Contact map reconstruction comparison between MBO and CMDS.

<p>A: Original contact map. Blue dot indicates the presence of a contact in the single-cell Hi-C data set for chromosome X (cell 1). B: Contact map obtained after 3D reconstruction using MBO, based on the contact map (in A) and then re-calculating the contacts. C: Reconstructed contact map, as in B, but using CMDS. D: Reconstructed 3D structure using MBO, corresponding to the contact map in B. E: Reconstructed 3D structure using CMDS, corresponding to the contact map in C. Each bead in D and E has a diameter of 150 nm. Lines represent connected beads with missing bead position information.</p

Reconstruction of chromosome X at different levels of observed information.

<p>A: Original chromosome X structure from [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004396#pcbi.1004396.ref018" target="_blank">18</a>], resampled at 600 kbp. B-F: Reconstructed 3D structures of chromosome X, with different ratios of observed distance information (20%, 10%, 5%, 2% and 1%, respectively). Information about the RMSD (in nm) and 1 − <i>ρ</i>, compared to the original structure (A) is given below each of the structures in A-F. G: Ratio of observed values as a function of the number of bins <i>n</i>, i.e. the size of the structure being reconstructed. The structures in B-F are highlighted (orange dots), and compared to an estimated curve showing the minimum ratio of observed values for complete reconstruction ([1-<i>ρ</i>]<1e-10; blue curve) or partial reconstruction ([1-<i>ρ</i>]<0.1; black curve). All data from [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004396#pcbi.1004396.ref018" target="_blank">18</a>] are shown as gray circles, and the X chromosome data sets from cell 1 and cell 2 are highlighted in green.</p

Example of generation of distance and weight matrices for the optimization procedure.

<p>A: Original chromosomal contact map (<i>C</i>_<i>ij</i>) based on chromosome X from cell 1. A blue dot indicates the presence of an observed interaction in the single-cell Hi-C data set. B: Distance matrix (<i>D</i>_<i>ij</i>) consisting of Euclidean distances (in <i>μ</i>m) corresponding to the contact map to the left after running the shortest-path algorithm. C: Corresponding weight matrix (<i>H</i>_<i>ij</i>), containing numbers between 0 and 1 giving the weight for each of the distances in the Euclidean distance matrix to the left. See the <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004396#sec008" target="_blank">Methods</a> section for details.</p

Computational time evaluations for the different algorithms.

<p>Computational time (in seconds) for reconstructing a single chromosome structure using three different algorithms CMDS (dark blue), ChromSDE (green), and the MBO algorithm (red) presented here. For comparison, the shortest path algorithm (light blue) is also shown. The computational time is shown as a function of structure size <i>n</i>, i.e. the number of bins in the structure.</p

Consistency comparison of reconstructed 3D genome models based on MBO and CMDS.

<p>Consistency of the structures obtained from reconstructing all chromosomes for cell 1 (left) and 2 (right) using MBO (blue) and CMDS (red). A: Reconstruction accuracy, given as the percent correct contacts when comparing original and reconstructed contacts maps for different chromosomes. B: Distance violation, given as the occurrence (in percent) of regions in the structures that are below the minimum distance (at 30 nm). C: Connectivity violation, given as the occurrence (in percent) of consecutive regions in the structures that are further away than the maximum distance (200 nm). Blue bars indicate the performance of MBO, while red bars indicate the performance of CMDS.</p

A PMF based on the radius of gyration.

<p>The goal is to adapt a distribution – which allows sampling of local structures – such that a given target distribution is obtained. For , we used the amino acid sequence of ubiquitin. Sampling from alone results in a distribution with an average of about 27 (triangles). Sampling using the correct expression (open circles), given by Eq. 8, results in a distribution that coincides with the target distribution (solid line). Not taking the reference state into account results in a significant shift towards higher (black circles).</p

Iterative estimation of a PMF.

<p>For each of the eight hydrogen bond categories (see text), the black bar to the right denotes the fraction of occurrence in the native structure of protein G. The gray bars denote the fractions of the eight categories in samples from each iteration; the first iteration is shown to the left in light gray. In the last iteration (iteration 6; dark gray bars, right) the values are very close to the native values for all eight categories. Note that hydrogen bonds between -strands are nearly absent in the first iteration (category ).</p

General statistical justification of PMFs.

<p>The goal is to combine a distribution over a fine grained variable (top right), with a probability distribution over a coarse grained variable (top left). could be, for example, embodied in a fragment library (), a probabilistic model of local structure () or an energy function (); could be, for example, the radius of gyration, the hydrogen bond network, or the set of pairwise distances. usually reflects the distribution of in known protein structures (PDB), but could also stem from experimental data (). Sampling from results in a distribution that differs from . Multiplying and does not result in the desired distribution for either (red box); the correct result requires dividing out the signal with respect to due to (green box). The <i>reference</i> distribution in the denominator corresponds to the contribution of the reference state in a PMF. If is only approximately known, the method can be applied iteratively (dashed arrow). In that case, one attempts to iteratively sculpt an energy funnel. The procedure is statistically rigorous provided and are proper probability distributions; this is usually not the case for conventional pairwise distance PMFs.</p

Illustration of the central idea presented in this article.

<p>In this example, the goal is to sample conformations with a given distribution for the radius of gyration , and a plausible local structure. could, for example, be derived from known structures in the Protein Data Bank (PDB, left box). is a probability distribution over local structure , typically embodied in fragment library (right box). In order to combine and in a meaningful way (see text), the two distributions are multiplied and divided by (formula at the bottom); is the probability distribution over the radius of gyration for conformations sampled solely from the fragment library (that is, ). The probability distribution will generate conformations with plausible local structures (due to ), while their radii of gyration will be distributed according to , as desired. This simple idea lies at the theoretical heart of the PMF expressions used in protein structure prediction.</p

Highest probability structures for each iteration.

<p>The structures with highest probability out of 50,000 samples for all six iterations (indicated by a number) are shown as cartoon representations. The N-terminus is shown in blue. The figure was made using PyMOL <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0013714#pone.0013714-Delano1" target="_blank">[64]</a>.</p