Subject: The Poisson-Boltzmann Equation: Analysis, A Posteriori Error Estimates and Applications
treated by: Svetoslav Nakov
Supervisor: Prof. Dr. Johannes Kraus (University of Duisburg-Essen)
Second Supervisor: Univ.-Prof. Dr. Dirk Praetorius (TU Wien)

The Poisson-Boltzmann equation (PBE) gives a mean field description of the electrostatic potential in a system of molecules in ionic solution. It is a commonly accepted and widely used approach to the modelling of the electrostatic fields in and around biological macromolecules such as proteins, RNA or DNA. The PBE is a semilinear elliptic equation with a nonlinearity of exponential type, a measure right hand-side, and jump discontinuities of its coefficients across complex surfaces that represent the molecular structures under study. These features of the PBE pose a number of challenges to its rigorous analysis and numerical solution.

Figure 4.24: mesh at refinement level i = 3.

This thesis is devoted to the existence and uniqueness analysis of the PBE and the derivation of a posteriori error estimates for the distance between its exact solution and any admissible approximation of it, measured either in global energy norms or in terms of a specific goal quantity represented in terms of a linear functional. These error estimates allow for the construction of adaptive finite element methods for the fully reliable and computationally efficient solution of the PBE in large systems with complicated molecular geometries and distribution of charges.

One of the main focuses of this work is the rigorous analysis of the Poisson-Boltzmann equation and its linearized version, the LPBE. The starting point is to give a weak formulation which is appropriate for elliptic equations with measure data, such as the delta distributions due to fixed point charges in the molecular regions. For this weak formulation we are able to show existence of a solution by means of 2-term and 3-term splittings, where the full potential is decomposed into a singular Coulomb potential and a more regular part, a particular representative of which can be defined by a weak formulation involving $H^1$ Sobolev spaces. In the case of the LPBE we are also able to show the uniqueness of the full electrostatic potential.

Another main goal of this thesis is the derivation of a posteriori error estimates for the linearized and fully nonlinear Poisson-Boltzmann equation. More precisely, we derive two types of a posteriori error estimates: global estimates for the error in the electrostatic potential, measured in the so-called energy norm, and goal-oriented error estimates for the electrostatic interaction between molecules. We apply the first type of error estimates to the study of the electrostatic potential in and around the insulin protein with PDB ID 1RWE, the Alexa 488 and 594 dyes, as well as the membrane protein-conducting channel SecYEG. In all these applications we obtain guaranteed and fully computable bounds on the relative errors in global energy norms. Moreover, we are able to establish a near best approximation result for the regular part of the electrostatic potential which is the basis for deriving a priori error estimates in energy norm for the finite element method. The second type of error estimates, also called goal-oriented a posteriori error estimates, are employed in the computation of the electrostatic interaction between the dyes Alexa 488 and Alexa 594 being either in their ground state or transition state. The latter configuration is related to the calculation of the efficiency of the Fröster resonance energy transfer (FRET) between the two dyes.

Figure 4.24: mesh at refinement level i = 3.
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