Physical Models of Viruses
Brain cancer cells
Bacteria & Cells:
•Do not form crystals.
•Have no particular symmetry.
•Exceptions.
E. coli.
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Rhinovirus
320 Angstroms
Molecular Weight: > 106
Symmetry Axes: 2 fold, 3 fold, & 5 fold.
Cryo-TEM
X-ray Crystallography
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Flock-House Virus
Genome: two RNA molecules
(4,800 bases)
“Capsid”:
180 identical proteins
Capsid Proteins: “Amphiphilic” molecules
Hydrophobic
Hydrophilic
Icosahedral Symmetry
Icosahedron: 20 identical equilateral triangular faces.
 15 two-fold rotation axes
 10 three-fold rotation axes
 6 five-fold rotation axes
Sir Aaron
Klug
Why are viruses icosahedral?
Francis Crick (1956):
1) The small size of the virus genome only allows only a single
protein type for the coat. The capsid must be made of many
identical units of a single protein type.
2) The viral shell should be symmetric, so these identical proteins
can occupy identical minimum energy environments. (That’s
how crystals are organized)
3) The viral shell should have optimal “information storage”
capacity (i.e., a low surface to volume ratio).
Mathematical Problem.
Divisions of a Spherical Surface into identical unit cells.
“Unit Cell”
Dodecahedron Icosahedron
Tetrahedron Cube Octahedron
Five “Platonic”Solids
Lowest surface/volume ratio:
Icosahedron 60 proteins
Actual capsids: # proteins equals 60 times an integer (“T Number”)
T-Number Classification: (Caspar & Klug, 1968)
-> Construct closed icosahedral shells from hexagonal sheets.
-> Insert 12 pentamers.
12 five-fold sites
# Capsomers: 10 T + 2 with T=1, 3, 4, 7,… (12 pentamers)
# Proteins: 60 T
Nearly all spherical viral shells follow T-Number classification!
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Buckminster Fuller
How is a Virus Assembled ?
Ceres & Zlotnick (2002): Hepatitis B Virus
30 nm
240 protein capsid
Law of Mass Action: minimize free energy F
formation energy
free energy
F
f
f
f
 (1  f )ln  (1  f )  ln   fc 0 (N )

N
N N
dimers
equilibrium
capsids: N = 120 (T = 4)
N = 90 (T = 3)
F
 0  f  NK N 1(1 f ) N
f
K  exp f
0
c

critical
concentration

1
f* 
2
* N 1
   NK  1
 2 
f: fraction of protein
in capsid form.
: concentration of
proteins.
(van der Schoot &
Kegel)
Capsid Protein Fraction
1
0.8
0.6
f
f
0
f
Fit: c / kBT : 1,000
0.4
0.2
0
0
1
2
c/c*
3
4
  
  * 
Protein concentration
Virus Life Cycle (Polio)
Spontaneous
Assembly
VIRAL PROTEINS
+
VIRAL RNA
Formation Energy.
Capsomer
*Treat capsomers as attractive disks. (hydrophobic edge)
*Pack maximum number of disks on the sphere: “Tammes Problem”.
Surface Area N disks
 N  
Surface Area Sphere
Measure of attractive energy.
*Maximum coverage of a flat surface: Hexagonal close-packing.
  0.91
*Maximum coverage of a sphere:
  0.91
Tammes shells with high coverage: N= 12, 24, 32, 48, or 72.
4-fold axis !
13 “Archimedean” Solids.
“Snub Cube”
Only Tammes shell with icosahedral symmetry is N=12!?
*Model Unrealistic?
Polyoma
virus
Aggregates of capsid proteins:
Spherical with N=12, N=24, & N=72.
(Salunke)
N=24: Snub-Cube!
What’s special about Polyoma ?
Cowpea Chlorotic Mottle Virus (“CCMV”)
TWO capsomer types:
Pentamers (12)
Hexamers (20)
Icosahedral Symmetry: Pentamers at 12 five-fold sites
Two-State Capsomer Model
Capsomer: Hexamer or Pentamer
2R6
Hexamer
2R5
*Size Difference R5 / R6 = 0.93
*Adjustable Energy Difference: E
*N5+N6=N fixed
Variable
Pentamer
*Numerical simulation
(Roya Zandi & David Reguera)
*Energy per capsomer versus the Number N of capsomers (E=0)
Capsid structure associated
with the minima of energy
Capsid Structures: “T-Number icosahedra”
Increase E ?
Tammes Structures!
Advantage icosahedral symmetry?
Structural view:
Apply 2-state model
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Distribution of Forces.
High Elastic Stress
T=9
Intermediate Stress
Low Stress
Five-fold sites: Focus of strong
radially outward force.
T=13
Genome release: Pentamer ejection (Tymovirus)
Elasticity Theory
(D.Nelson)
“Young’s Modulus”
Thickness
R
"FK #" 
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YhR
2

“Bending Modulus”
Small FK#: Spherical Shell.
Large FK#: Icosahedral Shell.
Cryo-TEM
(Baker)
5
5
Repeat Motif
“Capsomer”
Icosahedral
5
Rhinovirus
Slide 5
Spherical
Are viral capsids really miniature elastic shells ?
•Capsid proteins have a complex internal structure.
“Atomic Force Microscope” (AFM)
Force (F) versus Compression (x).
x
F(x)
Hollow Elastic Shell:
F(x) = kx
Yh 2
k  2.25
R
k: “spring constant”
Y: Young’s Modulus
h: Thickness
R. Radius
CCMV – empty capsid
J-P Michel&
C.Knobler
Force (nanoNewton)
1.2
indentation
z= 7 nm
1.0
F, nN
0.8
0.6
Glass F-Z curve
F(z) = kz
Virus F-Z curve
0.4
k=0.13 N/m
Tip-sample
contact
0.2
Yh 2
k  2.25
R
Tip approach
0.0
0
5
10
15
20
25
30
35
nm
Tip Displacementz,(nm)
40
45
50
55
60
Y≈ 0.1-1.0 GPa
•At pH 5, empty CCMV viral capsids obey elastic shell
theory for compressions of 10-15%
•To first approximation, capsids can be described as
uniform thin shells with a Young’s modulus of about 1 GPa
•They can be damaged by repeated compressions or by
compressions greater than 25%
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Does the RNA inside the capsid respond as a solid ?
Solid elastic spheres go non-linear under compression
(H. Hertz, 1880)
F(x)  x 3/2 R1/2
Does the RNA inside the capsid respond as a fluid ?
Capsid wall: semi-permeable --> Internal “Osmotic Pressure” P
F(x)  k  PR / 2 x
Osmotic Pressure
Fitted value: of order 100 Atm!
Self-Assembly by “Amphiphiles”
Hydrophilic
Hydrophobic
Vesicles
“Free Energy Minimization”