Computational methods for analysing multiscale 3D genome organization.

Recent progress in whole-genome mapping and imaging technologies has enabled the characterization of the spatial organization and folding of the genome in the nucleus. In parallel, advanced computational methods have been developed to leverage these mapping data to reveal multiscale three-dimensional (3D) genome features and to provide a more complete view of genome structure and its connections to genome functions such as transcription. Here, we discuss how recently developed computational tools, including machine-learning-based methods and integrative structure-modelling frameworks, have led to a systematic, multiscale delineation of the connections among different scales of 3D genome organization, genomic and epigenomic features, functional nuclear components and genome function. However, approaches that more comprehensively integrate a wide variety of genomic and imaging datasets are still needed to uncover the functional role of 3D genome structure in defining cellular phenotypes in health and disease.

Evaluating the role of the nuclear microenvironment in gene function by population-based modeling.

The nuclear folding of chromosomes relative to nuclear bodies is an integral part of gene function. Here, we demonstrate that population-based modeling-from ensemble Hi-C data-provides a detailed description of the nuclear microenvironment of genes and its role in gene function. We define the microenvironment by the subnuclear positions of genomic regions with respect to nuclear bodies, local chromatin compaction, and preferences in chromatin compartmentalization. These structural descriptors are determined in single-cell models, thereby revealing the structural variability between cells. We demonstrate that the microenvironment of a genomic region is linked to its functional potential in gene transcription, replication, and chromatin compartmentalization. Some chromatin regions feature a strong preference for a single microenvironment, due to association with specific nuclear bodies in most cells. Other chromatin shows high structural variability, which is a strong indicator of functional heterogeneity. Moreover, we identify specialized nuclear microenvironments, which distinguish chromatin in different functional states and reveal a key role of nuclear speckles in chromosome organization. We demonstrate that our method produces highly predictive three-dimensional genome structures, which accurately reproduce data from a variety of orthogonal experiments, thus considerably expanding the range of Hi-C data analysis.

Conformational analysis of chromosome structures reveals vital role of chromosome morphology in gene function.

The 3D conformations of chromosomes are highly variant and stochastic between single cells. Recent progress in multiplexed 3D FISH imaging, single cell Hi-C and genome structure modeling allows a closer analysis of the structural variations of chromosomes between cells to infer the functional implications of structural heterogeneity. Here, we introduce a two-step dimensionality reduction method to classify a population of single cell 3D chromosome structures, either from simulation or imaging experiment, into dominant conformational clusters with distinct chromosome morphologies. We found that almost half of all structures for each chromosome can be described by 5-10 dominant chromosome morphologies, which play a fundamental role in establishing conformational variation of chromosomes. These morphologies are conserved in different cell types, but vary in their relative proportion of structures. Chromosome morphologies are distinguished by the presence or absence of characteristic chromosome territory domains, which expose some chromosomal regions to varying nuclear environments in different morphologies, such as nuclear positions and associations to nuclear speckles, lamina, and nucleoli. These observations point to distinct functional variations for the same chromosomal region in different chromosome morphologies. We validated chromosome conformational clusters and their associated subnuclear locations with data from DNA-MERFISH imaging and single cell sci-HiC data. Our method provides an important approach to assess the variation of chromosome structures between cells and link differences in conformational states with distinct gene functions.

Integrative genome modeling platform reveals essentiality of rare contact events in 3D genome organizations.

A multitude of sequencing-based and microscopy technologies provide the means to unravel the relationship between the three-dimensional organization of genomes and key regulatory processes of genome function. Here, we develop a multimodal data integration approach to produce populations of single-cell genome structures that are highly predictive for nuclear locations of genes and nuclear bodies, local chromatin compaction and spatial segregation of functionally related chromatin. We demonstrate that multimodal data integration can compensate for systematic errors in some of the data and can greatly increase accuracy and coverage of genome structure models. We also show that alternative combinations of different orthogonal data sources can converge to models with similar predictive power. Moreover, our study reveals the key contributions of low-frequency ('rare') interchromosomal contacts to accurately predicting the global nuclear architecture, including the positioning of genes and chromosomes. Overall, our results highlight the benefits of multimodal data integration for genome structure analysis, available through the Integrative Genome Modeling software package.

Integrative approaches in genome structure analysis.

New technological advances in integrated imaging, sequencing-based assays, and computational analysis have revolutionized our view of genomes in terms of their structure and dynamics in space and time. These advances promise a deeper understanding of genome functions and mechanistic insights into how the nucleus is spatially organized and functions. These wide arrays of complementary data provide an opportunity to produce quantitative integrative models of nuclear organization. In this article, we highlight recent key developments and discuss the outlook for these fields.

Uncovering the Principles of Genome Folding by 3D Chromatin Modeling.

Our understanding of how genomic DNA is tightly packed inside the nucleus, yet is still accessible for vital cellular processes, has grown dramatically over recent years with advances in microscopy and genomics technologies. Computational methods have played a pivotal role in the structural interpretation of experimental data, which helped unravel some organizational principles of genome folding. Here, we give an overview of current computational efforts in mechanistic and data-driven 3D chromatin structure modeling. We discuss strengths and limitations of different methods and evaluate the added value and benefits of computational approaches to infer the 3D structural and dynamic properties of the genome and its underlying mechanisms at different scales and resolution, ranging from the dynamic formation of chromatin loops and topological associated domains to nuclear compartmentalization of chromatin and nuclear bodies.

Spectral Properties of Effective Dynamics from Conditional Expectations.

The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers-Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.

Coarse-graining molecular systems by spectral matching.

Coarse-graining has become an area of tremendous importance within many different research fields. For molecular simulation, coarse-graining bears the promise of finding simplified models such that long-time simulations of large-scale systems become computationally tractable. While significant progress has been made in tuning thermodynamic properties of reduced models, it remains a key challenge to ensure that relevant kinetic properties are retained by coarse-grained dynamical systems. In this study, we focus on data-driven methods to preserve the rare-event kinetics of the original system and make use of their close connection to the low-lying spectrum of the system's generator. Building on work by Crommelin and Vanden-Eijnden [Multiscale Model. Simul. 9, 1588 (2011)], we present a general framework, called spectral matching, which directly targets the generator's leading eigenvalue equations when learning parameters for coarse-grained models. We discuss different parametric models for effective dynamics and derive the resulting data-based regression problems. We show that spectral matching can be used to learn effective potentials which retain the slow dynamics but also to correct the dynamics induced by existing techniques, such as force matching.

Sparse learning of stochastic dynamical equations.

With the rapid increase of available data for complex systems, there is great interest in the extraction of physically relevant information from massive datasets. Recently, a framework called Sparse Identification of Nonlinear Dynamics (SINDy) has been introduced to identify the governing equations of dynamical systems from simulation data. In this study, we extend SINDy to stochastic dynamical systems which are frequently used to model biophysical processes. We prove the asymptotic correctness of stochastic SINDy in the infinite data limit, both in the original and projected variables. We discuss algorithms to solve the sparse regression problem arising from the practical implementation of SINDy and show that cross validation is an essential tool to determine the right level of sparsity. We demonstrate the proposed methodology on two test systems, namely, the diffusion in a one-dimensional potential and the projected dynamics of a two-dimensional diffusion process.

Rapid Calculation of Molecular Kinetics Using Compressed Sensing.

Recent methods for the analysis of molecular kinetics from massive molecular dynamics (MD) data rely on the solution of very large eigenvalue problems. Here we build upon recent results from the field of compressed sensing and develop the spectral oASIS method, a highly efficient approach to approximate the leading eigenvalues and eigenvectors of large generalized eigenvalue problems without ever having to evaluate the full matrices. The approach is demonstrated to reduce the dimensionality of the problem by 1 or 2 orders of magnitude, directly leading to corresponding savings in the computation and storage of the necessary matrices and a speedup of 2 to 4 orders of magnitude in solving the eigenvalue problem. We demonstrate the method on extensive data sets of protein conformational changes and protein-ligand binding using the variational approach to conformation dynamics (VAC) and time-lagged independent component analysis (TICA). Our approach can also be applied to kernel formulations of VAC, TICA, and extended dynamic mode decomposition (EDMD).

A Data-Driven Perspective on the Hierarchical Assembly of Molecular Structures.

Macromolecular systems are composed of a very large number of atomic degrees of freedom. There is strong evidence suggesting that structural changes occurring in large biomolecular systems at long time scale dynamics may be captured by models coarser than atomistic, although a suitable or optimal coarse-graining is a priori unknown. Here we propose a systematic approach to learning a coarse representation of a macromolecule from microscopic simulation data. In particular, the definition of effective coarse variables is achieved by partitioning the degrees of freedom both in the structural (physical) space and in the conformational space. The identification of groups of microscopic particles forming dynamical coherent states in different metastable states leads to a multiscale description of the system, in space and time. The application of this approach to the folding dynamics of two proteins provides a revised view of the classical idea of prestructured regions (foldons) that combine during a protein-folding process and suggests a hierarchical characterization of the assembly process of folded structures.

Investigating Molecular Kinetics by Variationally Optimized Diffusion Maps.

Identification of the collective coordinates that describe rare events in complex molecular transitions such as protein folding has been a key challenge in the theoretical molecular sciences. In the Diffusion Map approach, one assumes that the molecular configurations sampled have been generated by a diffusion process, and one uses the eigenfunctions of the corresponding diffusion operator as reaction coordinates. While diffusion coordinates (DCs) appear to provide a good approximation to the true dynamical reaction coordinates, they are not parametrized using dynamical information. Thus, their approximation quality could not, as yet, be validated, nor could the diffusion map eigenvalues be used to compute relaxation rate constants of the system. Here we combine the Diffusion Map approach with the recently proposed Variational Approach for Conformation Dynamics (VAC). Diffusion Map coordinates are used as a basis set, and their optimal linear combination is sought using the VAC, which employs time-correlation information on the molecular dynamics (MD) trajectories. We have applied this approach to ultra-long MD simulations of the Fip35 WW domain and found that the first DCs are indeed a good approximation to the true reaction coordinates of the system, but they could be further improved using the VAC. Using the Diffusion Map basis, excellent approximations to the relaxation rates of the system are obtained. Finally, we evaluate the quality of different metric spaces and find that pairwise minimal root-mean-square deviation performs poorly, while operating in the recently introduced kinetic maps based on the time-lagged independent component analysis gives the best performance.