MotivationExample: Src SH3 protein

Two-dimensional embedded free-energy landscape from Isomap and its gradient [2]

Manifold description

Manifold processing from scattered points forMolecular Simulations: A two-pendulum system

D. Millán, A. Rosolen, M. Arroyo| Laboratori de Càlcul Numèric | Universitat Politècnica de Catalunya

[1] N. Hori, G. Chikenji, S. Berry, and S. Takada, Folding energy landscape and network dynamics ofsmall globular proteins, PNAS, 106(1):73-78 (2009)[2] P. Das, M. Moll, H. Stamati, L. Kavraki, and C. Clementi, Low-dimensional, free-energy landscapes ofprotein-folding reactions by nonlinear dimensionality reduction, PNAS, 103(26):9885-9890(2006)[3] M. Arroyo and M. Ortiz, Local maximum-entropy approximation schemes: a seamless bridge betweenfinite elements and meshfree methods, Int. J. Numer. Meth. Engng., 65: 2167-2202 (2006)R

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For the last years, and probably for the foreseeable future, in the area of Biophysics and Biochemistry theresearch efforts have been predominantly directed towards the study of the transitions betweenconformations of large macro-molecules. We intend to develop a computational data-driven method toautomatically detect the dominant nonlinear reaction coordinate pathways and to compute useful quantities(e.g. free energy) along these trajectories allowing us to link experimental observables with simulationresults. This method avoids any kind of global representation of the reaction coordinates, and importantingredients are: (1) the automatic detection of the local geometric structure of the manifold by statisticallearning methods, (2) its local parameterization with smooth meshfree approximants defined on the output ofthe first step, (3) computing free energies along this manifold with enhanced sampling techniques. Similarideas can be applied to meshfree thin shell analysis.

Abst

ract

Two-pendulum systemRelated work

• Meshfree thin-shells analysis from scattered set of nodes

• Stiffness of the springs is very large, fixed temperature• Essential dynamics is described by

Structure Ensembles found by Principal Component Analysis (PCA) [1]

1º embedded dimension

2º

emb

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d di

men

sion

(θ1,θ2)• Non-linear local parameterizations

Parametric space from Isomap

Free-energy landscape at constante temperature

Configurational space from a Molecular Simulation

Embeddeddimensionestimates

Uniform load Original Punctual load Complex geometry

Local parameterization of around

Approximate patch

Numerical tangent plane

Local parametric space

• Numerical representation of a smooth d-manifold • Local description of the Euclidian structure by statistical linear/nonlinear learning methods

Parametric space from local weighted PCA

Physicalspace

Smoothmeshfree [3] approximants

θ1

2θ

1m

k1(θ1)

k2(θ2)

U(θ1,θ2)

2m

ExternalPotential