Effect of charges in solution

In this problem, you'll learn about how charge, say on DNA or a protein, affects its surrounding region. A charge in a vacuum produces a long range force on other charged objects. The electrostatic potential at a distance $r$ $r$ away falls off as $1 / r$ $1/r$ .

When the same charge is put in a solution containing mobile ions, the situation completely changes. Biological solvents contain a lot of salts such as NaCl, and these disassociate in water at typical concentrations. So you don't just have a fixed charge, you also have a lot of positively and negatively charged ions floating around as well, interacting strongly with each other as well as the fixed charge. This situation has been heavily analyzed and an approximate still very useful equation describing it is called the "Poisson Boltzmann" equation. There are many ways of deriving it Links to an external site., and it results in a nonlinear partial differential equation for the potential $V$ $V$ as a function of position $\bf r$ $\bf r$ . There are many papers with analysis of this equation for particular geometries, see for example an analysis for the case of cylinders Links to an external site. and of spheres Links to an external site.. In the case where there is just one kind of positive and one kind of negative ion, (e.g. just water and NaCl) you get:
$\nabla^2 V({\bf r}) = ( 8 \pi n z e/\epsilon) \sinh (z e V/k_B T)$ $\nabla^2 V({\bf r}) = ( 8 \pi n z e/\epsilon) \sinh (z e V/k_B T)$

where $n$ $n$ is the average number density of one kind of ion (far from fixed charges). $z e$ $ze$ is the charge per ion, $ε$ $\epsilon$ is the dielectric constant, and $T$ $T$ is the temperature. This is all in cgs units.

1. Change the scaling of $V$ $V$ and $\bf r$ $\bf r$ , to turn it into a dimensionless equation independent of the above parameters:

$\nabla^{2} v = \sinh (v)$ $\nabla^2 v = \sinh(v)$

where $v$ $v$ is a rescaled version of $V$ $V$ , and distances have been rescaled by a screening length. How does the screening length depend on $n$ $n$ ?

2.Because the above equation is nonlinear, it is often considered in the case where $v$ $v$ is small. In this case the right hand side can be approximated for $v ≪ 1$ $v\ll 1$ . Derive the resulting equation.

As an alternative to an analytical solution, the solution to the Poisson Boltzmann equation can be found numerically. Code is given Download Code is given

for both the one dimensional and two dimensional cases which we will use below.

The code divides space into discrete points, so that the potential v is (in 2d) an nx by ny array. The code to solve the Poisson Boltzmann equation is not particularly efficient but is relatively easy to understand. We turn the problem into one involving diffusion, at watch it go to a steady state, time-independent solution, so now $v({\bf r},t)$ $v({\bf r},t)$ obeys

$\frac{&partial; v}{&partial; t} = \nabla^{2} v - \sinh (v)$ $\frac{\partial v}{\partial t} = \nabla^2 v - \sinh(v)$

3. Show that in steady state, this reduces to the Poisson Boltzmann equation.

In addition, we can add fixed charge to the equation. In reduced units, this leads to the equation,

$\nabla^{2} v = \sinh (v) - ρ_{0} (r)$ $\nabla^2 v = \sinh(v) - \rho_0(r)$

where $ρ_{0} (r)$ $\rho_0(r)$ is proportional to the fixed charge density. This is easily incorporated into our numerical method and is done in the code.

In the code, in order to get a more fine grained solution, a length scale, length_scale, is introduced, so that we can properly examine the behavior of the potential for distances less than 1.

In the code, a single fixed charge q0 is introduced into the middle of the system, and the diffusion-like equation is evolved until it appears close to being time-independent. This is tested by taking considering how the solution changes in one time step. The "error" is computed by looking at a measure of the difference between these solutions. When it gets less than the tolerance, in this case 10^-5, the evolution is stopped.

Both the nonlinear and linear cases can be examined by commenting out the relevant line of code.

4. Run the code for both the linear and non-linear cases. The green curve is the analytical solution to the linearized equation. Can you see how to derive the analytical equation? How well does it agree? How well do the two curves agree when Q0 is changed?

5 Run the 2d code for both the linear and non-linear cases as was done for the 1d case.

Applications

What does the $p_{K a}$ $pK_a$ value refer to in chemistry? Why is it important in application to proteins?
How is the method discussed in this problem useful in calculating the $p_{K a}$ $pK_a$ ?
How do you expect the persistence length of single stranded RNA or DNA to depend on the ionic concentration? "Persistence length Links to an external site." is a standard term from polymer physics Links to an external site..