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Parikh vectors
5IFP_1 2DCA_1 5UAJ_1 Letter Amino acid
78 9 19 T Threonine
20 8 1 W Tryptophan
28 5 10 Q Glutamine
7 4 5 M Methionine
48 19 36 E Glutamic acid
100 35 20 G Glycine
49 14 24 I Isoleucine
46 11 16 K Lycine
94 23 17 S Serine
71 11 23 A Alanine
55 14 9 N Asparagine
61 15 30 V Valine
6 16 4 C Cysteine
40 9 4 F Phenylalanine
20 9 8 H Histidine
94 9 38 L Leucine
54 15 16 P Proline
59 11 5 Y Tyrosine
29 8 23 R Arginine
54 11 21 D Aspartic acid 

5IFP_1|Chain A|beta-galactosidase A|Aspergillus niger CBS 513.88 (425011)
>2DCA_1|Chain A|CATHEPSIN B|Bos taurus (9913)
>5UAJ_1|Chains A, B, G, H|DNA-directed RNA polymerase subunit alpha|Escherichia coli (strain K12) (83333)
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 \(w=f(c)\)
5IFP , Knot 369 1013 0.84 40 297 893 
MKLSSACAIALLAAQAAGASIKHRINGFTLTEHSDPAKRELLQKYVTWDDKSLFINGERIMIFSGEFHPFRLPVKELQLDIFQKVKALGFNCVSFYVDWALVEGKPGEYRADGIFDLEPFFDAASEAGIYLLARPGPYINAESSGGGFPGWLQRVNGTLRSSDKAYLDATDNYVSHVAATIAKYQITNGGPIILYQPENEYTSGCCGVEFPDPVYMQYVEDQARNAGVVIPLINNDASASGNNAPGTGKGAVDIYGHDSYPLGFDCANPTVWPSGDLPTNFRTLHLEQSPTTPYAIVEFQGGSYDPWGGPGFAACSELLNNEFERVFYKNDFSFQIAIMNLYMIFGGTNWGNLGYPNGYTSYDYGSAVTESRNITREKYSELKLLGNFAKVSPGYLTASPGNLTTSGYADTTDLTVTPLLGNSTGSFFVVRHSDYSSEESTSYKLRLPTSAGSVTIPQLGGTLTLNGRDSKIHVTDYNVSGTNIIYSTAEVFTWKKFADGKVLVLYGGAGEHHELAISTKSNVTVIEGSESGISSKQTSSSVVVGWDVSTTRRIIQVGDLKILLLDRNSAYNYWVPQLATDGTSPGFSTPEKVASSIIVKAGYLVRTAYLKGSGLYLTADFNATTSVEVIGVPSTAKNLFINGDKTSHTVDKNGIWSATVDYNAPDISLPSLKDLDWKYVDTLPEIQSSYDDSLWPAADLKQTKNTLRSLTTPTSLYSSDYGFHTGYLLYRGHFTATGNESTFAIDTQGGSAFGSSVWLNGTYLGSWTGLYANSDYNATYNLPQLQAGKTYVITVVIDNMGLEENWTVGEDLMKTPRGILNFLLAGRPSSAISWKLTGNLGGEDYEDKVRGPLNEGGLYAERQGFHQPEPPSQNWKSSSPLEGLSEAGIGFYSASFDLDLPKGWDVPLFLNIGNSTTPSPYRVQVYVNGYQYAKYISNIGPQTSFPVPEGILNYRGTNWLAVTLWALDSAGGKLESLELSYTTPVLTALGEVESVDQPKYKKRKGAYHHHHHH
2DCA , Knot 117 256 0.84 40 175 248 
LPESFDAREQWPNCPTIKEIRDQGSCGSCWAFGAVEAISDRICIHSNGRVNVEVSAEDMLTCCGGECGDGCNGGFPSGAWNFWTKKGLVSGGLYNSHVGCRPYSIPPCEHHVNGSRPPCTGEGDTPKCSKTCEPGYSPSYKEDKHFGCSSYSVANNEKEIMAEIYKNGPVEGAFSVYSDFLLYKSGVYQHVSGEIMGGHAIRILGWGVENGTPYWLVGNSWNTDWGDNGFFKILRGQDHCGIESEIVAGMPCTHQY
5UAJ , Knot 146 329 0.85 40 180 315 
MQGSVTEFLKPRLVDIEQVSSTHAKVTLEPLERGFGHTLGNALRRILLSSMPGCAVTEVEIDGVLHEYSTKEGVQEDILEILLNLKGLAVRVQGKDEVILTLNKSGIGPVTAADITHDGDVEIVKPQHVICHLTDENASISMRIKVQRGRGYVPASTRIHSEEDERPIGRLLVDACYSPVERIAYNVEAARVEQRTDLDKLVIEMETNGTIDPEEAIRRAATILAEQLEAFVDLRDVRQPEVKEEKPEFDPILLRPVDDLELTVRSANCLKAEAIHYIGDLVQRTEVELLKTPNLGKKSLTEIKDVLASRGLSLGMRLENWPPASIADE

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 \(c\).

\(|P_{f(5IFP_1)}(2) \setminus P_{f(2DCA_1)}(2)|=153\), \(|P_{f(2DCA_1)}(2) \setminus P_{f(5IFP_1)}(2)|=31\). 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 1:10100101111111011110100010110100000110001100010100001110100111101010110111001010110010111100101010111101011000101110101110110011101110111010100011111111001010100000101010000100111011000100111111001000000100110110110100100010011111111000101010011101011101010000111100101011101011001001010001001011101011000111111111000110001001100001010111101011111001101101010000001011000001000000010111011010110101011010001010000101011110001011110000000000000010110011010110111010101000010100001010011000101101001101011110111100001110000010110100011000000001111101000001101101011110000100011101100100111001001100111011011001010101101010101000101111100100111010000001000111010100011010110100101001001101000000011111010000001001001001000001100101100101010100001110001101110011101001101011010000010001101011000110111001110001011001100101110111110100110101010111000000101110011101000110010110001000011011001111100101010110110111110110000101001010101000100100111000111101110001001111011110011101001010000111011101001001000000110000000
Pair \(Z_2\) Length of longest common subsequence
5IFP_1,2DCA_1 184 4
5IFP_1,5UAJ_1 155 4
2DCA_1,5UAJ_1 175 3

Newick tree

 
[
	2DCA_1:93.51,
	[
		5IFP_1:77.5,5UAJ_1:77.5
	]:16.01
]

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 \((0.85)(0.8)(\frac{1269 }{\log_{20} 1269}-\frac{256}{\log_{20}256})=267.\)
Status Protein1 Protein2 d d1/2
Query variables 5IFP_1 2DCA_1 336 210
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]