monte carlo study of dna condensation phenomena russell hanson biology graduate student association...
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Monte Carlo study of DNA condensation phenomena
Russell HansonBiology Graduate Student Association
SymposiumOct 1, 2004
BGSA Symposium Oct 1, 2004 2
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)