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Parikh vectors
9BIP_1 1ZLP_1 4UGW_1 Letter Amino acid
8 5 14 Y Tyrosine
13 24 18 V Valine
14 41 31 A Alanine
4 2 4 C Cysteine
14 5 9 Q Glutamine
18 28 36 E Glutamic acid
16 13 28 I Isoleucine
22 15 20 R Arginine
13 10 15 N Asparagine
21 25 27 L Leucine
14 27 16 T Threonine
19 28 22 G Glycine
3 10 5 M Methionine
15 15 17 S Serine
3 2 11 W Tryptophan
18 13 23 D Aspartic acid
6 9 12 H Histidine
18 19 26 K Lycine
15 14 13 F Phenylalanine
7 13 16 P Proline 

9BIP_1|Chain A|miniGs|Homo sapiens (9606)
>1ZLP_1|Chains A, B|petal death protein|Dianthus caryophyllus (3570)
>4UGW_1|Chain A|NITRIC OXIDE SYNTHASE OXYGENASE|BACILLUS SUBTILIS (224308)
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 \(w=f(c)\)
9BIP , Knot 116 261 0.82 40 179 249 
GGSLEVLFQGPSGNSKTEDQRNEEKAQREANKKIEKQLQKDKQVYRATHRLLLLGADNSGKSTIVKQMRILHGGSGGSGGTSGIFETKFQVDKVNFHMFDVGGQRDERRKWIQCFNDVTAIIFVVDSSDYNRLQEALNLFKSIWNNRWLRTISVILFLNKQDLLAEKVLAGKSKIEDYFPEFARYTTPEDATPEPGEDPRVTRAKYFIRDEFLRISTASGDGRHYCYPHFTCAVDTENARRIFNDCRDIIQRMHLRQYELL
1ZLP , Knot 135 318 0.81 40 184 298 
MAPPNGTTNGETEVATQGSYTAVSTGRKTTMHRLIEEHGSVLMPGVQDALSAAVVEKTGFHAAFVSGYSVSAAMLGLPDFGLLTTTEVVEATRRITAAAPNLCVVVDGDTGGGGPLNVQRFIRELISAGAKGVFLEDQVWPKKCGHMRGKAVVPAEEHALKIAAAREAIGDSDFFLVARTDARAPHGLEEGIRRANLYKEAGADATFVEAPANVDELKEVSAKTKGLRIANMIEGGKTPLHTPEEFKEMGFHLIAHSLTAVYATARALVNIMKILKEKGTTRDDLDQMATFSEFNELISLESWYEMESKFKNFTPKAT
4UGW , Knot 153 363 0.82 40 217 348 
MEEKEILWNEAKAFIAACYQELGKAAEVKDRLADIKSEIDLTGSYVHTKEELEHGAKMAWRNSNRCIGRLFWNSLNVIDRRDVRTKEEVRDALFHHIETATNNGKIRPTITIFPPEEKGEKQVEIWNHQLIRYAGYESDGERIGDPASCSLTAACEELGWRGERTDFDLLPLIFRMKGDEQPVWYELPRSLVIEVPITHPDIEAFSDLELKWYGVPIISDMKLEVGGIHYNAAPFNGWYMGTEIGARNLADEKRYDKLKKVASVIGIAADYNTDLWKDQALVELNKAVLHSYKKQGVSIVDHHTAASQFKRFEEQAEEAGRKLTGDWTWLIPPISPAATHIFHRSYDNSIVKPNYFFQDKPYE

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 \(c\).

\(|P_{f(9BIP_1)}(2) \setminus P_{f(1ZLP_1)}(2)|=78\), \(|P_{f(1ZLP_1)}(2) \setminus P_{f(9BIP_1)}(2)|=83\). 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 1:110101110110100000000000010001000100010000010010001111110001000110010110110110110011100010100101011011100000001100100101111110000000100110110011000110010111110000111001111000100011011000010010101100101001001100011010010101000001010011000010011000001100101000011
Pair \(Z_2\) Length of longest common subsequence
9BIP_1,1ZLP_1 161 3
9BIP_1,4UGW_1 170 3
1ZLP_1,4UGW_1 171 4

Newick tree

 
[
	4UGW_1:86.77,
	[
		9BIP_1:80.5,1ZLP_1:80.5
	]:6.27
]

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 \((0.85)(0.8)(\frac{579 }{\log_{20} 579}-\frac{261}{\log_{20}261})=89.8\)
Status Protein1 Protein2 d d1/2
Query variables 9BIP_1 1ZLP_1 113 103
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]