THE ASYMPTOTIC PERIODICITY OF PIECEWISE MONOTONE SELFMAPS OF THE INTERVAL

Let f:I→I be a piecewise monotone interval map. The critical point set C(f) of f consists of all the preimages of its turning points. It is proved that the complex dynamical behaviors of f are all concentrated on the derived set of C(f). f is asymptotically periodic if and only if the derived set of C(f) is countable.

Multiple Interval Mapping for Whole Cocoon Weight and Related Economically Important Traits QTL in Silkworm(Bombyx mori)

A backcrossed population （BC1） derived from a cross between C100 and Dazao was obtained. The quantitative trait loci （QTLs） of the economically important traits for whole cocoon weight, cocoon shell weight, ratio of cocoon shell and weight of pupae, etc., were analyzed for the first time using the multiple interval mapping software WinQTLCart2.0. In total 40 QTLs were detected and contributed to 21 groups based on the constructed linkage map. According to the mapping results, 2, 2, 3, and 2 major QTLs explained over 20% of total phenotypic variations, whereas four QTLs, namely qCW-19, qSW-2, qCSR-4, and qPW-23, explained more than 30% of total phenotypic variations for whole cocoon weight, cocoon shell weight, ratio of cocoon shell and weight of pupae, respectively. Correlated traits QTLs often share the same location. Furthermore, most of the detected QTLs were closed to one-side marker. By using the very closed markers, positive QTLs can be aggregated, which can form a basis for molecular marker-assisted selection and breeding.

QTL Analysis of Agronomic Traits in Soybean

[Objective]The aim was to analyze QTL of agronomic traits in soybean and provide reference for a discussion on soybean genetic mechanism and genetic breeding. [Method]The composite interval mapping method was used for QTL location and genetic effects analysis on 5 quantitative traits including protein content,fat content,yield,100-grain weight and growth period. [Result]The control of these traits 4,4,1,2,5,a total of 16 QTL loci was detected. The genetic contribution rate was in 7.4%-33.7%,among which,a large main-effect QTL of the genetic contribution rate were located in linkage group I Satt562-Sat_219,Sat_219-Satt496,Sat_219-Satt496 interval of the three control protein content QTL sites,their genetic contribution rates were 29.15%,33.7 % and 31.67% respectively,all from the female parent Hefeng 25 plus minor gene; still in O linkage group Satt477-Satt331,Satt331-Satt153 interval of two control growing period QTL loci,their genetic contribution rates were up to 24.69% and 24.96%,also from the female parent Hefeng 25 plus minor gene. In addition,six QTL sites from M linkage group Satt175 （protein）,A1 linkage group Satt684 （oil）,F linkage group Satt348 （oil）,J linkage group Sat_412 （oil）,C1 linkage group Sat_416 （100-grain weight） and C1 linkage group Sat_416 （growth period） marks only 0.01 cm were detected. [Conclusion]QTL sites which had effects on the 5 important agronomic traits in soybean were located.

A Note on “Multi-separation, Centrifugality and Centripetality Imply Chaos”

In a recent paper from Trans. Amer. Math. Soc., 351(1)(1999), 343—351, the maps f and g from [0,n] to itself are constructed, which have periodic points of period 2a+3 . In this note we prove that they have no periodic points of period 2a+1.

Mapping of quantitative trait loci using the skew-normal distribution

In standard interval mapping (IM) of quantitative trait loci (QTL), the QTL effect is described by a normal mixture model. When this assumption of normality is violated, the most commonly adopted strategy is to use the previous model after data transformation. However, an appropriate transformation may not exist or may be difficult to find. Also this approach can raise interpretation issues. An interesting alternative is to consider a skew-normal mixture model in standard IM, and the resulting method is here denoted as skew-normal IM. This flexible model that includes the usual symmetric normal distribution as a special case is important, allowing continuous variation from normality to non-normality. In this paper we briefly introduce the main peculiarities of the skew-normal distribution. The maximum likelihood estimates of parameters of the skew-normal distribution are obtained by the expectation-maximization (EM) algorithm. The proposed model is illustrated with real data from an intercross experiment that shows a significant departure from the normality assumption. The performance of the skew-normal IM is assessed via stochastic simulation. The results indicate that the skew-normal IM has higher power for QTL detection and better precision of QTL location as compared to standard IM and nonparametric IM.

Identification of additional QTLs for flowering time by removing the effect of the maturity gene E1 in soybean

The adaptability of soybean to be grown at a wide range of latitudes is attributed to natural variation in the major genes and quantitative trait loci （QTLs） that control flowering time and maturity. Thus, the identification of genes controlling flowering time and maturity and the understanding of their molecular basis are critical for improving soybean productivity. However, due to the great effect of the major maturity gene E1 on flowering time, it is difficult to detect other small-effect QTLs. In this study, aiming to reduce the effect of the QTL, associated with the E1 gene, on the detection of other QTLs, we divided a population of 96 recombinant inbred lines （RILs） into two sub-populations： one with the E1 allele and another with the elns allele. Compared with the results of using all 96 recombinant inbred lines, additional QTLs for flowering time were identified in the sub-populations, two （qFT-B1 and qFT-H） in RILs with the E1 allele and one （qFT-J-2） in the RILs with the elnl allele, respectively. The three QTLs, qFT-B1, qFT-H and qFT-J-2 were true QTLs and played an important role in the regulation of growth period. Our data provides valuable information for the genetic mapping and gene cloning of traits controlling flowering time and maturity and will help a better understanding of the mechanism of photoperiod-regulated flowering and molecular breeding in soybean.

QTL detection for growth traits in Pinus elliottii var. elliottii and P. caribaea var. hondurensis

Quantitative trait locus （QTL） detection was carried out for growth traits in 122 F1 progenies of Pinus elliottiivar. elliottii （PEE） x tR caribaea var. hondurensis （PCH） hybrid. The PCH male parent linkage map contained 108 mark- ers in 16 linkage groups, while the PEE female parent contained 93 markers in 19 linkage groups. Sequence-related amplified polymorphism （SRAP）, microsatellite （SSR）, expressed sequence tag polymorphism （ESTP） and inter-simple sequence repeat （ISSR） were selected from an existing linkage map. Growth traits investigated were height at age five （HT05） and six years （HT06）, diameter at breast height at age five （DBH05） and six years （DBH06） and annual growth from age five to six years of height （AGHT） and diameter at breast height （AGDBH）. KruskaI-Wallis and interval mapping approaches were used to estimate levels of significance, the number of QTLs, the percentage of the phenotypic varia- tion explained by each of QTLs and their positions on the genetic linkage maps. Twenty six QTLs with significance levels p 〈 0.05 were detected on the parental maps for the six growth traits, which explained more than 15% of the phenotypic variation, suggesting an existence of major-effect genes. Several QTLs had the pleiotropy controlling more than one sin- gle growth trait. Overall, the proportion of phenotypic variation explained by QTLs ranged from 5.9% to 40.6% for HT05 and HT06, from 6.6% to 42.0% for DBH05 and DBH06 and from 5.8% to 22.3% for AGHT and AGDBH. The results from this study provide a basis for marker-aided selection （MAS） in PEE x PCH hybrids.

Identification of QTLs for Yield and Associated Traits in F_(2) Population of Rice

Identification of quantitative trait loci(QTLs)controlling yield and yield-related traits in rice was performed in the F_(2) mapping population derived from parental rice genotypes DHMAS and K343.A total of 30 QTLs governing nine different traits were identified using the composite interval mapping(CIM)method.Four QTLs were mapped for number of tillers per plant on chromosomes 1(2 QTLs),2 and 3;three QTLs for panicle number per plant on chromosomes 1(2 QTLs)and 3;four QTLs for plant height on chromosomes 2,4,5 and 6;one QTL for spikelet density on chromosome 5;four QTLs for spikelet fertility percentage(SFP)on chromosomes 2,3 and 5(2 QTLs);two QTLs for grain length on chromosomes 1 and 8;three QTLs for grain width on chromosomes1,3 and 8;three QTLs for 1000-grain weight(TGW)on chromosomes 1,4 and 8 and six QTLs for yield per plant(YPP)on chromosomes 2(3 QTLs),4,6 and 8.Most of the QTLs were detected on chromosome 2,so further studies on chromosome 2 could help unlock some new chapters of QTL for this cross of rice variety.Identified QTLs elucidating high phenotypic variance can be used for marker-assisted selection(MAS)breeding.Further,the exploitation of information regarding molecular markers tightly linked to QTLs governing these traits will facilitate future crop improvement strategies in rice.

The Statistical Power of Inclusive Composite Interval Mapping in Detecting Digenic Epistasis Showing Common F_2 Segregation Ratios

Epistasis is a commonly observed genetic phenomenon and an important source of variation of complex traits, which could maintain additive variance and therefore assure the long-term genetic gain in breeding. Inclusive composite interval mapping （ICIM） is able to identify epistatic quantitative trait loci （QTLs） no matter whether the two interacting QTLs have any additive effects. In this article, we conducted a simulation study to evaluate detection power and false discovery rate （FDR） of ICIM epistatic mapping, by considering F2 and doubled haploid （DH） populations, different F2 segregation ratios and population sizes. Results indicated that estimations of QTL locations and effects were unbiased, and the detection power of epistatic mapping was largely affected by population size, heritability of epistasis, and the amount and distribution of genetic effects. When the same likelihood of odd （LOD） threshold was used, detection power of QTL was higher in F2 population than power in DH population; meanwhile FDR in F2 was also higher than that in DH. The increase of marker density from 10 cM to 5 cM led to similar detection power but higher FDR. In simulated populations, ICIM achieved better mapping results than multiple interval mapping （MIM） in estimation of QTL positions and effect. At the end, we gave epistatic mapping results of ICIM in one actual population in rice （Oryza sativa L.）.

The Strongly Simple Cycles with Given Rotation Pairs of an Interval Map

In this paper, we introduce the notion of the strongly simple cycles with some rotation pair for interval maps and prove that, if an interval map has a cycle with given rotation pair, then it, has a strongly simple cycle with the same rotation pair.

On non-uniform hyperbolicity assumptions in one-dimensional dynamics

We give an essentially equivalent formulation of the backward contracting property,defined by Juan Rivera-Letelier,in terms of expansion along the orbits of critical values,for complex polynomials of degree at least 2 which are at most finitely renormalizable and have only hyperbolic periodic points,as well as all C3 interval maps with non-flat critical points.

Unique Ergodicity for Zero-entropy Dynamical Systems with the Approximate Product Property

We show that for every topological dynamical system with the approximate product property, zero topological entropy is equivalent to unique ergodicity. Equivalence of minimality is also proved under a slightly stronger condition. Moreover, we show that unique ergodicity implies the approximate product property if the system has periodic points.