Elias August

A model invalidation-based approach for elucidating biological signalling pathways, applied to the chemotaxis pathway in R. sphaeroides.

BackgroundDeveloping methods for understanding the connectivity of signalling pathways is a major challenge in biological research. For this purpose, mathematical models are routinely developed based on experimental observations, which also allow the prediction of the system behaviour under different experimental conditions. Often, however, the same experimental data can be represented by several competing network models.ResultsIn this paper, we developed a novel mathematical model/experiment design cycle to help determine the probable network connectivity by iteratively invalidating models corresponding to competing signalling pathways. To do this, we systematically design experiments in silico that discriminate best between models of the competing signalling pathways. The method determines the inputs and parameter perturbations that will differentiate best between model outputs, corresponding to what can be measured/observed experimentally. We applied our method to the unknown connectivities in the chemotaxis pathway of the bacterium Rhodobacter sphaeroides. We first developed several models of R. sphaeroides chemotaxis corresponding to different signalling networks, all of which are biologically plausible. Parameters in these models were fitted so that they all represented wild type data equally well. The models were then compared to current mutant data and some were invalidated. To discriminate between the remaining models we used ideas from control systems theory to determine efficiently in silico an input profile that would result in the biggest difference in model outputs. However, when we applied this input to the models, we found it to be insufficient for discrimination in silico. Thus, to achieve better discrimination, we determined the best change in initial conditions (total protein concentrations) as well as the best change in the input profile. The designed experiments were then performed on live cells and the resulting data used to invalidate all but one of the remaining candidate models.ConclusionWe successfully applied our method to chemotaxis in R. sphaeroides and the results from the experiments designed using this methodology allowed us to invalidate all but one of the proposed network models. The methodology we present is general and can be applied to a range of other biological networks.

Feedback control architecture and the bacterial chemotaxis network.

Bacteria move towards favourable and away from toxic environments by changing their swimming pattern. This response is regulated by the chemotaxis signalling pathway, which has an important feature: it uses feedback to ‘reset’ (adapt) the bacterial sensing ability, which allows the bacteria to sense a range of background environmental changes. The role of this feedback has been studied extensively in the simple chemotaxis pathway of Escherichia coli. However it has been recently found that the majority of bacteria have multiple chemotaxis homologues of the E. coli proteins, resulting in more complex pathways. In this paper we investigate the configuration and role of feedback in Rhodobacter sphaeroides, a bacterium containing multiple homologues of the chemotaxis proteins found in E. coli. Multiple proteins could produce different possible feedback configurations, each having different chemotactic performance qualities and levels of robustness to variations and uncertainties in biological parameters and to intracellular noise. We develop four models corresponding to different feedback configurations. Using a series of carefully designed experiments we discriminate between these models and invalidate three of them. When these models are examined in terms of robustness to noise and parametric uncertainties, we find that the non-invalidated model is superior to the others. Moreover, it has a ‘cascade control’ feedback architecture which is used extensively in engineering to improve system performance, including robustness. Given that the majority of bacteria are known to have multiple chemotaxis pathways, in this paper we show that some feedback architectures allow them to have better performance than others. In particular, cascade control may be an important feature in achieving robust functionality in more complex signalling pathways and in improving their performance.

A new computational tool for establishing model parameter identifiability.

We describe a novel method to establish a priori whether the parameters of a nonlinear dynamical system are identifiable--that is, whether they can be deduced from output data (experimental observations). This is an important question as usually identifiability is assumed, and parameters are sought without first establishing whether these can be inferred from a set of measurements. We highlight the connections between parameter identifiability and state observability. We show how observability criteria can be used to check for identifiability, and we use new, state of the art computational tools to implement our approach. Nonlinear dynamical systems are prevalent in systems biology, where they are often used to represent a biological system. Thus, examples from biology are used to illustrate our method.

Structured model reduction for dynamical networked systems

Mathematical models of networked systems usually take the form of large-scale, nonlinear differential equations. Model reduction is a commonly used technique for understanding and analyzing systems of this size, by producing simplified yet accurate descriptions for them. Most available reduction methods work well for linear system descriptions or small-scale nonlinear system descriptions but they usually involve a state transformation to ‘balance’ the system before truncation. However, linear or nonlinear state combinations destroy the system structure that is important for drawing conclusions about the original networked system from the reduction. In this paper we propose an algorithmic methodology for model order reduction of nonlinear systems, without inducing state transformations. A priority list of states to be collapsed according to the estimated worst-case 2-norm of the error between the outputs of the original and reduced systems is produced. The main advantage of the method is that the states of the reduced system are a subset of the states of the original system.

Finding invariant sets for biological systems using monomial domination

In this paper we present a novel approach to the analysis of nonnegative dynamical systems whose vector fields are polynomial or rational functions. Our analysis framework is based on results developed and presented in a previous study on general conditions that imply non-vanishing of polynomial functions on the positive orthant. This approach is due to the sparsity of the negative terms in the polynomial, which are then “dominated” by the positive terms. Particularly, we present a novel approach to find invariant sets of a dynamical system and one to aid the search for the number of possible equilibria of the system. To illustrate this approach we apply it to a model of a food web to check for overpopulation or extinction of species.

Trajectory enclosures for nonlinear systems with uncertain initial conditions and parameters

In this paper we provide a novel method to deal with uncertainties in initial value problems based on solving positivity conditions, by means of semidefinite programmes and sum of squares decompositions. More specifically, given a nonlinear dynamical system with uncertainties in initial conditions and parameter values, our method provides enclosures for state trajectories. Due to the fact that we compute ellipsoids rather than boxes to estimate the enclosures, our method is less affected by conservation effects. We demonstrate the proposed method to models from biology, on the tasks of parameter estimation, model robustness and model selection. These problems that arise from biological applications are particularly challenging due to the paucity of data as well as the presence of significant data noise. We illustrate the applicability of our method on a simple model of gene regulation and the mitogen activated protein kinase signalling pathway.

Using Noise for Model-Testing

For realistic models in molecular biology, you need to consider the noise in the cellular and intracellular environments. In this article, we present a novel approach for testing the validity of nonlinear models representing a biological system affected by noise. Our approach is based on results by Kushner and Øksendal and uses computational techniques that rely on efficient solvers. By providing analytically upper bounds for the exit probability of solution trajectories of a system from a particular set in the phase space, we can compare measurement data with this prediction and try to invalidate models with certain parameter values or noise properties. Thus, our approach complements the usual methods that are based on deterministic models. It is particularly useful in the field of reverse engineering in systems biology, when one seeks to determine model parameters and noise properties as we show in the Results section, where we applied the approach to examples of increasing complexity and to the Hog1 signalling pathway.

Feedback control architecture of the R. sphaeroides chemotaxis network

This paper investigates the chemotaxis behavior of the bacterium R. sphaeroides. We review the results of a recent study comparing different possible mathematical models of this bacteriums chemotaxis decision mechanisms. It was found that only one of the aforesaid models could explain the experimental chemotactic response data. From a control theoretic perspective, we show that, compared to the other models posed, this model exhibits better and more robust chemotactic performance. This decision mechanism parallels a feedback architecture that has been used extensively to improve performance in engineered systems. We suggest that this mechanism may play a role in maintaining the chemotactic performance of this and potentially other bacteria.

Solutions of Weakly Reversible Chemical Reaction Networks Are Bounded and Persistent

Abstract We present extensions to chemical reaction network theory which are relevant to the analysis of models of biochemical systems. We show that, for positive initial conditions, solutions of a weakly reversible chemical reaction network are bounded and remain in the positive orthant. Thus, weak reversibility implies persistence as conjectured by Martin Feinberg. Our result provides a qualitative criterion to establish that a biochemical network will not diverge or converge to the boundary, where some concentration levels are zero. It relies on checking structural properties of the graph of the reaction network solely. It can also be used to characterise certain bifurcations from stationary to oscillatory behaviour. We illustrate the use of our result through applications.

Determining interconnections in biochemical networks using linear programming

We present a methodology for efficient, robust determination of the interaction topology of networked dynamical systems using time series data collected from experiments, under the assumption that these networks are sparse, i.e., have much less edges than the full graph with the same vertex set. To achieve this, we minimize the 1-norm of the decision variables while keeping the data in close Euler fit, thus putting more emphasis on determining the interconnection pattern rather than the closeness of fit. First, we consider a networked system in which the interconnection strength enters in an affine way in the system dynamics. We demonstrate the ability of our method to identify a network structure through numerical examples. Second, we extend our approach to the case of gene regulatory networks, in which the system dynamics are much more complicated.