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Parikh vectors
6FWP_1 3HQU_1 5DRV_1 Letter Amino acid
23 12 7 P Proline
26 12 4 T Threonine
25 14 2 I Isoleucine
21 17 11 L Leucine
16 21 7 K Lycine
11 4 0 W Tryptophan
26 10 8 Y Tyrosine
23 14 6 A Alanine
14 5 9 H Histidine
11 7 6 Q Glutamine
8 4 5 M Methionine
17 9 8 F Phenylalanine
19 14 7 R Arginine
21 21 6 D Aspartic acid
21 12 9 E Glutamic acid
32 25 12 G Glycine
32 11 11 S Serine
13 17 16 V Valine
24 10 7 N Asparagine
2 7 1 C Cysteine 

6FWP_1|Chain A|Glycosyl hydrolase family 71|Bacteroides xylanisolvens XB1A (657309)
>3HQU_1|Chain A|Egl nine homolog 1|Homo sapiens (9606)
>5DRV_1|Chain A|Ras GTPase-activating protein-binding protein 2|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 \(w=f(c)\)
6FWP , Knot 160 385 0.82 40 222 361 
MGSSHHHHHHSSGLVPRGSHMDDNNPSNSENNGGNNNLGTELDYDTFCFYYDWYGSEAIDGQYRHWAHAIAPDPNGGSGQNPGTIPGTQESIASNFYPQLGRYSSSDPNILTKHMDMFVMARTGVLALTWWNEQDETEAKRIGLILDAADKKKIKVCFHLEPYPSRNVQNLRENIVKLITRYGNHPAFYRKDGKPLFFIYDSYLIEPSEWEKLLSPGGSITIRNTAYDALMIGLWTSSPTVQRPFILNAHFDGFYTYFAATGFTYGSTPTNWVSMQKWAKENGKIFIPSVGPGYIDTRIRPWNGSVIRTRTDGQYYDAMYRKAIEAGVSAISITSFNEWHEGSQIEPAVPYTSSEFTYLDYENREPDYYLTRTAYWVGKFRESKQ
3HQU , Knot 111 246 0.82 40 175 238 
PNGQTKPLPALKLALEYIVPCMNKHGICVVDDFLGKETGQQIGDEVRALHDTGKFTDGQLVSQKSDSSKDIRGDKITWIEGKEPGCETIGLLMSSMDDLIRHCNGKLGSYKINGRTKAMVACYPGNGTGYVRHVDNPNGDGRCVTCIYYLNKDWDAKVSGGILRIFPEGKAQFADIEPKFDRLLFFWSDRRNPHEVQPAYATRYAITVWYFDADERARAKVKYLTGEKGVRVELNKPSDSVGKDVF
5DRV , Knot 71 142 0.82 38 112 137 
GSHMVMEKPSPLLVGREFVRQYYTLLNKAPEYLHRFYGRNSSYVHGGVDASGKPQEAVYGQNDIHHKVLSLNFSECHTKIRHVDAHATLSDGVVVQVMGLLSNSGQPERKFMQTFVLAPEGSVPNKFYVHNDMFRYEDEVFG

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 \(c\).

\(|P_{f(6FWP_1)}(2) \setminus P_{f(3HQU_1)}(2)|=114\), \(|P_{f(3HQU_1)}(2) \setminus P_{f(6FWP_1)}(2)|=67\). 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 1:1100000000001111010010000100000011000110010000101000101001101000011011110101101001101110000110010101100000010110001011111001111101100000001001111101100001010101010100010010001101100010011100001011111000011010010011011101010001001111111000101001111010101100011101100100100110100110001011110111101000101101011000001000011000110111011010010010010010111100000100100000010001000101110100000
Pair \(Z_2\) Length of longest common subsequence
6FWP_1,3HQU_1 181 4
6FWP_1,5DRV_1 184 4
3HQU_1,5DRV_1 173 4

Newick tree

 
[
	6FWP_1:92.78,
	[
		3HQU_1:86.5,5DRV_1:86.5
	]:6.28
]

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 \((0.85)(0.8)(\frac{631 }{\log_{20} 631}-\frac{246}{\log_{20}246})=108.\)
Status Protein1 Protein2 d d1/2
Query variables 6FWP_1 3HQU_1 134 111
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]