|dc.description.abstracteng||Over the last decade, Cryo Electron Microscopy (cryo-EM) has emerged as a powerful and reliable technique to determine the three-dimensional (3D) structure of large macro-molecular assemblies at near atomic resolution. Tens of thousands of two-dimensional (2D) noisy images from copies of a macromolecule are captured at different unknown orientations to characterize the 3D volume of biological complexes. The reconstruction problem is typically formulated as a nonlinear, non-convex optimization task and multi-modality causes many of the conventional reconstruction techniques that work based on local optimization to become trapped in local modes with poor initialization. These difficulties necessitate the development of adequate statistical models to describe the whole reconstruction process. At the core of all algorithms proposed in this thesis, we use grid-free coarse grained representation of molecular densities rather than their atomic models in real space. To this
end, we employ radial basis functions with spherical Gaussian kernels. Each Gaussian kernel is centered on a particle characterized by a 3D position and a non-negative weight. The contribution of this dissertation can be divided into three major parts. The first contribution is introducing a new ab initio reconstruction method for estimating an initial model in cryo-EM based on a mixture of spherical Gaussian model. Here, we consider the full reconstruction problem when structure as well as orientations are unknown. We adopt Markov Chain Monte Carlo algorithms within a Bayesian framework to estimate the parameters of the model. Sampling from the posterior distribution is challenging due to the high-dimensionality and multi-modality. We address these difficulties by using Hamiltonian Monte Carlo and a global rotational sampling approach.
As we mentioned earlier, there are two major challenges in the reconstruction problems, unknown structure and unknown projection directions. If either of two quantities is known, then the problem boils down into two simpler tasks, which we address as two sub-problems in this thesis. The first sub-problem is the tomographic reconstruction problem that occurs when in cryo-EM reconstruction, the projection directions are known. The second contribution of the thesis is the development of three kernel-based algorithms to characterize the 3D volume from 2D tomographic images in real space. In our first method, the particle positions cab be located at a fixed grid, such as a hexagonal grid and only updates their weights by using an iterative non-negative least-squares algorithm. The other two algorithms are mesh-free approaches in which for the first method, we use Expectation Maximization to find the weighted particle positions and for the second method, we develop a fully Bayesian framework by assigning equally weighted particles. The second sub-problem is a rigid registration (pose estimation) problem, and it happens when in cryo-EM reconstruction, the structure is known. The last contribution of the thesis is dedicated to find a unifying framework for assessing the match between biomolecular structures. We propose a kernel-correlation to compute the rigid-transformation between two complexes. We use an upper bound method (sometimes augmented by a deterministic annealing or a global search strategy), to solve the kernel-correlation. Finally, we quantify our proposed approaches on various simulated and real datasets. This thesis is based on the following publications and manuscripts, respectively:
• Vakili N, Habeck M. Bayesian Random Tomography of Particle Systems. Frontiers in molecular biosciences. 2021 May 21;8:399.
• Vakili N, Habeck M. Kernel-based tomographic reconstruction on and off the grid, in preparation.
• Vakili N, Habeck M. Matching biomolecular structures by registration of point clouds, in preparation||de