Reconstruction from limited single-particle diffraction data via simultaneous determination of state, orientation, intensity, and phase

Significance

Single-particle diffraction is an emerging technique in which diffraction images are collected from individual particles, and is used to study molecular structure from multiple conformational states that may be inaccessible through other methods. However, determining structure from these experiments is challenging, since orientations and states of imaged particles are unknown and images are highly contaminated with noise. Furthermore, the number of useful images is often limited by achievable single-particle hit rates, currently around 0.1%. We introduce an iterative projection framework to simultaneously determine orientations, states, and molecular structure from limited single-particle data by leveraging structural constraints throughout the reconstruction. This framework offers a potential pathway to increasing the amount of information that can be extracted from single-particle diffraction.

Abstract

Free-electron lasers now have the ability to collect X-ray diffraction patterns from individual molecules; however, each sample is delivered at unknown orientation and may be in one of several conformational states, each with a different molecular structure. Hit rates are often low, typically around 0.1%, limiting the number of useful images that can be collected. Determining accurate structural information requires classifying and orienting each image, accurately assembling them into a 3D diffraction intensity function, and determining missing phase information. Additionally, single particles typically scatter very few photons, leading to high image noise levels. We develop a multitiered iterative phasing algorithm to reconstruct structural information from single-particle diffraction data by simultaneously determining the states, orientations, intensities, phases, and underlying structure in a single iterative procedure. We leverage real-space constraints on the structure to help guide optimization and reconstruct underlying structure from very few images with excellent global convergence properties. We show that this approach can determine structural resolution beyond what is suggested by standard Shannon sampling arguments for ideal images and is also robust to noise.

Single-particle X-ray diffraction aims to determine the structures of biological molecules from an ensemble of diffraction patterns, each collected from a single particle per shot. Particles are delivered to the beam at random orientations through either a liquid medium (1) or aerosolization (2). Large-scale experimental facilities, such as the Linac Coherent Light Source (3), have been developed to perform these experiments, with initial results indicating the feasibility of single-particle diffraction as a viable technique to address challenges in the health, material, and energy sciences (4, 5).

A fundamental challenge in single-particle imaging is that the orientation of each imaged particle is unknown and must be recovered to determine structural information. Additionally, many biological samples display conformational flexibility and may exist in one of many possible structural states. To account for varying structural states and avoid a loss of resolution due to averaging of states, the diffraction patterns may need to be classified to the correct state. Furthermore, single particles scatter very few photons; hence the images are heavily contaminated by shot noise, often with less than a photon per Shannon pixel at high scattering angles.

Current approaches typically determine states, orientations, and structures in separate consecutive steps. Classification work includes manifold mapping (6), spectral clustering (7), principal component analysis, and support vector machines (8). Orientation methods include common curve approaches (9–12), expectation maximization (13–15), and manifold embedding (16–19). Once images are classified, oriented, and assembled into a 3D intensity function, iterative phasing (20) is often used to determine molecular structure.

A major drawback of these serial approaches is that error committed in one step is locked in as one moves to the next, which may then lead to a further increase in error. Additionally, serial methods do not leverage prior knowledge about the underlying structure as constraints until the reconstruction of the molecule in the last step. Without these extra real-space constraints, classification and orientation may require a much greater number of images to arrive at a solution. Furthermore, the intensity function is typically smeared from the images onto a computational grid through inverse-distance weighted averaging, which can result in inaccuracies in the assembled intensity function unless a very fine grid is used and the imaged particle orientations densely sample the rotation group.

We present here an approach that reconstructs both single and multiple states, based on simultaneously determining the states and orientations of the images, the 3D intensity function, the complex phases, and the underlying structure. This approach is an extension of the multitiered iterative phasing (M-TIP) algorithm, originally introduced in ref. 21 for fluctuation X-ray scattering reconstruction. Our single-particle version of M-TIP is formulated in a polar framework, using spherical harmonics for fast orientational alignment and spectrally accurate assembly of a 3D intensity function from a sparse set of 2D images. We show that this approach can determine molecular structure with very few diffraction images by supplementing optimization with constraints on the electron densities, and is able to leverage the global convergence properties of iterative phasing schemes to seek the globally optimal states and orientations corresponding to the images.

Background

Here we formulate single-particle reconstruction using a spherical polar framework, which extends the framework used in refs. 22–24 to a curved Ewald sphere. We use $𝐫$ and $𝐪$ to denote real- and Fourier-space coordinates, respectively, and represent their magnitudes via $r=|𝐫|$ and $q=|𝐪|$. The 3D scattered intensity function $I$ of an object with electron density $\rho$ is given by $I(𝐪)={|\widehat{\rho }(𝐪)|}^2$, with Fourier transform $\widehat{\rho }$ given by

$$\widehat{\rho }(𝐪)={\int }_{{ℝ}^3}\rho (𝐫)e^{-2\pi i𝐪\cdot 𝐫}d𝐫.$$ [1]

The intensity function can be expanded in a spherical harmonic series for every resolution shell $q$ as¹

$$I(q,\theta ,\varphi )=\sum _{l=0}^{\mathrm{\infty }}\sum _{m=-l}^l{I}_{lm}(q)Y_l^m(\theta ,\varphi ),$$

where the spherical harmonics $Y_l^m$ have the form

$$Y_l^m(\theta ,\varphi )=P_l^m(\cos \theta )e^{im\varphi },$$

the $P_l^m$ are the normalized associated Legendre polynomials, and the spherical harmonic coefficients $I_{lm}$ are given by

$$I_{lm}(q)={\int }_0^{2\pi }{\int }_0^{\pi }I(q,\theta ,\varphi )Y_l^{m\ast }(\theta ,\varphi )\sin \theta d\theta d\varphi .$$

For a coordinate system rotated by $R\in SO(3)$, the rotated intensity function $I^{(R)}(𝐪)=I(R𝐪)$ can be expressed as

$$I^{(R)}(q,\theta ,\varphi )=\sum _{l=0}^{\mathrm{\infty }}\sum _{m=-l}^l\sum _{m\prime =-l}^l{D}_{lmm\prime }(R)Y_l^m(\theta ,\varphi )I_{lm\prime }(q),$$ [2]

where the $D_{lmm\prime }$ are the Wigner-D matrix elements.

In a single-particle imaging experiment, one collects a series of diffraction patterns $J^{(1)},\mathrm{\dots },J^{(K)}$, where the $k$th image is collected from an unknown sample orientation $R_k\in SO(3)$. Each image samples the 3D intensity function along the Ewald sphere and can be expressed in polar coordinates as

$$J^{(k)}(q,\varphi )=I^{(R_k)}(q,\theta (q),\varphi ),$$ [3]

where $\theta (q)=\pi /2$ for a flat Ewald sphere, $\theta (q)=\text{arccos}(q\lambda /2)$ for a curved Ewald sphere, and $\lambda$ is the X-ray wavelength.

The images may be expanded in a circular harmonic series

$$J^{(k)}(q,\varphi )=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_m^{(k)}(q)e^{im\varphi },$$

where the circular harmonic coefficients $J_m^{(k)}(q)$ are given by

$$J_m^{(k)}(q)=\frac{1}{2\pi }{\int }_0^{2\pi }{J}^{(k)}(q,\varphi )e^{-im\varphi }d\varphi .$$ [4]

The image circular harmonic coefficients can be related directly to the intensity spherical harmonic coefficients by applying the expressions in Eqs. 2 and 3 to Eq. 4, yielding

$$J_m^{(k)}(q)=\sum _{l=|m|}^{\mathrm{\infty }}\sum _{m\prime =-l}^l{D}_{lmm\prime }(R_k)P_l^m(\cos \theta (q))I_{lm\prime }(q).$$ [5]

Our goal is to determine the underlying structural model, encoded as the electron density $\rho$, from a series of single-particle diffraction images $J^{(k)}$; this requires determining the orientation $R_k$ of each image along with the intensity function $I$. Furthermore, if images are collected from a sample mixed over multiple conformational states, then one must also classify each image to the correct state and determine the electron density for each of the associated structures.

M-TIP

A class of “iterative phasing” techniques are commonly used to solve the classical phase problem, which is to determine the structural density $\rho$ from the 3D intensity function $I$, with the inclusion of additional real-space constraints on $\rho$ (20, 25, 26). Given a model and a constraint, these methods apply operators that project the model to the closest object satisfying the constraint. The constraints often include consistency with the intensity function, compact support, density bounds, and/or symmetry. Current single-particle reconstruction algorithms typically use iterative phasing reconstruction only as a last step, after images have been classified, oriented, and merged into a 3D intensity function.

Here we outline an alternative approach, based on M-TIP (21). In this single-particle version of M-TIP, we iteratively modify a set of models for the density, intensity function, orientations, and states to be consistent with specified constraints and the measured data. In particular, we develop a set of operators that choose the optimal state and orientations given a 3D intensity model, project the current intensity model to be consistent with the current state and orientation estimates, filter the intensities, project a density model to be consistent with the intensities, and enforce constraints on the real-space density model. These operators are applied several times in an iterative process that simultaneously determines the states, orientations, phases, and underlying structural densities.

The M-TIP procedure is formulated using a spherical harmonic basis, which allows for fast orientational alignment of the images and assembly of an accurate 3D intensity model from a sparse set of images. The real- and Fourier-space quantities are discretized on a spherical polar grid, where the radial nodes, for $q$ and $r$, are spaced uniformly, the $\theta$ nodes for each radial component lie on Gauss–Legendre quadrature nodes, and the $\varphi$ nodes for each $\theta$ component are spaced uniformly. The integration for the Fourier transform, spherical harmonic transform, and their inverses are numerically computed on this grid using the quadratures described in ref. 21. In particular, we note that the numerical integration scheme for the Fourier transform is a direct approximation of Eq. 1 that maps the values on the real-space grid directly to values on the Fourier-space grid, and vice versa for the inverse transform; thus a radial basis expansion is not required.

We begin by formulating the operators used to update estimates for the missing quantities and show how to use these operators in the M-TIP reconstruction algorithm. We first describe the M-TIP procedure for the single-state case and then show how to generalize this approach to the multistate case. We also describe an approach to modeling noise. An overview of the procedure is given in SI Text and in Figs. S1–S3.

Fig. S1.

Flowchart of the single-state single-particle M-TIP procedure. The “mod” superscript denotes temporary values that have been modified to be consistent with constraints. Bold arrows give the main flow of the algorithm, and dashed arrows show dependencies.

Fig. S3.

Flowchart of the noise-aware version of the single-state single-particle M-TIP procedure. The “mod” superscript denotes temporary values that have been modified to be consistent with constraints. Bold arrows give the main flow of the algorithm, and dashed arrows show dependencies. A similar modification can be performed for the noise-aware multistate procedure.

Real-Space Projectors.

The support projection operator $P_S$ projects a function to have support contained in a support set $S$,

$$(P_S\rho )(𝐫)=\{\begin{array}{cc}\rho (𝐫),\hfill & \text{if}\hspace{0.17em}𝐫\in S,\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}$$

This projector can also be extended to enforce nonnegativity $(P_{S+}\rho )(𝐫)=\max (P_S\rho (𝐫),0)$, as well as other constraints (21).

Magnitude Projector.

Given an intensity model $I$, the associated magnitude projection operator $P_M(I)$ projects a real-space density model $\rho$ so that its power spectrum, given by the squared modulus of its Fourier transform, matches the intensity model. This projector can be expressed in Fourier space as the following renormalization:

$$(\widehat{P_M(I)\rho })(𝐪)=\frac{\widehat{\rho }(𝐪)}{|\widehat{\rho }(𝐪)|}\sqrt{\max (I(𝐪),0)}.$$

Intensity Filtering.

To improve the stability of the iterative algorithm, we use an intensity filtering operator $\mathcal{G}$, which removes high-frequency oscillations in an intensity model $I$ by modifying it to be consistent with a set of real-space constraints. This process is accomplished by using a density model $\rho$, computed during the iterative procedure, applying the magnitude projector $P_M(I)$, projecting the result to satisfy the specified real-space constraints with $P_{S\ast }=P_S$ or $P_{S+}$, and then computing the resulting density’s power spectrum,

$$(\mathcal{G}(\rho )I)(𝐪)={|(\widehat{P_{S\ast }{P}_M(I)\rho })(𝐪)|}^2.$$

Orientation Matching.

Given a 3D intensity model $I$, orientation matching seeks to find, for each image, the orientation that best matches it to the corresponding Ewald-sphere slice of the intensity model. Determining the optimal orientation for an image $J$ can be formulated as

$$\underset{R\in SO(3)}{\text{arg}\hspace{0.17em}\text{min}}{\int }_0^{q_{\max }}{\int }_0^{2\pi }{[J(q,\varphi )-I^{(R)}(q,\theta (q),\varphi )]}^{2}w(q)d\varphi dq,$$ [6]

where $w$ is a weighting function that can be used to compensate for the decay in the signal and/or noise as a function of the resolution shell, e.g., $w(q)=q^{\alpha }$ for some $\alpha \in ℝ$.

After discretizing the integral and expanding the quantities into circular harmonic series, the result of Eq. 6 can be approximated by finding the orientation $R$ that maximizes

$$F(I,J,R)=\sum _q\sum _{l=0}^{L-1}\sum _{m=-l}^l\sum _{m\prime =-l}^l{D}_{lmm\prime }(R)P_{lq}^m{I}_{lm\prime }(q)J_m^{\ast }(q)w(q)-\frac{1}{2}\sum _q\sum _{l=0}^{L-1}\sum _{m\prime =-l}^l{D}_{l0m\prime }(R)P_{lq}^0{(I^2)}_{lm\prime }(q)w(q),$$

with $P_{lq}^m=P_l^m(\cos \theta (q))$ (see SI Text for derivation).

The fast Wigner-D transform can be used to efficiently evaluate $F$ for all rotations represented by their associated Euler angles on a Cartesian grid (27, 28). For an $O(L^3)$ Euler angle grid, this evaluation can be performed with $O(L^4)$ complexity. The optimal image orientation is then approximated by the Euler angles that maximize $F$ on this grid; this is similar to the approach in ref. 24, except that we use an objective function derived from the $L^2$ norm instead of an inner product, to prevent bias toward regions where $I$ is large.

We will compactly express this procedure via the orientation-matching operator $\mathcal{O}(I,J)$, which, given an intensity model $I$ and an image $J$, computes the orientation $R$ that best aligns the image to the intensity model, i.e.,

$$\mathcal{O}(I,J)=\underset{R}{\text{arg}\hspace{0.17em}\text{max}}\hspace{0.17em}F(I,J,R).$$

Intensity Synthesis.

Given a set of orientation estimates for the images, the goal of intensity synthesis is to project an intensity model to be as consistent as possible with the images, i.e., we would like to find the smallest perturbation of $I$ that is a least-squares solution to the linear system in Eq. 5 for each value of $q$. For a general linear system, the orthogonal projection of a vector $x_0$ to the closest least-squares solution $x$ to $Ax=b$ is given by $x=x_0+A^{†}(b-A{x}_0)$, where $A^{†}$ is the pseudoinverse of $A$. However, computing the pseudoinverses of the linear systems in Eq. 5 each time the orientations are updated would be very computationally expensive, as this would require $O(\min (K^2{L}^4,K{L}^5))$ time for each $q$.

In contrast, the pseudoinverse for projecting an intensity model to be consistent with a single image has an analytic expression. Specifically, the single-image intensity projection $P_J(R)$ for an image $J$ and orientation $R$ can be expressed in a spherical harmonic basis as²

$$\begin{array}{c}{(P_J(R)I)}_{lm\prime }(q)=I_{lm\prime }(q)+\mathrm{\Delta }{I}_{lm\prime }(q),\\ \mathrm{\Delta }{I}_{lm\prime }(q)=\sum _{m=-l}^l\frac{1}{c_m}{D}_{lm\prime m}^{\ast }(R)P_l^m(\cos \theta (q))\mathrm{\Delta }{J}_m(q),\\ \mathrm{\Delta }{J}_m(q)=J_m(q)-\sum _{l=|m|}^{L-1}\sum _{m\prime =-l}^l{D}_{lmm\prime }(R)P_l^m(\cos \theta (q))I_{lm\prime }(q),\\ c_m=\sum _{l=|m|}^{L-1}{|P_l^m(\cos \theta (q))|}^2.\end{array}$$

This expression can be computed in $O(L^3)$ time for each $q$.

Given a set of images $\text{𝑱}=\{J^{(1)},\mathrm{\dots },J^{(K)}\}$ and orientations $\text{𝑹}=\{R_1,\mathrm{\dots },R_K\}$, we form an approximation $\mathcal{A}(\text{𝑱},\text{𝑹})$ to the projection associated with the full pseudoinverse by applying the single-image projections in a random sequential order,

$$\mathcal{A}(\text{𝑱},\text{𝑹})I=P_{J^{({\sigma }_1)}}(R_{{\sigma }_1})\mathrm{\dots }{P}_{J^{({\sigma }_N)}}(R_{{\sigma }_N})I,$$

where $\sigma$ is a random $N$-tuple of the indices.

Iterative Schemes.

We now describe how to use the above operators for single-particle M-TIP reconstruction. We initialize the procedure with ${\rho }^{(0)}$, $I^{(0)}$, and ${\text{𝑹}}^{(0)}=\{R_1^{(0)},\mathrm{\dots },R_K^{(0)}\}$ and apply a combination of the previously defined operators to produce the iterates ${\rho }^{(n)}$, $I^{(n)}$, and ${\text{𝑹}}^{(n)}$ after $n$ steps. During each iteration, we filter the previous intensity model, update the image orientations by aligning the images to this filtered model, update the intensity model to match the images at their updated orientations, and then apply a step of an iterative phasing scheme to update the real-space density model.

A widely used iterative phasing scheme known as the hybrid input–output (HIO) scheme (26) employs a negative feedback term to prevent the iterates from getting stuck in local minima. Here we generalize HIO to the following globally optimizing M-TIP scheme for single-particle imaging:

$$\begin{array}{c}{\stackrel{\sim }{I}}^{(n+1)}=\mathcal{G}({\rho }^{(n)})I^{(n)},\\ R_k^{(n+1)}=\mathcal{O}({\stackrel{\sim }{I}}^{(n+1)},J^{(k)}),k=1,\mathrm{\dots },K,\\ I^{(n+1)}=\mathcal{A}(\text{𝑱},{\text{𝑹}}^{(n+1)}){\stackrel{\sim }{I}}^{(n+1)},\\ {\rho }^{(n+1)}(𝐫)=\{\begin{array}{cc}(P_M(I^{(n+1)}){\rho }^{(n)})(𝐫),\hfill & \text{if}\hspace{0.17em}𝐫\in {\mathcal{B}}_{S\ast }\hfill \\ {\rho }^{(n)}(𝐫)-\beta (P_M(I^{(n+1)}){\rho }^{(n)})(𝐫),\hfill & \text{otherwise},\hfill \end{array}\end{array}$$

where ${\mathcal{B}}_{S\ast }=\{𝐫:(P_{S\ast }\rho )(𝐫)=\rho (𝐫)\}$ and $\beta \in (\mathrm{0,1}]$.

Note that the HIO scheme allows for nonzero density values outside of the support region, which may cause high-frequency perturbations, or noise, in the intensity function. By applying the intensity filter $\mathcal{G}$ in the first step, our approach removes this noise from the intensity function before it is used in the orientational alignment and intensity synthesis steps.

To refine the solution, we use the following generalization of the error reducing (ER) algorithm (25), which is a local minimizer:

$$\begin{array}{c}{\stackrel{\sim }{I}}^{(n+1)}={|{\widehat{\rho }}^{(n)}|}^2,\\ R_k^{(n+1)}=\mathcal{O}({\stackrel{\sim }{I}}^{(n+1)},J^{(k)}),k=1,\mathrm{\dots },K,\\ I^{(n+1)}=\mathcal{A}(\text{𝑱},{\text{𝑹}}^{(n+1)}){\stackrel{\sim }{I}}^{(n+1)},\\ {\rho }^{(n+1)}=P_{S\ast }{P}_M(I^{(n+1)}){\rho }^{(n)}.\end{array}$$

In this case, the intensity filter is not needed, because ER directly enforces the support constraint on the density model.

To estimate the true support region, we use the shrinkwrap method (29), which periodically convolves the iterate with a Gaussian of width $\sigma$ and then applies a threshold $ϵ$; i.e., the updated support is taken to be

$$S=\{𝐫:\frac{1}{{(2\pi {\sigma }^2)}^{3/2}}{\int }_{{ℝ}^3}{\rho }^{(n)}(𝐬)e^{-{|𝐫-𝐬|}^2/2{\sigma }^2}d𝐬\ge ϵ\}.$$

In particular, we apply several iterations of the M-TIP HIO scheme, followed by several M-TIP ER iterations, perform shrinkwrap, then apply several more M-TIP ER iterations, and repeat this process until convergence is reached.

Multiple States.

If the imaged samples are present in multiple states, we then need to determine the structure corresponding to each state, requiring the images to be classified into the possible states. Here we assume that there is a known finite number $N_{\text{state}}$ of states. We denote the electron density and intensity models for the $s$th state as ${\rho }_s$ and $I_s$, respectively, and, for the $k$th image, we denote its estimated state as ${\mathcal{S}}_k\in \{1,\mathrm{\dots },N_{\text{state}}\}$. We now describe how to generalize the M-TIP procedure to determine the density associated to each state from a mixed distribution of images.

Given the set of intensity models $\text{𝑰}=\{I_1,\mathrm{\dots },I_{N_{\text{state}}}\}$ and an image $J$, the optimal orientation and state for $J$ is found by maximizing the orientation-matching objective function $F(I_s,J,R)$ over all orientations and states. In particular, the orientation-matching operator $\mathcal{O}$ gives the optimal orientation,

$$\mathcal{O}(\text{𝑰},J)=\underset{R}{\text{arg}\hspace{0.17em}\text{max}}\underset{s}{\max }F(I_s,J,R),$$

and the classification operator $\mathcal{C}$ gives the optimal state,

$$\mathcal{C}(\text{𝑰},J)=\underset{s}{\text{arg}\hspace{0.17em}\text{max}}\hspace{0.17em}\underset{R}{\max }F(I_s,J,R).$$

Given a set of images J, orientations R, and states $\mathcal{𝓢}$, the multistate intensity synthesis operator ${\mathcal{A}}_s(\text{𝑱},\text{𝑹},\mathcal{𝓢})$ now only uses the single-image projections for the images currently assigned to the $s$th state, i.e.,

$${\mathcal{A}}_s(\text{𝑱},\text{𝑹},\mathcal{𝓢})I_s=P_{J^{({\sigma }_{s1})}}(R_{{\sigma }_{s1}})\mathrm{\dots }{P}_{J^{({\sigma }_{s{N}_s})}}(R_{{\sigma }_{s{N}_s}})I_s,$$

where we now have that ${\sigma }_s$ is a random $N_s$-tuple of the indices associated to the $s$th state, i.e., ${\sigma }_{sj}\in \{k:{\mathcal{S}}_k=s\}$.

The multistate M-TIP HIO update step can now be described in terms of these modified operators and, during each iteration, is performed over all states $s=\mathrm{1,2},\mathrm{\dots },N_{\text{state}}$,

$$\begin{array}{c}{\stackrel{\sim }{I}}_s^{(n+1)}=\mathcal{G}({\rho }_s^{(n)})I_s^{(n)},\\ R_k^{(n+1)}=\mathcal{O}({\stackrel{\sim }{\text{𝑰}}}^{(n+1)},J^{(k)}),k=1,\mathrm{\dots },K,\\ {\mathcal{S}}_k^{(n+1)}=\mathcal{C}({\stackrel{\sim }{\text{𝑰}}}^{(n+1)},J^{(k)}),k=1,\mathrm{\dots },K,\\ I_s^{(n+1)}={\mathcal{A}}_s(\text{𝑱},{\text{𝑹}}^{(n+1)},{\mathcal{𝓢}}^{(n+1)}){\stackrel{\sim }{I}}_s^{(n+1)},\\ {\rho }_s^{(n+1)}(𝐫)=\{\begin{array}{cc}(P_M(I_s^{(n+1)}){\rho }_s^{(n)})(𝐫),\hfill & \text{if}\hspace{0.17em}𝐫\in {\mathcal{B}}_{S_s}\hfill \\ {\rho }_s^{(n)}(𝐫)-\beta (P_M(I_s^{(n+1)}){\rho }_s^{(n)})(𝐫),\hfill & \text{otherwise}.\hfill \end{array}\end{array}$$

The multistate M-TIP ER update is performed similarly, but without the intensity filter and with using the ER scheme to update the density. Here, shrinkwrap is performed on each state, yielding separate support estimates $S_1,S_2,\mathrm{\dots },S_{N_{\text{state}}}$.

A degeneracy common in classification algorithms occurs when a subset of the states become empty. To prevent this, images are randomly reshuffled among states if the number of images assigned to any state falls below a given threshold.

Noise.

Here we outline a strategy to model noise within M-TIP. Instead of fitting a model directly to noisy data, we will, instead, enforce only that the difference between the measured and model image values satisfy appropriate statistics. In particular, we will enforce that the weighted second-order moment of the measured image values about the model image values is bounded above by a weighted sum of the variances ${\sigma }^2(q)$, over each spherical shell, from a noise model, which we will assume can be estimated from the data. The following steps are applied to refine the reconstruction in the presence of noise after the first few M-TIP cycles; see SI Text for more details.

During each iteration, after the orientation matching step, we compute the set ${\text{𝑱}}^M$ of the model images from the current intensity model via Eq. 5. Then, we apply the noise projector $P_N$, which projects ${\text{𝑱}}^M$ to the closest set of values that satisfies the aforementioned second-order moment constraint; i.e., the projected quantity $\text{𝑱}=P_N{\text{𝑱}}^M$ is given by the solution to

$$\underset{\text{𝑱}}{\min }{||\text{𝑱}-{\text{𝑱}}^M||}_w^2,\text{subject to}{||\text{𝑱}-{\text{𝑱}}^D||}_w^2\le \sum _{q}w(q){\sigma }^2(q),$$

where ${\text{𝑱}}^D$ is the set of values given by the measured data and ${||\text{𝑱}||}_w^2=\frac{1}{K{N}_{\varphi }}{\sum }_{k,q,\varphi }{(J^{(k)}(q,\varphi ))}^{2}w(q)$. The solution to this optimization problem can be computed with Lagrange multipliers. For both the orientation matching and noise projector step, we choose a weight of the form $w(q)=q^{\alpha }/{\sigma }^2(q)$ to reduce the contribution from the noisier resolution shells. This projected quantity J is then used in place of the measured data in the M-TIP steps listed in Iterative Schemes and Multiple States. This procedure allows the various constraints used throughout the M-TIP procedure to filter out noise in the image ensemble.

As an additional noise filtering step, we set $J_m(q)=\mathrm{\hspace{0.17em}0}$ for $|m|$ larger than the number of Shannon angles at $q$, given by $2\pi Dq$, where $D$ estimates the diameter of the imaged particle.

SI Text

Outline of Supporting Information.

The main text introduces the M-TIP algorithm for determining structure from a set of single-particle diffraction images by simultaneously determining the states, orientations, and phases while enforcing constraints on the real-space structure. In this Supporting Information, we provide an overview of the M-TIP procedure, derive the single-image projection operator, list the parameters that we use in the reconstructions, describe the orientation error calculation, and present the reconstructions described in the main text.

The outline for the remainder of this document is as follows. In Overview of the Single-Particle M-TIP Scheme, we give an overview of the M-TIP algorithm along with flowchart descriptions in Figs. S1–S3. In Derivation of the Discretized Orientation-Matching Operator, we derive the discrete approximation of the orientation matching operator presented in Orientation Matching. In Derivation of the Single-Image Projection Operator, we derive the single-image intensity synthesis projector by directly computing the pseudoinverse of the linear system in Eq. 5. In Reconstruction Parameters, we list the parameters that we use in the reconstruction procedure. In Orientation Error Calculation, we describe how the orientation error statistics presented in the main text are computed. The reconstructions from clean simulated images described in the Results are shown in Figs. S4 and S5 and are compared against the molecular model in Figs. S6 and S7. Finally, the orientation and classification errors of the presented reconstructions are given in Tables S1 and S2, and examples of the noisy images used in the reconstructions from noisy data are given in Fig. S8.

Fig. S4.

Reconstruction of pRb bound to E2F from single-particle diffraction data, displayed as density isosurfaces, for different image counts $K$ and maximum scattering angles $\psi$. The density isosurfaces for $K=6$ with $\varphi \le \mathrm{\hspace{0.17em}30}°$ are visualized at a lower density level in order to show the bulk structures, which had a lower median density in these cases.

Fig. S5.

Reconstructions of the open and closed states SiaP bound from multistate single-particle diffraction data, displayed as density isosurfaces, for different image counts $K$ and maximum scattering angles $\psi$.

Fig. S6.

(Left) The 3D electron density and (Middle and Right) local details of the original and reconstructions of the single-state example obtained from 24, 12, and 6 images with maximum scattering angles of 15°, 60°, and 90°, respectively, are compared with the associated molecular model, indicating that the reconstructions are of sufficient quality to recognize major secondary structure elements such as alpha helices and, in some cases, even bulky side chains.

Fig. S7.

The 3D electron density (first and third columns) and local details (second and fourth columns) of the original and reconstructions of the multistate example obtained from 48 images with a maximum scattering angle of 30° are compared with the associated molecular models, indicating that the reconstructions are of sufficient quality to recognize major secondary structure elements such as alpha helices and, in some cases, even bulky side chains.

Table S1.

Orientation error (deg) of single-state reconstructions

No. of images	Maximum scattering angle
	90°	60°	30°	15°	0°
24	0.7	0.5	0.5	0.8	0.7
12	0.5	0.5	0.8	0.6	1.2
6	0.6	2.4	61.2	32.2	54.2

Table S2.

Classification (%) error of multistate reconstructions

No. of images per state	Maximum scattering angle
	90°	60°	30°	15°	0°
24	0	0	0	0	0
12	0	0	0	13	46
6	0	50	50	42	42

Fig. S8.

Selection of the simulated noisy images used in (Left) the single-state reconstruction and (Middle and Right) the multistate reconstructions. The color scale gives the number of photons recorded in a pixel, and gray corresponds to regions where no photon was recorded.

Overview of the Single-Particle M-TIP Scheme.

Here we provide an overview of the M-TIP reconstruction algorithm presented in the main text. The procedure for the single-state version of M-TIP is depicted graphically in Fig. S1 and is summarized below. We begin with random estimates for the density $\rho$ and the intensity function $I$ and perform the following steps:

• i)Filter the intensity function estimate to get $\stackrel{\sim }{I}=\mathcal{G}(\rho )I$ from the previous density estimate $\rho$ and intensity function estimate $I$. For the M-TIP ER iteration, the intensity function does not require filtering, and, instead, we compute these intensity functions directly from the density model as $\stackrel{\sim }{I}={|\widehat{\rho }|}^2$.
• ii)Align the images to the filtered intensity function to obtain the orientation estimates $R_1,\mathrm{\dots },R_K$, where $R_k=\mathcal{O}(\stackrel{\sim }{I},J)$. This step is performed using the spherical harmonic coefficients ${\stackrel{\sim }{I}}_{lm}$ of the filtered intensity function and circular harmonic coefficients $J_m^{(k)}$ of each image.
• iii)Modify the filtered intensity function $\stackrel{\sim }{I}$ to be consistent with the oriented images by applying the approximate intensity synthesis operator $\mathcal{A}$, and update the intensity function to be $I=\mathcal{A}(\text{𝑱},\text{𝑹})\stackrel{\sim }{I}$. This step is performed using the spherical harmonic coefficients ${\stackrel{\sim }{I}}_{lm}$ of the filtered intensity function and circular harmonic coefficients $J_m^{(k)}$ of each image.
• iv)Project the density $\rho$ so that the squared magnitude of its Fourier transform matches the new intensity function by applying the magnitude projector $P_M$ to get the modified density ${\rho }^{\text{mod}}=P_M(I)\rho$.
• v)Enforce real-space constraints on the density by applying either the support projector $P_S$ or the support-and-nonnegativity projector $P_{S+}$ to the modified density estimate ${\rho }^{\text{mod}}$ through either the ER or HIO scheme to obtain the updated density estimate $\rho$.
• vi)Repeat steps i–v until convergence.

The procedure for the multistate version of M-TIP is depicted graphically in Fig. S2 and is summarized below. We begin with random starting models for the densities ${\rho }_s$ and the intensity functions $I_s$ for each state $s=\mathrm{\hspace{0.17em}1},\mathrm{\dots },N_{\text{state}}$ and perform the following steps:

• i)Filter the intensity function estimates to get ${\stackrel{\sim }{I}}_s=\mathcal{G}({\rho }_s)I_s$ from the previous density estimates ${\rho }_s$ and intensity function estimates $I_s$. For the M-TIP ER iteration, the intensity functions do not require filtering, and, instead, we compute these intensity functions directly from the density models as ${\stackrel{\sim }{I}}_s={|{\widehat{\rho }}_s|}^2$.
• ii)Align the images to the filtered intensity function for each state. For each image, assign it to the state that it best matches after alignment and set its new orientations to be given by that alignment, i.e., the new orientation estimates $R_1,\mathrm{\dots },R_K$ are given by $R_k=\mathcal{O}(\stackrel{\sim }{\text{𝑰}},J^{(k)})$, and the new state estimates ${\mathcal{S}}_1,\mathrm{\dots },{\mathcal{S}}_K$ are given by ${\mathcal{S}}_k=\mathcal{C}(\stackrel{\sim }{\text{𝑰}},J^{(k)})$. This step is performed using the spherical harmonic coefficients ${({\stackrel{\sim }{I}}_s)}_{lm}$ of the filtered intensity function for each state and circular harmonic coefficients $J_m^{(k)}$ of each image.
• iii)Modify the filtered intensity functions ${\stackrel{\sim }{I}}_s$ for each state to be consistent with the oriented and classified images by applying the approximate intensity synthesis operators ${\mathcal{A}}_s$, yielding the updated intensity functions $I_s={\mathcal{A}}_s(\text{𝑱},\text{𝑹}){\stackrel{\sim }{I}}_s$. This step is performed using the spherical harmonic coefficients ${({\stackrel{\sim }{I}}_s)}_{lm}$ of the filtered intensity function for each state and circular harmonic coefficients $J_m^{(k)}$ of each image.
• iv)Project the density ${\rho }_s$ for each state so that the squared magnitude of its Fourier transform matches the new intensity function computed for that state, by applying the magnitude projector $P_M$ to get the modified densities ${\rho }_s^{\text{mod}}=P_M(I_s){\rho }_s$.
• v)Enforce real-space constraints on the densities by applying either the support projectors $P_{S_s}$ or the support-and-nonnegativity projectors $P_{S_s+}$ to the modified density function for each state through either the ER or HIO scheme to obtain the updated density estimates ${\rho }_s$.
• vi)Repeat steps i–v until convergence.

Fig. S2.

Flowchart of the multistate single-particle M-TIP procedure. The “mod” superscript denotes temporary values that have been modified to be consistent with constraints. Each step is performed over all states $s=1,\mathrm{\dots },N_{\text{state}}$. Bold arrows give the main flow of the algorithm, and dashed arrows show dependencies.

The procedure for the noise-aware single-state version of M-TIP is depicted graphically in Fig. S3 and is summarized below. The noise-aware extension of the multistate version of M-TIP can be performed in a similar fashion. In particular, we initialize the procedure by applying a few cycles of the single-state steps described above and then perform the following steps within each iteration.

• i)Filter the intensity function estimate to get $\stackrel{\sim }{I}=\mathcal{G}(\rho )I$ from the previous density estimate $\rho$ and intensity function estimate $I$. For the M-TIP ER iteration, the intensity function does not require filtering, and, instead, we compute these intensity functions directly from the density model as $\stackrel{\sim }{I}={|\widehat{\rho }|}^2$.
• ii)Align the images to the filtered intensity function to obtain the orientation estimates $R_1,\mathrm{\dots },R_K$, where $R_k=\mathcal{O}(\stackrel{\sim }{I},J)$. This step is performed using the spherical harmonic coefficients ${\stackrel{\sim }{I}}_{lm}$ of the filtered intensity function and circular harmonic coefficients $J_m^{(k)}$ of each image.
• iii)Compute the set of model images ${\text{𝑱}}^M$ from the current filtered intensity model $\stackrel{\sim }{I}$ via Eq. 5.
• iv)Apply the noise projector $P_N$ to modify the image estimates J to the set of values closest to the current model ${\text{𝑱}}^M$ that lie within the uncertainty of the measured data ${\text{𝑱}}^D$.
• v)Modify the filtered intensity function $\stackrel{\sim }{I}$ to be consistent with the oriented images by applying the approximate intensity synthesis operator $\mathcal{A}$, and update the intensity function to be $I=\mathcal{A}(\text{𝑱},\text{𝑹})\stackrel{\sim }{I}$. This step is performed using the spherical harmonic coefficients ${\stackrel{\sim }{I}}_{lm}$ of the filtered intensity function and circular harmonic coefficients $J_m^{(k)}$ of each image.
• vi)Project the density $\rho$ so that the squared magnitude of its Fourier transform matches the new intensity function by applying the magnitude projector $P_M$ to get the modified density ${\rho }^{\text{mod}}=P_M(I)\rho$.
• vii)Enforce real-space constraints on the density by applying either the support projector $P_S$ or the support-and-nonnegativity projector $P_{S+}$ to the modified density estimate ${\rho }^{\text{mod}}$ through either the ER or HIO scheme to obtain the updated density estimate $\rho$.
• viii)Repeat steps i–v until convergence.

Derivation of the Discretized Orientation-Matching Operator.

The orientation-matching objective function is given by

$$\underset{R\in SO(3)}{\text{arg}\hspace{0.17em}\text{min}}{\int }_0^{q_{\max }}{\int }_0^{2\pi }{[J(q,\varphi )-I^{(R)}(q,\theta (q),\varphi )]}^{2}w(q)d\varphi dq.$$

The discretized orientation-matching operator is given by

$$\mathcal{O}(I,J)=\underset{R\in SO(3)}{\text{arg}\hspace{0.17em}\text{min}}\sum _q{\int }_0^{2\pi }{[J(q,\varphi )-I^{(R)}(q,\theta (q),\varphi )]}^{2}w(q)d\varphi =\underset{R\in SO(3)}{\text{arg}\hspace{0.17em}\text{min}}\sum _q{\int }_0^{2\pi }[J^2(q,\varphi )+{(I^2)}^{(R)}(q,\theta (q),\varphi )-2J(q,\varphi )I^{(R)}(q,\theta (q),\varphi )]w(q)d\varphi .=\underset{R\in SO(3)}{\text{arg}\hspace{0.17em}\text{min}}\sum _q{\int }_0^{2\pi }[{(I^2)}^{(R)}(q,\theta (q),\varphi )-2J(q,\varphi )I^{(R)}(q,\theta (q),\varphi )]w(q)d\varphi .$$

Note that the $J^2$ term drops out because it does not depend on the orientation $R$. We multiply through by $-1/2$, so that the form of the optimization has a form similar to the traditional form used in correlation-based matching techniques, to get

$$\mathcal{O}(I,J)=\underset{R\in SO(3)}{\text{arg}\hspace{0.17em}\text{min}}\sum _q{\int }_0^{2\pi }[J(q,\varphi )I^{(R)}(q,\theta (q),\varphi )-\frac{1}{2}{(I^2)}^{(R)}(q,\theta (q),\varphi )]w(q)d\varphi .$$

By applying Parseval’s theorem, we get

$$\mathcal{O}(I,J)=\underset{R\in SO(3)}{\text{arg}\hspace{0.17em}\text{min}}\sum _q\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{J}_m^{\ast }(q)I_m^{(R)}(q,\theta (q))w(q)-\frac{1}{2}\sum _q{(I^2)}_0^{(R)}(q,\theta (q))w(q),$$

where $I_m^{(R)}(q,\theta (q))=\frac{1}{2\pi }{\int }_0^{2\pi }{I}^{(R)}(q,\theta (q),\varphi )e^{-im\varphi }d\varphi$ and ${(I^2)}_0^{(R)}(q,\theta (q))=\frac{1}{2\pi }{\int }_0^{2\pi }{(I^2)}^{(R)}(q,\theta (q),\varphi )d\varphi$.

By inserting Eq. 2 into the definitions of $I_m^{(R)}$ and ${(I^2)}_0^{(R)}$, we get

$$\mathcal{O}(I,J)=\underset{R\in SO(3)}{\text{arg}\hspace{0.17em}\text{min}}\sum _q\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}\sum _{l=|m|}^{\mathrm{\infty }}\sum _{m\prime =-l}^l{D}_{lmm\prime }(R)P_{lq}^m{I}_{lm}(q)J_m^{\ast }(q)w(q)-\frac{1}{2}\sum _q\sum _{l=0}^{\mathrm{\infty }}\sum _{m\prime =-l}^l{D}_{l0m\prime }(R)P_{lq}^0{(I^2)}_{lm\prime }(q)w(q).$$

We now bandlimit the terms in the above sum to $l<L$ and rearrange the summations to get

$$\mathcal{O}(I,J)=\underset{R\in SO(3)}{\text{arg}\hspace{0.17em}\text{min}}\sum _q\sum _{l=0}^{L-1}\sum _{m=-l}^l\sum _{m\prime =-l}^l{D}_{lmm\prime }(R)P_{lq}^m{I}_{lm\prime }(q)J_m^{\ast }(q)w(q)-\frac{1}{2}\sum _q\sum _{l=0}^{L-1}\sum _{m\prime =-l}^l{D}_{l0m\prime }(R)P_{lq}^0{(I^2)}_{lm\prime }(q)w(q),$$

which is the orientation-matching operator given in Orientation Matching.

Derivation of the Single-Image Projection Operator.

For a single image $J$ with estimated orientation $R$ and resolution shell $q$, the linear system in Eq. 5 gives

$$J_m(q)=\sum _{l=|m|}^{L-1}\sum _{m\prime =-l}^l{D}_{lmm\prime }(R)P_l^m(\cos \theta (q))I_{lm\prime }(q).$$

As stated in the main text, we assume that all sums over $l$ only include even terms.

If we set $b_m=J_m(q)$ and $x_{lm\prime }=I_{lm\prime }(q)$, then we have the linear system $Ax=b$ where the linear operator is of the form $A=PD$, where matrix multiplication by $P$ is given for $-L<m<L$ via

$$\begin{array}{c}{(Px)}_m=\sum _{l=|m|}^{L-1}{P}_{lq}^m{x}_{lm}\\ ={(P^m)}^T{x}^m,\end{array}$$

where $P^m={[P_{|m|q}^m,P_{(|m|+1)q}^m,\mathrm{\dots },P_{(L-1)q}^m]}^T$ and $x^m={[x_{|m|}^m,x_{|m|+1}^m,\mathrm{\dots },x_{L-1}^m]}^T$, and $D$ is a unitary matrix whose matrix multiplication is given for $0\le l<L$ and $-l\le m\le l$ by

$${(Dx)}_{lm}=\sum _{m\prime =-l}^l{D}_{lmm\prime }(R)x_{lm\prime }.$$

To derive the single-image projection operator, we need to compute $x_0+A^{†}(b-A{x}_0)$, where $A^{†}$ is the pseudoinverse of $A$. Because $D$ is unitary, we have that $A^{†}=D^{\ast }{P}^{†}$. Note that $P$ consists of $2L-1$ independent equations for $-L<m<L$. Therefore, the pseudoinverse of $P$ can be described by the pseudoinverse of each of these independent equations. In particular, the pseudoinverse of a vector $y$ is given by $y^T/{||y||}_2^2$, and, therefore, we have that

$$\begin{array}{c}{(P^{†}b)}_{lm}={\left({(P^{mT})}^{†}{b}_m\right)}_l\\ ={\left(\frac{P^m}{{||P^m||}_2^2}{b}_m\right)}_l\\ =\frac{1}{c_m}{P}_{lq}^m{b}_m,\end{array}$$

where $c_m={||P^m||}_2^2$, with $c_m^{-1}$ set to 0 when ${||P^m||}_2^2$ is 0.

Because $D$ is blockwise unitary, its inverse is given by

$${(D^{\ast }x)}_{lm\prime }=\sum _{m=-l}^l{D}_{lm\prime m}^{\ast }(R)x_{lm}.$$

Therefore, $x=x_0+A^{†}(b-A{x}_0)$ can be written for each $l$ and $m\prime$ as

$$\begin{array}{c}{x}_{lm\prime }={(x_0+\mathrm{\Delta }x)}_{lm\prime },\\ \mathrm{\Delta }{x}_{lm\prime }={(D^{\ast }{P}^{†}\mathrm{\Delta }b)}_{lm\prime }=\sum _{m=-l}^l\frac{1}{c_m}{D}_{lm\prime m}^{\ast }(R)P_{lq}^m\mathrm{\Delta }{b}_m,\\ \mathrm{\Delta }{b}_m={(b-A{x}_0)}_m=b_m-\sum _{l=|m|}^{L-1}\sum _{m\prime =-l}^l{D}_{lmm\prime }(R)P_{lq}^m{(x_0)}_{lm\prime },\end{array}$$

which is the formula given in Intensity Synthesis for ${(P_J(R)I)}_{lm\prime }(q)$ after substituting back $x_{lm\prime }=I_{lm\prime }(q)$ and $b_m=J_m(q)$.

Reconstruction Parameters.

The real- and Fourier-space quantities are discretized on a 3D polar grid as described in ref. 1. For the real-space grid, the radial nodes are given by $r_n=n/44$ for $n=\mathrm{\hspace{0.17em}0},\mathrm{\dots },43$, the longitudinal nodes for the $n$th radial node are given by the arccos of the order $L_n$ Gaussian quadrature points with $L_n=\lceil \pi n\rceil +7$, and the azimuthal angle is discretized over $2L_n-1$ equidistributed nodes. For the Fourier-space grid, the radial nodes are given by $q_n=\frac{n}{41}{q}_{\max }$ for $n=\mathrm{\hspace{0.17em}0},\mathrm{\dots },43$, the longitudinal nodes are given by the arccos of the order $L$ Gaussian quadrature points with $L=\lceil \pi (N-1)\rceil +7$, and the azimuthal angle is discretized over $2L-1$ equidistributed nodes. Note that the resolution shells corresponding to the last two Fourier-space radial nodes of the 3D grid are not sampled by the images, and the intensity values on these nodes are either allowed to float or set to zero during the reconstruction. This technique ultimately improves the reconstruction, as the support of the molecule can be better resolved on an oversampled real-space grid.

The images are sampled on a 2D polar grid with 42 radial nodes and $2L-1$ angular nodes. The Euler-angle grid used for the fast Wigner-D transform is based on the Z – Y – Z convention with angles $(\alpha ,\beta ,\gamma )$, where $\alpha$ and $\gamma$ are discretized uniformly over $2L-1$ nodes each and the $\beta$ nodes are given by the arccos of $L$ points uniformly distributed from $-1$ to $1$.

We set $\beta =0.5$, $\alpha =3$ in the weight functions, $\sigma$ to the length of a typical grid element, and $ϵ$ to ∼4% of the maximum density. We observed that a lower cutoff improved the performance of the noise projecter, and, as such, we reduced $ϵ$ to 2% of the maximum density during the cycles where the noise projector was applied. The density values used to initialize the reconstruction are sampled randomly from a uniform distribution within the initial support region. The initial intensity function is computed as $I^{(0)}=\mathcal{A}(J,{\text{𝑹}}^{(0)})\mathrm{𝟎}$ for the single-state reconstructions and $I_s^{(0)}={\mathcal{A}}_s(J,{\text{𝑹}}^{(0)},{\mathcal{𝓢}}^{(0)})\mathrm{𝟎}$ for the multistate reconstructions, with random initial orientations and state assignments and where $\mathrm{𝟎}$ is the zero function. For the intensity synthesis operators, we set $N$, or $N_s$, to be 8 times the number of images currently assigned to the associated state. The images for the multistate reconstructions were randomly reshuffled between states whenever the number of images assigned to any state fell below or equal to $K/4$, where $K$ is the total number of images. All of the reconstructions used the same seeds to initialize the random number generators.

For the single-image intensity synthesis projections, we observed that ∼96% of the $c_m$ values were in the range of $[.1,4.1]$, and that the remaining values, which occur for $m$ very close to $L$, decayed very rapidly. Therefore, to maintain stability of the pseudoinversion used in these single-image projections, we replace $1/c_m$ with $0$ whenever $c_m$ is less than $0.1$, which effectively allows the corresponding $J_m(q)$ values that are not well resolved on the grid to float.

For each reconstruction, we run M-TIP for 10 cycles, where each cycle consists of 60 iterations of the M-TIP HIO scheme followed by 60 iterations of the M-TIP ER scheme, an application of shrinkwrap, and then 60 more iterations of the M-TIP ER scheme. For the first 10 iterations of each step, the states and orientations are fixed, which prevents large changes in the computed quantities that can occur when moving between schemes. After the 10 cycles, the solution is refined by running 1,000 iterations of the M-TIP ER scheme. To ensure that the reconstructed density is mostly positive, the density is multiplied by the sign of the average density value before each application of shrinkwrap.

For the majority of the reconstructions, the only real-space constraint that we enforce is a support constraint. In particular, the densities of the target structures have negative density values at the simulated resolutions, which causes the nonnegativity constraint to decrease the resolution of the reconstructions. However, in some of the more difficult cases, where there is a lack of images and/or small scatterings angles are used, initially applying extra constraints, such as nonnegativity, can help to obtain a resolution sufficient for determining good initial estimates of the orientations and states. In these cases, we enforce a nonnegativity constraint for a set number of cycles, disable it toward the end of the reconstruction procedure, and also set $\widehat{\rho }$ to zero in the $q_{42}$ and $q_{43}$ resolution shells. For the single-state reconstructions, this procedure is performed for $K=\mathrm{\hspace{0.17em}12}$ with $\psi \le \mathrm{\hspace{0.17em}30}°$ and for $K=\mathrm{\hspace{0.17em}6}$, and the nonnegativity constraint is removed after the seventh cycle. For the multistate reconstructions, this procedure is done for $K=\mathrm{\hspace{0.17em}48}$ with $\psi =\mathrm{\hspace{0.17em}0}°$, $K=\mathrm{\hspace{0.17em}24}$ with $\psi \le \mathrm{\hspace{0.17em}15}°$, and $K=\mathrm{\hspace{0.17em}12}$ with $\psi \le \mathrm{\hspace{0.17em}60}°$, and the nonnegativity constraint is removed after the fifth cycle. For the reconstructions from noisy data, we enforced a nonnegativity constraint for the first five cycles, and used the noise projector only for the last seven cycles.

The above parameters were found through a combination of trial and error and established best practices. In general, the quality of the reconstructions is not very sensitive to the choice of parameters when a sufficient amount of data is available. However, an optimal selection of parameters may be necessary for the more difficult cases and can also decrease the number of iterations needed for convergence. In practice, these parameters could be chosen adaptively to minimize the discrepancy between the reconstructed intensity model and the data.

Orientation Error Calculation.

Here we show how to compute the orientation errors presented in Table S1. Because the reconstructions are only determined up to a global orthogonal transformation, the computed orientations $R_k$ must first be globally aligned to the true orientations ${\stackrel{\sim }{R}}_k$. For a curved Ewald sphere, this global alignment is given by the orthogonal matrix $Q_{\text{opt}}$, which is calculated as

$$Q_{\text{opt}}=\underset{Q\in O(3)}{\text{arg}\hspace{0.17em}\text{min}}\sum _{k=1}^K{||Q{R}_k-{\stackrel{\sim }{R}}_k||}_F^2=\underset{Q\in O(3)}{\text{arg}\hspace{0.17em}\text{min}}{||Q\text{𝑹}-\stackrel{\sim }{\text{𝑹}}||}_F^2,$$

where $\text{𝑹}=[R_1,\mathrm{\dots },R_k]$, $\stackrel{\sim }{\text{𝑹}}=[{\stackrel{\sim }{R}}_1,\mathrm{\dots },{\stackrel{\sim }{R}}_k]$, and $||\cdot |{|}_F$ is the Frobenius norm. This optimization problem is known as the orthogonal Procrustes problem, and its solution can be expressed in terms of the singular value decomposition $\stackrel{\sim }{\text{𝑹}}{\text{𝑹}}^T=\mathcal{U}\mathrm{\Sigma }{\mathcal{V}}^T$ via

$$Q_{\text{opt}}=\mathcal{U}{\mathcal{V}}^T.$$

The orientation error is then given by the average angle between $Q_{\text{opt}}{R}_k$ and ${\stackrel{\sim }{R}}_k$, which is calculated as

$$\frac{1}{K}\sum _{k=1}^K\text{arccos}\left(\frac{\text{tr}(Q_{\text{opt}}{R}_k{\stackrel{\sim }{R}}_k^T)-1}{2}\right).$$

For a flat Ewald sphere, Friedel symmetry induces two valid orientations for each image, corresponding to in-plane rotation by 180°. This symmetry introduces an extra complication in the computation of $Q_{\text{opt}}$, which is now given by

$$Q_{\text{opt}}=\underset{Q\in O(3)}{\text{arg}\hspace{0.17em}\text{min}}\sum _{k=1}^K\underset{{\theta }_k=0,\pi }{\min }{||Q{R}_k{R}_z({\theta }_k)-{\stackrel{\sim }{R}}_k||}_F^2,$$

where $R_z({\theta }_k)$ corresponds to in-plane rotation by ${\theta }_k$.

Unfortunately, without knowledge of the optimal ${\theta }_k$ values, the previous analysis used in the case of a curved Ewald sphere cannot be directly applied to compute $Q_{\text{opt}}$, and a brute force search over all choices of ${\theta }_k$ would result in exponential complexity. Instead, we begin by finding the orthogonal matrix $Q_v$ that best aligns the viewing angles of the calculated orientations to the true orientations. The viewing angles are given by the third column of the associated rotation matrices; i.e., the $i$th elements for the viewing angles of the $k$th image are given by ${(v_k)}_i={(R_k)}_{i3}$ and ${({\stackrel{\sim }{v}}_k)}_i={({\stackrel{\sim }{R}}_k)}_{i3}$. $Q_v$ is then found as

$$\begin{array}{c}{Q}_v=\underset{Q\in O(3)}{\text{arg}\hspace{0.17em}\text{min}}\sum _{k=1}^K{||Q{v}_k-{\stackrel{\sim }{v}}_k||}_2^2\\ =\underset{Q\in O(3)}{\text{arg}\hspace{0.17em}\text{min}}{||QV-\stackrel{\sim }{V}||}_2^2,\end{array}$$

where $V=[v_1,\mathrm{\dots },v_K]$ and $\stackrel{\sim }{V}=[{\stackrel{\sim }{v}}_1,\mathrm{\dots },{\stackrel{\sim }{v}}_K]$; this is another instance of the orthogonal Procrustes problem, whose solution is now given in terms of the singular value decomposition $\stackrel{\sim }{V}{V}^T={\mathcal{U}}_v{\mathrm{\Sigma }}_v{\mathcal{V}}_v^T$ as

$$Q_v={\mathcal{U}}_v{\mathcal{V}}_v^T.$$

We now use $Q_v$ to approximate the optimal choices of ${\theta }_k$ as

$${\theta }_k^{\text{opt}}=\underset{{\theta }_k=0,\pi }{\text{arg}\hspace{0.17em}\text{min}}{||Q_v{R}_k{R}_z({\theta }_k)-{\stackrel{\sim }{R}}_k||}_F^2.$$

These choices for the ${\theta }_k$ values are then used to approximate $Q_{\text{opt}}$ via the orthogonal Procrustes analysis described for the curved Ewald sphere with $R_k$ replaced by $R_k{R}_z({\theta }_k^{\text{opt}})$. For sufficiently accurate orientations, this process is guaranteed to select the optimal values for ${\theta }_k$ and, hence, computes the error exactly. Otherwise, this calculation will provide an upper bound on the orientation error.

Results

We apply our single-particle M-TIP algorithm to determine structure from simulated single-particle diffraction images for both single-state and multistate data. We begin by studying the effects of the curvature of the Ewald sphere, scattering angle, and image count with clean images, and then study the algorithm’s performance in the presence of shot noise. We simulate data with a fixed maximum resolution shell $q_{\max }$ and vary the maximum scattering angles $\psi$. Nonzero values of $\psi$ are related to the wavelength via $\lambda =q_{\text{max}}^{-1}\sqrt{2-2\cos (\psi )}$, and $\psi =\mathrm{\hspace{0.17em}0}$ corresponds to having a flat Ewald sphere.

The diffraction images are sampled on a 2D polar grid, with an oversampling factor of about 3 in the radial component, from random orientations and without the $q=\mathrm{\hspace{0.17em}0}$ information. Each reconstruction is initialized with a random density ${\rho }^{(0)}$ and a support estimate $S$ equal to a ball with a diameter slightly larger than that of the target structure. In each of the following examples, the data are calculated to a resolution of 5.5 Å. More details can be found in SI Text.

We study reconstruction quality by calculating resolution estimates, orientation error, and classification error. Resolution estimates are obtained by applying a 0.5 cutoff to Fourier shell correlations between the aligned reconstructions and target structures. Orientation error is calculated as the average angle between the calculated and true orientations after they are globally aligned (see SI Text). (For a flat Ewald sphere, Friedel symmetry induces two valid orientations for each image, corresponding to in-plane rotation by 180°. In this case, the global orientational alignment also determines the optimal choice for the in-plane rotation for each image.) The classification error is given by the percentage of incorrectly classified images.

The single-state M-TIP scheme is first tested on noise-free data simulated from a retinoblastoma protein (pRb) bound to E2F using Protein Data Bank entry 1O9K (30), which has a 75-Å diameter. Reconstruction statistics are shown in Table 1 and Table S1; reconstructions are shown in Fig. 1 and Fig. S4. In all except the most difficult cases, the orientation error was less than $0.8°$.

Table 1.

Resolution (Å) of single-state reconstructions

No. of images per state	Maximum scattering angle
	90°	60°	30°	15°	0^°
24	5.9	5.8	5.6	5.8	6.6
12	6.6	6.3	7.0	7.0	8.3
6	7.5	9.4	17.3	18.8	20.5

Fig. 1.

Reconstructions of pRb bound to E2F from noise-free single-particle diffraction data, displayed as density isosurfaces. (Left) Original and (Middle and Right) reconstructions from (Middle) 24 images with $\psi =15°$ and (Right) six images with $\psi =90°$.

The multistate M-TIP scheme is first tested on noise-free data simulated from the open and closed states of a sialic acid binding protein (SiaP) using biomolecules A and B from Protein Data Bank entry 2CEX (31), which have 68- and 65-Å diameters. Reconstruction statistics are shown in Table 2 and Table S2; reconstructions are shown in Fig. 2 and Fig. S5. In all except the most difficult cases, there was 0% classification error.

Table 2.

Resolution (Å) of multistate reconstructions

		Maximum scattering angle
No. of images	State	90°	60°	30°	15°	0^°
24	A	5.9	5.8	5.8	6.1	7.0
	B	6.1	6.1	5.6	6.1	6.8
12	A	6.3	6.4	6.4	7.8	9.4
	B	6.3	6.8	7.5	8.3	9.8
6	A	7.5	10.7	16.0	16.0	22.5
	B	8.0	11.8	16.0	17.3	22.5

Fig. 2.

Reconstructions of open and closed states of SiaP from multistate single-particle diffraction data, displayed as density isosurfaces. (Top) Original structures and (Bottom) reconstructions from 48 images with $\psi =30°$.

To assess the algorithm’s robustness to noise, we repeat the above single-state and multistate tests with simulated shot noise for $\psi ={\mathrm{\hspace{0.17em}60}}^{\circ }$ and 192 images per state. Noise is simulated directly on the polar images by generating Poisson statistics on $J(q,\varphi )\mathrm{\Delta }\mathrm{\Omega }(q)$, where the polar pixel solid angle is given by $\mathrm{\Delta }\mathrm{\Omega }(q)=q$, and then renormalizing by $\mathrm{\Delta }\mathrm{\Omega }(q)$. On average, there are ∼8,000 photons per image, with 0.07 photons per pixel and 0.25 photons per Shannon pixel at the image boundary; see Fig. S8 for sample images. We incorporate the noise projector into M-TIP, where variances for shot noise can be estimated from the angular averages of the data via ${\sigma }^2(q)=\mathrm{\hspace{0.17em}1}/[\mathrm{\Delta }\mathrm{\Omega }(q)K{N}_{\varphi }]{\sum }_{k,\varphi }{J}^{(k)}(q,\varphi )$. The reconstructions from simulated noisy images are given in Fig. 3. The single-state reconstruction achieved a resolution of 6.4 Å and an orientation error of 1.3°. Multistate reconstructions of open and closed states of SiaP achieved resolutions of 6.3 Å for state A and 6.8 Å for state B, and 0% classification error.

Fig. 3.

Reconstructions from noisy simulated single-particle diffraction data using 192 images per state with $\psi ={\mathrm{\hspace{0.17em}60}}^{\circ }$. (Left) Single-state reconstruction of pRB bound to E2F. (Middle and Right) Multistate reconstructions of (Middle) open and (Right) closed states of SiaP.

Each reconstruction took less than 8 min per image on 24 cores with a 2.3-GHz Haswell processor. The runtime could be improved by reducing the number of spherical harmonics used in the first few iterations and for low $q$, and also by using local optimization techniques during the final refinement.

Discussion

M-TIP is able to successfully orient and classify images, even with low image counts. Many of the reconstructions have sufficient quality to optically recognize the 3D arrangement of large secondary structural elements; see Figs. S6 and S7.

When an insufficient number of images are available, the algorithm is unable to determine an initial structure to sufficient resolution to distinguish between the different orientations and states. Nevertheless, in these cases, low-resolution structures are still found with resolutions similar to what can be obtained via small-angle X-ray scattering.

The presented algorithm is able to obtain resolutions beyond what is suggested by typical Shannon-angle estimates, which approximate the number of required images to obtain a resolution of $R$ for a particle with diameter $D$ as $\pi D/R$ (32). This resolution is possible because M-TIP leverages additional information that these estimates do not take into account, including curvature of the Ewald sphere, real-space constraints, and a trickle-down effect of high-resolution data, discussed below.

Note that a curved Ewald sphere increases the amount of information per image by up to a factor of 2, as inversely related points on the image now sample non-Friedel related areas in Fourier space. As a result, the reconstructions for a sufficiently curved Ewald sphere, corresponding to high maximum scattering angles, have quality similar to the reconstructions for a flat Ewald sphere at twice the image count.

Enforcement of real-space constraints may further reduce the number of required images. For instance, using a support constraint fills in missing intensity information in unsampled regions between images, an approach used extensively in limited-view tomography (33). Although Shannon-angle estimates take the particle diameter into account, a molecule’s true support is often sparse within its circumscribing sphere and, if found, can be used to reduce the degrees of freedom of the system. M-TIP is able to pick up on the true support during the reconstruction through the shrinkwrap procedure.

Additionally, there is a trickle-down effect: High-resolution data can help fill in missing low-resolution information via the real-space constraints; this can be seen from the fact that the information content depends linearly on the image count, whereas the degrees of freedom depend cubically on the resolution. Due to this effect, the resolutions of the reconstructions decrease slowly as a function of the image count, until there is insufficient information to orient and classify the images.

The results from noisy simulated data indicate that M-TIP maintains robustness in the presence of noise. M-TIP is able to leverage real-space constraints to filter out noise in the images through the noise projector, allowing for accurate reconstructions even though there may be an insufficient amount of data to remove the noise through standard averaging techniques.

Conclusions

We have shown that it is possible to determine structure from a sparse set of single-particle diffraction data by simultaneously determining states, orientations, intensities, and phases. Our results suggest that using a highly curved Ewald sphere, accurately representing the image data on a 3D grid, and leveraging real-space constraints throughout the reconstruction can greatly boost the amount of information that can be extracted from single-particle data, even in the presence of shot noise.

In practice, one will also need to determine missing experimental parameters, model varying scale factors, fill in gaps in the data, and correct for systematic issues. It may be possible to treat many of these effects within M-TIP via additional projections. Furthermore, to fully study conformational heterogeneity, the multistate framework will need to be extended to model a continuum, rather than a discrete number, of states.

The simulations shown here assume scattering from molecules in a vacuum. However, in practice, there is typically a bulk solvent layer around each particle, which may significantly alter the scattering profile at low resolution through Babinet’s principle and increase noise levels at higher resolution. The success of the presented approach with experimental data will rely heavily on proper evaporation of liquids around the particles, as well as other advances in both experimental delivery and modeling of systematic issues in the data.