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Parikh vectors
6SJM_1 2OKJ_1 2UZY_1 Letter Amino acid
8 17 1 M Methionine
14 30 18 V Valine
12 26 21 D Aspartic acid
19 34 14 E Glutamic acid
7 15 6 H Histidine
12 32 22 I Isoleucine
13 37 28 K Lycine
14 23 21 S Serine
4 9 1 C Cysteine
11 39 13 G Glycine
10 27 9 F Phenylalanine
9 25 19 T Threonine
2 8 2 W Tryptophan
4 18 5 Y Tyrosine
20 34 11 A Alanine
8 23 26 N Asparagine
5 22 12 Q Glutamine
33 46 43 L Leucine
13 20 13 P Proline
12 19 4 R Arginine 

6SJM_1|Chain A|Retinoic acid receptor RXR-alpha|Homo sapiens (9606)
>2OKJ_1|Chains A, B|Glutamate decarboxylase 1|Homo sapiens (9606)
>2UZY_1|Chains A, C|INTERNALIN B|LISTERIA MONOCYTOGENES (169963)
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 \(w=f(c)\)
6SJM , Knot 108 230 0.85 40 164 219 
SMDMPVERILEAELAVEPKTETYVEANMGLNPSSPNDPVTNICQAADKQLFTLVEWAKRIPHFSELPLDDQVILLRAGWNELLIASFSHRSIAVKDGILLATGLHVHRNSAHSAGVGAIFDRVLTELVSKMRDMQMDKTELGCLRAIVLFNPDSKGLSNPAEVEALREKVYASLEAYCKHKYPEQPGRFAKLLLRLPALRSIGLKCLEHLFFFKLIGDTPIDTFLMEMLE
2OKJ , Knot 209 504 0.86 40 272 483 
TDFSNLFARDLLPAKNGEEQTVQFLLEVVDILLNYVRKTFDRSTKVLDFHHPHQLLEGMEGFNLELSDHPESLEQILVDCRDTLKYGVRTGHPRFFNQLSTGLDIIGLAGEWLTSTANTNMFTYEIAPVFVLMEQITLKKMREIVGWSSKDGDGIFSPGGAISNMYSIMAARYKYFPEVKTKGMAAVPKLVLFTSEQSHYSIKKAGAALGFGTDNVILIKCNERGKIIPADFEAKILEAKQKGYVPFYVNATAGTTVYGAFDPIQEIADICEKYNLWLHVDAAWGGGLLMSRKHRHKLNGIERANSVTWNPHKMMGVLLQCSAILVKEKGILQGCNQMCAGYLFQPDKQYDVSYDTGDKAIQCGRHVDIFKFWLMWKAKGTVGFENQINKCLELAEYLYAKIKNREEFEMVFNGEPEHTNVCFWYIPQSLRGVPDSPQRREKLHKVAPKIKALMMESGTTMVGYQPQGDKANFFRMVISNPAATQSDIDFLIEEIERLGQDLHH
2UZY , Knot 117 289 0.76 40 151 258 
GAMETITVPTPIKQIFSDDAFAETIKDNLKKKSVTDAVTQNELNSIDQIIANNSDIKSVQGIQYLPNVTKLFLNGNKLTDIKPLANLKNLGWLFLDENKVKDLSSLKDLKKLKSLSLEHNGISDINGLVHLPQLESLYLGNNKITDITVLSRLTKLDTLSLEDNQISDIVPLAGLTKLQNLYLSKNHISDLRALAGLKNLDVLELFSQECLNKPINHQSNLVVPNTVKNTDGSLVTPEIISDDGDYEKPNVKWHLPEFTNEVSFIFYQPVTIGKAKARFHGRVTQPLKE

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 \(c\).

\(|P_{f(6SJM_1)}(2) \setminus P_{f(2OKJ_1)}(2)|=36\), \(|P_{f(2OKJ_1)}(2) \setminus P_{f(6SJM_1)}(2)|=144\). 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 1:01011100110101110100000101011101001001100100110001101101100110100111000111101110011110100001110011111011010000100111111100110011001001010000110101111101000110011010110001010101000000100110110111011110011100100111101110011001110110
Pair \(Z_2\) Length of longest common subsequence
6SJM_1,2OKJ_1 180 4
6SJM_1,2UZY_1 163 3
2OKJ_1,2UZY_1 183 3

Newick tree

 
[
	2OKJ_1:93.63,
	[
		6SJM_1:81.5,2UZY_1:81.5
	]:12.13
]

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 \((0.85)(0.8)(\frac{734 }{\log_{20} 734}-\frac{230}{\log_{20}230})=140.\)
Status Protein1 Protein2 d d1/2
Query variables 6SJM_1 2OKJ_1 178 127.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]