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Parikh vectors
6PKP_1 3WNR_1 2FZF_1 Letter Amino acid
5 3 31 E Glutamic acid
22 4 2 T Threonine
33 11 15 A Alanine
7 6 5 Q Glutamine
4 2 4 H Histidine
8 3 18 K Lycine
6 4 5 F Phenylalanine
9 8 6 P Proline
2 1 3 W Tryptophan
5 3 0 C Cysteine
13 4 4 D Aspartic acid
33 9 8 G Glycine
11 6 13 I Isoleucine
14 11 21 L Leucine
37 10 5 S Serine
19 8 6 V Valine
12 5 9 R Arginine
5 3 10 M Methionine
17 5 8 Y Tyrosine
17 8 2 N Asparagine 

6PKP_1|Chain A|Proteinase K|Parengyodontium album (37998)
>3WNR_1|Chains A, B, C|Macrophage migration inhibitory factor|Homo sapiens (9606)
>2FZF_1|Chains A, B|hypothetical protein|Pyrococcus furiosus (2261)
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 \(w=f(c)\)
6PKP , Knot 119 279 0.80 40 169 265 
AAQTNAPWGLARISSTSPGTSTYYYDESAGQGSCVYVIDTGIEASHPEFEGRAQMVKTYYYSSRDGNGHGTHCAGTVGSRTYGVAKKTQLFGVKVLDDNGSGQYSTIIAGMDFVASDKNNRNCPKGVVASLSLGGGYSSSVNSAAARLQSSGVMVAVAAGNNNADARNYSPASEPSVCTVGASDRYDRRSSFSNYGSVLDIFGPGTSILSTWIGGSTRSISGTSMATPHVAGLAAYLMTLGKTTAASACRYIADTANKGDLSNIPFGTVNLLAYNNYQA
3WNR , Knot 60 114 0.83 40 98 111 
PMFIVNTNVPRASVPDGFLSELTQQLAQATGKPPQYIAVHVVPDQLMAFGGSSEPCALCSLHSIGKIGGAQNRSYSKLLCGLLAERLRISPDRVYINYYDMNAANVGWNNSTFA
2FZF , Knot 80 175 0.78 38 113 157 
GSTIQEVREGLPIEKMADFSLEELLGMAIKAEIGAREFYKSLAEKIKIEALKEKINWLAEEEKKHEALLRKLYSQMFPGKEVVFPKEHIGPELQPVARELEKVQDIIDLIRWAMKAEEIAAEFYLKLEEMVKEEEKKRLMRYLADMERGHYYTLRAEYELLLNWEMYSQMMHIGP

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 \(c\).

\(|P_{f(6PKP_1)}(2) \setminus P_{f(3WNR_1)}(2)|=114\), \(|P_{f(3WNR_1)}(2) \setminus P_{f(6PKP_1)}(2)|=43\). 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 1:110001111110100001100000000011010010110011010010101010110000000001010100011011000011100001111011000101000011111011100000000101111010111100001001110100011111111100010100001100101001110000000001000101101111100110011110000101001101011111101101100011010001100100101001111010111000001
Pair \(Z_2\) Length of longest common subsequence
6PKP_1,3WNR_1 157 3
6PKP_1,2FZF_1 186 3
3WNR_1,2FZF_1 155 3

Newick tree

 
[
	6PKP_1:88.72,
	[
		3WNR_1:77.5,2FZF_1:77.5
	]:11.22
]

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 \((0.85)(0.8)(\frac{393 }{\log_{20} 393}-\frac{114}{\log_{20}114})=84.9\)
Status Protein1 Protein2 d d1/2
Query variables 6PKP_1 3WNR_1 105 74.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]