Yili Zheng

Dynamics in cryo EM reconstructions visualized with maximum-likelihood derived variance maps.

CryoEM data capture the dynamic character associated with biological macromolecular assemblies by preserving the various conformations of the individual specimens at the moment of flash freezing. Regions of high variation in the data set are apparent in the image reconstruction due to the poor density that results from the lack of superposition of these regions. These observations are qualitative and, to date, only preliminary efforts have been made to quantitate the heterogeneity in the ensemble of particles that are individually imaged. We developed and tested a quantitative method for simultaneously computing a reconstruction of the particle and a map of the space-varying heterogeneity of the particle based on an entire data set. The method uses a maximum likelihood algorithm that explicitly takes into account the continuous variability from one instance to another instance of the particle. The result describes the heterogeneity of the particle as a variance to be plotted at every voxel of the reconstructed density. The test, employing time resolved data sets of virus maturation, not only recapitulated local variations obtained with difference map analysis, but revealed a remarkable time dependent reduction in the overall particle dynamics that was unobservable with classical methods of analysis.

Explicit orthonormal fixed bases for spaces of functions that are totally symmetric under the rotational symmetries of a platonic solid

Explicit complete orthonormal fixed bases are computed for subspaces of the space of square-integrable functions on the sphere where the subspaces contain functions that are totally symmetric under the rotational symmetries of a Platonic solid. Each function in the fixed basis is a linear combination of spherical harmonics of fixed l. For each symmetry (icosahedra/dodecahedral, octahedral/cubic, tetrahedral), the calculation has three steps : First, a bilinear equation is derived for the coefficients in the linear combination by equating the expansion of a symmetrized δ function in both spherical harmonics and the fixed basis functions for the appropriate subspace. The equation is parameterized by the location (θ 0 , φ 0 ) of the δ function and must be satisfied for all locations. Second, the dependence on the δ-function location is expressed in a Fourier (φ 0 ) and a Taylor (θ 0 ) series and thereby a new system of bilinear equations is derived by equating selected coefficients. Third, a recursive solution of the new system is derived and the recursion is solved explicitly with the aid of symbolic computation. The results for the icosahedral case are important for structural studies of small spherical viruses.

Three-dimensional reconstruction of the statistics of heterogeneous objects from a collection of one projection image of each object.

An estimation problem for statistical reconstruction of heterogeneous three-dimensional objects from two-dimensional tomographic data (single-particle cryoelectron microscope images) is posed as the problem of estimating class probabilities, means, and covariances for a Gaussian mixture where both the mean and covariance are stochastically structured. Both discrete (i.e., classes) and continuous heterogeneity is included. A maximum likelihood solution computed by a generalized expectation-maximization algorithm is presented and demonstrated on experimental images of Flock House Virus.

A parallel software toolkit for statistical 3-D virus reconstructions from cryo electron microscopy images using computer clusters with multi-core shared-memory nodes

A statistical approach for computing 3-D reconstructions of virus particles from cryo electron microscope images and minimal prior information has been developed which can solve a range of specific problems. The statistical approach causes high computation and storage complexity in the software implementation. A parallel software toolkit is described which allows the construction of software targeted at commodity PC clusters which is modular, reusable, and user-transparent while at the same time delivering nearly linear speedup on practical problems.

Stochastic simulation of a T-cell signaling pathway

Understanding, quantifying, and controlling the networked T-cell signaling pathways will impact the understanding and potential treatment design for several immunologically-related and malignant diseases. Currently most T-cell signaling pathways have been mapped out in terms of their demonstrated interactions and participating elements; however, the quantification of these pathways and the identification of rate-limiting interactions, and controlling feedback mechanisms remain to be delineated. A mathematical model of the T-cell MAPK signaling pathway has been constructed to assist in this quantification and delineation. This model is a hybrid model comprising a stochastic module capturing T-cell receptor stimulation, and a deterministic module composed of nonlinear differential equations to simulate the subsequently induced MAPK activation. Upon T-cell receptor (TCR) engagement by a MHC-peptide complex, the affinity of the peptide-TCR bond and the surface density of engaged receptors modulates the phosphorylation state of the CD3 /spl zeta/-chain which in turn controls the induction of signaling pathways (including the MAPK). Our stochastic module incorporates these probabilistic events and integrates them with the deterministic module capturing the downstream MAPK signaling pathway through an initiation event. Simulations and analysis have helped delineate the signaling pathway.

3D signal reconstruction from noisy projection data for stochastic objects as a generalization of Gaussian mixture parameter estimation

A statistical estimation problem for determining 3-D reconstructions from a single 2-D projection image of each of multiple objects when the objects are heterogeneous is described. The method is based on a Gaussian mixture description of the heterogeneity and is motivated by cryo electron microscopy of biological objects.

Image reconstruction in electron microscopy

Maximum likelihood statistical algorithms are described for estimating the 3-D variation of the electron scattering intensity of biological objects from cryo electron microscopy images of multiple instances of the object. Three virus objects, two spherical and one helical, are considered. Solution of the maximum likelihood problem by expectation maximization algorithms or by direct maximization of the log likelihood requires large scale computing and end-to-end codesign of biological problem formulation, statistical models, algorithms, and software design and implementation have contributed to achieving practical results.

A fast algorithm for 3D reconstruction from unoriented projections and cryo electron microscopy of viruses

In a cryo electron microscopy experiment, the data is noisy 2-D projection images of the 3-D electron scattering intensity where the orientation of the projections is not known. In previous work we have developed a solution for this problem based on a maximum likelihood estimator that is computed by an expectation maximization algorithm. In the expectation maximization algorithm the expensive step is the expectation which requires numerical evaluation of 3- or 5-dimensional integrations of a square matrix of dimension equal to the number of Fourier series coeffcients used to describe the 3-D reconstruction. By taking advantage of the rotational properties of spherical harmonics, we can reduce the integrations of a matrix to integrations of a scalar. The key properties is that a rotated spherical harmonic can be expressed as a linear combination of the other harmonics of the same order and that the weights in the linear combination factor so that each of the three factors is a function of only one of the Euler angles describing the orientation of the projection.

Maximum likelihood 3D reconstruction of multiple viruses from mixtures of cryo electron microscope images

3-D reconstruction of the electron scattering intensity of a virus from cryo electron microscopy is essentially a 3-D tomography problem in which the orientation of the 2-D projections is unknown. Many biological problems concern mixtures of different types of virus particles or mixtures of different maturation states of the same type of virus particle. For a variety of reasons, especially low SNR, it can be very challenging to label the type or state shown in an individual image. Algorithms capable of computing multiple reconstructions, one for each type or state, based on images which are not labeled according to type or state, are described and demonstrated on experimental images.

3-D maximum likelihood reconstructions of viruses from cryo electron microscope images and parallel computation

A statistical method for computing 3-D reconstructions of virus particles from cryo electron microscope images and minimal prior information - particle symmetry and radii - is described. Two different approaches for parallel implementation of the algorithm are described based on parallel processing of different images and based on parallel numerical integration and numerical results showing nearly linear speedup are presented.