The dynamics of hybrid metabolic-genetic oscillators

Abstract

The synthetic construction of intracellular circuits is frequently hindered by a poor knowledge of appropriate kinetics and precise rate parameters. Here, we use generalized modeling (GM) to study the dynamical behavior of topological models of a family of hybrid metabolic-genetic circuits known as "metabolators." Under mild assumptions on the kinetics, we use GM to analytically prove that all explicit kinetic models which are topologically analogous to one such circuit, the "core metabolator," cannot undergo Hopf bifurcations. Then, we examine more detailed models of the metabolator. Inspired by the experimental observation of a Hopf bifurcation in a synthetically constructed circuit related to the core metabolator, we apply GM to identify the critical components of the synthetically constructed metabolator which must be reintroduced in order to recover the Hopf bifurcation. Next, we study the dynamics of a re-wired version of the core metabolator, dubbed the "reverse" metabolator, and show that it exhibits a substantially richer set of dynamical behaviors, including both local and global oscillations. Prompted by the observation of relaxation oscillations in the reverse metabolator, we study the role that a separation of genetic and metabolic time scales may play in its dynamics, and find that widely separated time scales promote stability in the circuit. Our results illustrate a generic pipeline for vetting the potential success of a circuit design, simply by studying the dynamics of the corresponding generalized model. The engineering of biological circuits, synthetically constructed in the laboratory and designed to exhibit novel behaviors, has matured rapidly as a discipline over the past decade. A major challenge for synthetic biology is understanding how different cellular subsystems, composed of distinct components and operating at widely different speeds, interface and work together to generate coherent cellular behaviors. In particular, metabolic pathways can rapidly harvest energy and nutrients for reproduction, while genetic regulatory networks slowly control these pathways in response to environmental changes. Here, we study a generalized version of the metabolator, a unique synthetic circuit composed of both metabolic and regulatory components, previously constructed experimentally and shown to exhibit oscillations. In addition to exploring multiple alternative network topologies, our generalized approach overcomes the inherent uncertainty associated with precise kinetic details of biological circuits. Through this analysis, we identify circuit components that are essential for producing sustained oscillations. Furthermore, we find that certain dynamical regimes of a rewired metabolator display unexpected oscillatory properties, such as the capacity to suddenly transition to high amplitude oscillations. This novel behavior could be tested experimentally, and lead to novel synthetic biology modules and applications.

Flux Imbalance Analysis and the Sensitivity of Cellular Growth to Changes in Metabolite Pools

“…We do not know the mechanism which induces relatively fast changes in growth-limiting metabolites, when compared to non-growth-limiting metabolites. Indeed, an exciting prospect for future work will be bridging our findings with well-established schools of metabolic theory, including metabolic control analysis [44], biochemical systems theory [45], and structural kinetic modeling [46], [47]. Compellingly, the dual of the FBA problem has also been suggested to constitute a window onto the thermodynamics of biochemical networks, with potential implications for understanding the energetics of metabolism [19].…”

The average enzyme principle

“…Under appropriate conditions [14], and in line with studies of hybrid metabolic‐genetic systems [18,19], we can write the Michaelis–Menten equation in which the enzyme concentration is a time‐dependent variable: Eq. (4) is still separable; in other words, enzyme concentration can be still isolated from the terms corresponding to substrate, yielding upon integration: where E avg is the average enzyme level during the time interval [0, ΔT ].…”

“…Under appropriate conditions [14], and in line with studies of hybrid metabolic-genetic systems [18,19], we can write the Michaelis-Menten equation in which the enzyme concentration is a time-dependent variable:…”

Mathematical model of galactose regulation and metabolic consumption in yeast

“…Complete mathematical model of the galactose network in the presence of glucose 2.4.1. Nine dimensional (9D) GAL model In the galactose network, metabolic reactions occur on a faster time scale than the rates of change of the proteins (Reznik et al, 2013). For example, transcription and translation occur with a time scale on the order of minutes, whereas transport via facilitated diffusion and phosphorylation occur at a rate greater than 500 times per minute (de Atauri et al, 2005).…”