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Parikh vectors
3KWG_1 8DGQ_1 8PMF_1 Letter Amino acid
7 7 0 H Histidine
13 18 0 I Isoleucine
12 22 0 L Leucine
6 16 0 P Proline
5 12 0 N Asparagine
8 22 3 G Glycine
7 12 0 K Lycine
8 8 0 M Methionine
6 11 0 F Phenylalanine
8 22 0 S Serine
1 16 0 Y Tyrosine
1 0 3 C Cysteine
9 21 0 E Glutamic acid
9 11 3 T Threonine
11 16 0 D Aspartic acid
3 10 0 Q Glutamine
2 4 0 W Tryptophan
8 18 0 V Valine
10 7 3 A Alanine
7 20 0 R Arginine 

3KWG_1|Chains A, B|Non-structural protein 1|Influenza A virus (381517)
>8DGQ_1|Chains A, B|Ras GTPase-activating protein 1|Homo sapiens (9606)
>8PMF_1|Chain A[auth B]|DNA|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 \(w=f(c)\)
3KWG , Knot 67 141 0.78 40 114 134 
MAHHHHHHVDDDDKMTMASTPASRYITDMTIEELSRDWFMLMPKQKVEGPLCIRIDQAIMDKNIMLKANFSVIFDRLETLILLRAFTEEGAIVGEISPLPSFPGHTIEDVKNAIGVLIGGLEANDNTVRVSKTLQRFAWGS
8DGQ , Knot 119 273 0.81 38 175 254 
GSTAPPTNQWYHGKLDRTIAEERLRQAGKSGSYLIRESDRRPGSFVLSFLSQMNVVNHFRIIAMSGDYYIGGRRFSSLSDLIGYYSHVSSLLKGEKLLYPVAPPEPVEDRRRVRAILPYTKVPDTDEISFLKGDMFIVHNELEDGWMWVTNLRTDEQGLIVEDLVEEVGREEDPHEGKIWFHGKISKQEAYNLLMTVGQVSSFLVRPSDNTPGDYSLYFRTNENIQRFKISPTPNNQFMMGGRYYNSIGDIIDHYRKEQIVEGYYLKEPVPMQ
8PMF , Knot 7 12 0.48 8 9 10 
CGCTAAACGGTT

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 \(c\).

\(|P_{f(3KWG_1)}(2) \setminus P_{f(8DGQ_1)}(2)|=53\), \(|P_{f(8DGQ_1)}(2) \setminus P_{f(3KWG_1)}(2)|=114\). 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 1:110000001000001011001100010010100100011111100010111010100111000111010101110010011110110001111101011101110010010011111111101000010100010011110
Pair \(Z_2\) Length of longest common subsequence
3KWG_1,8DGQ_1 167 3
3KWG_1,8PMF_1 121 2
8DGQ_1,8PMF_1 180 2

Newick tree

 
[
	8DGQ_1:93.95,
	[
		3KWG_1:60.5,8PMF_1:60.5
	]:33.45
]

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 \((0.85)(0.8)(\frac{414 }{\log_{20} 414}-\frac{141}{\log_{20}141})=81.9\)
Status Protein1 Protein2 d d1/2
Query variables 3KWG_1 8DGQ_1 102 77
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]