Continuous analogue to iterative optimization for PDE-constrained inverse problems.

The parameters of many physical processes are unknown and have to be inferred from experimental data. The corresponding parameter estimation problem is often solved using iterative methods such as steepest descent methods combined with trust regions. For a few problem classes also continuous analogues of iterative methods are available. In this work, we expand the application of continuous analogues to function spaces and consider PDE (partial differential equation)-constrained optimization problems. We derive a class of continuous analogues, here coupled ODE (ordinary differential equation)-PDE models, and prove their convergence to the optimum under mild assumptions. We establish sufficient bounds for local stability and convergence for the tuning parameter of this class of continuous analogues, the retraction parameter. To evaluate the continuous analogues, we study the parameter estimation for a model of gradient formation in biological tissues. We observe good convergence properties, indicating that the continuous analogues are an interesting alternative to state-of-the-art iterative optimization methods.

Data2Dynamics: a modeling environment tailored to parameter estimation in dynamical systems.

UNLABELLED: Modeling of dynamical systems using ordinary differential equations is a popular approach in the field of systems biology. Two of the most critical steps in this approach are to construct dynamical models of biochemical reaction networks for large datasets and complex experimental conditions and to perform efficient and reliable parameter estimation for model fitting. We present a modeling environment for MATLAB that pioneers these challenges. The numerically expensive parts of the calculations such as the solving of the differential equations and of the associated sensitivity system are parallelized and automatically compiled into efficient C code. A variety of parameter estimation algorithms as well as frequentist and Bayesian methods for uncertainty analysis have been implemented and used on a range of applications that lead to publications. AVAILABILITY AND IMPLEMENTATION: The Data2Dynamics modeling environment is MATLAB based, open source and freely available at http://www.data2dynamics.org. CONTACT: andreas.raue@fdm.uni-freiburg.de SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

Method of conditional moments (MCM) for the Chemical Master Equation: a unified framework for the method of moments and hybrid stochastic-deterministic models.

The time-evolution of continuous-time discrete-state biochemical processes is governed by the Chemical Master Equation (CME), which describes the probability of the molecular counts of each chemical species. As the corresponding number of discrete states is, for most processes, large, a direct numerical simulation of the CME is in general infeasible. In this paper we introduce the method of conditional moments (MCM), a novel approximation method for the solution of the CME. The MCM employs a discrete stochastic description for low-copy number species and a moment-based description for medium/high-copy number species. The moments of the medium/high-copy number species are conditioned on the state of the low abundance species, which allows us to capture complex correlation structures arising, e.g., for multi-attractor and oscillatory systems. We prove that the MCM provides a generalization of previous approximations of the CME based on hybrid modeling and moment-based methods. Furthermore, it improves upon these existing methods, as we illustrate using a model for the dynamics of stochastic single-gene expression. This application example shows that due to the more general structure, the MCM allows for the approximation of multi-modal distributions.

High-dimensional Bayesian parameter estimation: case study for a model of JAK2/STAT5 signaling.

In this work we present results of a detailed Bayesian parameter estimation for an analysis of ordinary differential equation models. These depend on many unknown parameters that have to be inferred from experimental data. The statistical inference in a high-dimensional parameter space is however conceptually and computationally challenging. To ensure rigorous assessment of model and prediction uncertainties we take advantage of both a profile posterior approach and Markov chain Monte Carlo sampling. We analyzed a dynamical model of the JAK2/STAT5 signal transduction pathway that contains more than one hundred parameters. Using the profile posterior we found that the corresponding posterior distribution is bimodal. To guarantee efficient mixing in the presence of multimodal posterior distributions we applied a multi-chain sampling approach. The Bayesian parameter estimation enables the assessment of prediction uncertainties and the design of additional experiments that enhance the explanatory power of the model. This study represents a proof of principle that detailed statistical analysis for quantitative dynamical modeling used in systems biology is feasible also in high-dimensional parameter spaces.

Analysis and simulation of division- and label-structured population models : a new tool to analyze proliferation assays.

In most biological studies and processes, cell proliferation and population dynamics play an essential role. Due to this ubiquity, a multitude of mathematical models has been developed to describe these processes. While the simplest models only consider the size of the overall populations, others take division numbers and labeling of the cells into account. In this work, we present a modeling and computational framework for proliferating cell populations undergoing symmetric cell division, which incorporates both the discrete division number and continuous label dynamics. Thus, it allows for the consideration of division number-dependent parameters as well as the direct comparison of the model prediction with labeling experiments, e.g., performed with Carboxyfluorescein succinimidyl ester (CFSE), and can be shown to be a generalization of most existing models used to describe these data. We prove that under mild assumptions the resulting system of coupled partial differential equations (PDEs) can be decomposed into a system of ordinary differential equations (ODEs) and a set of decoupled PDEs, which drastically reduces the computational effort for simulating the model. Furthermore, the PDEs are solved analytically and the ODE system is truncated, which allows for the prediction of the label distribution of complex systems using a low-dimensional system of ODEs. In addition to modeling the label dynamics, we link the label-induced fluorescence to the measure fluorescence which includes autofluorescence. Furthermore, we provide an analytical approximation for the resulting numerically challenging convolution integral. This is illustrated by modeling and simulating a proliferating population with division number-dependent proliferation rate.

Parameter identification, experimental design and model falsification for biological network models using semidefinite programming.

One of the most challenging tasks in systems biology is parameter identification from experimental data. In particular, if the available data are noisy, the resulting parameter uncertainty can be huge and should be quantified. In this work, a set-based approach for parameter identification in discrete time models of biochemical reaction networks from time series data is developed. The basic idea is to determine an outer approximation to the set of parameters for which trajectories are consistent with the available data. In order to approximate the set of consistent parameters (SCP) a feasibility problem is derived. This feasibility problem is used to verify that complete parameter sets cannot contain consistent parameters. This method is very appealing because instead of checking a finite number of distinct points, complete sets are analysed. With this approach, model falsification simply corresponds to showing that the SCP is empty. Besides parameter identification, a novel set-based method for experimental design is presented. This method yields reliable predictions on the information content of future measurements also for the case of very limited a priori knowledge and uncertain inputs. The properties of the method are presented using a discrete time model of the MAP kinase cascade.

Cell differentiation modeled via a coupled two-switch regulatory network.

Mesenchymal stem cells can give rise to bone and other tissue cells, but their differentiation still escapes full control. In this paper we address this issue by mathematical modeling. We present a model for a genetic switch determining the cell fate of progenitor cells which can differentiate into osteoblasts (bone cells) or chondrocytes (cartilage cells). The model consists of two switch mechanisms and reproduces the experimentally observed three stable equilibrium states: a progenitor, an osteogenic, and a chondrogenic state. Conventionally, the loss of an intermediate (progenitor) state and the entailed attraction to one of two opposite (differentiated) states is modeled as a result of changing parameters. In our model in contrast, we achieve this by distributing the differentiation process to two functional switch parts acting in concert: one triggering differentiation and the other determining cell fate. Via stability and bifurcation analysis, we investigate the effects of biochemical stimuli associated with different system inputs. We employ our model to generate differentiation scenarios on the single cell as well as on the cell population level. The single cell scenarios allow to reconstruct the switching upon extrinsic signals, whereas the cell population scenarios provide a framework to identify the impact of intrinsic properties and the limiting factors for successful differentiation.

Allosteric regulation of the access channels to the Rb+ occlusion sites of (Na+ + K+)-ATPase.

Previous work on the role of occluded Rb+ (a K+ substitute) in the reaction cycle of (Na+ + K+)-ATPase has focused on the kinetics of the dissociation of the enzyme-Rb+ complex at 20-24 degrees C. Doing experiments at 4 degrees C, we have made the following observations on the equilibrium binding levels and the kinetics of binding and release of Rb+. 1) The plot of bound Rb+ as a function of [Rb+] showed occupancy of high affinity sites, followed by binding to sites of lower affinity. The estimated number of Rb+ sites/active site was two to three, but a higher number was not ruled out. Release of bound Rb+ was slow and not monoexponential, the major portion being in a pool with a half-life of 4-5 h. Dissociation curves were identical at different levels of site occupancy. Rb+ binding also had fast and slow phases, requiring about 24 h to reach steady state at vastly different [Rb+]. These data suggest that (a) Rb+ occlusion sites are confined within the protein matrix and connected to the medium by narrow access channels that are heterogeneous in size, and (b) channel heterogeneity is distinct from differences in occlusion site affinities. 2) ATP, at a low affinity allosteric site, had no significant effect on the maximal level of bound Rb+ at any [Rb+], but it accelerated both the fast and the slow phases of Rb+ binding and release, and it increased the ratio of fast to slow phases. Evidently, ATP activates the channels (lowers the energy barrier for access) without altering binding site affinities. 3) Na+ was a competitive inhibitor of Rb+ at the occluded sites, but it also acted at an allosteric site to activate the access channels. Rb+ and K+ also had allosteric effects: although they did not affect the access channels directly, they blocked the allosteric effect of Na+. 4) Ouabain was an access channel inhibitor. It reduced the rates of binding and release of Rb+, blocked channel activation by ATP and Na+, but seemed to have no effect on the events at the occluded sites. The existence of heterogeneous access channels to the ion transport sites and the demonstration of channel regulation by the physiological ligands of the enzyme suggest the necessity of the inclusion of such allosteric mechanisms in the reaction cycle of (Na+ + K+)-ATPase.