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Parikh vectors
1JFT_1 7KAS_1 3CDG_1 Letter Amino acid
0 11 17 D Aspartic acid
0 1 14 H Histidine
0 47 5 I Isoleucine
0 36 16 S Serine
0 29 9 F Phenylalanine
4 34 19 A Alanine
0 23 15 Q Glutamine
0 66 21 L Leucine
0 22 8 K Lycine
0 19 21 R Arginine
0 18 12 Y Tyrosine
0 26 15 V Valine
0 14 3 M Methionine
0 24 14 P Proline
5 30 22 T Threonine
0 5 9 W Tryptophan
0 13 5 N Asparagine
4 3 4 C Cysteine
0 19 26 E Glutamic acid
4 40 18 G Glycine 

1JFT_1|Chain A[auth B]|5'-D(*TP*AP*CP*GP*CP*AP*AP*AP*CP*GP*TP*TP*TP*GP*CP*GP*T)-3'|
>7KAS_1|Chain A|Protein transport protein SEC61|Saccharomyces cerevisiae BY4741 (1247190)
>3CDG_1|Chains A, E[auth C]|HLA class I histocompatibility antigen, alpha chain E|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 \(w=f(c)\)
1JFT , Knot 9 17 0.50 8 9 13 
TACGCAAACGTTTGCGT
7KAS , Knot 193 480 0.82 40 218 444 
MSSNRVLDLFKPFESFLPEVIAPERKVPYNQKLIWTGVSLLIFLILGQIPLYGIVSSETSDPLYWLRAMLASNRGTLLELGVSPIITSSMIFQFLQGTQLLQIRPESKQDRELFQIAQKVCAIILILGQALVVVMTGNYGAPSDLGLPICLLLIFQLMFASLIVMLLDELLSKGYGLGSGISLFTATNIAEQIFWRAFAPTTVNSGRGKEFEGAVIAFFHLLAVRKDKKRALVEAFYRTNLPNMFQVLMTVAIFLFVLYLQGFRYELPIRSTKVRGQIGIYPIKLFYTSNTPIMLQSALTSNIFLISQILFQKYPTNPLIRLIGVWGIRPGTQGPQMALSGLAYYIQPLMSLSEALLDPIKTIVYITFVLGSCAVFSKTWIEISGTSPRDIAKQFKDQGMVINGKRETSIYRELKKIIPTAAAFGGATIGALSVGSDLLGTLGSGASILMATTTIYGYYEAAAKEGGFTKNLVPGFSDLM
3CDG , Knot 120 273 0.82 40 178 261 
SHSLKYFHTSVSRPGRGEPRFISVGYVDDTQFVRFDNDAASPRMVPRAPWMEQEGSEYWDRETRSARDTAQIFRVNLRTLRGYYNQSEAGSHTLQWMHGCELGPDRRFLRGYEQFAYDGKDYLTLNEDLRSWTAVDTAAQISEQKSNDASEAEHQRAYLEDTCVEWLHKYLEKGKETLLHLEPPKTHVTHHPISDHEATLRCWALGFYPAEITLTWQQDGEGHTQDTELVETRPAGDGTFQKWAAVVVPSGEEQRYTCHVQHEGLPEPVTLRW

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 \(c\).

\(|P_{f(1JFT_1)}(2) \setminus P_{f(7KAS_1)}(2)|=3\), \(|P_{f(7KAS_1)}(2) \setminus P_{f(1JFT_1)}(2)|=212\). 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 1:01010111010001010
Pair \(Z_2\) Length of longest common subsequence
1JFT_1,7KAS_1 215 3
1JFT_1,3CDG_1 179 2
7KAS_1,3CDG_1 192 3

Newick tree

 
[
	7KAS_1:10.72,
	[
		1JFT_1:89.5,3CDG_1:89.5
	]:16.22
]

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 \((0.85)(0.8)(\frac{497 }{\log_{20} 497}-\frac{17}{\log_{20}17})=150.\)
Status Protein1 Protein2 d d1/2
Query variables 1JFT_1 7KAS_1 189 97.5
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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