Abstract
The conformational and thermodynamic properties of disordered proteins are commonly described in terms of structural ensembles and free energy landscapes. To provide information on the transition rates between the different states populated by these proteins, it would be desirable to generalize this description to kinetic ensembles. Approaches based on the theory of stochastic processes can be particularly suitable for this purpose. Here, we develop a Markov state model and apply it to determine a kinetic ensemble of Aβ42, a disordered peptide associated with Alzheimer’s disease. Through the Google Compute Engine, we generated 315-µs all-atom molecular dynamics trajectories. Using a probabilistic-based definition of conformational states in a neural network approach, we found that Aβ42 is characterized by inter-state transitions on the microsecond timescale, exhibiting only fully unfolded or short-lived, partially folded states. Our results illustrate how kinetic ensembles provide effective information about the structure, thermodynamics and kinetics of disordered proteins.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 / 30 days
cancel any time
Subscribe to this journal
Receive 12 digital issues and online access to articles
$99.00 per year
only $8.25 per issue
Buy this article
- Purchase on SpringerLink
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
Data availability
Subsampled trajectory and intermediate data, as well as the trained neural network weights, analysis notebooks and source data, are available from https://zenodo.org/record/424732178. Source data are provided with this paper.
Code availability
Analysis notebooks, code and example data are available from https://github.com/vendruscolo-lab/ab42-kinetic-ensemble and https://zenodo.org/record/424732178.
References
Heller, G. T., Sormanni, P. & Vendruscolo, M. Targeting disordered proteins with small molecules using entropy. Trends Biochem. Sci. 40, 491–496 (2015).
Babu, M. M., van der Lee, R., de Groot, N. S. & Gsponer, J. Intrinsically disordered proteins: regulation and disease. Curr. Opin. Struct. Biol. 21, 432–440 (2011).
Sormanni, P. et al. Simultaneous quantification of protein order and disorder. Nat. Chem. Biol. 13, 339–342 (2017).
Berman, H. M. et al. The Protein Data Bank. Nucleic Acids Res. 28, 235–242 (2000).
Bonomi, M., Heller, G. T., Camilloni, C. & Vendruscolo, M. Principles of protein structural ensemble determination. Curr. Opin. Struct. Biol. 42, 106–116 (2017).
Lindorff-Larsen, K., Best, R. B., DePristo, M. A., Dobson, C. M. & Vendruscolo, M. Simultaneous determination of protein structure and dynamics. Nature 433, 128–132 (2005).
Fraser, J. S. et al. Accessing protein conformational ensembles using room-temperature X-ray crystallography. Proc. Natl Acad. Sci. USA 108, 16247–16252 (2011).
Bonomi, M. & Vendruscolo, M. Determination of protein structural ensembles using cryo-electron microscopy. Curr. Opin. Struct. Biol. 56, 37–45 (2019).
Van Kampen, N. G. Stochastic Processes in Physics and Chemistry (Elsevier, 2007); https://doi.org/10.1016/B978-0-444-52965-7.X5000-4
Prinz, J.-H. et al. Markov models of molecular kinetics: generation and validation. J. Chem. Phys. 134, 174105 (2011).
Chodera, J. D. & Noé, F. Markov state models of biomolecular conformational dynamics. Curr. Opin. Struct. Biol. 25, 135–144 (2014).
Husic, B. E. & Pande, V. S. Markov state models: from an art to a science. J. Am. Chem. Soc. 140, 2386–2396 (2018).
Pérez-Hernández, G., Paul, F., Giorgino, T., De Fabritiis, G. & Noé, F. Identification of slow molecular order parameters for Markov model construction. J. Chem. Phys. 139, 015102 (2013).
Schwantes, C. R. & Pande, V. S. Improvements in Markov state model construction reveal many non-native interactions in the folding of NTL9. J. Chem. Theory Comput. 9, 2000–2009 (2013).
Bowman, G. R. in An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation Vol. 797 (eds. Bowman, G. R. et al.) 7–22 (Springer, 2014).
Kohlhoff, K. J. et al. Cloud-based simulations on Google Exacycle reveal ligand modulation of GPCR activation pathways. Nat. Chem. 6, 15–21 (2014).
Voelz, V. A., Bowman, G. R., Beauchamp, K. & Pande, V. S. Molecular simulation of ab initio protein folding for a millisecond folder NTL9(1−39). J. Am. Chem. Soc. 132, 1526–1528 (2010).
Klus, S. et al. Data-driven model reduction and transfer operator approximation. J. Nonlinear Sci. 28, 985–1010 (2018).
Mardt, A., Pasquali, L., Wu, H. & Noé, F. VAMPnets for deep learning of molecular kinetics. Nat. Commun. 9, 5 (2018).
Mardt, A., Pasquali, L., Noé, F. & Wu, H. Deep learning Markov and Koopman models with physical constraints. In Proc. First Mathematical and Scientific Machine Learning Conference Vol. 107 (eds. Lu, J. & Ward, R.) 451–475 (PMLR, 2020).
Cummings, J., Lee, G., Ritter, A., Sabbagh, M. & Zhong, K. Alzheimer’s disease drug development pipeline: 2019. Alzheimers Dement. 5, 272–293 (2019).
Hardy, J. A. & Higgins, G. A. Alzheimer’s disease: the amyloid cascade hypothesis. Science 256, 184–185 (1992).
Knowles, T. P. J., Vendruscolo, M. & Dobson, C. M. The amyloid state and its association with protein misfolding diseases. Nat. Rev. Mol. Cell Biol. 15, 384–396 (2014).
Jack, C. R. et al. NIA-AA Research Framework: toward a biological definition of Alzheimer’s disease. Alzheimers Dement. 14, 535–562 (2018).
Cohen, S. I. A., Vendruscolo, M., Dobson, C. M. & Knowles, T. P. J. From macroscopic measurements to microscopic mechanisms of protein aggregation. J. Mol. Biol. 421, 160–171 (2012).
Heller, G. T. et al. Small-molecule sequestration of amyloid-β as a drug discovery strategy for Alzheimer’s disease. Sci. Adv. 6, eabb5924 (2020).
Michaels, T. C. T. et al. Thermodynamic and kinetic design principles for amyloid-aggregation inhibitors. Proc. Natl Acad. Sci. USA 117, 24251–24257 (2020).
Meng, F. et al. Highly disordered amyloid-β monomer probed by single-molecule FRET and MD simulation. Biophys. J. 114, 870–884 (2018).
Lin, Y.-S., Bowman, G. R., Beauchamp, K. A. & Pande, V. S. Investigating how peptide length and a pathogenic mutation modify the structural ensemble of amyloid-β monomer. Biophys. J. 102, 315–324 (2012).
Rosenman, D. J., Connors, C. R., Chen, W., Wang, C. & García, A. E. Aβ monomers transiently sample oligomer and fibril-like configurations: ensemble characterization using a combined MD/NMR approach. J. Mol. Biol. 425, 3338–3359 (2013).
Sgourakis, N. G. et al. Atomic-level characterization of the ensemble of the Aβ(1–42) monomer in water using unbiased molecular dynamics simulations and spectral algorithms. J. Mol. Biol. 405, 570–583 (2011).
Nasica-Labouze, J. et al. Amyloid β protein and Alzheimer’s disease: when computer simulations complement experimental studies. Chem. Rev. 115, 3518–3563 (2015).
Granata, D. et al. The inverted free energy landscape of an intrinsically disordered peptide by simulations and experiments. Sci. Rep. 5, 15449 (2015).
Zimmerman, M. I. & Bowman, G. R. FAST conformational searches by balancing exploration/exploitation trade-offs. J. Chem. Theory Comput. 11, 5747–5757 (2015).
Hellerstein, J. L., Kohlhoff, K. J. & Konerding, D. E. Science in the Cloud: accelerating discovery in the 21st century. IEEE Internet Comput. 16, 64–68 (2012).
Robustelli, P., Piana, S. & Shaw, D. E. Developing a molecular dynamics force field for both folded and disordered protein states. Proc. Natl Acad. Sci. USA 115, E4758–E4766 (2018).
Rahman, M. U., Rehman, A. U., Liu, H. & Chen, H.-F. Comparison and evaluation of force fields for intrinsically disordered proteins. J. Chem. Inf. Model. (2020); https://doi.org/10.1021/acs.jcim.0c00762
McGibbon, R. T. & Pande, V. S. Variational cross-validation of slow dynamical modes in molecular kinetics. J. Chem. Phys. 142, 124105 (2015).
Noé, F., Wu, H., Prinz, J.-H. & Plattner, N. Projected and hidden Markov models for calculating kinetics and metastable states of complex molecules. J. Chem. Phys. 139, 184114 (2013).
Röblitz, S. & Weber, M. Fuzzy spectral clustering by PCCA+: application to Markov state models and data classification. Adv. Data Anal. Classif. 7, 147–179 (2013).
Wu, H. & Noé, F. Variational approach for learning Markov processes from time series data. J. Nonlinear Sci. 30, 23–66 (2020).
Klambauer, G. et al. Self-normalizing neural networks. In Proc. Advances in Neural Information Processing Systems (eds Guyon, I. et al.) 971–980 (Curran Associates, 2017).
Kohlhoff, K. J., Robustelli, P., Cavalli, A., Salvatella, X. & Vendruscolo, M. Fast and accurate predictions of protein NMR chemical shifts from interatomic distances. J. Am. Chem. Soc. 131, 13894–13895 (2009).
Löhr, T., Jussupow, A. & Camilloni, C. Metadynamic metainference: convergence towards force field independent structural ensembles of a disordered peptide. J. Chem. Phys. 146, 165102 (2017).
Bonomi, M., Camilloni, C. & Vendruscolo, M. Metadynamic metainference: enhanced sampling of the metainference ensemble using metadynamics. Sci. Rep. 6, 31232 (2016).
Roche, J., Shen, Y., Lee, J. H., Ying, J. & Bax, A. Monomeric Aβ1–40 and Aβ1–42 peptides in solution adopt very similar Ramachandran map distributions that closely resemble random coil. Biochemistry 55, 762–775 (2016).
Konrat, R. NMR contributions to structural dynamics studies of intrinsically disordered proteins. J. Magn. Reson. 241, 74–85 (2014).
Dai, W., Sengupta, A. M. & Levy, R. M. First passage times, lifetimes and relaxation times of unfolded proteins. Phys. Rev. Lett. 115, 048101 (2015).
Levy, R. M., Dai, W., Deng, N.-J. & Makarov, D. E. How long does it take to equilibrate the unfolded state of a protein? Protein Sci. 22, 1459–1465 (2013).
Yan, Y., McCallum, S. A. & Wang, C. M35 oxidation induces Aβ40-like structural and dynamical changes in Aβ42. J. Am. Chem. Soc. 130, 5394–5395 (2008).
Hou, L. et al. Solution NMR studies of the Aβ(1−40) and Aβ(1−42) peptides establish that the Met35 oxidation state affects the mechanism of amyloid formation. J. Am. Chem. Soc. 126, 1992–2005 (2004).
Hou, L., Kang, I., Marchant, R. E. & Zagorski, M. G. Methionine 35 oxidation reduces fibril assembly of the amyloid Aβ-(1–42) peptide of Alzheimer’s disease. J. Biol. Chem. 277, 40173–40176 (2002).
Schütt, K. T., Sauceda, H. E., Kindermans, P.-J., Tkatchenko, A. & Müller, K.-R. SchNet—a deep learning architecture for molecules and materials. J. Chem. Phys. 148, 241722 (2018).
Fout, A., Byrd, J., Shariat, B. & Ben-Hur, A. Protein interface prediction using graph convolutional networks. In Proc. 31st International Conference on Neural Information Processing Systems 6533–6542 (NIPS, 2017).
Boomsma, W. & Frellsen, J. Spherical convolutions and their application in molecular modelling. In Proc. Advances in Neural Information Processing Systems 30 (eds. Guyon, I. et al.) 3433–3443 (Curran Associates, 2017).
Chodera, J. D. & Noé, F. Probability distributions of molecular observables computed from Markov models. II. Uncertainties in observables and their time-evolution. J. Chem. Phys. 133, 105102 (2010).
Olsson, S., Wu, H., Paul, F., Clementi, C. & Noé, F. Combining experimental and simulation data of molecular processes via augmented Markov models. Proc. Natl Acad. Sci. USA 114, 8265–8270 (2017).
Paul, A., Samantray, S., Anteghini, M. & Strodel, B. Thermodynamics and kinetics of the amyloid-β peptide revealed by Markov state models based on MD data in agreement with experiment. Preprint at https://doi.org/10.1101/2020.07.27.223487 (2020).
Bowman, G. R. & Pande, V. S. Protein folded states are kinetic hubs. Proc. Natl Acad. Sci. USA 107, 10890–10895 (2010).
Dill, K. A. & Chan, H. S. From Levinthal to pathways to funnels. Nat. Struct. Biol. 4, 10–19 (1997).
Abraham, M. J. et al. GROMACS: high performance molecular simulations through multi-level parallelism from laptops to supercomputers. SoftwareX 1–2, 19–25 (2015).
Bussi, G., Donadio, D. & Parrinello, M. Canonical sampling through velocity-rescaling. J. Chem. Phys. 126, 014101 (2007).
Berendsen, H. J. C., Postma, J. P. M., van Gunsteren, W. F., DiNola, A. & Haak, J. R. Molecular dynamics with coupling to an external bath. J. Chem. Phys. 81, 3684–3690 (1984).
MacKerell, A. D. et al. All-atom empirical potential for molecular modeling and dynamics studies of proteins. J. Phys. Chem. B 102, 3586–3616 (1998).
Jorgensen, W. L., Chandrasekhar, J., Madura, J. D., Impey, R. W. & Klein, M. L. Comparison of simple potential functions for simulating liquid water. J. Chem. Phys. 79, 926–935 (1983).
Essmann, U. et al. A smooth particle mesh Ewald method. J. Chem. Phys. 103, 8577–8593 (1995).
Hess, B. P-LINCS: a parallel linear constraint solver for molecular simulation. J. Chem. Theory Comput. 4, 116–122 (2008).
Vanommeslaeghe, K. et al. CHARMM general force field: a force field for drug-like molecules compatible with the CHARMM all-atom additive biological force fields. J. Comput. Chem. (2009); https://doi.org/10.1002/jcc.21367
Chollet, F. Keras (2015).
Abadi, M. et al. TensorFlow: a system for large-scale machine learning. In Proc. 12th USENIX Conference on Operating Systems Design and Implementation 265–283 (USENIX Association, 2016).
Head, T. et al. Scikit-Optimize/Scikit-Optimize: V0.5Rc1 (Zenodo, 2018); https://doi.org/10.5281/ZENODO.1157319
Tribello, G. A., Bonomi, M., Branduardi, D., Camilloni, C. & Bussi, G. PLUMED 2: new feathers for an old bird. Comput. Phys. Commun. 185, 604–613 (2014).
PLUMED Consortium Promoting transparency and reproducibility in enhanced molecular simulations. Nat. Methods 16, 670–673 (2019).
Noé, F. & Clementi, C. Kinetic distance and kinetic maps from molecular dynamics simulation. J. Chem. Theory Comput. 11, 5002–5011 (2015).
Arthur, D. & Vassilvitskii, S. k-means++: the advantages of careful seeding. In Proc. Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms 1027–1035 (Society for Industrial and Applied Mathematics, 2007).
Beauchamp, K. A. et al. MSMBuilder2: modeling conformational dynamics on the picosecond to millisecond scale. J. Chem. Theory Comput. 7, 3412–3419 (2011).
Trendelkamp-Schroer, B., Wu, H., Paul, F. & Noé, F. Estimation and uncertainty of reversible Markov models. J. Chem. Phys. 143, 174101 (2015).
Löhr, T., Kohlhoff, K., Heller, G. T., Camilloni, C. & Vendruscolo, M. A Kinetic Ensemble of the Alzheimer’s Aβ Peptide (Zenodo, 2020); https://doi.org/10.5281/zenodo.4247321, https://doi.org/10.1101/2020.05.07.082818
Acknowledgements
G.T.H. is supported by the Rosalind Franklin Research Fellowship at Newnham College. We thank S. Kearnes for useful feedback and suggestions for the manuscript and M. Bonomi for useful discussions and suggestions. We also acknowledge support from Google and the Google Accelerated Science team for providing access to the Google Cloud Platform for simulations and analysis.
Author information
Authors and Affiliations
Contributions
T.L., K.K., G.T.H., C.C. and M.V. designed the research plan. T.L. and K.K. performed the simulations. T.L., K.K., G.T.H., C.C. and M.V. analyzed the data. T.L., K.K., G.T.H., C.C. and M.V. wrote the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Peer review information Fernando Chirigati was the primary editor on this article and managed its editorial process and peer review in collaboration with the rest of the editorial team. Nature Computational Science thanks Christopher Lockhart, Andreas Mardt and Fanjie Meng for their contribution to the peer review of this work.
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data
Extended Data Fig. 1 Relaxation (implied) timescales for conventional discrete-state Markov state models, showing inability to construct a model with converging timescales.
a, Relaxation timescales for a 200-microstate model as a function of model lag time. b-d, Relaxation timescales for hidden Markov state models using 2, 4, and 6 output states respectively, as a function of model lag time. e-f, Relaxation timescales for 4 and 6-state Markov state models built using Perron cluster-cluster analysis (PCCA) from the 200-microstate model as a function of model lag time. The gray shading indicates the timescales for which the Koopman model can no longer resolve the relaxation timescales. Shaded areas indicate 95% confidence intervals of the sample mean of the 20 models.
Extended Data Fig. 2 Equilibrium distributions of the models.
a, 2, (b) 3, (c) 4, (d) 5, and (e) 6-state model equilibrium distributions. The whiskers, boxes and horizontal lines indicate 95th percentiles, quartiles, and the median values over all 20 models, respectively, the labels show the mean model values.
Extended Data Fig. 3 Relaxation (implied) timescales as a function of model lag time.
a, 2, (b) 3, (c) 5, and (d) 6-state model timescales. The gray shading indicates the timescales for which the Koopman model can no longer resolve the relaxation timescales. Shaded areas indicate 95th percentiles of the sample mean over 20 models.
Extended Data Fig. 4 Experimental validation and comparison to an existing ensemble.
a, Root-mean-square deviations between experimentally determined NMR chemical shifts and those back calculated using CamShift49; the deviations are smaller than the intrinsic CamShift errors. b, Comparison of the probability distributions of the radius of gyration computed for the current Markov state model (MSM, green) and the previously performed metadynamic metainference simulations (MI, purple)31.
Supplementary information
Supplementary Information
Supplementary Figs. 1–12.
Source data
Source Data Fig. 2
Statistical source data.
Source Data Fig. 3
Statistical source data.
Source Data Fig. 4
Statistical source data.
Source Data Extended Data Fig. 1
Statistical source data.
Source Data Extended Data Fig. 2
Statistical source data.
Source Data Extended Data Fig. 3
Statistical source data.
Source Data Extended Data Fig. 4
Statistical source data.
Rights and permissions
About this article
Cite this article
Löhr, T., Kohlhoff, K., Heller, G.T. et al. A kinetic ensemble of the Alzheimer’s Aβ peptide. Nat Comput Sci 1, 71–78 (2021). https://doi.org/10.1038/s43588-020-00003-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s43588-020-00003-w
This article is cited by
-
Stabilization of pre-existing neurotensin receptor conformational states by β-arrestin-1 and the biased allosteric modulator ML314
Nature Communications (2023)
-
An atlas of amyloid aggregation: the impact of substitutions, insertions, deletions and truncations on amyloid beta fibril nucleation
Nature Communications (2022)
-
Kinetics of amyloid β from deep learning
Nature Computational Science (2021)