Monte Carlo study of DNA condensation phenomena

Russell HansonBiology Graduate Student Association

SymposiumOct 1, 2004

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Outline

• Overview

• Poisson-Boltzmann equation

• Metropolis Monte Carlo

• Theoretical models: DNA condensation

• Experimental and theoretical results

• Conclusions and future directions

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Origin of Poisson-Boltzmann equation

k

kk

πρ

)(4

)(2 rr

Gauss’ law

02

4)cos(

qqkdqk

r

dAqk eeeE

Poisson equation

combine…

)(4

)(11

12

ii

i

i i

i qqrr

rrr

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Boltzmann distribution law)/()( TkW Bie r

For positively and negatively charged ions the work functions are:

)()(1 rr ceW )()(2 rr ceW

)/()( Tke BcMeM r )/()( Tke BcMeM r

Let ionic strengths, i.e. number of positive and negative ions, be equal M+=M-. +ec is a charge.

))(

sinh(2)( )/()()/()(

Tk

eMeeMeeMe

B

cc

Tkec

Tkec

BcBcr

r rr

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Poisson-Boltzmann equation

Tk

eMe

B

cc )(sinh

8)(4)(2 rrr

iii

B

c

c

B qTk

e

e

Tk)(4

)(sinh)())()(( 2 rr

rrrr

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Metropolis scheme

Trial move

Calculate energy

deltaE = Enew - Eold

if Enew < Eold

move

else Enew > Eold

if random variable[0:1] < exp(-deltaE/kT)

move

else

reject move

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Monte Carlo example using Metropolis condition for 1D Ising system: average

energy increasesas temperature increases

-120

-100

-80

-60

-40

-20

0

0.1

0.35 0.

60.

85 1.1

1.35 1.

61.

85 2.1

2.35 2.

62.

85

Temperature

Ave

rag

e E

ner

gy

ji

jiiji

i ssJsHE,2

1

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Theoretical models

• Mean field approach (Poisson-Boltzmann and Debye-Hückel)

• Explicit ions (Lyubartsev)

• Hydration (Rau and Parsegian)

• Cylindrical models:– line charge, helical, cylinder with major and

minor grooves, helicity involved in observed attraction

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Error in PB greater than 1% in close-packing region

(Petrov, 2004)

(Lyubartsev et al., 1995)

PM := Primitive model of DNA in cylindrical shell

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Trigonal and hexagonal cylindrical packing

(Lyubartsev et al., 1995)

Trigonal packing

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Theoretical and experimental results

(Rau & Parsegian, 1992)

(Lyubartsev et al., 1995)

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Theoretical and experimental results (p2)

Energy landscapes for interaction between two crossed B-DNA-like helices. Plotted interaxial angle ψ and difference in about-axis rotation angles φ2-φ1. (Kornyshev et al., 2000)

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Conclusions and future directions

• Force field improvement for DNA condensation in presence of ions

• Polymer physics of viral capsid DNA folding

• Future modeling(LaMarque et al., 2003)