The presence of nuclear cactus in the early Drosophila embryo may extend the dynamic range of the dorsal gradient

Identifying effective evolutionary strategies-based protocol for uncovering reaction kinetic parameters under the effect of measurement noises

BACKGROUND

The transition from explanative modeling of fitted data to the predictive modeling of unseen data for systems biology endeavors necessitates the effective recovery of reaction parameters. Yet, the relative efficacy of optimization algorithms in doing so remains under-studied, as to the specific reaction kinetics and the effect of measurement noises. To this end, we simulate the reactions of an artificial pathway using 4 kinetic formulations: generalized mass action (GMA), Michaelis-Menten, linear-logarithmic, and convenience kinetics. We then compare the effectiveness of 5 evolutionary algorithms (CMAES, DE, SRES, ISRES, G3PCX) for objective function optimization in kinetic parameter hyperspace to determine the corresponding estimated parameters.

RESULTS

We quickly dropped the DE algorithm due to its poor performance. Baring measurement noise, we find the CMAES algorithm to only require a fraction of the computational cost incurred by other EAs for both GMA and linear-logarithmic kinetics yet performing as well by other criteria. However, with increasing noise, SRES and ISRES perform more reliably for GMA kinetics, but at considerably higher computational cost. Conversely, G3PCX is among the most efficacious for estimating Michaelis-Menten parameters regardless of noise, while achieving numerous folds saving in computational cost. Cost aside, we find SRES to be versatilely applicable across GMA, Michaelis-Menten, and linear-logarithmic kinetics, with good resilience to noise. Nonetheless, we could not identify the parameters of convenience kinetics using any algorithm.

CONCLUSIONS

Altogether, we identify a protocol for predicting reaction parameters under marked measurement noise, as a step towards predictive modeling for systems biology endeavors.

ISRES+: an improved evolutionary strategy for function minimization to estimate the free parameters of systems biology models

MOTIVATION

Mathematical models in systems biology help generate hypotheses, guide experimental design, and infer the dynamics of gene regulatory networks. These models are characterized by phenomenological or mechanistic parameters, which are typically hard to measure. Therefore, efficient parameter estimation is central to model development. Global optimization techniques, such as evolutionary algorithms (EAs), are applied to estimate model parameters by inverse modeling, i.e. calibrating models by minimizing a function that evaluates a measure of the error between model predictions and experimental data. EAs estimate model parameters "fittest individuals" by generating a large population of individuals using strategies like recombination and mutation over multiple "generations." Typically, only a few individuals from each generation are used to create new individuals in the next generation. Improved Evolutionary Strategy by Stochastic Ranking (ISRES), proposed by Runnarson and Yao, is one such EA that is widely used in systems biology to estimate parameters. ISRES uses information at most from a pair of individuals in any generation to create a new population to minimize the error. In this article, we propose an efficient evolutionary strategy, ISRES+, which builds on ISRES by combining information from all individuals across the population and across all generations to develop a better understanding of the fitness landscape.

RESULTS

ISRES+ uses the additional information generated by the algorithm during evolution to approximate the local neighborhood around the best-fit individual using linear least squares fits in one and two dimensions, enabling efficient parameter estimation. ISRES+ outperforms ISRES and results in fitter individuals with a tighter distribution over multiple runs, such that a typical run of ISRES+ estimates parameters with a higher goodness-of-fit compared with ISRES.

AVAILABILITY AND IMPLEMENTATION

Algorithm and implementation: Github-https://github.com/gtreeves/isres-plus-bandodkar-2022.

A reaction-diffusion network model predicts a dual role of Cactus/IκB to regulate Dorsal/NFκB nuclear translocation in Drosophila

Dorsal-ventral patterning of the Drosophila embryo depends on the NFκB superfamily transcription factor Dorsal (Dl). Toll receptor activation signals for degradation of the IκB inhibitor Cactus (Cact), leading to a ventral-to-dorsal nuclear Dl gradient. Cact is critical for Dl nuclear import, as it binds to and prevents Dl from entering the nuclei. Quantitative analysis of cact mutants revealed an additional Cact function to promote Dl nuclear translocation in ventral regions of the embryo. To investigate this dual Cact role, we developed a predictive model based on a reaction-diffusion regulatory network. This network distinguishes non-uniform Toll-dependent Dl nuclear import and Cact degradation, from the Toll-independent processes of Cact degradation and reversible nuclear-cytoplasmic Dl flow. In addition, it incorporates translational control of Cact levels by Dl. Our model successfully reproduces wild-type data and emulates the Dl nuclear gradient in mutant dl and cact allelic combinations. Our results indicate that the dual role of Cact depends on the dynamics of Dl-Cact trimers along the dorsal-ventral axis: In the absence of Toll activation, free Dl-Cact trimers retain Dl in the cytoplasm, limiting the flow of Dl into the nucleus; in ventral-lateral regions, Dl-Cact trimers are recruited by Toll activation into predominant signaling complexes and promote Dl nuclear translocation. Simulations suggest that the balance between Toll-dependent and Toll-independent processes are key to this dynamics and reproduce the full assortment of Cact effects. Considering the high evolutionary conservation of these pathways, our analysis should contribute to understanding NFκB/c-Rel activation in other contexts such as in the vertebrate immune system and disease.

In Drosophila, Toll pathway establishes spatially distinct gene expression territories that define the embryonic dorsal-ventral axis. Toll activation leads to degradation of the IκB inhibitor Cactus, releasing the NFκB superfamily transcription factor Dorsal for nuclear entry. Recently, quantitative analysis of cact mutants revealed that Cact displays an additional function to promote Dl nuclear translocation in ventral regions of the embryo. To understand this novel activity, we developed a predictive theoretical model that shows that the kinetics of Dorsal-Cactus complex formation prior to their recruitment to Toll-signaling complexes is an essential regulatory hub. Cactus controls the balance between the recruitment of these complexes by active Toll receptor and association-dissociation events that generate free Dorsal for direct nuclear import.

Mechanism and implications of morphogen shuttling: Lessons learned from dorsal and Cactus in Drosophila

In a developing animal, morphogen gradients act to pattern tissues into distinct domains of cell types. However, despite their prevalence in development, morphogen gradient formation is a matter of debate. In our recent publication, we showed that the Dorsal/NF-κB morphogen gradient, which patterns the DV axis of the early Drosophila embryo, is partially established by a mechanism of facilitated diffusion. This mechanism, also known as "shuttling," occurs when a binding partner of the morphogen facilitates the diffusion of the morphogen, allowing it to accumulate at a given site. In this case, the inhibitor Cactus/IκB facilitates the diffusion of Dorsal/NF-κB. In the fly embryo, we used computation and experiment to not only show that shuttling occurs in the embryo, but also that it enables the viability of embryos that inherit only one copy of dorsal maternally. In this commentary, we further discuss our evidence behind the shuttling mechanism, the previous literature data explained by the mechanism, and how it may also be critical for robustness of development. Finally, we briefly provide additional experimental data pointing toward an interaction between Dorsal and BMP signaling that is likely affected by shuttling.

Robustness of the Dorsal morphogen gradient with respect to morphogen dosage

In multicellular organisms, the timing and placement of gene expression in a developing tissue assigns the fate of each cell in the embryo in order for a uniform field of cells to differentiate into a reproducible pattern of organs and tissues. This positional information is often achieved through the action of spatial gradients of morphogens. Spatial patterns of gene expression are paradoxically robust to variations in morphogen dosage, given that, by definition, gene expression must be sensitive to morphogen concentration. In this work we investigate the robustness of the Dorsal/NF-κB signaling module with respect to perturbations to the dosage of maternally-expressed dorsal mRNA. The Dorsal morphogen gradient patterns the dorsal-ventral axis of the early Drosophila embryo, and we found that an empirical description of the Dorsal gradient is highly sensitive to maternal dorsal dosage. In contrast, we found experimentally that gene expression patterns are highly robust. Although the components of this signaling module have been characterized in detail, how their function is integrated to produce robust gene expression patterns to variations in the dorsal maternal dosage is still unclear. Therefore, we analyzed a mechanistic model of the Dorsal signaling module and found that Cactus, a cytoplasmic inhibitor for Dorsal, must be present in the nucleus for the system to be robust. Furthermore, active Toll, the receptor that dissociates Cactus from Dorsal, must be saturated. Finally, the vast majority of robust descriptions of the system require facilitated diffusion of Dorsal by Cactus. Each of these three recently-discovered mechanisms of the Dorsal module are critical for robustness. These mechanisms synergistically contribute to changing the amplitude and shape of the active Dorsal gradient, which is required for robust gene expression. Our work highlights the need for quantitative understanding of biophysical mechanisms of morphogen gradients in order to understand emergent phenotypes, such as robustness.

Spatiotemporal control of gene expression boundaries using a feedforward loop

BACKGROUND

A feedforward loop (FFL) is commonly observed in several biological networks. The FFL network motif has been mostly studied with respect to variation of the input signal in time, with only a few studies of FFL activity in a spatially distributed system such as morphogen-mediated tissue patterning. However, most morphogen gradients also evolve in time.

RESULTS

We studied the spatiotemporal behavior of a coherent FFL in two contexts: (a) a generic, oscillating morphogen gradient and (b) the dorsal-ventral patterning of the early Drosophila embryo by a gradient of the NF-κB homolog dorsal with its early target Twist. In both models, we found features in the dynamics of the intermediate node-phase difference and noise filtering-that were largely independent of the parameterization of the models, and thus were functions of the structure of the FFL itself. In the dorsal gradient model, we also found that proper target gene expression was not possible without including the effect of maternal pioneer factor Zelda.

CONCLUSIONS

An FFL buffers fluctuation to changes in the morphogen signal ensuring stable gene expression boundaries.

A facilitated diffusion mechanism establishes the Drosophila Dorsal gradient

The transcription factor NF-κB plays an important role in the immune system, apoptosis and inflammation. Dorsal, a Drosophila homolog of NF-κB, patterns the dorsal-ventral axis in the blastoderm embryo. During this stage, Dorsal is sequestered outside the nucleus by the IκB homolog Cactus. Toll signaling on the ventral side breaks the Dorsal/Cactus complex, allowing Dorsal to enter the nucleus to regulate target genes. Fluorescent data show that Dorsal accumulates on the ventral side of the syncytial blastoderm. Here, we use modeling and experimental studies to show that this accumulation is caused by facilitated diffusion, or shuttling, of the Dorsal/Cactus complex. We also show that active Toll receptors are limiting in wild-type embryos, which is a key factor in explaining global Dorsal gradient formation. Our results suggest that shuttling is necessary for viability of embryos from mothers with compromised dorsal levels. Therefore, Cactus not only has the primary role of regulating Dorsal nuclear import, but also has a secondary role in shuttling. Given that this mechanism has been found in other, independent, systems, we suggest that it might be more prevalent than previously thought.

A novel function for the IκB inhibitor Cactus in promoting Dorsal nuclear localization and activity in the Drosophila embryo

The evolutionarily conserved Toll signaling pathway controls innate immunity across phyla and embryonic patterning in insects. In the Drosophila embryo, Toll is required to establish gene expression domains along the dorsal-ventral axis. Pathway activation induces degradation of the IκB inhibitor Cactus, resulting in a ventral-to-dorsal nuclear gradient of the NFκB effector Dorsal. Here, we investigate how cactus modulates Toll signals through its effects on the Dorsal gradient and on Dorsal target genes. Quantitative analysis using a series of loss- and gain-of-function conditions shows that the ventral and lateral aspects of the Dorsal gradient can behave differently with respect to Cactus fluctuations. In lateral and dorsal embryo domains, loss of Cactus allows more Dorsal to translocate to the nucleus. Unexpectedly, cactus loss-of-function alleles decrease Dorsal nuclear localization ventrally, where Toll signals are high. Overexpression analysis suggests that this ability of Cactus to enhance Toll stems from the mobilization of a free Cactus pool induced by the Calpain A protease. These results indicate that Cactus acts to bolster Dorsal activation, in addition to its role as a NFκB inhibitor, ensuring a correct response to Toll signals.