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Parikh vectors
6SMC_1 4MPR_1 7PFG_1 Letter Amino acid
27 24 10 E Glutamic acid
9 7 2 W Tryptophan
11 16 6 Y Tyrosine
14 25 7 N Asparagine
26 20 12 T Threonine
17 30 13 D Aspartic acid
14 6 1 C Cysteine
25 50 18 L Leucine
23 40 7 V Valine
30 13 2 K Lycine
7 14 5 M Methionine
14 76 25 A Alanine
12 23 10 R Arginine
15 21 16 Q Glutamine
15 38 8 G Glycine
12 18 3 H Histidine
13 25 9 I Isoleucine
12 15 8 F Phenylalanine
16 40 5 P Proline
32 35 15 S Serine 

6SMC_1|Chains A, B, C, D|Interleukin-12 subunit beta|Mus musculus (10090)
>4MPR_1|Chain A|Benzoylformate decarboxylase|Pseudomonas putida (303)
>7PFG_1|Chain A|Ferritin|Synechococcus sp. (strain CC9311) (64471)
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 \(w=f(c)\)
6SMC , Knot 152 344 0.86 40 217 323 
MCPQKLTISWFAIVLLVSPLMAMWELEKDVYVVEVDWTPDAPGETVNLTCDTPEEDDITWTSDQRHGVIGSGKTLTITVKEFLDAGQYTCHKGGETLSHSHLLLHKKENGIWSTEILKNFKNKTFLKCEAPNYSGRFTCSWLVQRNMDLKFNIKSSSSSPDSRAVTCGMASLSAEKVTLDQRDYEKYSVSCQEDVTCPTAEETLPIELALEARQQNKYENYSTSFFIRDIIKPDPPKNLQMKPLKNSQVEVSWEYPDSWSTPHSYFSLKFFVRIQRKKEKMKETEEGCNQKGAFLVEKTSTEVQCKGGNVCVQAQDRYYNSSCSKWACVPCRVRSGTKHHHHHH
4MPR , Knot 213 536 0.83 40 244 485 
MASVHGTTYELLRRQGIDTVFGNPGSNELPFLKDFPEDFRYILALQEACVVGIADGYAQASRKPAFINLHSAAGTGNAMGALSNAWNSHSPLIVTAGQQTRAMIGVEALLTNVDAANLPRPLVKWSYEPASAAEVPHAMSEAIHMASMAPQGPVYLSVPYDDWDKDADPQSHHLFDRHVSSSVRLNDQDLDILVKALNSASNPAIVLGPDVDAANANADCVMLAERLKAPVWVAPSAPRCPFPTRHPCFRGLMPAGIAAISQLLEGHDVVLVIGAPVFRYHQYDPGQYLKPGTRLISVTCDPLEAARAPMGDAIVADIGAMASALANLVEESSRQLPTAAPEPAKVDQDAGRLHPETVFDTLNDMAPENAIYLNESTSTTAQMWQRLNMRNPGSYYFCAAGGLGFALPAAIGVQLAEPERQVIAVIGDGSANYSISALWTAAQYNIPTIFVIMNNGTYGALRWFAGVLEAENVPGLDVPGIDFRALAKGYGVQALKADNLEQLKGSLQEALSAKGPVLIEVSTVSPVKRSHHHHHH
7PFG , Knot 88 182 0.83 40 135 174 
MATDVAQGPNGRALAESMNPDLLSAIQQHISIERYASVTYLAMSIWCAERELAGFYQFFDGEAKDEQSHAVHFTQYLIARSQSNDLQTLDAPRQNWDSLASLMATAFQMEADTTSSIQSVYALAERNSDTRTTVFLDPLIAAQIQSEDQFAYLLGRVKFANGDPTALLVIDNELRAGQTQRG

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 \(c\).

\(|P_{f(6SMC_1)}(2) \setminus P_{f(4MPR_1)}(2)|=77\), \(|P_{f(4MPR_1)}(2) \setminus P_{f(6SMC_1)}(2)|=104\). 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 1:10100101011111111011111101000101101010101110010100001000010100000011110100101010011011000000110010000111000001110001100100001100011000101000111000101010100000010001100111010100101000000000100000100101000111011101000000000000111001101011001010110000101010010010010001010111010000001000001000011111000000100011010101000000000001101100100100000000
Pair \(Z_2\) Length of longest common subsequence
6SMC_1,4MPR_1 181 6
6SMC_1,7PFG_1 204 3
4MPR_1,7PFG_1 167 4

Newick tree

 
[
	6SMC_1:10.35,
	[
		4MPR_1:83.5,7PFG_1:83.5
	]:16.85
]

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 \((0.85)(0.8)(\frac{880 }{\log_{20} 880}-\frac{344}{\log_{20}344})=144.\)
Status Protein1 Protein2 d d1/2
Query variables 6SMC_1 4MPR_1 177 146
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]