@article{MenzLatorreSchuetteetal.2012,
  author    = {Menz, Stephan and Latorre, Juan C. and Sch{\"u}tte, Christof and Huisinga, Wilhelm},
  title     = {Hybrid stochastic-deterministic solution of the chemical master equation},
  series = {Multiscale modeling \& simulation : a SIAM interdisciplinary journal},
  volume    = {10},
  journal   = {Multiscale modeling \& simulation : a SIAM interdisciplinary journal},
  number    = {4},
  publisher = {Society for Industrial and Applied Mathematics},
  address   = {Philadelphia},
  issn      = {1540-3459},
  doi       = {10.1137/110825716},
  pages     = {1232 -- 1262},
  year      = {2012},
  abstract  = {The chemical master equation (CME) is the fundamental evolution equation of the stochastic description of biochemical reaction kinetics. In most applications it is impossible to solve the CME directly due to its high dimensionality. Instead, indirect approaches based on realizations of the underlying Markov jump process are used, such as the stochastic simulation algorithm (SSA). In the SSA, however, every reaction event has to be resolved explicitly such that it becomes numerically inefficient when the system's dynamics include fast reaction processes or species with high population levels. In many hybrid approaches, such fast reactions are approximated as continuous processes or replaced by quasi-stationary distributions in either a stochastic or a deterministic context. Current hybrid approaches, however, almost exclusively rely on the computation of ensembles of stochastic realizations. We present a novel hybrid stochastic-deterministic approach to solve the CME directly. Our starting point is a partitioning of the molecular species into discrete and continuous species that induces a partitioning of the reactions into discrete-stochastic and continuous-deterministic processes. The approach is based on a WKB (Wentzel-Kramers-Brillouin) ansatz for the conditional probability distribution function (PDF) of the continuous species (given a discrete state) in combination with Laplace's method of integral approximation. The resulting hybrid stochastic-deterministic evolution equations comprise a CME with averaged propensities for the PDF of the discrete species that is coupled to an evolution equation of the related expected levels of the continuous species for each discrete state. In contrast to indirect hybrid methods, the impact of the evolution of discrete species on the dynamics of the continuous species has to be taken into account explicitly. The proposed approach is efficient whenever the number of discrete molecular species is small. We illustrate the performance of the new hybrid stochastic-deterministic approach in an application to model systems of biological interest.},
  language  = {en}
}
@article{vonKleistMenzStockeretal.2011,
  author    = {von Kleist, Max and Menz, Stephan and Stocker, Hartmut and Arasteh, Keikawus and Schuette, Christof and Huisinga, Wilhelm},
  title     = {HIV quasispecies dynamics during pro-active treatment switching impact on multi-drug resistance and resistance archiving in latent reservoirs},
  series = {PLoS one},
  volume    = {6},
  journal   = {PLoS one},
  number    = {3},
  publisher = {PLoS},
  address   = {San Fransisco},
  issn      = {1932-6203},
  doi       = {10.1371/journal.pone.0018204},
  pages     = {12},
  year      = {2011},
  abstract  = {The human immunodeficiency virus (HIV) can be suppressed by highly active anti-retroviral therapy (HAART) in the majority of infected patients. Nevertheless, treatment interruptions inevitably result in viral rebounds from persistent, latently infected cells, necessitating lifelong treatment. Virological failure due to resistance development is a frequent event and the major threat to treatment success. Currently, it is recommended to change treatment after the confirmation of virological failure. However, at the moment virological failure is detected, drug resistant mutants already replicate in great numbers. They infect numerous cells, many of which will turn into latently infected cells. This pool of cells represents an archive of resistance, which has the potential of limiting future treatment options. The objective of this study was to design a treatment strategy for treatment-naive patients that decreases the likelihood of early treatment failure and preserves future treatment options. We propose to apply a single, pro-active treatment switch, following a period of treatment with an induction regimen. The main goal of the induction regimen is to decrease the abundance of randomly generated mutants that confer resistance to the maintenance regimen, thereby increasing subsequent treatment success. Treatment is switched before the overgrowth and archiving of mutant strains that carry resistance against the induction regimen and would limit its future re-use. In silico modelling shows that an optimal trade-off is achieved by switching treatment at \& 80 days after the initiation of antiviral therapy. Evaluation of the proposed treatment strategy demonstrated significant improvements in terms of resistance archiving and virological response, as compared to conventional HAART. While continuous pro-active treatment alternation improved the clinical outcome in a randomized trial, our results indicate that a similar improvement might also be reached after a single pro-active treatment switch. The clinical validity of this finding, however, remains to be shown by a corresponding trial.},
  language  = {en}
}