Molecular attractions:

Transcription

1 Molecular attractions: a.) van der Waals interactions b.) electrostatic correlation interactions c.) polyelectrolyte bridging interactions Rudi Podgornik Laboratory of Physical and Structural Biology National Institute of Child Health and Human Development National Institutes of Health Bethesda, MD Department of Physics Faculty of Mathematics and Physics, University of Ljubljana Department of Theoretical Physics J. Stefan Institute, Ljubljana Slovenia TROISIÈME CYCLE DE LA PHYSIQUE EN SUISSE ROMANDE COURS DU SEMESTRE DE PRINTEMPS 2009

2 Plan of lectures: Three important sources of attractive interactions in soft matter systems. I will take time to introduce the general principles as well as introduce the audience to specific model systems: 1. van der Waals interactions, which are due to correlated electromagnetic field fluctuations in dielectrically inhomogeneous media, their role, magnitude and form in various systems, in particular in the interactions between carbon nanotubes 2. the electrostatic correlation interactions which are due to electrostatically strongly coupled mobile counterions between charged macromolecules, their role in DNA collapse in multivalent salts 3. polyelectrolyte bridging interactions that are due to elastic deformation of stretched polyelectrolyte chains between charged macromolecular interfaces, their role in organization of eucaryotic genome, specifically in the interactions between nucleosomal core particles. Bonus: Packing of DNA in simple bacteriophages. I will present a few ideas on how DNA can be packed within small viruses and what is the energetics of this packing from the point of view of continuum theories.

3 2. the electrostatic correlation interactions which are due to electrostatically strongly coupled mobile counterions between charged macromolecules and their role in DNA collapse in multivalent salts

4 Charges, electrostatic interactions and dimensionality 3D Coulomb s law and Poisson equation: Equal sign charges - repulsion, different sign charges - attraction. This form of CL is only valid for two charges in 3D. For 2 or 1 dimensional systems the Coulomb law is modified: Where now µ and σ are now the surface charge density and the linear charge density. 2D 1D

5 kt and electrostatic interactions (energy vs. entropy) Apart from ES energy, we also have thermal effects. Below this separation (Bjerrum s length Å in water) the + and - would tend to associate (Bjerrum pairs). Entropy! Minimize the free energy 1D 1D 2D 3D 2D 3D Counterion condensation.

6 The Poisson - Boltzmann theory - collective description + electrostatic energy ideal gas entropy minimize to get equilibrium Non-equilibrium free energy = (electrostatic energy) - k (ideal gas entropy) Electrostatic energy by itself: collaps to the surface Entropy by itself: uniform distribution of charges The two together: non-homogeneous ionic profile

7 The Poisson - Boltzmann equation Minimization of the free energy (electrostatic filed energy plus ideal entropy) The Poisson - Boltzmann equation and the electroneutrality BC The cell model. Poisson Boltzmann Inhomogeneous electrostatic potential! To solve the Poisson - Boltzmann eq. in cylindrical geometry one has to introduce a neutralizing cell. Has to satisfy BC at both boundaries.

8 The Poisson - Boltzmann equation: cylinder Solution of the cylindrical cell model Approximate form of the solution The Manning parameter.

9 The Poisson - Boltzmann equation : counterion condensation Osmotic coefficient The concentration of counterions at the outer cell boundary gives the osmotic pressure in the system. A fundamental insight of the PB theory. For DNA: For HA e.g. Q ~ On the average 3 out of 4 phosphates are neutralized by counterions. The rest are acting as free, osmotically active counterions. Onsager-Oosawa-Manning counterion condensation. Onsager in 1967 observed an anomaly in the partition function of a Coulomb gas in 2D. Manning realised in 1969 this is relevant for polyelectrolytes. In the 70 Oosawa, Manning and Dolar realised that the FKL solution gives the same behavior.

10 The Poisson - Boltzmann equation : screening Take symmetric Coulomb fluid = electrolyte. In equilibrium the average φ is zero, and the average charge density ρ: Electroneutrality But in the presence of an external field Together with Poisson equation this yields the Debye - Hueckel equation Screening. Debye length ~ 3.05 Å / M

11 The Poisson - Boltzmann equation: effect of salt In cylindrical cell geometry there are no analytical solutions at finite dilution. At infinite dilution: Tracy - Widom (1997) analytical solution. At best we remain with the linearized solution or the complete numerical solution. Osmotic pressure: The dominant behavior is exponential with Debye screening length (inverse κ0).

12 annus mirabilis for colloid science: DLVO theory Gouy (1910) Chapman (1913) Debye & Huckel (1913) Verwey & Overbeek (1948) Deryaguin & Landau (1941) disjoining pressure DLVO theory of colloid stability: Electrostatics plus van der Waals.

13 annus horribilis for colloid science Developments in the 80 s colloid science: Oosawa derives attractive interactions between DNAs (late 60 s early 70) Simulation of DLVO interactions (early 80 s - electric bilayer simulation Torrie and Valleau) Fundamental paper by Gulbrand, Jonsson, Wennerstrom and Linse (1984) Established that for planar surfaces the Interactions with divalent counterions can be attractive! They dubbed it the correlation effect because it stemms from a correlation term in the stress tensor. Probably the biggest advance in colloid science since DLVO.

14 A historical guide to the correlation effect Oosawa (1971) (counterion fluctuations) Gulbrand, Jonsson, Wennerstrom and Linse (1984) Ninham and Parsergian (75) (van der Waals interactions) Lyubartsev and Nordenskiold (1995) Gronbech-Jensen et al. (1997) (MC simulations) Rouzina and Bloomfield (1996) (checkerboard model) Podgornik et al. ( ) Attard et al. (1988) Podgornik and Parsegian (1999) (Gaussian fluctuations) Kjellander and Marcelja ( ) (inhomogeneous integral equations closure) Shklovskii et al. ( ) Lau and Pincus (2001) (Wigner crystal model) Netz and Moreira ( ) Naji and Netz ( ) (General analysis of Coulomb fluids) Kornyshev and Leikin ( ) (Debye-Hueckel-Bjerrum model)

15 Getting worse and worse... A pair of DNAs with poly-counterions: (Gronbech-Jensen et al. 1997) Screening. Debye length ~ 3.05 Å / M Hexagonal array of DNA with poly-counterions: (Lyubartsev and Nordenskiold, 1995) Attractions seem to be everywhere!

16 DNA helix Structure (B-form double-helix) grooves Charge: 2 e 0 / 3.4 Å ~ e 0 / nm 2 discrete charges Polipeptides: 0.6 e o / nm Membranes: e 0 / nm 2 a ~ 1 nm h ~ 0.17 nm DNA pitch p ~ 3.4 nm DNA length L ~ 50 nm to ~ μm DNA persistence length L ~ 50 nm p 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

17 annus reconciliationis for colloid science Bjerrum length Gouy - Chapman length Coulomb s law and kt Ratio between the Bjerrum and the Gouy - Chapman lengths. Bulk versus surface interactions. Weak coupling limit (Poisson - Boltzmann) Coupling parameter Strong coupling limit (Netz - Moreira) Collective description (Poisson - Boltzmann N description) Screened Debye-Hueckel vs. Single particle description (Strong Coupling 1 description) Z

18 Weak and strong coupling - Netz (2000) Weak coupling = Poisson - Boltzmann (MF) + fluctuations Weak and strong coupling refer to the value of the Netz coupling constant. Weak coupling - interactions in the symmetric case strictly repulsive and large (Confusion: Bowen & Sharif (1998). - fluctuation contribution in the symmetric case strictly attractive and small Strong coupling - interactions in the symmetric case mostly attractive and large - repulsive only at small separations - Wigner cristal heuristic model (Shklovskii 1999)

19 What is the strong coupling limit? Z + Z + Z + Z Electrostatic energy without mobile counterions Electrostatic energy of a single counterion Electrostatic energy of two counterions Electrostatic energy of a single counterion plus entropy of a trapped counterion. Netz (2000) Fixed constant surface charge! Strong coupling limit = virial expansion to first order.

20 Interaction between planar charged surfaces Weak coupling ( = Poisson - Boltzmann + Gaussian flucts) Total pressure (PB+ flucts) Poisson - Boltzmann (analytic) Fluctuational (analytic) Strong coupling (a lot easier to evaluate)

21 Interaction between cylinders Z + Z + Electrostatic energy of a single counterion Electrostatic energy without mobile counterions Critical Manning parameter. Asymptotic forms of the free energy: Naji and Netz, 2004.

22 Interaction between cylinders Q = 1. Q = 0.3 Equilibrium spacing. Close to the critical value of Q Naji and Netz, 2004.

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24 Interactions can be measured Setting the osmotic pressure and measuring the density of DNA. DNA equation of state! Differrent ionic conditions, different DNA length, type of ions... The Boyle experiment Osmotic stress method (Parsegian & Rand)

25 Osmotic stress method Π dv µdn Setting the osmotic pressure and measuring the density of DNA. DNA equation of state!

26 DNA osmotic pressure - equation of state. Bulk energy. Boyle experiment. Equivalence of osmotic pressure (PEG or DEX etc.) Podgornik et al Different regions of DNA density correspond to different mesophases.

27 Repulsions vs. attractions A reminder of the DNA - DNA interactions. The DNA-DNA interactions can be measured directly in osmotic stress experiments. Attraction energies: ~ 0.1 kt/ base pair. Podgornik et al Monovalent counterions Rau et al., Polyvalent counterions

28 DNA osmotic pressure - equation of state. Large density. 9.0 HPC schizophyllan rescaled charge density Na-DNA Na-Xanthan TMA-DNA (raw) TMA-DNA (rescaled) DDP bilayers Surface separation, [Å] C DNA [M] 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) Marcelja and Radic, Perturbation of water order parameter. Similar foces in ice. Bjerrum defects screen polarization. (Onsager - Dupuis theory)

29 Monovalent counterions Electrostatics can only be seen indirectly, as modified by the presence of conformational fluctuations. DNA in monovalent (NaCl) salt solution. Osmotic pressure for a 2D hexagonal array. PB does not seem to be working! Conformational fluctus: renormalized value of λ: Osmotically stressed subphase.

30 Conformational fluctuations Electrostatics can only be seen indirectly, as modified by the presence of conformational fluctuations. Only for counterion-only case does one see direct effects of electrostatics. Renormalized value of λ: λ (r) = 4 λ D. DNA in monovalent (NaCl) salt solution. Paradigmatic behavior for all monovalent salts. Factor 4 due to elasticity (fourth derivative) as well as the 1D nature of DNA (linear polymer).

31 Comparison with experiment In Manning theory E = = Measured E seems to be twice as large. Similar result in EF experiments (Schellman and Stigter, 1977). Big differences between Na, Li, K, Rb and 3 Me (NH 3 ). A very pleasing analogy between multilamellar lipid arrays and hexagonal DNA arrays in terms of the interplay between thermal fluctuations and interactions. However, no vdw interactins in DNA arrays, except

32 Polyvalent counterions Attraction is obviously there. Quantitative comparison still difficult. Monovalent salt + polyvalent counterions + specific ionic effects Co(NH 3 ) mM 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) Osmotically stressed subphase. And/or condensed.

33 How to understand this? Compare with the van der Waals isotherm! It is not an isotherm it is an iso-ionic strength line. van der Waals equation of state, For polyvalent counterions one obtains discontinuous jumps in the force curve - an indication of the underlying SC attractions. But also a repulsive part dependent on the ionic strength and the amount of polyvalents. attractions-repulsions

34 Measured correlation effect? The attraction seen in the experiments with polyvalent counterions could be the strong coupling electrostatics. Also corroborated by extensive simulations. A pair of DNAs with poly-counterions: (Gronbech-Jensen et al. 1997) Equation of state for polyvalent counterions shows thermodynamically unstable regions (just like van der Waals isotherms) that are due to attractive iteractions between DNA molecules.

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36 DNA condensation Hud & Downing (2001) Chattoraj et al. (1978) nm nm 2.4 nm

37 DNA condensation: how universal In most vertebrate sperm cells DNA is condensed by arginine-rich proteins into thousands of toroidal structures, each measuring ~ 100 nm in outside diameter. The DNA of some bacteriophages also is packaged into a single toroid, or spool, with similar dimensions. Thus, the toroid represents a fundamental morphology selected by nature for the high-density packaging of DNA (Hud & Downing 2001). Very few - luckily for us - (cat)ions induce DNA condensation. Mn 2+, Cd 2+, Co(NH3 )3+, polyamines such as sperimidine 3+, spermine 4+, protamine 21+, polylysine N+ Condensation is a complex phenomenon still not completely understood. Axial charge separation along DNA seems to be necessary for condensation Best condensing agents bind into the major or minor groove. Almost all divalent cations condense ssdna but not dsdna. Electrostatics is an important but not the only factor in DNA collapse (CoHex more efficient then spermidine!). Cerritelli et al.(1997). T7 DNA.

38 DNA condensation and Euler buckling Manning (1985): DNA condensation is Euler buckling due to self-interactions. Standard Euler-Kirchhoff elastic energy. Minimization of the elastic energy leads to a slew of different shapes: Kirchhoff kinematic analogy. These shapes can be calculated and classified (Nizette and Goriely (2002).

39 Manning buckling Euler buckling instability: Press on an elastic filament hard enough and it buckles. Manning: turn on attractions along the filament and it buckles too! Original Euler drawing. The role of compressional force is played by diminished (on addition of polyvalent counterions) electrostatic interactions. No correlation effect at that time!

40 Correlation interaction potential We assume that the simple salt and the polyvalent counterion interaction potantials are additive and pick a model for the spatial dependence of the latter. it should correspond to some general form of short ranged attraction! The box model (1): The exponential- inverse r model (2): Reality check The exponential-inverse r 2 model (3): Refrain from details, concentrate on the salient features of models (1), (2), and (3). Hansen et al

41 Manning buckling with correlation attractions Now construct a free energy functional with elastic and attractive interactions. Shape equation of the elastic filament (DNA): V(r-r ) Models (1), (2) and (3) give similar results. This is a mean-fleld like calculation. No thermal fluctuations. Just elasticity and interactions.

42 Buckling and thermal fluctuations Thermal fluctuations? Odijk (1998) - harmonic theory. Renormalization of the bending rigidity. A full non-linear theory (Hansen et al. 1999) takes us to: Interaction has a repuslive (WC) and an attractive part (SC). collapse? WLC Liverpool, Golestanian, Kardar (2000). Ha and Thirumalai (2002), Nguyen, Rouzina and Shklovski (1999).

43 Single molecule physics Many chains Single chain

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

45 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.

46 DNA - an Euler-Kirchhoffian filament or not? In classical elasticity (cylindrical Euler - Kirchhoff filament) bending rigidity ~ stretching modulus 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).

47 Interactions and elasticity L A constrained fit : L 0, K c, λ(k c ) L Bending rigidity a Podgornik et al. (2000). Rouzina (2002) 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.

48 Elastic softening and attractive interactions Mg 2+ and Co (NH 3 ) 6 3+ yield lower persistence length then polyamines spermidine 3+ and putrescine do not condense DNA! Elastic softening by Itself wil lprobably not drive DNA collapse. (hydration, helix secondary structure ) Baumann et al Force plateau when Enough poly+ to condense DNA. Attractions of kt/bp Consistent with Rau et al.

49 Flexible polyelectrolyte condensation simulations Existence of correlation attractions has important consequences For polyelectrolyte conformations. Extended Swedish polyelectrolyte (no hard cores etc). Khan and Jonsson, (1999): polyvalent counterions can collapse a flexible chain. A new problem: interactions and polyelectrolyte flexibility.

50 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 then in Khan and Jonsson. Stevens BJ (2001). This collapse is very different from a flexible chain.

51 DNA condensation simulations Simulations point to various eulerian precursors of the collapsed phase. toroidal? Euler (elastic) intermediates are clearly seen also in simulations of Schnurr, Gittes and MacKintosh PRE (2002). racquet-like We understand well only one side of the transition. The destabilization of the persistence length leading to a 1st order transition.

52 DNA condensation simulations Coil - racket - torus. Montesi et al

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54 Discovery of the double helix structure of DNA Nature, april 25, (Rupprecht and RP, 1994) Difractogram 51 (Franklin and Gosling, 1952). Based on the CCV theory of diffraction by helical molecules and X-ray diffraction experiment.

55 Discovery of DNA mesophases DNA in concentrated solutions makes ordered phases. Not that difficult to observe. cholesteric phase line hexatic phase Conmar Robinson in 1958 described the cholesteric phase of poly(benzyl-l-glutamate) in dimethylformamide and in 1961 of DNA. V. Luzzati et al. in 1961 observed indications of ordered phases of DNA in SAXS (small angle X-ray scattering) R. Rill et a. in 1981 saw ordered phases with short fragment DNA (146 bp about 50 nm) Livolant et al

56 mg/ml 15 DNA mesophase zoo Mesophases at different DNA density nm fragment (F. Livolant) blue phases cholesteric hexatic - deformed hexatic orthorhombic columnar hexagonal - hexagonal The exact positions of the phase boundaries do depend on the method of preparation (osmotic stress, controlled drying, condensation ) and the length of the DNA. Durand, Doucet, Livolant (1992) J. Physique 2, Pelta, Durand, Doucet, Livolant (1996) Biophys. J., 71, 48 RP et al. COSB. 8 (1998) 309. COCIS. 3 (1998) 534.

57 Methods of preparation The question is how to control and change the DNA density in solution. Dense ordered phases of DNA can be prepared by a variety of methods used in different experimental setups. osmoticant condensing agents Osmotic stress (membrane) Osmotic stress (PEG...PSI DNA) Controlled drying Condensation (polyamines...) While there are no qualitative differences between the nature of the phases and their boundaries, they have not been studied systematically.

58 Cholesteric phase mg/ml (cholesteric droplets) C. J. Barrett Durand, Doucet, Livolant (1992) J. Physique 2, The pitch of the cholesteric phase ~ µm. Cholesteric pitch: (Stanley et al. BJ 89 (2005))

59 Hexatic - deformed hexatic mg/ml A most mysterious phase! (Predicted by Toner, 1983 observed by RP et al ) Long range BO order ~ 0.6 mm Long range nematic order Liquid like positional order, λ PO No twisting of the hexatic order (cholesteric-hexatic coupling) Kamien and Levine, PRL (2000).

60 Columnar hexagonal - hexagonal mg/ml 2D 3D A 2D crystal. Scattering from a columnar hexagonal phase reveals hexagonal order in the equatorial reflections. Hexagonal phase is a real 3D crystal, order in all directions. Relative displacements of 1/6 DNA pitch along z-axis. Livolant et al., Nature (1989). Changes in DNA pitch as a function of density. This is the range of densities of the Franklin-Gosling experiment. Livolant and Leforestier, Prog. Pol. Sci. (1996).

61 Orthorhombic 670 mg/ml Orthorhombic phase is also a real 3D crystal. A distorted hexagonal lattice. Relative displacement of m1 and m2 is ~ 1/3 DNA pitch along z-axis. A (volume) discontinuous transition from the hexagonal to orthorhombic. The orthorhombic lattice parameters also change as a function of the DNA density. Durand, Doucet, Livolant, J. Phys. France II (1992). Here too the pitch of the molecule changes.

62 The relevance of high density mesophases.. organisms exposed to prolonged stress can circumvent their fate through formation of highly ordered DNA assemblies... Minsky, Nature D. Nelson, Nature (1995) Cryo-electron microscopy of ε15. The genome is (hexagonally) packed in coaxial coils in at least three outer layers. Jiang et al Type II SC: Tension Non-chiral Magnetic field Temperature London repulsion DNA: Bending Chiral Density Ionic strength Debye-Huckel repulsion

63 Why does DNA make ordered phases? (C. Holm and M. Deserno) S. Fraden, Purified TMV in aqueous buffer. Top phase is isotropic, bottom is nematic. Onsager (1949) excluded volume interactions. Length to thickness ratio. HA or ss-dna For stiff polymers the role of the length is played by the persistence length. Khokhlov-Semenov et al No order in flexible polymer solutions. ds-dna persistence length (0.2 M NaCl) 50nm E~300 MPa (plexiglass)

64 DNA persistence length Nature, february 7, Anton Peterlin ( ) Analysis of light scattering (Dotty and Bunce) on DNA solutions. Watson-Crick-Franklin double helix Nature, april 25, A. Peterlin was one of the founding fathers of the physics department of UNI-LJ and the first director of the J. Stefan Institute.

65 What and why - the order of DNA mesophases Why does the molecular pitch change? Overwinding of DNA coupled to ordering. Why is hexatic order not modified by the cholesteric order? Hexatic order should rotate and give a circular diffraction pattern. Why are there relative displacements in the z direction of nearest neighbors in hex and ortho phases? DNAs do not behave like featureless cylinders. Why the orthorhombic symmetry? If DNAs were cylinders, they would pack hexagonally. IP = % IP = %

66 DNA - a helical molecule Description of the order in a parent nematic phase of DNA molecules. translation rotation Initial position and initial phase. screw-like motion The CCV theory (1952). The relative azimuthal orientation: Three possible types of fluctuations in a nematic composed of helical molecules. The phase ψ is a characteristic of each molecule. ψ i ψ j de Gennes macroscopic order parameter. P=0 parent nematic phase P=1 corresponds to a screw-like phase (Manna, Lorman, RP and Zeks, 2007)

67 Screw-like phase The order of this phase can be described by a helical polar vector field. The molecules are chiral and helical. with An appropriate order parameter for the transition into this phase is The Landau free energy with the chiral (Lifshiz) term.minimization of this free energy gives th eequilibrium phases. The minimum of this free energy close to the transition point (P=0) is obtained as: The chiral term introduces a renormalization of the phase of the order parameter. Since it describes the helical molecule this means that its pitch is changed! Instead of twisting the hexatic order, the screw phase overwinds the molecule. The solution of the puzzle.

68 Modulation of the molecular pitch Remember this? Overwinding of DNA. So the screw-like order can induce a change in the pitch of the molecules. Renormalization of the phase of the order parameter of drives the renormalization of molecular pitch. with p given by the B-DNA value (3.4 nm). Of course if DNA would not be soft an enormous energy would be needed to accomplish this. What is actually the energy needed to change the pitch of DNA? The twist energy is given by C is the twist persistence length ~ 75 nm. Taking the above experimental data of F. Livolant one obtains the torque needed to get these changes in pitch of The measured energies of DNA twisting and bending are ~ pn nm and the measured overwinding torques are indeed of this magnitude (30 pn nm). DNA is fortunately soft enough.

69 A side benefit of the screw-like phase Experimentally seen displacement in the z-direction for 1/3 or 1/6 of the pitch. Hexagonal Orthorhombic There exists a coupling between the order parameter of the screw-like order and the symmetry of the lattice. Too technical to derive here, but displacements of both p/6 and p/3 can be derived for hexagonal and orthorhombic symmetries. Exactly as seen in experiments!

70 Direct visualization of the screw-like phase Ordered phases of bacterial flagella (Barry et al. PRL 2006). Three pitches: zero, 1.1 μm and 3.6 μm. Screw-like fluctuations observed directly. - periodicity - direction of molecular axis - nature of fluctuations - modulation of the density For DNA (because of the difference of scale) this experiment is not feasible. Corresponding order under Xed polarizers and fluorescent labeling. Resonant X-ray scattering?

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72 The Boyle experiment - osmotic stress method Osmotic stress method (Parsegian & Rand, 80) pressure PEG osmotic pressure Π dv µdn

73 Experimental evidence for the great electrostatic divide Electrostatic repulsion: not eactly PB but a lot stronger. Correlation attractions. Hydration attractions. ~ 0.1 kt/ base pair. Podgornik et al Rau et al., Monovalent counterions Polyvalent counterions Electrostatics can only be seen indirectly, as modified by the presence of conformational fluctuations. Renormalized value of λ.

74 The nature of DNA-DNA interactions DNA as a cylindrical cow. No molecular details! Not good at large densities. A pair of DNAs with polycounterions. Gronbech - Jensen et al Charge: 2 e 0 / 3.4 Å ~ e 0 / nm 2 Hexagonal array of DNA with poly-counterions. Lyubartsev and Nordenskiold, DNA is a very charged up molecule. Huge electrostatic interactions in salt solutions.

75 Detailed nature of ES interactions Non-monotonic interactions (Kornyshev-Leikin, 1997). Theory based on the linearized PB equation! (π/2) Optimal angle. Orientational dependence. Two DNA molecules with separated + and - charges along the length can attract each other at preferred mutual orientation.

76 Assumptions of the KL theory Detailed charge distribution on the DNA helices - SC approach. The grooves act as flexible ionophores that coordinate counterions in the duplex. Kornyshev - Leikin, Debye-Hueckel interaction: smeared charge smeared counterions Manning condensation Kornyshev-Leikin interaction: explicit charge explicit counterions linearized PB Hud and Plavec, Na + in the minor groove of helix. But this preference is not universal and is stronger for polyvalent counterions. Non-electrostatic interactions.

77 WC (PB) and SC calculations An ab initio calculation. Poisson-Boltzmann vs. strong coupling limit Strongly coupled counterion spatial distribution. Whereas in the KL theory the strongly coupled counterions are put in by hand, here all interactions are electrostatic. Qualitatively the results are the same. M. Kanduč (2007)

78 Simulations Realistic geometric models of DNA. Similar (but not exactly the same) conclusions. Allahyarov et al., R A R A R Dubious validity for large DNA density. No explicit water. Schematics of the orientational effect. Strand opposition. explicit DNA structure explicit counterions explicit salt ions different salt concentrations

79 A two-state KL model The interaction potential between two helices is not a monotonic function of separation any more. φ mutual azimuthal orientation of the DNAs The vectors joining axes with the points where 5-3 strand hits the plane are conventionally called «transverse polarization» vectors p Simplified Form of the Interaction Potential (Harreis et al. 2002) Critical distance R 0 ; for R > R 0 (large interaxial distances): φ = 0 (parallel orientation) for R < R 0 (small interaxial distances) φ 0 (equilibrium nonzero angle orientation) A two state model: I. Intermediate densities : Parallel alignment of transverse polarization vectors II. High densities : Preferred angle between transverse polarization vectors

80 Azimutal interactions in constrained DNA nematics Assume a DNA fiber where molecules are not allowed to fluctuate translationally. Only azimutal fluctuations are allowed. A proper structural description of DNA on this level is: The origin of the azimutal angle can be chosen arbitrarily, as the KL interaction depends only on the difference! This description induces a macroscopic order parameter: Transverze polarization (TP) vector. Lorman, RP and Zeks, PRL 2005.

81 Intermediate density hexatic - TP coupling ϕ = 0 Line hexatic phase as the parent phase. Three order parameters: Transverze polarization: Hexatic order: Deformation vector: Bond angle ϴ. Follows polarization. Curie principle. Construct the Landau free energy with appropriate invariants of thse two order parameters: Three types of phases from the minimization of FE: A. deformed hexatic in the direction of first neighbors (excluding the central) ϕ = 0 B. deformed hexatic in the direction of second neighbors (excluding the central) ϕ = π/6 C. deformed hexatic in general direction 0 < ϕ < π/6.

82 High density hexagonal - TP coupling ϕ 0 Again line hexatic phase as the parent phase. But a bit more complicated. The nearest neighbor TP vector are at a finite angle! This is possible only in periodic systems in terms of TP. Crystalline order is thus a consequence! Azimutal frustrations of the lattice. Start with e.g. A. structure deformed hexatic a. one dimensional crystal in e.g. x-direction, one angle of reorientation b. one dimensional crystal in general direction, one angle of reorientation c. two dimensional crystal in, two angles of reorientation Freezing of the bond orientational order at higher densities. Angular correlations. Observed already by Franklin in 1952, but not understood until now!

83 Is this real? Can one observe azimutal correlations predicted on the theory that is based on the twostate KL model? Kornyshev and Leikin (PRL, 2005) reanalyzed the old diffraction data by S.B. Zimmerman. From almost contact to 40 Å separation. - equatorial n=0 peaks change with DNA density, positional order - for small inter-chain correlations n 0 peaks should show no density variation Equatorial lines remain the same - B-DNA for all densities. Strong azimutal correlations of the type A, 2a or 2c.

84 A related problem of azimutal correlations The structural cascade of chromatin in eucaryots starts with the nucleosomal particle. From DNA to chromosomes. Many levels of as yet poorly understood organization but with interesting physics (Schiessel, 2002). Viruses: toroidal packing just as in DNA collapse in vitro. Bacteria (prokariotes): nucleoid, a loose DNA - protein gel Existence of a polar dyadic axis. A histone octamer (4 X 2) of 4 core histones: H2A, H2B, H3 and H4 147 bp DNA wrapped 1.75 times in a lefthanded helix, stable up to 0.75 M salt

85 The nucleosomal core particle (NCP) The elementary particle of chromatin organization. Missing just linker DNA and H1 histone. Folded vs. expanded N-tails (N-terminals of histone core proteins) Folded tails below 50 mm and expanded from 50 mm and above. Folded N-tails (Mangenot et al. 2002) Expanded N-tails RP and Licer (2006) Bridging interactions can lead to association of nominally equally charged colloids. Important also at large densities of NCPs where they make condensed mesophases... Muehlbacher, Schiessel and Holm, (2005).

86 NCP mesophases in ionic solutions S. Mangenot, A. Leforestier, D. Durand, and F. Livolant ( ). Intriguing texture and order of mesophases... lamellar phase inverted hexagonal phase orthorhombic phase X-ray scattering plus optical texture serve to identify the phases. They come in very exotic varieties...superb work of F. Livolant et al.

87 More NCP mesophases in ionic solutions S. Mangenot, A. Leforestier, D. Durand, and F. Livolant ( ). X-ray scattering plus optical texture serve to identify the phases. They come in different varieties: isotropic, lamellar, hexagonal, inverted hexagonal.

88 NCP mesophase phase diagram Changing the osmotic pressure (i.e. density) and ionic strength of the solution. Here the parent phase is assumed to be columnar hexagonal. Constructing the order parameter based on the existence of dyadic axis: Three order parameters out of polarization vector: And the density variation:

89 Lamellar phase of NCPs Starting from a parent hexagonal phase transitions into several other phases of lower symmetry are possible, e.g. a lamellar phase (very technical). Duality between the P ordering and the u displacement (plus chirality). Curie s principle. Writing down the free energy from invariants of the three order parameters that are defined via transverse polarization vector. lamellar phase inverted hexagonal phase orthorhombic phase Free energy minimization. (Manna, RP, Lorman and Zeks, 2007)

90 Inverted hexagonal and orthorhombic lattice Apart from the lamellar phase other phases and transitions between them are possible. lamellar phase inverted hexagonal orthorhombic phase Phase diagram. These last three phases have not been observed (yet). Nature of the 2D molecular displacement. Lorman, RP, Zeks EPL (2005).

91 From observing to understanding DNA mesophases Not that difficult to observe, but not simple to understand. cholesteric phase line hexatic phase Livolant et al Many details still need to be worked out, but the big picture is most probably correct.

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