Adriano Ferrari
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We performed Monte Carlo simulations on a minimal model of two charged polymers (infinite, and parallel to each other), with counterions free to move in the simulation box (overall neutral charge). First, both “polymers” held fixed at a separation R. Varying R and getting the potential of mean force yielded a potential which we found to be attractive beyond a certain distance. Second, one “polymer” was held fixed, the other held at a distance R with sinusoidal perturbations [;\delta;]. The growth or decay rate of the perturbation amplitude, as a function of frequency, tells us which frequencies were stable and which unstable. Finally, we initialized the “polymers” in these unstable modes, and let the system evolve freely. We found that the chains exhibit [;1/f;] noise fluctuations from linearity. We also found that introducing a relatively weak “symmetry-breaking” potential preferring one handedness over another, was enough to produce a stable helix.
Even if I only have 50 words each for the remaining sections, I am 1.5 times over the max word count.
Perhaps I need an intermediate state, such as a 140-268 word version??
Something.
Biopolymer systems are difficult to analyze theoretically. Even basic tenets of electrostatics such as “like-charges repel” can appear to be violated due to complicated inter-particle interactions and correlations. For instance, when a “polyelectrolyte’” polymer is placed in a solvent such as water, the resulting like-charged chains can attract each other.
This has a number of important implications, since many biopolymers are polyelectrolytes: DNA, RNA, F-actin fibers, microtubules, and aggregating viruses, to name a few. A great number of other polyelectrolytes (PEs) have various technological applications.
A polyelectrolyte is simply a polymer whose repeating units contain electrolytes or “salts”, which are neutral ionic compounds that dissociate into their charged components when immersed in water. The result is a charged polymer, surrounded by counterions.
While very common, this system is difficult to analyze theoretically
. As a result, interactions such as the like-charged attraction, are not very well understood.
This attractive force, which has been observed in experiment and simulations, and which has been attributed to numerous (sometimes competing) mechanisms, has been studied mostly for rigid rods, rigid finite-sized segments, rigid bundles, or completely free chains. However, each of these studies was restricted in its scope because of the complexity of the system.
There has not yet been a study of a single pair of infinite, flexible polyelectrolytes, and the behaviors they exhibit. This is what we address in this work.
The system we study consists of two chains, each of \(N_\ell\) discrete charges , and counterions of opposite charge .
The counterions were randomly (uniformly) distributed in a periodic box; their number and charge chosen to make the unit cell neutral.
The charged monomers were spaced a distance [;b;] from each other. The chains were parallel to the y-axis, and separated from each other by a distance R. The chains were straight, or sinusoidally perturbed, depending on whether we were studying the attractive force itself, or its effect on perturbed chains.
Each particle interacts with all others (and their periodic images), via the Lekner potential, which is the Coulomb potential for 1D periodic boundary conditions:
where \(K_0(\cdot)\) is the modified Bessel function of the second kind, which is an exponentially decaying function.
All particles also have a repulsive Lennard-Jones potential, to prevent overlap:
Finally, the chain particles are held together by a FENE potential . FootnoteEnd particles are attached to their images.
Here [;h;] is the spring constant. In all our simulations, we choose [;h = ??;] to approximate the spring constant of single-stranded DNA.
We explored this system using Monte Carlo. At every step, a random counterion was displaced by a vector [;\Delta x \cdot u;], where u is a unit vector in a uniform random direction.
If the chains were allowed to move, they were also displaced, but by a different step-size [;\Delta x_c;]. This allowed us to create two different time scales and, by setting [;\Delta x_c \ll \Delta x;], to average over many counterion configurations for each chain configuration.
In all cases, the chains were held fixed for at least 50k Monte Carlo sweeps, to allow the counterions to condense onto the chains, before allowing the chains to move. In order to speed up equilibration, all charges were initially scaled to [;q^\prime = q/100;], and then slowly “ramped up” to the full value. This slowly introduced the electrostatic interaction, and yielded much faster equilibration than “simulated annealing” methods.
We were mostly concerned with chain-chain interactions and chain fluctuations, so we measured the mean force between the two chains, their average positions, the root-mean-square amplitude about the mean, and the total energy of the system.
Our study has three parts. First, we need to establish that our model system does indeed reproduce the behavior of a real polyelectrolyte, including the like-charged attraction. Afterwards, we will explore how this interaction affects the polymer’s stability to deformation, and finally, what kinds of long-term structures form.
We chose a parameter set that models, as closely as possible, the conditions of DNA in solution. With this, we are able to find that, when the counterions are divalent or multivalent, the chains do attract each other. When we use monovalent counterions, there is no attraction. This behavior is as expected from experiments with DNA & counterion valencies.
However, the fact that “only” multivalent counterions induce attraction is a special case of the parameters of DNA. If we change the parameters to represent other polymers, temperatures, or solutions, monovalent counterions can also induce attraction.
With the same system, we varied the temperature from 200mK up to 300K, and found that above a critical temperature [;T_c;], the force was repulsive for all chain separations [;R;]. Below this temperature, the attractive force increased as the temperature decreased. We obtained [;T_c=169K;], which is also the critical temperature we would expect from setting the Manning parameter to 1.
This result unambiguously demonstrates that the counterion induced attraction is a zero-temperature effect.
Next we investigated this ground state, and found that the counterions arranged themselves in a zipper-like configuration. Directly computing the Lekner potential in one of these ground-state configurations shows that it is sufficient to create the attractive force that we see surviving up to higher temperatures.
Finally, we studied the force curve further, and found that, though it can be well-approximated in a narrow range by a combination of power laws, no fit can reproduce the slow transition from the attractive to repulsive regions. We have found that the form [;F(R)=[AK_0(R)^2+B\log (R)]\times\exp (-CR^2)-D;] fits the entire range, and so will use this to represent the force curve in later approximations.
#Result - Stability Diagram
So far we have studied straight chains, which model a perfectly rigid polymer. In real systems, even “stiff” polymers (such as F-actin) have some degree of flexibility. If a straight flexible polyelectrolyte begins to deform, will the attractive interaction cause these deformations to grow, or to decay? In other words, are polyelectrolytes stable or unstable against deformation?
How pairs (and bundles, by extension) of polyelectrolytes behave as they approach each other is an important area of study. It affects how polyelectrolytes form packing structures, it affects DNA condensation, and can lead to novel structurs such as F-actin grids.
We will use linear stability analysis to give us a first-order approximation to answer these questions.
The rate at which a given perturbation grows or decays is called the growth rate . For small perturbations, , where is the change in the force experienced by the displaced portion, and is the amplitude of that displacement.
We can decompose any deformation away from a straigh-line into a Fourier series, the modes of which are independent in this linear regime. The goal of this stability analysis is to find out when the growth rate is positive (unstable), leading the chains to buckle, and when it’s negative (stable), causing them to remain extended.
The most direct way of determining [;\Gamma;] is to perturb one chain, and then track how the amplitude (given by the rms deviation from straight) evolves over a short time. We track only the initial growth/decay of the amplitude for a given mode, because for longer times other modes will come into play, and wash out the effect.
Though we obtained valid results by tracking the amplitude, we found that the number of averages required was very high, and we were limited to an extremely short cut off time because tracking only the amplitude meant we would eventually pick up modes other than the initial one.
To solve this, we measured the fourier power spectrum of the displacement from the mean, as a function of time. This allowed us to follow the evolution of the initial excited mode, and track its growth or decay constant, independent of the other modes.
Figure 2: Stability of chains against deformation, using three different methods. Given a frequency of sinusoidal deformation, and a chain-chain separation R, this diagram tells us whether the deformation would grow (unstable, colored regions) or decay (stable, gray regions). The three methods used are (a) mean-field approximation with a smooth-fit force curve, (b) mean-field approximation using the actual force curve, and (c) the measured growth rate when the chains are allowed to move freely. All methods agree, within error, on the region of highest instability for our "DNA" parameter system ([;R \in [0.35, 0.87], k \lesssim 2.5;]).
If we initialize the system when the chains are in a region unstable to deformation, the system will quickly evolve to a 1/f noise spectrum. This is true whether we begin with a small perturbation in the most unstable mode, or with a white noise profile.
Figure 3: Spectrum of fluctuations of free chains in unstable region. Free chains eventually have a [;1/f;] noise profile, if initialized with white noise and allowed to move freely in the presence of counterions.
[;1/f;] noise, also called shot noise or flicker noise, appears everywhere. It has been explained as the result of self-organized criticality, or of the competition between a random driving force, and a stabilizing threshold (sand pile model).
Given that we have random fluctuations arising from finite temperature, combined with a fluctuation/perturbation-dependent instability, it is possible that this [;1/f;] behavior could have been predicted from the stability diagram alone. Exploration of this area is of interest for future reasearch.
To investigate the resulting structure of a pair of polyelectrolytes, we initialize the system in the aforementioned unstable states. First, we impose restrictions on the motion of the chain, and examine the structures as we lift these restrictions in turn.
Figure 4: Snapshots of the system after allowed to evolve freely for a long time. a) is the configuration for chains at a fixed [;R>R_eq;]. (b,c,d) are the results when the chain is restricted to a plane, for [;h=\{\infty, 1000,10\};] respectively. We can see in (c) that the counterions form charge density waves. In the case of very flexible chains (d), we have essentially an ionic crystal. (e & f) are the results of free chains of stiffness [;h=\{1000,10\};]. (g & h) shows chains with addition of a relatively small helical ``hinting'' potential, that breaks the handedness symmetry.
We begin by examining the ground-state configuration of a pair of rigid chains, when fixed at a separation greater than the equilibrium chain-chain distance (Fig. 4 (a)). The resulting configuration is similar to that of widely separated chains. This suggest that, though the chains are being attracted to each other at large distances, their counterion arrangements remain at their independent configurations until the chains are very close to each other.
If we then allow the chains to translate (but not to deform), we find that they immediately approach each other, and the counterions first stagger, and then zipper to arrange themselves in a line. The result is seen in (Figure 4, b), where there are twice as many counterions between the chains as outside them.
It should be noted that, though both chains and counterions were allowed to move in 3 dimensions, the final arrangement was entirely coplanar.
We then removed the rigidity constraint, but allowed the chains to move only in the xy-plane. Counterions, as always, are free to move in 3D space.
If the chains are relatively stiff (Fig. 4 c), we see that they form long wavelength perturbations (in this case, of the size of the box [;Ly;]), that are out of phase. In response to this undulation, the counterions form charge density waves, with higher density regions forming where the chains are closer.
If the chains are plane-restricted, but very flexible, the resulting arrangement (Fig. 4 d) is essentially an 2D ionic crystal.
We now remove the planar restriction on the chain monomers. In both the stiff and loose chains, the configuration was similar to the ``ionic crystal’’ planar versions. However, for the loose results (Fig. 4 f), we see that the long-wavelength modes are more pronounced.
Once we removed all restrictions, we found that some instances of our simulations resulted in helical configurations. We then added a potential , where is the dihedral angle between corresponding monomers on opposing chains, and the preferred angle. We found that even when was 100 times weaker than the other potentials, adding this slight preference for one chirality was enough to make the helical configuration stable in the long-term.
With no potentials besides electrostatics, a repulsive core, and chain forces, our model reproduces the key “unexpected” features of polyelectrolytes in solution. We were able to control many parameters of this system, to explore how it would behave in different circumstances. etc.
Figure 1: Our model system. [;N_\ell;] chain particles are used to represent each chain. Periodicity in [;y;] direction makes these infinitely long. We place [;N;] counterions in the box, and allow them to move. To measure the potential of mean force between the chains, they were held fixed at separation R (a). To measure stability, one chain was initialized with a sinusoidal perturbation (b). To see final structure, they were allowed to move freely (c).
Include figure of setup (with b spacing, R spacing, etc).
Code for Lennard Jones here
V(r,s)= 4.0*ep(030290932 9302 09 s9 s9 sc )
etc.