Tutorial: Symmetry Relaxation

A tutorial detailing symmetry relaxation in CryoSPARC, a tool to help improve the refinement of pseudosymmetric particles.

Introduction

Symmetry relaxation is an option available in Homogeneous and Non-Uniform refinement jobs that can help resolve pseudosymmetry and symmetry-mismatched complexes. This tutorial describes the specific case that symmetry relaxation attempts to address, and walks through how it can be applied to a dataset.

Alignment and Symmetry Mismatches

All refinement jobs in CryoSPARC require estimating the orientation from which each particle is viewed, relative to the density reference. This “alignment” step enables iterative refinement, and allows for successive reconstruction of an updated density map. This step is fairly costly though, as it requires solving a 5-dimensional search problem (3 pose dimensions and 2 shift dimensions) independently for every particle, via matching projections of the rotated and shifted density to particle images. In all refinements with global pose search, CryoSPARC uses a technique known as Branch and Bound (BnB) to accelerate the alignment step. This was described in our 2017 Nature Methods publication, cryoSPARC: algorithms for rapid unsupervised cryo-EM structure determination (Punjani et al., 2017).

Since poses and shifts range over a continuum of possible values, the search space of poses and shifts must be discretized, so that a projection-matching based likelihood calculation can be performed at all poses in the discrete space. Discretization of poses usually does not cause issues for asymmetric structures, for which each particle will have only one correct pose. Discretization isn’t problematic with perfectly symmetric structures either: each particle has a set of multiple correct poses, the number of which is equal to the symmetry order of the imposed symmetry group. Each of these correct poses is related to each other via symmetry transform. In this case, the alignment algorithm can find any of the optimal poses, and the entire set of symmetry-related poses will be used to reconstruct the molecule.

Where this strategy encounters difficulties is with nearly symmetric molecules. There are many terms used to refer different forms of approximate symmetries, including pseudosymmetry, symmetry-mismatches, or symmetry-breaking features. An excellent characterization of different forms of these symmetries is presented in Huiskonen, 2018. Pseudosymmetric particles may have several plausible poses at low resolution, and the correct pose may only be distinguishable if each of the symmetry-related poses are compared directly. The discretization of the pose search space used by CryoSPARC provides no guarantee that the symmetry-related poses are compared directly, unless symmetry relaxation is activated (as described in the subsequent section).

Note that there is no single processing strategy that will best resolve all types of symmetry-mismatched complexes, and optimal reconstruction of symmetry-mismatched complexes is an active area of research that is usually heavily informed by the specific sample being solved. For a thorough catalogue of different processing strategies used for resolving symmetry-mismatched molecules, refer to, Abrishami et al., 2021 published in Progress in Biophysics and Molecular Biology.

With symmetry-mismatched structures, keen attention to map details and quality is required. Simple resolution indicators like FSC are usually not the best indicator of optimal final results, and manual map inspection is strongly recommended.

Symmetry Relaxation

In effort to make CryoSPARC more robust to symmetry-mismatched complexes, we have updated the Homogeneous and Non-Uniform Refinement job types to include a tool known as symmetry relaxation. Symmetry relaxation is a modification to the orientation search procedure, which forces CryoSPARC to be more thorough during orientation search, to avoid placing particles into incorrect poses that are related to the true pose by a symmetry transform. It is recommended to enable symmetry relaxation when dealing with pseudosymmetric or symmetry-breaking molecules.

When symmetry relaxation is enabled, regular BnB alignment proceeds as normal for each particle, and the BnB-optimal pose is stored. After the BnB-optimal pose is found, symmetry relaxation imposes an extra step in which the alignment objective function is evaluated at all poses that are symmetry-related to the current optimal pose. The symmetry-related pose angles are computed analytically, and the objective function is evaluated at the current FSC resolution of the density. If any of the symmetry-related poses are found to have a better objective value, these new poses will be used to reconstruct the next iteration’s density map.

In CryoSPARC v4.4’s Homogeneous and Non-Uniform Refinement jobs, symmetry relaxation is made available via the Symmetry relaxation method parameter. There are three options available:

  • none: This disables symmetry relaxation. The input symmetry group will be enforced as usual, and each particle will be used during reconstruction N times, where N is the order of the symmetry group

  • maximization: This option enables symmetry relaxation. Once the BnB-optimal pose is found, the alignment objective will be evaluated over each of the N symmetry-related poses. The single best pose will carry forward to the reconstruction (”backprojection”) step.

  • marginalization: This option enables symmetry relaxation via marginalization. Once the BnB-optimal pose is found, the alignment objective will be evaluated over a small search radius covering each of the N symmetry-related modes. The single optimal mode will be carried forward to the reconstruction step, and each pose within the search radius will be weighted during backprojection by its normalized posterior probability.

Whether maximization or marginalization should be used depends on the size of the protein, size of the mask, and overall signal-to-noise ratio (SNR) of the dataset. For larger proteins, masks, and higher SNRs, maximization may be sufficient. For smaller proteins, masks, and lower SNRs, marginalization may produce better results. This advice is congruent with our general recommendation that marginalization is preferred when working with smaller proteins and lower SNRs.

Symmetry Relaxation Walk Through

In this walk-through, we will use a dataset with pseudo-icosahedral symmetry to illustrate how symmetry relaxation can help resolve symmetry breaking features. The density map we’ll be using is available on the Electron Microscopy Data Bank under entry #8254, “Phage Qbeta asymmetric reconstruction”. This structure was solved in Gorzelnik et al, 2016 and subsequently referred to in Huiskonen, 2018 as an example of a dataset containing a symmetry mismatch.

Since the raw movies or particles were not released, we will be using the solved density map to generate simulated particles within CryoSPARC v4.4 using the Simulate Data job, and then using these simulated particles to reconstruct the pseudo-icosahedral density. While the use of synthetic data is not representative of most workflows used in practice, we’re presenting this walk-through as it allows us to directly compare the faithfulness of each refinement method (with or without symmetry relaxation) both in terms of map quality and in terms of pose discrepancy from ground truth. With real datasets, we do not have access to ground truth orientations and thus cannot provide as robust of an analysis of the estimated latent variables.

Synthetic Data Generation

To begin, we’ll download the EMD-8254 volume and run an Import 3D Volumes job in CryoSPARC to import the volume into CryoSPARC. After the volume is imported, we need to ensure that the volume is aligned to CryoSPARC’s icosahedral symmetry axes conventions, which can be done via the symmetry alignment feature in the Volume Alignment Tools job. Even though we are working with an asymmetric volume, we intend on using symmetry relaxation under the icosahedral symmetry group, thus we require the volume to be (as best as possible) aligned to the icosahedral symmetry axes. This step can be done via connecting the imported volume to the Volume Alignment Tools job, and setting the following parameters:

  • Do symmetry alignment: True

  • Symmetry string: I

The output volume is now aligned to CryoSPARC’s icosahedral symmetry axes convention.

Next, we’ll generate the synthetic data using the Simulate Data job. Connect the aligned volume to Simulate Data, and simulate 20,000 particles at the default signal-to-noise ratio. We will also use Volume Tools to generate a mask (via thresholding, dilating, and padding) the aligned volume to be used downstream during refinement. Our workflow thus far is included below.

Initial 3D Reconstruction

Next, we will proceed with 3D reconstruction as usual: Ab-initio reconstruction followed by refinement. We run Ab-initio reconstruction with the following parameters:

  • Maximum resolution (Angstroms): 8

  • Initial resolution (Angstroms): 20

  • Symmetry: I*

*Icosahedral symmetry is applied at this point to ensure Ab-initio finds an output volume that is aligned to the symmetry axes, and to avoid the problem of “flattened” models with high-symmetry data.

In the next step, we’ll create an asymmetric reference from the Ab-initio output. Using Homogeneous Reconstruction Only, we’ll connect the particles from Ab-initio and set the following parameters:

  • Symmetry: I

  • Break Symmetry: True

This will take the input particle stack and reconstruct a volume where each particle’s pose has been randomly permuted by one of the icosahedral symmetry group’s operations. This step effectively “breaks” the perfect symmetry that may be present in the initial set of particle poses, and can be used on the outputs of a symmetry-enforced ab-initio reconstruction or refinement job.

Comparing Refinement Methods

Now we’ll compare the different methods of refinement to see how symmetry relaxation can help resolve symmetry mismatches. The three methods we’ll compare are:

  • Standard asymmetric refinement (SAR)

  • Symmetry relaxation (SR) via maximization

  • Symmetry relaxation (SR) via marginalization

Build three Homogeneous Refinement jobs. In all jobs, set the following parameters:

  • Initial lowpass resolution (A): 20

  • Force re-do GS split: False

  • Initialize noise model from images: True

For the standard asymmetric refinement, leave all other parameters as default.

For the symmetry relaxation via maximization and marginalization, set the following additional parameters:

  • Symmetry: I

  • Symmetry relaxation method: maximization or marginalization

Once the three jobs have completed, we can see that both symmetry relaxation (SR) methods recovered the correct asymmetric volume, whereas the SAR volume still appears fully symmetric.

Since we’re working with synthetic data, we can compare the estimated particle poses to their ground truth values, for each of the different methods. Below is a set of histograms in polar coordinates, displaying the number of particles with a given error between the ground truth pose and the pose found by Branch and Bound. We also investigated whether adding extra iterations to SAR helped; the plot below shows the results with 0, 1, and 2 extra iterations added.

Histograms concentrated near 0º (12 o’clock position) indicate the majority of particles had their poses correctly recovered. Manually forcing extra refinement iterations helped somewhat to reduce the number of misaligned particles (compare the orange and yellow histograms to the purple one), but both symmetry relaxed methods performed even better.

The tree view for this workflow is displayed below.

References

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