Ligand trapping in epithelial layers and cell cultures

Predictive Spatiotemporal Manipulation of Signaling Perturbations Using Optogenetics

Recently developed optogenetic methods promise to revolutionize cell biology by allowing signaling perturbations to be controlled in space and time with light. However, a quantitative analysis of the relationship between a custom-defined illumination pattern and the resulting signaling perturbation is lacking. Here, we characterize the biophysical processes governing the localized recruitment of the Cryptochrome CRY2 to its membrane-anchored CIBN partner. We develop a quantitative framework and present simple procedures that enable predictive manipulation of protein distributions on the plasma membrane with a spatial resolution of 5 μm. We show that protein gradients of desired levels can be established in a few tens of seconds and then steadily maintained. These protein gradients can be entirely relocalized in a few minutes. We apply our approach to the control of the Cdc42 Rho GTPase activity. By inducing strong localized signaling perturbation, we are able to monitor the initiation of cell polarity and migration with a remarkable reproducibility despite cell-to-cell variability.

Extended narrow escape problem: boundary homogenization-based analysis

Diffusion of particles in confined domains with absorbing spots on the otherwise reflecting boundaries is ubiquitous in nature and technology. Because of nonuniform boundary conditions, the problem of finding the mean first passage time (MFPT) of the particle to one of the spots is extremely complicated. We show how the difficulties can be overcome by means of boundary homogenization when the domain is a circular disk whose boundary contains n nonoverlapping identical absorbing arcs, which may occupy an arbitrary fraction of the boundary. We find the MFPT as a function of the fraction of the boundary occupied by the arcs (i) for n evenly spaced arcs and (ii) for two arcs arbitrarily located on the boundary. As the arc length tends to zero, our approximate solution reduces to the known asymptotic formula for the MFPT rigorously derived in studies devoted of the narrow escape problem.

Model of autocrine/paracrine signaling in epithelial layer: geometrical regulation of intercellular communication

An autocrine/paracrine signaling model in epithelial layers is described. The axially symmetric model of the epithelial layer explicitly considers the microvilli of the epithelial cells and the gaps between nearest neighbor microvilli. Ligand trapping site distribution functions and probability of autocrine signaling are calculated for different epithelial geometries and ligand sources by numerically solving the inhomogeneous stationary diffusion equation, the Poisson equation. In general, the global characteristics of the trapping site distribution curves are similar to the ones obtained for a planar epithelial model, and the superimposed small periodical changes of the curves reflect the details of the epithelial geometry. However, when ligands are emitted into a narrow gap between nearest neighbor microvilli the probability of local trapping is particularly high, causing a locally large deviation from the overall behavior of the trapping site distribution curves. If the microvilli of the cell are closely packed, then the probability of paracrine signaling is about 0.2. However, this probability jumps to about 0.5 if the cell is able to slightly loosen the tight packing, for example, by decreasing the diameter of the microvilli by only 2%. On the basis of our calculations, alteration of microvillus geometry represents a mechanism by which epithelial cells can efficiently regulate intercellular signaling.

Manipulation of signaling thresholds in "engineered stem cell niches" identifies design criteria for pluripotent stem cell screens

In vivo, stem cell fate is regulated by local microenvironmental parameters. Governing parameters in this stem cell niche include soluble factors, extra-cellular matrix, and cell-cell interactions. The complexity of this in vivo niche limits analyses into how individual niche parameters regulate stem cell fate. Herein we use mouse embryonic stem cells (mESC) and micro-contact printing (microCP) to investigate how niche size controls endogenous signaling thresholds. microCP is used to restrict colony diameter, separation, and degree of clustering. We show, for the first time, spatial control over the activation of the Janus kinase/signal transducer and activator of transcription pathway (Jak-Stat). The functional consequences of this niche-size-dependent signaling control are confirmed by demonstrating that direct and indirect transcriptional targets of Stat3, including members of the Jak-Stat pathway and pluripotency-associated genes, are regulated by colony size. Modeling results and empirical observations demonstrate that colonies less than 100 microm in diameter are too small to maximize endogenous Stat3 activation and that colonies separated by more than 400 microm can be considered independent from each other. These results define parameter boundaries for the use of ESCs in screening studies, demonstrate the importance of context in stem cell responsiveness to exogenous cues, and suggest that niche size is an important parameter in stem cell fate control.

Nuclear trapping shapes the terminal gradient in the Drosophila embryo

Patterning of the terminal regions of the Drosophila embryo relies on the gradient of phosphorylated ERK/MAPK (dpERK), which is controlled by the localized activation of the Torso receptor tyrosine kinase [1-4]. This model is supported by a large amount of data, but the gradient itself has never been quantified. We present the first measurements of the dpERK gradient and establish a new intracellular layer of its regulation. Based on the quantitative analysis of the spatial pattern of dpERK in mutants with different levels of Torso as well as the dynamics of the wild-type dpERK pattern, we propose that the terminal-patterning gradient is controlled by a cascade of diffusion-trapping modules. A ligand-trapping mechanism establishes a sharply localized pattern of the Torso receptor occupancy on the surface of the embryo. Inside the syncytial embryo, nuclei play the role of traps that localize diffusible dpERK. We argue that the length scale of the terminal-patterning gradient is determined mainly by the intracellular module.

Dynamics of maternal morphogen gradients in Drosophila

The first direct studies of morphogen gradients were done in the end of 1980s, in the early Drosophila embryo, which is patterned under the action of four maternally determined morphogens. Since the early studies of maternal morphogens were done with fixed embryos, they were viewed as relatively static signals. Several recent studies analyze dynamics of the anterior, dorsoventral, and terminal patterning signals. The results of these quantitative studies provide critical tests of classical models and reveal new modes of morphogen regulation and readout in one of the most extensively studied patterning systems.

Time and length scales of autocrine signals in three dimensions

A model of autocrine signaling in cultures of suspended cells is developed on the basis of the effective medium approximation. The fraction of autocrine ligands, the mean and distribution of distances traveled by paracrine ligands before binding, as well as the mean and distribution of the ligand lifetime are derived. Interferon signaling by dendritic immune cells is considered as an illustration.

An integrated agent-mathematical model of the effect of intercellular signalling via the epidermal growth factor receptor on cell proliferation

We have previously developed Epitheliome, a software agent representation of the growth and repair characteristics of epithelial cell populations, where cell behaviour is governed by a number of simple rules. In this paper, we describe how this model has been extended to incorporate an example of a molecular 'mechanism' behind a rule-in this case, how signalling by both endogenous and exogenous ligands of the epidermal growth factor receptor (EGFR) can impact on the proliferation of cell agents. We have developed a mathematical model representing release of endogenous ligand by cells, three-dimensional diffusion of the secreted molecules through a volume of cell culture medium, ligand-receptor binding, and bound receptor internalization and trafficking. Information relating to quantities of molecular species associated with each cell agent is frequently exchanged between the agent and signalling models, and the ratio of bound to free receptors determines cell cycle progression and hence the proliferative behaviour of the cell agents. We have applied this integrated model to examine the effect of plating density on tissue growth via autocrine/paracrine signalling. This predicts that cell growth is dependent on the concentration of exogenous ligand, but where this is limited, then growth becomes dependent on cell density and the availability of endogenous ligand. We have further modified the calcium concentration of the medium to modulate the formation of intercellular bonds between cells and shown that the increased propensity for cells to form colonies in physiological calcium does not result in significantly different patterns of receptor occupancy. In conclusion, our approach demonstrates that by combining agent-based and mathematical modelling paradigms, it is possible to probe the complex feedback relationship between the behaviour of individual cells and their interaction with one another and their environment.

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.

Monte Carlo simulations of VEGF binding to cell surface receptors in vitro

The vascular endothelial growth factor (VEGF) family binds multiple endothelial cell surface receptors. Our goal is to build comprehensive models of these interactions for the purpose of simulating angiogenesis. In view of low concentrations of growth factors in vivo and in vitro, stochastic modeling of molecular interactions may be necessary. Here, we compare Monte Carlo simulations of the stochastic binding of VEGF and two of its major receptors on cells in vitro to equivalent deterministic simulations. In the range of typical VEGF concentrations, the stochastic and deterministic models are in agreement. However, we observe significant variability in receptor binding, which may be linked to biological stochastic events, e.g., blood vessel sprout initiation. We study patches of cell surface of varying sizes to investigate spatial integration of the signal by the cell, which impacts directly the variability of binding, and find significant variability up to the single-cell level. Dimerization of VEGF receptors does not significantly alter the variability in ligand binding. A 'sliding window' approach demonstrated no reduction in the variability of binding by temporal integration. The variability is expected to be more prominent in in vivo situations where the number of ligand molecules available for binding is less.

Engineering gene networks to emulate Drosophila embryonic pattern formation

Pattern formation is essential in the development of higher eukaryotes. For example, in the Drosophila embryo, maternal morphogen gradients establish gap gene expression domain patterning along the anterior-posterior axis, through linkage with an elaborate gene network. To understand the evolution and behaviour of such systems better, it is important to establish the minimal determinants required for patterning. We have therefore engineered artificial transcription-translation networks that generate simple patterns, crudely analogous to the Drosophila gap gene system. The Drosophila syncytium was modelled using DNA-coated paramagnetic beads fixed by magnets in an artificial chamber, forming a gene expression network. Transient expression domain patterns were generated using various levels of network connectivity. Generally, adding more transcription repression interactions increased the “sharpness” of the pattern while reducing overall expression levels. An accompanying computer model for our system allowed us to search for parameter sets compatible with patterning. While it is clear that the Drosophila embryo is far more complex than our simplified model, several features of interest emerge. For example, the model suggests that simple diffusion may be too rapid for Drosophila-scale patterning, implying that sublocalisation, or “trapping,” is required. Second, we find that for pattern formation to occur under the conditions of our in vitro reaction-diffusion system, the activator molecules must propagate faster than the inhibitors. Third, adding controlled protease degradation to the system stabilizes pattern formation over time. We have reconstituted transcriptional pattern formation from purified substances, including phage RNA polymerases, ribonucleotides, and an eukaryotic translation extract. We anticipate that the system described here will be generally applicable to the study of any biological network with a spatial component.

To understand how patterns are established during early development, these authors have created an artificial system to mimic aspects of the early Drosophila embryo

Ligand accumulation in autocrine cell cultures

Cell-culture assays are routinely used to analyze autocrine signaling systems, but quantitative experiments are rarely possible. To enable the quantitative design and analysis of experiments with autocrine cells, we develop a biophysical theory of ligand accumulation in cell-culture assays. Our theory predicts the ligand concentration as a function of time and measurable parameters of autocrine cells and cell-culture experiments. The key step of our analysis is the derivation of the survival probability of a single ligand released from the surface of an autocrine cell. An expression for this probability is derived using the boundary homogenization approach and tested by stochastic simulations. We use this expression in the integral balance equations, from which we find the Laplace transform of the ligand concentration. We demonstrate how the theory works by analyzing the autocrine epidermal growth factor receptor system and discuss the extension of our methods to other experiments with cultured autocrine cells.