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In the well-known discrete modeling framework developed by R. Thomas, the structure of a biological regulatory network is captured in an interaction graph, which, together with a set of Boolean parameters, gives rise to a state transition graph describing all possible dynamical behaviors. For complex networks the analysis of the dynamics becomes more and more difficult, and efficient methods to carry out the analysis are needed. In this paper, we focus on identifying subnetworks of the system that govern the behavior of the system as a whole. We present methods to derive trajectories and attractors of the network from the dynamics suitable subnetworks display in isolation. In addition, we use these ideas to link the existence of certain structural motifs, namely circuits, in the interaction graph to the character and number of attractors in the state transition graph, generalizing and refining results presented in \cite{AB07}. Lastly, we show for a specific class of networks that all possible asymptotic behaviors of networks in that class can be derived from the dynamics of easily identifiable subnetworks.
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.
Boolean modeling frameworks have long since proved their worth for capturing and analyzing essential characteristics of complex systems.
Hybrid approaches aim at exploiting the advantages of Boolean formalisms while refining expressiveness. In this paper, we present a formalism that augments Boolean models with stochastic aspects. More specifically, biological reactions effecting a system in a given state are associated
with probabilities, resulting in dynamical behavior represented as a Markov chain. Using this approach, we model and analyze the cytokinin
response network of Arabidopsis thaliana with a focus on clarifying the character of an important feedback mechanism.
A well-known discrete approach to modeling biological regulatory networks is the logical framework developed by R. Thomas. The network structure is captured in an interaction graph, which, together with a set of Boolean parameters, gives rise to a state transition graph describing the dynamical behavior. Together with E. H. Snoussi, Thomas later extended the framework by including singular values representing the threshold values of interactions. A systematic approach was taken in \cite{AB07} to link circuits in the interaction graph with character and number of attractors in the state transition graph by using the information inherent in singular steady states. In this paper, we employ the concept of local interaction graphs to strengthen the results in \cite{AB07}. Using the local interaction graph of a singular steady state, we are able to construct attractors of the regulatory network from attractors of certain subnetworks. As a comprehensive generalization of the framework introduced in \cite{AB07}, we
drop constraints concerning the choice of parameter values to include so-called context sensitive networks.
A biological regulatory network can be modeled as a discrete function f that contains all available information on network component interactions. From f we can derive a graph representation of the network structure as well as of the dynamics of the system. In this paper we introduce a method to identify modules of the network that allow us to construct the behavior of f from the dynamics of the modules. Here, it proves useful to distinguish between dynamical and structural modules, and to define network modules combining aspects of both.
As a key concept we establish the notion of symbolic steady state, which basically represents a set of states where the behavior of f is in some sense predictable, and which gives rise to suitable network modules.
We apply the method to a regulatory network involved in T helper cell differentiation.
A discrete model of a biological regulatory network can be represented as a discrete function f that contains all available information on interactions between network components and the rules governing the evolution of the network in the discrete state space. Both the information on the structure as well as the dynamics of the system can be represented as directed graphs. Since the state space size grows exponentially with the number of network components, analysis of large networks is a complex problem.
In this paper, we introduce the notion of symbolic steady state that allows us to identify subnetworks that govern the dynamics of the original network in at least a subset of state space. We then state rules to explicitly construct attractors of the system from subnetwork attractors. A further application of the underlying concept allows us to formulate sufficient conditions for the existence of multiple attractors resp. a cyclic attractor based on the existence of positive resp. negative feedback circuits in the structure graph. All results are discussed for dynamics derived from f via the synchronous as well as the asynchronous update rule.
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.
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 .