Monte Carlo Simulation of Molecular Adsorption

1
Monte Carlo Simulation of Molecular Adsorption

• Paul Van Tassel
• Dept. of Chemical Engineering
• Yale University

2
Adsorption

• Definition excess mass in interfacial region
• Examples
• Adsorbed systems tend to be
• of high overall density
• characterized by large density gradients
• inhomogeneous
• highly constrained

Macromolecules on Surface
Simple Fluid in Pore

• Theoretical methods often insufficient
• Simulations challenging as well

3
Overview of Talk

• Introduce two adsorption problems
• Protein molecule on a surface
• Fluid in a templated porous material
• Discuss Monte Carlo methods
• Histogram re-weighting method
• Multicanonical sampling
• Jump-walking steps
• Configuration bias steps
• Discuss simulation results
• Kinetics / thermodynamics of surface-induced
  protein conformational transition
• Phase co-existence in templated porous material

4
Purpose of Talk

• to illustrate the use of advanced
• (and existing!) Monte Carlo methods
• to solve challenging adsorption problems

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Two Adsorption Problems

• Adsorbed protein undergoing conformational
  transition

folded state
denatured state

• Fluid adsorbed in disordered matrix

w/o matrix
T
fluid
w/ matrix
r
matrix
Common feature phase equilibrium
6
Probability Distributions
stable
Probability
Free Energy
meta-stable
Order Parameter
Order Parameter

• Define order parameter coupling two phases
• Energy
• Number of particles
• Free Energy - kT Ln Probability
• Probability max / free energy min are stable or
  metastable states

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Protein Conformational Change
Free Energy (F)
Order Parameter Internal Energy (E)

• Complete thermodynamic information from
  F(E)E-TS(E)
• Temperature at equilibrium
• Activation free energy gt kinetics of transition
  !
• Must determine S(E)k ln W(E), W is density of
  states

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Standard Monte Carlo
Metropolis, et al, (1953)
Activation barrier
Free Energy (F)
Meta-stable region
Stable region
Internal Energy (E)

• Sampling according to probability of given state
  n
• Resultant energy histogram
• Standard Monte Carlo focuses on stable (or
  metastable) region
• With special moves, both stable and metastable
  regions sampled
• Activation barrier poorly sampled!

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Multicanonical Monte Carlo
Berg and Neuhaus (1992), Lee (1993)
Probability
Multicanonical sampling
Boltzmann sampling
Boltzmann sampling probability Boltzmann
histogram Multicanonical sampling
probability Multicanonical histogram
Internal Energy
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Multicanonical Monte Carlo
Probability
Multicanonical sampling
Boltzmann sampling
Internal Energy
Multicanonical sampling probability

• Problem We do not know the density of states (W)
  a priori !
• Solution Obtain W iteratively.

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Multicanonical Monte Carlo
Begin with sampling probability Obtain
histogram Update sampling probability After
many iterations

• Sampling probability converges to inverse of
  density of states!

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Lattice Model Protein
Shakhnovich, et al (1993)
A
Contact energies eAA eBB lt 0 native eAB
eAA / 3 non-native
B

• Protein modeled as AB copolymer
• 27 segments occupy cubic lattice sites
• Bonds of unit length
• Nearest neighbor native (A-A, B-B) and
  non-native (A-B) interactions
• Folded state is cube of 28 native contacts

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Phase Diagram
Socci and Onuchic (1995)
A
Protein considered here
B
Eavg (eAA eAB ) / 2 D eAA - eAB
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Sampling Problem
Original Monte Carlo Moves

• Non-ergotic sampling
• Certain energy states appear in only some of the
  simulations
• Solution
• Configuration Bias steps
• Jump Walking steps

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Configuration Bias MC
Siepman and Frenkel (1992), de Pablo, et al (1992)

• N terminal segments of chain discarded and
  re-grown according to probability
• Sampling probability becomes

Energy of ith segment
Energy of ith segment in kth trial position
Hao and Scheraga (1994)
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Jump Walking MC
Frantz, et al (1990)

• Move to a conformation from a randomly selected
  pool of states

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Histogram and Entropy
Bulk Protein
free energy minima

• Histograms converge to flat profile
• Inflection point in S(E) gt thermodynamic
  transition

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Equal Affinity Surfaces
Entropy
Strongly adsorbing eAS eBS eAA Weakly
adsorbing eAS eBS eAB
Free Energy
folded structure

• Continuous, energetically driven transition on
  strongly adsorbing surface
• Activated, entropically driven transition on
  weakly adsorbing surface

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A Affinity Surfaces
Entropy
Strongly adsorbing eAS eAA eBS
eAB Weakly adsorbing eAS eAB eBS
0
Free Energy
folded structure

• Activated or continuous, energetically or
  entropically driven transitions possible on both
  strongly and weakly adsorbing surfaces

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Transition Properties
System Ttr DFact ______________________________
_______ Bulk 1.37 5.8 Weak A
Affinity 0.98 2.7 Strong A Affinity 0.39 3.2 W
eak Equal Affinity 0.92 5.7

• ? transition temperature with ? surface
  attraction
• A affinity surface characterized by lowest
  activation free energy

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Adsorbed Protein

• Lattice model adsorbed protein
• Multicanonicalconfiguration biasjump walking
  Monte Carlo method
• Obtain W(E) gt S(E) gt F(E)
• Obtain information on
• Equilibria (e.g. Ttr)
• Kinetics (e.g. DFact)

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Templated Porous Materials

• Material formed by gelation in the presence of a
  removable template

monomers
template
gelled material
material following template removal
(Ex TEOS)
(Ex micelle, organic)

• Pore space mimics template
• Porosity and pore size decoupled
• Ref Raman, Anderson, Brinker, Chem. Mat. (1996)

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Adsorption Model
New Model (inspired by formation process)
Equilibrated mixture
Quenched Configuration (gelation)
Template removal
Adsorbate addition
matrix
Previous Models (without template)
template
adsorbate
References Madden, Glandt (1986) Kaminski,
Monson (1991) Ford, Glandt (1994) Kierlick,
Tarjus, Rosinberg, Stell (1994).
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Phase Diagrams

• Temperature - density phase diagrams of fluids in
  non-templated disordered porous systems

Simulation Page and Monson (1996)
Perturbation Theory Ford and Glandt (1994)
Xe in porous glass Machim (1994)
gt porous matrix suppresses phase
transition multiple transitions possible
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Question

• What is the impact of a template on the phase
  behavior of a fluid in a porous medium?
• Answer via
• liquid state theory
• Monte Carlo simulation

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Liquid State Theory
Relate total (h) and direct (c) correlation
functions 0 matrix 0 template 1 fluid
(adsorbate) Inspired by previous equations for
system without template Madden, Glandt, 1986
Given, Stell, 1993 Tarjus, Kierlik, Rosinberg,
Stell, Monson 1994, 1997
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Liquid State Theory
7 equations and 14 unknowns (7 hs and 7 cs)
(hg-1) 7 additional relations closures
potential of mean force
pair potential
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Phase Diagrams
w/ template
w/o template
gt Templating enhances phase behavior gt
ORPAB2 theory predicts intermediate phase
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Histogram Reweighting Monte Carlo
Ferrenberg and Swendsen (1988)
Probability distribution of of particles (N)
relates to chemical potential (m) via
stable
Pm(N)
meta-stable
N
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Phase Diagrams
gt Simulation and ORPAB2 theory predict
intermediate phase gt Semi-quantitative
agreement
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Coexisting Phases
matrix
adsorbed fluid
T.75
Vapor Phase density.03
Intermediate Phase density.11
Liquid Phase density.33
Intermediate phase of fluid condensed in
largest pore regions
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Conclusions

• Monte Carlo methods useful for adsorption
  problems in chemical / biomolecular engineering
• Protein molecule on a surface
• Fluid in a templated porous material
• Methods allow for
• Stable/metastable phases gt equilibrium behavior
• Free energy trajectories gt kinetic behavior
• Future applications
• Conformational transitions / folding of proteins
• Water in intra- or extra-cellular space

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Acknowledgments

• Protein problem
• Victoria Castells
• Template problem
• Linghui Zhang
• Songyin Cheng
• Jeff Potoff
• Lev Sarkisov
• Funding NIH, NSF