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. Charge: 2 e 0 / 3.4 Å ~ e 0 / nm 2 grooves discrete charges Polipeptides: 0.6 e o / nm Membranes: 0.1-1 e 0 / nm 2 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 Coulomb s law and kt Gouy - Chapman length Ratio between the Bjerrum and the Gouy - Chapman lengths. Bulk versus surface interactions. Coupling parameter Weak coupling limit (Poisson - Boltzmann) Ξ 0 Strong coupling limit (Netz - Moreira) Ξ Collective description ( N description) vs. Single particle description ( 1 description) Z
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 without mobile counterions Electrostatic energy of a single counterion Electrostatic energy of two counterions 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 Collective description vs. one particle description. 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 Osmotic pressure Osmotic stress method (Parsegian & Rand) The Boyle experiment
Osmotic stress method Π dv µdn 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 Co(NH 3 ) 6 3+ 0mM 8mM Mn 2+ 5 o 20mM 12mM 50 o 35 o Polyvalent counterions + NaCl at 0.25 M: Co(NH 3 ) 6 Cl 3 counterion Co(NH 3 ) 6 3+ (Z = 3) MnCl 2 counterion Mn 2+ (Z = 2) Attraction is obviously there. Quantitative comparison still difficult. Monovalent salt + polyvalent counterions Osmotically stressed subphase. Or condensed.
Bjerrum defects screen polarization. (Onsager - Dupuis theory) 9.0 8.5 rescaled charge density Last few Angstroms... HPC schizophyllan Na-DNA Na-Xanthan TMA-DNA (raw) TMA-DNA (rescaled) DDP bilayers 8.2 8.0 log Π [dynes/cm 2 ] 8.0 7.5 7.0 Log[Π] [dynes/cm 2 ] 7.8 7.6 6.5 7.4 6.0 0 5 10 15 Surface separation, [Å] 20 25 7.2 1.0 1.2 1.4 C DNA [M] 1.6 1.8 2.0 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.
Conformational fluctuations Surprisingly the PB limit for finite salt does not work. What are we missing in this picture? Orientational order L p ~ 50 nm K C = k B T L p 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 10 7 TMV 0.1 M NaCl 10 6 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 Liquid crystalline mesophases. Livolant et al. (97) line hexatic
DNA phase diagram A B A B x-ray beam L L L P L c β β β 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? (D. Nelson. (1995)) E.Coli 630 m long 1 mm thick 25 cm T2 P~100 atm ρ~100 mg/ml (R. Cavenoff (1995)) (Kleinschmidt et al. (1962)) Vortex lines in II sc Tension Non-chiral Magnetic field Temperature London repulsion Bending Chiral density Ionic strength Debye-Huckel repulsion
Why orthorhombic phase at high density? Realistic geometric models of DNA.. Kornyshev - Leikin, 1998, 2000, 2002. Allahyarov et al., 2000-2004 R A R A R Schematics of the orientational effect. Strand opposition. from R= 24 Å out φ 0 = 18 0 and 0 0. explicit DNA structure 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 Many chains Single chain
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 Small force Entropic elasticity Hookeian elasticity Large force Entropic plus enthalpic 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 : L 0, K c, λ(k c ) L Bending rigidity Podgornik et al. 2000. Rouzina (2002) a a = 6.7 ± 0.7 Å (Manning a = 7.2 Å) L P ~ 48 nm l ~ 1200 pn a Stretching modulus Wenner, Williams, Rouzina and Bloomfield (2002). For ionic strengths: 1000, 500, 250, 100, 53.3, 25, 10, 2.6 mm.
Repulsions vs. attractions A reminder of the DNA - DNA interactions. Attraction energies: ~ 0.1 kt/ base pair. Correlation attractions. Hydration attractions. Podgornik et al. 2000 Monovalent counterions Rau et al., 1997. Polyvalent counterions
DNA condensation Hud & Downing (2001) Chattoraj et al. (1978). T4 DNA R ~ 1000 nm to 50 nm. 95-185 nm 35-85 nm 2.4 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.)
racquet-like Manning buckling with correlation attractions Shape equation of the elastic filament (DNA): V(r-r ) toroidal Euler (elastic) intermediates are clearly seen also in simulations of Schnurr et al. (2002). We understand well only one side of the transition. The destabilization of the persistence length leading to a 1st order transition.
DNA condensation simulations 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. DNA equation of state. PEG equation of state. DNA osmotic pressure insside balanced by PEG osmotic pressure outside. Bacteriophage λ with external PEG8000 solution. External osmotic pressure opposes ejection of viral DNA. Ejection regulation.
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