Brian Ingalls

Sensitivity analysis of stoichiometric networks: an extension of metabolic control analysis to non-steady state trajectories

A sensitivity analysis of general stoichiometric networks is considered. The results are presented as a generalization of Metabolic Control Analysis, which has been concerned primarily with system sensitivities at steady state. An expression for time-varying sensitivity coefficients is given and the Summation and Connectivity Theorems are generalized. The results are compared to previous treatments. The analysis is accompanied by a discussion of the computation of the sensitivity coefficients and an application to a model of phototransduction.

Control theory and systems biology

Issues of regulation and control are central to the study of biological and biochemical systems. Thus it is not surprising that the tools of feedback control theory--engineering techniques developed to design and analyze self-regulating systems--have proven useful in the study of these biological mechanisms. Such interdisciplinary work requires knowledge of the results, tools and techniques of another discipline, as well as an understanding of the culture of an unfamiliar research community. This volume attempts to bridge the gap between disciplines by presenting applications of systems and control theory to cell biology that range from surveys of established material to descriptions of new developments in the field. The first chapter offers a primer on concepts from dynamical systems and control theory, which allows the life scientist with no background in control theory to understand the concepts presented in the rest of the book. Following the introduction of ordinary differential equation-based modeling in the first chapter, the second and third chapters discuss alternative modeling frameworks. The remaining chapters sample a variety of applications, considering such topics as quantitative measures of dynamic behavior, modularity, stoichiometry, robust control techniques, and network identification. ContributorsDavid Angeli, Declan G. Bates, Eric Bullinger, Peter S. Chang, Domitilla Del Vecchio, Francis J. Doyle III, Hana El-Samad, Dirk Fey, Rolf Findeisen, Simone Frey, Jorge Goncalves, Pablo A. Iglesias, Brian P. Ingalls, Elling W. Jacobsen, Mustafa Khammash, Jongrae Kim, Eric Klavins, Eric C. Kwei, Thomas Millat, Jason E. Shoemaker, Eduardo D. Sontag, Stephanie R. Taylor, David Thorsley, Camilla Trane, Sean Warnick, Olaf Wolkenhauer

A small-gain theorem with applications to input/output systems, incremental stability, detectability, and interconnections

Abstract A general input-to-state stability (ISS)-type small-gain result is presented. It specializes to a small-gain theorem for ISS operators, and it also recovers the classical statement for ISS systems in state-space form. In addition, we highlight applications to incrementally stable systems, detectable systems, and to interconnections of stable systems.

Estimations of intrinsic and extrinsic noise in models of nonlinear genetic networks

We discuss two methods that can be used to estimate the impact of internal and external variability on nonlinear systems, and demonstrate their utility by comparing two experimentally implemented oscillatory genetic networks with different designs. The methods allow for rapid estimations of intrinsic and extrinsic noise and should prove useful in the analysis of natural genetic networks and when constructing synthetic gene regulatory systems.

Deterministic characterization of stochastic genetic circuits.

For cellular biochemical reaction systems where the numbers of molecules is small, significant noise is associated with chemical reaction events. This molecular noise can give rise to behavior that is very different from the predictions of deterministic rate equation models. Unfortunately, there are few analytic methods for examining the qualitative behavior of stochastic systems. Here we describe such a method that extends deterministic analysis to include leading-order corrections due to the molecular noise. The method allows the steady-state behavior of the stochastic model to be easily computed, facilitates the mapping of stability phase diagrams that include stochastic effects, and reveals how model parameters affect noise susceptibility in a manner not accessible to numerical simulation. By way of illustration we consider two genetic circuits: a bistable positive-feedback loop and a negative-feedback oscillator. We find in the positive feedback circuit that translational activation leads to a far more stable system than transcriptional control. Conversely, in a negative-feedback loop triggered by a positive-feedback switch, the stochasticity of transcriptional control is harnessed to generate reproducible oscillations.

Separation Principles for Input-Output and Integral-Input-to-State Stability

We present new characterizations of input-output-to-state stability. This is a notion of detectability formulated in the ISS (input-to-state stability) framework. Equivalent properties are presented in terms of asymptotic estimates of the state trajectories based on the magnitudes of the external input and output signals. These results provide a set of separation principles for input-output-to-state stability---characterizations of the property as conjunctions of weaker stability notions. When applied to the notion of integral ISS, these characterizations yield analogous results.

An infinite-time relaxation theorem for differential inclusions

The fundamental relaxation result for Lipschitz differential inclusions is the Filippov-Wažewski Relaxation Theorem, which provides approximations of trajectories of a relaxed inclusion on finite intervals. A complementary result is presented, which provides approximations on infinite intervals, but does not guarantee that the approximation and the reference trajectory satisfy the same initial condition.

Sequential Activation of Metabolic Pathways: a Dynamic Optimization Approach

The regulation of cellular metabolism facilitates robust cellular operation in the face of changing external conditions. The cellular response to this varying environment may include the activation or inactivation of appropriate metabolic pathways. Experimental and numerical observations of sequential timing in pathway activation have been reported in the literature. It has been argued that such patterns can be rationalized by means of an underlying optimal metabolic design. In this paper we pose a dynamic optimization problem that accounts for time-resource minimization in pathway activation under constrained total enzyme abundance. The optimized variables are time-dependent enzyme concentrations that drive the pathway to a steady state characterized by a prescribed metabolic flux. The problem formulation addresses unbranched pathways with irreversible kinetics. Neither specific reaction kinetics nor fixed pathway length are assumed.In the optimal solution, each enzyme follows a switching profile between zero and maximum concentration, following a temporal sequence that matches the pathway topology. This result provides an analytic justification of the sequential activation previously described in the literature. In contrast with the existent numerical approaches, the activation sequence is proven to be optimal for a generic class of monomolecular kinetics. This class includes, but is not limited to, Mass Action, Michaelis–Menten, Hill, and some Power-law models. This suggests that sequential enzyme expression may be a common feature of metabolic regulation, as it is a robust property of optimal pathway activation.

Sensitivity analysis: from model parameters to system behaviour

Sensitivity analysis addresses the manner in which model behaviour depends on model parametrization. Global sensitivity analysis makes use of statistical tools to address system behaviour over a wide range of operating conditions, whereas local sensitivity analysis focuses attention on a specific set of nominal parameter values. This narrow focus allows a complete analytical treatment and straightforward interpretation in the local case. Sensitivity analysis is a valuable tool for model construction and interpretation, and can be applied in medicine and biotechnology to predict the effect of interventions.

Estimating the stochastic bifurcation structure of cellular networks.

High throughput measurement of gene expression at single-cell resolution, combined with systematic perturbation of environmental or cellular variables, provides information that can be used to generate novel insight into the properties of gene regulatory networks by linking cellular responses to external parameters. In dynamical systems theory, this information is the subject of bifurcation analysis, which establishes how system-level behaviour changes as a function of parameter values within a given deterministic mathematical model. Since cellular networks are inherently noisy, we generalize the traditional bifurcation diagram of deterministic systems theory to stochastic dynamical systems. We demonstrate how statistical methods for density estimation, in particular, mixture density and conditional mixture density estimators, can be employed to establish empirical bifurcation diagrams describing the bistable genetic switch network controlling galactose utilization in yeast Saccharomyces cerevisiae. These approaches allow us to make novel qualitative and quantitative observations about the switching behavior of the galactose network, and provide a framework that might be useful to extract information needed for the development of quantitative network models.