Deep learning enhanced Markov State Models (MSMs) Wei Wang Feb 20, 2019

Outline General protocol of building MSM Challenges with MSM VAMPnets Time-lagged auto-encoder

Revisit the protocol of building MSM

Need a lot of expertise in biology & machine learning Wang, Cao, Zhu, Huang WIREs Comput. Mol. Sci., e1343, (2017)

Criterion to choose a model: slowest dynamics Choose the MSM that best captures the slowest transitions of the system Wang, Cao, Zhu, Huang WIREs Comput. Mol. Sci., e1343, (2017)

Choose the one with slowest transition Timescales (μs) Da, Pardo, Xu, Zhang, Gao, Wang, Huang,  Nature Communications., 7, 11244, (2016)

Perform this cumbersome work: search Propose good clustering algorithms & features Parametric search using good strategies http://msmbuilder.org/osprey/1.1.0

Challenges: parametric space is too large: Collective Variable (CV) Need to propose good features http://homepages.laas.fr/jcortes/algosb13/sutto-ALGO13-META.pdf

Challenges: parametric space is too large: CV Need to propose good features http://homepages.laas.fr/jcortes/algosb13/sutto-ALGO13-META.pdf

Challenges: parametric space is too large: CV Need to propose good features, otherwise will worsen the clustering stage Truth tICA Wehmeyera and Noe, J. Chem. Phys. 148, 241703 (2018)

Challenges: parametric space is too large: clustering Zhang et al., Methods in Enzymology, 578, 343-371 (2016)

Essence of these operations Linearlly/Nonlinearlly transform the protein configurations into the state vectors: 𝒙 𝑡 𝟑𝑵 → 𝑝 1 , 𝑝 2 ,…, 𝑝 𝑀 , 𝑗=1 𝑀 𝑝 𝑀 =1 (1, 0, 0, 0) (0, 0, 1, 0) Husic and Pande, J. Am. Chem. Soc. 2018, 140, 2386−2396

Deep learning can greatly help: powerful In the mathematical theory of artificial neural networks, the universal approximation theorem states that a feed-forward network with a single hidden layer containing a finite number of neurons can approximate continuous functions on compact subsets of Rn, under mild assumptions on the activation function. Deep learning has been widely applied in numerous fields Dog: 0.99 Cat: 0.01 https://en.wikipedia.org/wiki/Universal_approximation_theorem

Deep learning can greatly help MSM Dog: 0.99 Cat: 0.01 Macro1: 0.990 Macro2: 0.005 Macro3: 0.005

Outline General protocol of building MSM Challenges with MSM VAMPnets Time-lagged auto-encoder

VAMPnets for deep learning of molecular kinetics VAMPnets: employ the variational approach for Markov processes (VAMP) to develop a deep learning framework for molecular kinetics using neural networks, encodes the entire mapping from molecular coordinates to Markov states, thus combining the whole data processing pipeline in a single end-to-end framework. coordinates state vector Related to the implied timescale plot, maximize it Noe et al., 9, 5, 2018, Nature Communications

Understanding VAMPnets The basic structure of neural network What is VAMP score

Basic structure of neural network

Where can we get the weights? Forward propagation Where can we get the weights?

Backpropagation to update the weights Define a objective function 𝜖= 𝑖 𝑦 𝑡𝑟𝑢𝑒 − 𝑦 𝑝𝑟𝑒𝑑 2 Weights are updated following the largest gradient direction http://www.saedsayad.com/images/ANN_4.png

Backpropagation to update the weights https://independentseminarblog.files.wordpress.com/2017/12/giphy.gif

Backpropagation to update the weights In VAMPnets, it is VAMP-2 score Define a objective function 𝜖= 𝑖 𝑦 𝑡𝑟𝑢𝑒 − 𝑦 𝑝𝑟𝑒𝑑 2 Weights are updated following the largest gradient direction http://www.saedsayad.com/images/ANN_4.png

VAMP-2 score: objective function 𝜒(𝑥): state vector, e.g., 𝜒 𝑥 =(0,1,0) if x belongs to state 2 Noe et al., 9, 5, 2018, Nature Communications

VAMP-2 score: related to TPM Sum of eigenvalues of T(𝜏) 2 Related to the implied timescale plot, we want to maximize it 𝜒(𝑥): state vector, e.g., 𝜒 𝑥 =(0,1,0) if x belongs to state 2 Noe et al., 9, 5, 2018, Nature Communications

VAMPnets: example on alanine dipeptide Try to lump to 6 states Output: 6 probabilities 10 heavy atoms xyz for 10 heavy atoms Noe et al., 9, 5, 2018, Nature Communications

VAMPnets: example on alanine dipeptide Visualizing the outputs (soft assignments) Once we have the state vectors, we can calculate TPM, and get the kinetics Noe et al., 9, 5, 2018, Nature Communications

Comparison with traditional way to build MSM Advantages No need to worry about features to do tICA and the clustering algorithms Inputs are simple: aligned trajectories Find the variationally optimal one Disadvantages Easy to overfit the data Easy to be trapped in local optimal Alanine dipeptide Noe et al., 9, 5, 2018, Nature Communications

Outline General protocol of building MSM Challenges with MSM VAMPnets Time-lagged auto-encoder

Other application of deep learning in MSM: CV Improve PCA/tICA through nonlinear transformation trained by (time-lagged) auto-encoder PCA/tICA: find the direction that maximizes the variance/time- lagged covariance matrix.

PCA: minimizing reconstruction error http://alexhwilliams.info/itsneuronalblog/2016/03/27/pca/

PCA: Linear version of auto-encoder Original data Reconstructed data  Wehmeyer and Noe, J. Chem. Phys. 148, 241703 (2018)

Improving tICA using time-lagged auto-encoder D,E are constant matrix in tICA Current frame Next frame  Wehmeyer and Noe, J. Chem. Phys. 148, 241703 (2018)

Improving tICA using time-lagged auto-encoder D,E are constant matrix in tICA 𝝉=𝟑  Wehmeyer and Noe, J. Chem. Phys. 148, 241703 (2018)

Time-lagged autoencoder improves over tICA Villin  Wehmeyer and Noe, J. Chem. Phys. 148, 241703 (2018)

Summary Deep learning improves MSM in reducing the number of prior knowledge However, deep learning may overfit the data when our sampling is not enough