Konstantin Fackeldey

Coupling Molecular Dynamics and Continua with Weak Constraints

One of the most challenging problems in dynamic concurrent multiscale simulations is the reflectionless transfer of physical quantities between the different scales. In particular, when coupling molecular dynamics and finite element discretizations in solid body mechanics, often spurious wave reflections are introduced by the applied coupling technique. The reflected waves are typically of high frequency and are arguably of little importance in the domain where the finite element discretization drives the simulation. In this work, we provide an analysis of this phenomenon. Based on the gained insight, we derive a new coupling approach, which neatly separates high and low frequency waves. Whereas low frequency waves are permitted to bridge the scales, high frequency waves can be removed by applying damping techniques without affecting the coupled share of the solution. As a consequence, our new method almost completely eliminates unphysical wave reflections and deals in a consistent way with waves of arbit...

Investigation of the Ergopeptide Epimerization Process

Ergopeptides, like ergocornine and a-ergocryptine, exist in an S- and in an R-configuration. Kinetic experiments imply that certain configurations are preferred depending on the solvent. The experimental methods are explained in this article. Furthermore, computational methods are used to understand this configurational preference. Standard quantum chemical methods can predict the favored configurations by using minimum energy calculations on the potential energy landscape. However, the explicit role of the solvent is not revealed by this type of methods. In order to better understand its influence, classical mechanical molecular simulations are applied. It appears from our research that “folding” the ergopeptide molecules into an intermediate state (between the S- and the R-configuration) is mechanically hindered for the preferred configurations.

Solving Frictional Contact Problems with Multigrid Efficiency

The construction of fast and reliable solvers for contact problems with friction is even nowadays a challenging task. It is well known that contact problems with Coulomb friction have the weak form of a quasi-variational inequality [KO88, HHNL88, NJH80]. For small coefficients of friction, a solution can be obtained by means of a fixed point iteration in the boundary stresses [NJH80]. This fixed point approach is often used for the construction of numerical methods, since in each iteration step only a constrained convex minimization problem has to be solved [DHK02, LPR91]. Unfortunately, the convergence speed of the discrete fixed point iteration deteriorates for smaller meshsizes. Here, we present a nonlinear multigrid method which removes the outer fixed point iteration and gives rise to a highly efficient solution method for frictional contact problems with Coulomb friction and other local friction laws in 2 and 3 space dimensions. The numerical cost is comparable to frictionless contact problems. Our method is based on monotone multigrid methods, see [KK01], and does not require any regularization of the non penetration condition or of the friction law. Therefore, the results are highly accurate. Using the basis transformation given in [WK00], our method can also be applied to two body contact problems.

Computing the Minimal Rebinding Effect Included in a Given Kinetics

In this paper we show that the binding kinetics of a molecular system can be identified by a projection of a continuous process onto a finite number of macro states. We thus interpret binding kinetics as a projection. When projecting onto nonoverlapping macro states the Markovianity is spoiled. As a consequence, the description of, e.g., a receptor-ligand system by a two state kinetics, is not accurate. By assigning a degree of membership to each state, we abandon the nonoverlapping approach. This overlap is crucial for a correct mapping of binding effects by Markov state models with regard to their long time behavior. It enables us to describe the highly discussed rebinding effect, where the spatial arrangement of the system has the be included. By introducing a “degree of fuzziness,” we have an indicator for the strength of the rebinding effect such that the minimal rebinding effect can be derived from an optimization problem. The fuzziness also includes some new paradigms for molecular kinetics. These ...

Mixed-Integer Programming for Cycle Detection in Nonreversible Markov Processes

In this paper, we present a new, optimization-based method to exhibit cyclic behavior in non-reversible stochastic processes. While our method is general, it is strongly motivated by discrete simulations of ordinary differential equations representing non-reversible biological processes, in particular molecular simulations. Here, the discrete time steps of the simulation are often very small compared to the time scale of interest, i.e., of the whole process. In this setting, the detection of a global cyclic behavior of the process becomes difficult because transitions between individual states may appear almost reversible on the small time scale of the simulation. We address this difficulty using a mixed-integer programming model that allows us to compute a cycle of clusters with maximum net flow, i.e., large forward and small backward probability. For a synthetic genetic regulatory network consisting of a ring-oscillator with three genes, we show that this approach can detect the most productive overall cycle, outperforming classical spectral analysis methods. Our method applies to general non-equilibrium steady state systems such as catalytic reactions, for which the objective value computes the effectiveness of the catalyst.

Local Refinements in Classical Molecular Dynamics Simulations

Quantum mechanics provide a detailed description of the physical and chemical behavior of molecules. However, with increasing size of the system the complexity rises exponentially, which is prohibitive for efficient dynamical simulation. In contrast, classical molecular dynamics procure a coarser description by using less degrees of freedom. Thus, it seems natural to seek for an adequate trade-off between accurateness and computational feasibility in the simulation of molecules. Here, we propose a novel method, which combines classical molecular simulations with quantum mechanics for molecular systems. For this we decompose the state space of the respective molecule into subsets, by employing a meshfree partition of unity. We show, that this partition allows us to localize an empirical force field and to run locally constrained classical trajectories. Within each subset, we compute the energy on the quantum level for a fixed number of spatial states (ab initio points). With these energy values from the ab initio points we have a local scattered data problem, which can be solved by the moving least squares method.

Stability of Energy Transfer in the Weak Coupling Method

In this paper we are concerned with a weak coupling technique for the concurrent simulation of multiscale phenomena. In particular we focus on the construction of an initial embedding of discrete atomic data fields in an appropriate subspace \( \mathcal{H}_N \left( \Omega \right) \subset L^2 \left( \Omega \right) \) which provides the foundation for the proposed coupling technique in a function space setting. Thus, we must consider the regularity of the coupling information and the stability of the resulting basis.