On the relationship between cell cycle analysis with ergodic principles and age-structured cell population models.

Cyclic processes, in particular the cell cycle, are of great importance in cell biology. Continued improvement in cell population analysis methods like fluorescence microscopy, flow cytometry, CyTOF or single-cell omics made mathematical methods based on ergodic principles a powerful tool in studying these processes. In this paper, we establish the relationship between cell cycle analysis with ergodic principles and age structured population models. To this end, we describe the progression of a single cell through the cell cycle by a stochastic differential equation on a one dimensional manifold in the high dimensional dataspace of cell cycle markers. Given the assumption that the cell population is in a steady state, we derive transformation rules which transform the number density on the manifold to the steady state number density of age structured population models. Our theory facilitates the study of cell cycle dependent processes including local molecular events, cell death and cell division from high dimensional "snapshot" data. Ergodic analysis can in general be applied to every process that exhibits a steady state distribution. By combining ergodic analysis with age structured population models we furthermore provide the theoretic basis for extensions of ergodic principles to distribution that deviate from their steady state.

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.

Death wins against life in a spatially extended model of the caspase-3/8 feedback loop.

Apoptosis is an important physiological process which enables organisms to remove unwanted or damaged cells. A mathematical model of the extrinsic pro-apoptotic signaling pathway has been introduced by Eissing et al. (2007) and a bistable behavior with a stable death state and a stable life state of the reaction system has been established. In this paper, we consider a spatial extension of the extrinsic pro-apoptotic signaling pathway incorporating diffusion terms and make a model-based, numerical analysis of the apoptotic switch in the spatial dimension. For the parameter regimes under consideration it turns out that for this model diffusion homogenizes rapidly the concentrations which afterward are governed by the original reaction system. The activation of effector-caspase 3 depends on the space averaged initial concentration of pro-caspase 8 and pro-caspase 3 at the beginning of the process.

Design of biomolecular network modifications to achieve adaptation.

A biomolecular network is called adaptive if its output returns to the original value after a transient response even under a persisting stimulus. The conditions for adaptation have been investigated thoroughly with systems theory approaches in the literature and it is easy to check whether they are satisfied in the linear approximation. In contrast, it is in general not easy to modify a non-adaptive network model such that it gains adaptive behaviour, especially for medium- and large-scale networks. The authors present a systematic approach based on the notion of kinetic perturbations to construct adaptive biomolecular network models from non-adaptive ones. An advantage of kinetic perturbations in this application is that neither the stoichiometry nor the steady state of the system is changed. Furthermore, the method covers both parameter and network structure modifications and can be applied to any reaction rate formalism and even to medium-scale or partially unknown models. The approach is exemplified at a small- and a medium-sized biomolecular network, illustrating its potential to systematically evaluate the different network modifications for adaptation. The proposed method will be useful either in iterative model building to construct mathematical models of adaptive biomolecular networks, or in synthetic biology where it can be applied to design or modify synthetic networks for adaptation.

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.

Describing force-induced bone growth and adaptation by a mathematical model.

This work proposes a mathematical model that qualitative describes the process of mechanically force-induced bone growth and adaptation. The mathematical model includes osteocytes as the key interfacing layer connecting tissue, cellular and molecular signaling levels. Specifically, in the presence of an increase in the mechanical stimuli, osteocytes respond by mechano-transduction releasing the local factors nitric oxide (NO) and prostaglandin E(2) (PGE(2)). These local factors act as the signaling recruitment signals for bone cells progenitors and influence the coupling activity among osteoblasts and osteoclasts during the process of bone remodeling. The model is in agreement with qualitative observations found in the literature concerning the process of bone adaptation and the cellular interactions during a local bone remodeling cycle induced by mechanical stimulation.

Robustness properties of apoptosis models with respect to parameter variations and intrinsic noise.

Analyses of different robustness aspects for models of the direct signal transduction pathway of receptor-induced apoptosis is presented. Apoptosis is a form of programmed cell death, removing unwanted cells within multicellular organisms to maintain a proper balance between cell reproduction and death. Its signalling pathway includes an activation feedback loop that generates bistable behaviour, where the two steady states can be seen as 'life' and 'death'. Inherent robustness, widely recognised in biological systems, is of major importance in apoptosis signalling, as it guarantees the same cell fate for similar conditions. First, the influence of the stochastic nature of reactions indicating a role for inhibition reactions as noise filters and justifying a deterministic approach in the further analyses is evaluated. Second, the robustness of the bistable threshold with respect to parameter changes is evaluated by statistical methods, showing the need to balance both the forward and the back part of the activation loop. These analyses can also discriminate between the models favouring the model consistent with novel biological findings. The parameter robustness analyses are also applicable to other signal transduction networks, as several have been shown to display bistable behaviour. These methods therefore have a range of possible applications in systems biology not only to measure robustness, but also for model discrimination.

Reduction of mathematical models of signal transduction networks: simulation-based approach applied to EGF receptor signalling.

Biological systems and, in particular, cellular signal transduction pathways are characterised by their high complexity. Mathematical models describing these processes might be of great help to gain qualitative and, most importantly, quantitative knowledge about such complex systems. However, a detailed mathematical description of these systems leads to nearly unmanageably large models, especially when combining models of different signalling pathways to study cross-talk phenomena. Therefore, simplification of models becomes very important. Different methods are available for model reduction of biological models. Importantly, most of the common model reduction methods cannot be applied to cellular signal transduction pathways. Using as an example the epidermal growth factor (EGF) signalling pathway, we discuss how quantitative methods like system analysis and simulation studies can help to suitably reduce models and additionally give new insights into the signal transmission and processing of the cell.