Physics of DNA R. Podgornik Laboratory of Physical and Structural - - PowerPoint PPT Presentation
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 -
Laboratory of Physical and Structural Biology National Institute of Child Health and Human Development National Institutes of Health Bethesda, MD
DNA is not the proverbial spherical cow, or in this case a cylindrical one.
helix discrete charges
grooves
Charge: 2 e0 / 3.4 Å ~ e0 / nm2 Polipeptides: 0.6 eo / nm Membranes: 0.1 - 1 e0 / nm2 Structure (B-form)
Collective description (“N” description) vs. Single particle description (“1” description) Weak coupling limit (Poisson - Boltzmann)
Ξ➝ 0
Strong coupling limit (Netz - Moreira)
Ξ➝ ∞
Ratio between the Bjerrum and the Gouy - Chapman lengths. Bulk versus surface interactions. Bjerrum length Gouy - Chapman length Coulomb’s law and kT Coupling parameter
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
Electrostatic energy without mobile counterions Electrostatic energy
Electrostatic energy
Collective description vs. one particle description. repulsion + 2 X attraction = attraction
between DNAs (late 60’s)
Wennerstrom and Linse (1984)
DNA
correlation effect
Hexagonal array of DNA poly-counterions: (Lyubartsev and Nordenskiold, 1995) A pair of DNAs with poly-counterions: (Gronbech-Jensen et al. 1997)
The Boyle experiment Osmotic pressure Osmotic stress method (Parsegian & Rand)
Setting the osmotic pressure and measuring the density of DNA
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 + NaCl at 0.25 M:
3+ (Z = 3)
Co(NH3)6 3+ Mn2+ 0mM 8mM 12mM 20mM 5o 35o 50o
Attraction is obviously there. Quantitative comparison still difficult. Monovalent salt + polyvalent counterions Osmotically stressed subphase. Or condensed.
2
9.0 8.5 8.0 7.5 7.0 6.5 6.0
log Π [dynes/cm
2]
25 20 15 10 5
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] 2.0 1.8 1.6 1.4 1.2 1.0
CDNA[M]
Marcelja and Radic, 1984. Perturbation of water order parameter. Similar foces in ice. Bjerrum defects screen polarization. (Onsager - Dupuis theory)
Surprisingly the PB limit for finite salt does not work. What are we missing in this picture? Orientational order…
DNA is a flexible molecule. E ~ 300 MPa (plexiglass) At room temperature big conformational fluctuations.
Elastic 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)
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!
µ−tubules 0.1 M NaCl 107 TMV 0.1 M NaCl 106 F actin 0.1 M NaCl 10000 Schizophyllan water 200 Xanthan 0.1 M NaCl 120 ds-DNA 0.2 M NaCl 50 Spectrin 0.1 M NaCl 15 ss-DNA 0.2 M NaCl 3 Hyaluronic acid 0.2 M NaCl 1 Long 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.
A B L L L P L
β β’ α c β’
A B
x-ray beam
(Livolant, Leforestier, Rill, Robinson, Strzelecka, Podgornik, Strey …)
Durand, Doucet, Livolant (1992) J. Physique 2, 1769-178 Pelta, Durand, Doucet, Livolant (1996) Biophys. J., 71, 48-63 3
(Predicted by Toner, 1983)
An anomalous “3D” hexatic phase! (Podgornik et al. 1999)
(R. Cavenoff (1995)) (Kleinschmidt et al. (1962)) (D. Nelson. (1995)) E.Coli T2 P~100 atm ρ~100 mg/ml 630 m long 1 mm thick 25 cm Vortex lines in II sc Tension Non-chiral Magnetic field Temperature London repulsion Bending Chiral density Ionic strength Debye-Huckel repulsion
Realistic geometric models of DNA. .
Schematics of the orientational
A A R R R
Kornyshev - Leikin, 1998, 2000, 2002. Allahyarov et al., 2000 - 2004
For the non-parallel orientation state a hexatic (hexagonal) phase becomes a distorted (orthorhombic or monoclinic) crystal! (Rosalind Franklin, 1952).
Lattice distortions alleviate frustrations: In a lattice the configurations are frustrated
Hexagonal lattice (Lorman, Podgornik , Zeks 2001)
Single chain Many chains
(Baumann, Smith, Bloomfield, Bustamante 1997) Force curve fit to model (a 4 par fit) elastic constants
Entropic plus enthalpic Hookeian elasticity Entropic elasticity Hookeian elasticity Small force Large force stretching bending 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 = 1
4 λ R 2
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.
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)
A 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 nm 35-85 nm 2.4 nm
T4 DNA R ~ 1000 nm to 50 nm.
Euler buckling instability: Press on an elastic filament hard enough and it buckles. The role of compressional force is played by diminished (on addition of polyvalent counterions) electrostatic interactions. No correlation effect at that time! (Manning, 1985.) Kirchhoff kinematic analogy.
Shape 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
1st order transition.
V(r-r’)
Elastic, Euler-like, states are important for DNA collapse. Stiff polymers have a different Collapse 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.
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 insside balanced by PEG osmotic pressure outside. DNA equation of state. PEG equation of state. External osmotic pressure
Ejection regulation.