Mats Jirstrand

Systems Biology Toolbox for MATLAB: a computational platform for research in systems biology

We present a Systems Biology Toolbox for the widely used general purpose mathematical software MATLAB. The toolbox offers systems biologists an open and extensible environment, in which to explore ideas, prototype and share new algorithms, and build applications for the analysis and simulation of biological and biochemical systems. Additionally it is well suited for educational purposes. The toolbox supports the Systems Biology Markup Language (SBML) by providing an interface for import and export of SBML models. In this way the toolbox connects nicely to other SBML-enabled modelling packages. Models are represented in an internal model format and can be described either by entering ordinary differential equations or, more intuitively, by entering biochemical reaction equations. The toolbox contains a large number of analysis methods, such as deterministic and stochastic simulation, parameter estimation, network identification, parameter sensitivity analysis and bifurcation analysis.

Kinetic models in industrial biotechnology - Improving cell factory performance

An increasing number of industrial bioprocesses capitalize on living cells by using them as cell factories that convert sugars into chemicals. These processes range from the production of bulk chemicals in yeasts and bacteria to the synthesis of therapeutic proteins in mammalian cell lines. One of the tools in the continuous search for improved performance of such production systems is the development and application of mathematical models. To be of value for industrial biotechnology, mathematical models should be able to assist in the rational design of cell factory properties or in the production processes in which they are utilized. Kinetic models are particularly suitable towards this end because they are capable of representing the complex biochemistry of cells in a more complete way compared to most other types of models. They can, at least in principle, be used to in detail understand, predict, and evaluate the effects of adding, removing, or modifying molecular components of a cell factory and for supporting the design of the bioreactor or fermentation process. However, several challenges still remain before kinetic modeling will reach the degree of maturity required for routine application in industry. Here we review the current status of kinetic cell factory modeling. Emphasis is on modeling methodology concepts, including model network structure, kinetic rate expressions, parameter estimation, optimization methods, identifiability analysis, model reduction, and model validation, but several applications of kinetic models for the improvement of cell factories are also discussed.

Comparison of approaches for parameter identifiability analysis of biological systems

MOTIVATION Modeling of dynamical systems using ordinary differential equations is a popular approach in the field of Systems Biology. The amount of experimental data that are used to build and calibrate these models is often limited. In this setting, the model parameters may not be uniquely determinable. Structural or a priori identifiability is a property of the system equations that indicates whether, in principle, the unknown model parameters can be determined from the available data. RESULTS We performed a case study using three current approaches for structural identifiability analysis for an application from cell biology. The approaches are conceptually different and are developed independently. The results of the three approaches are in agreement. We discuss strength and weaknesses of each of them and illustrate how they can be applied to real world problems. AVAILABILITY AND IMPLEMENTATION For application of the approaches to further applications, code representations (DAISY, Mathematica and MATLAB) for benchmark model and data are provided on the authors webpage. CONTACT andreas.raue@fdm.uni-freiburg.de.

An Efficient Method for Structural Identifiability Analysis of Large Dynamic Systems

Ordinary differential equation models often contain a large number of parameters that must be determined from measurements by parameter estimation. For a parameter estimation procedure to be successful, there must be a unique set of parameters that can have produced the measured data. This is not the case if a model is not structurally identifiable with the given set of outputs selected as measurements. We describe the implementation of a recent probabilistic semi-numerical method for testing local structural identifiability based on computing the rank of a numerically instantiated Jacobian matrix (observability/identifiability matrix). To obtain this, matrix parameters and initial conditions are specialized to random integer numbers, inputs are specialized to truncated random integer coefficient power series, and the corresponding output of the state space system is computed in terms of a truncated power series, which then is utilized to calculate the elements of a Jacobian matrix. To reduce the memory requirements and increase the speed of the computations all operations are done modulo a large prime number. The method has been extended to handle parametrized initial conditions and is demonstrated to be capable of handling systems in the order of a hundred state variables and equally many parameters on a standard desktop computer.

A method for zooming of nonlinear models of biochemical systems

BackgroundModels of biochemical systems are typically complex, which may complicate the discovery of cardinal biochemical principles. It is therefore important to single out the parts of a model that are essential for the function of the system, so that the remaining non-essential parts can be eliminated. However, each component of a mechanistic model has a clear biochemical interpretation, and it is desirable to conserve as much of this interpretability as possible in the reduction process. Furthermore, it is of great advantage if we can translate predictions from the reduced model to the original model.ResultsIn this paper we present a novel method for model reduction that generates reduced models with a clear biochemical interpretation. Unlike conventional methods for model reduction our method enables the mapping of predictions by the reduced model to the corresponding detailed predictions by the original model. The method is based on proper lumping of state variables interacting on short time scales and on the computation of fraction parameters, which serve as the link between the reduced model and the original model. We illustrate the advantages of the proposed method by applying it to two biochemical models. The first model is of modest size and is commonly occurring as a part of larger models. The second model describes glucose transport across the cell membrane in bakers yeast. Both models can be significantly reduced with the proposed method, at the same time as the interpretability is conserved.ConclusionsWe introduce a novel method for reduction of biochemical models that is compatible with the concept of zooming. Zooming allows the modeler to work on different levels of model granularity, and enables a direct interpretation of how modifications to the model on one level affect the model on other levels in the hierarchy. The method extends the applicability of the method that was previously developed for zooming of linear biochemical models to nonlinear models.

Bridging the gaps in systems biology

Abstract Systems biology aims at creating mathematical models, i.e., computational reconstructions of biological systems and processes that will result in a new level of understanding—the elucidation of the basic and presumably conserved “design” and “engineering” principles of biomolecular systems. Thus, systems biology will move biology from a phenomenological to a predictive science. Mathematical modeling of biological networks and processes has already greatly improved our understanding of many cellular processes. However, given the massive amount of qualitative and quantitative data currently produced and number of burning questions in health care and biotechnology needed to be solved is still in its early phases. The field requires novel approaches for abstraction, for modeling bioprocesses that follow different biochemical and biophysical rules, and for combining different modules into larger models that still allow realistic simulation with the computational power available today. We have identified and discussed currently most prominent problems in systems biology: (1) how to bridge different scales of modeling abstraction, (2) how to bridge the gap between topological and mechanistic modeling, and (3) how to bridge the wet and dry laboratory gap. The future success of systems biology largely depends on bridging the recognized gaps.

Bile canalicular dynamics in hepatocyte sandwich cultures

Many substances are hepatotoxic due to their ability to cause intrahepatic cholestasis. Therefore, there is a high demand for in vitro systems for the identification of cholestatic properties of new compounds. Primary hepatocytes cultivated in collagen sandwich cultures are known to establish bile canaliculi which enclose secreted biliary components. Cholestatic compounds are mainly known to inhibit bile excretion dynamics, but may also alter canalicular volume, or hepatocellular morphology. So far, techniques to assess time-resolved morphological changes of bile canaliculi in sandwich cultures are not available. In this study, we developed an automated system that quantifies dynamics of bile canaliculi recorded in conventional time-lapse image sequences. We validated the hepatocyte sandwich culture system as an appropriate model to study bile canaliculi in vitro by showing structural similarity measured as bile canaliculi length per hepatocyte to that observed in vivo. Moreover, bile canalicular excretion kinetics of CMFDA (5-chloromethylfluorescein diacetate) in sandwich cultures resembled closely the kinetics observed in vivo. The developed quantification technique enabled the quantification of dynamic changes in individual bile canaliculi. With this technique, we were able to clearly distinguish between sandwich cultures supplemented with dexamethasone and insulin from control cultures. In conclusion, the automated quantification system offers the possibility to systematically study the causal relationship between disturbed bile canalicular dynamics and cholestasis.

Heterogeneous kinetics of AKT signaling in individual cells are accounted for by variable protein concentration

In most solid cancers, cells harboring oncogenic mutations represent only a sub-fraction of the entire population. Within this sub-fraction the expression level of mutated proteins can vary significantly due to cellular variability limiting the efficiency of targeted therapy. To address the causes of the heterogeneity, we performed a systematic analysis of one of the most frequently mutated pathways in cancer cells, the phosphatidylinositol 3 kinase (PI3K) signaling pathway. Among others PI3K signaling is activated by the hepatocyte growth factor (HGF) that regulates proliferation of hepatocytes during liver regeneration but also fosters tumor cell proliferation. HGF-mediated responses of PI3K signaling were monitored both at the single cell and cell population level in primary mouse hepatocytes and in the hepatoma cell line Hepa1_6. Interestingly, we observed that the HGF-mediated AKT responses at the level of individual cells is rather heterogeneous. However, the overall average behavior of the single cells strongly resembled the dynamics of AKT activation determined at the cell population level. To gain insights into the molecular cause for the observed heterogeneous behavior of individual cells, we employed dynamic mathematical modeling in a stochastic framework. Our analysis demonstrated that intrinsic noise was not sufficient to explain the observed kinetic behavior, but rather the importance of extrinsic noise has to be considered. Thus, distinct from gene expression in the examined signaling pathway fluctuations of the reaction rates has only a minor impact whereas variability in the concentration of the various signaling components even in a clonal cell population is a key determinant for the kinetic behavior.

Stochastic differential equations as a tool to regularize the parameter estimation problem for continuous time dynamical systems given discrete time measurements.

In this paper we consider the problem of estimating parameters in ordinary differential equations given discrete time experimental data. The impact of going from an ordinary to a stochastic differential equation setting is investigated as a tool to overcome the problem of local minima in the objective function. Using two different models, it is demonstrated that by allowing noise in the underlying model itself, the objective functions to be minimized in the parameter estimation procedures are regularized in the sense that the number of local minima is reduced and better convergence is achieved. The advantage of using stochastic differential equations is that the actual states in the model are predicted from data and this will allow the prediction to stay close to data even when the parameters in the model is incorrect. The extended Kalman filter is used as a state estimator and sensitivity equations are provided to give an accurate calculation of the gradient of the objective function. The method is illustrated using in silico data from the FitzHugh-Nagumo model for excitable media and the Lotka-Volterra predator-prey system. The proposed method performs well on the models considered, and is able to regularize the objective function in both models. This leads to parameter estimation problems with fewer local minima which can be solved by efficient gradient-based methods.

Yeast AMP-activated protein kinase monitors glucose concentration changes and absolute glucose levels

Background: Little is known about the signaling dynamics of AMP-activated protein kinase. Results: We define the dynamics of yeast AMPK signaling under different glucose concentrations. Conclusion: The Snf1-Mig1 signaling system monitors glucose concentration changes and absolute glucose levels to adjust the metabolism to a wide range of conditions. Significance: This description of AMPK signaling dynamics will stimulate studies defining the integration of signaling and metabolism. Analysis of the time-dependent behavior of a signaling system can provide insight into its dynamic properties. We employed the nucleocytoplasmic shuttling of the transcriptional repressor Mig1 as readout to characterize Snf1-Mig1 dynamics in single yeast cells. Mig1 binds to promoters of target genes and mediates glucose repression. Mig1 is predominantly located in the nucleus when glucose is abundant. Upon glucose depletion, Mig1 is phosphorylated by the yeast AMP-activated kinase Snf1 and exported into the cytoplasm. We used a three-channel microfluidic device to establish a high degree of control over the glucose concentration exposed to cells. Following regimes of glucose up- and downshifts, we observed a very rapid response reaching a new steady state within less than 1 min, different glucose threshold concentrations depending on glucose up- or downshifts, a graded profile with increased cell-to-cell variation at threshold glucose concentrations, and biphasic behavior with a transient translocation of Mig1 upon the shift from high to intermediate glucose concentrations. Fluorescence loss in photobleaching and fluorescence recovery after photobleaching data demonstrate that Mig1 shuttles constantly between the nucleus and cytoplasm, although with different rates, depending on the presence of glucose. Taken together, our data suggest that the Snf1-Mig1 system has the ability to monitor glucose concentration changes as well as absolute glucose levels. The sensitivity over a wide range of glucose levels and different glucose concentration-dependent response profiles are likely determined by the close integration of signaling with the metabolism and may provide for a highly flexible and fast adaptation to an altered nutritional status.