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Parikh vectors
1DSN_1 3NKJ_1 8WLB_1 Letter Amino acid
11 4 17 N Asparagine
27 4 29 G Glycine
4 1 7 H Histidine
24 8 24 L Leucine
21 4 29 V Valine
12 4 14 T Threonine
10 0 4 Y Tyrosine
28 3 7 R Arginine
12 0 1 C Cysteine
1 1 10 M Methionine
19 5 10 F Phenylalanine
22 5 11 P Proline
24 3 16 S Serine
31 5 40 A Alanine
16 5 18 D Aspartic acid
16 2 10 Q Glutamine
12 0 15 I Isoleucine
5 1 3 W Tryptophan
18 5 16 E Glutamic acid
20 7 28 K Lycine 

1DSN_1|Chain A|LACTOFERRIN|Homo sapiens (9606)
>3NKJ_1|Chain A|Villin-1|Gallus gallus (9031)
>8WLB_1|Chain A|Allose ABC transporter|Enterobacter cloacae (550)
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 \(w=f(c)\)
1DSN , Knot 143 333 0.83 40 189 321 
RRRRSVQWCAVSQPEATKCFQWQRNMRRVRGPPVSCIKRDSPIQCIQAIAENRADAVTLSGGFIYEAGLAPYKLRPVAAEVYGTERQPRTHYYAVAVVKKGGSFQLNELQGLKSCHTGLRRTAGWNVPIGTLRPFLNWTGPPEPIEAAVARFFSASCVPGADKGQFPNLCRLCAGTGENKCAFSSQEPYFSYSGAFKCLRDGAGDVAFIRESTVFEDLSDEAERDEYELLCPDNTRKPVDKFKDCHLARVPSHAVVARSVNGKEDAIWNLLRQAQEKFGKDKSPKFQLFGSPSGQKDLLFKDSAIGFSRVPPRIDSGLYLGSGYFTAIQNLRK
3NKJ , Knot 38 67 0.79 34 63 65 
PTKLETFPLDVLVNTAAEDLPRGVDPSRKENHLSDEDFKAVFGMTRSAFANGPLWKQQNLKKEKGLF
8WLB , Knot 130 309 0.80 40 168 290 
MGSSHHHHHHSSGLVPRGSHMAAEYAVVLKTLSNPFWVDMKKGIEDEAKTLGVSVDIFASPSEGDFQSQLQLFEDLSNKKYKGIAFAPLSSVNLVMPVARAWQKGLYLVNLDEKIDMDNLKKAGGNVEGFVTTDNVAVGAKGADFIINKLGAEGGEVAIIEGKAGNASGEARRNGATEAFKKANQIKLVASQPADWDRIKALDVATNVLQRNPNLKAFYCANDTMAMGVAQAVANAGKIGKVLVVGTDGIPEARKMVEAGQMTATVAQNPADIGATGLKLMVDAAKTGKVIPLEKTPEFKLVDSILVTK

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 \(c\).

\(|P_{f(1DSN_1)}(2) \setminus P_{f(3NKJ_1)}(2)|=148\), \(|P_{f(3NKJ_1)}(2) \setminus P_{f(1DSN_1)}(2)|=22\). 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 1:000001010110010100010100010010111100100001100101110001011010111100111110010111101010000100000111110011010100101100000110001110111101011101011101101111011010011110010110100101101000011000010100011100100111011110000110010001000000110100000110010000110110011110010100011101100100011000010101110101000111000111100111010011011010101100100
Pair \(Z_2\) Length of longest common subsequence
1DSN_1,3NKJ_1 170 4
1DSN_1,8WLB_1 161 5
3NKJ_1,8WLB_1 153 4

Newick tree

 
[
	1DSN_1:84.77,
	[
		8WLB_1:76.5,3NKJ_1:76.5
	]:8.27
]

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 \((0.85)(0.8)(\frac{400 }{\log_{20} 400}-\frac{67}{\log_{20}67})=103.\)
Status Protein1 Protein2 d d1/2
Query variables 1DSN_1 3NKJ_1 131 77
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]