Browsing by Author "Bohner, Matthias Ulrich"

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Open Access
Using umbrella integration to find minimum free energy paths
(2015) Bohner, Matthias Ulrich; Kästner, Johannes (Prof. Dr.)
The analysis of the detailed mechanism of chemical reactions is a key task of computational chemistry. The detailed knowledge may help to improve known processes or even contribute to the development of new ones. In this way ecological and economic demands can be reduced. Furthermore, the course of a reaction path also plays an important role in the action of drugs. Understanding the binding of the active ingredient to receptors, such as proteins, can make it possible to optimize drugs by reducing side effects, or even to find new effects. Unfortunately, chemical reactions are usually too fast to observe intermediate states by experimental methods. This is where theoretical chemistry joins the game. In theoretical chemistry we combine the coordinates of all atoms with the configuration space. The potential energy forms a hypersurface in this space. Minima represent stable or metastable states. Saddle points represent transition states which are the most unfavourable configurations occurring on the most favourable path between two minima. If the path of a reaction is known, all intermediate states can be observed by theoretical methods. However, these calculations are usually computationally costly. This is the reason why, in contrast to experimental methods, it is in general impossible to sample the whole configuration space. This would in most cases exceed the available computational resources. It is therefore necessary to use techniques enabling a reaction path to be found without sampling the whole configuration space. In the case of a thermodynamic ensemble, e.g. the contents of a test tube, statistical information has to be included. The corresponding potential is the so-called free energy landscape, for which some degrees of freedom of the configuration space are thermostatistically integrated out. Consequently this function includes statistical and energetic properties. The free energy can in general only be calculated by statistical simulations (Monte--Carlo or molecular dynamics). Unfortunately, the transition states in which we have a special interest are rarely sampled. Special methods have to be applied to get sufficient sampling as well in areas of these rare events. In this work a non-physical quadratic potential is used to bias the equation of motion of the particles while doing molecular dynamics simulations. In this way unfavourable areas in the configuration space can also be sampled sufficiently. This technique is called umbrella sampling. The bias is applied to one or more coordinates which describe the reaction and therefore are called reaction coordinates. An umbrella sampling run will result in a distribution of the reaction coordinates. The expectation value of this distribution will be located close to the minimum of the bias function without in general corresponding to it. The difference between the minimum of the bias function and the expectation value can be used to calculate the gradient of the underlying free energy surface. Similarly, the covariance of the distribution of the reaction coordinate can be used to calculate the Hessian of the free energy surface. This method of interpreting the data gained by umbrella sampling is called umbrella integration. At first these values are used for an iterative search of the saddle points, which represent the transition states. From these configurations, free energy paths can be constructed by following the gradient down to the minima. This algorithm was successfully tested for the alanine dipeptide system. This simple method has the disadvantages that it works serially and that one needs good initial guesses for the saddle points in order to find them. Therefore, in a second part of the work, the established method of nudged elastic band optimization (NEB) is extended for use in the free energy surface. NEB optimization searches for a reaction path. This path is discretized into a number configurations, so-called images. For the sake of equal distribution of the images along the path a non-physical spring force between the images is used. The force, which is actually minimized during the NEB optimisation, consists of the projection of the real force of the underlying potential perpendicular to the path, and the projection of the spring force parallel to the path. An optimizer is developed which archives quadratic convergence of NEB optimizations in the noise-free potential energy surface of some test systems. This optimiser uses gradients and Hessians at each step. For the free energy surface both values can be calculated by umbrella integration as mentioned above. NEB optimizations within the free energy are performed in this work in the following way: at first a guess path is assumed, e.g. a straight line between two points in the configuration space, usually minima. This path is discretized into a number of images. Molecular dynamics umbrella sampling simulations are performed on each image. The gradient and Hessian from the umbrella integration are fed into the newly developed NEB optimizer. This way one does not need good starting guesses for the saddle points but an interpolation between the much more easily accessible minima is sufficient. Furthermore, the need for independent molecular dynamics runs at each image makes the method intrinsically parallel. The whole method is applied to the well-studied alanine dipeptide system and compared with the results from the serial method. Subsequently the algorithm is applied to a much more costly system of binding a ligand to its receptor in water.