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Parikh vectors
1KAW_1 6QGU_1 8PKA_1 Letter Amino acid
8 37 33 L Leucine
9 18 20 T Threonine
4 3 3 W Tryptophan
6 14 10 N Asparagine
9 21 26 R Arginine
9 28 28 E Glutamic acid
24 17 29 G Glycine
6 17 21 K Lycine
4 11 14 M Methionine
1 21 7 F Phenylalanine
6 24 33 S Serine
9 32 30 A Alanine
13 30 30 V Valine
10 11 17 Q Glutamine
4 23 20 D Aspartic acid
1 11 12 H Histidine
4 16 30 I Isoleucine
4 10 19 P Proline
4 8 15 Y Tyrosine
0 8 7 C Cysteine 

1KAW_1|Chains A, B, C, D|SINGLE-STRANDED DNA BINDING PROTEIN|Escherichia coli (562)
>6QGU_1|Chains A, B|Phosphodiesterase|Trypanosoma brucei (5691)
>8PKA_1|Chain A|Cysteine desulfurase|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 \(w=f(c)\)
1KAW , Knot 68 135 0.82 38 102 129 
ASRGVNKVILVGNLGQDPEVRYMPNGGAVANITLATSESWRDKATGEMKEQTEWHRVVLFGKLAEVASEYLRKGSQVYIEGQLRTRKWTDQSGQDRYTTEVVVNVGGTMQMLGGRQGGGAPAGGNIGGGQPQGGW
6QGU , Knot 152 360 0.82 40 210 350 
GSHMASELNEHRATLFNKNVPSRAVKRVTAITKVEREAVLVCELPSFDVTDVEFDLFRARESTDKPLDVAAAIAYRLLLGSGLPQKFGCSDEVLLNFILQCRKKYRNVPYHNFYHVVDVCQTIHTFLYRGNVYEKLTELECFVLLITALVHDLDHMGLNNSFYLKTESPLGILSSASGNTSVLEVHHCNLAVEILSDPESDVFDGLEGAERTLAFRSMIDCVLATDMAKHGSALEAFLASAADQSSDEAAFHRMTMEIILKAGDISNVTKPFDISRQWAMAVTEEFYRQGDMEKERGVEVLPMFDRSKNMELAKGQIGFIDFVAAPFFQKIVDACLQGMQWTVDRIKSNRAQWERVLETR
8PKA , Knot 172 404 0.85 40 225 386 
MSLRPLYMDVQATTPLDPRVLDAMLPYLINYYGNPHSRTHAYGWESEAAMERARQQVASLIGADPREIIFTSGATESNNIAIKGVARFYRSRKKHLITTQTEHKCVLDSCRSLEAEGFQVTYLPVQKSGIIDLKELEAAIQPDTSLVSVMTVNNEIGVKQPIAEIGRICSSRKVYFHTDAAQAVGKIPLDVNDMKIDLMSISGHKIYGPKGVGAIYIRRRPRVRVEALQSGGGQERGMRSGTVPTPLVVGLGAACEVAQQEMEYDHKRISKLSERLIQNIMKSLPDVVMNGDPKHHYPGCINLSFAYVEGESLLMALKDVALSSGSACTSASLEPSYVLRAIGTDEDLAHSSIRFGIGRFTTEEEVDYTVEKCIQHVKRLREMSPLWEMVQDGIDLKSIKWTQH

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 \(c\).

\(|P_{f(1KAW_1)}(2) \setminus P_{f(6QGU_1)}(2)|=41\), \(|P_{f(6QGU_1)}(2) \setminus P_{f(1KAW_1)}(2)|=149\). 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 1:100110011111011001010011011111010110000100010101000001001111101101100010010010101010000100001000000011101110101111001111111101111010111
Pair \(Z_2\) Length of longest common subsequence
1KAW_1,6QGU_1 190 3
1KAW_1,8PKA_1 193 4
6QGU_1,8PKA_1 175 4

Newick tree

 
[
	1KAW_1:98.35,
	[
		6QGU_1:87.5,8PKA_1:87.5
	]:10.85
]

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 \((0.85)(0.8)(\frac{495 }{\log_{20} 495}-\frac{135}{\log_{20}135})=106.\)
Status Protein1 Protein2 d d1/2
Query variables 1KAW_1 6QGU_1 139 94
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]