Steady state and (bi-) stability evaluation of simple protease signalling networks

Signal transduction networks are complex, as are their mathematical models. Gaining a deeper understanding requires a system analysis. Important aspects are the number, location and stability of steady states. In particular, bistability has been recognised as an important feature to achieve molecular switching. This paper compares different model structures and analysis methods particularly useful for bistability analysis. The biological applications include proteolytic cascades as, for example, encountered in the apoptotic signalling pathway or in the blood clotting system. We compare three model structures containing zero-order, inhibitor and cooperative ultrasensitive reactions, all known to achieve bistability. The combination of phase plane and bifurcation analysis provides an illustrative and comprehensive understanding of how bistability can be achieved and indicates how robust this behaviour is. Experimentally, some so-called “inactive” components were shown to have a residual activity. This has been mostly ignored in mathematical models. Our analysis reveals that bistability is only mildly affected in the case of zero-order or inhibitor ultrasensitivity. However, the case where bistability is achieved by cooperative ultrasensitivity is severely affected by this perturbation

Network-level dynamics of diffusively coupled cells

We study molecular dynamics within populations of diffusively coupled cells under the assumption of fast diffusive exchange. As a technical tool, we propose conditions on boundedness and ultimate boundedness for systems with a singular perturbation, which extend the classical asymptotic stability results for singularly perturbed systems. Based on these results, we show that with common models of intracellular dynamics, the cell population is coordinated in the sense that all cells converge close to a common equilibrium point. We then study a more specific example of coupled cells which behave as bistable switches, where the intracellular dynamics are such that cells may be in one of two equilibrium points. Here, we find that the whole population is bistable in the sense that it converges to a population state where either all cells are close to the one equilibrium point, or all cells are close to the other equilibrium point. Finally, we discuss applications of these results for the robustness of cellular decision making in coupled populations

The smallest chemical reaction system with bistability

<p>Abstract</p> <p>Background</p> <p>Bistability underlies basic biological phenomena, such as cell division, differentiation, cancer onset, and apoptosis. So far biologists identified two necessary conditions for bistability: positive feedback and ultrasensitivity.</p> <p>Results</p> <p>Biological systems are based upon elementary mono- and bimolecular chemical reactions. In order to definitely clarify all necessary conditions for bistability we here present the corresponding minimal system. According to our definition, it contains the minimal number of (i) reactants, (ii) reactions, and (iii) terms in the corresponding ordinary differential equations (decreasing importance from i-iii). The minimal bistable system contains two reactants and four irreversible reactions (three bimolecular, one monomolecular).</p> <p>We discuss the roles of the reactions with respect to the necessary conditions for bistability: two reactions comprise the positive feedback loop, a third reaction filters out small stimuli thus enabling a stable 'off' state, and the fourth reaction prevents explosions. We argue that prevention of explosion is a third general necessary condition for bistability, which is so far lacking discussion in the literature.</p> <p>Moreover, in addition to proving that in two-component systems three steady states are necessary for bistability (five for tristability, etc.), we also present a simple general method to design such systems: one just needs one production and three different degradation mechanisms (one production, five degradations for tristability, etc.). This helps modelling multistable systems and it is important for corresponding synthetic biology projects.</p> <p>Conclusion</p> <p>The presented minimal bistable system finally clarifies the often discussed question for the necessary conditions for bistability. The three necessary conditions are: positive feedback, a mechanism to filter out small stimuli and a mechanism to prevent explosions. This is important for modelling bistability with simple systems and for synthetically designing new bistable systems. Our simple model system is also well suited for corresponding teaching purposes.</p

Optimality principles in the regulation of metabolic networks.

One of the challenging tasks in systems biology is to understand how molecular networks give rise to emergent functionality and whether universal design principles apply to molecular networks. To achieve this, the biophysical, evolutionary and physiological constraints that act on those networks need to be identified in addition to the characterisation of the molecular components and interactions. Then, the cellular “task” of the network—its function—should be identified. A network contributes to organismal fitness through its function. The premise is that the same functions are often implemented in different organisms by the same type of network; hence, the concept of design principles. In biology, due to the strong forces of selective pressure and natural selection, network functions can often be understood as the outcome of fitness optimisation. The hypothesis of fitness optimisation to understand the design of a network has proven to be a powerful strategy. Here, we outline the use of several optimisation principles applied to biological networks, with an emphasis on metabolic regulatory networks. We discuss the different objective functions and constraints that are considered and the kind of understanding that they provide

Bridging Time Scales in Cellular Decision Making with a Stochastic Bistable Switch

Cellular transformations which involve a significant phenotypical change of the cell's state use bistable biochemical switches as underlying decision systems. In this work, we aim at linking cellular decisions taking place on a time scale of years to decades with the biochemical dynamics in signal transduction and gene regulation, occuring on a time scale of minutes to hours. We show that a stochastic bistable switch forms a viable biochemical mechanism to implement decision processes on long time scales. As a case study, the mechanism is applied to model the initiation of follicle growth in mammalian ovaries, where the physiological time scale of follicle pool depletion is on the order of the organism's lifespan. We construct a simple mathematical model for this process based on experimental evidence for the involved genetic mechanisms. Despite the underlying stochasticity, the proposed mechanism turns out to yield reliable behavior in large populations of cells subject to the considered decision process. Our model explains how the physiological time constant may emerge from the intrinsic stochasticity of the underlying gene regulatory network. Apart from ovarian follicles, the proposed mechanism may also be of relevance for other physiological systems where cells take binary decisions over a long time scale.Comment: 14 pages, 4 figure

A feedback approach to bifurcation analysis in biochemical networks with many parameters

Feedback circuits in biochemical networks which underly cellular signaling pathways are important elements in creating complex behavior. A specific aspect thereof is how stability of equilibrium points depends on model parameters. For biochemical networks, which are modelled using many parameters, it is typically very difficult to estimate the influence of parameters on stability. Finding parameters which result in a change in stability is a key step for a meaningful bifurcation analysis. We describe a method based on well known approaches from control theory, which can locate parameters leading to a change in stability. The method considers a feedback circuit in the biochemical network and relates stability properties to the control system obtained by loop--breaking. The method is applied to a model of a MAPK cascade as an illustrative example

Computational study of the mechanism of Bcl-2 apoptotic switch

Programmed cell death - apoptosis is one of the most studied biological phenomenon of recent years. Apoptotic regulatory network contains several significant control points, including probably the most important one - Bcl--2 apoptotic switch. There are two proposed hypotheses regarding its internal working - the indirect activation and direct activation models. Since these hypotheses form extreme poles of full continuum of intermediate models, we have constructed more general model with these two models as extreme cases. By studying relationship between model parameters and steady-state response ultrasensitivity we have found optimal interaction pattern which reproduces behavior of Bcl-2 apoptotic switch. Our results show, that stimulus-response ultrasensitivity is negatively related to spontaneous activation of Bcl-2 effectors - subgroup of Bcl-2 proteins. We found that ultrasensitivity requires effector's activation, mediated by another subgroup of Bcl-2 proteins - activators. We have shown that the auto-activation of effectors forms ultrasensitivity enhancing feedback loop, only if mediated by monomers, but not by oligomers. Robustness analysis revealed that interaction pattern proposed by direct activation hypothesis is able to conserve stimulus-response dependence and preserve ultrasensitivity despite large changes of its internal parameters. This ability is strongly reduced as for the intermediate to indirect side of the models. Computer simulation of the more general model presented here suggest, that stimulus-response ultrasensitivity is an emergent property of the direct activation model, that cannot originate within model of indirect activation. Introduction of indirect-model-specific interactions does not provide better explanation of Bcl-2 functioning compared to direct model

Bioactive hydrogels for tissue engineering

Modern tissue engineering (TE) scaffolds are expected to actively promote tissue repair as well as meeting the traditional requirements of non-toxicity, degradability and structural integrity. This thesis presents two novel bioactive hydrogel systems for bone and cartilage TE. A series of alginate hydrogels were developed in which all or a fraction of the calcium normally used for crosslinking alginate was replaced by bioactive strontium and/or zinc ions. Strontium was chosen for its ability to stimulate bone formation, while zinc is essential for alkaline phosphatase activity. Due to an interaction between the crosslinking ion and alginate type, the hydrogel properties could be tailored independently of the crosslinking ion used – meaning that varying biological and materials requirements can be accommodated. Strontium release from alginate gels was of a physiologically relevant magnitude, and alkaline phosphatase protein activity in Saos-2 cells was highest in strontium gels. Secondly, a biomimetic strategy for transforming growth factor beta (TGF-β) presentation and release was evaluated. TGF-β in vivo is secreted as part of an inactive latent complex, which is sequestered in a stable form within extracellular matrix until released by cells. TGF-β was therefore incorporated into poly(ethylene glycol)-hyaluronic acid hydrogels in its latent form. When compared to free TGF-β, advantages were demonstrated in terms of lower protein adsorption to tissue culture plastic and relative biological inactivity. The latter implies that high doses may be loaded into TE scaffolds without exposing cells to excessive quantities of active growth factor, with TGF-β bioavailability then being controlled by gradual activation by cells. Increased metabolic activity and ECM deposition by bovine chondrocytes were seen after almost five weeks in culture with a single initial loading of LTGF-β. These innovations correspond to current TE trends, which seek to use biomimetic principles to evoke regenerative responses from transplanted or host cells, but to do so using technically and commercially feasible means

The Impact of Time Delays on the Robustness of Biological Oscillators and the Effect of Bifurcations on the Inverse Problem

Differential equation models for biological oscillators are often not robust with respect to parameter variations. They are based on chemical reaction kinetics, and solutions typically converge to a fixed point. This behavior is in contrast to real biological oscillators, which work reliably under varying conditions. Moreover, it complicates network inference from time series data. This paper investigates differential equation models for biological oscillators from two perspectives. First, we investigate the effect of time delays on the robustness of these oscillator models. In particular, we provide sufficient conditions for a time delay to cause oscillations by destabilizing a fixed point in two-dimensional systems. Moreover, we show that the inclusion of a time delay also stabilizes oscillating behavior in this way in larger networks. The second part focuses on the inverse problem of estimating model parameters from time series data. Bifurcations are related to nonsmoothness and multiple local minima of the objective function

Probabilistic reasoning and inference for systems biology

One of the important challenges in Systems Biology is reasoning and performing hypotheses testing in uncertain conditions, when available knowledge may be incomplete and the experimental data may contain substantial noise. In this thesis we develop methods of probabilistic reasoning and inference that operate consistently within an environment of uncertain knowledge and data. Mechanistic mathematical models are used to describe hypotheses about biological systems. We consider both deductive model based reasoning and model inference from data. The main contributions are a novel modelling approach using continuous time Markov chains that enables deductive derivation of model behaviours and their properties, and the application of Bayesian inferential methods to solve the inverse problem of model inference and comparison, given uncertain knowledge and noisy data. In the first part of the thesis, we consider both individual and population based techniques for modelling biochemical pathways using continuous time Markov chains, and demonstrate why the latter is the most appropriate. We illustrate a new approach, based on symbolic intervals of concentrations, with an example portion of the ERK signalling pathway. We demonstrate that the resulting model approximates the same dynamic system as traditionally defined using ordinary differential equations. The advantage of the new approach is quantitative logical analysis; we formulate a number of biologically significant queries in the temporal logic CSL and use probabilistic symbolic model checking to investigate their veracity. In the second part of the thesis, we consider the inverse problem of model inference and testing of alternative hypotheses, when models are defined by non-linear ordinary differential equations and the experimental data is noisy and sparse. We compare and evaluate a number of statistical techniques, and implement an effective Bayesian inferential framework for systems biology based on Markov chain Monte Carlo methods and estimation of marginal likelihoods by annealing-melting integration. We illustrate the framework with two case studies, one of which involves an open problem concerning the mediation of ERK phosphorylation in the ERK pathway