Novel Statistical methods for extracting information from genetic data

这个数学建模项目涉及对从植物中提取的遗传数据的分析，以及开发新的数学和统计工具来从数据中提取信息。该项目涉及将统计物理学和信息论的思想应用于关于基因-表型变异的数据。在这两个学科中，熵和微态分布的概念在描述宏观量的理论中是有用的。我们考虑从总体中抽取一个样本，然后按表型(即可观察到的特征，如身高或体重)进行排序。通过分析基因类型(AA、AA、AA)的顺序，我们得到了一个有效领域，它量化了一个特定的单核苷酸多态(SNP-一个微小的基因突变)对特定表型的影响程度。这为分析基因-表型互作提供了另一种方法，该方法比经典的基因组广谱关联研究更有效，并且不依赖于Fisher在1901所做的统计假设。在本博士项目中，这一新方法将被应用于拟南芥，以了解个体的遗传状态如何影响观察到的表型，如拟南芥对离子的吸收如何影响植物的生长。数据来自Sian Bray进行的离子研究，他研究一系列离子在植物中的传输。我们通常关注三种双等位基因状态以及这些状态对一系列连续的表型值的影响。然后，我们将考虑多个基因对单个表型的影响。该方法量化了基因-表型依赖性的强度；虽然许多SNP对表型没有显著影响，但有些会，并在不同程度上影响。我们将研究连锁不平衡，即当一个人沿着染色体移动到附近的SNPs时，相互作用的强度是如何变化的。通常，在有一个SNP具有很强的表型效应的地方，其他与染色体接近的SNP也被认为对表型有显著影响。我们将研究这种相互作用的范围，并构建模型来解释这种相互作用随着距离最重要的SNP的距离而衰减的速度有多快。虽然最终目标是解释植物数据和其他真实数据集，但新方法将在合成数据上进行测试，以测试其演绎能力。我们将研究该模型的概括性，例如，表型可能仅作为“入库”数据可用，即在一系列范围(例如10-20、20-30、30-40等)中每个基因型(AA、AA、AA)的数量。在这种情况下，数据的‘力量’和信息较少，因此可能不太可能被标记为显著的，但由于它更平滑，它可能会给出一种更简单的场强形式。我们还将考虑多等位基因系统-即存在三个以上等位基因的系统。随着排序列表的可能排列的数量变得越来越大，这种高维系统的数学分析明显更加复杂。

This mathematical modelling project involves a combination of analysis of genetic data taken from plants and the development of novel mathematical and statistical tools to extract information from the data.This project involves applying of ideas from statistical physics and information theory to data on genotype-phenotype variations. In both disciplines, the concepts of entropy and the distribution of microstates is useful in theories which describe macroscopic quantities. We consider a sample from the population, which is then ranked by phenotype (that is, an observable characteristic, such as height or weight). By analysing the respective order of genotypes (AA,Aa,aa), we derive an effective field which quantifies how strongly one particular single nucleotide polymorphism (SNP - a minor genetic mutation) influences a particular phenotype. This provides an alternative method for analysing genotype-phenotype interactions, which is more powerful than the classic genome wide association studies (GWAS), and does not rely on the statistical assumptions made by Fisher in 1901.In this PhD project this new method will be applied to Arabidopsis to understand how the genetic states of individuals influence observed phenotypes such as how ions uptake in Arabidopsis influence plant growth. Data comes from ionomic studies performed by Sian Bray, who studies the transport of a range of ions in plants. We will typically be concerned with three bi-allelic states and the impact of these states on a continuous range of phenotype values. We will then consider the effects of multiple genes on a single phenotype.The method quantifies the strength of the genotype-phenotype dependency; whilst many SNPs will have no significant effect on phenotype, some will, and to varying extents. We will investigate linkage disequilibrium, that is, how the strength of interaction varies as one moves along the chromosomes to nearby SNPs. Typically, where there is a SNP with a strong phenotype effect, other SNPs which are close by on the chromosome are also seen to have significant impact on phenotype. We will investigate the range of this interaction, and construct models to explain how rapidly this decays with distance from the most significant SNP. Whilst the ultimate goal is to interpret the plant-data, and other real data sets, new methods will be tested on synthetic data, in order to test their deductive power. We will investigate generalisations of the model, for example, the phenotypes may only available as `binned' data, that is numbers of each genotype (AA,Aa,aa) in a sequence of ranges (eg 10-20, 20-30, 30-40 etc). In this case the data has less 'power' and information, so might be less likely to be flagged as significant, but since it is smoother, it may give a simpler form for the field-strength.We will also consider multi-allelic systems - that is, where more than three alleles are present. The mathematical analysis of such higher-dimensional systems is significantly more complicated, as the number of possible arrangements of the ranked list becomes increasingly large.