Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

A kinetic ensemble of the Alzheimer’s Aβ peptide

A preprint version of the article is available at bioRxiv.

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

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Illustration of a kinetic ensemble of a protein and the training methodology.
Fig. 2: Determination of the states in the kinetic ensemble of Aβ42.
Fig. 3: Structural properties of Aβ42 in the kinetic ensemble.
Fig. 4: Populations and mean first-passage times in the kinetic ensemble of Aβ42.

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

  1. Heller, G. T., Sormanni, P. & Vendruscolo, M. Targeting disordered proteins with small molecules using entropy. Trends Biochem. Sci. 40, 491–496 (2015).

    Article  Google Scholar 

  2. 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).

    Article  Google Scholar 

  3. Sormanni, P. et al. Simultaneous quantification of protein order and disorder. Nat. Chem. Biol. 13, 339–342 (2017).

    Article  Google Scholar 

  4. Berman, H. M. et al. The Protein Data Bank. Nucleic Acids Res. 28, 235–242 (2000).

    Article  Google Scholar 

  5. Bonomi, M., Heller, G. T., Camilloni, C. & Vendruscolo, M. Principles of protein structural ensemble determination. Curr. Opin. Struct. Biol. 42, 106–116 (2017).

    Article  Google Scholar 

  6. 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).

    Article  Google Scholar 

  7. Fraser, J. S. et al. Accessing protein conformational ensembles using room-temperature X-ray crystallography. Proc. Natl Acad. Sci. USA 108, 16247–16252 (2011).

    Article  Google Scholar 

  8. Bonomi, M. & Vendruscolo, M. Determination of protein structural ensembles using cryo-electron microscopy. Curr. Opin. Struct. Biol. 56, 37–45 (2019).

    Article  Google Scholar 

  9. Van Kampen, N. G. Stochastic Processes in Physics and Chemistry (Elsevier, 2007); https://doi.org/10.1016/B978-0-444-52965-7.X5000-4

  10. Prinz, J.-H. et al. Markov models of molecular kinetics: generation and validation. J. Chem. Phys. 134, 174105 (2011).

    Article  Google Scholar 

  11. Chodera, J. D. & Noé, F. Markov state models of biomolecular conformational dynamics. Curr. Opin. Struct. Biol. 25, 135–144 (2014).

    Article  Google Scholar 

  12. Husic, B. E. & Pande, V. S. Markov state models: from an art to a science. J. Am. Chem. Soc. 140, 2386–2396 (2018).

    Article  Google Scholar 

  13. 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).

    Article  Google Scholar 

  14. 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).

    Article  Google Scholar 

  15. 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).

  16. Kohlhoff, K. J. et al. Cloud-based simulations on Google Exacycle reveal ligand modulation of GPCR activation pathways. Nat. Chem. 6, 15–21 (2014).

    Article  Google Scholar 

  17. 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).

    Article  Google Scholar 

  18. Klus, S. et al. Data-driven model reduction and transfer operator approximation. J. Nonlinear Sci. 28, 985–1010 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  19. Mardt, A., Pasquali, L., Wu, H. & Noé, F. VAMPnets for deep learning of molecular kinetics. Nat. Commun. 9, 5 (2018).

    Article  Google Scholar 

  20. 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).

  21. Cummings, J., Lee, G., Ritter, A., Sabbagh, M. & Zhong, K. Alzheimer’s disease drug development pipeline: 2019. Alzheimers Dement. 5, 272–293 (2019).

    Article  Google Scholar 

  22. Hardy, J. A. & Higgins, G. A. Alzheimer’s disease: the amyloid cascade hypothesis. Science 256, 184–185 (1992).

    Article  Google Scholar 

  23. 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).

    Article  Google Scholar 

  24. Jack, C. R. et al. NIA-AA Research Framework: toward a biological definition of Alzheimer’s disease. Alzheimers Dement. 14, 535–562 (2018).

    Article  Google Scholar 

  25. 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).

    Article  Google Scholar 

  26. 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).

    Article  Google Scholar 

  27. Michaels, T. C. T. et al. Thermodynamic and kinetic design principles for amyloid-aggregation inhibitors. Proc. Natl Acad. Sci. USA 117, 24251–24257 (2020).

    Article  Google Scholar 

  28. Meng, F. et al. Highly disordered amyloid-β monomer probed by single-molecule FRET and MD simulation. Biophys. J. 114, 870–884 (2018).

    Article  Google Scholar 

  29. 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).

    Article  Google Scholar 

  30. 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).

    Article  Google Scholar 

  31. 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).

    Article  Google Scholar 

  32. Nasica-Labouze, J. et al. Amyloid β protein and Alzheimer’s disease: when computer simulations complement experimental studies. Chem. Rev. 115, 3518–3563 (2015).

    Article  Google Scholar 

  33. Granata, D. et al. The inverted free energy landscape of an intrinsically disordered peptide by simulations and experiments. Sci. Rep. 5, 15449 (2015).

    Article  Google Scholar 

  34. Zimmerman, M. I. & Bowman, G. R. FAST conformational searches by balancing exploration/exploitation trade-offs. J. Chem. Theory Comput. 11, 5747–5757 (2015).

    Article  Google Scholar 

  35. 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).

    Article  Google Scholar 

  36. 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).

    Article  Google Scholar 

  37. 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

  38. McGibbon, R. T. & Pande, V. S. Variational cross-validation of slow dynamical modes in molecular kinetics. J. Chem. Phys. 142, 124105 (2015).

    Article  Google Scholar 

  39. 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).

    Article  Google Scholar 

  40. 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).

    Article  MathSciNet  MATH  Google Scholar 

  41. Wu, H. & Noé, F. Variational approach for learning Markov processes from time series data. J. Nonlinear Sci. 30, 23–66 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  42. 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).

  43. 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).

    Article  Google Scholar 

  44. 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).

    Article  Google Scholar 

  45. Bonomi, M., Camilloni, C. & Vendruscolo, M. Metadynamic metainference: enhanced sampling of the metainference ensemble using metadynamics. Sci. Rep. 6, 31232 (2016).

    Article  Google Scholar 

  46. 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).

    Article  Google Scholar 

  47. Konrat, R. NMR contributions to structural dynamics studies of intrinsically disordered proteins. J. Magn. Reson. 241, 74–85 (2014).

    Article  Google Scholar 

  48. Dai, W., Sengupta, A. M. & Levy, R. M. First passage times, lifetimes and relaxation times of unfolded proteins. Phys. Rev. Lett. 115, 048101 (2015).

    Article  Google Scholar 

  49. 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).

    Article  Google Scholar 

  50. 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).

    Article  Google Scholar 

  51. 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).

    Article  Google Scholar 

  52. 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).

    Article  Google Scholar 

  53. 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).

    Article  Google Scholar 

  54. 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).

  55. 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).

  56. 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).

    Article  Google Scholar 

  57. 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).

    Article  Google Scholar 

  58. 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).

  59. Bowman, G. R. & Pande, V. S. Protein folded states are kinetic hubs. Proc. Natl Acad. Sci. USA 107, 10890–10895 (2010).

    Article  Google Scholar 

  60. Dill, K. A. & Chan, H. S. From Levinthal to pathways to funnels. Nat. Struct. Biol. 4, 10–19 (1997).

    Article  Google Scholar 

  61. Abraham, M. J. et al. GROMACS: high performance molecular simulations through multi-level parallelism from laptops to supercomputers. SoftwareX 1–2, 19–25 (2015).

    Article  Google Scholar 

  62. Bussi, G., Donadio, D. & Parrinello, M. Canonical sampling through velocity-rescaling. J. Chem. Phys. 126, 014101 (2007).

    Article  Google Scholar 

  63. 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).

    Article  Google Scholar 

  64. 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).

    Article  Google Scholar 

  65. 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).

    Article  Google Scholar 

  66. Essmann, U. et al. A smooth particle mesh Ewald method. J. Chem. Phys. 103, 8577–8593 (1995).

    Article  Google Scholar 

  67. Hess, B. P-LINCS: a parallel linear constraint solver for molecular simulation. J. Chem. Theory Comput. 4, 116–122 (2008).

    Article  Google Scholar 

  68. 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

  69. Chollet, F. Keras (2015).

  70. 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).

  71. Head, T. et al. Scikit-Optimize/Scikit-Optimize: V0.5Rc1 (Zenodo, 2018); https://doi.org/10.5281/ZENODO.1157319

  72. 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).

    Article  Google Scholar 

  73. PLUMED Consortium Promoting transparency and reproducibility in enhanced molecular simulations. Nat. Methods 16, 670–673 (2019).

    Article  Google Scholar 

  74. Noé, F. & Clementi, C. Kinetic distance and kinetic maps from molecular dynamics simulation. J. Chem. Theory Comput. 11, 5002–5011 (2015).

    Article  Google Scholar 

  75. 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).

  76. Beauchamp, K. A. et al. MSMBuilder2: modeling conformational dynamics on the picosecond to millisecond scale. J. Chem. Theory Comput. 7, 3412–3419 (2011).

    Article  Google Scholar 

  77. Trendelkamp-Schroer, B., Wu, H., Paul, F. & Noé, F. Estimation and uncertainty of reversible Markov models. J. Chem. Phys. 143, 174101 (2015).

    Article  Google Scholar 

  78. 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

Download references

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

Authors

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

Correspondence to Michele Vendruscolo.

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.

Source data

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.

Source data

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.

Source data

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.

Source data

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s43588-020-00003-w

This article is cited by

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing