Electrostatic theory of viral self-assembly
Tao Hu, Rui Zhang, B. I. Shklovskii
Department of Physics
University of Minnesota
Outline

Opimization of viral structure stability

Kinetics of viral self-assembly
Icosahedral viruses
Self-assembly from a solution of
capsid proteins and ss RNA.
 The number of capsid proteins is
60T, T=1, 3, 4, 7……
 The absolute value of ss RNA
charges is approximately twice
larger than the total capsid charge.

CCMV (T=3)
http://viperdb.scripps.edu/
P. van der Schoot and R. Bruinsma,
Phys. Rev. E 71, 061928 (2005).
For many viruses the capsid protein has long Nterminal tail, which carriers all positive charge.
ss RNA
N-terminal tail
capsid protein
See also:
V. A. Belyi and M. Muthukumar,
PNAS 103, 17174 (2006)
Virus
Qr
qp
qt
Nt  r  t Nd Nt
Brome Mosaic
Cowpea Chlorotic Mottle
Nodamura
Pariacoto
Sesbania Mosaic
Rice Yellow Mottle
Southern Bean Mosaic
Cocksfoot Mottle
Average
Cucumber Mosaic
Tomato Aspermy
Carnation Mottle
Tobacco Necrosis
Tomato Bushy Stunt
Average
3030
2980
4540
4322
4149
4450
4136
4082
3944
3214
3391
4003
3700
4776
3817
10
7
13
13
6
17
16
15
12
15
14
10
9
12
12
9
9
18
14
6
12
14
13
12
12
12
11
10
13
12
48
49
52
47
57
52
58
54
52
66
67
81
79
92
77
2.8
2.8
1.5
1.8
5.0
2.3
2.2
2.2
2.6
2.9
2.9
3.9
4.1
3.7
3.5
0.44
0.63
0.79
0.61
0.89
0.79
0.89
0.88
0.74
0.59
0.55
1.00
0.90
0.91
0.79
R
(exp)
1.7
2.4
1.9
1.8
3.8
1.5
1.4
1.5
2.0
1.2
1.3
2.2
2.3
2.2
1.8
R
(theor)
2.8
2.8
1.8
1.9
5.0
2.3
2.2
2.2
2.6
2.9
2.9
3.9
4.1
3.7
3.5
N-terminal tail overcharged by ss RNA
(with total length S)
 X r  L t   rs 
 rs 
2
Free energy: F ( X )  L
 ln    ( S  X ) r ln  
L

 b
b
2
Optimize F ( X ) with respect to X
At  r /  t  2 ss RNA wraps around the N-terminal tail
X  (t / r  1 / 2) L
At r / t  2 both polymers are stretched
ss RNA
N-terminal tail
Viruses are most stable when the
total contour length of ss RNA is
close to the total length of the tails.
Virus
Qr
qp
qt
Nt  r  t Nd Nt
Brome Mosaic
Cowpea Chlorotic Mottle
Nodamura
Pariacoto
Sesbania Mosaic
Rice Yellow Mottle
Southern Bean Mosaic
Cocksfoot Mottle
Average
Cucumber Mosaic
Tomato Aspermy
Carnation Mottle
Tobacco Necrosis
Tomato Bushy Stunt
Average
3030
2980
4540
4322
4149
4450
4136
4082
3944
3214
3391
4003
3700
4776
3817
10
7
13
13
6
17
16
15
12
15
14
10
9
12
12
9
9
18
14
6
12
14
13
12
12
12
11
10
13
12
48
49
52
47
57
52
58
54
52
66
67
81
79
92
77
2.8
2.8
1.5
1.8
5.0
2.3
2.2
2.2
2.6
2.9
2.9
3.9
4.1
3.7
3.5
0.44
0.63
0.79
0.61
0.89
0.79
0.89
0.88
0.74
0.59
0.55
1.00
0.90
0.91
0.79
R
(exp)
1.7
2.4
1.9
1.8
3.8
1.5
1.4
1.5
2.0
1.2
1.3
2.2
2.3
2.2
1.8
R
(theor)
2.8
2.8
1.8
1.9
5.0
2.3
2.2
2.2
2.6
2.9
2.9
3.9
4.1
3.7
3.5
Very long tails
 t* ~ 2 t ,
*
Rtheor
~ Rtheor / 2
The ss RNA speeds up the self-assembly when
the capsid proteins are at excess
Without antenna:
J 3  4Dcr
With antenna:
J  4DcR
For a typical T=3 virus
ss RNA ~ 3000 bases ~ 2100 nm
radius of antenna R ~ 60 nm
size of capsid protein r ~ 4 nm
number of proteins M = 60T =180
R
r
 a   0 (r / R)   0 / 15
 0  M / 4cDr
The ss RNA tail slows down the self-assembly
when the ss RNA molecules are at excess
Kinetic trap
Assembly time  a as a function of ratio X  c / Mc R ,
 0  M / 4cDr is the assembly time without the
antenna effect.
excess of RNA
stoichiometry
excess of protein
Assembly from short RNA
CF can grow only via CF-CF collisions and merging.
Typical size of fragment grows with time:
n  n(t ),
c(n)  c / n,
 (n)  1 / 4D(n)r (n)c(n)  n / 4Drc ,
 a  M (n) / n  M / 4Drc   0 .
Excess of capsid proteins or of ss RNA plays NO role!
Summary
Viruses are most stable when the total
contour length of ss RNA is close to the
total length of the tails.
 When capsid proteins are at excess, the
ss RNA antenna speeds up self-assembly.
 When ss RNA molecules are at excess,
the ss RNA tail slows down self-assembly.
