I co-organize the One World Cryo-EM online seminar series about mathematics and algorithms in cryo-electron microscopy. Please contact me if you are interested in speaking or getting involved in our activities around the seminar.
I am looking for extraordinary postdocs and extraordinary graduate students (at Yale, any graduate program). More information about postdoc positions is available here.
Office hours available here and on Canvas (internal Yale system).
Structural biology and cryo-EM: inverse problems and unsupervised learning, applications of representation theory, numerical analysis, and data organization to imaging of molecules
Computational biology: fast search algorithms, statistics of DNA, sequencing, organization of biological data
CRYO-EM (a little out of date – more updates coming soon)
Cryo-electron microscopy (cryo-EM) is a method for imaging molecules without crystallization. The Nobel Prize in Chemistry 2017 was awarded to Jacques Dubochet, Joachim Frank and Richard Henderson “for the development of cryo-electron microscopy, which both simplifies and improves the imaging of biomolecules.” I work on various problems of alignment, classification and signal processing that are motivated by application in cryo-EM with many other applications. I am particularly interested in heterogeneity, i.e. imaging of mixtures of different types of molecules.
I work on “hyper-molecules” which represent heterogeneous molecules as higher-dimension objects. The movie below is an example of a reconstruction of a continuously heterogeneous object, using the approach described in this paper and this paper.
This is one of several approaches that I am developing for the heterogeneity problem in cryo-EM, and for other aspects of cryo-EM. For more information on my work in cryo-EM, see project page.
The Truncated Fourier Transform and its eigenfunctions, Prolate Spheroidal Wave Functions (PSWF) and Generalized Prolate Spheroidal Functions (GPSF) (also known as Slepian Functions) are frequently encountered in mathematics, physics, signal processing, optics and other areas. Surprisingly, very few resources and code for the numerical computation of GPSFs and their eigenvalues are publicly available. Our sample implementation and associated paper are available at http://github.com/lederman/prol. The code also contains an experimental “open-source proof,” which is code for analytical proofs of some of the results that appear in this paper.
The Laplace Transform and Grunbaum Functions
The Laplace transform is frequently encountered in mathematics, physics, engineering and other areas. However, the spectral properties of the Laplace transform tend to complicate its numerical treatment; therefore, the closely related “Truncated” Laplace Transforms are often used in applications.
Lower bounds on the truncated Fourier transform and truncated Laplace transform: see paper.
GEOMETRY OF DATA
Alternating Diffusion, a method for recovering the common variable in multi-sensor experiments, is discussed in this paper, this technical report and this project webpage. A different approach to the common variable recovery problem, which also constructs representations that are invariable to unknown transformations, is discussed in this paper.
Additional Application: Assembly. The algorithm is also used to construct approximate overlap graphs. These graph are used for fast assembly. Unlike other algorithms, this algorithm allows errors in the reads, so no error-correction is necessary prior to the construction of the graph. See: paper.
@article{herreros_approximating_2021,
title = {Approximating deformation fields for the analysis of continuous heterogeneity of biological macromolecules by 3D Zernike polynomials},
author = {D Herreros and Roy R Lederman and J Krieger and A Jiménez-Moreno and M Martínez and D Myška and D Strelak and J Filipovic and I Bahar and J M Carazo and C O S Sanchez},
url = {https://journals.iucr.org/m/issues/2021/06/00/eh5012/},
doi = {10.1107/S2052252521008903},
issn = {2052-2525},
year = {2021},
date = {2021-11-01},
urldate = {2021-10-26},
journal = {IUCrJ},
volume = {8},
number = {6},
abstract = {A new tool based on 3D Zernike polynomials is presented that allows the study of the continuous heterogeneity of biological macromolecules, revealing the structural relationships present among different states by the approximation of deformation fields.},
note = {Number: 6
Publisher: International Union of Crystallography},
keywords = {cryo-EM, heterogeneity, Zernike},
pubstate = {published},
tppubtype = {article}
}
A new tool based on 3D Zernike polynomials is presented that allows the study of the continuous heterogeneity of biological macromolecules, revealing the structural relationships present among different states by the approximation of deformation fields.
Calero, David Herreros; Lederman, Roy R; Krieger, James; Myška, David; Strelak, David; Filipovic, Jiri; Bahar, Ivet; Carazo, Jose Maria; Sorzano, Carlos Oscar
@article{calero_continuous_2021,
title = {Continuous heterogeneity analysis of CryoEM images through Zernike polynomials and spherical harmonics},
author = {David Herreros Calero and Roy R Lederman and James Krieger and David Myška and David Strelak and Jiri Filipovic and Ivet Bahar and Jose Maria Carazo and Carlos Oscar Sorzano},
url = {https://www.cambridge.org/core/journals/microscopy-and-microanalysis/article/continuous-heterogeneity-analysis-of-cryoem-images-through-zernike-polynomials-and-spherical-harmonics/2A8C58651F413C8A0D66071CB4BC9AAD},
doi = {10.1017/S1431927621006176},
issn = {1431-9276, 1435-8115},
year = {2021},
date = {2021-08-01},
urldate = {2021-08-03},
journal = {Microscopy and Microanalysis},
volume = {27},
number = {S1},
pages = {1680--1682},
abstract = {//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS1431927621006176/resource/name/firstPage-S1431927621006176a.jpg},
note = {Publisher: Cambridge University Press},
keywords = {cryo-EM, heterogeneity, Zernike},
pubstate = {published},
tppubtype = {article}
}
@article{lederman_hyper-molecules_2020,
title = {Hyper-molecules: on the representation and recovery of dynamical structures for applications in flexible macro-molecules in cryo-EM},
author = {Roy R Lederman and Joakim Andén and Amit Singer},
url = {https://iopscience.iop.org/article/10.1088/1361-6420/ab5ede},
doi = {10.1088/1361-6420/ab5ede},
issn = {0266-5611, 1361-6420},
year = {2020},
date = {2020-04-01},
urldate = {2020-08-13},
journal = {Inverse Problems},
volume = {36},
number = {4},
pages = {044005},
keywords = {cryo-EM, heterogeneity, HyperMolecules, MCMC, Variational inference},
pubstate = {published},
tppubtype = {article}
}
@article{bandeira_non-unique_2020,
title = {Non-unique games over compact groups and orientation estimation in cryo-EM},
author = {Afonso S Bandeira and Yutong Chen and Roy R Lederman and Amit Singer},
url = {https://iopscience.iop.org/article/10.1088/1361-6420/ab7d2c},
doi = {10.1088/1361-6420/ab7d2c},
issn = {0266-5611, 1361-6420},
year = {2020},
date = {2020-01-01},
urldate = {2020-08-13},
journal = {Inverse Problems},
volume = {36},
number = {6},
pages = {064002},
keywords = {Algorithms, cryo-EM, Non-unique games, Representation Theory},
pubstate = {published},
tppubtype = {article}
}
@article{lederman_representation_2020,
title = {A representation theory perspective on simultaneous alignment and classification},
author = {Roy R Lederman and Amit Singer},
url = {http://www.sciencedirect.com/science/article/pii/S1063520319301034},
doi = {10.1016/j.acha.2019.05.005},
issn = {1063-5203},
year = {2020},
date = {2020-01-01},
urldate = {2021-01-22},
journal = {Applied and Computational Harmonic Analysis},
volume = {49},
number = {3},
pages = {1001--1024},
abstract = {Single particle cryo-electron microscopy (EM) is a method for determining the 3-D structure of macromolecules from many noisy 2-D projection images of individual macromolecules whose orientations and positions are random and unknown. The problem of orientation assignment for the images motivated work on multireference alignment. The recent non-unique games framework provides a representation theoretic approach to alignment over compact groups, and offers a convex relaxation with certificates of global optimality in some cases. One of the great opportunities in cryo-EM is studying heterogeneous samples, containing two or more distinct conformations of molecules. Taking advantage of this opportunity presents an algorithmic challenge: determining both the class and orientation of each particle. We generalize multireference alignment to a problem of alignment and classification, and propose to extend non-unique games to the problem of simultaneous alignment and classification with the goal of simultaneously classifying cryo-EM images and aligning them within their classes.},
keywords = {Algorithms, Alignment, Classification, cryo-EM, Graph-cut, heterogeneity, Heterogeneous multireference alignment, Representation Theory, Rotation group, SDP, Synchronization},
pubstate = {published},
tppubtype = {article}
}
Single particle cryo-electron microscopy (EM) is a method for determining the 3-D structure of macromolecules from many noisy 2-D projection images of individual macromolecules whose orientations and positions are random and unknown. The problem of orientation assignment for the images motivated work on multireference alignment. The recent non-unique games framework provides a representation theoretic approach to alignment over compact groups, and offers a convex relaxation with certificates of global optimality in some cases. One of the great opportunities in cryo-EM is studying heterogeneous samples, containing two or more distinct conformations of molecules. Taking advantage of this opportunity presents an algorithmic challenge: determining both the class and orientation of each particle. We generalize multireference alignment to a problem of alignment and classification, and propose to extend non-unique games to the problem of simultaneous alignment and classification with the goal of simultaneously classifying cryo-EM images and aligning them within their classes.
@inproceedings{boumal_heterogeneous_2018,
title = {Heterogeneous multireference alignment: A single pass approach},
author = {N Boumal and T Bendory and Roy R Lederman and A Singer},
doi = {10.1109/CISS.2018.8362313},
year = {2018},
date = {2018-01-01},
booktitle = {2018 52nd Annual Conference on Information Sciences and Systems (CISS)},
pages = {1--6},
abstract = {Multireference alignment (MRA) is the problem of estimating a signal from many noisy and cyclically shifted copies of itself. In this paper, we consider an extension called heterogeneous MRA, where K signals must be estimated, and each observation comes from one of those signals, unknown to us. This is a simplified model for the heterogeneity problem notably arising in cryo-electron microscopy. We propose an algorithm which estimates the K signals without estimating either the shifts or the classes of the observations. It requires only one pass over the data and is based on low-order moments that are invariant under cyclic shifts. Given sufficiently many measurements, one can estimate these invariant features averaged over the K signals. We then design a smooth, non-convex optimization problem to compute a set of signals which are consistent with the estimated averaged features. We find that, in many cases, the proposed approach estimates the set of signals accurately despite non-convexity, and conjecture the number of signals K that can be resolved as a function of the signal length L is on the order of √L.},
keywords = {bispectrum, concave programming, cryo-EM, cyclic shifts, Discrete Fourier transforms, estimation theory, expectation-maximization, Gaussian mixture models, heterogeneity, heterogeneous MRA, Heterogeneous multireference alignment, Multireference alignment, Noise measurement, non-convex optimization, nonconvex optimization problem, Optimization, Reliability, signal estimation, signal processing, Signal resolution, Signal to noise ratio, single pass approach, Standards},
pubstate = {published},
tppubtype = {inproceedings}
}
Multireference alignment (MRA) is the problem of estimating a signal from many noisy and cyclically shifted copies of itself. In this paper, we consider an extension called heterogeneous MRA, where K signals must be estimated, and each observation comes from one of those signals, unknown to us. This is a simplified model for the heterogeneity problem notably arising in cryo-electron microscopy. We propose an algorithm which estimates the K signals without estimating either the shifts or the classes of the observations. It requires only one pass over the data and is based on low-order moments that are invariant under cyclic shifts. Given sufficiently many measurements, one can estimate these invariant features averaged over the K signals. We then design a smooth, non-convex optimization problem to compute a set of signals which are consistent with the estimated averaged features. We find that, in many cases, the proposed approach estimates the set of signals accurately despite non-convexity, and conjecture the number of signals K that can be resolved as a function of the signal length L is on the order of √L.
@techreport{lederman_numerical_2017,
title = {Numerical Algorithms for the Computation of Generalized Prolate Spheroidal Functions},
author = {Roy R Lederman},
url = {https://arxiv.org/abs/1710.02874v1},
year = {2017},
date = {2017-10-01},
urldate = {2020-08-13},
abstract = {Generalized Prolate Spheroidal Functions (GPSF) are the eigenfunctions of the
truncated Fourier transform, restricted to D-dimensional balls in the spatial
domain and frequency domain. Despite their useful properties in many
applications, GPSFs are often replaced by crude approximations. The purpose of
this paper is to review the elements of computing GPSFs and associated
eigenvalues. This paper is accompanied by open-source code.},
keywords = {Algorithms, cryo-EM, Fourier Transform, Numerical Analysis, Prolate, Slepian, Software},
pubstate = {published},
tppubtype = {techreport}
}
Generalized Prolate Spheroidal Functions (GPSF) are the eigenfunctions of the
truncated Fourier transform, restricted to D-dimensional balls in the spatial
domain and frequency domain. Despite their useful properties in many
applications, GPSFs are often replaced by crude approximations. The purpose of
this paper is to review the elements of computing GPSFs and associated
eigenvalues. This paper is accompanied by open-source code.
@techreport{lederman_continuously_2017,
title = {Continuously heterogeneous hyper-objects in cryo-EM and 3-Đ movies of many temporal dimensions},
author = {Roy R Lederman and Amit Singer},
url = {http://arxiv.org/abs/1704.02899},
year = {2017},
date = {2017-04-01},
urldate = {2020-08-13},
number = {arXiv:1704.02899 [cs]},
abstract = {Single particle cryo-electron microscopy (EM) is an increasingly popular method for determining the 3-D structure of macromolecules from noisy 2-D images of single macromolecules whose orientations and positions are random and unknown. One of the great opportunities in cryo-EM is to recover the structure of macromolecules in heterogeneous samples, where multiple types or multiple conformations are mixed together. Indeed, in recent years, many tools have been introduced for the analysis of multiple discrete classes of molecules mixed together in a cryo-EM experiment. However, many interesting structures have a continuum of conformations which do not fit discrete models nicely; the analysis of such continuously heterogeneous models has remained a more elusive goal. In this manuscript, we propose to represent heterogeneous molecules and similar structures as higher dimensional objects. We generalize the basic operations used in many existing reconstruction algorithms, making our approach generic in the sense that, in principle, existing algorithms can be adapted to reconstruct those higher dimensional objects. As proof of concept, we present a prototype of a new algorithm which we use to solve simulated reconstruction problems.},
note = {arXiv: 1704.02899},
keywords = {Computer Science - Computer Vision and Pattern Recognition, cryo-EM, heterogeneity, HyperMolecules},
pubstate = {published},
tppubtype = {techreport}
}
Single particle cryo-electron microscopy (EM) is an increasingly popular method for determining the 3-D structure of macromolecules from noisy 2-D images of single macromolecules whose orientations and positions are random and unknown. One of the great opportunities in cryo-EM is to recover the structure of macromolecules in heterogeneous samples, where multiple types or multiple conformations are mixed together. Indeed, in recent years, many tools have been introduced for the analysis of multiple discrete classes of molecules mixed together in a cryo-EM experiment. However, many interesting structures have a continuum of conformations which do not fit discrete models nicely; the analysis of such continuously heterogeneous models has remained a more elusive goal. In this manuscript, we propose to represent heterogeneous molecules and similar structures as higher dimensional objects. We generalize the basic operations used in many existing reconstruction algorithms, making our approach generic in the sense that, in principle, existing algorithms can be adapted to reconstruct those higher dimensional objects. As proof of concept, we present a prototype of a new algorithm which we use to solve simulated reconstruction problems.
@techreport{lederman_representation_2016,
title = {A Representation Theory Perspective on Simultaneous Alignment and Classification},
author = {Roy R Lederman and Amit Singer},
url = {http://arxiv.org/abs/1607.03464},
year = {2016},
date = {2016-07-01},
urldate = {2021-01-22},
number = {arXiv:1607.03464 [cs, math]},
abstract = {One of the difficulties in 3D reconstruction of molecules from images in single particle Cryo-Electron Microscopy (Cryo-EM), in addition to high levels of noise and unknown image orientations, is heterogeneity in samples: in many cases, the samples contain a mixture of molecules, or multiple conformations of one molecule. Many algorithms for the reconstruction of molecules from images in heterogeneous Cryo-EM experiments are based on iterative approximations of the molecules in a non-convex optimization that is prone to reaching suboptimal local minima. Other algorithms require an alignment in order to perform classification, or vice versa. The recently introduced Non-Unique Games framework provides a representation theoretic approach to studying problems of alignment over compact groups, and offers convex relaxations for alignment problems which are formulated as semidefinite programs (SDPs) with certificates of global optimality under certain circumstances. In this manuscript, we propose to extend Non-Unique Games to the problem of simultaneous alignment and classification with the goal of simultaneously classifying Cryo-EM images and aligning them within their respective classes. Our proposed approach can also be extended to the case of continuous heterogeneity.},
note = {arXiv: 1607.03464},
keywords = {Algorithms, Computer Science - Computer Vision and Pattern Recognition, cryo-EM, Mathematics - Optimization and Control, Representation Theory},
pubstate = {published},
tppubtype = {techreport}
}
One of the difficulties in 3D reconstruction of molecules from images in single particle Cryo-Electron Microscopy (Cryo-EM), in addition to high levels of noise and unknown image orientations, is heterogeneity in samples: in many cases, the samples contain a mixture of molecules, or multiple conformations of one molecule. Many algorithms for the reconstruction of molecules from images in heterogeneous Cryo-EM experiments are based on iterative approximations of the molecules in a non-convex optimization that is prone to reaching suboptimal local minima. Other algorithms require an alignment in order to perform classification, or vice versa. The recently introduced Non-Unique Games framework provides a representation theoretic approach to studying problems of alignment over compact groups, and offers convex relaxations for alignment problems which are formulated as semidefinite programs (SDPs) with certificates of global optimality under certain circumstances. In this manuscript, we propose to extend Non-Unique Games to the problem of simultaneous alignment and classification with the goal of simultaneously classifying Cryo-EM images and aligning them within their respective classes. Our proposed approach can also be extended to the case of continuous heterogeneity.