Noise reduction in the intracellular pom1p gradient by a dynamic clustering mechanism

Chemical gradients can generate pattern formation in biological systems. In the fission yeast Schizosaccharomyces pombe, a cortical gradient of pom1p (a DYRK-type protein kinase) functions to position sites of cytokinesis and cell polarity and to control cell length. Here, using quantitative imaging, fluorescence correlation spectroscopy, and mathematical modeling, we study how its gradient distribution is formed. Pom1p gradients exhibit large cell-to-cell variability, as well as dynamic fluctuations in each individual gradient. Our data lead to a two-state model for gradient formation in which pom1p molecules associate with the plasma membrane at cell tips and then diffuse on the membrane while aggregating into and fragmenting from clusters, before disassociating from the membrane. In contrast to a classical one-component gradient, this two-state gradient buffers against cell-to-cell variations in protein concentration. This buffering mechanism, together with time averaging to reduce intrinsic noise, allows the pom1p gradient to specify positional information in a robust manner.     

Shaping a Morphogen Gradient for Positional Precision

Morphogen gradients, which provide positional information to cells in a developing tissue, could in principle adopt any nonuniform profile. To our knowledge, how the profile of a morphogen gradient affects positional precision has not been well studied experimentally. Here, we compare the positional precision provided by the Drosophila morphogenetic protein Bicoid (Bcd) in wild-type (wt) embryos with embryos lacking an interacting cofactor. The Bcd gradient in the latter case exhibits decreased positional precision around mid-embryo compared with its wt counterpart. The domain boundary of Hunchback (Hb), a target activated by Bcd, becomes more variable in mutant embryos. By considering embryo-to-embryo, internal, and measurement fluctuations, we dissect mathematically the relevant sources of fluctuations that contribute to the error in positional information. Using this approach, we show that the defect in Hb boundary positioning in mutant embryos is directly reflective of an altered Bcd gradient profile with increasing flatness toward mid-embryo. Furthermore, we find that noise in the Bcd input signal is dominated by internal fluctuations but, due to time and spatial averaging, the spatial precision of the Hb boundary is primarily affected by embryo-to-embryo variations. Our results demonstrate that the positional information provided by the wt Bcd gradient profile is highly precise and necessary for patterning precision.     

Precision of sensing cell length via concentration gradients

Unicellular organisms are typically found to have a characteristic cell size. To achieve a homeostatic distribution of cell sizes over many generations requires that cell length is actively sensed and regulated. However, the mechanisms by which cell size is controlled remain poorly understood. Recent experiments in fission yeast have shown that cell length is controlled in part by polar gradients of the protein Pom1 together with localized measurement of concentration at midcell. Dilution as the cell grows leads to a reduction in the midcell protein concentration, which lifts a block on mitosis. Here we analyze the precision of this mechanism for length sensing in the presence of inevitable intrinsic noise in the processes leading to formation and measurement of this gradient. We find that the use of concentration gradients allows for more robust length sensing than a comparable spatially uniform system, and allows for reliable length determination even if the average protein concentration throughout the cell remains constant as the cell grows. Optimal values for the gradient decay length and receptor dissociation constant emerge from maximizing sensitivity while minimizing the impact of density fluctuations.     

Sociality of an agent during morphogenetic canalization: Asynchronous updating with potential resonance

Canalization is a typical self-organization process leading to complementarity between parts and the whole. In the field of developmental biology, concerns about morphogenesis canalization are often framed as the French flag problem, questioning how each cell knows its own position in the whole system. Although chemical gradients have been suggested to provide positional information, there is no direct evidence that gradients are used to gain positional information. The chemical gradient hypothesis is based on the assumption that agents (e.g., cells) in a domain that locally interact with each other can generate a chemical gradient thanks to a global reference point. Instead of a chemical gradient, we here propose a model based on agents that are equipped with sociality that is based not on a global reference but rather on the ability to sense other neighboring agents, or potential resonance. The interaction among the agents with sociality, developed from undifferentiated types or tokens, is implemented using asynchronous updating automata equipped with potential resonance. We show that these automata can generate a French flag pattern that is very robust against perturbations without positional information by comparing automata with synchronous updating and asynchronous automata without potential resonance.     

Implications of diffusion and time-varying morphogen gradients for the dynamic positioning and precision of bistable gene expression boundaries

The earliest models for how morphogen gradients guide embryonic patterning failed to account for experimental observations of temporal refinement in gene expression domains. Following theoretical and experimental work in this area, dynamic positional information has emerged as a conceptual framework to discuss how cells process spatiotemporal inputs into downstream patterns. Here, we show that diffusion determines the mathematical means by which bistable gene expression boundaries shift over time, and therefore how cells interpret positional information conferred from morphogen concentration. First, we introduce a metric for assessing reproducibility in boundary placement or precision in systems where gene products do not diffuse, but where morphogen concentrations are permitted to change in time. We show that the dynamics of the gradient affect the sensitivity of the final pattern to variation in initial conditions, with slower gradients reducing the sensitivity. Second, we allow gene products to diffuse and consider gene expression boundaries as propagating wavefronts with velocity modulated by local morphogen concentration. We harness this perspective to approximate a PDE model as an ODE that captures the position of the boundary in time, and demonstrate the approach with a preexisting model for Hunchback patterning in fruit fly embryos. We then propose a design that employs antiparallel morphogen gradients to achieve accurate boundary placement that is robust to scaling. Throughout our work we draw attention to tradeoffs among initial conditions, boundary positioning, and the relative timescales of network and gradient evolution. We conclude by suggesting that mathematical theory should serve to clarify not just our quantitative, but also our intuitive understanding of patterning processes.     

Different designs of kinase-phosphatase interactions and phosphatase sequestration shapes the robustness and signal flow in the MAPK cascade

Background

In developmental biology, there has been a recent focus on the robustness of morphogen gradients as possible providers of positional information. It was shown that functional morphogen gradients present strong biophysical constraints and lack of robustness to noise. Here we explore how the details of the mechanism which underlies the generation of a morphogen gradient can influence those properties.

Results

We contrast three gradient-generating mechanisms, (i) a source-decay mechanism; and (ii) a unidirectional transport mechanism; and (iii) a so-called reflux-loop mechanism. Focusing on the dynamics of the phytohormone auxin in the root, we show that only the reflux-loop mechanism can generate a gradient that would be adequate to supply functional positional information for the Arabidopsis root, for biophysically reasonable kinetic parameters.

Conclusions

We argue that traits that differ in spatial and temporal time-scales can impose complex selective pressures on the mechanism of morphogen gradient formation used for the development of the particular organism.     

Morphogen gradient formation and vesicular trafficking

Morphogens are secreted signaling molecules which form spatial concentration gradients while moving away from a restricted source of production. A simple model of gradient formation postulates that the morphogens dilute as they diffuse between cells. In this review we discuss recent data supporting the idea that movement of the morphogen could also occur via vesicular trafficking through the cells. We explore the implications of these results for the control of gradient formation and the determination of the gradient slope which ultimately encodes the coordinates of positional information.     

Establishing positional information through gradient dynamics: A lesson from the Hedgehog signaling pathway

A long standing question in developmental biology is how morphogen gradients establish positional information during development. Although the existence of gradients and their role in developmental patterning is no longer in doubt, the ability of cells to respond to different morphogen concentrations has been controversial. In the Drosophila wing disc, Hedgehog (Hh) forms a concentration gradient along the anterior-posterior axis and establishes at least three different gene expression patterns. In a recent study, we challenged the prevailing idea that Hh establishes positional information in a dose-dependent manner and proposed a model in which dynamics of the gradient, resulting from the Hh gene network architecture, determines pattern formation in the wing disc. In this Extra View, we discuss further the methodology used in this study, highlight differences between this and other models of developmental patterning, and also present some questions that remain to be answered in this system.Key words: Hedgehog, developmental patterning, morphogen, dynamics, mathematical modeling     

Determination of sensory bristles and pattern formation in Drosophila: I. A model

A model is presented to explain the formation of the pattern of sensory bristles in Drosophila. The model is based on the idea that precision and reproducibility in pattern formation can be achieved by averaging out of many moderately accurate responses to positional cues. According to this model, the determination of bristles in imaginal discs occurs in two steps. First, large numbers of imaginal cells synthesize a freely diffusible inducer, the chaetogen. Second, cells in which the concentration of this chaetogen reaches a threshold are induced to differentiate into a bristle apparatus. Induced cells prevent neighboring cells from being induced too. The synthesis of chaetogen is supposed to be a probabilistic response of cells to positional cues, so that a cell located in one region of the disc is more likely to have its chaetogen gene turned on than a cell located in another region. Various probability distributions are shown to generate the various bristle patterns observed in the adult: precisely located bristles (e.g., thoracic macrochaetes), evenly spaced bristles (e.g., tergal microchaetes), and rows of bristles (e.g., thoracic microchaetes). In the particular case of the precisely located bristles, we show that (i) the distribution of chaetogen concentration in the tissue presents a unique maximum even when a large number of contiguous cells are all engaged in the synthesis of chaetogen; (ii) the position of the maximum is largely unaffected by statistical fluctuations in the decision of each cell to synthesize or not to synthesize the chaetogen; (iii) different maxima can be reproducibly generated even when the corresponding populations of chaetogen-producing cells overlap.     

A theory of measurement error and its implications for spatial and temporal gradient sensing during chemotaxis

In order that cells respond to environmental cues, they must be able to measure ambient ligand concentration. Concentrations fluctuate, however, because of thermal noise, and one can readily show that estimates based on concentration values at a particular moment will be subject to substantial error. Cells are therefore expected to average their estimates over some limited time period. In this paper we assume that a cell uses fractional receptor occupancy as a measure of ambient ligand concentration and develop general expressions for the error a cell makes because the length of the averaging period is necessarily limited. Our analysis is general, relieving many of the assumptions underlying the seminal work of Berg and Purcell. The most important formal difference is our inclusion of occupancy-dependent dissociation--a phenomenon that has been well-documented for many systems. In addition, our formulation permits signal averaging to begin before chemical equilibrium has been established and it allows binding kinetics to be nonlinear (i.e., biomolecular rather than pseudo-first-order). The results are applied to spatial and temporal concentration gradients. In particular we estimate the minimum averaging times required for cells to detect such gradients under typical in vitro conditions. These estimates involve assigning numerical values to receptor ligand rate constants. If the rate constants are at their maximum possible values (limited only by center of mass diffusion), then either temporal or spatial gradients can be detected in minutes or less. If, however, as suggested by experiments, the rate constants are several orders of magnitude below their diffusion-limited values, then under typical constant gradient conditions the time required to detect a spatial gradient is prohibitively long, whereas temporal gradients can still be detected in reasonable lengths of time. This result was obtained for large cells such as lymphocytes, as well as for the smaller, bacterial cells. The ratio of averaging times for the two mechanisms--amounting to several orders of magnitude--is well beyond what could be reconciled by limitations of the calculation, and strongly suggests heavy reliance on temporal sensing mechanisms under typical in vitro conditions with constant spatial gradients.     

A theory of measurement error and its implications for spatial and temporal gradient sensing during chemotaxis. II. The effects of non-equilibrated ligand binding

Cells generally chemotax along a direction in which their receptor occupancy gradient--whether spatial or temporal--is maximum. Occupancy differentials are, however, often so small as to be masked by thermal noise; i.e., by fluctuations inherent in the stochastic nature of ligand binding. Such fluctuations therefore impose a fundamental limit on the sensitivity of a cell's ability to detect a chemoattractant gradient. In order to pursue the implications of this limit, fluctuation theories have been developed. The theories assume that the signal is some function of the receptor occupancy gradient, allow an estimate of the standard deviation about the mean signal, and permit an evaluation of, among other things, the extent to which a receptor defect can impair an effective response. Previous theories have assumed an equilibrated ligand-receptor interaction. In this paper we introduce a generalized definition of a signal caused by a receptor occupancy gradient that allows us to develop a non-equilibrium theory of thermal noise. We show that previous formulations are a special case of the current development. More specifically, we find the following. Swimming cells subject to Brownian tumbling must generally average their signals over a very long time period to achieve a signal-to-noise ratio less than or equal to 1. Spatial gradient detection is possible with ligand-receptor equilibrium constants less than 10(3)M-1, but since such ligands are rare, theory predicts that tumbling cells will generally not detect gradients by measuring spatial occupancy differentials. These conclusions hold irrespective of whether chemical equilibrium is achieved. For crawling cells not subject to Brownian tumbling, a range of affinities exists in which spatial or temporal gradient detection is possible. In general a spatial mechanism is more efficient for low affinity ligands (dissociation times less than 0.3 s), whereas a temporal mechanism is more efficient for higher K. In this case the detection of gradients in slowly dissociating ligand will be facilitated if signal processing begins prior to chemical equilibration. An important new parameter is indicated by the theory. The definitions of a temporal gradient signal is based on estimating and comparing average occupancy over two time intervals displaced by a time t1. The theory predicts an optimal t1, of order milliseconds, that leads to the shortest minimum averaging time. For t1 values at and longer than the optimum, and for all averaging times exceeding some minimum, the cell will detect a temporal signal.(ABSTRACT TRUNCATED AT 400 WORDS)     

A stochastic model for leukocyte random motility and chemotaxis based on receptor binding fluctuations

Two central features of polymorphonuclear leukocyte chemosensory movement behavior demand fundamental theoretical understanding. In uniform concentrations of chemoattractant, these cells exhibit a persistent random walk, with a characteristic "persistence time" between significant changes in direction. In chemoattractant concentration gradients, they demonstrate a biased random walk, with an "orientation bias" characterizing the fraction of cells moving up the gradient. A coherent picture of cell movement responses to chemoattractant requires that both the persistence time and the orientation bias be explained within a unifying framework. In this paper, we offer the possibility that "noise" in the cellular signal perception/response mechanism can simultaneously account for these two key phenomena. In particular, we develop a stochastic mathematical model for cell locomotion based on kinetic fluctuations in chemoattractant/receptor binding. This model can simulate cell paths similar to those observed experimentally, under conditions of uniform chemoattractant concentrations as well as chemoattractant concentration gradients. Furthermore, this model can quantitatively predict both cell persistence time and dependence of orientation bias on gradient size. Thus, the concept of signal "noise" can quantitatively unify the major characteristics of leukocyte random motility and chemotaxis. The same level of noise large enough to account for the observed frequency of turning in uniform environments is simultaneously small enough to allow for the observed degree of directional bias in gradients.     

Suppression of positional errors in biological development

Cells in developing embryos behave according to their positions in the organism, and therefore seem to be receiving 'positional information'. A widespread view of the mechanism for this is that each cell responds locally to the concentration level of some extracellular chemical which is distributed in a spatial gradient. For molecules conveying and receiving the positional signal, concentrations are likely to be low enough that, per individual cell, only a few thousand molecules may be involved. Fluctuations to be expected in these numbers (Poisson distribution) could readily lead to errors up to a few percent of embryo length in the reading of position. This is an intolerable level of error for some developmental pattern-forming events. Embryos must have means of suppressing such errors. We maintain that this requires communication between cells, and illustrate this by using the reaction part of two well-known Turing-type reaction-diffusion models as the local gradient reader. We show that switching on diffusion in these models leads to adequate suppression of positional errors.     

Polarized signaling endosomes coordinate BDNF-induced chemotaxis of cerebellar precursors

During development, neural precursors migrate in response to positional cues such as growth factor gradients. However, the mechanisms that enable precursors to sense and respond to such gradients are poorly understood. Here we show that cerebellar granule cell precursors (GCPs) migrate along a gradient of brain-derived neurotrophic factor (BDNF), and we demonstrate that vesicle trafficking is critical for this chemotactic process. Activation of TrkB, the BDNF receptor, stimulates GCPs to secrete BDNF, thereby amplifying the ambient gradient. The BDNF gradient stimulates endocytosis of TrkB and associated signaling molecules, causing asymmetric accumulation of signaling endosomes at the subcellular location where BDNF concentration is maximal. Thus, regulated BDNF exocytosis and TrkB endocytosis enable precursors to polarize and migrate in a directed fashion along a shallow BDNF gradient.     

Smiling Gurken gradient: An expansion of the Gurken gradient

Morphogen gradients provide unique positional information within a tissue. Cells that are sensitive to the concentration of the morphogen integrate this signal and develop an appropriately distinct cell fate. A morphogen gradient is usually generated by a restricted source and shaped by the speed of diffusion and stability of the signaling molecule. In addition, the availability of receptor and Heparan Sulfate Proteoglycans (HSPGs) help to shape the gradient. We have shown that overexpression of Dally-like protein (Dlp) causes an expansion of Gurken distribution and a loss of cell fates which are specified by high levels of epidermal growth factor receptor (Egfr) signaling. In this article, we discuss how D-Cbl mediated Egfr endocytosis and the levels of Dlp affect the shape of the Gurken gradient.     

Requirements for Rapid Plasmid ColE1 Copy Number Adjustments: A Mathematical Model of Inhibition Modes and RNA Turnover Rates

The random distribution of ColE1 plasmids between the daughter cells at cell division introduces large copy number variations. Statistic variation associated with limited copy number in single cells also causes fluctuations to emerge spontaneously during the cell cycle. Efficient replication control out of steady state is therefore important to tame such stochastic effects of small numbers. In the present model, the dynamic features of copy number control are divided into two parts: first, how sharply the replication frequency per plasmid responds to changes in the concentration of the plasmid-coded inhibitor, RNA I, and second, how tightly RNA I and plasmid concentrations are coupled. Single (hyperbolic)- and multiple (exponential)-step inhibition mechanisms are compared out of steady state and it is shown how the response in replication frequency depends on the mode of inhibition. For both mechanisms, sensitivity of inhibition is “bought” at the expense of a rapid turnover of a replication preprimer, RNA II. Conventional, single-step, inhibition kinetics gives a sloppy replication control even at high RNA II turnover rates, whereas multiple-step inhibition has the potential of working with unlimited precision. When plasmid concentration changes rapidly, RNA I must be degraded rapidly to be “up to date” with the change. Adjustment to steady state is drastically impaired when the turnover rate constants of RNA I decrease below certain thresholds, but is basically unaffected for a corresponding increase. Several features of copy number control that are shown to be crucial for the understanding of ColE1-type plasmids still remain to be experimentally characterized. It is shown how steady-state properties reflect dynamics at the heart of regulation and therefore can be used to discriminate between fundamentally different copy number control mechanisms. The experimental tests of the predictions made require carefully planned assays, and some suggestions for suitable experiments arise naturally from the present work. It is also discussed how the presence of the Rom protein may affect dynamic qualities of copy number control.     

Smiling Gurken gradient-an expansion of the Gurken gradient

Morphogen gradients provide unique positional information within a tissue. Cells that are sensitive to the concentration of the morphogen integrate this signal and develop an appropriately distinct cell fate. A morphogen gradient is usually generated by a restricted source and shaped by the speed of diffusion and stability of the signaling molecule. In addition, the availability of receptor and Heparan Sulfate Proteoglycans (HSPGs) help to shape the gradient. We have shown that over-expression of Dally-like protein (Dlp) causes an expansion of Gurken distribution and a loss of cell fates which are specified by high levels of epidermal growth factor receptor (Egfr) signaling. In this article, we discuss how D-Cbl mediated Egfr endocytosis and the levels of Dlp affect the shape of the Gurken gradient.