In the recent decades, antibacterial peptides have occupied a strategic position for pharmaceutical drug applications and became subject of intense research activities since they are used to strengthen the immune system of all living organisms by protecting them from pathogenic bacteria. This work proposes a simple and easy statistical/computational method through a peptide polarity index measure by which an antibacterial peptide subgroup can be efficiently identified, that is, characterized by a high toxicity to bacterial membranes but presents a low toxicity to mammal cells. These peptides also have the feature not to adopt to an alpha-helicoidal structure in aqueous solution. The double-blind test carried out to the whole Antimicrobial Peptide Database (November 2011) showed an accuracy of 90% applying the polarity index method for the identification of such antibacterial peptide groups.
The increasing resistance of pathogen agents towards multiple drugs has oriented parts of the investigation in bioinformatics to fast and efficient techniques that can predict the remarkable impact of antibacterial peptide action. These techniques can help to enhance the sometimes cumbersome chemical synthetic approach as well as the subsequent trial and error experiments to identify the peptide performance.
Among the proposed various classifications of peptides, one of it refers to the alpha-helicoidal versus beta-sheet conformation that the peptides can adopt in aqueous solution. This classification refers to the predominance of certain amino acids in the linear sequence of the peptides such as proline-arginine, cathelicidin, or cysteine. It is important to note that such classification appears to be without any influence on the toxicity or selectivity of the peptide once it got in contact with the target membrane [
Although nature was used as the main source of peptides with antibacterial properties in the past [
To obtain efficient antibacterial peptides by measuring the potential action of each altered peptide with the-above described methods would result in a possibility combination that exceeds by far the capacity of the known verification methods in the laboratory. For instance, the number of possible peptides to be formed from one peptide with 8 amino acids in length would be 208 = 25,600,000,000 peptides. This is the reason why contemporary technique profiles to construct antibacterial peptides are the result of joint computational and/or mathematical methods to simulate peptide variations and then to evaluate and qualify these variations to eventually determine if the peptide complies with the required purposes. However, these methods with the aim to simulate the properties of the peptides as well as to evaluate their performance respecting all possible combinatorics are highly complex in their mathematical/computational model design.
In this paper, we present a statistical method that can be attributed to a single physical-chemical property, which is easy to computerize and that efficiently identifies antibacterial peptide subgroups for its highly selective toxicity to bacteria, hereinafter referred to as “Selective Cationic Amphipathic Antibacterial Peptides” (SCAAPs). A SCAAP is characterized by being less than 60 amino acids in length, not adopting an alpha-helicoidal structure in neutral aqueous solution, and showing a therapeutic index higher than 75 [
Our method determines an index that we call polarity index that uses the existent 20 proteic amino acid classification differentiated by its side chain R that divides them in four types and three categories [
20 proteinogenic amino acid classification differentiated by their side chain according to their polarity [
Symbol | Category | 1-letter code |
---|---|---|
P− | polar | D, E, Y |
N | neutral | C, G, N, Q, S, T |
P+ | basic hydrophilic | H, K, R |
NP | non polar residues | A, F, I, L, M, P, V, W |
Peptides can be expressed linearly as an amino acid sequence [ isoelectric point [ hydrophobic moment [
Note that the original parameter values [
A statistical-computational method was designed based only on one physicochemical property: polarity, which quickly and efficiently discerns if a peptide falls into the category of SCAAP or not. The verification was carried out by evaluating the IP and HM physicochemical properties.
The polarity index method uses the 20 amino acid classification differentiated by their side chains that fall into four polarity groups: [P+] polar, [N] neutral, [P+] basic hydrophilic, and [NP] nonpolar residues (Table
From these four groups, a polarity
The method consists of the following steps. Creating a Generating a Comparing the incidences from both
The
The
SCAAP subjects [
Entry | Peptide | Sequence | IP | HM | TI |
---|---|---|---|---|---|
1 | (KIAKKIA)2NH2 | KIAKKIAKIAKKIA | 11.5 | 0.48 | 86.2 |
2 | (KLGKKLG)3NH2 | KLGKKLGKLGKKLGKLGKKLG | 11.7 | 0.49 | 98.3 |
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4 | Melittin | GIGAVLKVLTTGLPALISWIKRKRQQ | 12.6 | 0.46 | 500.0 |
5 | Magainin 2 | GIGKFLHSAKKFGKAFVGEIMNS | 10.8 | 0.56 | 75.0 |
6 | CA(1-13)M(1-13)NH2 | KWKLFKKIEKVGQGIGAVLKVLTTGL | 11.1 | 0.53 | 400.0 |
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In both the
1894 peptides registered in the Antimicrobial Peptide Database (APD) [
The verification of peptides found in the single-action database on Gram+/Gram– bacteria was carried out by validating both the isoelectric point (IP) and hydrophobic moment (HM) in the ranges stated (see Section
Due to the importance of detecting possible peptide pathogenic action, the use of computer programs that evaluate peptic sequences to predict their action on different pathogen agents such as fungi, virus, mammalian cells, and Gram+/Gram– bacteria has become a standard practice among different research groups. The polarity index method is one of these computer programs, but it differs in measuring exclusively one physicochemical property to identify a SCAAP.
The
Number of matches in a typical SCAAP sequence in each peptide database with single or multiple action on fungi, viruses, mammalian cells, Gram+/Gram− bacteria, cancer cells, insects, parasites, and sperms (see also Section
Total | Action | Fungi | Viruses | Mammalian cells | Bacteria | Cancer cells | Insects | Parasites | Sperms |
---|---|---|---|---|---|---|---|---|---|
879 | Unique | 0/77 | 0/22 | 0/10 | 51/743 | 1/16 | 0/2 | 0/9 | 0/0 |
2644 | Multiple | 62/638 | 7/122 | 20/205 | 76/1489 | 21/121 | 3/20 | 5/40 | 1/9 |
Note that the polarity index method only identified SCAAP subjects basically in the bacterial group. Whereas SCAAP subjects identified from the multiple pathogenic action peptide database were fungi (62/638), viruses (7/122), mammalian cells (20/205), Gram+/Gram+ bacteria (76/1489), cancer cells (21/121), insects (3/20), parasites (5/40), and sperms (1/9) (Table
SCAAP subjects identified by the polarity index method in APD Gram+/Gram− bacteria database [
No. | Peptide | Sequence | IP | HM | Status | Reference |
---|---|---|---|---|---|---|
1 | Clavanin D (sea squirt, tunicate, invertebrates, animals) | AFKLLGRIIHHVGNFVYGFSHVF | 10.85 | 0.54 | [ | |
2 | Palustrin-1b (frog, amphibians, animals; XXU) | ALFSILRGLKKLGNMGQAFVNCKIYKKC | 10.80 | 0.49 | [ | |
3 | Palustrin-1d (frog, amphibians, animals; XXU) | ALSILKGLEKLAKMGIALTNCKATKKC | 10.50 | 0.35 | [ | |
4 | Palustrin-1c (frog, amphibians, animals; XXU) | ALSILRGLEKLAKMGIALTNCKATKKC | 10.60 | 0.35 | [ | |
5 | Brevinin-1PRc (frog, amphibians, animals; XXU) | FFPMLAGVAARVVPKVICLITKKC | 10.50 | 0.38 | [ | |
6 | Brevinin-1Be (frog, amphibians, animals; XXU) | FLPAIVGAAAKFLPKIFCVISKKC | 10.30 | 0.43 | [ | |
7 | Brevinin-1HSa (frog, amphibians, animals; XXU) | FLPAVLRVAAKIVPTVFCAISKKC | 10.50 | 0.40 | [ | |
8 | Brevinin-1Ba (frog, amphibians, animals; XXU) | FLPFIAGMAAKFLPKIFCAISKKC | 10.30 | 0.50 | [ | |
9 | Brevinin-1Bc (frog, amphibians, animals; XXU) | FLPFIAGVAAKFLPKIFCAISKKC | 10.30 | 0.49 | [ | |
10 | RANATUERIN 4 (ranatuerin-4, frog, amphibians, animals; XXU) | FLPFIARLAAKVFPSIICSVTKKC | 10.50 | 0.46 | [ | |
11 | Phylloseptin-H11 (PLS-H11, Phylloseptin-13, PS-13; frog, amphibians, animals; XXA) | FLSL IPHAINAVGVHAKHF | 9.65 | 0.36 | [ | |
12 | Phylloseptin-H5 (phylloseptin-7, PLS-H5, PS-7, XXA, frog, amphibians, animals) | FLSLIPHAINAVSAIAKHF | 9.65 | 0.45 | [ | |
13 | Phylloseptin-H2 (PLS-H2, Phylloseptin-2, PS-2) (XXA, frog, amphibians, animals) | FLSLIPHAINAVSTLVHHF | 7.80 | 0.46 | X | [ |
14 | Phylloseptin-B1 (PLS-B1, PBN1; frog, amphibians, animals; XXA) | FLSLIPHIVSGVAALAKHL | 9.65 | 0.46 | [ | |
15 | Papilosin (tunicate, ascidian, invertebrates, sea animals) | GFWKKVGSAAWGGVKAAAKGAAVGGLNALAKHIQ | 11.40 | 0.32 | [ | |
16 | SMAP-34 (sheep myeloid antimicrobial peptide-34; OaMAP34, ovine cathelicidin, sheep, ruminant, animals) | GLFGRLRDSLQRGGQKILEKAERIWCKIKDIFR | 10.43 | 0.48 | [ | |
17 | Caerin 1.17 (frog, amphibians, animals; XXA) | GLFSVLGSVAKHLLPHVAPIIAEKL | 9.50 | 0.49 | [ | |
18 | Caerin 1.18 (frog, amphibians, animals; XXA) | GLFSVLGSVAKHLLPHVVPVIAEKL | 9.50 | 0.50 | [ | |
19 | Fallaxidin 3.2 (XXA, frog, amphibians, animals) | GLLDFAKHVIGIASKL | 9.50 | 0.49 | [ | |
20 | Fallaxidin 3.1 (XXA, frog, amphibians, animals) | GLLDLAKHVIGIASKL | 9.50 | 0.48 | [ | |
21 | Dahlein 5.2 (frog, amphibians, animals) | GLLGSIGNAIGAFIANKLKPK | 11.10 | 0.52 | [ | |
22 | Caerin 1.2 (XXA, frog, amphibians, animals) | GLLGVLGSVAKHVLPHVVPVIAEHL | 7.02 | 0.49 | X | [ |
23 | Caerin 1.4 (XXA, frog, amphibians, animals) | GLLSSLSSVAKHVLPHVVPVIAEHL | 7.02 | 0.48 | X | [ |
24 | Palustrin-2SIb (frog, amphibians, animals; XXU) | GLWNSIKIAGKKLFVNVLDKIRCKVAGGCKTSPDVE | 10.10 | 0.36 | [ | |
25 | XPF (the xenopsin precursor fragment, African clawed frog, amphibians, animals) | GWASKIGQTLGKIAKVGLKELIQPK | 11.00 | 0.40 | [ | |
26 | Pleurocidin (fish, animals) | GWGSFFKKAAHVGKHVGKAALTHYL | 11.00 | 0.34 | [ | |
27 | Cecropin (insects, invertebrates, animals) | GWLKKIGKKIERVGQNTRDATVKGLEVAQQAANVAATVR | 11.30 | 0.36 | [ | |
28 | Pm_mastoparan PMM (insects, invertebrates, animals; XXA; derivatives) | INWKKIASIGKEVLKAL | 10.80 | 0.37 | [ | |
29 | Hinnavin II (Hin II, insects, invertebrates, animals; JJsn) | KWKIFKKIEHMGQNIRDGLIKAGPAVQVVGQAATIYKG | 10.12 | 0.45 | [ | |
30 | Ostrich AvBD2 (Ostrich avian beta defensin 2, ostricacin-1, OSP-1, birds, animals; BBL) | LFCRKGTCHFGGCPAHLVKVGSCFGFRACCKWPWDV | 8.94 | 0.33 | X | [ |
31 | Clavanin D (sea squirt, tunicate, invertebrates, animals) | LFKLLGKIIHHVGNFVHGFSHVF | 10.80 | 0.56 | [ | |
32 | Enterocin Q (EntQ, class 2d bacteriocins; leaderless, that is, no signal peptide, bacteria) | MNFLKNGIAKWMTGAELQAYKKKYGCLPWEKISC | 10.00 | 0.39 | [ | |
33 | Temporin-1Lb (Temporin 1Lb, frog, amphibians, animals) | NFLGTLINLAKKIM | 10.80 | 0.41 | [ | |
34 | Bovine beta-defensin 6 (bBD-6,cow, ruminant, animals) | QGVRNHVTCRIYGGFCVPIRCPGRTRQIGTCFGRPVKCCRRW | 11.30 | 0.37 | [ | |
35 | mBD-4 (mBD4, mouse beta-defensin 4, or Defb4, animals; 3S=S) | QIINNPITCMTNGAICWGPCPTAFRQIGNCGHFKVRCCKIR | 8.91 | 0.33 | X | [ |
36 | ChBac5 (Pro-rich; Arg-rich, goat cathelicidin, ruminant, animals) | RFRPPIRRPPIRPPFNPPFRPPVRPPFRPPFRPPFRPPIGPFP | 13.45 | 0.50 | [ | |
37 | Cyclic dodecapeptide (OaDode, ovine cathelicidin, sheep, ruminant, animals) | RICRIIFLRVCR | 12.00 | 0.36 | [ | |
38 | Bactenecin (cyclic dodecapeptide, bovine cathelicidin, cow, cattle, ruminant, animals; BBMm; JJsn; derivatives: Bac2A) | RLCRIVVIRVCR | 12.00 | 0.33 | [ | |
39 | RL-37 (RL37, cathelicidin, Old World monkey, primates, animals) | RLGNFFRKVKEKIGGGLKKVGQKIKDFLGNLVPRTAS | 11.90 | 0.59 | [ | |
40 | BACTENECIN 7 (bac-7, bac 7; bac7; Pro-rich; cow cathelicidin, ruminant, animals; BBL, SeqAR, BBPP) | RRIRPRPPRLPRPRPRPLPFPRPGPRPIPRPLPFPRPGPRPIPRPLPFPRPGPRPIPRPL | 13.20 | 0.25 | [ | |
41 | Bac4 (Pro-rich, Arg-rich; cow cathelicidin, ruminant, animals) | RRLHPQHQRFPRERPWPKPLSLPLPRPGPRPWPKPL | 12.90 | 0.23 | [ | |
42 | Hyphancin IIIE (insects, invertebrates, animals) | RWKFFKKIERVGQNVRDGLIKAGPAIQVLGAAKAL | 11.80 | 0.41 | [ | |
43 | Cecropin B (insects, invertebrates, animals) | RWKIFKKIEKMGRNIRDGIVKAGPAIEVLGSAKAI | 11.40 | 0.42 | [ | |
44 | Hyphancin IIID (insects, invertebrates, animals) | RWKIFKKIERVGQNVRDGIIKAGPAIQVLGTAKAL | 11.80 | 0.41 | [ | |
45 | Hyphancin IIIG (insects, invertebrates, animals) | RWKVFKKIEKVGRHIRDGVIKAGPAITVVGQATAL | 11.80 | 0.38 | [ | |
46 | Hyphancin IIIF (insects, invertebrates, animals) | RWKVFKKIEKVGRNIRDGVIKAGPAIAVVGQAKAL | 11.80 | 0.39 | [ | |
47 | Phylloseptin-H4 (Phylloseptin-6, PLS-H4, PS-6, XXA, frog, amphibians, animals) | SLIPHAINAVSAIAKHF | 9.65 | 0.47 | [ | |
48 | Pep5 (Lantibiotic, type 1, class 1 bacteriocin, Gram-positive bacteria; XXT3; XXW3) | TAGPAIRASVKQCQKTLKATRLFTVSCKGKNGCK | 11.10 | 0.27 | [ | |
49 | Clavanin C (sea squirt, tunicate, invertebrates, animals) | VFHLLGKIIHHVGNFVYGFSHVF | 9.55 | 0.47 | [ | |
50 | Andropin (insects, invertebrates, animals) | VFIDILDKVENAIHNAAQVGIGFAKPFEKLINPK | 7.50 | 0.45 | X | [ |
51 | Clavanin A (urochordates, sea squirts, and sea pork, tunicate, invertebrates, animals) | VFQFLGKIIHHVGNFVHGFSHVF | 9.71 | 0.61 | [ |
The APD database information integrity verification [
All different peptide classifications achieved over the decades seem to be directed to validate the peptide action and toxicity. However, it appears that these two characteristics are intrinsically related to the space where the peptide interacts as well as to the structural form of the subject membrane. Missing peptide specificity in the studied isolated peptides indicates that nature avoids peptide specificity in order not to favor certain pathogen agents in their blocking action.
Most peptides found experimentally show multiple actions on pathogen agents. Thus it appears that the detection and prediction of antibacterial peptides—in our case SCAAP—is more related to general, nonspecific peptide profiles that are well known for their antibacterial action. For that reason and as given in the present case, more efficient algorithms should rather evaluate fundamental characteristics of such peptides and search for small differences among them.
The design of bioinformatical algorithms to detect antimicrobial peptides is basically of two types. Based on a system of differential equations [ The inclusion of multiple peptide characteristics without affecting its complexity [
Our polarity index method falls in the latter category and is characterized by the following. Effectively excluding multiple action peptides, with a margin of error less than 10% and single-action peptides with a margin of error less than 6%. Its efficiency to identify SCAAP subjects which is higher than 90%. The simplicity of the computational method which is easy to implement for massive parallel processing in GPUs [ Its straightforwardness by measuring the peptide polarity exclusively and from this information effectively classifying its pathogenic action.
The algorithm involved in this method allows simple modifications to identify in a general level peptide groups by their pathogenic action and in a more specific level to refine the peptide search and identification as in the group used here.
The polarity index method uses the amino acid polarity classification; however there are other types of classifications [
As this method is a simple mathematical and computational algorithm, it does not demand heavy computational resources as processing memory or speed; therefore it can be used to explore peptide regions. These peptide regions can be worked out by evaluating massively all possible peptide combinations with the same length [
The statistical/computacional polarity index method is an effective algorithm to find potential antibacterial peptides from a public domain database. These peptides have been denominated “Selective Cationic Amphipathic Antibacterial Peptides” (SCAAP). The method features a high efficiency to exclude peptides that exhibit single pathogenic action on other pathogens than bacteria, and it is equally efficient to exclude multiple-action peptides. In summary, the polarity index method is an adaptable and efficient method to detect and predict SCAAPs and it is a useful analysis and modeling tool for biological sequences using a single physicochemical property.
The polarity index computational implementation is listed in the Appendix section.
Author Carlos Polanco 2011.
Program Detection of SCAAP by Polarity-index method.
Operating System: GNU Linux Fedora 14
Compilation: gfortran program. f
Execution: ./a.out AEVAPAPAAAAPAKAPKKKAAAKPKKAGPS
implicit none
character * 1 arreglo(100), arreglo3(500)
character * 500 backup
character * 1 convert
integer convertN, tipo2
integer base(16), candidato(16), aciertos2, aciertos0
integer aciertost, aciertos3, aciertos4, aciertos14, aciertos24
integer aciertos34, aciertos44, aciertos04, aciertos1, aciertos5
integer x1, x2, x3, x4,
real tipo1
real comodin
double precision matriz(4, 4)
double precision total, peso(4, 4)
equivalence (arreglo3, backup)
open (2, file = “candidate0.dat”)
format (f8.4, 1x, I2)
format (A3)
Relative frequency position of pairs of amino acid in the
candidate SCAAP
peso (4, 4) = 0.272727281/0.272727281
peso (1, 4) = 0.209790215/0.272727281
peso (4, 1) = 0.164335668/0.272727281
peso (1, 1) = 0.087412588/0.272727281
peso (4, 3) = 0.083916083/0.272727281
peso (3, 3) = 0.062937066/0.272727281
peso (3, 4) = 0.059440561/0.272727281
peso (3, 1) = 0.024475524/0.272727281
peso (2, 1) = 0.006993007/0.272727281
peso (1, 3) = 0.006993007/0.272727281
peso (4, 2) = 0.006993007/0.272727281
peso (2,4) = 0.003496503/0.272727281
peso (2, 3) = 0.003496503/0.272727281
peso (1, 2) = 0.003496503/0.272727281
peso (3, 2) = 0.003496503/0.272727281
peso (2, 2) = 0.000000000/0.272727281
Position of pairs of amino acid in the candidate SCAAP
base(1) = 16
base(2) = 4
base(3) = 13
base(4) = 15
base(5) = 12
base(6) = 1
base(7) = 11
base(8) = 9
base(9) = 3
base(10)= 14
base(11)= 6
base(12)= 8
base(13)= 2
base(14)= 7
base(15)= 5
base(16)= 10
do
do
matriz
enddo
enddo
x1 = 0
x2 = 0
x3 = 0
x4 = 0
total = 0
Command to gets the peptide (sequence of amino acid in letter-code)
call getarg (1, backup)
do
if (arreglo3(i). ne. “ ”)
enddo
do
arreglo
enddo
Procedure to determine the relative frequency
distribution of amino acid in the sequence
do
if (arreglo(i).eq. “1”) x1 = x1 + 1
if (arreglo(i).eq. “2”) x2 = x2 + 1
if (arreglo(i).eq. “3”) x3 = x3 + 1
if (arreglo(i).eq. “4”) x4 = x4 + 1
if (arreglo(i).eq. “0”) goto 100
if (arreglo(i).ne. “0”) total = total +1
matriz (convertN (arreglo
enddo
do
do
write
enddo
enddo
close(1)
close(2)
call system (“sort −r candidate0.dat > candidate1.dat”)
open (3, file = “candidate1.dat”)
open (4, file = “candidate0.dat”)
do
read (3, *) tipo1, tipo2
write (4, *) tipo2
enddo
close(3)
close(4)
open (2, file = “candidate0.dat”)
Procedure to evaluate if the sequence of peptide is or
not candidate SCAAP
do
read (2, *, END = 101) candidato
enddo
call parte04 (base, candidato, aciertos0)
call parte14 (base, candidato, aciertos1)
call parte54 (base, candidato, aciertos5)
if ((aciertos0.eq.1). and.(aciertos1.eq.3).and. (aciertos5.eq.1))then
write (6, 52) “Yes”
else
write (6, 52) “No”
call system (“rm candidate0.dat”)
call system (“rm candidate1.dat”)
stop
end
Subroutines and functions
Verification of position 1
subroutine parte04(base, candidato, aciertos0)
integer base(16), candidato(16), aciertos0
aciertos0 = 0
if (candidato(1).eq. base(1)) aciertos0 = aciertos0 + 1
return
end
Verification of positions 2, 3 and 4
subroutine parte14(base, candidato, aciertos1)
integer base(16), candidato(16), aciertos1
aciertos1 = 0
do
if (candidato(
enddo
return
end
Verification of position 16
subroutine parte54 (base, candidato, aciertos5)
integer base(16), candidato(16), aciertos5
aciertos5 = 0
if (candidato(16).eq. base(16)) aciertos5 = aciertos5 + 1
return
end
Conversion letters to the corresponding groups of polarity (in numbers)
character function convert(tipo)
character * 1 tipo
if (tipo.eq. “A”) convert = “4”
if (tipo.eq. “C”) convert = “3”
if (tipo.eq. “D”) convert = “2”
if (tipo.eq. “E”) convert = “2”
if (tipo.eq. “F”) convert = “4”
if (tipo.eq. “G”) convert = “3”
if (tipo.eq. “H”) convert = “1”
if (tipo.eq. “I”) convert = “4”
if (tipo.eq. “K”) convert = “1”
if (tipo.eq. “L”) convert = “4”
if (tipo.eq. “M”) convert = “4”
if (tipo.eq. “N”) convert = “3”
if (tipo.eq. “P”) convert = “4”
if (tipo.eq. “Q”) convert = “3”
if (tipo.eq. “R”) convert = “1”
if (tipo.eq. “S”) convert = “3”
if (tipo.eq. “T”) convert = “3”
if (tipo.eq. “V”) convert = “4”
if (tipo.eq. “W”) convert = “4”
if (tipo.eq. “Y”) convert = “2”
if (tipo.eq. “X”) convert = “0”
return
end
Conversion number in code-letters to numbers in code-numbers
integer function convertN(tipo)
character * 1 tipo
if (tipo.eq. “1”) convertN = 1
if (tipo.eq. “2”) convertN = 2
if (tipo.eq. “3”) convertN = 3
if (tipo.eq. “4”) convertN = 4
return
end
We declare that we do not have any financial and personal relationship with other people or organizations that could inappropriately influence (bias) our work.
Experiments conception and design were done by C. Polanco and J. L. Samaniego. Experimental performance was made by C. Polanco. Data analysis was made by T. Buhse. Results discussion was made by: T. Buhse, F. G. Mosqueira, A. Negron-Mendoza, S. Ramos-Bernal, and J. A. Castanon-Gonzalez.
The authors acknowledge the support given by the Departamento de Computo and the Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de México, and by Concepcion Celis Juarez for proofreading the paper.