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Parikh vectors
2ISQ_1 4ZGN_1 3WSU_1 Letter Amino acid
1 7 11 W Tryptophan
36 19 40 A Alanine
9 10 18 R Arginine
2 6 2 C Cysteine
37 11 40 G Glycine
9 7 7 M Methionine
16 19 19 P Proline
11 14 12 N Asparagine
7 7 8 Q Glutamine
26 17 13 I Isoleucine
10 25 28 D Aspartic acid
24 27 5 K Lycine
20 29 16 S Serine
7 12 8 Y Tyrosine
22 17 25 V Valine
25 28 15 E Glutamic acid
2 13 23 H Histidine
27 34 29 L Leucine
11 19 8 F Phenylalanine
18 17 19 T Threonine 

2ISQ_1|Chain A|Cysteine synthase|Arabidopsis thaliana (3702)
>4ZGN_1|Chain A|Cell division cycle protein 123|Schizosaccharomyces pombe (284812)
>3WSU_1|Chains A, B|Beta-mannanase|Streptomyces thermolilacinus (285540)
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 \(w=f(c)\)
2ISQ , Knot 135 320 0.81 40 177 299 
SRIAKDVTELIGNTPLVYLNNVAEGCVGRVAAKLEMMEPCSSVKDRIGFSMISDAEKKGLIKPGESVLIEPTSGNTGVGLAFTAAAKGYKLIITMPASMSTERRIILLAFGVELVLTDPAKGMKGAIAKAEEILAKTPNGYMLQQFENPANPKIHYETTGPEIWKGTGGKIDGFVSGIGTGGTITGAGKYLKEQNANVKLYGVEPVESAILSGGKPGPHKIQGIGAGFIPSVLNVDLIDEVVQVSSDESIDMARQLALKEGLLVGISSGAAAAAAIKLAQRPENAGKLFVAIFPSFGERYLSTVLFDATRKEAEAMTFEA
4ZGN , Knot 150 338 0.86 40 206 321 
GSSHHHHHHSSGLVPRGSHMTLILTKNQVLHCQFSSWYSLFRKLTPKAKVIKPIPATVLKYLHEDSIYVEQPMNTVEEVDSEEDEESAPAYYPEREAIQLIEKAIKELGGAVVPKLNWSTPKDALWITTTGSLKCTTAEEVLLLLKSSDFVAHDLNHAFDDCKDFDNADGSVPKDFSFELVLKEWFPMHASTEFRCFVKSKRLIAFCQRDDNYYEFLKENIDCYEKLISDLLKKLDTFPDPDFVFDVYIHKDRAWLIDINPFYPRTDGLLFSWSELESMNSENMKPEIRLIPKGSMPSTGSAKYYTNRVPFDMIAASEGENLLEFAQKWQDLTNKSNE
3WSU , Knot 141 346 0.79 40 188 320 
GSSHHHHHHSSGLVPRGSHMRTAPPGSAGQPPDAAVTGLHVQGGRLLEGNGNDFVMRGVNHAHTWYPGQTRSLADIKALGANTVRVVLSDGHRWTRNGPADVAAVIDRCKANRLICVLEVHDTTGYGEEPAAGTLDHAADYWISLMDVLAGQEDYVIVNIGNEPWGNTDPAGWTAPTIAAVKKLRAAGLAHTLMIDAPNWGQDWQGVMRADARSVYEADPTGNLLFSIHMYSVFDTAAEIDDYLEAFVDAGLPLVIGEFGGPPDQWGDPDEDTMLAAAERLRLGYLAWSWSGNTDPVLDLAIGFDPDRLSGWGQRVFHGVHGIGETSREATVFGKLAALEHHHHHH

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 \(c\).

\(|P_{f(2ISQ_1)}(2) \setminus P_{f(4ZGN_1)}(2)|=70\), \(|P_{f(4ZGN_1)}(2) \setminus P_{f(2ISQ_1)}(2)|=99\). 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 1:00110010011100111010011010110111010110100010001110110010001110110011101001001111110111010011101110100000111111110111001101101111010011100101011001001101010000011011010110101110111011010111001000010101011011001110110111001011111111011010110011010000010110011100111111001111111101100100110111111101100010011101000010110101
Pair \(Z_2\) Length of longest common subsequence
2ISQ_1,4ZGN_1 169 4
2ISQ_1,3WSU_1 169 4
4ZGN_1,3WSU_1 166 20

Newick tree

 
[
	2ISQ_1:84.99,
	[
		4ZGN_1:83,3WSU_1:83
	]:1.99
]

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 \((0.85)(0.8)(\frac{658 }{\log_{20} 658}-\frac{320}{\log_{20}320})=93.5\)
Status Protein1 Protein2 d d1/2
Query variables 2ISQ_1 4ZGN_1 118 113
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]