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.

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.

Discovery of Membrane-Permeating Cyclic Peptides via mRNA Display.

Small synthetic peptides capable of crossing biological membranes represent valuable tools in cell biology and drug delivery. While several cell-penetrating peptides (CPPs) of natural or synthetic origin have been reported, no peptide is currently known to cross both cytoplasmic and outer embryonic membranes. Here, we describe a method to engineer membrane-permeating cyclic peptides (MPPs) with broad permeation activity by screening mRNA display libraries of cyclic peptides against embryos at different developmental stages. The proposed method was demonstrated by identifying peptides capable of permeating Drosophila melanogaster (fruit fly) embryos and mammalian cells. The selected peptide cyclo[Glut-MRKRHASRRE-K*] showed a strong permeation activity of embryos exposed to minimal permeabilization pretreatment, as well as human embryonic stem cells and a murine fibroblast cell line. Notably, in both embryos and mammalian cells, the cyclic peptide outperformed its linear counterpart and the control MPPs. Confocal microscopy and single cell flow cytometry analysis were utilized to assess the degree of permeation both qualitatively and quantitatively. These MPPs have potential application in studying and nondisruptively controlling intracellular or intraembryonic processes.

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

In a developing embryo, the spatial distribution of a signaling molecule, or a morphogen gradient, has been hypothesized to carry positional information to pattern tissues. Recent measurements of morphogen distribution have allowed us to subject this hypothesis to rigorous physical testing. In the early Drosophila embryo, measurements of the morphogen Dorsal, which is a transcription factor responsible for initiating the earliest zygotic patterns along the dorsal-ventral axis, have revealed a gradient that is too narrow to pattern the entire axis. In this study, we use a mathematical model of Dorsal dynamics, fit to experimental data, to determine the ability of the Dorsal gradient to regulate gene expression across the entire dorsal-ventral axis. We found that two assumptions are required for the model to match experimental data in both Dorsal distribution and gene expression patterns. First, we assume that Cactus, an inhibitor that binds to Dorsal and prevents it from entering the nuclei, must itself be present in the nuclei. And second, we assume that fluorescence measurements of Dorsal reflect both free Dorsal and Cactus-bound Dorsal. Our model explains the dynamic behavior of the Dorsal gradient at lateral and dorsal positions of the embryo, the ability of Dorsal to regulate gene expression across the entire dorsal-ventral axis, and the robustness of gene expression to stochastic effects. Our results have a general implication for interpreting fluorescence-based measurements of signaling molecules.

Size-dependent regulation of dorsal-ventral patterning in the early Drosophila embryo.

How natural variation in embryo size affects patterning of the Drosophila embryo dorsal-ventral (DV) axis is not known. Here we examined quantitatively the relationship between nuclear distribution of the Dorsal transcription factor, boundary positions for several target genes, and DV axis length. Data were obtained from embryos of a wild-type background as well as from mutant lines inbred to size select embryos of smaller or larger sizes. Our data show that the width of the nuclear Dorsal gradient correlates with DV axis length. In turn, for some genes expressed along the DV axis, the boundary positions correlate closely with nuclear Dorsal levels and with DV axis length; while the expression pattern of others is relatively constant and independent of the width of the Dorsal gradient. In particular, the patterns of snail (sna) and ventral nervous system defective (vnd) correlate with nuclear Dorsal levels and exhibit scaling to DV length; while the pattern of intermediate neuroblasts defective (ind) remains relatively constant with respect to changes in Dorsal and DV length. However, in mutants that exhibit an abnormal expansion of the Dorsal gradient which fails to scale to DV length, only sna follows the Dorsal distribution and exhibits overexpansion; in contrast, vnd and ind do not overexpand suggesting some additional mechanism acts to refine the dorsal boundaries of these two genes. Thus, our results argue against the idea that the Dorsal gradient works as a global system of relative coordinates along the DV axis and suggest that individual targets respond to changes in embryo size in a gene-specific manner.

Image analysis and empirical modeling of gene and protein expression.

Protein gradients and gene expression patterns are major determinants in the differentiation and fate map of the developing embryo. Here we discuss computational methods to quantitatively measure the positions of gene expression domains and the gradients of protein expression along the dorsal-ventral axis in the Drosophila embryo. Our methodology involves three layers of data. The first layer, or the primary data, consists of z-stack confocal images of embryos processed by in situ hybridization and/or antibody stainings. The secondary data are relationships between location, usually an x-axis coordinate, and fluorescent intensity of gene or protein detection. Tertiary data comprise the optimal parameters that arise from fits of the secondary data to empirical models. The tertiary data are useful to distill large datasets of imaged embryos down to a tractable number of conceptually useful parameters. This analysis allows us to detect subtle phenotypes and is adaptable to any set of genes or proteins with a canonical pattern. For example, we show how insights into the Dorsal transcription factor protein gradient and its target gene ventral-neuroblasts defective (vnd) were obtained using such quantitative approaches.

Optimal Performance Objectives in the Highly Conserved Bone Morphogenetic Protein Signaling Pathway.

Throughout development, complex networks of cell signaling pathways drive cellular decision-making across different tissues and contexts. The transforming growth factor ß (TGF-ß) pathways, including the BMP/Smad pathway, play crucial roles in these cellular responses. However, as the Smad pathway is used reiteratively throughout the life cycle of all animals, its systems-level behavior varies from one context to another, despite the pathway connectivity remaining nearly constant. For instance, some cellular systems require a rapid response, while others require high noise filtering. In this paper, we examine how the BMP- Smad pathway balances trade-offs among three such systems-level behaviors, or "Performance Objectives (POs)": response speed, noise amplification, and the sensitivity of pathway output to receptor input. Using a Smad pathway model fit to human cell data, we show that varying non-conserved parameters (NCPs) such as protein concentrations, the Smad pathway can be tuned to emphasize any of the three POs and that the concentration of nuclear phosphatase has the greatest effect on tuning the POs. However, due to competition among the POs, the pathway cannot simultaneously optimize all three, but at best must balance trade-offs among the POs. We applied the multi-objective optimization concept of the Pareto Front, a widely used concept in economics to identify optimal trade-offs among various requirements. We show that the BMP pathway efficiently balances competing POs across species and is largely Pareto optimal. Our findings reveal that varying the concentration of NCPs allows the Smad signaling pathway to generate a diverse range of POs. This insight identifies how signaling pathways can be optimally tuned for each context.

Mesoderm migration in Drosophila is a multi-step process requiring FGF signaling and integrin activity.

Migration is a complex, dynamic process that has largely been studied using qualitative or static approaches. As technology has improved, we can now take quantitative approaches towards understanding cell migration using in vivo imaging and tracking analyses. In this manner, we have established a four-step model of mesoderm migration during Drosophila gastrulation: (I) mesodermal tube formation, (II) collapse of the mesoderm, (III) dorsal migration and spreading and (IV) monolayer formation. Our data provide evidence that these steps are temporally distinct and that each might require different chemical inputs. To support this, we analyzed the role of fibroblast growth factor (FGF) signaling, in particular the function of two Drosophila FGF ligands, Pyramus and Thisbe, during mesoderm migration. We determined that FGF signaling through both ligands controls movements in the radial direction. Thisbe is required for the initial collapse of the mesoderm onto the ectoderm, whereas both Pyramus and Thisbe are required for monolayer formation. In addition, we uncovered that the GTPase Rap1 regulates radial movement of cells and localization of the beta-integrin subunit, Myospheroid, which is also required for monolayer formation. Our analyses suggest that distinct signals influence particular movements, as we found that FGF signaling is involved in controlling collapse and monolayer formation but not dorsal movement, whereas integrins are required to support monolayer formation only and not earlier movements. Our work demonstrates that complex cell migration is not necessarily a fluid process, but suggests instead that different types of movements are directed by distinct inputs in a stepwise manner.

Quantitative imaging of the Dorsal nuclear gradient reveals limitations to threshold-dependent patterning in Drosophila.

The NF-kappaB-related transcription factor, Dorsal, forms a nuclear concentration gradient in the early Drosophila embryo, patterning the dorsal-ventral (DV) axis to specify mesoderm, neurogenic ectoderm, and dorsal ectoderm cell fates. The concentration of nuclear Dorsal is thought to determine these patterning events; however, the levels of nuclear Dorsal have not been quantified previously. Furthermore, existing models of Dorsal-dependent germ layer specification and patterning consider steady-state levels of Dorsal relative to target gene expression patterns, yet both Dorsal gradient formation and gene expression are dynamic. We devised a quantitative imaging method to measure the Dorsal nuclear gradient while simultaneously examining Dorsal target gene expression along the DV axis. Unlike observations from other insects such as Tribolium, we find the Dorsal gradient maintains a constant bell-shaped distribution during embryogenesis. We also find that some classical Dorsal target genes are located outside the region of graded Dorsal nuclear localization, raising the question of whether these genes are direct Dorsal targets. Additionally, we show that Dorsal levels change in time during embryogenesis such that a steady state is not reached. These results suggest that the multiple gene expression outputs observed along the DV axis do not simply reflect a steady-state Dorsal nuclear gradient. Instead, we propose that the Dorsal gradient supplies positional information throughout nuclear cycles 10-14, providing additional evidence for the idea that compensatory combinatorial interactions between Dorsal and other factors effect differential gene expression along the DV axis.

Experimental and Analytical Framework for "Mix-and-Read" Assays Based on Split Luciferase.

The use of immunodetection assays including the widely used enzyme-linked immunosorbent assay (ELISA) in applications such as point-of-care detection is often limited by the need for protein immobilization and multiple binding and washing steps. Here, we describe an experimental and analytical framework for the development of simple and modular "mix-and-read" enzymatic complementation assays based on split luciferase that enable sensitive detection and quantification of analytes in solution. In this assay, two engineered protein binders targeting nonoverlapping epitopes on the target analyte were each fused to nonactive fragments of luciferase to create biosensor probes. Binding proteins to two model targets, lysozyme and Sso6904, were isolated from a combinatorial library of Sso7d mutants using yeast surface display. In the presence of the analyte, probes were brought into close proximity, reconstituting enzymatic activity of luciferase and enabling detection of low picomolar concentrations of the analyte by chemiluminescence. Subsequently, we constructed an equilibrium binding model that relates binding affinities of the binding proteins for the target, assay parameters such as the concentrations of probes used, and assay performance (limit of detection and concentration range over which the target can be quantified). Overall, our experimental and analytical framework provides the foundation for the development of split luciferase assays for detection and quantification of various targets.

Quantitative models of developmental pattern formation.

Pattern formation in developing organisms can be regulated at a variety of levels, from gene sequence to anatomy. At this level of complexity, mechanistic models of development become essential for integrating data, guiding future experiments, and predicting the effects of genetic and physical perturbations. However, the formulation and analysis of quantitative models of development are limited by high levels of uncertainty in experimental measurements, a large number of both known and unknown system components, and the multiscale nature of development. At the same time, an expanding arsenal of experimental tools can constrain models and directly test their predictions, making the modeling efforts not only necessary, but feasible. Using a number of problems in fruit fly development, we discuss how models can be used to test the feasibility of proposed patterning mechanisms and characterize their systems-level properties.

Quantifying the Gurken morphogen gradient in Drosophila oogenesis.

Quantitative information about the distribution of morphogens is crucial for understanding their effects on cell-fate determination, yet it is difficult to obtain through direct measurements. We have developed a parameter estimation approach for quantifying the spatial distribution of Gurken, a TGFalpha-like EGFR ligand that acts as a morphogen in Drosophila oogenesis. Modeling of Gurken/EGFR system shows that the shape of the Gurken gradient is controlled by a single dimensionless parameter, the Thiele modulus, which reflects the relative importance of ligand diffusion and degradation. By combining the model with genetic alterations of EGFR levels, we have estimated the value of the Thiele modulus in the wild-type egg chamber. This provides a direct characterization of the shape of the Gurken gradient and demonstrates how parameter estimation techniques can be used to quantify morphogen gradients in development.

Argos inhibits epidermal growth factor receptor signalling by ligand sequestration.

The epidermal growth factor receptor (EGFR) has critical functions in development and in many human cancers. During development, the spatial extent of EGFR signalling is regulated by feedback loops comprising both well-understood activators and less well-characterized inhibitors. In Drosophila melanogaster the secreted protein Argos functions as the only known extracellular inhibitor of EGFR, with clearly identified roles in multiple stages of development. Argos is only expressed when the Drosophila EGFR (DER) is activated at high levels, and downregulates further DER signalling. Although there is ample genetic evidence that Argos inhibits DER activation, the biochemical mechanism has not been established. Here we show that Argos inhibits DER signalling without interacting directly with the receptor, but instead by sequestering the DER-activating ligand Spitz. Argos binds tightly to the EGF motif of Spitz and forms a 1:1 (Spitz:Argos) complex that does not bind DER in vitro or at the cell surface. Our results provide an insight into the mechanism of Argos function, and suggest new strategies for EGFR inhibitor design.

Formation, interpretation, and regulation of the Drosophila Dorsal/NF-κB gradient.

The morphogen gradient of the transcription factor Dorsal in the early Drosophila embryo has become one of the most widely studied tissue patterning systems. Dorsal is a Drosophila homolog of mammalian NF-κB and patterns the dorsal-ventral axis of the blastoderm embryo into several tissue types by spatially regulating upwards of 100 zygotic genes. Recent studies using fluorescence microscopy and live imaging have quantified the Dorsal gradient and its target genes, which has paved the way for mechanistic modeling of the gradient. In this review, we describe the mechanisms behind the initiation of the Dorsal gradient and its regulation of target genes. The main focus of the review is a discussion of quantitative and computational studies of the Dl gradient system, including regulation of the Dl gradient. We conclude with a discussion of potential future directions.

Tunable self-cleaving ribozymes for modulating gene expression in eukaryotic systems.

Advancements in the field of synthetic biology have been possible due to the development of genetic tools that are able to regulate gene expression. However, the current toolbox of gene regulatory tools for eukaryotic systems have been outpaced by those developed for simple, single-celled systems. Here, we engineered a set of gene regulatory tools by combining self-cleaving ribozymes with various upstream competing sequences that were designed to disrupt ribozyme self-cleavage. As a proof-of-concept, we were able to modulate GFP expression in mammalian cells, and then showed the feasibility of these tools in Drosophila embryos. For each system, the fold-reduction of gene expression was influenced by the location of the self-cleaving ribozyme/upstream competing sequence (i.e. 5' vs. 3' untranslated region) and the competing sequence used. Together, this work provides a set of genetic tools that can be used to tune gene expression across various eukaryotic systems.

The engineering principles of combining a transcriptional incoherent feedforward loop with negative feedback.

BACKGROUND: Regulation of gene expression is of paramount importance in all living systems. In the past two decades, it has been discovered that certain motifs, such as the feedforward motif, are overrepresented in gene regulatory circuits. Feedforward loops are also ubiquitous in process control engineering, and are nearly always structured so that one branch has the opposite effect of the other, which is a structure known as an "incoherent" feedforward loop in biology. In engineered systems, feedforward control loops are subject to several engineering constraints, including that (1) they are finely-tuned so that the system returns to the original steady state after a disturbance occurs (perfect adaptation), (2) they are typically only implemented in the combination with negative feedback, and (3) they can greatly improve the stability and dynamical characteristics of the conjoined negative feedback loop. On the other hand, in biology, incoherent feedforward loops can serve many purposes, one of which may be perfect adaptation. It is an open question as to whether those that achieve perfect adaptation are subject to the above engineering principles. RESULTS: We analyzed an incoherent feedforward gene regulatory motif from the standpoint of the above engineering principles. In particular, we showed that an incoherent feedforward loop Type 1 (I1-FFL), from within a gene regulatory circuit, can be finely-tuned for perfect adaptation after a stimulus, and that the robustness of this behavior is increased by the presence of moderate negative feedback. In addition, we analyzed the advantages of adding a feedforward loop to a system that already operated under negative feedback, and found that the dynamical properties of the combined feedforward/feedback system were superior. CONCLUSIONS: Our analysis shows that many of the engineering principles used in engineering design of feedforward control are also applicable to feedforward loops in biological systems. We speculate that principles found in other domains of engineering may also be applicable to analogous structures in biology.