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Parikh vectors
5BRD_1 1FMO_1 3ZKN_1 Letter Amino acid
18 35 13 K Lycine
13 14 18 Y Tyrosine
22 15 12 R Arginine
21 16 17 N Asparagine
8 2 6 C Cysteine
10 15 13 Q Glutamine
15 8 1 H Histidine
38 22 36 G Glycine
14 25 23 F Phenylalanine
28 20 24 V Valine
11 19 17 D Aspartic acid
39 32 32 L Leucine
16 6 7 M Methionine
19 13 20 P Proline
3 6 7 W Tryptophan
27 23 30 A Alanine
26 26 21 E Glutamic acid
18 21 27 I Isoleucine
22 17 36 S Serine
13 15 26 T Threonine 

5BRD_1|Chains A, B|Glucokinase 1, putative|Trypanosoma cruzi (strain CL Brener) (353153)
>1FMO_1|Chain A[auth E]|CAMP-DEPENDENT PROTEIN KINASE|Mus musculus (10090)
>3ZKN_1|Chains A, B|BETA-SECRETASE 2|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 \(w=f(c)\)
5BRD , Knot 159 381 0.82 40 217 358 
MGRGSHHHHHHGMAMNIKELSLHELCEELKTPAWNVPLTFVGDVGGTSARMGFVREGKNDSVHACVTRYSMKRKDITELIEFFNEIIELMPASVIKRVKAGVINVPGPVTGGAVGGPFNNLKGIARLSDYPKALFPPGRSAILNDLEAGGFGVLAVSDAHVFSEYFGVMWEGTQWRTCEQEPAGSVIGRGRCLVLAPGTGLGSSLIYYNPMNQQHIVVPLELGSQTIPMRKDIDYIQTLHAELKLLPNYENMVSGAGLEFHYRQVVRGSRPPCSAGEIAKLASEGDANACKAMKKYHEYLMRVGSEASMALLPLTIVLVGDNIVNNAFFYRNPQNLKEMHREALNHEMERFGFQSRVTYLRQKKLLNLNLMGCYRCGLDLS
1FMO , Knot 152 350 0.84 40 209 337 
GNAAAAKKGSEQESVKEFLAKAKEDFLKKWETPSQNTAQLDQFDRIKTLGTGSFGRVMLVKHKESGNHYAMKILDKQKVVKLKQIEHTLNEKRILQAVNFPFLVKLEFSFKDNSNLYMVMEYVAGGEMFSHLRRIGRFSEPHARFYAAQIVLTFEYLHSLDLIYRDLKPENLLIDQQGYIQVTDFGFAKRVKGRTWTLCGTPEYLAPEIILSKGYNKAVDWWALGVLIYEMAAGYPPFFADQPIQIYEKIVSGKVRFPSHFSSDLKDLLRNLLQVDLTKRFGNLKNGVNDIKNHKWFATTDWIAIYQRKVEAPFIPKFKGPGDTSNFDDYEEEEIRVSINEKCGKEFTEF
3ZKN , Knot 164 386 0.84 40 220 370 
ANFLAMVDNLQGDSGRGYYLEMLIGTPPQKLQILVDTGSSNFAVAGTPHSYIDTYFDTERSSTYRSKGFDVTVKYTQGSWTGFVGEDLVTIPKGFNTSFLVNIATIFESENFFLPGIKWNGILGLAYATLAKPSSSLETFFDSLVTQANIPNVFSMQMCGAGLPVAGSGTNGGSLVLGGIEPSLYKGDIWYTPIKEEWYYQIEILKLEIGGQSLNLDCREYNADKAIVDSGTTLLRLPQKVFDAVVEAVARASLIPEFSDGFWTGSQLACWTNSETPWSYFPKISIYLRDENSSRSFRITILPQLYIQPMMGAGLNYECYRFGISPSTNALVIGATVMEGFYVIFDRAQKRVGFAASPCAEIAGAAVSEISGPFSTEDVASNCVPA

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 \(c\).

\(|P_{f(5BRD_1)}(2) \setminus P_{f(1FMO_1)}(2)|=91\), \(|P_{f(1FMO_1)}(2) \setminus P_{f(5BRD_1)}(2)|=83\). 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 1:110100000001111010010100100010011101110111011100101111001000010101000010000100110110011011110110010111101111101111111100101110100010111111001110010111111111001011000111110100100000011101110100111111011100110001100001111101100011100010010010101011100001101111010000110100110011011011001010100110000001101100101111110111110011001110001001001000110001001110001001000011010111000011010
Pair \(Z_2\) Length of longest common subsequence
5BRD_1,1FMO_1 174 4
5BRD_1,3ZKN_1 161 4
1FMO_1,3ZKN_1 145 3

Newick tree

 
[
	5BRD_1:87.25,
	[
		3ZKN_1:72.5,1FMO_1:72.5
	]:14.75
]

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 \((0.85)(0.8)(\frac{731 }{\log_{20} 731}-\frac{350}{\log_{20}350})=104.\)
Status Protein1 Protein2 d d1/2
Query variables 5BRD_1 1FMO_1 131 126.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]