Physics of DNA

R. Podgornik

Laboratory of Physical and Structural Biology

National Institute of Child Health and Human Development

National Institutes of Health

Bethesda, MD

- DNA as a polyelectrolyte- electrostatic interactions- correlation effect- equation of state- fluctuation effect- DNA mesophases - orientational interactions- interactions and order- DNA elasticity- anomalous elastic moduli- DNA collapse

DNA

DNA is not the proverbial spherical cow, or in this case a cylindrical one.• it is a RH double helix

• it has lots of discrete structural (phosphate) charges (pH > 1) • it has lots of room to accommodate small counterions

helix

discrete charges

groovesCharge:

2 e0 / 3.4 Å~ e0 / nm2

Polipeptides: 0.6 eo / nm Membranes: 0.1 - 1 e0 / nm2

Structure (B-form)R. Franklin, photo 51.

• a ~ 1 nm• h(DNA) = 1.7 Å

• DNA length from 50 nm to ~ µm

The great electrostatic divide

Collective description(“N” description)

vs.Single particle description

(“1” description)

Weak coupling limit(Poisson - Boltzmann)

Ξ➝ 0

Strong coupling limit(Netz - Moreira)

Ξ➝ ∞

Z

Ratio between the Bjerrum and the Gouy - Chapman lengths. Bulk versus surface interactions.

Bjerrum length Gouy - Chapman length

Coulomb’s lawandkT

Coupling parameter

The weak coupling limit (collective description)

+

electrostatic energy

Non-equilibrium free energy = (electrostatic energy) - k (ideal gas entropy)Minimization yields the Poisson - Boltzmann equation.

ideal gas entropy minimize to get equilibrium

Screening. Debye length ~ 3.05 Å /√M

The strong coupling limit (one particle description)

+

Electrostatic energywithout mobile counterions

Electrostatic energyof a single counterion

Z+Z

Z

Z

+ …

Electrostatic energyof two counterions

Collective description vs. one particle description.

repulsion + 2 X attraction = attraction

• Oosawa derives attractive interactions between DNAs (late 60’s)

• Simulation of DLVO interactions (early 80’s -el. bilayer Torrie and Valleau (1980))

• Fundamental paper by Gulbrand, Jonsson, Wennerstrom and Linse (1984)

• ‘90 realisation of the correlation effect in DNA

• 2000-2004 quantitative theories of the correlation effect

Simulations

Hexagonal array of DNA poly-counterions: (Lyubartsev and Nordenskiold, 1995)

A pair of DNAs with poly-counterions:(Gronbech-Jensen et al. 1997)

Experiments

The Boyle experimentOsmotic pressure

Osmotic stress method (Parsegian & Rand)

Π dV −µdN

Osmotic stress method

Setting the osmotic pressure and measuring the density of DNA

Experiment vs. theory monovalent counterions

DNA in monovalent (NaCl) salt solution.Osmotic pressure for a 2D hexagonal array.

PB does not seem to be working!Osmotically stressed subphase.

Polyvalent counterions

Polyvalent counterions + NaCl at 0.25 M: • Co(NH3)6Cl3 counterion Co(NH3)6

3+ (Z = 3)• MnCl2 counterion Mn2+ (Z = 2)

Co(NH3)63+

Mn2+0mM

8mM

12mM20mM

5o

35o50o

Attraction is obviously there.Quantitative comparison still difficult.

Monovalent salt + polyvalent counterionsOsmotically stressed subphase.

Or condensed.

Last few Angstroms ....

2

9.0

8.5

8.0

7.5

7.0

6.5

6.0

log

Π [d

ynes

/cm

2]

2520151050

Surface separation, [Å]

rescaled charge density

HPC schizophyllan

Na-DNA Na-Xanthan TMA-DNA (raw)

TMA-DNA (rescaled) DDP bilayers

Commonality of forces among charged, neutral, cylindrical and planar molecules in salt solution and distilled water. Charges:DNA 1e/1.75 Å, xanthan 4 e/ 15 Å, DDP 1e/55 Å (Leikin et al.

1993)

8.2

8.0

7.8

7.6

7.4

7.2

Log[

Π] [dynes/cm2]

2.01.81.61.41.21.0

CDNA[M]

Marcelja and Radic, 1984.Perturbation of water order parameter.

Similar foces in ice.

Bjerrum defects screen polarization.(Onsager - Dupuis theory)

Conformational fluctuations Surprisingly the PB limit for finite salt does not work.

What are we missing in this picture? Orientational order…

• Lp ~ 50 nm• KC = kBT Lp

DNA is a flexible molecule.

E ~ 300 MPa (plexiglass) At room temperature big conformational fluctuations.

Conformational fluctuationsElastic energy of the DNA

Consequences:bumping into the hard wall of its nearest neighbors.

This is the Odijk interaction (1986). Similar to Helfrich interaction between surfaces.Long range interaction (short range → thermal undulations → long range)

Now assume a soft Debye - Hueckel potential:

Fluctuation renormalization of interactions! (Podgornik et al. 1989)

Conformational fluctuations

DNA in monovalent (NaCl) salt solution. Paradigmatic behavior for all monovalent salts.

Electrostatics can only be seen indirectly,as modified by the presence of

conformational fluctuations.

Renormalized value of λ:

λ(r) = 4 λD.

Factor 4 due to elasticity(fourth derivative) as well as the 1D

nature of DNA (linear polymer).

Liquid disorder!

DNA Elasticity and mesophases

µ−tubules 0.1 M NaCl 107

TMV 0.1 M NaCl 106

F actin 0.1 M NaCl 10000Schizophyllan water 200Xanthan 0.1 M NaCl 120ds-DNA 0.2 M NaCl 50Spectrin 0.1 M NaCl 15ss-DNA 0.2 M NaCl 3 Hyaluronic acid 0.2 M NaCl 1Long Alkanes 0.5

Persistence length of a semiflexible polymer

Livolant et al. (97)

cholesteric

line hexatic

E~300 Mpa (plexiglass)

Onsager’s argument valid also for polymers.

Liquid crystalline mesophases.

DNA phase diagram

A B

L

LLP

L

ββ’

α

c

β’

A B

x-ray beam

(Livolant, Leforestier, Rill, Robinson, Strzelecka, Podgornik, Strey …)

Durand, Doucet, Livolant (1992) J. Physique 2, 1769-178Pelta, Durand, Doucet, Livolant (1996) Biophys. J., 71, 48-63 3

The line hexatic phase

(Predicted by Toner, 1983)

• Long range BO order ~ 0.6 mm• Long range nematic order• Liquid like positional order, λPO

An anomalous “3D” hexatic phase!(Podgornik et al. 1999)

Why is this relevant?

(R. Cavenoff (1995)) (Kleinschmidt et al. (1962))

(D. Nelson. (1995))

E.Coli

T2P~100 atmρ~100 mg/ml

630 m long1 mm thick25 cm

Vortex lines in II sc

TensionNon-chiralMagnetic fieldTemperatureLondon repulsion

BendingChiraldensityIonic strengthDebye-Huckel repulsion

Why orthorhombic phase at high density?Realistic geometric models of DNA..

• explicit DNA structure• explicit counterions• explicit salt ions• different salt concentrations

• from R= 24 Å out• φ0 = 180 and 0 0.

Schematics of the orientational effect. Strand opposition.

A AR R R

Kornyshev - Leikin, 1998, 2000, 2002.Allahyarov et al., 2000 - 2004

• distorted hexatic phase A• 1D crystallization (1)• 2D crystallization (2a, 2b, 2c)

For the non-parallel orientation state a hexatic (hexagonal) phase becomes a distorted (orthorhombic or monoclinic) crystal! (Rosalind Franklin, 1952).

Lattice frustrations due to orientational interactions

Lattice distortions alleviate frustrations:

In a lattice the configurations are frustrated

• nearest neighbors in optimal config.• not all are happy

Hexagonal lattice

(Lorman, Podgornik , Zeks 2001)

Single chainMany chains

Single molecule physics

(Baumann, Smith, Bloomfield, Bustamante 1997)

Force curve fit to model (a 4 par fit) elastic constants

Measuring DNA elasticity

Bending and stretching

Entropic plus enthalpic Hookeian elasticity

Entropic elasticityHookeian elasticity

Small force Large force

stretchingbending external force

The experiment gives us both moduli Kc as well as λ(0). ds-DNA is not very stretchable, but it is not rigid either.

Lowering the ionic strength increases the measured persistence length, but seems to reduce DNA’s elastic stretch modulus, contradicting the elastic rod model. Bustamante et al. (2000).

KC =14 λ R

2

DNA - an Euler-Kirchhoffian filament or not?

In classical elasticity(cylindrical Euler - Kirchhoff filament)

Bending is just local stretching.Landau and Lifshitz, 1995.

Since variations in ionic strength are involved, we assume that the foul play is due to

electrostatics.

Interactions and elasticity

L

L’

a

a’

Bending rigidity

Stretching modulus

Wenner, Williams, Rouzina and Bloomfield (2002). For ionic strengths: 1000, 500, 250, 100, 53.3, 25, 10, 2.6

mM.

Podgornik et al. 2000. Rouzina (2002)

a = 6.7 ± 0.7 Å (Manning a = 7.2 Å)LP ~ 48 nm l ~ 1200 pN

A constrained fit : L0, Kc, λ(Kc)

Repulsions vs. attractionsA reminder of the DNA - DNA interactions.

Rau et al., 1997.Podgornik et al. 2000

Monovalent counterions Polyvalent counterions

Attraction energies:~ 0.1 kT/ base pair.

Correlation attractions.Hydration attractions.

Chattoraj et al. (1978).

Hud & Downing (2001)

95-185 nm35-85 nm2.4 nm

DNA condensation

T4 DNA R ~ 1000 nm to 50 nm.

Euler bucklingEuler buckling instability: Press on an elastic filament hard enough and it buckles.

The role of compressional force is played by diminished (on addition of polyvalentcounterions) electrostatic interactions. No correlation effect at that time!

(Manning, 1985.)

Kirchhoffkinematic analogy.

Manning buckling with correlation attractionsShape equation of the elastic filament (DNA):

Euler (elastic) intermediates are clearly seen also in simulations of

Schnurr et al. (2002).

toroidal

racquet-like

We understand “well” only one side of the transition. The destabilization

of the persistence length leading to a 1st order transition.

V(r-r’)

Elastic, Euler-like, states are important for DNA collapse. Stiff polymers have a differentCollapse pathway (originates in the buckling transition) then flexible polymers.

There might be a whole slew of Euler-like intermediate states that lead to DNA collapse.Much more ordered collapsed state than for flexible polymers.

Stevens BJ (2001). This collapse is very different from a flexible chain.

DNA condensation simulations

Organization of ds-DNA inside the viral capsid shows nematic or hexatic- like order with ~25 Å separation, similar to toroidal aggregates.

Cerritelli et al. (1997). T7 DNA.

Osmotic pressure inside the capsid ~ 100 atm (Champagne bottle ~ 5 atm).

Harnessing the DNA spring.Evilevitch et al. 2003.

Bacteriophage λ with external PEG8000 solution.

DNA osmotic pressure inssidebalanced by

PEG osmotic pressure outside.

DNA equation of state.PEG equation of state.

External osmotic pressureopposes ejection of viral DNA.

Ejection regulation.

FINIS