Sensitivity and robustness in covalent modification cycles with a bifunctional converter enzyme

Phenotypic heterogeneity in human genetic diseases: ultrasensitivity-mediated threshold effects as a unifying molecular mechanism

Phenotypic heterogeneity is very common in genetic systems and in human diseases and has important consequences for disease diagnosis and treatment. In addition to the many genetic and non-genetic (e.g., epigenetic, environmental) factors reported to account for part of the heterogeneity, we stress the importance of stochastic fluctuation and regulatory network topology in contributing to phenotypic heterogeneity. We argue that a threshold effect is a unifying principle to explain the phenomenon; that ultrasensitivity is the molecular mechanism for this threshold effect; and discuss the three conditions for phenotypic heterogeneity to occur. We suggest that threshold effects occur not only at the cellular level, but also at the organ level. We stress the importance of context-dependence and its relationship to pleiotropy and edgetic mutations. Based on this model, we provide practical strategies to study human genetic diseases. By understanding the network mechanism for ultrasensitivity and identifying the critical factor, we may manipulate the weak spot to gently nudge the system from an ultrasensitive state to a stable non-disease state. Our analysis provides a new insight into the prevention and treatment of genetic diseases.

The inverse correlation between robustness and sensitivity to autoregulation in two-component systems

Two-component systems (TCS) are signal transduction systems in bacteria and many other organisms that relay the sensory signal to genetic components. TCS consist of two proteins: a histidine kinase and a response regulator that the histidine kinase activates. This seemingly simple machinery can generate complex regulatory dynamics that enables the level of gene expression that matches the input signal: many TCS response regulators act on their own genes as transcription factors, resulting in a positive autoregulation mechanism. This regulation, in return, modulates the transcription factor activity as a function of the input signal. Positive autoregulation does not necessarily result in positive feedback. Sensitivity to autoregulation is quantified as the output level amplification resulting from the positive autoregulation mechanism. Another structural property of these systems is formally characterized as "robustness": in a robust TCS, the output of the system is solely a function of the input signal. Thus, a robust TCS remains insensitive to fluctuations in the concentrations of its protein components and, this way, maintains the precision in the output transcription factor activity in response to input stimulus. In this paper, we show with a formal model that TCS operate on a spectrum of inverse correlation between robustness and sensitivity to autoregulation. Our model predicts that the modulation by positive autoregulation is a function of loss in TCS robustness, for example, by spontaneous dephosphorylation of the histidine kinase. Consequently, the loss in robustness provides a proportional modulation by positive autoregulation to widen the response range with a scaled amplification of the output. At the other end of the spectrum, in the presence of a strictly robust TCS machinery, amplification of the transcription factor activity by autoregulation is diminished. We show that our results are in agreement with published experimental results. Our results suggest that these TCS evolve to converge at a trade-off between robustness and positive autoregulation.

A two-way switch for inositol pyrophosphate signaling: Evolutionary history and biological significance of a unique, bifunctional kinase/phosphatase

The inositol pyrophosphates (PP-InsPs) are a unique subgroup of intracellular signals with diverse functions, many of which can be viewed as reflecting an overarching role in metabolic homeostasis. Thus, considerable attention is paid to the enzymes that synthesize and metabolize the PP-InsPs. One of these enzyme families - the diphosphoinositol pentakisphosphate kinases (PPIP5Ks) - provides an extremely rare example of separate kinase and phosphatase activities being present within the same protein. Herein, we review the current state of structure/function insight into the PPIP5Ks, the separate specialized activities of the two metazoan PPIP5K genes, and we describe a phylogenetic analysis that places PPIP5K evolutionary origin within the Excavata, the very earliest of eukaryotes. These different aspects of PPIP5K biology are placed in the context of a single, overriding question. Why are they bifunctional: i.e., what is the particular significance of the ability to turn PP-InsP signaling on or off from two separate 'switches' in a single protein?

When More Is Less: Dual Phosphorylation Protects Signaling Off State against Overexpression

Kinases in signaling pathways are commonly activated by multisite phosphorylation. For example, the mitogen-activated protein kinase Erk is activated by its kinase Mek by two consecutive phosphorylations within its activation loop. In this article, we use kinetic models to study how the activation of Erk is coupled to its abundance. Intuitively, Erk activity should rise with increasing amounts of Erk protein. However, a mathematical model shows that the signaling off state is robust to increasing amounts of Erk, and Erk activity may even decline with increasing amounts of Erk. This counterintuitive, bell-shaped response of Erk activity to increasing amounts of Erk arises from the competition of the unmodified and single phosphorylated form of Erk for access to its kinase Mek. This shows that phosphorylation cycles can contain an intrinsic robustness mechanism that protects signaling from aberrant activation e.g., by gene expression noise or kinase overexpression after gene duplication events in diseases like cancer.

Analysis of network motifs in cellular regulation: Structural similarities, input-output relations and signal integration

Much of the complexity of regulatory networks derives from the necessity to integrate multiple signals and to avoid malfunction due to cross-talk or harmful perturbations. Hence, one may expect that the input-output behavior of larger networks is not necessarily more complex than that of smaller network motifs which suggests that both can, under certain conditions, be described by similar equations. In this review, we illustrate this approach by discussing the similarities that exist in the steady state descriptions of a simple bimolecular reaction, covalent modification cycles and bacterial two-component systems. Interestingly, in all three systems fundamental input-output characteristics such as thresholds, ultrasensitivity or concentration robustness are described by structurally similar equations. Depending on the system the meaning of the parameters can differ ranging from protein concentrations and affinity constants to complex parameter combinations which allows for a quantitative understanding of signal integration in these systems. We argue that this approach may also be extended to larger regulatory networks.

GlnK Facilitates the Dynamic Regulation of Bacterial Nitrogen Assimilation

Ammonium assimilation in Escherichia coli is regulated by two paralogous proteins (GlnB and GlnK), which orchestrate interactions with regulators of gene expression, transport proteins, and metabolic pathways. Yet how they conjointly modulate the activity of glutamine synthetase, the key enzyme for nitrogen assimilation, is poorly understood. We combine experiments and theory to study the dynamic roles of GlnB and GlnK during nitrogen starvation and upshift. We measure time-resolved in vivo concentrations of metabolites, total and posttranslationally modified proteins, and develop a concise biochemical model of GlnB and GlnK that incorporates competition for active and allosteric sites, as well as functional sequestration of GlnK. The model predicts the responses of glutamine synthetase, GlnB, and GlnK under time-varying external ammonium level in the wild-type and two genetic knock-outs. Our results show that GlnK is tightly regulated under nitrogen-rich conditions, yet it is expressed during ammonium run-out and starvation. This suggests a role for GlnK as a buffer of nitrogen shock after starvation, and provides a further functional link between nitrogen and carbon metabolisms.

Operating regimes of covalent modification cycles at high enzyme concentrations

The Goldbeter-Koshland model has been a paradigm for ultrasensitivity in biological networks for more than 30 years. Despite its simplicity the validity of this model is restricted to conditions when the substrate is in excess over the converter enzymes - a condition that is easy to satisfy in vitro, but which is rarely satisfied in vivo. Here, we analyze the Goldbeter-Koshland model by means of the total quasi-steady state approximation which yields a comprehensive classification of the steady state operating regimes under conditions when the enzyme concentrations are comparable to or larger than that of the substrate. Where possible we derive simple expressions characterizing the input-output behavior of the system. Our analysis suggests that enhanced sensitivity occurs if the concentration of at least one of the converter enzymes is smaller (but not necessarily much smaller) than that of the substrate and if that enzyme is saturated. Conversely, if both enzymes are saturated and at least one of the enzyme concentrations exceeds that of the substrate the system exhibits concentration robustness with respect to changes in that enzyme concentration. Also, depending on the enzyme's saturation degrees and the ratio between their maximal reaction rates the total fraction of phosphorylated substrate may increase, decrease or change nonmonotonically as a function of the total substrate concentration. The latter finding may aid the interpretation of experiments involving genetic perturbations of enzyme and substrate abundances.

Intimate connections: Inositol pyrophosphates at the interface of metabolic regulation and cell signaling

Inositol pyrophosphates are small, diffusible signaling molecules that possess the most concentrated three-dimensional array of phosphate groups in Nature; up to eight phosphates are crammed around a six-carbon inositol ring. This review discusses the physico-chemical properties of these unique molecules, and their mechanisms of action. Also provided is information on the enzymes that regulate the levels and hence the signaling properties of these molecules. This review pursues the idea that many of the biological effects of inositol pyrophosphates can be rationalized by their actions at the interface of cell signaling and metabolism that is essential to cellular and organismal homeostasis.

The Significance of the Bifunctional Kinase/Phosphatase Activities of Diphosphoinositol Pentakisphosphate Kinases (PPIP5Ks) for Coupling Inositol Pyrophosphate Cell Signaling to Cellular Phosphate Homeostasis

Proteins responsible for P_i homeostasis are critical for all life. In Saccharomyces cerevisiae, extracellular [P_i] is "sensed" by the inositol-hexakisphosphate kinase (IP6K) that synthesizes the intracellular inositol pyrophosphate 5-diphosphoinositol 1,2,3,4,6-pentakisphosphate (5-InsP₇) as follows: during a period of P_i starvation, there is a decline in cellular [ATP]; the unusually low affinity of IP6Ks for ATP compels 5-InsP₇ levels to fall in parallel (Azevedo, C., and Saiardi, A. (2017) Trends. Biochem. Sci. 42, 219-231. Hitherto, such P_i sensing has not been documented in metazoans. Here, using a human intestinal epithelial cell line (HCT116), we show that levels of both 5-InsP₇ and ATP decrease upon [P_i] starvation and subsequently recover during P_i replenishment. However, a separate inositol pyrophosphate, 1,5-bisdiphosphoinositol 2,3,4,6-tetrakisphosphate (InsP₈), reacts more dramatically (i.e. with a wider dynamic range and greater sensitivity). To understand this novel InsP₈ response, we characterized kinetic properties of the bifunctional 5-InsP₇ kinase/InsP₈ phosphatase activities of full-length diphosphoinositol pentakisphosphate kinases (PPIP5Ks). These data fulfil previously published criteria for any bifunctional kinase/phosphatase to exhibit concentration robustness, permitting levels of the kinase product (InsP₈ in this case) to fluctuate independently of varying precursor (i.e. 5-InsP₇) pool size. Moreover, we report that InsP₈ phosphatase activities of PPIP5Ks are strongly inhibited by P_i (40-90% within the 0-1 mm range). For PPIP5K2, P_i sensing by InsP₈ is amplified by a 2-fold activation of 5-InsP₇ kinase activity by P_i within the 0-5 mm range. Overall, our data reveal mechanisms that can contribute to specificity in inositol pyrophosphate signaling, regulating InsP₈ turnover independently of 5-InsP₇, in response to fluctuations in extracellular supply of a key nutrient.

Invariants reveal multiple forms of robustness in bifunctional enzyme systems

Experimental and theoretical studies have suggested that bifunctional enzymes catalyzing opposing modification and demodification reactions can confer steady-state concentration robustness to their substrates. However, the types of robustness and the biochemical basis for them have remained elusive. Here we report a systematic study of the most general biochemical reaction network for a bifunctional enzyme acting on a substrate with one modification site, along with eleven sub-networks with more specialized biochemical assumptions. We exploit ideas from computational algebraic geometry, introduced in previous work, to find a polynomial expression (an invariant) between the steady state concentrations of the modified and unmodified substrate for each network. We use these invariants to identify five classes of robust behavior: robust upper bounds on concentration, robust two-sided bounds on concentration ratio, hybrid robustness, absolute concentration robustness (ACR), and robust concentration ratio. This analysis demonstrates that robustness can take a variety of forms and that the type of robustness is sensitive to many biochemical details, with small changes in biochemistry leading to very different steady-state behaviors. In particular, we find that the widely-studied ACR requires highly specialized assumptions in addition to bifunctionality. An unexpected result is that the robust bounds derived from invariants are strictly tighter than those derived by ad hoc manipulation of the underlying differential equations, confirming the value of invariants as a tool to gain insight into biochemical reaction networks. Furthermore, invariants yield multiple experimentally testable predictions and illuminate new strategies for inferring enzymatic mechanisms from steady-state measurements.

Reciprocal regulation as a source of ultrasensitivity in two-component systems with a bifunctional sensor kinase

Two-component signal transduction systems, where the phosphorylation state of a regulator protein is modulated by a sensor kinase, are common in bacteria and other microbes. In many of these systems, the sensor kinase is bifunctional catalyzing both, the phosphorylation and the dephosphorylation of the regulator protein in response to input signals. Previous studies have shown that systems with a bifunctional enzyme can adjust the phosphorylation level of the regulator protein independently of the total protein concentrations – a property known as concentration robustness. Here, I argue that two-component systems with a bifunctional enzyme may also exhibit ultrasensitivity if the input signal reciprocally affects multiple activities of the sensor kinase. To this end, I consider the case where an allosteric effector inhibits autophosphorylation and, concomitantly, activates the enzyme's phosphatase activity, as observed experimentally in the PhoQ/PhoP and NRII/NRI systems. A theoretical analysis reveals two operating regimes under steady state conditions depending on the effector affinity: If the affinity is low the system produces a graded response with respect to input signals and exhibits stimulus-dependent concentration robustness – consistent with previous experiments. In contrast, a high-affinity effector may generate ultrasensitivity by a similar mechanism as phosphorylation-dephosphorylation cycles with distinct converter enzymes. The occurrence of ultrasensitivity requires saturation of the sensor kinase's phosphatase activity, but is restricted to low effector concentrations, which suggests that this mode of operation might be employed for the detection and amplification of low abundant input signals. Interestingly, the same mechanism also applies to covalent modification cycles with a bifunctional converter enzyme, which suggests that reciprocal regulation, as a mechanism to generate ultrasensitivity, is not restricted to two-component systems, but may apply more generally to bifunctional enzyme systems.

Bacteria often use two-component systems to sense and respond to environmental changes, which involves autophosphorylation of a sensor kinase and phosphotransfer to a cognate response regulator. However, despite conservation of this ‘classical’ scheme there exist substantial variations in the mechanism of phosphotransfer among systems. Also, many sensor kinases exhibit phosphatase activity raising the question whether such a bifunctional architecture enables special regulatory properties in the response behavior to input signals. According to previous studies, classical two-component systems are unlikely to produce sigmoidal response curves (ultrasensitivity) if the sensor protein is bifunctional. Here, I argue that this is not necessarily true if the input stimulus (allosteric effector) reciprocally affects multiple activities of the sensor kinase, as it seems to be common for bifunctional enzymes. To this end, I propose and analyze an extension of the experimentally well-supported Batchelor-Goulian model which shows that ultrasensitivity requires a high-affinity effector and saturation of the phosphatase activity. The underlying mechanism involves sequestration of the effector by the sensor kinase which restricts the occurrence of ultrasensitivity to sufficiently low effector concentrations. Hence, this operating regime might be useful to sense effector limitations or to amplify weak input signals.

A fundamental trade-off in covalent switching and its circumvention by enzyme bifunctionality in glucose homeostasis

Covalent modification provides a mechanism for modulating molecular state and regulating physiology. A cycle of competing enzymes that add and remove a single modification can act as a molecular switch between "on" and "off" and has been widely studied as a core motif in systems biology. Here, we exploit the recently developed "linear framework" for time scale separation to determine the general principles of such switches. These methods are not limited to Michaelis-Menten assumptions, and our conclusions hold for enzymes whose mechanisms may be arbitrarily complicated. We show that switching efficiency improves with increasing irreversibility of the enzymes and that the on/off transition occurs when the ratio of enzyme levels reaches a value that depends only on the rate constants. Fluctuations in enzyme levels, which habitually occur due to cellular heterogeneity, can cause flipping back and forth between on and off, leading to incoherent mosaic behavior in tissues, that worsens as switching becomes sharper. This trade-off can be circumvented if enzyme levels are correlated. In particular, if the competing catalytic domains are on the same protein but do not influence each other, the resulting bifunctional enzyme can switch sharply while remaining coherent. In the mammalian liver, the switch between glycolysis and gluconeogenesis is regulated by the bifunctional 6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase (PFK-2/FBPase-2). We suggest that bifunctionality of PFK-2/FBPase-2 complements the metabolic zonation of the liver by ensuring coherent switching in response to insulin and glucagon.