB_1,4OBH_1 141 3

Newick tree

 
[
	6GOB_1:73.48,
	[
		2FLK_1:59.5,4OBH_1:59.5
	]:13.98
]

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 \((0.85)(0.8)(\frac{258 }{\log_{20} 258}-\frac{129}{\log_{20}129})=40.5\)
Status Protein1 Protein2 d d1/2
Query variables 2FLK_1 6GOB_1 55 53
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]