The probability of two loci, separated by a certain genome length, being in contact can be inferred using the chromosome conformation capture (3C) method and related Hi-C experiments. How to go from the contact map, a matrix listing the mean contact probabilities between a large number of pairs of loci, to an ensemble of three-dimensional structures is an open problem. A solution to this problem, without assuming an assumed energy function, would be the first step in understanding the way nature has solved the packaging of chromosomes in tight cellular spaces. We created a theory, based on polymer physics characteristics of chromosomes and the maximum entropy principles, referred to as HIPPS (Hi-C-polymer-physics-structures) method, that allows us to calculate the 3D structures solely from Hi-C contact maps. The first step in the HIPPS method is to relate the mean contact probability ($⟨p_{ij}⟩$) between loci $i$ and $j$ and the average spatial distance $⟨{\overline{r}}_{ij}⟩$. This is a difficult problem to solve because the cell population is heterogeneous, which means that a given contact exists only in a small unknown fraction of cells. Despite the population heterogeneity, we first prove that there is a theoretical lower bound connecting $⟨p_{ij}⟩$ and $⟨{\overline{r}}_{ij}⟩$ via a power-law relation. We show, using simulations of a precisely solvable model, that the overall organization is accurately captured by constructing the distance map from the contact map even if the cell population is highly heterogeneous, thus justifying the use of the lower bound. In the second step, the mean distance matrix, with elements $⟨{\overline{r}}_{ij}⟩\prime \mathrm{s}$, is used as a constraint in the maximum entropy principle to obtain the joint distribution of spatial positions of the loci. Using the two steps, we created an ensemble of 3D structures for the 23 chromosomes from lymphoblastoid cells using the measured contact maps as inputs. The HIPPS method shows that conformations of chromosomes are heterogeneous even in a single cell type. The differences in the conformational heterogeneity of the same chromosome in different cell types (normal as well as cancerous cells) can also be quantitatively discerned using our theory. We validate the method by showing that the calculated volumes of the 23 chromosomes from the predicted 3D structures are in good agreement with experimental estimates. Because the method is general, the 3D structures for any species may be calculated directly from the contact map without the need to assume a specific polymer model, as is customarily done.

中文翻译：

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从Hi-C接触图到人间相染色体的三维组织

可以使用染色体构象捕获（3C）方法和相关的Hi-C实验来推断两个基因座相隔一定基因组长度的可能性。如何从接触图，列出大量成对基因座之间的平均接触概率的矩阵到三维结构的集合，是一个悬而未决的问题。在不假设能量函数的前提下，解决该问题的方法将是理解自然界解决紧密细胞空间中染色体包装的方式的第一步。我们基于染色体的聚合物物理特征和最大熵原理创建了一个理论，称为HIPPS（Hi-C-聚合物物理结构）方法，该理论使我们能够仅从Hi-C接触图计算3D结构。 。$⟨p_{\mathrm{一世}Ĵ}⟩$）之间的位点 $\mathrm{一世}$ 和 $Ĵ$ 和平均空间距离 $⟨{\overline{\mathrm{[R}}}_{\mathrm{一世}Ĵ}⟩$。这是一个很难解决的问题，因为细胞群是异质的，这意味着给定的接触仅存在于一小部分未知的细胞中。尽管总体上存在异质性，但我们首先证明存在理论上的下界连接$⟨p_{\mathrm{一世}Ĵ}⟩$ 和 $⟨{\overline{\mathrm{[R}}}_{\mathrm{一世}Ĵ}⟩$通过幂律关系。我们显示，使用精确可解模型的模拟，即使细胞群体高度异质，通过从接触图构建距离图也可以准确地捕获整个组织，从而证明使用下限是合理的。第二步，平均距离矩阵，包含元素$⟨{\overline{\mathrm{[R}}}_{\mathrm{一世}Ĵ}⟩\prime \mathrm{s}$用作最大熵原理的约束，以获得基因座空间位置的联合分布。使用这两个步骤，我们使用测量的接触图作为输入，为来自淋巴母细胞的23条染色体创建了3D结构的整体。HIPPS方法显示，即使在单个细胞类型中，染色体的构象也是异质的。相同染色体在不同细胞类型（正常细胞和癌细胞）中构​​象异质性的差异也可以使用我们的理论进行定量识别。我们通过显示从预测的3D结构计算出的23条染色体的体积与实验估计值很好地验证了该方法。因为方法是通用的，