The biophysical nature of cells: Potential cell behaviours revealed by analytical and computational studies of cell surface mechanics

Background The biophysical characteristics of cells determine their shape in isolation and when packed within tissues. Cells can form regular or irregular epithelial structures, round up and form clusters, or deform and attach to substrates. The acquired shape of cells and tissues is a consequence of (i) internal cytoskeletal processes, such as actin polymerisation and cortical myosin contraction, (ii) adhesion molecules within the cell membrane that interact with substrates and neighbouring cells, and (iii) processes that regulate cell volume. Although these processes seem relatively simple, when combined they unleash a rich variety of cellular behaviour that is not readily understandable outside a theoretical framework. Methods We perform a mathematical analysis of a commonly used class of model formalisms that describe cell surface mechanics using an energy-based approach. Predictions are then confirmed through comparison with the computational outcomes of a Vertex model and 2D and 3D simulations of the Cellular Potts model. Results The analytical study reveals the complete possible spectrum of single cell behaviour and tissue packing in both 2D and 3D, by taking the typical core elements of cell surface mechanics into account: adhesion, cortical tension and volume conservation. We show that from an energy-based description, forces and tensions can be derived, as well as the prediction of cell behaviour and tissue packing, providing an intuitive and biologically relevant mapping between modelling parameters and experiments. Conclusions The quantitative cellular behaviours and biological insights agree between the analytical study and the diverse computational model formalisms, including the Cellular Potts model. This illustrates the generality of energy-based approaches for cell surface mechanics and highlights how meaningful and quantitative comparisons between models can be established. Moreover, the mathematical analysis reveals direct links between known biophysical properties and specific parameter settings within the Cellular Potts model

A sigmoid functional response emerges when Cytotoxic T Lymphocytes start killing fresh target cells

Cytotoxic T lymphocyte (CTL)-mediated killing involves the formation of a synapse with a target cell, followed by delivery of perforin and granzymes. Previously, we derived a general functional response for CTL killing while considering that CTLs form stable synapses (i.e., single-stage) and that the number of conjugates remains at steady state. However, the killing of target cells sometimes requires multiple engagements (i.e., multistage). To study how multistage killing and a lack of steady state influence the functional response, we here analyze a set of differential equations as well as simulations employing the cellular Potts model, in both cases describing CTLs that kill target cells. We find that at steady state the total killing rate (i.e., the number of target cells killed by all CTLs) is well described by the previously derived double saturation function. Compared to single-stage killing, the total killing rate during multistage killing saturates at higher CTL and target cell densities. Importantly, when the killing is measured before the steady state is approached, a qualitatively different functional response emerges for two reasons: First, the killing signal of each CTL gets diluted over several targets and because this dilution effect is strongest at high target cell densities; this can result in a peak in the dependence of the total killing rate on the target cell density. Second, the total killing rate exhibits a sigmoid dependence on the CTL density when killing is a multistage process, because it takes typically more than one CTL to kill a target. In conclusion, a sigmoid dependence of the killing rate on the CTLs during initial phases of killing may be indicative of a multistage killing process. Observation of a sigmoid functional response may thus arise from a dilution effect and is not necessarily due to cooperative behavior of the CTLs

Mathematical modeling and experimental validation of the spatial distribution of boron in the root of arabidopsis thaliana identify high boron accumulation in the tip and predict a distinct root tip uptake function

Boron, an essential micronutrient, is transported in roots of Arabidopsis thaliana mainly by two different types of transporters, BORs and NIPs (nodulin26-like intrinsic proteins). Both are plasma membrane localized, but have distinct transport properties and patterns of cell type-specific accumulation with different polar localizations, which are likely to affect boron distribution. Here, we used mathematical modeling and an experimental determination to address boron distributions in the root. A computational model of the root is created at the cellular level, describing the boron transporters as observed experimentally. Boron is allowed to diffuse into roots, in cells and cell walls, and to be transported over plasma membranes, reflecting the properties of the different transporters. The model predicts that a region around the quiescent center has a higher concentration of soluble boron than other portions. To evaluate this prediction experimentally, we determined the boron distribution in roots using laser ablation-inductivity coupled plasma-mass spectrometry. The analysis indicated that the boron concentration is highest near the tip and is lower in the more proximal region of the meristem zone, similar to the pattern of soluble boron distribution predicted by the model. Our model also predicts that upward boron flux does not continuously increase from the root tip toward the mature region, indicating that boron taken up in the root tip is not efficiently transported to shoots. This suggests that root tip-absorbed boron is probably used for local root growth, and that instead it is the more mature root regions which have a greater role in transporting boron toward the shoots

Growth Based Morphogenesis of Vertebrate Limb Bud

Many genes and their regulatory relationships are involved in developmental phenomena. However, by chemical information alone, we cannot fully understand changing organ morphologies through tissue growth because deformation and growth of the organ are essentially mechanical processes. Here, we develop a mathematical model to describe the change of organ morphologies through cell proliferation. Our basic idea is that the proper specification of localized volume source (e.g., cell proliferation) is able to guide organ morphogenesis, and that the specification is given by chemical gradients. We call this idea “growth-based morphogenesis.” We find that this morphogenetic mechanism works if the tissue is elastic for small deformation and plastic for large deformation. To illustrate our concept, we study the development of vertebrate limb buds, in which a limb bud protrudes from a flat lateral plate and extends distally in a self-organized manner. We show how the proportion of limb bud shape depends on different parameters and also show the conditions needed for normal morphogenesis, which can explain abnormal morphology of some mutants. We believe that the ideas shown in the present paper are useful for the morphogenesis of other organs

Spiral breakup in excitable tissue due to lateral instability

In a simple two-component model of excitable tissue a spiral wave is found to break up into a large number of small spirals. More specifically, we used a modified FitzHugh-Nagumo model. Spiral breakup is found to occur when the recovery variable diffuses at a high rate. The breakup is caused by lateral instability of the wave front. We analyzed this instability using quasi-one-dimensional computations and found that it could be connected to a negative eikonal-curvature relation for the parameter values at which spiral breakup occurs

Modelling Dictyostelium discoideum morphogenesis: The culmination

The culmination of the morphogenesis of the cellular slime mould Dictyostelium discoideum involves complex cell movements which transform a mound of cells into a globule of spores on a slender stalk. We show that cyclic AMP signalling and differential adhesion, combined with cell differentiation and slime production, are sufficient to produce the morphogenetic cell movements which lead to culmination. We have simulated the process of culmination using a hybrid cellular automata/partial differential equation model. With our model we have been able to reproduce the main features that occur during culmination, namely the straight downward elongation of the stalk, its anchoring to the substratum and the formation of the long thin stalk topped by the spore head. We conclude that the cyclic AMP signalling system is responsible for the elongation and anchoring of the stalk, but in a roundabout way: pressure waves that are induced by the chemotaxis towards cyclic AMP squeeze the stalk through the cell mass. This mechanism forces the stalk to elongate precisely in the direction opposite to that of the chemotactically moving cells. The process turns out to be ‘guided’ by inactive ‘pathfinder’ cells, which form the tip of the stalk. We show that the entire development is enacted by means of the aforementioned building blocks. This means that no global gradients or different modes of chemotaxis are needed to complete the culmination

Juicy stories on female reproductive tissue development: Coordinating the hormone flows

In the past 20–30 years, developmental biologists have made tremendous progress in identifying genes required for the specification of individual cell types of an organ and in describing how they interact in genetic networks. In comparison, very little is known about the mechanisms that regulate tissue polarity and overall organ patterning. Gynoecia and fruits from members of the Brassicaceae family of flowering plants provide excellent model systems to study organ patterning and tissue specification because they become partitioned into distinct domains whose formation is determined by polarity establishment both at a cellular and whole tissue level. Interactions among key regulators of Arabidopsis gynoecium and fruit development have revealed a network of upstream transcription factor activities required for such tissue differentiation. Regulation of the plant hormone auxin is emerging as both an immediate downstream output and input of these activities, and here we aim to provide an overview of the current knowledge regarding the link between auxin and female reproductive development in plants. In this review, we will also demonstrate how available data can be exploited in a mathematical modeling approach to reveal and understand the feedback regulatory circuits that underpin the polarity establishment, necessary to guide auxin flows

Mathematical model for spatial segregation of the Rho-family GTPases based on inhibitory crosstalk

Cdc42, Rac, and Rho are small GTPases known to play a central role in signal transduction to the actin cytoskeleton. These proteins regulate cell motility, by affecting nucleation, uncapping, and depolymerization of actin filaments, and acto-myosin contractility. Studies of crosstalk and mutual feedbacks in these three proteins have led to a number of proposals for their interaction. At the same time, observations of the spatio-temporal dynamics of Rho-family proteins give evidence of spatial polarization and mutual exclusion between Cdc42/Rac and Rho. In this paper, we formulate a mathematical model to account for such observations, based on the known underlying biology of these proteins. We first investigate which of the crosstalk schemes proposed in the literature is consistent with observed dynamics, and then derive a simple model that can correctly describe these dynamics (assuming crosstalk is mediated via Rho GEFs). We show that cooperativity is an essential ingredient in the interactions of the proteins. The co-occurrence of a stable rest state with the possibility of fast spatial segregation can be related to bistability in a set of underlying ODEs in which the inactive forms of these proteins are fixed at a constant level. We show that the fast diffusion of the inactive forms is essential for stabilizing the transition fronts in the PDE formulation of the model, leading to robust spatial polarization, rather than traveling waves