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Parikh vectors
6ZKG_1 1ULP_1 7XHF_1 Letter Amino acid
5 21 8 A Alanine
1 3 8 R Arginine
1 5 8 Q Glutamine
17 12 11 L Leucine
4 2 0 W Tryptophan
23 14 15 V Valine
26 18 10 G Glycine
0 0 7 H Histidine
16 3 3 I Isoleucine
12 0 6 M Methionine
4 2 11 N Asparagine
8 9 5 E Glutamic acid
14 4 8 F Phenylalanine
3 10 5 P Proline
11 9 7 S Serine
4 12 7 D Aspartic acid
1 2 1 C Cysteine
5 0 7 K Lycine
10 20 5 T Threonine
10 6 7 Y Tyrosine 

6ZKG_10|Chain J|NADH-ubiquinone oxidoreductase chain 6|Ovis aries (9940)
>1ULP_1|Chain A|ENDOGLUCANASE C|Cellulomonas fimi (1708)
>7XHF_1|Chains A, B|Ras GTPase-activating protein-binding protein 1|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 \(w=f(c)\)
6ZKG , Knot 82 175 0.80 38 114 170 
MMTYIVFILSIIFVMGFVGFSSKPSPIYGGLGLIVSGGVGCGIVLNFGGSFLGLMVFLIYLGGMMVVFGYTTAMATEQYPEVWVSNKVVLGTFITGLLMEFLMVYYVLKDKEVEIVFKFNGMGDWVIYDTGDSGFFSEEAMGIAALYSYGTWLVIVTGWSLLIGVVVIMEITRGN
1ULP , Knot 67 152 0.73 34 95 143 
ASPIGEGTFDDGPEGWVAYGTDGPLDTSTGALCVAVPAGSAQYGVGVVLNGVAIEEGTTYTLRYTATASTDVTVRALVGQNGAPYGTVLDTSPALTSEPRQVTETFTASATYPATPAADDPEGQIAFQLGGFSADAWTLCLDDVALDSEVEL
7XHF , Knot 69 139 0.81 38 114 135 
MVMEKPSPLLVGREFVRQYYTLLNQAPDMLHRFYGKNSSYVHGGLDSNGKPADAVYGQKEIHRKVMSQNFTNCHTKIRHVDAHATLNDGVVVQVMGLLSNNNQALRRFMQTFVLAPEGSVANKFYVHNDIFRYQDEVFG

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 \(c\).

\(|P_{f(6ZKG_1)}(2) \setminus P_{f(1ULP_1)}(2)|=74\), \(|P_{f(1ULP_1)}(2) \setminus P_{f(6ZKG_1)}(2)|=55\). 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 1:1100111110111111111100010110111111101111011110111011111111101111111110001110000101110001111011011110111100110000101110101110111000100111000111111100010111110110111111111010010
Pair \(Z_2\) Length of longest common subsequence
6ZKG_1,1ULP_1 129 3
6ZKG_1,7XHF_1 138 4
1ULP_1,7XHF_1 133 3

Newick tree

 
[
	7XHF_1:68.81,
	[
		6ZKG_1:64.5,1ULP_1:64.5
	]:4.31
]

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 \((0.85)(0.8)(\frac{327 }{\log_{20} 327}-\frac{152}{\log_{20}152})=53.4\)
Status Protein1 Protein2 d d1/2
Query variables 6ZKG_1 1ULP_1 68 61
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]