MatSE 385 Project Abstract

Polyelectrolyte Simulation

Camilo GuaquetaMatthew WongChunguang Xia

Atomic scale simulation

Introduction -- Polyelectrolyte

+

+

+

Counterion

Charged rod, polyelectrolyte

Solvent solution with certain

dielectric

•Two macromolecules with like charge would naturally repel each other.

•However, attraction has been observed in the presence of counterions.

•Example: DNA (a stiff, highly negatively charged polyelectrolyte) can be condensed by multivalent counterions

Past Simulation -- MDM. Deserno, A. Arnold, and C. Holm, Macromolecules 36, 249 (2003).

Bulk System Simulation Pair of Charged Rods Simulation

Cubic unit cell with polyelectrolyte at the body diagonal and distribution of counterions

Cubic unit cell with pair of polyelectrolyte parallel to the z-axis and distribution of counterions

Osmotic coefficient vs. counterion density Force between two rods vs. Bjerrum length

Past Simulation – MD 2

x Electrostatic contribution

Excluded volume contribution

Past Simulation -- BDN. Gronbeck-Jensen, R. J. Mashl, R.F. Bruinsma, and W.M. Gelbart, Phys. Rev. Lett 78, 2477 (1997).

Past Simulation – MCE. Allahyarov, I.D'Amico, H.Lowen, Phys. Rev. Lett. 81, 1334 (1998).

Counterion Depletion Zone theory:

Induced by the combined effect of the macroions excluded volume and the Coulombic interactions

Leading to an imbalanced pressure from the counterions acting onto the macroions surfaces

Past Simulation – ExperimentalT.E. Angelini, H. Liang, W. Wriggers, G.C.L. Wong, PNAS, 100, 8634 (2003).

Experimental Work on cytoskeletal filamentous actin:

Observable counterions organization using synchrotron x-ray diffraction

Discovered symmetry-breaking collective countion cehamism for generating attraction

Experimental Setup

Non-cubic unit cell to form hexagonal lattice with one polyelectrolyte and fix amount of counterions per unit cell

•Study polyelectrolyte interactions dependence on valency

•Replicate the experimental observation that multivalent ions are necessary for condensation

•Compare bulk systems experiments from simulation and reference

•Extend pair of charge rods experiments from reference to a bulk system simulation

•Compare bulk system behavior of simple linear charged rods to experiments with complex molecules, such as DNA and actin

The Ewald Sum

The Ewald sum is a technique for efficiently summing the interaction between an ion and all its periodic images. each ion is effectively neutralized (at long range) by the superposition of a spherical gaussian cloud of opposite charge centered on the ion. The combined assembly of point ions and gaussian charges becomes the Real Space part of the Ewald, which is now short-ranged superimpose a second set of gaussian charges, with the same charges as the original point ions and again centered on the point ions (so nullifying the effect of the first set of gaussians). The potential due to these gaussians is obtained from Poisson's equation and is solved as a Fourier series in Reciprocal Space. correction terms, such as the self-energy and surface dipole.

The Ewald Sum (continued)

The expression for Coulomb energy is:

In our model the fourth and fifth summation is set to be zero, because we use the rigid body molecular and the whole system has zero net charge.

The Ewald Sum (continued)

Here:

Pressure

As the definitions from M.P. ALLEN and D.J. TILDESLEY’s book, Computer Simulation of Liquids, we define the internal virial :

the instantaneous pressure:

here , , , , , and are vector of particle i and j, force between particle i and j, volume of system, density of system, Boltzman constant , temperature of system and number of the particle in system, respectively.

In MOLDY The internal stress is computed using a novel approach, using the extension of the Verlet neighbor list algorithm developed by Bekker.

w

N

i

N

ijijij

frw1 13

1

Vwp TkB /

ijr

ijf V

Bk T N

Pressure(continued)

n

N

i

N

ijnjinji frW

1 1),(),(

N

i nnii ngfr

1

Bekker showed that the virial tensor is:

where is the lattice vector between the central MD cell and its periodic images and is also used as an index to represent a sum over all image cells and the expression refers to the periodic image of particle in image cell . The quantity :

n

),( nj j

n ng

Advantage:•This is easily evaluated in the inner loops of the link-cell force calculation by summing separately the pair forces on all particles in each image cell. •Also it is not necessary to perform the sum of contributions within the inner loop of the force calculation since it depends only on the site co-ordinates and the total force acting on each site.

Pressure(continued)

The instantaneous pressure tensor is:

)(11

WvvmV

Pii

N

ii

here is the mass and the velocity of the i-th particle, and the volume of the system. And the tensorial direct product is defined as:

im i

v V

jiijvuvu )(

Finally, according to the equipartition principle the instantaneous pressure is one-third of the trace of instantaneous pressure tensor, that is:

)(3

1 PTracep

Set up

Hexagonal lattice obtained by periodically replicating extruded parallelogram

Polyelectrolyte modeled as a linear molecule of spheres with diameter s=7 A and charge -1, at a linear separation of 1.042 s.

Counterions have diameter s.

Short-range interaction given by B*exp(-C * r). We chose this form because it closely mimics the STLJ often used in the literature

Potential Comparison

Results

Microcanonical (constant energy) ensemble

Time step (0.01ps) chosen to conserve energy, and also small enough given relaxation time of the counterions

250000 timesteps per simulation. Errors were always less than 5% of the mean.

Densities depend on box volume --> osmotic coefficient more useful than osmotic pressure.

Results (continued)

Conclusion:

•Noisy results, large errors are an intrinsic problem with computing the pressure via the virial theorem.

•Not enough time to run sufficiently long results

•Did not explore the more dilute systems because the box size is large, and thus the runs are long

•Leftmost point is a theoretical limit

Conclusion:

•Qualitatively the trends are correct:

•Trivalent solutions cause condensation

•Monovalent solutions do not

•Divalent osmotic coefficient is lower than monovalent

•Quantitative results for trivalent ions are in fair agreement with Deserno et al

•Regime of condensation is smaller in our case

•Might be due to ensemble difference, or finite-size effects

Conclusion:

•Results suggest that our approach is a valid way of studying like-charge attraction in the polyelectrolyte system.

•However, more careful analysis (longer runs to reduce errors, and explore dilute systems) is necessary.