Sparse approximations of protein structure from noisy random projections

Abstract

Single-particle electron microscopy is a modern technique that biophysicists employ to learn the structure of proteins. It yields data that consist of noisy random projections of the protein structure in random directions, with the added complication that the projection angles cannot be observed. In order to reconstruct a three-dimensional model, the projection directions need to be estimated by use of an ad-hoc starting estimate of the unknown particle. In this paper we propose a methodology that does not rely on knowledge of the projection angles, to construct an objective data-dependent low-resolution approximation of the unknown structure that can serve as such a starting estimate. The approach assumes that the protein admits a suitable sparse representation, and employs discrete $L^1$-regularization (LASSO) as well as notions from shape theory to tackle the peculiar challenges involved in the associated inverse problem. We illustrate the approach by application to the reconstruction of an E. coli protein component called the Klenow fragment.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS479 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fake uniformity in a shape inversion formula

“…As shown in (Panaretos, V. andKonis, K., 2011, Lemma 5.1, p. 2588), three orthogonal views suffice to reconstruct the Gram matrix. While, as noted in (Panaretos, V. and Konis, K., 2011, pp. 2588-2589, it is impossible to ensure that the views selected from the available imagery are indeed orthogonal, the procedure they developed fared well enough in their practical example.…”

“…In the context of devising statistical procedures from Proposition 1 in conjunction with (suitable versions of) the law of large numbers and central limit theorem, two issues stand out, as stated in (Panaretos, V. and Konis, K., 2011, Sec.s 5.1-5.2, pp. 2586-2589.…”

“…This should be carefully distinguished from the notion of orientation defined around (4.1). The notion of 'view' adopted in Panaretos, V. and Konis, K. (2011), as well as here, is relevant only to two-dimensional images. By definition, the viewing direction is given by the vector w n ∈ S 2 , which is unique up to sign, in the representation HU n = I − w n w T n , where H(x 1 , x 2 , x 3 ) = (x 1 , x 2 , 0).…”

“…The 'raw view' of the particle ρ (or its landmarks) is the image of ρ (or its landmarks) under the map HU n . We may choose to omit the vanishing third (x 3 ) coordinate, leaving us with what is called a 'profile' in Panaretos, V. (2009); Panaretos, V. and Konis, K. (2011). The view of the projected particle (or its landmarks) is obtained when the 'raw view' (in R 3 ) is identified with (i) the image of any rotation that leaves the projection plane of H invariant; (ii) the image obtained from reversal of the viewing direction along the x 3 axis.…”

Fake uniformity in a shape inversion formula

“…In the context of devising statistical procedures from Proposition 1 in conjunction with (suitable versions of) the law of large numbers and central limit theorem, two issues stand out, as stated in Panaretos and Konis (2011), Sections 5.1-5.2, pages 2586-2589. We give them in reverse order.…”