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Parikh vectors
6SDU_1 8IXB_1 5HLT_1 Letter Amino acid
4 21 11 R Arginine
6 37 20 E Glutamic acid
7 20 17 F Phenylalanine
33 21 14 S Serine
18 34 11 V Valine
9 36 13 A Alanine
5 16 14 Q Glutamine
10 27 23 D Aspartic acid
0 12 4 C Cysteine
11 19 5 H Histidine
7 32 21 L Leucine
13 19 28 K Lycine
3 10 5 M Methionine
15 20 10 P Proline
19 16 25 N Asparagine
11 4 2 W Tryptophan
11 19 15 Y Tyrosine
13 30 10 T Threonine
16 27 28 I Isoleucine
26 37 12 G Glycine 

6SDU_1|Chains A, B, C, D|Type III effector NopAA|Sinorhizobium fredii USDA 257 (1185652)
>8IXB_1|Chains A, D[auth E], G[auth F], J[auth H]|Tubulin alpha-1A chain|Mus musculus (10090)
>5HLT_1|Chains A, B|Restriction endonuclease R.BpuJI|Bacillus pumilus (1408)
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 \(w=f(c)\)
6SDU , Knot 106 237 0.81 38 153 224 
MHHHHHHITSLYKKAGMPIWSSHAPYGSFSRDGYSWNNDVWGPRPGPQTISVSGVNRWSVWSDQPNTPGIKSYPHVAFNIGKPLSSINTLSSSFNQEVPTGGAWDVAYDIWDSSNKHEIMLWTNYTGNSDGSGNVKPISYHYAPSGAAIPVYSNVNVGGATWNVFEGEGPDGHKVISLLRTSKTNSGTVDIKSILQWIKSKGYFGDIEVGSVQYGVEITSSPGGKNFNFNNWSVTSK
8IXB , Knot 192 457 0.85 40 257 430 
MRECISIHVGQAGVQIGNACWELYCLEHGIQPDGQMPSDKTIHHHHHHGGGDDSFNTFFSETGAGKHVPRAVFVDLEPTVIDEVRTGTYRQLFHPEQLITGKEDAANNYARGHYTIGKEIIDLVLDRIRKLADQCTGLQGFLVFHSFGGGTGSGFTSLLMERLSVDYGKKSKLEFSIYPAPQVSTAVVEPYNSILTTHTTLEHSDCAFMVDNEAIYDICRRNLDIERPTYTNLNRLIGQIVSSITASLRFDGALNVDLTEFQTNLVPYPRIHFPLATYAPVISAEKAYHEQLSVAEITNACFEPANQMVKCDPRHGKYMACCLLYRGDVVPKDVNAAIATIKTKRTIQFVDWCPTGFKVGINYQPPTVVPGGDLAKVQRAVCMLSNTTAIAEAWARLDHKFDLMYAKRAFVHWYVGEGMEEGEFSEAREDMAALEKDYEEVGVDSVEGEGEEEGEEY
5HLT , Knot 127 288 0.83 40 178 271 
GSHMNYNPEEQFRCTIIRGKAKNMLDNLLPAYANIIDDICPCDKASFVKDFNNRLIEILGEETTKKTLDNHRTEIAGKLFGMFYEDDEVIFPSGRTNKYIEDSDQPAFFKDICFKFQFPNGMDKLDKVIEKVGAKIQIRQFPYILQVLLTADNNNIQLSKDDIAYYVLNSLQVLQGKIKPIEVIEKIIEDRSNDITKKVRHPGKETSYSMQHIREQLNYLELANLIRIDGNLVKLNYREAENINYIAQFWGNKPEFNAYKYDFTSEDDKKSFFKDWQQYYSNVNSHKS

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 \(c\).

\(|P_{f(6SDU_1)}(2) \setminus P_{f(8IXB_1)}(2)|=44\), \(|P_{f(8IXB_1)}(2) \setminus P_{f(6SDU_1)}(2)|=148\). 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 1:100000010010001111110001101010001001000111101110010101100101100010011100010111011011001001000100011011110110011000000011110000100010101011000011011111100010111101011010110100110110000000101010011011000101101011010011010001110010100101000
Pair \(Z_2\) Length of longest common subsequence
6SDU_1,8IXB_1 192 6
6SDU_1,5HLT_1 187 5
8IXB_1,5HLT_1 169 4

Newick tree

 
[
	6SDU_1:97.93,
	[
		5HLT_1:84.5,8IXB_1:84.5
	]:13.43
]

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 \((0.85)(0.8)(\frac{694 }{\log_{20} 694}-\frac{237}{\log_{20}237})=127.\)
Status Protein1 Protein2 d d1/2
Query variables 6SDU_1 8IXB_1 164 122
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]