^{*}

Conceived and designed the experiments: XSH. Performed the experiments: XSH. Analyzed the data: XSH. Contributed reagents/materials/analysis tools: XSH. Wrote the paper: XSH.

One crucial feature of zygotic linkage disequilibrium (LD) analysis is its direct use of diploid genotyping data, irrespective of the type of mating system. Previous theories from an evolutionary perspective mainly focus on gametic LD, but the equivalent development for zygotic LD is not available. Here I study the evolution of zygotic LD and the covariances between gametic and zygotic LDs or between distinct zygotic LDs in a finite local population under constant immigration from a continent population. I derive the analytical theory under genetic hitchhiking effects or in a neutral process. Results indicate that zygotic LDs (diploid level) are more informative than gametic LD (haploid level) in indicating the effects of different evolutionary forces. Zygotic LDs may be greater than or comparable to gametic LD under the epistatic selection process, but smaller than gametic LD under the non epistatic selection process. The covariances between gametic and zygotic LDs are strongly affected by the mating system, linkage distance, and genetic drift effects, but weakly affected by seed and pollen flow and natural selection. The covariances between different zygotic LDs are generally robust to the effects of gene flow, selection, and linkage distance, but sensitive to the effects of genetic drift and mating system. Consistent patterns exist for the covariances between the zygotic LDs for the two-locus genotypes with one common genotype at one locus or without any common genotype at each locus. The results highlight that zygotic LDs can be applied to detecting natural population history.

Zygotic linkage disequilibrium (LD) refers to the difference between the joint genotypic frequency at two loci and the product of genotypic frequencies at each locus

Previously relevant theories emphasize the joint frequency of double heterozygotes or double homozygotes in a neutral process, or the joint descent measures for a population with a mixed mating system

In flowering plants, three distinct processes in a life cycle are involved in changing zygotic LD and its relationship with gametic LD in a local population. One process is the asymmetric immigration through haploid pollen flow and diploid seed flow from a source population. Pollen flow directly generates gametic LD, but indirectly affects zygotic LD since each pollen grain only carries one gamete in fusion with ovules in the recipient population. Seed flow can generate both zygotic and gametic LDs since each seed carries two gametes into the recipient population simultaneously.

The second process influencing zygotic LD in plants is the mating system

The third process influencing zygotic LD in plants is selection in either the gametophyte or the sporophyte stage, or in both stages. Selection against heterozygote or epistatic selection at the sporophyte stage can directly change zygotic LD, but indirectly changes gametic LD

Here I examine how different driving forces (mating system, genetic drift, migration, and natural selection) affect zygotic LD from the evolutionary perspective, complementary to the existing statistical issues. An island-continent model is considered, with an emphasis on the evolution of zygotic LD in the finite island population. I begin by presenting an exact model and use Monte Carlo (MC) simulations to evaluate the evolution of zygotic LD under different evolutionary forces in the population with a mixed mating system. I then derive analytical theories in two specific cases (genetic hitchhiking effects and a neutral process) under random mating, and validate the theories through MC simulations. Through the analytical and simulation results, I explore the evolutionary properties of zygotic LDs and discuss their potential utility.

Consider an island population, with constant immigration rates of pollen (

Consider two diallelic nuclear loci, with alleles

In the continent population, let

Let

After genetic drift, the numbers of distinct genotypes follow a multinomial distribution. Here, the genetic sampling of

Genetic drift at each generation can cause the associations between gametic and zygotic LDs or between different zygotic LDs due to their sharing of some alleles or genotypes. There are four types of covariances between gametic and zygotic LDs,

For example, the covariance between _{,} described in the section of Analytical Theory). Similarly, multiple independent simulations can be used to evaluate the expectations of these high-order LDs.

Note that the above general model can reduce to specific models with different numbers of evolutionary forces (e.g., the model with a single evolutionary force). Also, I only concentrate on the covariances between allelic frequencies, or between genotypic frequencies, or between gametic and zygotic LDs, or between different zygotic LDs. The expectations of their normalized values, like the square of normalized gametic LD,

MC simulations are used to examine how different evolutionary forces change zygotic LDs and other types of covariances in the plant species of a mixed mating system. Suppose that the island population initially has the same genetic composition as the continent population. For simplicity, notation for the alleles and subscripts in the above exact model is changed as

_{AB}_{AaBB}_{AABb}_{AB}_{AABb}_{AB}_{AaBB}_{AABB}_{AaBB}_{AABB}_{AaBB}_{AaBB}_{AaBb}_{AABb}_{AaBb}_{AB}_{AABB}

Average steady-state gametic and zygotic LDs (a) and their standard deviations (b); average steady-state covariances between gametic and zygotic LDs (c) and their standard deviations (d); and average steady-state covariances between distinct zygotic LDs (e) and their standard deviations (f). Results are obtained from 10000 independent simulation runs. Parameter settings are the recombination rate = 5%, the immigration rate of pollen _{P}_{S}_{AB}_{Ab}_{aB}_{ab}_{AABB}_{AABb}_{AaBB}_{AAbb}_{AaBb}_{aaBB}_{Aabb}_{aaBb}_{aabb}

The steady-state covariances between gametic and zygotic LDs (_{AB}_{AABb}_{AB}_{AaBB}_{AB}_{AABB}_{AB}_{AaBb}_{AABB}_{AaBB}_{AABB}_{AABb}_{AaBB}_{AaBb}_{AABb}_{AaBb}_{AABB}_{AaBb}_{AaBB}_{AABb}

The steady-state _{AB}_{AABB}_{AB}_{AB}_{AaBb}_{AaBB}_{AABb}_{AB}_{AABB}_{AB}_{AaBb}

The above examples indicate that plants with distinct mating systems have different zygotic LDs and other covariances in a local population even under the same impacts of immigration. Both zygotic and gametic LDs are sensitive to the pattern of mating system. Predominant outcrossing species have weaker covariances between zygotic and gametic LDs, but stronger covariances between distinct zygotic LDs than do the predominant selfing species.

_{AB}_{AB}_{AABB}_{AaBb}_{AaBB}_{AABb}_{AB}_{AABb}_{AB}_{AaBB}_{AB}_{AaBb}_{AB}_{AABB}

Average steady-state gametic and zygotic LDs (a) and their standard deviations (b); average steady-state covariances between gametic and zygotic LDs (c) and their standard deviations (d); and average steady-state covariances between distinct zygotic LDs (e) and their standard deviations (f). Results are obtained from 10000 independent simulation runs. Parameter settings are the selfing rate = 5%, the recombination rate = 5%, the effective population size = 50, the immigration rate of seeds _{S}_{AB}_{Ab}_{aB}_{ab}_{AABB}_{AABb}_{AaBB}_{AAbb}_{AaBb}_{aaBB}_{Aabb}_{aaBb}_{aabb}

Seed flow has greater effects than pollen flow on zygotic LDs and other covariances (_{AB}_{AABB}_{AaBb}_{AB}_{AABB}_{AB}_{AaBb}_{AABB}_{AaBb}_{AaBB}_{AABb}_{AaBB}_{AABb}_{AB}_{AaBB}_{AB}_{AABb}_{AABB}_{AaBB}_{AABB}_{AABb}_{AaBB}_{AaBb}_{AABb}_{AaBb}

Average steady-state gametic and zygotic LDs (a) and their standard deviations (b); average steady-state covariances between gametic and zygotic LDs (c) and their standard deviations (d); and average steady-state covariances between distinct zygotic LDs (e) and their standard deviations (f). Results are obtained from 10000 independent simulation runs. Parameter settings are the selfing rate = 5%, the effective population size = 50, the immigration rate of pollen _{P}_{AB}_{Ab}_{aB}_{ab}_{AABB}_{AABb}_{AaBB}_{AAbb}_{AaBb}_{aaBB}_{Aabb}_{aaBb}_{aabb}

These examples indicate that a local plant population generally exhibits robust responses to the impacts of immigration of pollen and seeds in terms of zygotic LDs, or the covariances between gametic and zygotic LDs, or the covariances between distinct zygotic LDs. Seed and pollen flow have small effects on high-order LDs in a local population.

_{AB}_{AABB}_{AaBB}_{AABb}_{AaBb}_{AB}_{AABB}_{AB}_{AaBb}_{AB}_{AABb}_{AB}_{AaBB}_{AABB}_{AaBb}_{AaBB}_{AABb}_{AABB}_{AaBB}_{AABB}_{AABb}_{AaBB}_{AaBb}_{AABb}_{AaBb}

Average steady-state gametic and zygotic LDs (a) and their standard deviations (b); average steady-state covariances between gametic and zygotic LDs (c) and their standard deviations (d); and average steady-state covariances between distinct zygotic LDs (e) and their standard deviations (f). Results are obtained from 10000 independent simulation runs. Parameter settings are the selfing rate = 5%, the immigration rate of seeds _{S}_{P}_{AB}_{Ab}_{aB}_{ab}_{AABB}_{AABb}_{AaBB}_{AAbb}_{AaBb}_{aaBB}_{Aabb}_{aaBb}_{aabb}

The examples indicate that a small local population can affect zygotic LDs, and has large effects on the covariances between gametic and zygotic LDs or between distinct zygotic LDs. These high-order covariances are more informative than gametic LD in signaling the effects of population demographic dynamics.

_{AB}

Gametic selection | Zygotic selection | Gametic and zygotic selection | |

0.5780± 0.1221 | 0.5893±0. 1217 | 0.6521±0.1135 | |

0.5822±0.1224 | 0.5919±0.1209 | 0.6559±0.1131 | |

0.0755±0.0497 | 0.0746±0.0495 | 0.0730±0.0463 | |

0.0619±0.0495 | 0.0629±0.0499 | 0.0722±0.0519 | |

–0.0338±0.0427 | –0.0355±0.0433 | –0.0468±0.0456 | |

–0.0342±0.0428 | –0.0358±0.0431 | –0.0471±0.0456 | |

0.0437±0.0468 | 0.0438±0.0467 | 0.0492±0.0471 | |

3.619×10^{−4}±1.101×10^{−4} | 3.679×10^{−4}±1.107×10^{−4} | 4.035×10^{−4}±1.099×10^{−4} | |

–1.813×10^{−4}±1.341×10^{−4} | –1.887×10^{−4}±1.350×10^{−4} | –2.348×10^{−4}±1.329×10^{−4} | |

–1.840×10^{−4}±1.329×10^{−4} | –1.904×10^{−4}±1.338×10^{−4} | –2.368×10^{−4}±1.313×10^{−4} | |

2.691×10^{−4}±1.615×10^{−4} | 2.689×10^{−4}±1.616×10^{−4} | 2.880×10^{−4}±1.508×10^{−4} | |

–6.525×10^{−4}±2.086×10^{−4} | –6.653×10^{−4}±2.064×10^{−4} | –7.266×10^{−4}±1.877×10^{−4} | |

–6.550×10^{−4}±2.088×10^{−4} | –6.665×10^{−4}±2.056×10^{−4} | –7.280×10^{−4}±1.874×10^{−4} | |

5.107×10^{−4}±2.043×10^{−4} | 5.256×10^{−4}±2.025×10^{−4} | 6.088×10^{−4}±1.870×10^{−4} | |

5.500×10^{−4}±2.083×10^{−4} | 5.636×10^{−4}±2.059×10^{−4} | 6.428×10^{−4}±1.855×10^{−4} | |

–6.816×10^{−4}±2.344×10^{−4} | –6.945×10^{−4}±2.297×10^{−4} | –7.631×10^{−4}±2.035×10^{−4} | |

–6.723×10^{−4}±2.338×10^{−4} | –6.892×10^{−4}±2.316×10^{−4} | –7.562×10^{−4}±2.051×10^{−4} |

Three selection schemes are: _{AB}_{Ab}_{aB}_{ab}_{AABB}_{AABb}_{AaBB}_{AAbb}_{AaBb}_{aaBB}_{Aabb}_{aaBb}_{aabb}_{AB}_{Ab}_{aB}_{ab}_{AABB}_{AABb}_{AaBB}_{AAbb}_{AaBb}_{aaBB}_{Aabb}_{aaBb}_{aabb}_{P}_{S}_{d}

To assess the effects of epistatic selection, I use Dobzhansky-Muller’s incompatibility model _{AABB}_{aabb}_{AaBB}_{aaBb}_{AABb}_{Aabb}_{aaBB}_{AAbb}_{AaBb}_{AABB}_{aabb}_{AaBB}_{aaBb}_{AABb}_{Aabb}_{aaBB}_{AAbb}_{AaBb}_{AABB}_{aabb}_{AaBB}_{aaBb}_{AABb}_{Aabb}_{aaBB}_{AAbb}_{AaBb}

Results indicate that epistatic selection can change the relative gametic and zygotic LDs (_{AaBB}_{AaBb}_{AB}_{AB}_{AABB}_{AaBb}_{AaBB}_{AABb}_{AB}_{AABB}_{AB}_{AABb}_{AB}_{AaBb}_{AB}_{AaBB}_{AABB}_{AABb}_{AABB}_{AaBb}_{AaBB}_{AABb}_{AABB}_{AaBB}_{AaBB}_{AaBb}_{AABb}_{AaBb}

Case I | Case II | Case III | |

0.5068± 0.1280 | 0.3090±0. 2127 | 0.2673±0.2343 | |

0.5078±0.1292 | 0.7573±0.1937 | 0.8170±0.2009 | |

0.0788±0.0510 | 0.0329±0.0218 | 0.0165±0.0113 | |

0.0533±0.0462 | 0.0219±0.0236 | 0.0100±0.0112 | |

–0.0235±0.0394 | 0.0257±0.0337 | 0.0310±0.0248 | |

–0.0231±0.0397 | –0.0172±0.0205 | –0.0079±0.0095 | |

0.0436±0.0474 | –0.0120±0.0312 | –0.0245±0.0215 | |

3.217×10^{−4}±1.149×10^{−4} | 1.368×10^{−4}±1.195×10^{−4} | 0.556×10^{−4}±7.466×10^{−4} | |

–1.384×10^{−4}±1.353×10^{−4} | 0.6650×10^{−4}±1.158×10^{−4} | 8.250×10^{−4}±6.508×10^{−4} | |

–1.399×10^{−4}±1.341×10^{−4} | –0.9688×10^{−4}±0.8927×10^{−4} | –0.377×10^{−4}±0.467×10^{−4} | |

2.692×10^{−4}±1.642×10^{−4} | 0.1426×10^{−4}±1.194×10^{−4} | –0.499×10^{−4}±0.505×10^{−4} | |

–5.511×10^{−4}±2.298×10^{−4} | –0.928×10^{−4}±1.275×10^{−4} | 0.1252×10^{−4}±0.537×10^{−4} | |

–5.518×10^{−4}±2.289×10^{−4} | –1.987×10^{−4}±1.753×10^{−4} | –0.6276×10^{−4}±0.823×10^{−4} | |

3.942×10^{−4}±2.105×10^{−4} | 0.8561×10^{−4}±1.252×10^{−4} | –0.092×10^{−4}±0.486×10^{−4} | |

4.386×10^{−4}±2.212×10^{−4} | 0.8903×10^{−4}±1.284×10^{−4} | –0.091×10^{−4}±0.489×10^{−4} | |

–5.628×10^{−4}±2.536×10^{−4} | –5.495×10^{−4}±2.362×10^{−4} | –2.307×10^{−4}±1.544×10^{−4} | |

–5.611×10^{−4}±2.513×10^{−4} | –0.7594×10^{−4}±1.251×10^{−4} | 0.128×10^{−4}±0.4867×10^{−4} |

Three selection schemes are: _{AABB}_{aabb}_{AaBB}_{aaBb}_{AABb}_{Aabb}_{aaBB}_{AAbb}_{AaBb}_{AABB}_{aabb}_{AaBB}_{aaBb}_{AABb}_{Aabb}_{aaBB}_{AAbb}_{AaBb}_{AABB}_{aabb}_{AaBB}_{aaBb}_{AABb}_{Aabb}_{aaBB}_{AAbb}_{AaBb}_{P}_{S}_{d}

The above examples indicate that zygotic and gametic LDs have different responding patterns to natural selection. The cumulative selection can enhance zygotic LDs and other covariances in the additive-viability selection model. One striking result is that epistatic selection at the diploid level can produce zygotic LDs that are greater than or comparable to gametic LD. This pattern can be used to detect the epistatic selection process in natural populations.

To further understand the evolution of zygotic LDs, I derive the analytical theory in a linear-additive-viability model with weak selection and random mating (

With the weak selection, all items containing the second or higher orders of selection coefficients are neglected. The immigration rates of seeds and pollen are assumed to be small. The items containing the second or higher orders of the migration rate (

From Eqs. (A1) ∼ (A5) in

Let

Notation

In Eq. (3), there are seven items with the average change coefficients

With the diffusion model, the expectations of zygotic LDs and the covariances between gametic and zygotic LDs or between different zygotic LDs can be calculated in theory. However, the algebraic deduction remains complicated when the joint effects of selection, migration, and genetic drift are considered. Here, I consider two specific cases. One case is that locus

The steady-state equation for allelic frequency at locus

The steady-state allelic frequency can be numerically calculated from the above cubic equation, given the condition of _{a}

To calculate the expectations of the steady-state zygotic LDs and other types of covariances from

Substitution of

where

Expectations of the remaining nine functions can be numerically calculated using

With the availability of the above fourteen expectations, the expectations of some lower or the same order functions can be indirectly calculated. For instance, I can obtain _{….}

Eqs. (5) and (8) indicate that effects of seed and pollen flow are compounded in generating gametic LD, but can be separated in generating zygotic LDs.

The expectations of the steady-state variances of any zygotic LDs can be calculated using Fisher’s delta method by omitting all items containing

The expectation of any steady-state covariance between gametic and different zygotic LDs can be calculated using Fisher’s delta method by omitting all items with

Similarly, expectations of other covariances in

Simulations confirm that the above analytical model performs well. For instance, consider the same parameter settings as in _{a}^{th} generation; data not shown here), reflecting the equilibrium among the effects of migration, genetic drift, and genetic hitchhiking. All analytical predictions are distributed within the ranges of one-standard deviations of the simulation results (

MC simulations | Analytical model | |

0.7165±0.0716 | 0.7071 | |

0.5916±0.0915 | 0.5509 | |

0.0777±0.0355 | 0.0536 | |

0.0732±0.0405 | 0.0465 | |

–0.0533±0.0347 | –0.0317 | |

–0.0367±0.0317 | –0.0169 | |

0.0419±0.0375 | 0.0202 | |

2.011×10^{−4}±4.323×10^{−5} | 2.018×10^{−4} | |

–1.463×10^{−4}±3.994×10^{−5} | –1.389×10^{−4} | |

–1.059×10^{−4}± 5.793×10^{−5} | –7.188×10^{−5} | |

1.541×10^{−4}±6.185×10^{−5} | 1.032×10^{−4} | |

–3.918×10^{−4}±7.381×10^{−5} | –4.125×10^{−4} | |

–3.496×10^{−4}±8.167×10^{−5} | –3.514×10^{−4} | |

3.140×10^{−4}±7.641×10^{−5} | 3.123×10^{−4} | |

3.278×10^{−4}±7.803×10^{−5} | 3.217×10^{−4} | |

–3.310×10^{−4}±9.101×10^{−5} | –3.366×10^{−4} | |

–4.654×10^{−4}±5.826×10^{−5} | –5.191×10^{−4} |

Parameter settings are the immigration rate of pollen _{P}_{S}_{aO}_{aP}_{a}_{d}

_{B}_{AABB}_{AaBb}_{AaBB}_{AABb}_{AB}_{AABB}_{AABB}_{AaBb}_{AaBB}_{AABb}_{AABB}_{AABb}_{AaBB}_{AaBb}_{AABB}_{AaBB}_{AABb}_{AaBb}

Gametic and zygotic LDs (a); covariances between gametic and zygotic LDs (b); and covariances between distinct zygotic LDs (c). Results are obtained from the analytical model in the section of Analytical Theory. Parameter settings are the immigration rate of pollen _{P}_{S}_{aO}_{aP} = _{a}

The above results indicate that the gametic LD can have a similarly changing pattern to some zygotic LDs with the selection pressure. This provides the genetic basis of using zygotic LDs to describe genetic hitchhiking effects at the diploid level. Furthermore, the covariances between gametic and zygotic LDs or between distinct zygotic LDs are informative to indicate genetic hitchhiking effects.

Letting

The _{st}

Substituting

Eq. (12) indicates that the expectation of gametic LD is equal to zero in the absence of LD in migrants, such as in a completely isolated population. Eq. (13) indicates that the expectation of joint allele frequencies at two loci is related to the gametic LD in migrants (

Expectations of the remaining forty-four functions can be calculated in the following steps. Substitution of

The above order of

Once the expectations of the above fifty-four functions are available, the expectations of lower or the same order functions can be indirectly calculated. For instance, I can obtain

For instance, the expectation of steady-state zygotic LDs for the genotype with double heterozygotes (

Simulations confirm that the above analytical model performs well. The gametic and zygotic LDs and other covariances between two neutral loci can quickly reach steady-state distributions, reflecting the equilibrium among the effects of migration, recombination, and genetic drift. All analytical results are distributed within the range of one standard deviation of the simulation results (_{AABb}_{AaBB}

MC simulations | Analytical model | |

0.5079±0.0898 | 0.5 | |

0.5102±0.0913 | 0.5 | |

0.0974±0.0399 | 0.0615 | |

0.0623±0.0363 | 0.0360 | |

–0.0251±0.0312 | –0.0100 | |

–0.0256±0.0312 | –0.0100 | |

0.0462±0.0383 | 0.0200 | |

1.671×10^{−4}±3.842×10^{−5} | 1.689×10^{−4} | |

–0.909×10^{−4}±4.71×10^{−5} | –0.634×10^{−4} | |

–0.916×10^{−4}±4.67×10^{−5} | –0.634×10^{−4} | |

1.774×10^{−4}±5.876×10^{−5} | 1.269×10^{−4} | |

–3.164×10^{−4}±8.517×10^{−5} | –2.921×10^{−4} | |

–3.167×10^{−4}±8.527×10^{−5} | –2.921×10^{−4} | |

2.318×10^{−4}±8.527×10^{−5} | 2.193×10^{−4} | |

2.575×10^{−4}±8.78×10^{−5} | 2.267×10^{−4} | |

–2.977×10^{−4}±1.002 ×10^{−4} | –3.237×10^{−4} | |

–2.949×10^{−4}±0.990×10^{−4} | –3.237×10^{−4} |

Parameter settings are the immigration rate of pollen _{P}_{S}_{d}

_{AB}_{AABB}_{AABB}_{AaBb}_{AaBB}_{AABb}

Gametic and zygotic LDs (a); covariances between gametic and zygotic LDs (b); and covariances between distinct zygotic LDs (c). Results are obtained from the analytical model in the section of Analytical Theory. Parameter settings are the immigration rate of pollen _{P}_{S}_{aO}_{aP}_{a}

The above results indicate that a neutral process can generate a similar pattern between zygotic and gametic LDs along chromosomes, with strong LDs within short distances and weak LDs within long distances. The covariances between gametic and zygotic LDs or between distinct zygotic LDs are relatively insensitive to the linkage distance.

In this study, I have developed the evolutionary theory of zygotic LDs in a local plant population, complementing the previous theories that mainly focus on the statistical issues

It is important to understand that in a pure drift process, LD is transient in a completely isolated population of random mating. Expectations of both gametic and zygotic LDs are zero although the expectations of their squared values are nonzero

Zygotic LDs in the finite population are calculated by synthesizing the theories of Robertson

Note that the theory only addresses the constant immigration of seeds and pollen. In reality, a frequent situation is the stochastic migration of seeds and pollen due to the influences of biotic and abiotic factors

Also, note that a plant mating system in a natural population may exhibit a dynamic property

Apart from the above assumptions, the theory suggests several useful implications

The occurrence of a weak gametic LD combined with strong zygotic LDs suggests epistatic interactions at the diploid level (e.g., postzygotic isolation due to the Dobzhansky-Muller incompatibility

Second, the theory provides a genetic basis of using zygotic LDs for QTL mapping that has been recently addressed

Third, the theory aids in predicting the effects of seed and pollen flow on zygotic LDs in a local population. Previous studies use gametic LD to estimate gene flow in a specific case, such as in hybrid zones

Fourth, the theory aids in assessing the selection mode (additive or epistatic) in the gametophyte and sporophyte stages in generating gametic and zygotic LDs. “Bulmer effects” mainly emphasize the impacts of selection on gametic LD

In addition, the genotypic interaction on fitness may arise from the dominance by dominance effects for _{AaBb}_{AABb}_{AaBB}_{AABB}

Finally, it is of interest to discuss the utility of the covariances between distinct zygotic LDs since few studies have examined such high-order LDs

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I sincerely appreciate three anonymous reviewers for very helpful comments that substantially improved the presentation of this article.