Linear approaches to intramolecular Förster resonance energy transfer probe measurements for quantitative modeling

Efficient parameter estimation for ODE models of cellular processes using semi-quantitative data

MOTIVATION

Quantitative dynamical models facilitate the understanding of biological processes and the prediction of their dynamics. The parameters of these models are commonly estimated from experimental data. Yet, experimental data generated from different techniques do not provide direct information about the state of the system but a nonlinear (monotonic) transformation of it. For such semi-quantitative data, when this transformation is unknown, it is not apparent how the model simulations and the experimental data can be compared.

RESULTS

We propose a versatile spline-based approach for the integration of a broad spectrum of semi-quantitative data into parameter estimation. We derive analytical formulas for the gradients of the hierarchical objective function and show that this substantially increases the estimation efficiency. Subsequently, we demonstrate that the method allows for the reliable discovery of unknown measurement transformations. Furthermore, we show that this approach can significantly improve the parameter inference based on semi-quantitative data in comparison to available methods.

AVAILABILITY AND IMPLEMENTATION

Modelers can easily apply our method by using our implementation in the open-source Python Parameter EStimation TOolbox (pyPESTO) available at https://github.com/ICB-DCM/pyPESTO.

POLARISED views and FRETting about probe modulation assays: Learning from High Throughput Screening

Fluorescent probe modulation assays are a widely used approach to monitor displacement or stabilisation of fluorescently labelled tool ligands by test compounds. These assays allow an optical read-out of probe-receptor binding and can be used to detect compounds that compete with the labelled ligand, either directly or indirectly. Probes for both orthosteric and allosteric sites are often employed. The method can also be used to identify test compounds that may stabilise the ternary complex, offering an opportunity to discover novel molecular glues. The utility of these fluorescence-based assays within high-throughput screening has been facilitated by the use of streptavidin labelled terbium as a donor and access to a range of different acceptor fluorophores. During 2023, the High-throughput Screening group at AstraZeneca carried out 8 high-throughput screens using these approaches. In this manuscript we will present the types of assays used, an overview of the timelines for assay development and screening, the application of orthogonal artefact methods to aid hit finding and the results of the screens in terms of hit rate and the number of compounds identified with IC50 values of better than 30 µM. Learning across the development, execution and analysis of these screens will be presented.

Efficient gradient-based parameter estimation for dynamic models using qualitative data

MOTIVATION

Unknown parameters of dynamical models are commonly estimated from experimental data. However, while various efficient optimization and uncertainty analysis methods have been proposed for quantitative data, methods for qualitative data are rare and suffer from bad scaling and convergence.

RESULTS

Here, we propose an efficient and reliable framework for estimating the parameters of ordinary differential equation models from qualitative data. In this framework, we derive a semi-analytical algorithm for gradient calculation of the optimal scaling method developed for qualitative data. This enables the use of efficient gradient-based optimization algorithms. We demonstrate that the use of gradient information improves performance of optimization and uncertainty quantification on several application examples. On average, we achieve a speedup of more than one order of magnitude compared to gradient-free optimization. In addition, in some examples, the gradient-based approach yields substantially improved objective function values and quality of the fits. Accordingly, the proposed framework substantially improves the parameterization of models from qualitative data.

AVAILABILITY AND IMPLEMENTATION

The proposed approach is implemented in the open-source Python Parameter EStimation TOolbox (pyPESTO). pyPESTO is available at https://github.com/ICB-DCM/pyPESTO. All application examples and code to reproduce this study are available at https://doi.org/10.5281/zenodo.4507613.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Oncogenic mutant RAS signaling activity is rescaled by the ERK/MAPK pathway

Activating mutations in RAS are present in ~ 30% of human tumors, and the resulting aberrations in ERK/MAPK signaling play a central role in oncogenesis. However, the form of these signaling changes is uncertain, with activating RAS mutants linked to both increased and decreased ERK activation in vivo. Rationally targeting the kinase activity of this pathway requires clarification of the quantitative effects of RAS mutations. Here, we use live‐cell imaging in cells expressing only one RAS isoform to quantify ERK activity with a new level of accuracy. We find that despite large differences in their biochemical activity, mutant KRAS isoforms within cells have similar ranges of ERK output. We identify roles for pathway‐level effects, including variation in feedback strength and feedforward modulation of phosphatase activity, that act to rescale pathway sensitivity, ultimately resisting changes in the dynamic range of ERK activity while preserving responsiveness to growth factor stimuli. Our results reconcile seemingly inconsistent reports within the literature and imply that the signaling changes induced by RAS mutations early in oncogenesis are subtle.

Efficient gradient-based parameter estimation for dynamic models using qualitative data

Motivation

Unknown parameters of dynamical models are commonly estimated from experimental data. However, while various efficient optimization and uncertainty analysis methods have been proposed for quantitative data, methods for qualitative data are rare and suffer from bad scaling and convergence.

Results

Here, we propose an efficient and reliable framework for estimating the parameters of ordinary differential equation models from qualitative data. In this framework, we derive a semi-analytical algorithm for gradient calculation of the optimal scaling method developed for qualitative data. This enables the use of efficient gradient-based optimization algorithms. We demonstrate that the use of gradient information improves performance of optimization and uncertainty quantification on several application examples. On average, we achieve a speedup of more than one order of magnitude compared to gradient-free optimization. In addition, in some examples, the gradient-based approach yields substantially improved objective function values and quality of the fits. Accordingly, the proposed framework substantially improves the parameterization of models from qualitative data.

Availability and implementation

The proposed approach is implemented in the open-source Python Parameter EStimation TOolbox (pyPESTO). pyPESTO is available at https://github.com/ICB-DCM/pyPESTO. All application examples and code to reproduce this study are available at https://doi.org/10.5281/zenodo.4507613.

Supplementary information

Supplementary data are available at Bioinformatics online.

Parameterization of mechanistic models from qualitative data using an efficient optimal scaling approach

Quantitative dynamical models facilitate the understanding of biological processes and the prediction of their dynamics. These models usually comprise unknown parameters, which have to be inferred from experimental data. For quantitative experimental data, there are several methods and software tools available. However, for qualitative data the available approaches are limited and computationally demanding. Here, we consider the optimal scaling method which has been developed in statistics for categorical data and has been applied to dynamical systems. This approach turns qualitative variables into quantitative ones, accounting for constraints on their relation. We derive a reduced formulation for the optimization problem defining the optimal scaling. The reduced formulation possesses the same optimal points as the established formulation but requires less degrees of freedom. Parameter estimation for dynamical models of cellular pathways revealed that the reduced formulation improves the robustness and convergence of optimizers. This resulted in substantially reduced computation times. We implemented the proposed approach in the open-source Python Parameter EStimation TOolbox (pyPESTO) to facilitate reuse and extension. The proposed approach enables efficient parameterization of quantitative dynamical models using qualitative data.

Temporal perturbation of ERK dynamics reveals network architecture of FGF2/MAPK signaling

Stimulation of PC-12 cells with epidermal (EGF) versus nerve (NGF) growth factors (GFs) biases the distribution between transient and sustained single-cell ERK activity states, and between proliferation and differentiation fates within a cell population. We report that fibroblast GF (FGF2) evokes a distinct behavior that consists of a gradually changing population distribution of transient/sustained ERK signaling states in response to increasing inputs in a dose response. Temporally controlled GF perturbations of MAPK signaling dynamics applied using microfluidics reveal that this wider mix of ERK states emerges through the combination of an intracellular feedback, and competition of FGF2 binding to FGF receptors (FGFRs) and heparan sulfate proteoglycan (HSPG) co-receptors. We show that the latter experimental modality is instructive for model selection using a Bayesian parameter inference. Our results provide novel insights into how different receptor tyrosine kinase (RTK) systems differentially wire the MAPK network to fine-tune fate decisions at the cell population level.

Temporal perturbation of ERK dynamics reveals network architecture of FGF2/MAPK signaling

Stimulation of PC-12 cells with epidermal (EGF) versus nerve (NGF) growth factors (GFs) biases the distribution between transient and sustained single-cell ERK activity states, and between proliferation and differentiation fates within a cell population. We report that fibroblast GF (FGF2) evokes a distinct behavior that consists of a gradually changing population distribution of transient/sustained ERK signaling states in response to increasing inputs in a dose response. Temporally controlled GF perturbations of MAPK signaling dynamics applied using microfluidics reveal that this wider mix of ERK states emerges through the combination of an intracellular feedback, and competition of FGF2 binding to FGF receptors (FGFRs) and heparan sulfate proteoglycan (HSPG) co-receptors. We show that the latter experimental modality is instructive for model selection using a Bayesian parameter inference. Our results provide novel insights into how different receptor tyrosine kinase (RTK) systems differentially wire the MAPK network to fine-tune fate decisions at the cell population level.

Akt regulation of glycolysis mediates bioenergetic stability in epithelial cells

Cells use multiple feedback controls to regulate metabolism in response to nutrient and signaling inputs. However, feedback creates the potential for unstable network responses. We examined how concentrations of key metabolites and signaling pathways interact to maintain homeostasis in proliferating human cells, using fluorescent reporters for AMPK activity, Akt activity, and cytosolic NADH/NAD⁺ redox. Across various conditions, including glycolytic or mitochondrial inhibition or cell proliferation, we observed distinct patterns of AMPK activity, including both stable adaptation and highly dynamic behaviors such as periodic oscillations and irregular fluctuations that indicate a failure to reach a steady state. Fluctuations in AMPK activity, Akt activity, and cytosolic NADH/NAD⁺ redox state were temporally linked in individual cells adapting to metabolic perturbations. By monitoring single-cell dynamics in each of these contexts, we identified PI3K/Akt regulation of glycolysis as a multifaceted modulator of single-cell metabolic dynamics that is required to maintain metabolic stability in proliferating cells.

Two New FRET Imaging Measures: Linearly Proportional to and Highly Contrasting the Fraction of Active Molecules

We developed two new FRET imaging measures for intramolecular FRET biosensors, called linearly proportional (LP) and highly contrasting (HC) measures, which can be easily calculated by the fluorescence intensities of donor and acceptor as a ratio between their weighted sums. As an alternative to the conventional ratiometric measure, which non-linearly depends on the fraction of active molecule, we first developed the LP measure, which is linearly proportional to the fraction of active molecules. The LP measure inherently unmixes bleed-through signals and is robust against fluorescence noise. By extending the LP measure, we furthermore designed the HC measure, which provides highly contrasting images of the molecular activity, more than the ratiometric measure. In addition to their advantages, these measures are insensitive to the biosensor expression level, which is a fundamental property of the ratiometric measure. Using artificial data and FRET imaging data, we showed that the LP measure effectively represents the fraction of active molecules and that the HC measure improves visual interpretability by providing high contrast images of molecular activity. Therefore, the LP and HC measures allow us to gain more quantitative and qualitative insights from FRET imaging than the ratiometric measure.

Frequency modulation of ERK activation dynamics rewires cell fate

Transient versus sustained ERK MAP kinase (MAPK) activation dynamics induce proliferation versus differentiation in response to epidermal (EGF) or nerve (NGF) growth factors in PC‐12 cells. Duration of ERK activation has therefore been proposed to specify cell fate decisions. Using a biosensor to measure ERK activation dynamics in single living cells reveals that sustained EGF/NGF application leads to a heterogeneous mix of transient and sustained ERK activation dynamics in distinct cells of the population, different than the population average. EGF biases toward transient, while NGF biases toward sustained ERK activation responses. In contrast, pulsed growth factor application can repeatedly and homogeneously trigger ERK activity transients across the cell population. These datasets enable mathematical modeling to reveal salient features inherent to the MAPK network. Ultimately, this predicts pulsed growth factor stimulation regimes that can bypass the typical feedback activation to rewire the system toward cell differentiation irrespective of growth factor identity.

Receptor Level Mechanisms Are Required for Epidermal Growth Factor (EGF)-stimulated Extracellular Signal-regulated Kinase (ERK) Activity Pulses

In both physiological and cell culture systems, EGF-stimulated ERK activity occurs in discrete pulses within individual cells. Many feedback loops are present in the EGF receptor (EGFR)-ERK network, but the mechanisms driving pulsatile ERK kinetics are unknown. Here, we find that in cells that respond to EGF with frequency-modulated pulsatile ERK activity, stimulation through a heterologous TrkA receptor system results in non-pulsatile, amplitude-modulated activation of ERK. We further dissect the kinetics of pulse activity using a combination of FRET- and translocation-based reporters and find that EGFR activity is required to maintain ERK activity throughout the 10-20-minute lifetime of pulses. Together, these data indicate that feedbacks operating within the core Ras-Raf-MEK-ERK cascade are insufficient to drive discrete pulses of ERK activity and instead implicate mechanisms acting at the level of EGFR.

Frequency-modulated pulses of ERK activity transmit quantitative proliferation signals

The EGF-stimulated ERK/MAPK pathway is a key conduit for cellular proliferation signals and a therapeutic target in many cancers. Here, we characterize two central quantitative aspects of this pathway: the mechanism by which signal strength is encoded and the response curve relating signal output to proliferation. Under steady-state conditions, we find that ERK is activated in discrete, asynchronous pulses with frequency and duration determined by extracellular concentrations of EGF spanning the physiological range. In genetically identical sister cells, cell-to-cell variability in pulse dynamics influences the decision to enter S phase. While targeted inhibition of EGFR reduces the frequency of ERK activity pulses, inhibition of MEK reduces their amplitude. Continuous response curves measured in multiple cell lines reveal that proliferation is effectively silenced only when ERK pathway output falls below a threshold of ~10%, indicating that high-dose targeting of the pathway is necessary to achieve therapeutic efficacy.