Computer Simulationsof DNA Topology at the

atomic level

Sarah HarrisPolymer IRC

School of Physics and AstronomyLeeds University

Introduction

1) An introduction to biological thermodynamics

2) A description of Molecular Dynamics using DNAas a working example

3) DNA supercoiling

4) Peptide aggregation into amyloid-like fibrils

The Structure of Duplex DNA

N

N

N

N

HN

O

H

H

H

N

O

NH

N

Guanine Cytosine

N

N

HNN

N

H O

H N

N

O

Adenine Thymine

CH3

2 H-bonds

3 H-bonds

Majorgroove

Minorgroove

Helix diameter 20Å

Hel

ical

rep

eat

10 b

ase

pair

s / 3

4Å

Classical (MD) with AMBER

-

-

-

-

-

-

H-Bonds

CovalentBonds

ElectrostaticRepulsion

Van DerWaalsforces

( )

( )

[ ]

!

!

!

!

!

+""

#

$

%%

&

'

((

)

*

++

,

-.

((

)

*

++

,

-

+.+

+.+

.=

ChargesPartial

Atoms

612

Dihedrals

Angles

2

Bonds

2

Total

cos(12

ij

ji

ij

ij

ij

ij

ij

eq

eqr

r

qq

r

R

r

R

V

K

rrKU

/

/

012

333

Bonds

Angles

Dihedrals

Van derWaals

Electrostatics

MD represents all of the atoms as classical “balls” possessing atomtypes which mimic the chemistry of the biomolecule.

Contents of the Simulation Cell

iii) Enough water molecules tosurround the macromolecule.

The simulation cell contains:

Implicit solvent models cannot reproduce structural waterand also neglect friction and solvent damping

i) The DNA

ii) Sufficient counterions(Na+/K+) to neutralise the

system.

The Spine of Hydration

Different DNA sequences have subtlydifferent structures.

For example ~ a run of AT bases willgive the DNA a particularly narrowminor groove ~ this is responsible forthe “Spine of Hydration”

The precise position of chemicalgroups (ie H-bonds) also depends onthe DNA sequence.

The spine of hydration in A-tract DNA

DNA Dynamics

Global, large amplitude quasi-harmonicoscillations over ns timescales.

High frequency, low amplitude localmodes over ps timescales

DNA is a highly flexible molecule with hybrid solid/liquid likecharacteristics

Topoisomerase

DNA gyrase

Supercoiled DNA

RelaxedDNA

DNA Supercoiling and Packing

In chromosomes, 1-20cm of DNA is densely packaged into only~1-10μm

Bacterial DNA is compacted by supercoiling the circular plasmidsinto higher order helical structures.

Cellular DNA is generally under-wound ~ only thermophiles containenzymes which can over-wind DNA

Topoisomerases

Topoisomerases controlsupercoiling by changingthe number of times thetwo strands wrap around

each other

Energy (ATP) isrequired to increase

levels of torsional stressand induce supercoiling

Topoisomerase I (single strand breaks)

Looped and Circular DNA Structures

The ability of DNA to form eitherlooped or circular DNA structuresdepends on the bending rigidity of

the duplex.

DNA circles as small as 89base pairs have been reported

experimentally.

The CAP/DNA complex1. Cloutier T. E. & Widom J. (2005)

PNAS 102, 3645-3650.2. Du at al (2005) PNAS 102, 5397-

5402.

Simulations of 90bp DNA Circles

8 helical turnsAlternating AT

A circle containing ~ 90bp + solvent consists of ~150,000 atoms1ns takes ~ 48hrs on 32 processors of the Oxford NGS node.

90bp of DNA possesses 8.5 helical turns when torsionally relaxed

~ 100Å

Undertwisted DNA Circles

When the DNA is under-wound by 2.5 turns the DNA locallymelts to relieve torsional stress

6 helical turns: Alternating AT

Overtwisted DNA Circles

DNA overtwisted by 1.5 turns undergoes a buckle instability toform a supercoiled structure which relaxes when one strand is

nicked.

10 helical turns: Alternating GC

Supercoiling occurs spontaneously as it is thermodynamically favourable

Twisting, Supercoiling and Transcription

Under-winding is thought topromote cellular processes which

require strand separation (egtranscription, translation,

replication) by destablising theduplex.

Bacteria placed under environmental stress respond using dramaticchanges in the levels of supercoiling to alter gene expression.

Melting

Untwisting

Untwisting critically weakensthe van der Waals interactionsbetween stacked bases and lead

to strand separation

Destabilisation and the Helical Repeat

Under-wound DNA structures denaturewhenever the minor groove is on theinside of the circle.

Overwound circles preferentiallydenature at the apices of the supercoil

Bending the DNA into a circular or supercoiled structure breaks thehelical symmetry of the linear molecule.

90 base pairs-1.5 helical

turns

+1.5 turnsCould this be used to in vivo tofine tune genetic control?

The Importance of the Environment 1

Under-winding lead to negative supercoiling only for larger circles(>118 base pairs) in high salt conditions (1M)

Reducing the salt to0.01M leads to

rapid unwindingdue to electrostaticrepulsion between

crossed DNAstrands

178 base pairs at -2 turns in 1M salt

Simulations use an implicit solvent model

~170Å

The Importance of the Environment 2

The 178 base paircircles at 1M saltforms a supercoilwith two crossing

points.

When the salt isreduced to 0.01M

the circle refolds toform a structure

containing only onecrossing point.

178 base pairs over-wound by 2 helical

turns in 1M salt

“Frustrated” DNA Circles

Supercoiling relievestorsional stress but

requires sharp bendsat the apices

This bending stressis larger for smaller

circles

118 base pairs at -2 helical turns in

1M salt

These terms almostbalance for circles of

118 base pairs.

This system undergoes large thermal fluctuations between circular andsupercoiled structures

~ 130Å

A Phase Diagram for Nanocircle Topology 1

C

S+D

C

D

S+

C

0.0

0.1

0.2

0.3

-0.1

-0.2

-0.3

100 200

SuperhelicalDensity

Circle Size (bp)

C/D

Low Salt

A Phase Diagram for Nanocircle Topology 2

S+S++

S-

0.0

0.1

0.2

0.3

-0.1

-0.2

-0.3

100 200

SuperhelicalDensity

Circle Size(bp)

C S-

DNot investigated

C/D

C

DNot investigated

C

High Salt

Thermodynamics and Supercoiling

Supercoiling is easier when:

i) The circles are larger ~ bending energy at theapices is reduced (end loops can be larger)

ii) The salt concentration is high ~ better electrostaticscreening

iii) Circles are over-wound rather than under-wound

Can we use our theoretical methods to quantify thechanges in enthalpy and entropy when supercoils form?

Harris, Laughton, Liverpool Nucleic Acids Res. 36, 21-29, 2008

Liverpool, Harris, Laughton Phys. Rev. Lett. 100, 238103, 2008

Thermodynamics from Experiment

Experimental measurements show that 178 base pairsnegatively supercoils in high salt, but that topoisomerase can

only supercoil sequences larger than 148 base pairs1

Larger DNA sequences supercoil more easily experimentally2,as we also observe in the simulations

Future calculations aim to make a quantitative comparison

1. Bates, A.D. and Maxwell, A. (1989) DNA Gyrase Can Supercoil DNA Circles as Small as 174Base-Pairs. EMBO, 8, 1861-1866.

2. Fogg, J.M., Kolmakova, N., Rees, I., Magonov, S., Hansma, H., Perona, J.J. and Zechiedrich, E.L.(2006) Exploring writhe in supercoiled minicircle DNA. J. Phys. Cond. Mat., 18, S145-S159

Calculating the Enthaply Change

The enthalpy change should be straightforward toestimate from the molecular dynamics forcefield.

Can we estimate the relative contributions frombending/crossing electrostatics?

Usually, we use GB/SA – but may have to use moreexpensive PB methods due to the high salt

concentration.

But ~ what happens when there is ordered water?

Quasiharmonic analysis calculates a covariance matrix of atomicpositions and then diagonalises it to find the eigenvectors andeigenvalues.

The entropy is given by:

!"

=

+=63

1

2

ln2

1

0

N

i

iTotalTotalxkSS

The Schlitter entropy: !"

=##$

%&&'

(+=

63

1

2

2

2

1ln2

1N

i

i

kTekS )

h

Can these methods be extended to systems as large asnanocircles?

The Entropy from Quasiharmonic Analysis

Current Work ~ “Programmable” Sequences

GCGC…

GGGG…

As might be expected, the alternating GCGC…sequences (which are less rigid than GGGG…)

lie at the apices of the writhed circle

Current Work ~ Mixed Sequences

AT rich regions have atendency to denature

during writhing ~ is thecorrect for these

superhelical densities ~or an artefact of our

equilibration protocols?

Trial run in explicit solvent ~ 1million atoms ~128 processors

on Hector supercomputer

Some Concluding remarks…

Entropy/dynamics/physics is important in determining thethermodynamic and biomechanical properties of

biomolecules as well as internal energy/structure/chemistry

Thermal fluctuations can produce interesting effects (egentropic allostery, frustration) at the nanoscale that have no

macroscopic analogue.

The lowest free energy state is often determined by a delicatebalance of large but compensating terms.

This enables biomolecules to be highly responsive to changesin the environment ~ as is necessary for living systems

Acknowledgements

DNA StretchingZara Sands (AstraZenica)Pavel Hobza (Chemistry, Prague)Jan Rezac (Chemistry, Prague)BBSRC

DNA CirclesJon MitchellTannie Liverpool (Maths, Bristol)Ramin Golestanian (Physics, Sheffield)Agnes Noy (Scientific Park, Barcelona)Tony Maxwell (Biological Chemistry, Norwich)Andy Bates (Biological Sciences, Liverpool)EPSRC

GeneralAlan Real

(Supercomputing,Leeds)

The National GridService

Entropic AllosteryCharlie Laughton (Pharmacy, Nottingham)Mark Searle (Chemistry, Nottingham)Modesto Orozco (Scientific Park, Barcelona)Tom McLeish (Physics, Leeds)Hedvika Toncrova (Physics, Leeds)BBSRC

Computational BiophysicsCurrent Projects

i) MD Simulations of Amyloid-like aggregates Josh Berryman Sheena Radford

Binbin Liu Amalia Aggeli/Tom McLeish

ii) DNA mechanics, topology and packagingJon Mitchell Charlie Laughton/Tannie Liverpool

iii) Multi-scale models of charge transfer through DNADavid Reha William Barford

iv) Melting kinetics of RNA aptamers Jonathan Westwood Peter Stockley

v) Calculation of protein dielectric constantsGeorge Patargias John Harding