Positive and negative feedback loops are ubiquitous regulatory features of biological systems in which the system output reinforces or opposes the system input, respectively. Quantitative models are increasingly being used to study the function and dynamic properties of complicated, feedback-laden biological systems. These models can be broadly classified by the extent to which they represent specific molecular details of the network. At one extreme are the exhaustive models that dynamically track quantities of virtually all biomolecules in a system, often using differential equations based either on known or assumed reaction stoichiometries and rates. At the other end of the modeling spectrum is the minimalist approach, which aims to fit and predict a system’s input-output dynamics with only a few key parameters, each potentially the distillation of a large group of reactions.
Examples of exhaustive and minimalist modeling approaches illustrate their unique advantages and disadvantages. For instance, an exhaustive model of EGF-receptor regulation (Schoeberl et al., 2002
) can impressively predict the dynamics of 94 specific network elements, but it requires nearly 100 parameters—some of which are not easily measured biochemically—and could suffer from the omission of important reactions not yet biologically identified. By contrast, minimalist models frequently lack such potentially desirable reaction- and network-specific details, yet they excel at providing intuitive and general insights into the dynamic properties of recurrent system architectures. For instance, two elegant studies of bacterial chemotaxis—a system said to “perfectly adapt” because abrupt changes in the amount of ligand only transiently affect the tumbling frequency, whereas steady-state tumbling is notably independent of the ligand concentration—highlighted a general feature of all perfectly adapting systems (Barkai and Leibler, 1997
; Yi et al., 2000
). Specifically, it was shown that a negative feedback loop implementing “integral feedback” is both necessary and sufficient for robust perfect adaptation in any biological system (Yi et al., 2000
). Mathematically, a dynamic variable (e.g., x
) is an “integrator” if its rate of change is independent of the variable itself (e.g., if the dx/dt
equations contain no terms involving x
, respectively), and integral feedback describes a negative-feedback loop that contains at least one integrator. Biologically, a biomolecule acts as an integrator if its rate equation is not a function of the biomolecule concentration itself; such a situation arises if, say, the synthesis and degradation reactions are saturated (Supplemental Data
). By providing specific mechanistic constraints that apply to any perfectly adapting system, these two studies underscored the function and significance of perfect adaptation in homeostatic regulation and demonstrated the power of the minimalist modeling approach.
Both exhaustive and minimalist modeling tactics have been successfully applied to the osmosensing network in the budding yeast Saccharomyces cerevisiae
(Klipp et al., 2005
; Mettetal et al., 2008
). The core of this network is a highly conserved mitogen-activated protein kinase (MAPK) cascade, one of several such cascades in yeast that regulate processes ranging from mating to invasive growth while being remarkably robust to cross-talk despite their many shared components (Hohmann, 2002
; Schwartz and Madhani, 2004
). Yeast cells maintain an intracellular osmolarity in excess of the extracellular osmolarity, thereby creating positive turgor pressure across the cell wall and membrane that is required for many processes including budding itself. Sudden drops in turgor pressure, potentially caused by an upward spike in the external osmolyte concentration, are detected by membrane proteins such as Sln1 (Reiser et al., 2003
), which rapidly initiates a MAPK cascade culminating in the dual phosphorylation of the MAPK Hog1 (). Upon dual phosphorylation, the normally cytoplasmic and inactive Hog1 becomes activated and translocates to the nucleus (Ferrigno et al., 1998
), where it plays direct and indirect roles in a broad transcriptional response (O’Rourke and Herskowitz, 2004
). Glycerol-producing factors are among the activated genes, and they facilitate osmoadaptation through the increase of intracellular osmolarity (Hohmann et al., 2007
). In fact, glycerol accumulation has been shown to comprise 95% of the internal osmolarity recovery (Reed et al., 1987
). The subsequent restoration of turgor pressure leads to nuclear export of Hog1, which is dephosphorylated by several nuclear and cytoplasmic phosphatases.
Hog1 translocates to the nucleus in response to hyperosmotic shock
Non-transcriptional mechanisms also play an important role in the hyperosmotic-shock response. Some are independent of Hog1 (e.g., rapid closure of Fps1 channels, which otherwise allow passive leakage of glycerol (Luyten et al., 1995
; Tamás et al., 2000
)), but others involve feedback mediated by the Hog1 pathway (Dihazi et al., 2004
; Proft and Struhl, 2004
; Westfall et al., 2008
). For instance, in response to hyperosmotic stress, the glycolytic protein Pfk26, which stimulates production of glycerol precursors, was found to be activated via phosphorylation at MAPK consensus sites in a Hog1-dependent manner (Dihazi et al., 2004
). Additionally, in a recent study by Thorner and colleagues, it was shown that Hog1 sequestered in the cytoplasm can still mount an effective osmotic response (Westfall et al., 2008
Biochemical characterization of most systems is rarely so rich to be deemed exhaustive, nor so minimal to consider a system as a black box. Thus, models combining elements from both approaches can be quite useful, such as a recent study that started with the exhaustive osmoadaptation model (Klipp et al., 2005
) and abstracted several elements to yield a reduced representation (Gennemark et al., 2006
). Here we take the reverse strategy, starting instead with the minimalist model () and then using biological measurements and engineering principles to better understand systems-level dynamics and their relation with network topology.
In this study, we monitor the single-cell dynamics at high temporal resolution of both cell volume and Hog1 nuclear enrichment simultaneously in response to hyperosmotic stress. We observe perfect adaptation of Hog1 nuclear enrichment in response to step inputs of osmolyte; this adaptation occurs with very low cell-to-cell variability and is robust to the signaling fidelity of the MAPK cascade. From extensive theory developed in control engineering, we know that perfect adaptation in this feedback system requires an integral-feedback mechanism. We refine the position(s) of integrator(s) in our network by generating a range of putative network configurations and systematically rejecting those inconsistent with our data. Facilitating this process of elimination is our observation that perfect adaptation requires Hog1 kinase activity but not new protein production, suggesting that Hog1 may implement integral-feedback via a yet-unknown role in protein-protein interactions that increase the internal osmolyte concentration. Measurements of glycerol accumulation suggest that this crucial role for Hog1 kinase activity upregulates glycerol synthesis but does not otherwise regulate its leakage. Finally, in an experiment imposing severe constraints on the possible valid network configuration, we show that neither cell volume nor Hog1 nuclear enrichment perfectly adapts in response to a ramp input of salt. Together, our results establish that the system’s negative feedback loop contains exactly one effective integrating mechanism. Though the loop may branch such that this effective integrator is composed of multiple integrating reactions arranged in parallel, we can reject the possibility that the net feedback loop contains two or more effective integrating mechanisms arranged in series.