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Parikh vectors
4ESO_1 9OFH_1 7FZL_1 Letter Amino acid
4 8 2 H Histidine
14 47 8 I Isoleucine
9 24 14 K Lycine
5 14 5 M Methionine
18 32 12 T Threonine
10 32 11 D Aspartic acid
8 14 1 P Proline
15 40 6 R Arginine
17 44 9 E Glutamic acid
23 37 5 L Leucine
22 33 17 V Valine
0 3 2 W Tryptophan
34 33 6 A Alanine
9 20 4 N Asparagine
0 3 2 C Cysteine
6 17 2 Q Glutamine
30 42 11 G Glycine
12 28 6 F Phenylalanine
16 43 10 S Serine
3 13 2 Y Tyrosine 

4ESO_1|Chains A, B, C, D|Putative oxidoreductase|Sinorhizobium meliloti (266834)
>9OFH_1|Chains A, B, C, D, E, F|Circadian clock oscillator protein KaiC|Synechococcus elongatus (32046)
>7FZL_1|Chain A|Fatty acid-binding protein, adipocyte|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 \(w=f(c)\)
4ESO , Knot 112 255 0.81 36 157 244 
MVMGNYQGKKAIVIGGTHGMGLATVRRLVEGGAEVLLTGRNESNIARIREEFGPRVHALRSDIADLNEIAVLGAAAGQTLGAIDLLHINAGVSELEPFDQVSEASYDRQFAVNTKGAFFTVQRLTPLIREGGSIVFTSSVADEGGHPGMSVYSASKAALVSFASVLAAELLPRGIRVNSVSPGFIDTPTKGVAGITEAERAEFKTLGDNITPMKRNGTADEVARAVLFLAFEATFTTGAKLAVDGGLGQKLSTAA
9OFH , Knot 209 527 0.82 40 238 485 
DYKDDDDKMTSAEMTSPNNNSEHQAIAKMRTMIEGFDDISHGGLPIGRSTLVSGTSGTGKTLFSIQFLYNGIIEFDEPGVFVTFEETPQDIIKNARSFGWDLAKLVDEGKLFILDASPDPEGQEVVGGFDLSALIERINYAIQKYRARRVSIDSVTSVFQQYDASSVVRRELFRLVARLKQIGATTVMTTERIEEYGPIARYGVEEFVSDNVVILRNVLEGERRRRTLEILKLRGTSHMKGEYPFTITDHGINIFPLGAMRLTQRSSNVRVSSGVVRLDEMCGGGFFKDSIILATGATGTGKTLLVSRFVENACANKERAILFAYEESRAQLLRNAYSWGMDFEEMERQNLLKIVCAYPESAGLEDHLQIIKSEINDFKPARIAIDSLSALARGVSNNAFRQFVIGVTGYAKQEEITGLFTNTSDQFMGAHSITDSHIEEITDTIILLQYVEIRGEMSRAINVFKMRGSWHDKAIREFMISDKGPDIKDSFRNFERIISGSPTRITVDEKSELSRIVRGVQEKGPES
7FZL , Knot 69 135 0.83 40 104 129 
GSHMCDAFVGTWKLVSSENFDDYMKEVGVGFATRKVAGMAKPNMIISVNGDVITIKSESTFKNTEISFILGQEFDEVTADDRKVKSTITLDGGVLVHVQKWDGKSTTIKRKREDDKLVVECVMKGVTSTRVYERA

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 \(c\).

\(|P_{f(4ESO_1)}(2) \setminus P_{f(9OFH_1)}(2)|=31\), \(|P_{f(9OFH_1)}(2) \setminus P_{f(4ESO_1)}(2)|=112\). 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 1:111100010011111100111110100110111011101000001101000111010110001101001111111110011110110101110010110010010000011100011110100101110011011100011001101110100100111101101111011101101001011110010011111001001010011001011000101001101111111010100110111011110010011
Pair \(Z_2\) Length of longest common subsequence
4ESO_1,9OFH_1 143 3
4ESO_1,7FZL_1 141 3
9OFH_1,7FZL_1 180 4

Newick tree

 
[
	9OFH_1:84.56,
	[
		4ESO_1:70.5,7FZL_1:70.5
	]:14.06
]

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 \((0.85)(0.8)(\frac{782 }{\log_{20} 782}-\frac{255}{\log_{20}255})=145.\)
Status Protein1 Protein2 d d1/2
Query variables 4ESO_1 9OFH_1 178 132.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]