Refine
Language
- English (10)
Keywords
- systems biology (3)
- cow (2)
- hormone patterns (2)
- reproduction (2)
- affine invariant Gauss-Newton algorithm (1)
- bioparkin (1)
- differential equations (1)
- graphical user interface (1)
- gyncycle (1)
- metastable conformation (1)
Application Area
- A (10)
The complexity of molecular kinetics can be reduced significantly by a restriction to metastable conformations which are almost invariant sets of molecular dynamical systems. With the Robust Perron Cl uster Analysis PCCA+, developed by Weber and Deuflhard, we have a tool available which can be used to identify these conformations from a transition probability matrix. This method can also be applied to the corresponding transition rate matrix which provides important information concerning transition pathways of single molecules. In the present paper, we explain the relationship between these tw o concepts and the extraction of conformation kinetics from transition rates. Moreover, we show how transition rates can be approximated and conclude with numerical examples.
Whenever the invariant stationary density of metastable dynamical systems decomposes into almost invariant partial densities, its computation as eigenvector of some transition probability matrix is an ill-conditioned problem. In order to avoid this computational difficulty, we suggest to apply an aggregation/disaggregation method which only addresses wellconditioned sub-problems and thus results in a stable algorithm. In contrast to existing methods, the aggregation step is done via a sampling algorithm which covers only small patches of the sampling space. Finally, the theoretical analysis is illustrated by two biomolecular examples.
Wigner functions are functions on classical phase space, which are in one-to-one correspondence to square integrable functions on configuration space. For molecular quantum systems, classical transport of Wigner functions provides the basis of asymptotic approximation methods in the high energy regime. The article addresses the sampling of Wigner functions by Monte Carlo techniques. The approximation step is realized by an adaption of the Metropolis algorithm for real-valued functions with disconnected support. The quadrature, which computes values of the Wigner function, uses importance sampling with a Gaussian weight function. The numerical experiments combine the sampling with a surface hopping algorithm for non-adiabatic quantum dynamics. In agreement with theoretical considerations, the obtained results show an accuracy of two to four percent.
Bovine fertility is the subject of extensive research in animal sciences,
especially because fertility of dairy cows has declined during the last
decades. The regulation of estrus is controlled by the complex interplay
of various organs and hormones. Mathematical modeling of the bovine
estrous cycle could help in understanding the dynamics of this complex
biological system. In this paper we present a mechanistic mathematical
model of the bovine estrous cycle that includes the processes of follicle
and corpus luteum development and the key hormones that interact to
control these processes. The model generates successive estrous cycles of
21 days, with three waves of follicle growth per cycle. The model contains
12 differential equations and 54 parameters. Focus in this paper is on
development of the model, but also some simulation results are presented,
showing that a set of equations and parameters is obtained that describes
the system consistent with empirical knowledge. Even though the majority
of the mechanisms that are included in the model are based on relations
that in literature have only been described qualitatively (i.e. stimulation
and inhibition), the output of the model is surprisingly well in line with
empirical data. This model of the bovine estrous cycle could be used
as a basis for more elaborate models with the ability to study effects of
external manipulations and genetic differences.
This study presents a differential equation model for the feedback mechanisms between
Gonadotropin-releasing Hormone (GnRH), Follicle-Stimulating Hormone (FSH), Luteinizing
Hormone (LH), development of follicles and corpus luteum, and the production of estradiol
(E2), progesterone (P4), inhibin A (IhA), and inhibin B (IhB) during the female menstrual
cycle. In contrast to other models, this model does not involve delay differential equations
and is based on deterministic modelling of the GnRH pulse pattern, which allows for faster
simulation times and efficient parameter identification. These steps were essential to tackle
the task of developing a mathematical model for the administration of GnRH analogues. The
focus of this paper is on model development for GnRH receptor binding and the integration of
a pharmacokinetic/pharmacodynamic model for the GnRH agonist Nafarelin and the GnRH
antagonist Cetrorelix into the menstrual cycle model. The final mathematical model describes
the hormone profiles (LH, FSH, P4, E2) throughout the menstrual cycle in 12 healthy women.
Moreover, it correctly predicts the changes in the cycle following single and multiple dose
administration of Nafarelin or Cetrorelix at different stages in the cycle.
Modelling, parameter identification, and simulation play an important rôle in Systems Biology. In recent years, various software packages have been established for scientific use in both licencing types, open source as well as commercial. Many of these codes are based on inefficient and mathematically outdated algorithms. By introducing the package BioPARKIN recently developed at ZIB, we want to improve this situation significantly. The development of the software BioPARKIN involves long standing mathematical ideas that, however, have not yet entered the field of Systems Biology, as well as new ideas and tools that are particularly important for the analysis of the dynamics of biological networks. BioPARKIN originates from the package PARKIN, written by P.Deuflhard and U.Nowak, that has been applied successfully for parameter identification in physical chemistry for many years.
Our model of the bovine estrous cycle is a set of ordinary differential equations which generates hormone profiles of successive estrous cycles with several follicular waves per cycle. It describes the growth and decay of the follicles and the corpus luteum, as well as the change of the key substances over time. In this work we describe recent improvements of this model, including the introduction of new components, and elimination of time delays. We validate our model by showing that the simulations agree with observations from synchronization studies and with measured progesterone data after a single dose administration of synthetic prostaglandin F2alpha.
Particle methods have become indispensible in conformation dynamics to compute transition rates in protein folding, binding processes and molecular design, to mention a few. Conformation dynamics requires at a decomposition of a molecule's position space into metastable conformations. In this paper, we show how this decomposition can be obtained via the design of either ``soft'' or ``hard'' molecular conformations. We show, that the soft approach results in a larger metastabilitiy of the decomposition and is thus more advantegous. This is illustrated by a simulation of Alanine Dipeptide.
Markov state models have become very popular for the description of conformation dynamics of molecules over
long timescales. The construction of such models requires a partitioning of the configuration space such that the discretization
can serve as an approximation of metastable conformations. Since the computational complexity for the construction of a
Markov state model increases quadratically with the number of sets, it is desirable to obtain as few sets as necessary. In this
paper we propose an algorithm for the adaptive refinement of an initial coarse partitioning. A spectral clustering method is
applied to the final partitioning to detect the metastable conformations. We apply this method to the conformation analysis of
a model tri-peptide molecule, where metastable beta- and gamma-turn conformations can be identified.
Motivation. Modelling, parameter identification, and simulation play an important role in systems biology. Usually, the goal is to determine parameter values that minimise the difference between experimental measurement values and model predictions in a least-squares sense. Large-scale biological networks, however, often suffer from missing data for parameter identification. Thus, the least-squares problems are rank-deficient and solutions are not unique. Many common optimisation methods ignore this detail because they do not take into account the structure of the underlying inverse problem. These algorithms simply return a “solution” without additional information on identifiability or uniqueness. This can yield misleading results, especially if parameters are co-regulated and data are noisy.
Results. The Gauss-Newton method presented in this paper monitors the numerical rank of the Jacobian and converges locally, for the class of adequate problems, to a solution that is unique within the subspace of identifiable parameters. This method has been implemented in BioPARKIN, a software package that combines state-of-the-art numerical algorithms with compliance to system biology standards, most importantly SBML, and an accessible interface.
Availability. The software package BioPARKIN is available for download at http://bioparkin.zib.de .