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Parikh vectors
4IFM_1 8GYP_1 1KZI_1 Letter Amino acid
1 16 13 E Glutamic acid
4 71 28 L Leucine
0 19 13 F Phenylalanine
5 21 13 V Valine
1 20 13 R Arginine
3 19 21 D Aspartic acid
0 9 5 C Cysteine
0 20 14 P Proline
2 16 14 T Threonine
2 15 9 Y Tyrosine
7 24 13 A Alanine
0 11 11 N Asparagine
2 19 12 K Lycine
0 6 7 W Tryptophan
1 11 16 Q Glutamine
7 10 18 G Glycine
0 12 11 H Histidine
6 21 16 I Isoleucine
2 9 7 M Methionine
3 43 10 S Serine 

4IFM_1|Chain A|PF1 FILAMENTOUS BACTERIOPHAGE|Pseudomonas phage Pf1 (2011081)
>8GYP_1|Chains A, B, C, D, E, F, G|Pannexin-3|Homo sapiens (9606)
>1KZI_1|Chains A, B|Thymidylate synthase|Escherichia coli (562)
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 \(w=f(c)\)
4IFM , Knot 26 46 0.72 28 40 44 
GVIDTSAVESAITDGQGDMKAIGGYIVGALVILAVAGLIYSMLRKA
8GYP , Knot 159 392 0.80 40 216 362 
MSLAHTAAEYMLSDALLPDRRGPRLKGLRLELPLDRIVKFVAVGSPLLLMSLAFAQEFSSGSPISCFSPSNFSIRQAAYVDSSCWDSLLHHKQDGPGQDKMKSLWPHKALPYSLLALALLMYLPVLLWQYAAVPALSSDLLFIISELDKSYNRSIRLVQHMLKIRQKSSDPYVFWNELEKARKERYFEFPLLERYLACKQRSHSLVATYLLRNSLLLIFTSATYLYLGHFHLDVFFQEEFSCSIKTGLLSDETHVPNLITCRLTSLSIFQIVSLSSVAIYTILVPVIIYNLTRLCRWDKRLLSVYEMLPAFDLLSRKMLGCPINDLNVILLFLRANISELISFSWLSVLCVLKDTTTQKHNIDTVVDFMTLLAGLEPSKPKHLTNSACDEHP
1KZI , Knot 124 264 0.87 40 186 256 
MKQYLELMQKVLDEGTQKNDRTGTGTLSIFGHQMRFNLQDGFPLVTTKRCHLRSIIHELLWFLQGDTNIAYLHENNVTIWDEWADENGDLGPVYGKQWRAWPTPDGRHIDQITTVLNQLKNDPDSRRIIVSAWNVGELDKMALAPCHAFFQFYVADGKLSCQLYQRSCDVFLGLPFNIASYALLVHMMAQQCDLEVGDFVWTGGDTHLYSNHMDQTHLQLSREPRPLPKLIIKRKPESIFDYRFEDFEIEGYDPHPGIKAPVAI

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 \(c\).

\(|P_{f(4IFM_1)}(2) \setminus P_{f(8GYP_1)}(2)|=13\), \(|P_{f(8GYP_1)}(2) \setminus P_{f(4IFM_1)}(2)|=189\). 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 1:1110001100110010101011110111111111111110011001
Pair \(Z_2\) Length of longest common subsequence
4IFM_1,8GYP_1 202 3
4IFM_1,1KZI_1 178 3
8GYP_1,1KZI_1 188 4

Newick tree

 
[
	8GYP_1:10.25,
	[
		4IFM_1:89,1KZI_1:89
	]:11.25
]

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 \((0.85)(0.8)(\frac{438 }{\log_{20} 438}-\frac{46}{\log_{20}46})=122.\)
Status Protein1 Protein2 d d1/2
Query variables 4IFM_1 8GYP_1 155 85.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]

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