Phase plane dynamics of ERK phosphorylation.

The extracellular signal-regulated kinase (ERK) controls multiple critical processes in the cell and is deregulated in human cancers, congenital abnormalities, immune diseases, and neurodevelopmental syndromes. Catalytic activity of ERK requires dual phosphorylation by an upstream kinase, in a mechanism that can be described by two sequential Michaelis-Menten steps. The estimation of individual reaction rate constants from kinetic data in the full mechanism has proved challenging. Here, we present an analytically tractable approach to parameter estimation that is based on the phase plane representation of ERK activation and yields two combinations of six reaction rate constants in the detailed mechanism. These combinations correspond to the ratio of the specificities of two consecutive phosphorylations and the probability that monophosphorylated substrate does not dissociate from the enzyme before the second phosphorylation. The presented approach offers a language for comparing the effects of mutations that disrupt ERK activation and function in vivo. As an illustration, we use phase plane representation to analyze dual phosphorylation under heterozygous conditions, when two enzyme variants compete for the same substrate.

Dynamics of an incoherent feedforward loop drive ERK-dependent pattern formation in the early Drosophila embryo.

Positional information in development often manifests as stripes of gene expression, but how stripes form remains incompletely understood. Here, we use optogenetics and live-cell biosensors to investigate the posterior brachyenteron (byn) stripe in early Drosophila embryos. This stripe depends on interpretation of an upstream ERK activity gradient and the expression of two target genes, tailless (tll) and huckebein (hkb), that exert antagonistic control over byn. We find that high or low doses of ERK signaling produce transient or sustained byn expression, respectively. Although tll transcription is always rapidly induced, hkb converts graded ERK inputs into a variable time delay. Nuclei thus interpret ERK amplitude through the relative timing of tll and hkb transcription. Antagonistic regulatory paths acting on different timescales are hallmarks of an incoherent feedforward loop, which is sufficient to explain byn dynamics and adds temporal complexity to the steady-state model of byn stripe formation. We further show that 'blurring' of an all-or-none stimulus through intracellular diffusion non-locally produces a byn stripe. Overall, we provide a blueprint for using optogenetics to dissect developmental signal interpretation in space and time.

Dynamics of an incoherent feedforward loop drive ERK-dependent pattern formation in the early Drosophila embryo.

Positional information in developing tissues often takes the form of stripes of gene expression that mark the boundaries of a particular cell type or morphogenetic process. How stripes form is still in many cases poorly understood. Here we use optogenetics and live-cell biosensors to investigate one such pattern: the posterior stripe of brachyenteron (byn) expression in the early Drosophila embryo. This byn stripe depends on interpretation of an upstream signal - a gradient of ERK kinase activity - and the expression of two target genes tailless (tll) and huckebein (hkb) that exert antagonistic control over byn . We find that high or low doses of ERK signaling produce either transient or sustained byn expression, respectively. These ERK stimuli also regulate tll and hkb expression with distinct dynamics: tll transcription is rapidly induced under both low and high stimuli, whereas hkb transcription converts graded ERK inputs into an output switch with a variable time delay. Antagonistic regulatory paths acting on different timescales are hallmarks of an incoherent feedforward loop architecture, which is sufficient to explain transient or sustained byn dynamics and adds temporal complexity to the steady-state model of byn stripe formation. We further show that an all-or-none stimulus can be 'blurred' through intracellular diffusion to non-locally produce a stripe of byn gene expression. Overall, our study provides a blueprint for using optogenetic inputs to dissect developmental signal interpretation in space and time.

Putting in the Erk: Growth factor signaling and mesoderm morphogenesis.

It has long been known that FGF signaling contributes to mesoderm formation, a germ layer found in triploblasts that is composed of highly migratory cells that give rise to muscles and to the skeletal structures of vertebrates. FGF signaling activates several pathways in the developing mesoderm, including transient activation of the Erk pathway, which triggers mesodermal fate specification through the induction of the gene brachyury and activates morphogenetic programs that allow mesodermal cells to position themselves in the embryo. In this review, we discuss what is known about the generation and interpretation of transient Erk signaling in mesodermal tissues across species. We focus specifically on mechanisms that translate the level and duration of Erk signaling into cell fate and cell movement instructions and discuss strategies for further interrogating the role that Erk signaling dynamics play in mesodermal gastrulation and morphogenesis.

On the parameter combinations that matter and on those that do not: data-driven studies of parameter (non)identifiability.

We present a data-driven approach to characterizing nonidentifiability of a model's parameters and illustrate it through dynamic as well as steady kinetic models. By employing Diffusion Maps and their extensions, we discover the minimal combinations of parameters required to characterize the output behavior of a chemical system: a set of effective parameters for the model. Furthermore, we introduce and use a Conformal Autoencoder Neural Network technique, as well as a kernel-based Jointly Smooth Function technique, to disentangle the redundant parameter combinations that do not affect the output behavior from the ones that do. We discuss the interpretability of our data-driven effective parameters, and demonstrate the utility of the approach both for behavior prediction and parameter estimation. In the latter task, it becomes important to describe level sets in parameter space that are consistent with a particular output behavior. We validate our approach on a model of multisite phosphorylation, where a reduced set of effective parameters (nonlinear combinations of the physical ones) has previously been established analytically.

A synthetic gene circuit for imaging-free detection of signaling pulses.

Cells employ intracellular signaling pathways to sense and respond to changes in their external environment. In recent years, live-cell biosensors have revealed complex pulsatile dynamics in many pathways, but studies of these signaling dynamics are limited by the necessity of live-cell imaging at high spatiotemporal resolution. Here, we describe an approach to infer pulsatile signaling dynamics from a single measurement in fixed cells using a pulse-detecting gene circuit. We computationally screened for circuits with the capability to selectively detect signaling pulses, revealing an incoherent feedforward topology that robustly performs this computation. We implemented the motif experimentally for the Erk signaling pathway using a single engineered transcription factor and fluorescent protein reporter. Our "recorder of Erk activity dynamics" (READer) responds sensitively to spontaneous and stimulus-driven Erk pulses. READer circuits open the door to permanently labeling transient, dynamic cell populations to elucidate the mechanistic underpinnings and biological consequences of signaling dynamics.

Temporal integration of inductive cues on the way to gastrulation.

Markers for the endoderm and mesoderm germ layers are commonly expressed together in the early embryo, potentially reflecting cells' ability to explore potential fates before fully committing. It remains unclear when commitment to a single-germ layer is reached and how it is impacted by external signals. Here, we address this important question in Drosophila, a convenient model system in which mesodermal and endodermal fates are associated with distinct cellular movements during gastrulation. Systematically applying endoderm-inducing extracellular signal-regulated kinase (ERK) signals to the ventral medial embryo-which normally only receives a mesoderm-inducing cue-reveals a critical time window during which mesodermal cell movements and gene expression are suppressed by proendoderm signaling. We identify the ERK target gene huckebein (hkb) as the main cause of the ventral furrow suppression and use computational modeling to show that Hkb repression of the mesoderm-associated gene snail is sufficient to account for a broad range of transcriptional and morphogenetic effects. Our approach, pairing precise signaling perturbations with observation of transcriptional dynamics and cell movements, provides a general framework for dissecting the complexities of combinatorial tissue patterning.

Inference of Multisite Phosphorylation Rate Constants and Their Modulation by Pathogenic Mutations.

Multisite protein phosphorylation plays a critical role in cell regulation [1-3]. It is widely appreciated that the functional capabilities of multisite phosphorylation depend on the order and kinetics of phosphorylation steps, but kinetic aspects of multisite phosphorylation remain poorly understood [4-6]. Here, we focus on what appears to be the simplest scenario, when a protein is phosphorylated on only two sites in a strict, well-defined order. This scenario describes the activation of ERK, a highly conserved cell-signaling enzyme. We use Bayesian parameter inference in a structurally identifiable kinetic model to dissect dual phosphorylation of ERK by MEK, a kinase that is mutated in a large number of human diseases [7-12]. Our results reveal how enzyme processivity and efficiencies of individual phosphorylation steps are altered by pathogenic mutations. The presented approach, which connects specific mutations to kinetic parameters of multisite phosphorylation mechanisms, provides a systematic framework for closing the gap between studies with purified enzymes and their effects in the living organism.