Molecular attractions:

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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 2007 Taiwan International Workshop on Biological Physics and Complex Systems (BioComplex Taiwan 2007)

2 Conceptual introduction to physics of viruses: phenomenology of viruses - bacteriophages elastic theory of viral capsids stability of viral capsids DNA packing in bacteriophages DNA nematic nanodrop theory DNA encaspidation Mostly description of work with Antonio Šiber, IP and V. Adrian Parsegian, NIH. EPJE (2008) PRL submitted (2008)...

3 What are viruses? 1. Acellular (nucleic acid with protein capsid +/- membrane envelope) 2. Obligate intracellular parasites 3. No ATP generating system 4. No Ribosomes or means of Protein Synthesis - Nucleic acids : DNA or RNA; single stranded vs double stranded; linear vs circular; one or more pieces (segmented genome) - Capsid: helical, icosahedral (complex protection, attachment, enzymatic) - Envelope derived from host membrane lipids and virus proteins - crystallization of a virus first reported in the 1930s. - first atomic resolution structure of a virus was 1978, tomato bushy stunt virus. Although some viruses are very fragile & are essentially unable to survive outside the protected host cell environment, many are able to persist for long periods, in some cases for years in hostile conditions.

4 TMV HIV influenza Hepatitis A Rhinovirus (common cold) Bacterophage T7 (bacterial virus) Ebola

Bacteriophages. - assemble the particle utilizing only the information available from the components which make up the particle itself (capsid + genome).

capsid DNA The three families of tailed dsdna viruses (phages) that infect bacteria.

5 Bacteriophages. - assemble the particle utilizing only the information available from the components which make up the particle itself (capsid + genome). - form regular geometric shapes, even though the proteins from which they are made are irregularly shaped. capsid DNA The three families of tailed dsdna viruses (phages) that infect bacteria. a, Myoviruses, contractile tails, are typically lytic and often have relatively broad host ranges. b, Podoviruses, short non-contractile tail, are also typically lytic and have very narrow host ranges. c, Siphoviruses, long non-contractile tails. Relatively broad host range, and many are capable of integrating into the host genome. Scale bar, 50 nm Curtis A. Suttle Nature (2005).

particle. Crick & Watson, Nature (1956).

6 Crick-Watson hypothesis Protein subunits in a virus capsid are multiply redundant, i.e. present in many copies per particle. Damage to one subunit may render that subunit non-functional, but does not destroy the infectivity of the whole particle. Crick & Watson, Nature (1956). Crick &Watson (1956), were the first to suggest that virus capsids are composed of numerous identical protein sub-units arranged either in helical or cubic (=icosahedral) symmetry after seeing EMs.

Cubic (icosahedral) symmetry An alternative way of building a virus capsid is to arrange

within. 20 equilateral triangles arranged into a sphere. As simple as it comes.

Folding a sheet of local hexagonal symmetry into a sphere.

7 Cubic (icosahedral) symmetry An alternative way of building a virus capsid is to arrange protein subunits in the form of a hollow quasispherical structure, enclosing the genome within. 20 equilateral triangles arranged into a sphere. As simple as it comes. 60 identical subunits form a capsid. 3 protein subunits per triangular face. Most have more. Folding a sheet of local hexagonal symmetry into a sphere. bacteriophage ΦX 174 Packing of triangles into a sphere : tetrahedron, octahedron and icosahedron. Sheets with hexagonal symmetry into spheres. No way!

protein subunits is to use rotational symmetry &

around the circumference of a circle to form a

of one another to form a cylinder, with the

8 Helical symmetry. The simplest way to arrange multiple, identical protein subunits is to use rotational symmetry & to arrange the irregularly shaped proteins around the circumference of a circle to form a disc. Multiple discs can then be stacked on top of one another to form a cylinder, with the virus genome coated by the protein shell or contained in the hollow centre of the cylinder.

9 Physical principles of viral shapes. Crick & Watson, Nature (1956). Crick &Watson (1956), were the first to suggest that virus capsids are composed of numerous identical protein sub-units arranged either in helical or cubic (=icosahedral) symmetry after seeing EMs. Caspar & Klug, (1962). Principle of quasi-equivalence. Triangulation number T. 10 (T-1)

10 Fivefold defects make a sphere out of a hexagonal sheet. Folding of hexagonal sheet into a geodesic dome (Buckminster Fuller, 1960). Pentamers and hexamers. P. Ziherl

11 Cationic lipids (single chain) below chain freezing. CTAOH, CTABr. 1 micron in size! Dubois et al Altschuler et al The Thomson problem.

12 Not always icosahedral: HIV cores. HIV-1. Welker et al fivefold defects needed to close the shape (7 top + 5 bottom). Quantization of cone angles: 112.9º (P=1), 83.6º (P=2), 60º (P=3)38.9º (P=4), 19.2º (P=5) Ganser et al

13 A zoo of icosahedral viruses, Baker et al Each has a different triangulation number.

coat protein were incubated together, virus particles formed.

First observation of a self-assembly of a biological particle!

14 Viruses are equilibrium structures! In 1955, Fraenkel-Conrat & Williams showed that mixtures of purified tobacco mosaic virus (TMV) RNA & coat protein were incubated together, virus particles formed. One TMV virus: 1 RNA protein molecules. A two-molecule virus. Very simple! First observation of a self-assembly of a biological particle! F = W - TS = minimum (driven by physics only!) - assemble the particle utilizing only the information available from the components which make up the particle itself (capsid + genome). - form regular geometric shapes, even though the proteins from which they are made are irregularly shaped.

15 Shape universality and size variability. A pronounced difference in the details of the shape between small and large viruses. spherical vs. faceted Why?

16 Continuum theory of viral shapes. Continuum theory of viral shapes. Föppl - von Karman equations (1907) 2D elasticity curvature energy Föppl - von Karman number: Horribly non-linear, difficult to solve. Larger values of γ > 154 lead to pronounced faceting. The triangulation indices are (6,6). Lidmar et al., 2003.

17 Solutions of the continuum theory of viral shapes. (2,2) (4,4) Systematic solutions... γ = (6,6) (8,8) Sharpening of the edges. γ = 45, 176, 393, 694. Lidmar et al., 2003.

18 Comparison with the real world of viruses. Fitting the solution of Föppl - von Karman equation to real virus shape Bacteriophage HK97 (full virus and cross-section on the r.h.s.) The best fit occurs at γ = Lidmar et al., 2003.

19 Stability and collapse of viral capsids. Osmotically stressing viral capsids. At a critical value of the osmotic pressure. Evilevitch Two dimensionless parameters: Siber and Podgornik, 2008.

20 High packing density. Bacteriophage T2 ~ 630 m long ~ 1 mm thick pack into 25 cm 6000 times compaction. P ~ 100 atm ρ ~ 100 mg/ml (Champagne at 5-6 atm) Similar type of packing: bacteriophage T2 bacteriophage φ27 herpes simplex chicken pox shingles (Kleinschmidt et al., 1962)

21 Cerritelli et al. Cell 91 (1997) 271. T7 bacteriophage. Organization of ds-dna inside the viral capsid nematic or hexatic-like order with ~25 Å separation.

$Scattering of X-rays from P22 phage heads. Diffraction ring corresponds to 25 Å.$ $Packing models based on the X-ray diffraction and electron densitometry data: -$

22 Details of viral genome packing. Earnshaw & Harrison, Nature Scattering of X-rays from P22 phage heads. Diffraction ring corresponds to 25 Å. Model of packing from densitometry traces. Packing models based on the X-ray diffraction and electron densitometry data: - ball of string - coaxial spool - ordered chain folding

Direct experimental observation of DNA packing Cryo-electron

The genome packed in coaxial coils in at least three outer

core and into the portal complex. Jiang et al. 2006.

23 Direct experimental observation of DNA packing Cryo-electron microscopy, epsilon15. The genome packed in coaxial coils in at least three outer layers, terminal 90 nucleotides extend through the protein core and into the portal complex. Jiang et al Molecular mechanics (simulations). Arsuaga et al Numerical minimization of single layer. Slosar and Podgornik

) liquid crystal model with local parallel

24 Models of viral packing a.) concentric shell or toroidal winding (Earnshaw & Casjens 1980) b.) spiral fold model (Black et al. 1985) c.) liquid crystal model with local parallel packing (Lepault et al. 1987) d.) ball of yarn Earnshaw et al. (1987) (Lepault et al. 1987) Cryomicrographs of T4 bacteriophages. Optical diffraction of the capsid.

25 Molecular simulations - consensus. More detailed computer generated spooling of DNA inside the capsid. Arsuaga et al A completely disordered spool of 10 kb in a spherical volume of ~ 190 Å. A thermally annealed spool of 10 kb in a spherical volume of ~ 190 Å, from an initial ordered configuration. A completely ordered spool of 10 kb in a spherical volume of ~ 190 Å.

26 Computer simulations of DNA packing inside the capsid Simulation of a stiff chain within a spherical enclosure. Review by Angelescu and Linse (2008). Various computer models give similar results for DNA packing within bacteriophages.

27 Molecular simulations. Optimal packing of a relaxed closed circular DNA 10 kb into a sphere with substantial free volume. The initial structure was axially spooled along the full length of the molecule. The outer region consists of two coaxially spooled layers, containing approximately 7.5 kb. The cavity inside these layers is occupied by the second coil (red). The structure is not knotted. Arsuaga et al

Odijk-Gelbart inverse spool A mechanical or nanomechanical theory of

Klug and Ortiz, 2003. Odijk and Slok, 2003. Purohit et al. 2005.

for the DNA inside the inverse spool. But the spool itself is assumed.

28 Odijk-Gelbart inverse spool A mechanical or nanomechanical theory of viral packing. Started with Grosberg in 79. Grosberg, Klug and Ortiz, Odijk and Slok, Purohit et al Different authors differ on the details of the free energy expression for the DNA inside the inverse spool. But the spool itself is assumed. Odijk & Slork Total (free) energy = bending energy + interaction energy EXPERIMENTS? a

balance (osmotic stress technique, Parsegian et al. 80).

29 Boyle experiment in viro Compressing the DNA in solution or in a capsid by a piston or equivalently by an osmotic balance (osmotic stress technique, Parsegian et al. 80). Pressure as a function of volume or equivalently of density.

30 DNA osmotic pressure - equation of state. The equation of state of DNA in the bulk is its osmotic pressure as a function of DNA density for any given (temperature, ionic strength, nature of salts...) condition. Podgornik et al Equivalence of osmotic pressure (PEG or DEX etc.) Different regions of DNA density correspond to different mesophases.

Podgornik et al. 2000 Rau et al., 1997. Monovalent counterions.

31 Monovalent vs. polyvalent counterions Electrostatic repulsion. Fluctuation enhanced. Correlation attraction. ~ 0.1 kt/ bp. Podgornik et al Rau et al., Monovalent counterions. Polyvalent counterions. Electrostatics can only be observed masqued by fluctuations.

32 DNA osmotic pressure - phase diagram. As observed by F. Livolant Durand, Doucet, Livolant (1992) J. Physique 2, Pelta, Durand, Doucet, Livolant (1996) Biophys. J., 71,

polymerase 15 to 20 pn. DNA polymerase 35 pn myosin (contracts muscle fibers) 5 pn.

33 Energetics of viral packing Bacteriophage φ27 portal motor: 57 to 60 pn of force. Scaled up to human dimensions lift six aircraft carriers DNA pressure 60 atm (10 X Champagne bottle) RNA polymerase 15 to 20 pn. DNA polymerase 35 pn myosin (contracts muscle fibers) 5 pn. The motor has a 10 nm diameter ring of RNA between two protein rings very intriguing and different from other motors Bustamante et al., kt ~ 9.1 nm pn.

Optical tweezers Bustamante et al. 2001. Final pressure in the capsid 6 MPa.

34 Packing forces and packing speed. Direct experimental observation! 6.6 μm of DNA take ~ 5.5 min to pack. Total work done ~ kt. Stalling force of the portal motor of φ27. Optical tweezers Bustamante et al Final pressure in the capsid 6 MPa. Young modulus of the capsid ~ 100 MPa (aluminum alloy) Cocking of the DNA trigger followed by passive emission. Nature of DNA packing inside the capsid?

Osmotic equilibrium in viruses The energetics of genome packing. Boyle experiment. Grayson et al. 2005. Ejection % for EMBL3, lambda c160 and lambda c 221 bacteriophages.

35 Osmotic equilibrium in viruses The energetics of genome packing. Boyle experiment. Grayson et al Ejection % for EMBL3, lambda c160 and lambda c 221 bacteriophages. Approximate length of the genome is 37.7 kbp and 48.5 kbp. Main features of the experiment are captured by the inverse spool model. Again equivalence of osmotic pressure (PEG or DEX etc.) Evilevitch et al. 2008

36 Continuum nematic nanodrop model of a virus. Klug et al Jiang et al elastic constants (bare & interaction) persistence length of DNA, ~ 50 nm. total free energy density

37 Equilibrium local osmotic pressure and the inverse spool Thermodynamic equilibrium is 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.

38 Packing symmetry and loading curve The inverse method (from elasticity theory) : assume a director profile and its symmetry. Angelescu, Linse (2008). Siber et al. (2008). Cylindrically symmetric spooling inside viral capsid Define the amount of DNA within the capsid as: This we call the osmotic loading or osmotic encapsidation curves.

39 Density profile and loading curve. 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.

$Enacapsidated DNA fraction Monovalent salt! NaCl Good fit for small concentrations. Results of the continuum LC drop model compared with Evilevitch et al. data.$

40 Enacapsidated DNA fraction Monovalent salt! NaCl Good fit for small concentrations. Results of the continuum LC drop model compared with Evilevitch et al. data. Siber et al A deconvolution of the bulk osmotic pressure via the osmotic equilibrium equation. Grayson et al. (2006) data for bacteriophage YcI60 (48.5 kbp).

41 Viral DNA equation of state Encapsidated fraction as a function of external osmotic pressure. DNA elastic constant. E~300 MPa (plexiglass) Different DNA loading curves for mono and polyvalent salts. Small (almost negligible) effects of DNA elasticity.

42 Comparison monovalent salt vs. polyvalent salt... Inhibition pressure (NaCl) Jumps in the osmotic pressure. Attractive interactions (like van der Waals isotherms) Jumps in OP lead to jumps in loading. (MnCl ) 2 Comparison of monovalent salt and divalent salt. Inhibition pressure lowering!

43 A hot dog and viruses on the side, please!

44 Osmotic pressure and self assembly of RNA viruses In DNA bacteriophages it is large and positive. It was more than fifty years ago since Fraenkel- Conrat and Williams demonstrated that fully infectious tobacco mosaic viruses could be created simply by mixing the viral RNA molecules together with the viral proteins. Under the right conditions ph and salinity, the virusesrna in the optimal case are to an excellent approximation formed spontaneously, i.e., without any special external impetus. This suggests that the process of spontaneous assembly of simple viruses can be understood by relatively simple thermodynamics. Not all viruses self-assemble in in vitro conditions, but many simple viruses containing ssrna molecule do.

structure were represented as spheres of radius 3.4 Å which is the experimental resolution.

45 Complexation free energy This is One quarter of the cucumber mosaic virus capsid strain FNY. The image was constructed by applying the group of icosahedral transformations to the RCSB Protein Databank entry 1F15 and all atoms in the resulting structure were represented as spheres of radius 3.4 Å which is the experimental resolution. They were colored in accordance with their distance from the geometrical center of the capsid, so that the atoms that are farthest away from the center are orange, while those that are closest to the center and belonging to the capsid protein tails are light blue.

Our results show that the spatial distribution of protein charge determines the important features of the energetics of viruses with regard to salt concentration.

46 Our results show that the spatial distribution of protein charge determines the important features of the energetics of viruses with regard to salt concentration. We conclude that the delocalization of the charge density on the protein tails may contribute to the robustness of the viral assembly and we speculate that it may offer an evolutionary advantage to such viruses. Influence of N-tails

47 Viral osmotic pressure and ss-rna bridging Intriguingly, in the range of polyelectrolyte lengths for which the filled viruses are more stable than the empty ones, osmotic pressures are negative inward, i.e., the electrostatic forces act to decrease the radius of the capsid. Osmotic pressures vanish close to the border of feasibility of spontaneous self-assembly of filled capsids and change the sign afterwards. The typical magnitudes of the pressures are about 0.5 atm at physiological salt conditions, but we have found even smaller pressures for capsids of larger radii. Very similar results are also found for the capsid with the charges delocalized on the protein tails.

49 FINIS

Physics of DNA. R. Podgornik. Laboratory of Physical and Structural Biology. National Institute of Child Health and Human Development

Physics of DNA. R. Podgornik. Laboratory of Physical and Structural Biology. National Institute of Child Health and Human Development 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