Mathematical Modeling of Photo- and Thermomorphogenesis in Plants.

Increased day lengths and warm conditions inversely affect plant growth by directly modulating nuclear phyB, ELF3, and COP1 levels. Quantitative measures of the hypocotyl length have been key to gaining a deeper understanding of this complex regulatory network, while similar quantitative data are the foundation for many studies in plant biology. Here, we explore the application of mathematical modeling, specifically ordinary differential equations (ODEs), to understand plant responses to these environmental cues. We provide a comprehensive guide to constructing, simulating, and fitting these models to data, using the law of mass action to study the evolution of molecular species. The fundamental principles of these models are introduced, highlighting their utility in deciphering complex plant physiological interactions and testing hypotheses. This brief introduction will not allow experimentalists without a mathematical background to run their own simulations overnight, but it will help them grasp modeling principles and communicate with more theory-inclined colleagues.

Feedback control of organ size precision is mediated by BMP2-regulated apoptosis in the Drosophila eye.

Biological processes are intrinsically noisy, and yet, the result of development-like the species-specific size and shape of organs-is usually remarkably precise. This precision suggests the existence of mechanisms of feedback control that ensure that deviations from a target size are minimized. Still, we have very limited understanding of how these mechanisms operate. Here, we investigate the problem of organ size precision using the Drosophila eye. The size of the adult eye depends on the rates at which eye progenitor cells grow and differentiate. We first find that the progenitor net growth rate results from the balance between their proliferation and apoptosis, with this latter contributing to determining both final eye size and its variability. In turn, apoptosis of progenitor cells is hampered by Dpp, a BMP2/4 signaling molecule transiently produced by early differentiating retinal cells. Our genetic and computational experiments show how the status of retinal differentiation is communicated to progenitors through the differentiation-dependent production of Dpp, which, by adjusting the rate of apoptosis, exerts a feedback control over the net growth of progenitors to reduce final eye size variability.

Mathematical models of nitrogen-fixing cell patterns in filamentous cyanobacteria.

The Anabaena genus is a model organism of filamentous cyanobacteria whose vegetative cells can differentiate under nitrogen-limited conditions into a type of cell called a heterocyst. These heterocysts lose the possibility to divide and are necessary for the filament because they can fix and share environmental nitrogen. In order to distribute the nitrogen efficiently, heterocysts are arranged to form a quasi-regular pattern whose features are maintained as the filament grows. Recent efforts have allowed advances in the understanding of the interactions and genetic mechanisms underlying this dynamic pattern. Here, we present a systematic review of the existing theoretical models of nitrogen-fixing cell differentiation in filamentous cyanobacteria. These filaments constitute one of the simplest forms of multicellular organization, and this allows for several modeling scales of this emergent pattern. The system has been approached at three different levels. From bigger to smaller scale, the system has been considered as follows: at the population level, by defining a mean-field simplified system to study the ratio of heterocysts and vegetative cells; at the filament level, with a continuous simplification as a reaction-diffusion system; and at the cellular level, by studying the genetic regulation that produces the patterning for each cell. In this review, we compare these different approaches noting both the virtues and shortcomings of each one of them.

Terminal heterocyst differentiation in the Anabaena patA mutant as a result of post-transcriptional modifications and molecular leakage.

The Anabaena genus is a model organism of filamentous cyanobacteria whose vegetative cells can differentiate under nitrogen-limited conditions into a type of cell called heterocyst. These heterocysts lose the possibility to divide and are necessary for the colony because they can fix and share environmental nitrogen. In order to distribute the nitrogen efficiently, heterocysts are arranged to form a quasi-regular pattern whose features are maintained as the filament grows. Recent efforts have allowed advances in the understanding of the interactions and genetic mechanisms underlying this dynamic pattern. However, the main role of the patA and hetF genes are yet to be clarified; in particular, the patA mutant forms heterocysts almost exclusively in the terminal cells of the filament. In this work, we investigate the function of these genes and provide a theoretical model that explains how they interact within the broader genetic network, reproducing their knock-out phenotypes in several genetic backgrounds, including a nearly uniform concentration of HetR along the filament for the patA mutant. Our results suggest a role of hetF and patA in a post-transcriptional modification of HetR which is essential for its regulatory function. In addition, the existence of molecular leakage out of the filament in its boundary cells is enough to explain the preferential appearance of terminal heterocysts, without any need for a distinct regulatory pathway.

COP1 dynamics integrate conflicting seasonal light and thermal cues in the control of Arabidopsis elongation.

As the summer approaches, plants experience enhanced light inputs and warm temperatures, two environmental cues with an opposite morphogenic impact. Key components of this response are PHYTOCHROME B (phyB), EARLY FLOWERING 3 (ELF3), and CONSTITUTIVE PHOTOMORPHOGENIC 1 (COP1). Here, we used single and double mutant/overexpression lines to fit a mathematical model incorporating known interactions of these regulators. The fitted model recapitulates thermal growth of all lines used and correctly predicts thermal behavior of others not used in the fit. While thermal COP1 function is accepted to be independent of diurnal timing, our model shows that it acts at temperature signaling only during daytime. Defective response of cop1-4 mutants is epistatic to phyB-9 and elf3-8, indicating that COP1 activity is essential to transduce phyB and ELF3 thermosensory function. Our thermal model provides a unique toolbox to identify best allelic combinations enhancing climate change resilience of crops adapted to different latitudes.

Multiple Layered Control of the Conjugation Process of the Bacillus subtilis Plasmid pLS20.

Bacterial conjugation is the main horizontal gene transfer route responsible for the spread of antibiotic resistance, virulence and toxin genes. During conjugation, DNA is transferred from a donor to a recipient cell via a sophisticated channel connecting the two cells. Conjugation not only affects many different aspects of the plasmid and the host, ranging from the properties of the membrane and the cell surface of the donor, to other developmental processes such as competence, it probably also poses a burden on the donor cell due to the expression of the large number of genes involved in the conjugation process. Therefore, expression of the conjugation genes must be strictly controlled. Over the past decade, the regulation of the conjugation genes present on the conjugative Bacillus subtilis plasmid pLS20 has been studied using a variety of methods including genetic, biochemical, biophysical and structural approaches. This review focuses on the interplay between RcopLS20, RappLS20 and Phr*pLS20, the proteins that control the activity of the main conjugation promoter P c located upstream of the conjugation operon. Proper expression of the conjugation genes requires the following two fundamental elements. First, conjugation is repressed by default and an intercellular quorum-signaling system is used to sense conditions favorable for conjugation. Second, different layers of regulation act together to repress the P c promoter in a strict manner but allowing rapid activation. During conjugation, ssDNA is exported from the cell by a membrane-embedded DNA translocation machine. Another membrane-embedded DNA translocation machine imports ssDNA in competent cells. Evidences are reviewed indicating that conjugation and competence are probably mutually exclusive processes. Some of the questions that remain unanswered are discussed.

The turning point and end of an expanding epidemic cannot be precisely forecast.

Epidemic spread is characterized by exponentially growing dynamics, which are intrinsically unpredictable. The time at which the growth in the number of infected individuals halts and starts decreasing cannot be calculated with certainty before the turning point is actually attained; neither can the end of the epidemic after the turning point. A susceptible-infected-removed (SIR) model with confinement (SCIR) illustrates how lockdown measures inhibit infection spread only above a threshold that we calculate. The existence of that threshold has major effects in predictability: A Bayesian fit to the COVID-19 pandemic in Spain shows that a slowdown in the number of newly infected individuals during the expansion phase allows one to infer neither the precise position of the maximum nor whether the measures taken will bring the propagation to the inhibition regime. There is a short horizon for reliable prediction, followed by a dispersion of the possible trajectories that grows extremely fast. The impossibility to predict in the midterm is not due to wrong or incomplete data, since it persists in error-free, synthetically produced datasets and does not necessarily improve by using larger datasets. Our study warns against precise forecasts of the evolution of epidemics based on mean-field, effective, or phenomenological models and supports that only probabilities of different outcomes can be confidently given.

Novel regulatory mechanism of establishment genes of conjugative plasmids.

The principal route for dissemination of antibiotic resistance genes is conjugation by which a conjugative DNA element is transferred from a donor to a recipient cell. Conjugative elements contain genes that are important for their establishment in the new host, for instance by counteracting the host defense mechanisms acting against incoming foreign DNA. Little is known about these establishment genes and how they are regulated. Here, we deciphered the regulation mechanism of possible establishment genes of plasmid p576 from the Gram-positive bacterium Bacillus pumilus. Unlike the ssDNA promoters described for some conjugative plasmids, the four promoters of these p576 genes are repressed by a repressor protein, which we named Reg576. Reg576 also regulates its own expression. After transfer of the DNA, these genes are de-repressed for a period of time until sufficient Reg576 is synthesized to repress the promoters again. Complementary in vivo and in vitro analyses showed that different operator configurations in the promoter regions of these genes lead to different responses to Reg576. Each operator is bound with extreme cooperativity by two Reg576-dimers. The X-ray structure revealed that Reg576 has a Ribbon-Helix-Helix core and provided important insights into the high cooperativity of DNA recognition.

Formation and maintenance of nitrogen-fixing cell patterns in filamentous cyanobacteria.

Cyanobacteria forming one-dimensional filaments are paradigmatic model organisms of the transition between unicellular and multicellular living forms. Under nitrogen-limiting conditions, in filaments of the genus Anabaena, some cells differentiate into heterocysts, which lose the possibility to divide but are able to fix environmental nitrogen for the colony. These heterocysts form a quasiregular pattern in the filament, representing a prototype of patterning and morphogenesis in prokaryotes. Recent years have seen advances in the identification of the molecular mechanism regulating this pattern. We use these data to build a theory on heterocyst pattern formation, for which both genetic regulation and the effects of cell division and filament growth are key components. The theory is based on the interplay of three generic mechanisms: local autoactivation, early long-range inhibition, and late long-range inhibition. These mechanisms can be identified with the dynamics of hetR, patS, and hetN expression. Our theory reproduces quantitatively the experimental dynamics of pattern formation and maintenance for wild type and mutants. We find that hetN alone is not enough to play the role as the late inhibitory mechanism: a second mechanism, hypothetically the products of nitrogen fixation supplied by heterocysts, must also play a role in late long-range inhibition. The preponderance of even intervals between heterocysts arises naturally as a result of the interplay between the timescales of genetic regulation and cell division. We also find that a purely stochastic initiation of the pattern, without a two-stage process, is enough to reproduce experimental observations.

Cux1 Enables Interhemispheric Connections of Layer II/III Neurons by Regulating Kv1-Dependent Firing.

Neuronal subtype-specific transcription factors (TFs) instruct key features of neuronal function and connectivity. Activity-dependent mechanisms also contribute to wiring and circuit assembly, but whether and how they relate to TF-directed neuronal differentiation is poorly investigated. Here we demonstrate that the TF Cux1 controls the formation of the layer II/III corpus callosum (CC) projections through the developmental transcriptional regulation of Kv1 voltage-dependent potassium channels and the resulting postnatal switch to a Kv1-dependent firing mode. Loss of Cux1 function led to a decrease in the expression of Kv1 transcripts, aberrant firing responses, and selective loss of CC contralateral innervation. Firing and innervation were rescued by re-expression of Kv1 or postnatal reactivation of Cux1. Knocking down Kv1 mimicked Cux1-mediated CC axonal loss. These findings reveal that activity-dependent processes are central bona fide components of neuronal TF-differentiation programs and establish the importance of intrinsic firing modes in circuit assembly within the neocortex.

Synchronization dynamics in the presence of coupling delays and phase shifts.

In systems of coupled oscillators, the effects of complex signaling can be captured by time delays and phase shifts. Here, we show how time delays and phase shifts lead to different oscillator dynamics and how synchronization rates can be regulated by substituting time delays by phase shifts at a constant collective frequency. For spatially extended systems with time delays, we show that the fastest synchronization can occur for intermediate wavelengths, giving rise to novel synchronization scenarios.

Wnt-regulated dynamics of positional information in zebrafish somitogenesis.

How signaling gradients supply positional information in a field of moving cells is an unsolved question in patterning and morphogenesis. Here, we ask how a Wnt signaling gradient regulates the dynamics of a wavefront of cellular change in a flow of cells during somitogenesis. Using time-controlled perturbations of Wnt signaling in the zebrafish embryo, we changed segment length without altering the rate of somite formation or embryonic elongation. This result implies specific Wnt regulation of the wavefront velocity. The observed Wnt signaling gradient dynamics and timing of downstream events support a model for wavefront regulation in which cell flow plays a dominant role in transporting positional information.

Lateral inhibition and neurogenesis: novel aspects in motion.

Neuronal production in metazoans is tightly controlled by Delta/Notch-dependent signals regulating lateral inhibition. It is currently thought that lateral inhibition takes place in clusters of precursors with equal capacity to trigger and receive Notch-dependent inhibitory signals. However, this view neglects crucial dynamical aspects of the process. In this review, we discuss two of these dynamic factors, whose alterations yield dysfunctions in neurogenesis. First, precursors show variable neurogenic capacity as they go through the cell cycle. Second, differentiating precursors are in direct contact with non-neurogenic cells at the wavefront of expanding neurogenic domains. We discuss the mechanisms adopted by Metazoa to prevent these dysfunctions in the lateral inhibitory process, which include cell cycle synchronization occurring in the invertebrate neural epithelium and during primary neurogenesis in anamniotes, interkinetic nuclear movement in the vertebrate neuroepithelium and generalized Delta expression ahead of the neurogenic wavefront. The emerging concept is that lateral inhibition during neurogenesis occurs in dynamic clusters of precursors and requires specific mechanisms to avoid distortions resulting from the interaction between neurogenic and non-neurogenic precursors. The advance in visualizing Notch dynamics with real-time imaging at cellular and subcellular levels will notably contribute to our understanding of these novel "aspects of motion" in neurogenesis.

Collective modes of coupled phase oscillators with delayed coupling.

We study the effects of delayed coupling on timing and pattern formation in spatially extended systems of dynamic oscillators. Starting from a discrete lattice of coupled oscillators, we derive a generic continuum theory for collective modes of long wavelengths. We use this approach to study spatial phase profiles of cellular oscillators in the segmentation clock, a dynamic patterning system of vertebrate embryos. Collective wave patterns result from the interplay of coupling delays and moving boundary conditions. We show that the phase profiles of collective modes depend on coupling delays.

Topology and dynamics of the zebrafish segmentation clock core circuit.

During vertebrate embryogenesis, the rhythmic and sequential segmentation of the body axis is regulated by an oscillating genetic network termed the segmentation clock. We describe a new dynamic model for the core pace-making circuit of the zebrafish segmentation clock based on a systematic biochemical investigation of the network's topology and precise measurements of somitogenesis dynamics in novel genetic mutants. We show that the core pace-making circuit consists of two distinct negative feedback loops, one with Her1 homodimers and the other with Her7:Hes6 heterodimers, operating in parallel. To explain the observed single and double mutant phenotypes of her1, her7, and hes6 mutant embryos in our dynamic model, we postulate that the availability and effective stability of the dimers with DNA binding activity is controlled in a "dimer cloud" that contains all possible dimeric combinations between the three factors. This feature of our model predicts that Hes6 protein levels should oscillate despite constant hes6 mRNA production, which we confirm experimentally using novel Hes6 antibodies. The control of the circuit's dynamics by a population of dimers with and without DNA binding activity is a new principle for the segmentation clock and may be relevant to other biological clocks and transcriptional regulatory networks.

Regulation of neuronal differentiation at the neurogenic wavefront.

Signaling mediated by the Delta/Notch system controls the process of lateral inhibition, known to regulate neurogenesis in metazoans. Lateral inhibition takes place in equivalence groups formed by cells having equal capacity to differentiate, and it results in the singling out of precursors, which subsequently become neurons. During normal development, areas of active neurogenesis spread through non-neurogenic regions in response to specific morphogens, giving rise to neurogenic wavefronts. Close contact of these wavefronts with non-neurogenic cells is expected to affect lateral inhibition. Therefore, a mechanism should exist in these regions to prevent disturbances of the lateral inhibitory process. Focusing on the developing chick retina, we show that Dll1 is widely expressed by non-neurogenic precursors located at the periphery of this tissue, a region lacking Notch1, lFng, and differentiation-related gene expression. We investigated the role of this Dll1 expression through mathematical modeling. Our analysis predicts that the absence of Dll1 ahead of the neurogenic wavefront results in reduced robustness of the lateral inhibition process, often linked to enhanced neurogenesis and the presence of morphological alterations of the wavefront itself. These predictions are consistent with previous observations in the retina of mice in which Dll1 is conditionally mutated. The predictive capacity of our mathematical model was confirmed further by mimicking published results on the perturbation of morphogenetic furrow progression in the eye imaginal disc of Drosophila. Altogether, we propose that Notch-independent Delta expression ahead of the neurogenic wavefront is required to avoid perturbations in lateral inhibition and wavefront progression, thus optimizing the neurogenic process.

Optimal cellular mobility for synchronization arising from the gradual recovery of intercellular interactions.

Cell movement and intercellular signaling occur simultaneously during the development of tissues, but little is known about how movement affects signaling. Previous theoretical studies have shown that faster moving cells favor synchronization across a population of locally coupled genetic oscillators. An important assumption in these studies is that cells can immediately interact with their new neighbors after arriving at a new location. However, intercellular interactions in cellular systems may need some time to become fully established. How movement affects synchronization in this situation has not been examined. Here, we develop a coupled phase oscillator model in which we consider cell movement and the gradual recovery of intercellular coupling experienced by a cell after movement, characterized by a moving rate and a coupling recovery rate, respectively. We find (1) an optimal moving rate for synchronization and (2) a critical moving rate above which achieving synchronization is not possible. These results indicate that the extent to which movement enhances synchrony is limited by a gradual recovery of coupling. These findings suggest that the ratio of time scales of movement and signaling recovery is critical for information transfer between moving cells.

Computational approaches to developmental patterning.

Computational approaches are breaking new ground in understanding how embryos form. Here, we discuss recent studies that couple precise measurements in the embryo with appropriately matched modeling and computational methods to investigate classic embryonic patterning strategies. We include signaling gradients, activator-inhibitor systems, and coupled oscillators, as well as emerging paradigms such as tissue deformation. Parallel progress in theory and experiment will play an increasingly central role in deciphering developmental patterning.

Patterning embryos with oscillations: structure, function and dynamics of the vertebrate segmentation clock.

The segmentation clock is an oscillating genetic network thought to govern the rhythmic and sequential subdivision of the elongating body axis of the vertebrate embryo into somites: the precursors of the segmented vertebral column. Understanding how the rhythmic signal arises, how it achieves precision and how it patterns the embryo remain challenging issues. Recent work has provided evidence of how the period of the segmentation clock is regulated and how this affects the anatomy of the embryo. The ongoing development of real-time clock reporters and mathematical models promise novel insight into the dynamic behavior of the clock.