Department of Mathematics

Abstract Evolutionary game theory has been used to predict the effect on sperm expenditure of a trade–off between the value of a mating and the cost of its acquisition. In particular, GA Parker has predicted that if two males 'know'whether they are first or second to mate, but these roles are assigned randomly, then sperm numbers should be the same for both males whether the 'raffle'for fertilization is fair or unfair.

Abstract There are many examples of individuals forming coalitions to obtain or protect a valuable resource. We present an analytical model of coalition formation in which individuals seek alliances if they judge themselves too weak to secure the resource alone. We allow coalition seeking to carry an investment cost (θ) and let contest outcomes depend probabilistically on the relative fighting strengths of contesting parties, with effective coalition strength directly proportional to combined partner strength.

Summary Game-theoretic analysis is used to demonstrate that a self-monitoring, cooperative agreement among residents to conserve a wildlife resource may be cheaper by at least an order of magnitude for a government to sustain if the community incentive structure separates bene ts for not harvesting and bonuses for arrests made from payments for monitoring per se.

Abstract We develop a game–theoretic model to predict the effect of size–dependent contest outcomes on optimal–clutch–size decisions. We consider the case where larger individuals develop from smaller clutches and, as adults, are advantaged in competition for limiting resources. The relationship between fitness and size thus depends on the sizes of other members of the population.

Abstract RA Fisher's sex ratio theory predicts that if sons and daughters cost fixed amounts of resources to raise and parents have fixed amounts to invest, then the numerical sex ratio of a panmictic population will evolve to be inversely proportional to relative cost. However, the theory assumes control by both parents. We show that allowing one parent to control the sex ratio biases it further from parity than Fisher's theory predicts.

Cacao (Theobroma cacao L.) is an important cash crop in tropical regions around the world and has a rich agronomic history in South America. As a key component in the cosmetic and confectionary industries, millions of people worldwide use products made from cacao, ranging from shampoo to chocolate. An Illumina Infinity II array was created using 13,530 SNPs identified within a small diversity panel of cacao. Of these SNPs, 12,643 derive from variation within annotated cacao genes. The genotypes of 3,072 trees were obtained, including two mapping populations from Ecuador. High-density linkage maps for these two populations were generated and compared to the cacao genome assembly. Phenotypic data from these populations were combined with the linkage maps to identify the QTLs for yield and disease resistance.

Cacao is an important crop, its beans are key raw materials for the chocolate and cosmetic industries. Ceratocystis wilt of cacao (CWC) caused by Ceratocystis cacaofunesta is a lethal disease for the crop. Therefore, the selection of resistant cacao varieties is one of the viable ways to minimize losses in cacao production. In this paper, we described the identification of a major QTL associated with CWC in an F1 mapping population from a cross between a resistant, "TSH 1188," and a susceptible genotype, "CCN 51." A set of 266 trees were genotyped using 3,526 single nucleotide polymorphic markers and then multiple QTL mapping analyses were performed. Two QTLs were identified on chromosomes IV and VI. The major QTL was located at 20 cM from the top position of chromosome VI, accounting for more than 60% of the phenotypic variation. The favorable allele T1, with haplotype GTT, came from the "TSH 1188" parent. It was evident that the haplotype combination T1C2 on chromosome VI was the most significant for resistance, since 93% of resistant trees had this haplotype. The major QTL converged to a genomic region of 739.4 kb that harbored nine candidate genes, including two major classes of resistance genes, which would make them the primary candidates involved in the resistance to CWC. The haplotypes detected are now used to improve the efficiency and precision of the selection of resistant trees in cacao breeding.

Cacao () is a globally important crop, and its yield is severely restricted by disease. Two of the most damaging diseases, witches' broom disease (WBD) and frosty pod rot disease (FPRD), are caused by a pair of related fungi: and , respectively. Resistant cultivars are the most effective long-term strategy to address diseases, but efficiently generating resistant and productive new cultivars will require robust methods for screening germplasm before field testing. Marker-assisted selection (MAS) and genomic selection (GS) provide two potential avenues for predicting the performance of new genotypes, potentially increasing the selection gain per unit time. To test the effectiveness of these two approaches, we performed a genome-wide association study (GWAS) and GS on three related populations of cacao in Ecuador genotyped with a 15K single nucleotide polymorphism (SNP) microarray for three measures of WBD infection (vegetative broom, cushion broom, and chirimoya pod), one of FPRD...

A comprehensive understanding of the genetic basis of target traits in any crop is critical to design breeding strategies for the development and release of new improved varieties. In this study, 34 cacao families were evaluated for vigor and yield related traits over the course of 6 years in Costa Rica. Linear mixed models provided the variance components for the partitioning of additive and non-additive effects. Heritabilities of yield over time ranged from 0.085 to 0.576, from 0.127 to 0.399 for vigor, and 0.141 to 0.146 for disease resistance traits. Significant (p < 0.001) general combining abilities were observed for ICS-43 and LcTeen-37 with negative effect on average yield (−0.674, −0.690), respectively. Specific combining abilities for yield had significant (p < 0.001) positive effect from the cross GU-154-L x UF-273 Type 2 (0.703) and strong negative interaction between ICS-43 and LF-1 (−0.975). A weighted index was used to select the top performers while providing the corresponding genetic gains. At an 1% selection intensity, yield component gains ranged from 17.8 to 331.9%. Agronomic traits such as branch angle, trunk diameter and jorquette height had lower genetic gains and lower heritabilities. In addition, the parents in this study were genotyped with a 96-SNP marker off-typing set and a significant positive correlation of 0.39 (p = 0.019) was found between genetic distance and specific combining abilities for yield. Preliminary comparison of clonal parents vs. seedlings yield in the family with the highest SCA suggest for the first time presence of heterobeltiosis in cacao.

Breeding programs of cacao (Theobroma cacao L.) trees share the many challenges of breeding long-living perennial crops, and genetic progress is further constrained by both the limited understanding of the inheritance of complex traits and the prevalence of technical issues, such as mislabeled individuals (off-types). To better understand the genetic architecture of cacao, in this study, 13 years of phenotypic data collected from four progeny trials in Bahia, Brazil were analyzed jointly in a multisite analysis. Three separate analyses (multisite, single site with and without off-types) were performed to estimate genetic parameters from statistical models fitted on nine important agronomic traits (yield, seed index, pod index, % healthy pods, % pods infected with witches broom, % of pods other loss, vegetative brooms, diameter, and tree height). Genetic parameters were estimated along with variance components and heritabilities from the multisite analysis, and a trial was fingerprinted with low-density SNP markers to determine the impact of off-types on estimations. Heritabilities ranged from 0.37 to 0.64 for yield and its components and from 0.03 to 0.16 for disease resistance traits. A weighted index was used to make selections for clonal evaluation, and breeding values estimated for the parental selection and estimation of genetic gain. The impact of off-types to breeding progress in cacao was assessed for the first time. Even when present at <5% of the total population, off-types altered selections by 48%, and impacted heritability estimations for all nine of the traits analyzed, including a 41% difference in estimated heritability for yield. These results show that in a mixed model analysis, even a low level of pedigree error can significantly alter estimations of genetic parameters and selections in a breeding program.

The main ingredients of chocolate are usually cocoa powder, cocoa butter, and sugar. Both the powder and the butter are extracted from the beans of the cacao tree (Theobroma cacao L.). The cocoa butter represents the fat in the beans and possesses a unique fatty acid profile that results in chocolate's characteristic texture and mouthfeel. Here, we used a linkage mapping population and phenotypic data of 3,292 samples from 420 progeny which led to the identification of 27 quantitative trait loci (QTLs) for fatty acid composition and six QTLs for fat content. Progeny showed extensive variation in fat levels and composition, with the level of palmitic acid negatively correlated to the sum of stearic acid, oleic acid, and linoleic acid. A major QTL explaining 24% of the relative level of palmitic acid was mapped to the distal end of chromosome 4, and those higher levels of palmitic acid were associated with the presence of a haplotype from the "TSH 1188" parent in the progeny. Within this region of chromosome 4 is the Thecc1EG017405 gene, an orthologue and isoform of the stearoyl-acyl carrier protein (ACP) desaturase (SAD) gene in plants, which is involved in fatty acid biosynthesis. Besides allelic differences, we also show that climate factors can change the fatty acid composition in the beans, including a significant positive correlation between higher temperatures and the higher level of palmitic acid. Moreover, we found a significant pollen donor effect from the variety "SIAL 70" which was associated with decreased palmitic acid levels.

BackgroundTheobroma cacao L. cultivar Matina 1-6 belongs to the most cultivated cacao type. The availability of its genome sequence and methods for identifying genes responsible for important cacao traits will aid cacao researchers and breeders.ResultsWe describe the sequencing and assembly of the genome of Theobroma cacao L. cultivar Matina1-6. The genome of the Matina 1-6 cultivar is 445 Mbp, which is significantly larger than a sequenced Criollo cultivar, and more typical of other cultivars. The chromosome-scale assembly, version 1.1, contains 711 scaffolds covering 346.0 Mbp, with a contig N50 of 84.4 kbp, a scaffold N50 of 34.4 Mbp, and an evidence-based gene set of 29,408 loci. Version 1.1 has 10x the scaffold N50 and 4x the contig N50 as Criollo, and includes 111 Mb more anchored sequence. The version 1.1 assembly has 4.4% gap sequence, while Criollo has 10.9%. Through a combination of haplotype, association mapping and gene expression analyses, we leverage this robust refere...

Economically, cacao (Theobroma cacao L.) is a major tropical commodity for the Americas; however, severe losses due to Moniliophthora roreri (Cif. and Par.), which causes frosty pod rot (FPR), and Phytophthora spp., which causes black pod rot (BPR), have reduced cacao production in the Americas. The objectives of this study are to (i) re-confirm the QTL using different marker set; (ii) discover new QTL associated with FPR and BPR resistance using SNP markers; and (iii) find genes in the candidate QTL regions. At CATIE in Turrialba, Costa Rica, an F1 mapping population of cacao was obtained by crossing “POUND 7,” a clone moderately susceptible to FPR and resistant to BPR, with “UF 273,” resistant to FPR and highly susceptible to BPR. A total of 179 F1 progeny were fingerprinted with 5149 SNP markers and a dense linkage map composed of 10 linkage groups was developed using 2910 polymorphic SNP markers. Also segregating F1 trees were screened for resistance to FPR and BPR diseases. Sev...

Tamarind ( Tamarindus indica L.) is a large evergreen tree belonging to the family Fabaceae (Leguminosae). Its fruits are eaten fresh, processed into a paste, used as a spice, made into a beverage, and used for tanning hides. Although this commercially important species has a wide geographical distribution, research on tamarind is sparse. Hence, the purpose of our research was to characterize 13 genotypes of tamarind at the Subtropical Horticultural Research Station in Miami, FL, USA, and to determine whether physical and pomological traits of this collection in fact reflect the genetic diversity within this fruit tree species. This work was conducted at SHRS in Miami, Florida. The pomological diversity of 13 tamarind genotypes was analyzed based on 18 qualitative and quantitative traits. Principal components analysis (PCA) conducted on quantitative traits showed an important degree of variability of about 72% for the first three principal components. A dendrogram was constructed ba...

We employ a MACSYMA program for the lbrmal, perturbation, multiple bifurcation analysis of a system of nonlinear PDEs arising in double-diffusive convection. The program requires, as input, the system of PDEs to be solved, the form of the asymptotic expansion of the solution and of the bifurcation parameter, the solution of the adjoint linearized system and the desired order of the approximation. The program is automatic and requires no user intervention once started. It outputs the asymptotic solution and its amplitude equation, which describes the bifurcation of the solution branches. A significant amount of hand calculation time is saved by the computer analysis of the problem. The program and its extensions opens up a number of interesting problems which were heretofore intractable due to unwieldy algebra.

Bacteriophages initiate infection by releasing their double-stranded DNA into the cytosol of their bacterial host. However, what controls and sets the timescales of DNA ejection? Here we provide evidence from stochastic simulations which shows that the topology and organization of DNA packed inside the capsid plays a key role in determining these properties. Even with similar osmotic pressure pushing out the DNA, we find that spatially ordered DNA spools have a much lower effective friction than disordered entangled states. Such spools are only found when the tendency of nearby DNA strands to align locally is accounted for. This topological or conformational friction also depends on DNA knot type in the packing geometry and slows down or arrests the ejection of twist knots and very complex knots. We also find that the family of (2, 2k+1) torus knots unravel gradually by simplifying their topology in a stepwise fashion. Finally, an analysis of DNA trajectories inside the capsid shows that the knots formed throughout the ejection process mirror those found in gel electrophoresis experiments for viral DNA molecules extracted from the capsids.

Icosahedral bacteriophages pack their double-stranded DNA genomes to near-crystalline density and achieve one of the highest levels of DNA condensation found in nature. Despite numerous studies, some essential properties of the packaging geometry of the DNA inside the phage capsid are still unknown. We present a different approach to the problems of randomness and chirality of the packed DNA. We recently showed that most DNA molecules extracted from bacteriophage P4 are highly knotted because of the cyclization of the linear DNA molecule confined in the phage capsid. Here, we show that these knots provide information about the global arrangement of the DNA inside the capsid. First, we analyze the distribution of the viral DNA knots by high-resolution gel electrophoresis. Next, we perform Monte Carlo computer simulations of random knotting for freely jointed polygons confined to spherical volumes. Comparison of the knot distributions obtained by both techniques produces a topological proof of nonrandom packaging of the viral DNA. Moreover, our simulations show that the scarcity of the achiral knot 41 and the predominance of the torus knot 51 over the twist knot 52 observed in the viral distribution of DNA knots cannot be obtained by confinement alone but must include writhe bias in the conformation sampling. These results indicate that the packaging geometry of the DNA inside the viral capsid is writhe-directed.

When linear double-stranded DNA is packed inside bacteriophage capsids, it becomes highly compacted. However, the phage is believed to be fully effective only if the DNA is not entangled. Nevertheless, when DNA is extracted from a tailless mutant of the P4 phage, DNA is found to be cyclic and knotted (probability of 0.95). The knot spectrum is very complex, and most of the knots have a large number of crossings. We quantified the frequency and crossing numbers of these knots and concluded that, for the P4 tailless mutant, at least half the knotted molecules are formed while the DNA is still inside the viral capsid rather than during extraction. To analyze the origin of the knots formed inside the capsid, we compared our experimental results to Monte Carlo simulations of random knotting of equilateral polygons in confined volumes. These simulations showed that confinement of closed chains to tightly restricted volumes results in high knotting probabilities and the formation of knots with large crossing numbers. We conclude that the formation of the knots inside the viral capsid is driven mainly by the effects of confinement. ¶ To whom reprint requests should be addressed.

The product of Xer recombination at directly repeated psi sites on a circular unknotted DNA molecule is a right-hand four-noded catenane. Here, we use tangle equations to analyze the topological changes associated with Xer recombination at psi. This mathematical method allows computation of all possible topological pathways consistent with the experimental data. We give a rigorous mathematical proof that, under reasonable biological assumptions, there are only three solutions to the tangle equations. One of the solutions corresponds to a synaptic complex with antiparallel alignment of recombination core sites, the other two correspond to parallel alignment of cores. We show that all three solutions can be unified into a single three-dimensional model for Xer recombination. Thus the three distinct mathematical solutions do not necessarily represent distinct three-dimensional pathways, and in this case the three distinct tangle solutions are different planar projections of the same three-dimensional configuration.