SelfAvoiding Random Walk and Long Polymer Molecules Spring 2005

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Self-Avoiding Random Walk and Long Polymer
MoleculesSpring 2005

• By Yaohang Li, Ph.D.
• Department of Computer Science
• North Carolina AT State University
• yaohang_at_ncat.edu

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Review

• Last Class
• Global Optimization
• Hill-Climbing Algorithms
• Metropolis Method
• Simulated Annealing
• Simulated Tempering
• Accelerated Simulated Tempering
• Parallel Tempering
• This Class
• Long Polymer Molecules
• Self-avoiding Walks
• Next Class
• Protein Structure Prediction

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Introduction

• Polymers
• Macromolecules
• very large
• thousands, sometimes even millions of times
  larger than a single water molecule
• can be seen under an electron microscope
• Nature of Polymers
• Made up of long chains of monomer units
• Connected by bonds
• Example
• DNA and RNA
• nucleotides
• Protein
• Amino acids
• Polyethylene
• CH2

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Properties of Polymers

• Hydrophobic
• The attraction between monomers is stronger than
  their attraction to the molecules of the
  surrounding solvent, e.g., water
• Hydrophilic
• The attraction between monomers is weaker than
  their attraction to the molecules of the
  surrounding solvent, e.g., water
• Non Self-intersect
• No two monomers can occupy the same place
• excluded volume

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Solvent

• Low Temperature (or in a poor solvent)
• The attractive interactions between monomers pull
  the polymer into a dense ball-like configuration
• globule
• High Temperature (or in good solvent)
• The interactions are mediated by the solvent
  molecules
• Typical configurations are open coils
• Phase Transition
• Coil-Globule transition

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Abstraction of Polymer

• Real Polymer
• the monomers occupy positions in continuous space
• bonds btw. monomers are constrained to have only
  certain angles
• depending on the nature of the monomers
• Simplification
• Embed the polymer into discrete space
• Require that the monomers exist at integer
  coordinates
• only a lattice spacing apart

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Radius

• Average size of a polymer containing n monomers
• Radius of gyration
• average distance of a monomer from the polymers
  center of mass
• ltRn2gt Anv
• v is the critical exponent
• in the swollen phase v ? 0.588
• in the collapse phase v1/3
• A is unknown
• use linear regression

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Early Solution

• Goal
• Estimate ltRn2gt
• Method
• Generate unrestricted random walks
• Accept if no interception
• Not accept if interception
• Problem
• Not efficient

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Self-avoiding Random Walk

• Self-avoiding Random Walk
• Walk on 2D or 3D lattice
• Explore the geometric properties of linear
  polymers in good solvent
• Constraint random walk (dont allow to go
  backward)
• Introduced by Orr
• Analysis of Self-avoiding Random Walks
• At first glance, the model is far too simple
• Phenomenon of universality
• Many quantities are not dependent on the specific
  details of thesystem
• They are determined only by its universality
  class
• All systems in the same universality class share
  the same dominant asymptotic behavior

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A Picture is Worth a Thousand Words
3D Walk
2D Walk
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Self-avoiding Random Walk Algorithm
include ltiostream.hgt include ltstdlib.hgt include
ltmath.hgt void do_walk (int maxstep, int nstep,
double rsquared ) const int MAXSTEP20
int map MAXSTEP2MAXSTEP20 // start
point int completed0 int x MAXSTEP
int y MAXSTEP int npoint 1
mapxy npoint do int xnewx
int ynewy switch ( (int)(4
(double)rand()/(RAND_MAX1.0)) )
case 0 xnew- 1 break case 1
xnew 1 break case 2 ynew- 1
break case 3 ynew 1 break
if ( mapxnewynew 0 )
npoint mapxnewynew npoint
x xnew y ynew
if ( npoint maxstep1 )completed1
else if ( mapxnewynew ! npoint-1 )
completed1 while (
!completed )
// Print window centred on map for ( int
i5 ilt2MAXSTEP-5 i ) for ( int j5
j lt 2MAXSTEP-5 j )
cout.width(3) cout ltlt mapij
cout ltlt endl nstep
npoint-1 rsquared pow( x-MAXSTEP,2.0)
pow( y-MAXSTEP, 2.0 ) int main() int
maxstep20,nstep double rsquared
srand(987654321) for (int i1 ilt10 i )
do_walk(maxstep,nstep,rsquared)
cout ltlt endl ltlt "Nsteps " ltltnstep ltlt " Rsquared
" ltltrsquaredltltendl return 0
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Output of Self-avoiding Random Walk
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Biased Random Walk

• Problems of self-avoiding random walk
• Have to reject many terminated walks in order to
  have unbiased statistics
• Unlikely to produce long polymer
• Inefficiency
• Biased Random Walk
• Basic Idea
• Instead of abandoning a walk when an illegal step
  is attempted, we go back and pick one of the
  possible legal steps
• Enable a walk to make a full distance

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Biased Random Walk Algorithm

• Weight Factor W(N)
• Initially 1
• 3 possibilities
• No further steps are possible, we have reached a
  dead end
• Abandon this walk
• All steps, other than going directly backwards
  are possible
• proceed as normal, set W(N) W(N-1)
• Only m steps are possible
• Randomly choose one of the possible steps
• set W(N)m/3W(N)

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Output of Biased Random Walk
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Applets for Self-avoiding Random Walks

• http//polymer.bu.edu/java/java/saw/sawapplet.html
  

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Summary

• Long Polymer Molecule
• Self-avoiding Random Walk
• Biased Random Walk

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What I want you to do?

• Review Slides
• Read the UNIX handbook if you are not familiar
  with UNIX
• Review basic probability/statistics concepts
• Work on your Assignment 4 and 5
• Prepare for your presentation topic and term paper