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Parikh vectors
3RFU_1 8IQP_1 3BEE_1 Letter Amino acid
68 16 0 G Glycine
35 13 5 K Lycine
9 3 1 W Tryptophan
7 7 3 Y Tyrosine
76 9 4 V Valine
16 3 6 N Asparagine
23 11 5 Q Glutamine
46 11 6 E Glutamic acid
77 17 15 L Leucine
46 10 6 I Isoleucine
27 5 2 M Methionine
19 6 3 F Phenylalanine
82 5 12 A Alanine
28 8 2 R Arginine
27 9 6 D Aspartic acid
6 3 0 C Cysteine
24 8 2 H Histidine
35 8 2 P Proline
48 19 6 S Serine
37 8 7 T Threonine 

3RFU_1|Chains A, B, C, D|Copper efflux ATPase|Legionella pneumophila subsp. pneumophila (272624)
>8IQP_1|Chains A[auth L], C[auth l]|M9 VL-SARAH|Mus musculus (10090)
>3BEE_1|Chains A, B|Putative YfrE protein|Vibrio parahaemolyticus RIMD 2210633 (223926)
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 \(w=f(c)\)
3RFU , Knot 279 736 0.83 40 268 648 
MKHDHHQGHTHSGKGHACHHEHNSPKTQQASSKMEGPIVYTCPMHPEIRQSAPGHCPLCGMALEPETVTVSEVVSPEYLDMRRRFWIALMLTIPVVILEMGGHGLKHFISGNGSSWIQLLLATPVVLWGGWPFFKRGWQSLKTGQLNMFTLIAMGIGVAWIYSMVAVLWPGVFPHAFRSQEGVVAVYFEAAAVITTLVLLGQVLELKAREQTGSAIRALLKLVPESAHRIKEDGSEEEVSLDNVAVGDLLRVRPGEKIPVDGEVQEGRSFVDESMVTGEPIPVAKEASAKVIGATINQTGSFVMKALHVGSDTMLARIVQMVSDAQRSRAPIQRLADTVSGWFVPAVILVAVLSFIVWALLGPQPALSYGLIAAVSVLIIACPCALGLATPMSIMVGVGKGAQSGVLIKNAEALERMEKVNTLVVDKTGTLTEGHPKLTRIVTDDFVEDNALALAAALEHQSEHPLANAIVHAAKEKGLSLGSVEAFEAPTGKGVVGQVDGHHVAIGNARLMQEHGGDNAPLFEKADELRGKGASVMFMAVDGKTVALLVVEDPIKSSTPETILELQQSGIEIVMLTGDSKRTAEAVAGTLGIKKVVAEIMPEDKSRIVSELKDKGLIVAMAGDGVNDAPALAKADIGIAMGTGTDVAIESAGVTLLHGDLRGIAKARRLSESTMSNIRQNLFFAFIYNVLGVPLAAGVLYPLTGLLLSPMIAAAAMALSSVSVIINALRLKRVTL
8IQP , Knot 84 179 0.81 40 132 168 
MSKIKGHHHHHHGGMDVLMTQTPLSLPVSLGDQASISCRSSQSIVHSNGNTYLEWYLQKPGQSPKLLIYKVSNRFSGVPDRFSGSGSGTDFTLKISRVEAEDLGVYYCFQGSHVPWTFGGGTKLEIKRGSDYEFLKSWTVEDLQKRLLALDPMMEQEIEEIRQKYQCKRQPILDAIEAK
3BEE , Knot 48 93 0.78 36 75 90 
SNAVTATQLAAKATTLYYLHKQAMTDEVSLLLEQALQLEPYNEAALSLIANDHFISFRFQEAIDTWVLLLDSNDPNLDRVTIIESINKAKKLM

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 \(c\).

\(|P_{f(3RFU_1)}(2) \setminus P_{f(8IQP_1)}(2)|=171\), \(|P_{f(8IQP_1)}(2) \setminus P_{f(3RFU_1)}(2)|=35\). 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 1:1000000100001010100000001000010001011110001101010001110011011110100101001101001010001111111011111101110110011010100110111101111111111100110010010101101111111111100111111111110110000111110101111100111110110101000010110111011100100100010000101001111011010110011101010010011000110101111100101011110100010111011011000111011011001000011100110010111111111111101111111110111001111110111110101111101101111110110011110010110010010011100010100101010011000110001111111100000011101110110001101101011011010111101010011110101100011001111001001010110111111010011111100110000100110100011011110100000101111011100111011100000110010001111111101100111110101111110100111001110110101011101001000010010001111110011111111111011011110111111111100101110110100101
Pair \(Z_2\) Length of longest common subsequence
3RFU_1,8IQP_1 206 4
3RFU_1,3BEE_1 219 4
8IQP_1,3BEE_1 139 3

Newick tree

 
[
	3RFU_1:11.00,
	[
		8IQP_1:69.5,3BEE_1:69.5
	]:46.50
]

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 \((0.85)(0.8)(\frac{915 }{\log_{20} 915}-\frac{179}{\log_{20}179})=203.\)
Status Protein1 Protein2 d d1/2
Query variables 3RFU_1 8IQP_1 251 155
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]