Molecular attractions: a.) van der Waals interactions b.) electrostatic correlation interactions c.) polyelectrolyte bridging interactions

Rudi Podgornik

Laboratory of Physical and Structural BiologyNational Institute of Child Health and Human Development

National Institutes of HealthBethesda, MD

Department of PhysicsFaculty of Mathematics and Physics, University of Ljubljana

Department of Theoretical PhysicsJ. Stefan Institute, Ljubljana Slovenia

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

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)...

- 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

What are viruses?1. Acellular (nucleic acid with protein capsid +/- membrane envelope)2. Obligate intracellular parasites3. No ATP generating system4. No Ribosomes or means of Protein Synthesis

- 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.

TMV

Hepatitis A

HIV

Bacterophage T7(bacterial virus)

Ebola

influenza

Rhinovirus(common cold)

Bacteriophages.

capsid

DNA

- 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.

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).

Crick & Watson, Nature (1956).

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 (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.

Crick-Watson hypothesis

An alternative way of building a virus capsid is to arrange protein subunits in the form of a hollow quasi-spherical 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.

Cubic (icosahedral) symmetry

Sheets with hexagonal symmetry into spheres. No way!

Folding a sheet of local hexagonal symmetry into a sphere.

bacteriophage ΦX 174

Packing of triangles into a “sphere”:tetrahedron, octahedron and icosahedron.

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.

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)

Folding of hexagonal sheet into a geodesic dome (Buckminster Fuller, 1960).

Pentamers and hexamers.

Fivefold defects make a “sphere” out of a hexagonal sheet.

P. Ziherl

Cationic lipids (single chain) below chain freezing.

Dubois et al. 2001.

CTAOH, CTABr.1 micron in size!

Altschuler et al. 1997.The Thomson problem.

HIV-1. Welker et al. 2000.

Ganser et al. 1999.

Not always icosahedral: HIV cores.

12 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)

A zoo of icosahedral viruses, Baker et al. 1999.

Each has a different triangulation number.

In 1955, Fraenkel-Conrat & Williams showed that mixtures of purified tobacco mosaic virus (TMV) RNA & coat protein were incubated together, virus particles formed.

- 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.

One TMV virus:1 RNA + 2130 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!)

Viruses are equilibrium structures!

A pronounced difference in the details of the shape between small

and large viruses.

spherical vs. facetedWhy?

Shape universality and size variability.

Continuum theory of viral shapes. Föppl - von Karman equations (1907)

2D elasticity curvature energy

Föppl - von Karman number:

Larger values of γ > 154 lead to pronounced faceting.

The triangulation indices are (6,6).

Horribly non-linear, difficult to solve.

Lidmar et al., 2003.

Continuum theory of viral shapes.

(2,2) (4,4)

(6,6) (8,8)

γ = 45, 176, 393, 694. Lidmar et al., 2003.

γ = 8000000.

Sharpening of the edges.

Systematic solutions...

Solutions of the continuum theory of viral shapes.

Fitting the solution of Föppl - von Karman equation to real virus shape

Lidmar et al., 2003.

Bacteriophage HK97 (full virus and cross-section on the r.h.s.) The best fit occurs at γ = 1480.

Comparison with the real world of viruses.

Siber and Podgornik, 2008.

Stability and collapse of viral capsids.Osmotically stressing viral capsids.

At a critical value of the osmotic pressure.Evilevitch 2008.

Two dimensionless parameters:

(Kleinschmidt et al., 1962)

Bacteriophage T2

P ~ 100 atmρ ~ 100 mg/ml

(Champagne at 5-6 atm)

~ 630 m long~ 1 mm thick

pack into 25 cm

6000 times compaction.

Similar type of packing:

bacteriophage T2 bacteriophage φ27 herpes simplex chicken pox shingles

High packing density.

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.

Earnshaw & Harrison, Nature 1977.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

Details of viral genome packing.

Direct experimental observation of DNA packingCryo-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. 2006.

Molecular mechanics

(simulations).Arsuaga et al.

2002.

Numerical minimization of

single layer.Slosar and

Podgornik. 2006.

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)

Cryomicrographs of T4 bacteriophages.

Optical diffraction of the capsid.

(Lepault et al. 1987)

Molecular simulations - consensus.

A completely disordered spool of 10 kb in a spherical

volume of ~ 190 Å.

A completely ordered 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.

More detailed computer generated spooling of DNA inside the capsid.Arsuaga et al. 2002.

Review by Angelescu and Linse (2008).

Various computer models give similar results for DNA packing within bacteriophages.

Computer simulations of DNA packing inside the capsidSimulation of a stiff chain within a spherical enclosure.

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. 2002.

Molecular simulations.

Odijk-Gelbart inverse spool

Grosberg, 1979. Klug and Ortiz, 2003. Odijk and Slok, 2003. Purohit et al. 2005.

Different authors differ on the details of the free energy expression for the DNA inside the inverse spool. But the spool itself is assumed.

Total (free) energy = bending energy + interaction energy

authorsauthors

Odijk & Slork 2003.

A mechanical or nanomechanical theory of viral packing. Started with Grosberg in ‘79.

EXPERIMENTS?

“Boyle” experiment in viroCompressing 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.

Equivalence of osmotic pressure

(PEG or DEX etc.)Podgornik et al. 2001.

DNA osmotic pressure - equation of state.

Different regions of DNA density correspond to different mesophases.

The equation of state of DNA in the bulk is its osmotic pressure as a function of DNA densityfor any given (temperature, ionic strength, nature of salts...) condition.

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

Monovalent counterions. Polyvalent counterions.

Electrostatic repulsion.Fluctuation enhanced.

Correlation attraction.~ 0.1 kT/ bp.

Electrostatics can only be observed masqued by fluctuations.

Monovalent vs. polyvalent counterions

DNA osmotic pressure - phase diagram.

Durand, Doucet, Livolant (1992) J. Physique 2, 1769-178Pelta, Durand, Doucet, Livolant (1996) Biophys. J., 71, 48-63 3

As observed by F. Livolant

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

kT ~ 9.1 nm pN.Bustamante et al., 2001.

Energetics of viral packing

Packing forces and packing speed.

Stalling force of the portal motor of φ27.

6.6 μm of DNA take ~ 5.5 min to pack.Total work done ~ 20000 kT.

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?

Optical tweezers Bustamante et al. 2001.

Direct experimental observation!

Osmotic equilibrium in viruses

Grayson et al. 2005.

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.

Boyle experiment.

Again equivalence of osmotic pressure

(PEG or DEX etc.)Evilevitch et al. 2008

The energetics of genome packing.

Continuum nematic nanodrop model of a virus.

Klug et al. 2005.

elastic constants(bare & interaction)

total free energy density

persistence length of DNA, ~ 50 nm.

Jiang et al. 2006.

Equilibrium local osmotic pressure and the inverse spool

interaction pressure curvature pressure total pressure

Inverse spool! Derived from nanomechanics.Quadratic depletion at the center.

No need to assume the depletion at the core.

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.

Thermodynamic equilibrium is given by:

Two asymptotic forms of the solution.

Cylindrically symmetric spooling inside viral capsid

Packing symmetry and loading curve

This we call the osmotic loading or osmotic encapsidation curves.

The inverse method (from elasticity theory) : assume a director profile and its symmetry.

Define the amount of DNA within the capsid as:

Angelescu, Linse (2008). Siber et al. (2008).

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

Grayson et al. (2006) data for bacteriophage YcI60 (48.5 kbp).

Results of the continuum LC drop model comparedwith Evilevitch et al. data.

Siber et al. 2008.

A deconvolution of the bulk osmotic pressurevia the osmotic equilibrium equation.

Monovalent salt!NaCl

Good fit for small concentrations.

Viral DNA equation of stateEncapsidated fraction as a function of external osmotic pressure.

Small (almost negligible) effects of DNA elasticity.

DNA elastic constant.E~300 MPa (plexiglass)

Different DNA loading curves for mono and poly-

valent salts.

Comparison monovalent salt vs. polyvalent salt ......

Comparison of monovalent salt and divalent salt. Inhibition pressure lowering!

Jumps in the osmotic pressure. Attractive interactions

(like van der Waals isotherms) Jumps in OP leadto jumps in loading.

(NaCl)

(MnCl )

Inhibition pressure

2

A hot dog and viruses on the side, please!

Osmotic pressure and self assembly of RNA virusesIn 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.

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.

Complexation free energy

Influence of N-tails

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.

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.

FINIS