Deblurring and Recovering Conformational States in 3D Single Particle Electron

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14335

Résumé

In this paper we study two forms of blurring effects that may appear in the reconstruction of 3D Electron Microscopy (EM), specifically in single particle reconstruction from random orientations of large multi-unit biomolecular complexes. We model the blurring effects as being due to independent contributions from: (1) variations in the conformation of the biomolecular complex; and (2) errors accumulated in the reconstruction process. Under the assumption that these effects can be separated and treated independently, we show that the overall blurring effect can be expressed as a special form of a convolution operation of the 3D density with a kernel defined on SE(3), the Lie group of rigid body motions in 3D. We call this form of convolution mixed spatial-motional convolution.We discuss the ill-conditioned nature of the deconvolution needed to deblur the reconstructed 3D density in terms of parameters associated with the unknown probability in SE(3). We provide an algorithm for recovering the conformational information of large multi-unit biomolecular complexes (essentially deblurring) under certain biologically plausible prior structural knowledge about the subunits of the complex in the case the blurring kernel has a special form.

Deblurring and Recovering Conformational States in 3D Single Particle Electron

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In this paper we study two forms of blurring effects that may appear in the reconstruction of 3D Electron Microscopy (EM), specifically in single particle reconstruction from random orientations of large multi-unit biomolecular complexes. We model the blurring effects as being due to independent contributions from: (1) variations in the conformation of the biomolecular complex; and (2) errors accumulated in the reconstruction process. Under the assumption that these effects can be separated and treated independently, we show that the overall blurring effect can be expressed as a special form of a convolution operation of the 3D density with a kernel defined on SE(3), the Lie group of rigid body motions in 3D. We call this form of convolution mixed spatial-motional convolution.We discuss the ill-conditioned nature of the deconvolution needed to deblur the reconstructed 3D density in terms of parameters associated with the unknown probability in SE(3). We provide an algorithm for recovering the conformational information of large multi-unit biomolecular complexes (essentially deblurring) under certain biologically plausible prior structural knowledge about the subunits of the complex in the case the blurring kernel has a special form.

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A Methodology for Deblurring and Recovering Conformational States of Biomolecular Complexes from Single Particle Electron Microscopy Bijan Afsari and Gregory S. Chirikjian The Johns Hopkins University bijan@cis.jhu.edu October 30, 2015 B. Afsari and G. Chirikjian (JHU) Deblurring in Cryo-EM October 30, 2015 1 / 12 Introduction and Motivation Cryo-Electron Microscopy (EM) is becoming very popular in structural biology as a favorite imaging modlity to glean information about large biomolecular complexes in their native states. An example of 3-dimensional EM. Its resolution is lower than crystallography but alleviates the need for crystalization which is a lentghty and (in some cases) undesirable process. Challenges: Extremely high noise, 3D volume reconstruction is very complicated and complexes can be in different states within the sample. We introduce a methodology to (possibly) deal with these difficulties in certain applications. Remark on the terminology: “single particle” means “ not crystallography.” Cryo-EM is a prominent example of single particle EM. B. Afsari and G. Chirikjian (JHU) Deblurring in Cryo-EM October 30, 2015 2 / 12 Basics of Cryo-EM D density map is built from projections of the complex at different n SAXS the average of scattering intensities of various copies of the mple lead to a 1D profile. rom biosaxs.com) (b) Cryo-EM (from stat.uchicago.edu) HU) Cross-Validation of SAXS and Cryo-EM November 9, 2015 2 / 13 Copies of a complex lie at random orientation and positions in a thin frozen sample. Noise is extremely high (e.g., SNR = 1/100 is common!) Thousands of images (projections) are classified (clustered) to similar classes all coming from similar views. Images within a class are aligned and averaged to reduce noise. As opposed to standard tomography, projection angles are unknown. Using the “method of common lines” the unknown projection angles can be found and standard “back-projection method” is used to reconstruct the 3D density map. B. Afsari and G. Chirikjian (JHU) Deblurring in Cryo-EM October 30, 2015 3 / 12 Blurring Effects The final 3D structure can be blurred due to high noise and copies of the complex being at different conformational states (conformational blurring). Both lead to errors in classification, etc. In general, these effects are very complicated to model. B. Afsari and G. Chirikjian (JHU) Deblurring in Cryo-EM October 30, 2015 4 / 12 Blurring Effects (cont’ed) In general, these effects are very complicated to model. We use (unknown) SE(3)-kernels to model such blurring effects. SE(3): special Linear/Euclidean group, Lie group of rigid body motions in the space. 3D density map ρ : R3 → R and SE(3)-kernel f : SE(3) → R: ˜ρ(r) = (f ρ)(r) := SE(3) f (g)ρ(g−1 · r)dg, (1) where R3 r → g · r = R r + t, (2) is the standard SE(3) action and (R, t) ∈ SO(3) × R3 is the representation of g ∈ SE(3). Group action version of the convolution theorem for the prob. density of the sum of two random variables. As an SE(3)-density: mean and covariance for f can be defined. Gaussian SE(3) kernel is only determined by the covariance and mean (to be revisited). B. Afsari and G. Chirikjian (JHU) Deblurring in Cryo-EM October 30, 2015 5 / 12 The Case of Multiunit Large Complexes In general, deconvolution/deblurring is highly ill-posed. We consider a scenario in which the complex has N subunits of known shape. Subunits (modeled as rigid bodies) maybe known using crystallography and only their relative positions (∈ SE(3)) maybe unknown and of interest (e.g., relative positions may correlate with the biological function of the complex). ρ(r) = N i=1 ρi (r), (3) where ρi : R3 → R is the 3D density of subunit i. The blurred 3D reconstructed denisty: ˜ρ(r) = N i=1 ((k ∗ fi ) ρi )(r), (4) where fi is motional blurring kernel w.r.t. subunit 1 and k is reconstruction blurring kernel. Here ∗ is the standard SE(3) convolution (k ∗ fi )(g) := SE(3) k(h)fi (h−1 ◦ g)dh . We will treat k ∗ fi as a single SE(3) kernel. B. Afsari and G. Chirikjian (JHU) Deblurring in Cryo-EM October 30, 2015 6 / 12 Blurring a Single Subunit y = Rr + t, r ∈ R3 , E{r} = 0, E{rr } = Cr, g = R t 0 1 ∈ SE(3), (5) If r is Euclidean Gaussian ∼ N(0, Cr) and g is SE(3) Gaussian, what is the density of y? We use this simple model to model each subunit as an ellipsoid under an SE(3) kernel. Rigid body modeling (e.g., with ellipsoids) is common in structural biology. The goal is to find the unknown SE(3) kernel from the observed y and known Cr. B. Afsari and G. Chirikjian (JHU) Deblurring in Cryo-EM October 30, 2015 7 / 12 Parameterization of Gaussian SE(3) Kernels Mean: g ∼ f a prob. density on SE(3) its (algebraic or bi-invariant) mean µg E{log(µ−1 g g)} = SE(3) log(µ−1 g g)f (g)dg = 0 (6) The associated covariance Σg matrix (6 × 6) Σg := E{vec(Ωg )vec(Ωg ) } = SE(3) vec(Ωg )vec(Ωg ) f (g)dg (7) where vec : se(n) → R6 is an isomorphism between se(3) and R6 and Ωg = log(gµ−1 g ). Total number of parameters: 6 + 21 = 27. B. Afsari and G. Chirikjian (JHU) Deblurring in Cryo-EM October 30, 2015 8 / 12 Matching the Moments y = Rr + t, r ∈ R3 , E{r} = 0, E{rr } = Cr, g = R t 0 1 ∈ SE(3), (8) If g is with mean µg = µR µt 0 1 and covariance Σg then E{y} =E{t} 2nd = (I + 1 2 E{Ω2 R })µt (9a) Cy = E{RCrR } + Ct 2nd = ˜Cr + E{ΩR ˜CrΩR } + 1 2 E{Ω2 R } ˜Cr + 1 2 ˜CrE{Ω2 R }+ E{ΩR µt µt ΩR } + E{ωt ωt } (9b) where ˜Cr = µR CrµR and the expectations of quantities quadratic in ΩR and ωt can be expressed in terms of the SE(3) covariance of g, Σg. In these equations E{y} and Cy can be measured and Cr is known and µg and Σg are unknown. Still underdetermined! B. Afsari and G. Chirikjian (JHU) Deblurring in Cryo-EM October 30, 2015 9 / 12 A Simplified Model: Isotropic Blurring Kernel At the Lie algebra level we assume that all rotations and translations are statistically independent (diagonal covariance matrix). We assume that all rotational variances are σ2 R and all translations variances are σ2 t . Then the equations simplify as: E{y} = E{t} 2nd = (1 − σ2 R )µt (10a) Cy 2nd = µRCrµR + σ2 R tr(Cr)I3 − 3µRCrµR + σ2 R µt 2 I3 − µtµt + σ2 t I3, (10b) Still nonlinear but not severely! Still can be somewhat ill-posed: 9 equations and 8 variables. The effect isotropic blurring (banana shape density): B. Afsari and G. Chirikjian (JHU) Deblurring in Cryo-EM October 30, 2015 10 / 12 Solving the Inverse Problem and an Example F(µR , σ2 R , σ2 t ) = µR CrµR + σ2 R tr(Cr)I3 − 3µR CrµR + σ2 R µt 2 I3 − µt µt + σ2 t I3 − Cy 2 F , (11) We find µt from the first eq. assuming σ2 R and solve the regularized minimization min µR ∈SO(3),σ2 R ,σ2 t Fr (µR , σ2 R , σ2 t ; λR , λt ) (12) where Fr (µR , σ2 R , σ2 t ; λR , λt ) = F(µR , σ2 R , σ2 t ) + λR (σ2 R )2 + λt (σ2 t )2 and λR , λt > 0 are small regularization weights. Repeat till convergence. Example: Blurring two bodies with kernels (µi , σ2 Ri , σ2 ti ). Error in means µi and ti are small but for the variances is high! B. Afsari and G. Chirikjian (JHU) Deblurring in Cryo-EM October 30, 2015 11 / 12 Conclusions In certain applications in structural biology one has prior information about the subunits of a complex under study and the relative motion (position) between subunits may be of interest. We gave a plausibility study as to how this information can be used in a inverse problem setting to remove the blurring effects in cryo-EM. The blurring can be due to high noise and conformational variations within the copies of the complex in the sample imaged. The method was based on parameterization of SE(3) kernels and 2nd order moment matching. Higher order moments and information from other imaging modalities such as Small Angle X-ray Scattering (SAXS) can be used to further regularize the inverse problem. Acknowledgment National Institute of General Medical Sciences of the National Institutes of Health under award number R01GM113240. Thank You! B. Afsari and G. Chirikjian (JHU) Deblurring in Cryo-EM October 30, 2015 12 / 12