Electrostatic effects are very important in many phenomena in chemistry, materials science and biology. First, what is electrostatics? It is the study of electric fields in matter where there is no time dependence. In what we are looking at in this lecture that implies we are examining equilibrium phenomena. Consider a charge transfer reaction in ionic solution. The quantity we could in principle compute is the free energy profile during the coarse of the reaction assuming the solvent and the ions in solution relax very quickly on the time scale of the reaction. A key problem in chemistry, still not fully understood, is the structure of the proton in water. Also, what is the structure of the `double layer' that is so important in electrochemistry? In materials science, the structure of colloid solutions is very important. Colloids are particles in the size range of mm which have charges on the surface and then counterions and perhaps salt ions in the solution. Another related problem pertains to the structure of polyelectrolytes in solution. Polyelectrolytes are polymers with charges along the chains. How does their radius of gyration depend on the salt concentration? In biology, electrostatics is a key component in determining the structure of proteins, nucleic acids, and membranes. So they are REALLY important! For example, what is the free energy profile if we bring a restriction enzyme up to DNA in the presence of salt ions? What are the ionization states or pKa's of the ionizable groups on a protein, and how do they depend on other charged groups on the protein? What forces stabilize a membrane? Is electrostatics important in protein folding (a controversial topic!)? Where does a drug molecule prefer to bind on DNA or a membrane or an enzyme?
The second question we should ask is: based on all these fancy simulation methods we have discussed in the course (namely molecular dynamics and Monte Carlo) for classical modeling of large systems, why not just simulate everything at the atomic level, compute free energies and leave it at that? Well, consider the interaction of two medium sized proteins in water with a salt concentration of say 1M. This would certainly require a simulation box of about 200Å on a side. This implies about 70 waters on a side and 703 is 343,000. Add to that a large number of ions (maybe 10,000). You need to include the electrostatics properly for periodic boundary conditions,which is costly computationally. So you can see what's coming here, this is a huge, huge simulation and you need to run a long, long time to get good average quantities. Therefore, this is not feasible on supercomputers let alone our workstations, even with the fast workstations of today.
Therefore, we have to consider more approximate treatments of solvent and ion effects. People do make these approximations, and they are rather severe approximations. The only rather amazing thing is that they actually work to a fair degree of accuracy, and now people are studying just why they work so well. They also fail under many circumstances so it is important to understand when they fail and not blindly use the codes where they don't apply.
I will largely follow review articles written by Rob Coalson and myself[1] and Honig and Nicholls[2]. Also, I will discuss an article on salt effects on protein-DNA binding by Honig's group[3] and one on electrostatic binding of proteins to membranes[4]. There is an article by another of the people in this field on numerical methods for solving PB equations[5]. By now there is a huge literature in this field, so you can see me if you want more references to specific problems. The literature ranges from highly specialized theoretical methods to very applied calculations on systems like those in the listed references.
This document is available at:
http://bessie.che.uc.edu/tlb/teach/grad/chem981s99/pb.html
So what are the typical approximations in dielectric continuum theory? First, we discard the water completely and simply assume it is a continuum liquid with dielectric constant e = 78. If we have say a protein system, then we assign a different dielectric constant to the interior of the protein (usually somewhere between 4 and 20, and this is controversial too), so we need to allow for spacially varying dielectric constant. Then we treat the ions in the solution in an average way which also is a continuum model. We say that the charge density of the ions in the solution is proportional to the Boltzmann factor of the electrostatic potential at a point times the charge of the ion. This is what is called a mean field approximation. We also assume the ions are infinitely small in size. The effects of finite ion size become apparent at high concentrations, like above 1 M. We treat the fixed ions, for example those located on the protein, discretely just as we would in a simulation. That is, those ions remain fixed in space.
To summarize, the assumptions are:
Once we make these assumptions, we can solve the problem by solving a complicated nonlinear partial differential equation, the Poisson-Boltzmann equation:
| (1) |
What do we get when we solve this equation? We solve for the electrostatic potential at every point. The variables are as follows: e is the dielectric constant, f is the electrostatic potential, rf is the fixed ion distribution, [`n]± are the equilibrium concentrations of the ions in the absence of any field, b is 1/kT, and v is an exlcuded volume potential which keeps the mobile ions out of the inside of the molecule. From the electrostatic potential, we can then determine the approximate concentrations of positive and negative ions at all points in space, and importantly we can calculate the mean field free energy for the mobile ions moving in the potential of the fixed ions. This free energy can be used to calculate electrostatic effects on binding constants.
The Honig formula for the free energy is:
| (2) |
where rm is the mobile ion concentration which is:
| (3) |
and DP is:
| (4) |
Here E is the electric field and D is the dielectric displacement. Both can be obtained from the potential.
I have some problems with this formula but there is no question it is correct for the limit of infinite system size. The problem is of course we never do calculations on an infinite system. The expression derived initially by Coalson and Duncan and discussed in our review is size consistent and I believe is the physically `correct' form, and I have derived the connection between the two which shows how Coalson/Duncan goes over to Honig for infinite system size. At any rate, the point is that once you have the potential and the charge distributions, you can calculate free energies, not just potential energies. These free energies are approximations but are quite good under certain circumstances.
The PB level of theory is quite good for:
It begins to break down for:
The main causes of the breakdown are:
Still Honig and coworkers have gotten reasnable results for divalent ion systems, which are obviously important in biophysics, so that's good news.
There is a lower level of theory which is correct for very low concentrations of ions and small electric potentials which is called Debye-Hückel theory. This level results from making the approximation:
| (5) |
that is linearizing the Boltzmann factor. It has limited validity for the systems we are considering here. However, an important parameter which comes out of this theory is k, the inverse Debye length:
| (6) |
Here we have assumed both positive and negative ions have the same magnitude of charge. The inverse of k gives the approximate length over which a charge is screened in the ionic solution.
We have to discretize the PB equation on a grid in space to efficiently solve the differential equation for a large domain. Often people use simple finite difference representations for the second derivatives (one dimension here):
| (7) |
which leads to the following update equation for the potential:
| (8) |
There are fancy methods of solution of these equations which we don't need to get into. Some of the names are conjugate gradients, steepest descents, multigrid, etc. Underlying these approaches is the fact that the free energy is a minimum for the potential which satisfies the PB equation, that is there is a variational principle.
A small subset of the types of problems that I have seen addressed with PB methods is as follows: