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Parikh vectors
4PKG_1 4KKC_1 2ZAV_1 Letter Amino acid
12 11 9 N Asparagine
6 3 3 C Cysteine
19 12 24 K Lycine
12 27 9 F Phenylalanine
27 25 22 T Threonine
29 53 16 A Alanine
22 10 19 D Aspartic acid
28 14 19 E Glutamic acid
26 64 33 L Leucine
17 18 4 M Methionine
19 19 23 P Proline
23 11 20 S Serine
16 9 9 Y Tyrosine
21 30 25 V Valine
30 37 23 I Isoleucine
4 6 2 W Tryptophan
18 21 13 R Arginine
11 13 5 Q Glutamine
28 57 36 G Glycine
9 6 8 H Histidine 

4PKG_1|Chain A|Actin, alpha skeletal muscle|Oryctolagus cuniculus (9986)
>4KKC_1|Chains A, B|H(+)/Cl(-) exchange transporter ClcA|Escherichia coli (83333)
>2ZAV_1|Chains A, B|Arginase-1|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 \(w=f(c)\)
4PKG , Knot 160 377 0.84 40 220 362 
MCDEDETTALVCDNGSGLVKAGFAGDDAPRAVFPSIVGRPRHQGVMVGMGQKDSYVGDEAQSKRGILTLKYPIEHGIITNWDDMEKIWHHTFYNELRVAPEEHPTLLTEAPLNPKANREKMTQIMFETFNVPAMYVAIQAVLSLYASGRTTGIVLDSGDGVTHNVPIYEGYALPHAIMRLDLAGRDLTDYLMKILTERGYSFVTTAEREIVRDIKEKLCYVALDFENEMATAASSSSLEKSYELPDGQVITIGNERFRCPETLFQPSFIGMESAGIHETTYNSIMKCDIDIRKDLYANNVMSGGTTMYPGIADRMQKEITALAPSTMKIKIIAPPERKYSVWIGGSILASLSTFQQMWITKQEYDEAGPSIVHRKCF
4KKC , Knot 173 446 0.78 40 205 400 
MRRRQLIRQLLERDKTPLAILFMAAVVGTLVGLAAVAFDKGVAWLQNQRMGALVHTADNYPLLLTVAFLCSAVLAMFGYFLVRKYAPEAGGSGIPEIEGALEDQRPVRWWRVLPVKFFGGLGTLGGGMVLGRAGPTVQIGGNIGRMVLDIFRLKGDEARHTLLATGAAAGLAAAFNAPLAGILFIIEEMRPQFRYTLISIKAVFIGVIMSTIMYRIFNHEVALIDVGKLSDAPLNTLWLYLILGIIFGIFGPIFNKWVLGMQDLLHRVHGGNITKWVLMGGAIGGLCGLLGFVAPATSGGGFNLIPIATAGNFSMGMLVFIFVARVITTLLCFSSGAPGGIFAPMLALGTVLGTAFGMVAVELFPQYHLEAGTFAIAGMGALLAASIRAPLTGIILVLEMTDNYQLILPMIITGLGATLLAQFTGGKPLYSAILARTLAKQEAEQK
2ZAV , Knot 141 322 0.84 40 193 311 
MSAKSRTIGIIGAPFSKGQPRGGVEEGPTVLRKAGLLEKLKEQECDVKDYGDLPFADIPNDSPFQIVKNPRSVGKASEQLAGKVAEVKKNGRISLVLGGDHSLAIGSISGHARVHPDLGVIWVDAHTDINTPLTTTSGNLHGQPVSFLLKELKGKIPDVPGFSWVTPCISAKDIVYIGLRDVDPGEHYILKTLGIKYFSMTEVDRLGIGKVMEETLSYLLGRKKRPIHLSFDVDGLDPSFTPATGTPVVGGLTYREGLYITEEIYKTGLLSGLDIMEVNPSLGKTPEEVTRTVNTAVAITLACFGLAREGNHKPIDYLNPPK

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 \(c\).

\(|P_{f(4PKG_1)}(2) \setminus P_{f(4KKC_1)}(2)|=91\), \(|P_{f(4KKC_1)}(2) \setminus P_{f(4PKG_1)}(2)|=76\). 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 1:10000000111000101110111110011011110111010001111111000001100100001110100110011100100100110001000101110001011001110101000010011100101111011101110101010001111001011000111001011101110101110010001101100010011001000110010001001110100011011000010000011010110110001001001101011110011100000001100010100010100110110010111100100010111100101011111000001111101110100100111000000011101100001
Pair \(Z_2\) Length of longest common subsequence
4PKG_1,4KKC_1 167 4
4PKG_1,2ZAV_1 161 3
4KKC_1,2ZAV_1 162 4

Newick tree

 
[
	4KKC_1:82.83,
	[
		4PKG_1:80.5,2ZAV_1:80.5
	]:2.33
]

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 \((0.85)(0.8)(\frac{823 }{\log_{20} 823}-\frac{377}{\log_{20}377})=120.\)
Status Protein1 Protein2 d d1/2
Query variables 4PKG_1 4KKC_1 150 142
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]