Molecular biophysics

MOLECULAR BIOPHYSICS Rudi Podgornik Department of physics, Faculty of mathematics and physics, University of Ljubljana and Department of theoretical physics, J Stefan Institute, 1000 Ljubljana, Slovenia Sunday, May 27, 12 Physical properties of the DNA molecule from the level of the base pairs all the way to the organization of DNA in viruses and chromatin Interactions within the DNA molecule as well as between the DNA molecules in aqueous solutions at various environmental conditions Not chemistry, not biology, but physics, with as few equations as is humanely possible for a theoretical physicist Sunday, May 27, 12 STRUCTURE OF DNA (X-ray scattering, Structure factor of a continuous single helix, Scattering intensity of an orientationally averaged helix, Structure factor of a discrete helix, Scattering intensity of a double helix, Details of B-DNA structure), BASE-PAIR INTERACTIONS AND DNA MELTING (A model for primary stabilizing interactions, The Peyrard-Bishop-Dauxois model of DNA melting, The DNA melting temperature, Observing DNA melting), MECHANICS AND STATISTICAL MECHANICS OF DNA (Elastic deformation energy, Elastic equation of state, The Kirchhoff kinematic analogy, The Kratky-Porod model, Light scattering from a Kratky-Porod filament in solution, Elastic response of a Kratky-Porod filament, The limit of small stretching force, The limit of large stretching force, Extensible semiflexible chain, An approximate elastic equation of state for DNA), ELECTROSTATICS OF DNA and DNA -DNA interactions (Poisson-Boltzmann theory, Counterion distribution, manning condensation, Salt screening, Strong coupling theory, Correlation attraction, Osmotic stress method, Hydration force, Force equilibria with polyvalent counterions), DNA COLLAPSE AND DNA MESOPHASES (Collapse of a single DNA molecule, The DNA toroidal globule, Nematic LC transition in a DNA solution, Elastic energy of a DNA hexagonal columnar LC, Cell model of a DNA array, Osmotic pressure of a DNA array, Electrostatic part of the osmotic pressure, Equation of state of a DNA array, Fluctuations and positional order in a DNA array), DNA ORGANIZATION IN CHROMATIN AND VIRUSES (Nucleosomes, Caspar-Klug theory and elaborations, Continuum elasticity of viral capsids, Viral capsids under mechanical stress, Osmotic encapsulation of DNA, The inverse spool model) Sunday, May 27, 12 LECTURE STRUCTURE OF DNA (X-ray scattering, Structure factor of a continuous single helix, Scattering intensity of an orientationally averaged helix, Single and double helix, Scattering intensity of a double helix, Details of the double helical B-DNA structure) Sunday, May 27, 12 Sunday, May 27, 12 Thomas Young, hieroglyphs and interference The double life of Dr Thomas Younga (1773-1829) Interference, 1820 Light is a wave and not a flow of particles as Newton thought! Sunday, May 27, 12 Hieroglyphs can be read, 1814 Cartouches on the Rosetta stone They are not pictograms but have phonetic values! Interference Light is a wave And therefore light waves interfer with one another Just like waves on the surface of water Total amplitude is a sum of partial amplitudes of all waves Interferencof waves passing through two slits Interference of many sources of waves Surface or volume patterns Sunday, May 27, 12 Interference on multiple slits The separation between the slits is reflected in the interference pattern The exact relationship is given by the Fourier transform Huygens’ principle: every single slit acts as a source of a spherical wave so that the total amplitude is: The Fourier transform of the distribution of slits Therefore the intensity of light is: The intensity is given by the square of the Fourier transform of the slit distribution (numerically) Sunday, May 27, 12 X-rays November 1895, Wilhelm Röntgen discovers X-rays at the University of Wurzburg Thinks they are longitudinal light waves Wavelength ~ Å Wilhelm Conrad Röntgen (1845-1923) Max von Laue 1912 discovers diffraction of X-rays by crystals Nobel prize for physics 1915 Max Theodor Felix von Laue (1879-1960) Interference of many sources of waves Surface or volume patterns Sunday, May 27, 12 Scattering of X-rays In 1912 William Lawrence Bragg (1890-1971) understands why light waves passing through crystals produce intricate patterns Von Laue (1912) ZnS (zinc sulfide crystal) Sunday, May 27, 12 10 Equilibrium local osmotic pressure and the inverse spool The total osmotic pressure for an inverse spool (cylindrical symmetry) is then given by: interaction pressure curvature pressure total pressure Osmotic pressure (measurable) as opposed to chemical potential is the main variable Depletion of the polymer (DNA) at the center of the capsid due to high bending energy Two asymptotic forms of the solution Inverse spool! Derived from nanomechanics Quadratic depletion at the center No need to assume the depletion at the core (Odijk & Slork) Sunday, May 27, 12 303 DNA density profile inside capsid DNA density profile for monovalent and polyvalent salt extracted from the bulk DNA equation of state Monovalent and polyvalent salt density profiles show marked differences Density jumps in the polyvalent case The difference should be experimentally observable Sunday, May 27, 12 304 Bulk osmotic pressure: mono vs polyvalent counterions Big differences also in the encapsidation curves In many component systems of highly asymmetric electrolytes one gets vdW-form osmotic pressure curves Sunday, May 27, 12 305 Viral DNA equation of state: loading curve Encapsidated fraction as a function of external osmotic pressure This is now the connection between the in vitro and in viro Small (almost negligible) effects of DNA elasticity as embodied in the persistence length Sunday, May 27, 12 306 Free DNA toroid formation Log(Osmotic pressure) (dynes/cm^2) For sufficiently large polyvalent counterion concentration collapse of DNA Depending on the details the collapse can be into a toroidal structure Hud & Downing (2001) λ phage w CoHex 95-185 nm, 35-85 nm, 2.7 nm Toroid formation in solution, ‘70s Evdokhimov et a (1972) was the only one (?) who observed torus formation with PEG Sunday, May 27, 12 307 Free DNA toroid formation - calculation Previous mesoscopic theory of DNA nanodrop has to be upgraded leading to a minimization of the elastic plus surface energy (Ubbink & Odijk 1996): For spontaneous aggregation of DNA the total osmotic pressure has to be p=0! In order for this problem to be solvable, the interaction osmotic pressure has to have an attractive branch - attractive interactions Siber, Losdorfer & Podgornik (2010) α = 0.15, 0.5, 20 (outer, middle and inner toroid) The outer radii are approximately 160 nm, 120 nm and 50 nm, respectively Sunday, May 27, 12 308 Encapsidated DNA toroid formation Formation of DNA toroids inside confining bacteriophage capsids A careful examination of Figure in Jeembaeva et al (2008), led us to recognize the presence of toroids in partially DNA-filled Lambda phage particles Livolant, 2009 In T5 bacteriophage: Leforestier & Livolant, 2009 This problem is more complicated since there are additional interactions between the surface (inside) moieties of the capsid proteins and the DNA toroid Not much is known about these interactions Sunday, May 27, 12 309 Condensation in viro 2.88 nm nm ! T5 bacteriophage capsids containing a ds DNA fragment of about 17µm (50,000bp - initial 38,7 µm) after partial ejection of their genome In the presence of monovalent cations (a,a’) the DNA chain occupies the entire volume of the capsid (a’) DNA toroids are formed by addition of polyamines (Spm4+) (b-b”)) or PEG (c-c”) in the buffer (15% solution of PEG 6000, osmotic pressure 3.2 atm) The polycations can go through the capsid whereas PEG stays outside Scale bar= 20nm Sunday, May 27, 12 310 Details of the experiment 2.88 nm nm Receptor + 100mM NaCl, 1mM CaCl2 and 1mM MgCl2 mM Spm Sunday, May 27, 12 311 Calculating the shape of DNA condensed in viro We add the surface free energy to the total free energy We assume that the capsid DNA toroidal condensate interactions are very short range: with the surface free energy (σ can be positive or negative) and the total length of DNA: By minimizing the total free energy we get the shape of the DNA condensate with appropriate constraints A straightforward generalization of the Odijk-Ubbink calculation Ubbink, J and Odijk T.,Europhys Lett 33, 353 (1996) Sunday, May 27, 12 312 Deformation of DNA condensed in viro a) The shapes of DNA toroids in the [α, σ] parameter space Blue dashed lines in a) indicate the inner surface of the spherical container with R = 40 nm The volume of the DNA in these calculations was fixed to V = 1.4 10^4 nm^3 b) Three-dimensional rendering of the DNA toroid with α = 1, σ = c) Three-dimensional rendering of the DNA toroid with α = 10, σ = 100 Sunday, May 27, 12 313 Shape of a collapsed encapsidated DNA aggregate Because the available volume inside the viral capsid is smaller then the “natural” size of the collapsed DNA toroid, the aggregate has to interact sterically with the capsid walls! Attractive interactions Repulsive interactions Sunday, May 27, 12 314 Spm vs PEG condensed DNA in viro DNA - T5 capsid interactions drastically change the shape of condensed DNA From convex toroidal to non-convex toroidal (Spm) (PEG) Sunday, May 27, 12 315 Fundamental difference between PEG and Spm Why? The semi-permeable capsid is rigid and the volume of the capsid is kept to a good approximation a constant After being collapsed, DNA occupies only a fraction of the capsid volume and two compartments are created inside the capsid (left): one containing DNA with its associated water and counterions and another one devoid of DNA In Rau, Parsegian et al experiments (right) no semi-permeable membrane and therefore only one compartment: the DNA solution The existence of the second compartment could possibly be a consequence of the negative hydrostatic pressure inside the capsid caused by its impenetrability to the external PEG but we can not say anything more definitive at this point Sunday, May 27, 12 316 EST MODUS IN REBUS SUNT CERTI FINES DENIQUE Sunday, May 27, 12 317 ... Ferdinand Perutz (1914-2002) in John Cowdery Kendrew (1917-1997) Nobel prize for chemistry in 1962 molecular models because we have only 2D pattern of waves Sunday, May 27, 12 12 Scattering of X-rays... Watson-Crick model Sunday, May 27, 12 27 Much more then a single helix? Pauling is very good at molecular modeling Protein structural motifs are his work In the beginning of 1953 Pauling proposes