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Parikh vectors
5DDB_1 1UYP_1 3PWC_1 Letter Amino acid
24 30 14 K Lycine
22 17 7 P Proline
34 33 24 G Glycine
18 21 17 I Isoleucine
9 8 2 M Methionine
39 37 16 V Valine
27 22 2 R Arginine
24 26 6 D Aspartic acid
7 5 12 C Cysteine
60 36 13 L Leucine
19 25 4 F Phenylalanine
19 17 10 Y Tyrosine
45 12 15 A Alanine
22 5 9 Q Glutamine
33 42 6 E Glutamic acid
12 11 3 H Histidine
35 28 34 S Serine
19 23 10 T Threonine
8 11 4 W Tryptophan
13 23 15 N Asparagine 

5DDB_1|Chain A|Menin|Homo sapiens (9606)
>1UYP_1|Chains A, B, C, D, E, F|BETA-FRUCTOSIDASE|THERMOTOGA MARITIMA (243274)
>3PWC_1|Chain A|Cationic trypsin|Bos taurus (9913)
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 \(w=f(c)\)
5DDB , Knot 198 489 0.83 40 247 466 
GGSSSMGLKAAQKTLFPLRSIDDVVRLFAAELGREEPDLVLLSLVLGFVEHFLAVNRVGLTYFPVADLSIIAALYARFTAQIRGAVDLSLYPREGGVSSRELVKKVSDVIWNSLSRSYFKDRAHIQSLFSFITGTKLDSSGVAFAVVGACQALGLRDVHLALSEDHAWVVFGPNGEQTAEVTWHGKGNEDRRGQTVNAGVAERSWLYLKGSYMRCDRKMEVAFMVCAINPSIDLHTDSLELLQLQQKLLWLLYDLGHLERYPMALGNLADLEELEPTPGRPDPLTLYHKGIASAKTYYRDEHIYPYMYLAGYHCRNRNVREALQAWADTATVIQDYNYCREDEEIYKEFFEVANDVIPNLLKEAASLLEAGSQGSALQDPECFAHLLRFYDGICKWEEGSPTPVLHVGWATFLVQSLGRFEGQVRQKVRIVSVPAPAASPPPEGPVLTFQSEKMKGMKELLVATKINSSAIKLQLTAQSQVQMKKQKVS
1UYP , Knot 179 432 0.83 40 215 410 
LFKPNYHFFPITGWMNDPNGLIFWKGKYHMFYQYNPRKPEWGNICWGHAVSDDLVHWRHLPVALYPDDETHGVFSGSAVEKDGKMFLVYTYYRDPTHNKGEKETQCVVMSENGLDFVKYDGNPVISKPPEEGTHAFRDPKVNRSNGEWRMVLGSGKDEKIGRVLLYTSDDLFHWKYEGAIFEDETTKEIECPDLVRIGEKDILIYSITSTNSVLFSMGELKEGKLNVEKRGLLDHGTDFYAAQTFFGTDRVVVIGWLQSWLRTGLYPTKREGWNGVMSLPRELYVENNELKVKPVDELLALRKRKVFETAKSGTFLLDVKENSYEIVCEFSGEIELRMGNESEEVVITKSRDELIVDTTRSGVSGGEVRKSTVEDEATNRIRAFLDSCSVEFFFNDSIAFSFRIHPENVYNILSVKSNQVKLEVFELENIWL
3PWC , Knot 100 223 0.80 40 142 215 
IVGGYTCGANTVPYQVSLNSGYHFCGGSLINSQWVVSAAHCYKSGIQVRLGEDNINVVEGNEQFISASKSIVHPSYNSETYNNDIMLIKLKSAASLNSRVASISLPTSCASAGTQCLISGWGNTKSSGTSYPDVLKCLKAPILSDSSCKSASSFIITSNMFCAGYLEGGKDACQGDSGGPVVCSGKLQGIVSWGEGCAQKNKPGIYTKVCNYVSWIKQTIASN

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 \(c\).

\(|P_{f(5DDB_1)}(2) \setminus P_{f(1UYP_1)}(2)|=99\), \(|P_{f(1UYP_1)}(2) \setminus P_{f(5DDB_1)}(2)|=67\). 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 1:110001110110001111001001101111011000101111011111100111100111001111010111110101010101110101010011100001100100111001000010001010011011010010001111111110011110010111000011111110100010101010100000100101111000110101001000001011111011010101000010110100011111001101000111110110100101011010110100011101000000001010101110000000100110111001011000000000001000110110011101100110110110010110010011011010011001001010111011110111001101010100010110111111011101111010000101100111100100011010101000101000010
Pair \(Z_2\) Length of longest common subsequence
5DDB_1,1UYP_1 166 5
5DDB_1,3PWC_1 197 4
1UYP_1,3PWC_1 181 3

Newick tree

 
[
	3PWC_1:98.14,
	[
		5DDB_1:83,1UYP_1:83
	]:15.14
]

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 \((0.85)(0.8)(\frac{921 }{\log_{20} 921}-\frac{432}{\log_{20}432})=129.\)
Status Protein1 Protein2 d d1/2
Query variables 5DDB_1 1UYP_1 166 153.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]