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

DNA helix Structure (B-form) R. Franklin, photo 51. grooves Charge: 2 e0 / 3.4 Å ~ e0 / nm2 discrete charges Polipeptides: 0.6 eo / nm Membranes: 0.1 - 1 e0 / nm2 • a ~ 1 nm • h(DNA) = 1.7 Å • DNA length from 50 nm to ~ µm 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

The great electrostatic divide Bjerrum length Gouy - Chapman length Coulomb’s law and kT Ratio between the Bjerrum and the Gouy - Chapman lengths. Bulk versus surface interactions. Coupling parameter Weak coupling limit Strong coupling limit (Poisson - Boltzmann) (Netz - Moreira) Ξ ➝ 0 Ξ ➝ ∞ Collective description Z (“N” description) vs. Single particle description (“1” description)

The weak coupling limit (collective description) + electrostatic energy ideal gas entropy minimize to get equilibrium Non-equilibrium free energy = (electrostatic energy) - k (ideal gas entropy) Minimization yields the Poisson - Boltzmann equation. Screening. Debye length ~ 3.05 Å / √ M

The strong coupling limit (one particle description) Z Z + + … + Z Z Electrostatic energy Electrostatic energy Electrostatic energy of a single counterion of two counterions without mobile counterions Collective description vs. one particle description. • 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 repulsion + 2 X attraction = attraction

Simulations A pair of DNAs with poly-counterions: (Gronbech-Jensen et al. 1997) Hexagonal array of DNA poly-counterions: (Lyubartsev and Nordenskiold, 1995)

Experiments The Boyle experiment Osmotic pressure Osmotic stress method (Parsegian & Rand)

Osmotic stress method Π dV − µ dN 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 3+ Co(NH3)6 Mn2+ 0mM 5o 8mM 20mM 12mM 50o 35o Polyvalent counterions + NaCl at 0.25 M: 3+ (Z = 3) • Co(NH3)6Cl3 counterion Co(NH3)6 • MnCl2 counterion Mn2+ (Z = 2) Attraction is obviously there. Osmotically stressed subphase. Quantitative comparison still difficult. Or condensed. Monovalent salt + polyvalent counterions

Last few Angstroms .... 9.0 HPC schizophyllan 8.2 Na-DNA Na-Xanthan TMA-DNA (raw) 8.5 TMA-DNA (rescaled) rescaled charge density 8.0 DDP bilayers 8.0 2 ] Log[ Π ] [dynes/cm 2 ] log Π [dynes/cm 7.8 7.5 7.6 7.0 7.4 6.5 7.2 6.0 0 5 10 15 20 25 1.0 1.2 1.4 1.6 1.8 2.0 Surface separation, [Å] C DNA [M] Commonality of forces among charged, neutral, cylindrical and 2 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) 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 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)

Conformational fluctuations Electrostatics can only be seen indirectly, as modified by the presence of conformational fluctuations. Renormalized value of λ : λ (r) = 4 λ D. Factor 4 due to elasticity (fourth derivative) as well as the 1D nature of DNA (linear polymer). Liquid disorder! DNA in monovalent (NaCl) salt solution. Paradigmatic behavior for all monovalent salts.

DNA Elasticity and mesophases Persistence length of a semiflexible polymer 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 E~300 Mpa (plexiglass) Onsager’s argument valid also for polymers. cholesteric line hexatic Liquid crystalline mesophases. Livolant et al. (97)

DNA phase diagram A B A B x-ray beam L c L β L β ’ P β ’ L α (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

The line hexatic phase (Predicted by Toner, 1983) • Long range BO order ~ 0.6 mm • Long range nematic order • Liquid like positional order, λ PO An anomalous “3D” hexatic phase! (Podgornik et al. 1999)

Why is this relevant? E.Coli 630 m long P~100 atm 1 mm thick ρ ~100 mg/ml T2 25 cm (Kleinschmidt et al. (1962)) (R. Cavenoff (1995)) Vortex lines in II sc Tension Bending Non-chiral Chiral Magnetic field density Temperature Ionic strength London repulsion Debye-Huckel repulsion (D. Nelson. (1995))

Why orthorhombic phase at high density? Realistic geometric models of DNA. Kornyshev - Leikin, 1998, 2000, 2002. . Allahyarov et al., 2000 - 2004 A R A R R Schematics of the orientational effect. Strand opposition. • from R= 24 Å out • explicit DNA structure • φ 0 = 180 and 0 0. • explicit counterions • explicit salt ions • different salt concentrations

Lattice frustrations due to orientational interactions In a lattice the configurations are frustrated Hexagonal lattice • nearest neighbors in optimal config. • not all are happy (Lorman, Podgornik , Zeks 2001) Lattice distortions alleviate frustrations: • 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).

Single molecule physics Single chain Many chains

Measuring DNA elasticity (Baumann, Smith, Bloomfield, Bustamante 1997) Force curve fit to model (a 4 par fit) elastic constants

Bending and stretching bending external force stretching Large force Small force Entropic plus enthalpic Entropic elasticity Hookeian elasticity Hookeian elasticity The experiment gives us both moduli Kc as well as λ (0). ds-DNA is not very stretchable, but it is not rigid either.

DNA - an Euler-Kirchhoffian filament or not? In classical elasticity (cylindrical Euler - Kirchhoff filament) K C = 1 4 λ R 2 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. 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).

Interactions and elasticity L A constrained fit : L0, Kc, λ (Kc) L’ Bending rigidity Podgornik et al. 2000. Rouzina (2002) a a = 6.7 ± 0.7 Å (Manning a = 7.2 Å) LP ~ 48 nm l ~ 1200 pN Wenner, Williams, Rouzina and Bloomfield (2002). For a’ ionic strengths: 1000, 500, 250, 100, 53.3, 25, 10, 2.6 mM. Stretching modulus

Repulsions vs. attractions A reminder of the DNA - DNA interactions. Attraction energies: Correlation attractions. ~ 0.1 kT/ base pair. Hydration attractions. Podgornik et al. 2000 Rau et al., 1997. Monovalent counterions Polyvalent counterions

DNA condensation Hud & Downing (2001) 95-185 nm 35-85 nm Chattoraj et al. (1978). 2.4 nm T4 DNA R ~ 1000 nm to 50 nm.

Euler buckling Euler buckling instability: Press on an elastic filament hard enough and it buckles. Kirchhoff kinematic analogy. The role of compressional force is played by diminished (on addition of polyvalent counterions) electrostatic interactions. No correlation effect at that time! (Manning, 1985.)