From Molecular Dynamics to Conformation Dynamics in Drug Design

Abstract

Drug design. Computational drug design deals with the following type of problem: given some virus (say, by NMR of some crystalized large protein molecule with a small up to moderate size active site), design a molecule such that it inhibits the virus to do its harmful work. In order to find out whether a suggested molecule really acts as a drug, reliable information about the dynamics of the molecular system virus-drug is required. Molecular dynamics. If we ignore (for the purpose of simplification) any quantum effects (such as tunneling), we are left with a set of Hamiltonian differential equations originating from Newtons laws of motion. While the kinetic energy part of the Hamiltonian is easy to establish, the potential energy part is usually obtained from large data bases. However, the arising Hamiltonian initial value problems usually turn out to be ill-conditioned after psec time spans - with an exponentially increasing condition number. Therefore, numerical long term integration, often called molecular modelling, will not supply information about the dynamics; rather it supplies information about ensemble averages via temporal averages on the theoretical basis of either a mathematical or a physical ergodicity hypothesis. Conformational dynamics. Consequently whenever dynamical information is redry needed, as in drug design, only short term numerical integration should be applied. A rather recent approach to compute essential features of the dynamical behavior, called conformation dynamics, has recently been developed by the author, SCHUTTE, and coworkers. The key idea is borrowed from nonlinear dynamics: rather than computing trajectories, which realize a point concept in phase space, one directly aims at an identification of so-called almost invariant set of the associated Hamiltonian system thus obviously realizing a set concept. These sets turn out to be interpretable as metastable chemical conformations. Moreover, by computing these mathematical objects, additional information about the life spans and decay patterns of conformations comes up naturally. In other words: the ill-posed trajectory dynamics is replaced by the well-posed conformation dynamics. In a fisrst step, its actual computation had been done following mainly ideas of DELLNITZ ET BL. who had mainly tackled hyperbolic dynamical systems (which Hamiltonian systems are definitely not!): they had indicated that essential dynamical features could be computed via the solution of some cluster eigenvalue problem for the so-called PERRON-FROBENIUS operator defined by the how of the system. In this way one obtains information about almost invariant sets, within which the dynamical systems remains for a long time, once it is in there. In other words, this approach realizes some nonlinear multiscale decoupling in the framework of statistics. In the context of Hamiltonian systems, the (non-selfadjoint) Perron-Frobenius operator had to be replaced by some different (selfadjoint) Markov operator, which meant that almost invariant sets in phase space (including position variables and generalized momenta) were replaced by almost invariant sets in position space only. Hybrid Monte Carlo methods. Galerkin discretization of that selfadjoint operator in some weighted Hilbert space naturally brings up so-called hybrid Monte Carlo (HMC) methods, which may be understood as a compromise between molecular modelling (on short time scales only, where the initial value problem still is well-posed!) and Markov chain modelling via classical Monte Carlo. Along this line transition matrices for so-called nearly uncoupled Markov chains are created. In order to help avoid the undesirable critical slowing doum or trapping within local minima, a adaptive temperature variant of HMC has been designed: it permits to increase temperature for the generalized momenta only and to reweight this non-canonical ensemble to the canonical ensemble afterwards. Finally multigrid methods in connection with the Perron cluster of eigenvalues and the HMC process are under present investigation. Finally, in order to generate discretization boxes for the Markov operator, self-organizing maps due to KOHONEN are applied and combined efficiently with the Perron cluster problem. Virtual Lab. The whole approach aims at the substitution of experiments in chemical labs by simulations in a virtual lab. Numerical experiments on moderate size biomolecules including virus inhibitors for HIV and influenza have been presented.