Persistent topology for cryo‐EM data analysis

Abstract

In this work, we introduce persistent homology for the analysis of cryo-electron microscopy (cryo-EM) density maps. We identify the topological fingerprint or topological signature of noise, which is widespread in cryo-EM data. For low signal to noise ratio (SNR) volumetric data, intrinsic topological features of biomolecular structures are indistinguishable from noise. To remove noise, we employ geometric flows which are found to preserve the intrinsic topological fingerprints of cryo-EM structures and diminish the topological signature of noise. In particular, persistent homology enables us to visualize the gradual separation of the topological fingerprints of cryo-EM structures from those of noise during the denoising process, which gives rise to a practical procedure for prescribing a noise threshold to extract cryo-EM structure information from noise contaminated data after certain iterations of the geometric flow equation. To further demonstrate the utility of persistent homology for cryo-EM data analysis, we consider a microtubule intermediate structure (EMD-1129). Three helix models, an alpha-tubulin monomer model, an alphaand beta-tubulin model, and an alpha-and beta-tubulin dimer model, are constructed to fit the cryo-EM data. The least square fitting leads to similarly high correlation coefficients, which indicates that structure determination via optimization is an ill-posed inverse problem. However, these models have dramatically different topological fingerprints. Especially, linkages or connectivities that discriminate one model from another one, play little role in the traditional density fitting or optimization, but are very sensitive and crucial to topological fingerprints. The intrinsic topological features of the microtubule data are identified after topological denoising. By a comparison of the topological fingerprints of the original data and those of three models, we found that the third model is topologically favored. The present work offers persistent homology based new strategies for topological denoising and for resolving ill-posed inverse problems.

Object-oriented persistent homology

“…Recently, we have introduced molecular topological fingerprints (MTFs) as a quantitative tool for revealing topology-function relationships in protein folding, 86 modeling and prediction of the stability of proteins 86 and nano particles, 85 and resolving ill-posed inverse problems in cryo-electron microscopic (cryo-EM) structure determination. 88 We have proposed resolution based persistent homology 89 and multidimensional persistence 87 for biomolecules.…”

Multiscale Gaussian network model (mGNM) and multiscale anisotropic network model (mANM)

“…35,52 The continuous rigidity function, which can be regarded as the density distribution function (density estimator) of a biomolecule, plays many important roles beyond the scope of flexibility study. 55 For instance, it can be used to generate biomolecular surface representations, 56,57 which reduce to the Gaussian surface if an appropriate kernel is used. In fact, rigidity function can be applied to decipher the atomic information from the experimental electron density data.…”

“…In fact, rigidity function can be applied to decipher the atomic information from the experimental electron density data. 47,51,57 Second, protein multiscale collective motions can be captured by using multiple kernels in our FRI method, called multiscale FRI or multikernel FRI (mFRI). 36 This approach significantly improve the accuracy of FRI Bfactor predictions.…”

Multiresolution persistent homology for excessively large biomolecular datasets

“…37 We have utilized persistent homology to resolve ill-posed inverse problems in cryo-EM structural fitting. 38 Figure 1 illustrates the multiscale features of a virus particle. To understand the physical and biological properties of viruses and other macromolecular complexes, we need to have appropriate multiscale and multiresolution descriptions.…”

“…36 Additionally, persistent homology analysis using protein based coarse-grained representation has been introduced in our recent work for studying multiprotein complexes. 38 Nevertheless, coarse-grained persistent homology might suffer from inconsistency due to the ambiguity in choosing the coarse-grained particle.…”