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Parikh vectors
6HIN_1 1TLY_1 4GZL_1 Letter Amino acid
14 25 7 N Asparagine
7 0 5 C Cysteine
21 9 10 E Glutamic acid
58 15 21 L Leucine
30 8 12 P Proline
13 14 2 W Tryptophan
40 11 17 V Valine
20 15 13 A Alanine
28 23 12 D Aspartic acid
15 8 5 Q Glutamine
18 30 15 G Glycine
15 11 13 K Lycine
21 18 7 F Phenylalanine
30 18 12 S Serine
17 21 9 Y Tyrosine
28 9 8 R Arginine
35 13 10 I Isoleucine
23 12 13 T Threonine
10 13 9 H Histidine
7 5 4 M Methionine 

6HIN_1|Chains A, B, C, D, E|5-hydroxytryptamine receptor 3A|Mus musculus (10090)
>1TLY_1|Chains A, B|Nucleoside-specific channel-forming protein tsx|Escherichia coli (562)
>4GZL_1|Chains A, B|Ras-related C3 botulinum toxin substrate 1|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 \(w=f(c)\)
6HIN , Knot 183 450 0.82 40 230 419 
PALLRLSDHLLANYKKGVRPVRDWRKPTTVSIDVIMYAILNVDEKNQVLTTYIWYRQYWTDEFLQWTPEDFDNVTKLSIPTDSIWVPDILINEFVDVGKSPNIPYVYVHHRGEVQNYKPLQLVTACSLDIYNFPFDVQNCSLTFTSWLHTIQDINITLWRSPEEVRSDKSIFINQGEWELLEVFPQFKEFSIDISNSYAEMKFYVIIRRRPLFYAVSLLLPSIFLMVVDIVGFCLPPDSGERVSFKITLLLGYSVFLIIVSDTLPATAIGTPLIGVYFVVCMALLVISLAETIFIVRLVHKQDLQRPVPDWLRHLVLDRIAWILCLGEQPMAHRPPATFQANKTDDCSGSDLLPAMGNHCSHVGGPQDLEKTPRGRGSPLPPPREASLAVRGLLQELSSIRHFLEKRDEMREVARDWLRVGYVLDRLLFRIYLLAVLAYSITLVTLWSIW
1TLY , Knot 124 278 0.83 38 177 268 
AENDKPQYLSDWWHQSVNVVGSYHTRFGPQIRNDTYLEYEAFAKKDWFDFYGYADAPVFFGGNSDAKGIWNHGSPLFMEIEPRFSIDKLTNTDLSFGPFKEWYFANNYIYDMGRNKDGRQSTWYMGLGTDIDTGLPMSLSMNVYAKYQWQNYGAANENEWDGYRFKIKYFVPITDLWGGQLSYIGFTNFDWGSDLGDDSGNAINGIKTRTNNSIASSHILALNYDHWHYSVVARYWHDGGQWNDDAELNFGNGNFNVRSTGWGGYLVVGYNFHHHHHH
4GZL , Knot 96 204 0.83 40 150 194 
MGSSHHHHHHSSGLVPRGSHMENLYFQGQAIKCVVVGDGAVGKTCLLISYTTNAFPGEYIPTVFDNYSANVMVDGKPVNLGLWDTAGLEDYDRLRPLSYPQTDVFLICFSLVSPASFENVRAKWYPEVRHHCPNTPIILVGTKLDLRDDKDTIEKLKEKKLTPITYPQGLAMAKEIGAVKYLECSALTQRGLKTVFDEAIRAVL

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 \(c\).

\(|P_{f(6HIN_1)}(2) \setminus P_{f(1TLY_1)}(2)|=121\), \(|P_{f(1TLY_1)}(2) \setminus P_{f(6HIN_1)}(2)|=68\). 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 1:111101000111000011011001001001010111011101000001100011000010001101010010010010110001111011100110110010110101000101000011011010010100111010000101001100100101011001001000001110010101101110100101010000101010111000111011011110111111011110111001001010101111001111110001110111011111011101111110110011110110000100111011001110011111011001110011101010000000100111111000001111001000101010111110010111011100100100110000010011001101101100111010111111001011011011
Pair \(Z_2\) Length of longest common subsequence
6HIN_1,1TLY_1 189 4
6HIN_1,4GZL_1 172 4
1TLY_1,4GZL_1 171 6

Newick tree

 
[
	6HIN_1:91.90,
	[
		4GZL_1:85.5,1TLY_1:85.5
	]:6.40
]

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 \((0.85)(0.8)(\frac{728 }{\log_{20} 728}-\frac{278}{\log_{20}278})=124.\)
Status Protein1 Protein2 d d1/2
Query variables 6HIN_1 1TLY_1 159 128.5
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)],
\[ \delta= \alpha \mathrm{min} + (1-\alpha) \mathrm{max}= \begin{cases} d &\alpha=0,\\ d_1/2 &\alpha=1/2 \end{cases} \]