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CROP BREEDING & GENETICS

Optimization of the Marker-Based Procedures for Pyramiding Genes from Multiple Donor Lines: II. Strategies for Selecting the Objective Homozygous Plant

^a Marker-Assisted Rice Breeding Research Team, National Institute of Crop Science, Tsukuba 305-8518, Japan
^b Dep. of Biotechnology, Kyoto Sangyo Univ., Kyoto 603-8555, Japan

^* Corresponding author (yonezaw{at}cc.kyoto-su.ac.jp).

	ABSTRACT

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS NUMERICAL CALCULATIONS RESULTS AND DISCUSSION REFERENCES

 
For extended application of marker-based plant breeding, strategies are discussed for selecting a high-degree gene-pyramided line from among progeny of a multiparentally produced heterozygous plant (root genotype). A strategy with combined use of haplo-diploidization and crossing between selected plants will be highly efficient; selection starts with haplo-diploidized plants raised from the root genotype, and in the absence of a plant with the objective marker genotype, two plants with the best complementary genotypes are crossed to produce a hybrid, which in turn is haplo-diploidized for the next round of selection. In this strategy, even a plant having as many as 20 target markers can be obtained at an almost perfect certainty in about three rounds of selection with a maximum of 200 tested plants per round. When haplo-diploidized plants are unavailable, a plant with the most promising marker genotype should be selected and self-fertilized in each generation, or in the absence of any promising plant, two plants with the best complementary genotypes are crossed for the next round of selection. In this strategy, the number of tested plants in the first two generations counts when the markers are codominant, whereas the rounds of selection counts when the markers are dominant. Of various supplementary measures for this strategy, backcrossing the root genotype with one of the donors could be useful when the donor has more than 70% of all targeted markers.

Abbreviations: CGP, a pair of plants with complementary marker genotypes • CPP, a pair of plants with complementary marker phenotypes • CRO, expected number of crossings performed in a selection strategy • GEN, expected rounds of selection performed in a selection strategy • GNP, expected number of plants tested per selection round • HF₂, a selection strategy with combined use of F₂ enrichment and haplo-diploidization • IG, a plant with the objective homozygous marker genotype • PG, a plant with promising marker genotype • PP, a plant with promising marker phenotype • PRO, the probability of success; RHC, a selection strategy with recurrent use of haplo-diploidization and crossing between the best complementary genotypes • RSC, a recurrent selection strategy with selfing or crossing of selected plants • SH, a selection strategy with a single round of haplo-diploidization and selection • TPN, expected total number of tested plants in a selection strategy

Optimization of the Marker-Based Procedures for Pyramiding Genes from Multiple Donor Lines: II. Strategies for Selecting the Objective Homozygous Plant

^a Marker-Assisted Rice Breeding Research Team, National Institute of Crop Science, Tsukuba 305-8518, Japan
^b Dep. of Biotechnology, Kyoto Sangyo Univ., Kyoto 603-8555, Japan

^* Corresponding author (yonezaw{at}cc.kyoto-su.ac.jp).

For extended application of marker-based plant breeding, strategies are discussed for selecting a high-degree gene-pyramided line from among progeny of a multiparentally produced heterozygous plant (root genotype). A strategy with combined use of haplo-diploidization and crossing between selected plants will be highly efficient; selection starts with haplo-diploidized plants raised from the root genotype, and in the absence of a plant with the objective marker genotype, two plants with the best complementary genotypes are crossed to produce a hybrid, which in turn is haplo-diploidized for the next round of selection. In this strategy, even a plant having as many as 20 target markers can be obtained at an almost perfect certainty in about three rounds of selection with a maximum of 200 tested plants per round. When haplo-diploidized plants are unavailable, a plant with the most promising marker genotype should be selected and self-fertilized in each generation, or in the absence of any promising plant, two plants with the best complementary genotypes are crossed for the next round of selection. In this strategy, the number of tested plants in the first two generations counts when the markers are codominant, whereas the rounds of selection counts when the markers are dominant. Of various supplementary measures for this strategy, backcrossing the root genotype with one of the donors could be useful when the donor has more than 70% of all targeted markers.

Abbreviations: CGP, a pair of plants with complementary marker genotypes • CPP, a pair of plants with complementary marker phenotypes • CRO, expected number of crossings performed in a selection strategy • GEN, expected rounds of selection performed in a selection strategy • GNP, expected number of plants tested per selection round • HF₂, a selection strategy with combined use of F₂ enrichment and haplo-diploidization • IG, a plant with the objective homozygous marker genotype • PG, a plant with promising marker genotype • PP, a plant with promising marker phenotype • PRO, the probability of success; RHC, a selection strategy with recurrent use of haplo-diploidization and crossing between the best complementary genotypes • RSC, a recurrent selection strategy with selfing or crossing of selected plants • SH, a selection strategy with a single round of haplo-diploidization and selection • TPN, expected total number of tested plants in a selection strategy

	INTRODUCTION

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS NUMERICAL CALCULATIONS RESULTS AND DISCUSSION REFERENCES

 
INCREASING NUMBERS of DNA markers that are closely linked with desirable trait genes have been detected and validated in various crop plants in recent years (Charlson et al., 2005; Landi et al., 2005; Gordon et al., 2006; Jena et al., 2006; Liu et al., 2006). The practical application of these markers will be extended if they are accumulated into single plant genomes to construct high-degree gene-pyramided lines.

When obtaining a gene-pyramided homozygous line via assembling markers from multiple donor lines, a plant that has all the markers in a heterozygous state must be produced in any stage within the program. Any gene accumulation program, therefore, can be separated in two parts: the process (named step I in Ishii and Yonezawa, 2007) for producing such a heterozygous genotype, and the process (named step II) in which a plant that has all the targeted markers in a homozygous state is selected from among the progeny of the heterozygous genotype produced via step I. Because the efficiency of steps I and II depends on different parameters, optimization of these steps can be discussed separately. The efficiency of step I depends on the schedule of crossing between donor lines, the optimization of which was discussed in Ishii and Yonezawa (2007). The efficiency of step II, on the other hand, depends on the selection procedures for the progeny plants of the heterozygous genotype obtained in step I. Optimum procedures for step II are discussed in this paper.

Marker-assisted selection strategies have been discussed in various contexts, such as introgressing useful genes via backcrossing (Hospital et al., 1992; Tanksley and Nelson, 1996), improving population mean (Lande and Thompson, 1990; Whittaker et al., 1995; Bernardo, 1999; Liu et al., 2003), heightening the frequency of a favorable allele at each locus concerned (Luo et al., 1997; Hospital et al., 2000), selecting lines with a desired trait in early generations (Monforte et al., 1996; Igartua et al., 2000), and identifying pairs of recombinant inbred lines or haplo-diploidized lines with desirable crossing potentials (van Berloo and Stam, 1998; Charmet et al., 1999). It was assumed in these studies that the target population is initiated biparentally and both detection of useful markers and selection assisted by the detected makers are performed, in either parallel or tandem, within the breeding program concerned. In the context of our discussion, the selection is based on markers that have already been validated in each donor line, and the objective of selection is to obtain a plant with a particular marker genotype that carries all the markers in a homozygous state.

Selection strategies in such a context have been discussed by a number of research teams. Howes et al. (1998) compared the effectiveness of a number of different strategies using the haplo-diploidization method, having proposed two strategies named RF₂Sel and RDHS (explained later) in which haplo-diploidization, combined with random intercrossing between selected plants, is used. The recurrent selection strategy employed in the discussion of Charmet et al. (1999) could also be used in the present context of selection. When assembling a group of markers that are located on the same linkage block, Servin et al. (2004) suggested crossing the initial heterozygous genotype with a "blank parent" (a line carrying no target marker) or a "founding parent" (a donor line carrying a single marker). Meanwhile, Bonnett et al. (2005) proposed combined use of F₂ enrichment (elimination of genotypes that are homozygous for untargeted or null markers at one or more loci concerned) and generation advancement after F₂ to increase the homozygosity of plants. The purpose of our discussion is to provide some additional guidelines for practitioners via comparing the effectiveness of various strategies and supplementary measures suggested hitherto. Our discussion is based on the stochastic calculations using more realistic selection models and a wider range of conditions of related parameters than those employed previously.

	MATERIALS AND METHODS

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS NUMERICAL CALCULATIONS RESULTS AND DISCUSSION REFERENCES

 
Selection Strategies
Four strategies of marker-based selection, recurrent selection with crossing between selected plants (RSC), recurrent haplo-diploidization and crossing (RHC), haplo-diploidization of F₂ plants (HF₂), and a single round of haplo-diploidization and selection (SH) are discussed; workflows in the former three strategies are described in Fig. 1 . Each of these strategies starts with a population generated via self-fertilization or haplo-diploidization of a plant (denoted I₀ in Fig. 1), which has been produced via a schedule of crossing in step I mentioned before and carries all target markers in a heterozygous state. In what follows, the individual I₀ will be referred to as "root genotype," and the homozygous genotype to be obtained via step II, as "ideal genotype" or "ideotype," following the terminology of Servin et al. (2004). Unless otherwise stated, the target markers are assumed to be either codominant or dominant and independently inherited, each of which is perfectly linked with a desirable trait gene.

View larger version (33K): [in this window] [in a new window]   Figure 1. Diagrammatical definition of the selection strategies to be compared. I0 represents an initial heterozygous plant (root genotype) that has been produced via a schedule of crossing between multiple donor lines (Ishii and Yonezawa, 2007). For simplicity, generations in all strategies were designated by the same symbol Gi (i = 1,..., T). Selection procedures with codominant and dominant markers were written in roman and italic letters, respectively. No difference exists between codominance and dominance in strategy recurrent haplo-diploidization and crossing (RHC). Marker genotyping may be performed for haploids, only ones with a desirable genotype being haplo-diploidized. Symbols are defined as follows: Ni = a maximum permissible number of plants genotyped in generation Gi, IG = a plant with objective homozygous marker genotype, PG = a plant with a promising genotype that has a potential to leave IG in its progeny, CGP = a pair of plants with the best complementary genotypes to leave IG, PP = a plant with a promising marker phenotype (defined when the markers are dominant), and CPP = a pair of plants with complementary marker phenotypes.

Selection in RHC starts with a population of haplo-diploidized plants produced from the root genotype I₀, and ends at a generation when an IG is found (success), or neither IG nor CGP is found (failure). Otherwise, two plants with the best CGP are crossed to produce a hybrid, which is haplo-diploidized for the next round of selection. In strategy RHC, selection is performed in T rounds at the maximum, and the type of dominance, codominance versus dominance, makes no difference because all tested plants are homozygous. Strategy RHC is of the same type as RDHS of Howes et al. (1998) and the recurrent selection strategy adopted in Charmet et al. (1999); in all of these strategies, selection is performed recurrently with haplo-diploidized plants. These strategies differ in some procedural parameters, most importantly, in the number of plants selected per round; in RHC, only two plants with the best complementary marker genotypes are selected and crossed for the next round of selection, whereas, in the strategies of Howes et al. (1998) and Charmet et al. (1999), multiple plants are selected and crossed in multiple pairs. Not only haplo-diploidized plants selected but also one of the two parents used for the initiation of the population are incorporated in the crossing in the strategy of Charmet et al. (1999). Strategy RHC is more practicable (resource-saving) than the previously discussed ones. The idea of recurrent selection with haplo-diploidized plants traces back to the theory of Fouilloux (1980).

In strategy HF₂ of Fig. 1, selection starts with a population (G₁) produced via self-fertilization of I₀. The selection ends at generation G₁ when an IG (success) or none of IG, PG, and CGP (failure) is detected. Otherwise, a plant with the best PG is haplo-diploidized to raise a population for the second (final) round of selection, or in the absence of PG, two plants with the best CGP are crossed to produce a hybrid plant, which in turn is haplo-diploidized to raise plants for the final round of selection. When the markers are dominant, the selection ends at G₁ when neither PP nor CPP is found (a failure). Otherwise, PP found first or a hybrid plant of the best CPP is haplo-diploidized for the final selection.

Strategy HF₂ resembles RF₂Sel of Howes et al. (1998) and DH of Bonnett et al. (2005) that was defined under item "with F₂ enrichment for all marker loci" (cf., items 4.2 and 6.2 in their Tables 4 and 6, respectively). In all of these strategies, selection starts with F₂ population. In RF₂Sel, all F₂ plants with desirable marker genotypes (having all target markers in either homozygous or heterozygous state) are selected and randomly intercrossed to give rise to a population, from which plants with desirable genotypes are again selected and haplo-diploidized for the final round of selection. In the DH with F₂ enrichment of Bonnett et al. (2005), all F₂ plants with desirable marker genotypes are haplo-diploidized for the final selection. Both of these strategies will be impracticably resource-consuming because many plants are treated for crossing and/or chromosome doubling. Our HF₂ is practicable and used here as a check for examining the efficiency of combined use of F₂ enrichment and haplo-diploidization.

Efficiency Indicators
The efficiency of our selection strategies is evaluated based on the following five efficiency indicators:

PRO: the probability that a selection program ends with a successful result, that is, an IG is obtained within permissible numbers of generations (T) and genotyped plants per generation (N_i; symbol N is used when N_i is the same across generations),

GEN: the expected number of generations in which selection is performed, being calculated across all possible results of selection, whether ending with a successful or unsuccessful result under the pre-assigned conditions for T and N,

CRO: the expected number of crossings between selected plants (CGP) performed in the selection program,

TNP: the expected total number of plants genotyped in the selection program, and

GNP: the average number of plants genotyped per generation, being calculated by TNP/GEN.

With given conditions for N_i and T, a selection strategy with a higher value of PRO and smaller values of GEN, CRO, TNP, and GNP will be more efficient. Of these five indicators, PRO should be the most important because the expense is wasted when the selection fails. Our discussion therefore will be based mainly on PRO.

Discussion in the previous studies for step II was based on the expected segregation proportion of IG, the expected number (average of all possible numbers) of plants required to be genotyped for finding an IG, or the number of plants required for obtaining IG with a 0.95 or higher probability (Howes et al., 1998; Charmet et al., 1999; Servin et al., 2004; Bonnett et al., 2005). Either of these indicators will be good to rank the superiority of different strategies, but each of them has a weakness of its own; the expected segregation proportion does not directly indicate what the breeder wants to know; the expected plant number is rather small, which may bring about an undeserved optimism for the success; and the plant number for such a high probability of success is impractically large when many markers are involved, bringing about an unproductive reluctance to use marker-based breeding. Such weaknesses can be avoided if an indicator is used in which econo-technical limitations imposed on the program are considered. There should be maximum permissible sizes of N and T in any actual selection programs, and because of random drift, the selection is not always performed in scheduled rounds, ending at any round before the scheduled final round, when IG is found (a success) or no plant with a desirable genotype is found (a failure). Our indicators were introduced to conform to this reality.

Stochastic simulations are used to calculate our indicators. Ten thousand repeated runs are made for each set of conditions of the genetic and procedural variables concerned. The PRO is obtained as the proportion of runs that ended with a successful result. The values of the other indicators are obtained as the average across all repeated runs, whether ending in a success or failure. With 10,000 runs, calculations of the indicators are not completely immune from chance error, but exact enough to capture major points. The computer programs for the calculations are written in Fortran90, and SGI Altix3700 is used to generate random digits in each necessary step in the simulation. The programs are available on request.

	NUMERICAL CALCULATIONS

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS NUMERICAL CALCULATIONS RESULTS AND DISCUSSION REFERENCES

 
Comparison of the Strategies
Our discussion is directed first to the cases where all of the target markers are codominant and independently inherited. Calculations for N = 200 and T = 5 (Table 1 ) show that the superiority of the strategies changes, more or less, with the number of markers concerned (m). When m = 5, all strategies, RSC, HF₂, SH, and RHC, give almost a perfect probability of success (PRO = 1). Of these four, SH and RHC could be preferred because they require much fewer GEN and TNP than the other strategies. Usefulness of SH is lost when m = 10; PRO of SH is as low as 0.183, whereas those of the other three strategies are still almost perfect. Of the three strategies, HF₂ and RHC are superior to RSC because of much fewer GEN and TNP.

When the markers are dominant, PRO of RSC and HF₂ decrease markedly compared with those with codominant markers; PRO of RSC is as low as 0.340 even when m = 5, and that of HF₂ is 0.774 when m = 10 (Table 1). Of course, the effectiveness of RHC is unchanged whether the target markers are codominant or dominant.

Our calculations, therefore, confirm the superiority of RHC. Breeders may be rather reluctant to use RHC because it requires an extra expense of producing haplo-diploidized plants. This expense, however, could be readily recovered; as the calculations of Table 2 show, RHC gives a sufficiently high probability of success even when N = 50 and 100, unless m exceeds 15. Strategies HF₂ and SH will not be useful, and much less efficient than RSC when as many as 15 codominant markers are involved (Table 1). The CRO of strategy RSC is negligibly small unless m exceeds 10, indicating that crossing between selected plants is rarely needed.

Supplementary Measures for Increasing the Efficiency of RSC Strategy
Strategy RSC is the only choice when haplo-diploidization method is not available. There are a number of possible measures for improving the efficiency of RSC. Increasing either N or T or incorporating some rounds of generation advancement is the simplest and most practicable measure. Some more refined measures, such as backcrossing the root genotype I₀ with one of the donor lines (Servin et al., 2004; Bonnett et al., 2005) and crossing I₀ with a blank parent (Servin et al., 2004), have been suggested previously. The effectiveness of these measures is examined here.

Calculations in Table 4 show that, with codominant markers, increase in the population size N causes a marked increase in PRO when m = 15. With fewer markers (m 10), the contribution of increased N occurs as a decrease in GEN because PRO is almost perfect even when N is as small as 50. Increase in N provides no noticeable advantage when the markers are dominant.

A weighted rather than uniform allocation of population sizes across generations is expected to be useful when many codominant markers are involved. Comparison of uniform versus weighted allocations, under the same total tested plant number N₁ + N₂ + ... + N₅ = 500 (Table 5 ), reveals that PRO is significantly improved with weighting for earlier generations, that is, from 0.795 to 0.982 (under VN-2) when m = 15 and from 0.524 to 0.851 (under VN-2) when m = 20, although TNP and GNP increase to some extent. Our calculations, therefore, substantiate the importance of population sizes of the first and second generations, N₁ and N₂. It is also known from Table 5 that PRO is maximized with VN-2, not with VN-1 and VN-3, indicating existence of an optimum allocation of N₁ relative to N₂.

Advantage of generation advancement was not recognized in our calculations. With codominant markers, PRO decreased when RSC was started after any rounds of generation advancement (data not presented), indicating that advancing generations without selection is of no practical use. Generation advancement had some advantage when the markers are dominant (Table 7 ); when m equaled 5 and RSC was performed after two rounds of generation advancement, PRO increased from 0.331 (Table 4) to 0.446 (Table 7) under N = 100 and from 0.340 (Table 4) to 0.447 (Table 7) under N = 200. These increases would not be cost-effective. With m 10, PRO rather decreases when generation advancement is used, reflecting the fact that, because of random drift, desirable genotypes are subject to a high risk of failure during generation advancement.

The backcross should be advantageous for a locus where both of the heterozygous plant and the chosen donor have the targeted marker, but disadvantageous for a locus where the donor does not carry the targeted marker. For efficient backcrossing, therefore, the donor should have a sufficiently high proportion of the targeted markers. Calculations of Table 8 demonstrate that the proportion (f) must be higher than 0.7. When this condition is satisfied, the advantage of the backcross occurs in different modes depending on the kinds of dominance (codominance versus dominance) as well as the number of markers (m) involved. With codominant markers, the advantage occurs as a reduction in TNP and GNP when m = 6 (at a proportion of 0.83), as a reduction in GEN, TNP, and GNP when m = 10, and as a large improvement in PRO and a reduction in CRO when m = 15. No noticeable increase in PRO occurs with m = 6 and 10 because it is almost unity even without the backcross. With dominant markers, PRO is improved with all the three magnitudes of m, being accompanied by a substantial decrease in TNP and GNP when m = 10 and 15.

	RESULTS AND DISCUSSION

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS NUMERICAL CALCULATIONS RESULTS AND DISCUSSION REFERENCES

 
Our calculations showed that a selection strategy with combined use of haplo-diploidization and crossing between selected plants, that is, RHC, is highly efficient when producing a high-degree gene-pyramided line (m > 10); this strategy is efficient whether the markers are codominant or dominant, giving a sufficiently high probability of success under a quite affordable resource expense (GEN, CRO, TNP, and GNP) even when more than 20 markers are targeted. Strategies using haplo-diploidization of F₂ plants, such as our HF₂, RF₂Sel of Howes et al. (1998), and DH of Bonnett et al. (2005), are not useful. When haplo-diploidized plants are not available, strategy RSC should be used such that a maximum permissible number of plants is tested in the first two generations when most markers are codominant, or, the largest possible number of generations is used with relatively few plants (fewer than 100 when m < 20) being tested per generation when the markers are dominant. Incorporating generation advancement for a higher homozygosity is not advantageous. Backcrossing the root genotype with one of the donors will be useful when the donor has a sufficiently high proportion of the objective markers, i.e., higher than 70% with independently inherited markers (Table 8).

Howes et al. (1998) predicted that combining more than 12 target markers might not be feasible whether RDHS or RF₂Sel is used, because these strategies require a prohibitively large number of tested plants. The number 12 seems to have been adopted as a guideline in many practical breeding projects. Strategy RDHS is similar to our RHC under T = 2. Calculations of RHC under T = 2 and N = 200 showed that combining 12 target markers was never difficult but achievable with an almost perfect certainty (PRO = 0.998) and at quite an affordable resource expense (GEN = 1.952, CRO = 0.952, TNP = 209.1, and GNP = 107.4). The gap between the prediction of Howes et al. (1998) and ours could be ascribed to the difference in the procedures for raising a population for the second (final) round of selection. In RDHS, plants having "most of the target markers" are selected and randomly intercrossed (from the context of their discussion, a fairly large number of plants seem to be selected though not explicitly described), whereas only two plants with the best complementary marker genotypes are crossed in our RHC. Our calculations, therefore, show that intercrossing between many selected plants is not advantageous, emphasizing that efforts should be concentrated on a single best or two best complementary plants. Strategy RHC is highly effective, enabling one to obtain a plant even carrying 25 target markers, with PRO = 0.916, GEN = 3.037, CRO = 2.037, TNP = 439.1, and GNP = 174.6, when performed under N = 200 and T = 5.

The guideline condition f > 0.7 mentioned before, the reality of which was also confirmed with analytical calculations (not presented), indicates that the supplementary backcrossing is of rather limited application. This condition would not often be satisfied when the root genotype is produced with multiple donor lines; no single donor would have such a high proportion of targeted markers when three or more donors are involved. The application of the backcross may be even more restricted when linked markers are involved; in the genome of a multiparentally produced root genotype, coupling rather than repulsion linkages will dominate, which may be broken by backcrossing. Crossing the root parent with a blank parent or a parent that carries a single marker (Servin et al., 2004) would not be of general use either; such a crossing will be advantageous for assembling markers that are strongly linked in repulsion phase, but not for markers that are independent or linked in coupling phase. As suggested by Servin et al. (2004), a blank parent may effectively be employed in some stage in step I rather than in step II. The major concern of the discussion of Servin et al. (2004) was for the procedures of step I. We must confess that that we did not notice their work at the writing of our previous paper, Ishii and Yonezawa (2007).

Throughout this paper, the effectiveness of strategy RSC has been discussed only for the cases where all target markers are codominant or dominant. Both codominant and dominant markers may be targeted in the same selection program. Our calculations (not presented) showed that the effectiveness of selection decreased largely when only a few among many markers are dominant; PRO under N = 200 and T = 5 decreases from 1 to 0.640 when only two out of 10 independent target markers are dominant, and from 0.524 to 0.211 when four out of 20 markers are dominant, emphasizing the importance of developing codominant markers or technologies of haplo-diploidization. The problem of dominant markers does not occur when RHC is used.

	ACKNOWLEDGMENTS

 
The authors would like to thank an anonymous reviewer for valuable comments and suggestions. His or her suggestions, especially the reminder of some important papers, were highly helpful for widening the scope of our discussion.

	NOTES

TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS NUMERICAL CALCULATIONS RESULTS AND DISCUSSION REFERENCES

 
All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permission for printing and for reprinting the material contained herein has been obtained by the publisher.

Received for publication November 28, 2006.

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TOP NOTES ABSTRACT INTRODUCTION MATERIALS AND METHODS NUMERICAL CALCULATIONS RESULTS AND DISCUSSION REFERENCES

 

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