Toroidal DNA condensates

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Toroidal DNA condensates
with a new twist !
Igor M. Kulic, Denis Andrienko, and Markus Deserno
Max-Planck-Institut für Polymerforschung, Mainz,
Germany
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Outline

• Polymer collapse
• The role of stiffness
• Elasticity and local structure
• Global aspects
• Relation to experiments

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Polymer collapse
In sufficiently poor solvent flexible polymers
collapse to a dense globule.
Poor solvent
(surface tension)
Entropy!
Energy!
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Stiff polymer collapse
For stiff polymers chain bending also matters!
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Stiff polymer collapse
For stiff polymers chain bending also matters!
Local structure can be characterized by a smooth
field of tangent vectors.
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Problem!
Such a vector field cannot exist on a sphere!
One cant comb a sphere!
At least two (energetically expensive) defects
must exist on the surface.
Solution
Who said that the condensate must be spherical?
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Toroidal DNA condensates
N.V. Hud and K.H. Downing, PNAS 98, 14925 (2001)
O. Lambert, L. Letellier, W. M. Gelbart, and
J.-L. Rigaud, PNAS 97,7248 (2000)
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Scaling for the torus
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Scaling for the torus
shrinks if solvent gets poorer or chain longer!
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Subtleties
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Suggestion
Precessing loops could form a torus at
essentially constant bending energy (spirograph
motif)
N.V. Hud, K.H. Downing, and R. Balhorn, PNAS 92,
3581 (1995).
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Bits and pieces
Aim Describe the elastic energy of a toroidal
condensate which is not just circumferentially
wound.
Idea First chop the polymer into pieces and
study the elasticity of the resulting nematic
liquid crystal.
splay
twist
bend
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Nematic energy functional
Aim Describe the elastic energy of a toroidal
condensate which is not just circumferentially
wound.
Idea First chop the polymer into pieces and
study the elasticity of the resulting nematic
liquid crystal.
splay
twist
bend
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Nematic energy functional
The path of the polymer will later be recovered
as the integral curves of this nematic field.
Uniform polymer density ? splay must vanish!
Plug this into the Frank functional and minimize!
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Twist-bend-instability
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Twist-bend-instability
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Structural phase diagram
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Variational ansatz
1. Assume that field has no radial component.
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Variational ansatz
We now have a Landau expansion for the energy in
terms of a scalar order parameter!
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Phase boundary
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Phase boundary
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Global aspects (1)
Incompressibility cylindrical symmetry ?
Hamiltonian flow
This is exactly our variational ansatz!
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Global aspects (2)
Twist ? DNA heavily entangled with itself!
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Global aspects (2)
Twist ? DNA heavily entangled with itself!
Rough estimate of total threading by integrating
?? over the torus cross-section L15?m, ?1.5,
?0.1 ? about 30 threadings in total
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Relation to experiment
No direct evidence yet, but
Several experimental findings exist which can be
rationalized in the light of the twist-bend
scenario
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Giant T4 phage
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Giant T4 phage, disrupted
W.C. Earnshaw, J. King, S.C. Harison, F.A.
Eiserling, Cell 14, 559 (1978).
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Giant T4 phage, disrupted
W.C. Earnshaw, J. King, S.C. Harison, F.A.
Eiserling, Cell 14, 559 (1978).
? Interpretation as plectonemic supercoiling
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Why supercoiling?
Our interpretation Twist-Writhe exchange!
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Why supercoiling?
Our interpretation Twist-Writhe exchange!
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Why supercoiling?
Our interpretation Twist-Writhe exchange!
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Acknowledgements

• Igor Kulic
• Denis Andrienko
• Helmut Schießel
• Kurt Kremer
• DFG (Emmy-Noether grant)