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D'Arcy Thompson emphasised the importance of surface tension as a potential driving force in establishing cell shape and topology within tissues. Leaf epidermal pavement cells grow into jigsaw-piece shapes, highly deviating from such classical forms. We investigate the topology of developing Arabidopsis leaves composed solely of pavement cells. Image analysis of around 50,000 cells reveals a clear and unique topological signature, deviating from previously studied epidermal tissues. This topological distribution is established early during leaf development, already before the typical pavement cell shapes emerge, with topological homeostasis maintained throughout growth and unaltered between division and maturation zones. Simulating graph models, we identify a heuristic cellular division rule that reproduces the observed topology. Our parsimonious model predicts how and when cells effectively place their division plane with respect to their neighbours. We verify the predicted dynamics through in vivo tracking of 800 mitotic events, and conclude that the distinct topology is not a direct consequence of the jigsaw piece-like shape of the cells, but rather owes itself to a strongly life history-driven process, with limited impact from cell-surface mechanics., Summary: Development of the Arabidopsis leaf epidermis topology is driven by deceptively simple rules of cell division, independent of surface tension, cell size and, often complex, cell shape.

Leaves display a remarkable range of forms, from flat sheets with simple outlines to cup-shaped traps. Although much progress has been made in understanding the mechanisms of planar leaf development, it is unclear whether similar or distinctive mechanisms underlie shape transformations during development of more complex curved forms. Here, we use 3D imaging and cellular and clonal analysis, combined with computational modelling, to analyse the development of cup-shaped traps of the carnivorous plant Utricularia gibba. We show that the transformation from a near-spherical form at early developmental stages to an oblate spheroid with a straightened ventral midline in the mature form can be accounted for by spatial variations in rates and orientations of growth. Different hypotheses regarding spatiotemporal control predict distinct patterns of cell shape and size, which were tested experimentally by quantifying cellular and clonal anisotropy. We propose that orientations of growth are specified by a proximodistal polarity field, similar to that hypothesised to account for Arabidopsis leaf development, except that in Utricularia, the field propagates through a highly curved tissue sheet. Independent evidence for the polarity field is provided by the orientation of glandular hairs on the inner surface of the trap. Taken together, our results show that morphogenesis of complex 3D leaf shapes can be accounted for by similar mechanisms to those for planar leaves, suggesting that simple modulations of a common growth framework underlie the shaping of a diverse range of morphologies., Many plant and animal organs derive from tissue sheets, but how are they shaped to create the diversity of forms observed in nature? This study uses a combination of imaging and mathematical modelling to show how carnivorous plant traps shape themselves in 3D by a growth framework oriented by tissue polarity, similar to that found in planar leaves.

The phytohormone auxin is implied in steering various developmental decisions during plant morphogenesis in a concentration-dependent manner. Auxin maxima have been shown to maintain meristematic activity, for example, of the root apical meristem, and position new sites of outgrowth, such as during lateral root initiation and phyllotaxis. More recently, it has been demonstrated that sites of auxin minima also provide positional information. In the developing Arabidopsis fruit, auxin minima are required for correct differentiation of the valve margin. It remains unclear, however, how this auxin minimum is generated and maintained. Here, we employ a systems biology approach to model auxin transport based on experimental observations. This allows us to determine the minimal requirements for its establishment. Our simulations reveal that two alternative processes—which we coin “flux-barrier” and “flux-passage”—are both able to generate an auxin minimum, but under different parameter settings. Both models are in principle able to yield similar auxin profiles but present qualitatively distinct patterns of auxin flux. The models were tested by tissue-specific inducible ablation, revealing that the auxin minimum in the fruit is most likely generated by a flux-passage process. Model predictions were further supported through 3D PIN localization imaging and implementing experimentally observed transporter localization. Through such an experimental–modeling cycle, we predict how the auxin minimum gradually matures during fruit development to ensure timely fruit opening and seed dispersal.

A synthesis is presented of recent work by the authors and others on the formation of localised patterns, isolated spots, or sharp fronts in models of natural processes governed by reaction–diffusion equations. Contrasting with the well-known Turing mechanism of periodic pattern formation, a general picture is presented in one spatial dimension for models on long domains that exhibit sub-critical Turing instabilities. Localised patterns naturally emerge in generalised Schnakenberg models within the pinning region formed by bistability between the patterned state and the background. A further long-wavelength transition creates parameter regimes of isolated spots which can be described by semi-strong asymptotic analysis. In the species-conservation limit, another form of wave pinning leads to sharp fronts. Such fronts can also arise given only one active species and a weak spatial parameter gradient. Several important applications of this theory within natural systems are presented, at different lengthscales: cellular polarity formation in developmental biology, including root-hair formation, leaf pavement cells, keratocyte locomotion; and the transitions between vegetation states on continental scales. Philosophical remarks are offered on the connections between different pattern formation mechanisms and on the benefit of subcritical instabilities in the natural world.

A developing plant organ exhibits complex spatiotemporal patterns of growth, cell division, cell size, cell shape, and organ shape. Explaining these patterns presents a challenge because of their dynamics and cross-correlations, which can make it difficult to disentangle causes from effects. To address these problems, we used live imaging to determine the spatiotemporal patterns of leaf growth and division in different genetic and tissue contexts. In the simplifying background of the speechless (spch) mutant, which lacks stomatal lineages, the epidermal cell layer exhibits defined patterns of division, cell size, cell shape, and growth along the proximodistal and mediolateral axes. The patterns and correlations are distinctive from those observed in the connected subepidermal layer and also different from the epidermal layer of wild type. Through computational modelling we show that the results can be accounted for by a dual control model in which spatiotemporal control operates on both growth and cell division, with cross-connections between them. The interactions between resulting growth and division patterns lead to a dynamic distributions of cell sizes and shapes within a deforming leaf. By modulating parameters of the model, we illustrate how phenotypes with correlated changes in cell size, cell number, and organ size may be generated. The model thus provides an integrated view of growth and division that can act as a framework for further experimental study., Organ morphogenesis involves two coordinated processes: growth of tissue and increase in cell number through cell division. Both processes have been analysed individually in many systems and shown to exhibit complex patterns in space and time. However, it is unclear how these patterns of growth and cell division are coordinated in a growing leaf that is undergoing shape changes. We have addressed this problem using live imaging to track growth and cell division in the developing leaf of the mustard plant Arabidopsis thaliana. Using subsequent computational modelling, we propose an integrated model of leaf growth and cell division, which generates dynamic distributions of cell size and shape in different tissue layers, closely matching those observed experimentally. A key aspect of the model is dual control of spatiotemporal patterns of growth and cell division parameters. By modulating parameters in the model, we illustrate how phenotypes may correlate with changes in cell size, cell number, and organ size.

Quantifying cell morphology is fundamental to the statistical study of cell populations, and can help unravel mechanisms underlying cell and tissue morphogenesis. Current methods, however, require extensive human intervention, are highly parameter sensitive, or produce metrics that are difficult to interpret biologically. We therefore developed a method, lobe contribution elliptical Fourier analysis (LOCO-EFA), which generates from digitalised two-dimensional cell outlines meaningful descriptors that can be directly matched to morphological features. This is shown by studying well-defined geometric shapes as well as actual biological cells from plant and animal tissues. LOCO-EFA provides a tool to phenotype efficiently and objectively populations of cells, here demonstrated by applying it to the complex shaped pavement cells of Arabidopsis thaliana wild-type and speechless leaves, and Drosophila amnioserosa cells. To validate our method's applicability to large populations, we analysed computer-generated tissues. By controlling in silico cell shape, we explored the potential impact of cell packing on individual cell shape, quantifying through LOCO-EFA deviations between the specified shape of single cells in isolation and the resultant shape when they interact within a confluent tissue., Summary: A novel method for automated and quantitative assessment of complex cell shapes, including features such as protrusion number and amplitude, is demonstrated for Arabidopsis, Drosophila and in silico cells.

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.

Nutrient uptake by roots often involves substrate-dependent regulated nutrient transporters. For robust uptake, the system requires a regulatory circuit within cells and a collective, coordinated behaviour across the tissue. A paradigm for such systems is boron uptake, known for its directional transport and homeostasis, as boron is essential for plant growth but toxic at high concentrations. In Arabidopsis thaliana, boron uptake occurs via diffusion facilitators (NIPs) and exporters (BORs), each presenting distinct polarity. Intriguingly, although boron soil concentrations are homogenous and stable, both transporters manifest strikingly swift boron-dependent regulation. Through mathematical modelling, we demonstrate that slower regulation of these transporters leads to physiologically detrimental oscillatory behaviour. Cells become periodically exposed to potentially cytotoxic boron levels, and nutrient throughput to the xylem becomes hampered. We conclude that, while maintaining homeostasis, swift transporter regulation within a polarised tissue context is critical to prevent intrinsic traffic-jam like behaviour of nutrient flow., eLife digest Every multicellular organism, including all plants and animals, faces the challenge of taking up the nutrients it needs and distributing them throughout its body. Plants absorb many nutrients including nitrogen and boron from the soil into their roots, often using tightly controlled processes that require energy to work. Plant roots contain several distinct layers of cells and the nutrients need to cross these layers to reach a channel at the centre of the root known as the xylem, which transports the nutrients to other parts of the plant. Plants need boron to grow. However, high levels of this nutrient are toxic so plants have evolved to change the rate at which they absorb boron to optimize growth in different environments. When there is little boron in the soil, certain transporter proteins move to the surface of root cells to bring boron into the root more effectively. On the other hand, when plants grow in soils with high boron, their root cells have fewer of these transporters on their surfaces to prevent too much boron entering the plant. This regulation of boron uptake appears logical, except for one detail: at any given location, the amount of boron in the soil is relatively stable and changes only very slowly. Why do plants invest energy in responding rapidly to the supply of a nutrient that changes so slowly in nature? Sotta et al. used mathematics and experimental approaches to study boron uptake in a plant known as Arabidopsis. The work reveals that the plants ability to rapidly alter how efficiently boron moves into root cells actually serves to avoid internal “traffic jams” in boron transport. If the numbers of transporter proteins on the surface of root cells changed more slowly, individual cells would occasionally experience high levels of boron that would interfere with the movement of boron further into the root, causing a jam. Furthermore, these ‘peaks’ of boron could damage the individual cells they affect. The findings of Sotta et al. reveal that, by being able to rapidly change the numbers of certain transporter proteins on the surface of root cells, plants can ensure they receive a steady supply of boron. This work suggests that to develop artificial systems that can adapt to changing surroundings, researchers will need to engineer solutions like those found in plants in order to avoid similar traffic jams in the systems. Along with considering how plants interact with their environment, studying how they avoid internal traffic jams in nutrient uptake may help efforts to alter plants, including crops, so that they grow better in harsh environments.

The maintenance of boundaries between neighboring groups of distinct cell types is vital during development of multicellular organisms, as groups of cells with distinct functions must be kept physically separated to guarantee correct control of organ and body growth and function. In the Arabidopsis root, the transition zone is a developmental boundary in the meristem that separates dividing from differentiating cells. Here, we infer that a well-defined and tightly controlled minimum of the hormone auxin acts as a signal to establish the position of the transition zone by controlling the developmental switch from cell division to cell differentiation. We provide the mechanistic and genetic basis of how another hormone, cytokinin, controls and positions this auxin minimum, thus regulating root size.

Significance The maintenance of boundaries between neighboring groups of distinct cell types is vital during development of multicellular organisms, as groups of cells with distinct functions must be kept physically separated to guarantee correct control of organ and body growth and function. In the Arabidopsis root, the transition zone is a developmental boundary in the meristem that separates dividing from differentiating cells. Here, we infer that a well-defined and tightly controlled minimum of the hormone auxin acts as a signal to establish the position of the transition zone by controlling the developmental switch from cell division to cell differentiation. We provide the mechanistic and genetic basis of how another hormone, cytokinin, controls and positions this auxin minimum, thus regulating root size.

D’Arcy Thompson emphasised the importance of surface tension as a potential driving force in establishing cell shape and topology within tissues. Leaf epidermal pavement cells grow into jigsaw-piece shapes, highly deviating from such classical forms. We investigate the topology of developing Arabidopsis leaves composed solely of pavement cells. Image analysis of around 50,000 cells reveals a clear and unique topological signature, deviating from previously studied epidermal tissues. This topological distribution is however established early during leaf development, already before the typical pavement cell shapes emerge, with topological homestasis maintained throughout growth and unaltered between division and maturation zones. Simulating graph models, we identify a heuristic cellular division rule that reproduces the observed topology. Our parsimonious model predicts how and when cells effectively place their division plane with respect to their neighbours. We verify the predicted dynamics through in vivo tracking of 800 mitotic events, and conclude that the distinct topology is not a direct consequence of the jigsaw-like shape of the cells, but rather owes itself to a strongly life-history-driven process, with limited impact from cell surface mechanics.Summary statementDevelopment of the Arabidopsis leaf epidermis topology is driven by deceptively simple rules of cell division, independent of surface tension, cell size and, often complex, cell shape.

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. Figure 1. Download figure to PowerPoint Veronica A. Grieneisen, Athanasius F.M. Maree, Lars Ostergaard (Corresponding authors)

The development of outgrowths from plant shoots depends on formation of epidermal sites of cell polarity convergence with high intracellular auxin at their centre. A parsimonious model for generation of convergence sites is that cell polarity for the auxin transporter PIN1 orients up auxin gradients, as this spontaneously generates convergent alignments. Here we test predictions of this and other models for the patterns of auxin biosynthesis and import. Live imaging of outgrowths from kanadi1 kanadi2 Arabidopsis mutant leaves shows that they arise by formation of PIN1 convergence sites within a proximodistal polarity field. PIN1 polarities are oriented away from regions of high auxin biosynthesis enzyme expression, and towards regions of high auxin importer expression. Both expression patterns are required for normal outgrowth emergence, and may form part of a common module underlying shoot outgrowths. These findings are more consistent with models that spontaneously generate tandem rather than convergent alignments. DOI: http://dx.doi.org/10.7554/eLife.18165.001, eLife digest Plants, unlike animals, are able to grow and develop throughout their lives. New leaves and flowers are made from outgrowths that constantly form at the tip of growing shoots. Groups of cells in the outer layer of the shoot tip arrange a protein called PIN1 so that it is more abundant on the cell surfaces that face towards the centre of the group. PIN1 transports a hormone called auxin out of plant cells and this “convergent” arrangement of PIN1 increases the levels of auxin in cells at the centre of the group, leading to the formation of a new outgrowth. However, it is not clear what causes these cells to position their PIN1 proteins in this way. Several hypotheses have been proposed to explain how convergent patterns of PIN1 form. For example, according to the “up-the-gradient” hypothesis, PIN1 is allocated to the end of a cell that is next to a cell with a higher level of auxin. Abley et al. have now compared predictions from computer models with new experimental data from a plant called Arabidopsis to evaluate three hypotheses for how convergent PIN1 patterns form. A computer model based on the up-the-gradient hypothesis naturally creates convergent PIN1 patterns, even if each cell starts off with the same level of auxin. On the other hand, models based on two other hypotheses generate tandem alignments of PIN1 so that auxin is transported in the same direction along lines of cells. Next, Abley et al. tested these models using mutant Arabidopsis plants that develop outgrowths from the lower surface of their leaves. These outgrowths form in a similar way to outgrowths at the growing shoot tip, but in a simpler context. The experiments show that the patterns of where auxin is produced in growing leaves were more compatible with the tandem alignment models than the up-the-gradient model. This suggests that plants use a tandem alignment mechanism to form convergences of PIN1 proteins that generate the local increases in auxin needed to make new outgrowths. This study only examined a single layer of cells on the plant surface. Other cell layers also show highly organised patterns of PIN1 proteins, so a future challenge is to extend the approach to study the entire 3D structure of new shoot outgrowths. DOI: http://dx.doi.org/10.7554/eLife.18165.002

The visualization of the dynamic behaviour of and interactions between immune cells using time-lapse video microscopy has an important role in modern immunology. To draw robust conclusions, quantification of such cell migration is required. However, imaging experiments are associated with various artefacts that can affect the estimated positions of the immune cells under analysis, which form the basis of any subsequent analysis. Here, we describe potential artefacts that could affect the interpretation of data sets on immune cell migration. We propose how these errors can be recognized and corrected, and suggest ways to prevent the data analysis itself leading to biased results.

Cell aggregates are a tool for in vitro studies of morphogenesis, cancer invasion, and tissue engineering. They respond to mechanical forces as a complex rather than simple liquid. To change an aggregate's shape, cells have to overcome energy barriers. If cell shape fluctuations are active enough, the aggregate spontaneously relaxes stresses (“fluctuation-induced flow”). If not, changing the aggregate's shape requires a sufficiently large applied stress (“stress-induced flow”). To capture this distinction, we develop a mechanical model of aggregates based on their cellular structure. At stress lower than a characteristic stress τ*, the aggregate as a whole flows with an apparent viscosity η*, and at higher stress it is a shear-thinning fluid. An increasing cell–cell tension results in a higher η* (and thus a slower stress relaxation time t c ). Our constitutive equation fits experiments of aggregate shape relaxation after compression or decompression in which irreversibility can be measured; we find t c of the order of 5 h for F9 cell lines. Predictions also match numerical simulations of cell geometry and fluctuations. We discuss the deviations from liquid behavior, the possible overestimation of surface tension in parallel-plate compression measurements, and the role of measurement duration.

Macrophages play an important role in clearing apoptotic debris from tissue. Defective or reduced clearance, seen, for instance, in non-obese diabetic (NOD) mice, has been correlated with initiation of autoimmune (Type 1) diabetes (T1D) (O'Brien BA, Huang Y, Geng X, Dutz JP, Finegood DT. Diabetes 51: 2481–2488, 2002). To validate such a link, it is essential to quantify the reduced clearance (for example, by comparison to BALB/c control mice) and to determine which elements of that clearance are impaired. Recently, we fit data for the time course of in vitro macrophage feeding experiments to basic models of macrophage clearance dynamics, thus quantifying kinetics of uptake and digestion of apoptotic cells in both mouse strains (Marée AFM, Komba M, Dyck C, Łabeçki M, Finegood DT, Edelstein-Keshet L. J Theor Biol 233: 533–551, 2005). In the cycle of modeling and experimental investigation, we identified the importance of 1) measuring short-, intermediate-, and long-time data (to increase the accuracy of parameter fits), and 2) designing experiments with distinct observable regimes, including engulfment-only and digestion-only phases. Here, we report on new results from experiments so designed. In comparing macrophages from the two strains, we find that NOD macrophage engulfment of apoptotic cells is 5.5 times slower than BALB/c controls. Significantly, our new data demonstrate that digestion is at least two times slower in NOD, in contrast with previous conclusions. Moreover, new data enable us to detect an acceleration in engulfment (after the first engulfment) in both strains, but much smaller in NOD macrophages.

Hayashi and Carthew (Nature 431 [2004], 647) have shown that the packing of cone cells in the Drosophila retina resembles soap bubble packing, and that changing E- and N-cadherin expression can change this packing, as well as cell shape. The analogy with bubbles suggests that cell packing is driven by surface minimization. We find that this assumption is insufficient to model the experimentally observed shapes and packing of the cells based on their cadherin expression. We then consider a model in which adhesion leads to a surface increase, balanced by cell cortex contraction. Using the experimentally observed distributions of E- and N-cadherin, we simulate the packing and cell shapes in the wildtype eye. Furthermore, by changing only the corresponding parameters, this model can describe the mutants with different numbers of cells, or changes in cadherin expression., revised manuscript; 8 pages, 6 figures; supplementary information not included

In the early phases of an immune response, T cells of appropriate antigen specificity become activated by antigen-presenting cells in secondary lymphoid organs. Two-photon microscopy imaging experiments have shown that this stimulation occurs in distinct stages during which T cells exhibit different motilities and interactions with dendritic cells (DCs). In this paper, we utilize the Cellular Potts Model, a model formalism that takes cell shapes and cellular interactions explicitly into account, to simulate the dynamics of, and interactions between, T cells and DCs in the lymph node paracortex. Our three-dimensional simulations suggest that the initial decrease in T-cell motility after antigen appearance is due to "stop signals" transmitted by activated DCs to T cells. The long-lived interactions that occur at a later stage can only be explained by the presence of both stop signals and a high adhesion between specific T cells and antigen-bearing DCs. Furthermore, our results indicate that long-lasting contacts with T cells are promoted when DCs retract dendrites that detect a specific contact at lower velocities than other dendrites. Finally, by performing long simulations (after prior fitting to short time scale data) we are able to provide an estimate of the average contact duration between T cells and DCs.

An auxin maximum is positioned along the xylem axis of the Arabidopsis root tip. The pattern depends on mutual feedback between auxin and cytokinins mediated by the PIN class of auxin efflux transporters and AHP6, an inhibitor of cytokinin signalling. This interaction has been proposed to regulate the size and the position of the hormones’ respective signalling domains and specify distinct boundaries between them. To understand the dynamics of this regulatory network, we implemented a parsimonious computational model of auxin transport that considers hormonal regulation of the auxin transporters within a spatial context, explicitly taking into account cell shape and polarity and the presence of cell walls. Our analysis reveals that an informative spatial pattern in cytokinin levels generated by diffusion is a theoretically unlikely scenario. Furthermore, our model shows that such a pattern is not required for correct and robust auxin patterning. Instead, auxin-dependent modifications of cytokinin response, rather than variations in cytokinin levels, allow for the necessary feedbacks, which can amplify and stabilise the auxin maximum. Our simulations demonstrate the importance of hormonal regulation of auxin efflux for pattern robustness. While involvement of the PIN proteins in vascular patterning is well established, we predict and experimentally verify a role of AUX1 and LAX1/2 auxin influx transporters in this process. Furthermore, we show that polar localisation of PIN1 generates an auxin flux circuit that not only stabilises the accumulation of auxin within the xylem axis, but also provides a mechanism for auxin to accumulate specifically in the xylem-pole pericycle cells, an important early step in lateral root initiation. The model also revealed that pericycle cells on opposite xylem poles compete for auxin accumulation, consistent with the observation that lateral roots are not initiated opposite to each other., Author Summary After moving onto land, plants developed vascular tissues to support their weight and transport water and nutrients. Vascular tissue consists of xylem, which makes up wood, and phloem, which gives rise to the innermost bark. In the model species Arabidopsis thaliana, these tissues form in the growing root tip in a radial pattern consisting of a xylem axis and two phloem poles. Xylem is thought to be positioned by negative interactions between two plant hormones, auxin and cytokinins. Cytokinins activate exporters which pump auxin out of cells, while auxin activates a gene which blocks cytokinin response. This leads auxin to accumulate in some cells which become xylem cells. We developed a computational model which includes only the essential processes but allows them to interact in a realistic spatial context. Using this model we show that these interactions can produce the expected auxin pattern even without a pattern in cytokinin distribution, contrary to expectations based on observed patterns in cytokinin signalling. Furthermore, we learned that hormonal regulation fine-tunes the exporters’ activity, and auxin importers play an important role. The regulatory network not only ensures correct formation of the vasculature but may also position root branches on alternating sides of the xylem.

Antigen therapy remains a promising strategy for prevention and treatment of autoimmune diseases, but translating this strategy to clinical therapy has been largely unsuccessful. We have shown that development of autoimmune diabetes in non-obese diabetic (NOD) mice involves prevalent recruitment of CD8+ T cells recognizing epitopes of islet-specific glucose-6-phosphatase catalytic subunit-related protein (IGRP). Administration of peptide analogs of IGRP206-214, the dominant epitope, reduced disease incidence but only under conditions that led to selective deletion of high-avidity T-cell clones. Peptide types or doses that resulted in elimination of all IGRP206-214-reactive T cells, regardless of avidity, promoted the recruitment of sub-dominant epitope-specific T cells and failed to prevent disease development. Here, we mathematically model competition of IGRP-reactive T-cell clones during spontaneous disease, and in response to peptide treatment. Based on realistic T-cell activation, proliferation and differentiation parameter values, our model shows that progression of spontaneous disease is characterized by (i) initial expansion of all IGRP206-214-reactive T-cell clones (irrespective of avidity) and (ii) slow replacement of T-cell clones recognizing peptide/MHC with low avidity by their high-avidity counterparts. This model helps understand the paradoxical outcomes of IGRP-based peptide treatment experiments. Furthermore, it predicts that slight deviations in dose or peptide affinity can lead to treatment failure or disease progression. This will occur if the treatment (i) increases the imbalance between competing IGRP206-214-reactive T-cell clones such that it favors rapid takeover of high-avidity clones or (ii) deletes all IGRP206-214-reactive clones, thereby creating a vacuum that promotes the recruitment of pathogenic sub-dominant specificities. Our data and model urge caution in the application of peptide therapy in autoimmunity.

A wave of apoptosis (programmed cell death) occurs normally in pancreatic β-cells of newborn mice. We previously showed that macrophages from non-obese diabetic (NOD) mice become activated more slowly and engulf apoptotic cells at a lower rate than macrophages from control (Balb/c) mice. It has been hypothesized that this low clearance could result in secondary necrosis, escalating inflammation and self-antigen presentation that later triggers autoimmune, Type 1 diabetes (T1D). We here investigate whether this hypothesis could offer a reasonable and parsimonious explanation for onset of T1D in NOD mice. We quantify variants of theCopenhagen model(Freiesleben De Blasioet al. 1999Diabetes48, 1677), based on parameters from NOD and Balb/c experimental data. We show that the original Copenhagen model fails to explain observed phenomena within a reasonable range of parameter values, predicting an unrealistic all-or-none disease occurrence for both strains. However, if we take into account that, in general, activated macrophages produce harmful cytokinesonlywhen engulfing necrotic (but not apoptotic) cells, then the revised model becomes qualitatively and quantitatively reasonable. Further, we show that known differences between NOD and Balb/c mouse macrophage kinetics are large enough to account for the fact that an apoptotic wave can trigger escalating inflammatory response in NOD, but not Balb/c mice. In Balb/c mice, macrophages clear the apoptotic wave so efficiently, that chronic inflammation is prevented.

Macrophages from animals prone to autoimmune (type 1) diabetes differ from those of diabetes-resistant animals in processing and clearing apoptotic cells. Using in vitro time-course assays of the number of engulfed apoptotic cells observed within macrophages, we quantified these differences in non-obese diabetic (NOD) versus Balb/c mice. Simple models lead to several elementary parameter estimation techniques. We used these to compute approximate rates of macrophage engulfment and digestion of apoptotic cells from basic features of the data (such as initial rise-times, phagocytic index and percent phagocytosis). Combining these estimates with full fitting of a sequence of model variants to the data, we find that macrophages from normal (Balb/c) mice engulf apoptotic cells up to four times faster than macrophages from the diabetes-prone (NOD) mice. Further, Balb/c macrophages appear to undergo an activation step before achieving their high engulfment rate. In NOD macrophages, we did not see evidence for this activation step. Rates of digestion of engulfed apoptotic cells by macrophages are similar in both types. Since macrophage clearance is an important mechanism of disposal of self-antigen, these macrophage defects could potentially be a factor in predisposition to type 1 diabetes.

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. MPEG movies of the simulations are available on-line: http://www-binf.bio. uu.nl/stan/bmb.

Steady-state levels of HIV-1 viraemia in the plasma vary more than a 1,000-fold between HIV-positive patients and are thought to be influenced by several different host and viral factors such as host target cell availability, host anti-HIV immune response and the virulence of the virus. Previous mathematical models have taken the form of classical ecological food-chain models and are unable to account for this multifactorial nature of the disease. These models suggest that the steady-state viral load (i.e. the set-point) is determined by immune response parameters only. We have devised a generalized consensus model in which the conventional parameters are replaced by so-called 'process functions'. This very general approach yields results that are insensitive to the precise form of the mathematical model. Here we applied the approach to HIV-1 infections by estimating the steady-state values of several process functions from published patient data. Importantly, these estimates are generic because they are independent of the precise form of the underlying processes. We recorded the variation in the estimated steady-state values of the process functions in a group of HIV-1 patients. We developed a novel model by providing explicit expressions for the process functions having the highest patient-to-patient variation in their estimated values. Small variations from patient to patient for several parameters of the new model collectively accounted for the large variations observed in the steady-state viral burden. The novel model remains in full agreement with previous models and data.

The relative fitness of viral variants has previously been defined as the slope of the logarithmic ratio of the genotype or phenotype frequencies in time plots of pairwise competition experiments. Developing mathematical models for such experiments by employing the conventional coefficient of selection s, we demonstrate that this logarithmic ratio gives the fitness difference, rather than the relative fitness. This fitness difference remains proportional to the actual replication rate realized in the particular experimental setup and hence cannot be extrapolated to other situations. Conversely, the conventional relative fitness (1 + s) should be more generic. We develop an approach to compute the generic relative fitness in conventional competition experiments. This involves an estimation of the total viral replication during the experiment and requires an estimate of the average lifetime of productively infected cells. The novel approach is illustrated by estimating the relative fitness, i.e., the relative replication rate, of a set of zidovudine-resistant human immunodeficiency virus type 1 variants. A tool for calculating the relative fitness from observed changes in viral load and genotype (or phenotype) frequencies is publically available on the website athttp://www-binf.bio.uu.nl/∼rdb/fitness.html. Differences in the in vitro replication rate (or fitness) between viral variants can be estimated experimentally by pairwise competition experiments in tissue culture. The outcome of such an experiment is typically depicted in a logarithmic time plot of the ratio of the genotype or phenotype frequencies (7). On a logarithmic scale the ratio tends to change linearly in time, and the rate of change (i.e., the slope of the line) has previously been defined as the relative fitness (7). According to population genetics theory, the relative fitness (1 + s) of a variant represents its relative contribution to the next generation. The parameter s is defined as the coefficient of selection. The intertwined concepts of relative fitness (1 + s) and selection coefficients are traditionally employed in systems with discrete generations. They are equally valid for populations growing continuously, however, when time is scaled with respect to the generation time (11). Developing conventional population genetics models for pairwise competition experiments, we show that the above-mentioned slope in a logarithmic time plot provides the absolute fitness difference between the two variants rather than the generic relative fitness (1 +s) of one with respect to the other. As the fitness difference remains proportional to the replication rate realized in the particular experimental setup, viral strains having similar selection coefficients s may have large fitness differences. On the other hand, variants differing markedly in the selection coefficients will yield almost horizontal lines in logarithmic time plots when the realized replication rate in the experiment is sufficiently low. This has indeed caused confusion in the literature (see Discussion). Previous work on the fitness of human immunodeficiency virus type 1 (HIV-1) variants has indeed adopted the concept of a selection coefficient s from population genetics (2, 4). It is unfortunate, therefore, that the slope of the logarithmic time plot of the ratio of two variants in competition experiments has also been called a relative fitness. We here demonstrate that this slope gives the (absolute) fitness difference, and we develop a novel approach for estimating the generic relative fitness (1 +s) by competition experiments.

Tissue cell polarity plays a major role in plant and animal development. We propose that a fundamental building block for tissue cell polarity is the process of intracellular partitioning, which can establish individual cell polarity in the absence of asymmetric cues. Coordination of polarities may then arise through cell-cell coupling, which can operate directly, through membrane-spanning complexes, or indirectly, through diffusible molecules. Polarity is anchored to tissues through organisers located at boundaries. We show how this intracellular partitioning-based framework can be applied to both plant and animal systems, allowing different processes to be placed in a common evolutionary and mechanistic context.

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.

We examined the responses to transparent motion of complex cells in cat area 17 which show directional selectivity to moving random pixel arrays (RPAs). The response to an RPA moving in the cell's preferred direction is inhibited when a second RPA is transparently moving in another direction. The inhibition by the second pattern is quantified as a function of its direction. The response to a pattern moving in the preferred direction is never completely suppressed, not even when a second pattern is moving transparently in the opposite direction.To the extent that supra-spontaneous firing rates signal the presence of the optimal velocity vector, these cells therefore still signal the presence of this line-label stimulus despite additional opposing, or otherwise directed, motion components. The results confirm previous suggestions that, for the computation of motion energy in cat area 17 complex cells, a full opponent stage is not plausible. Furthermore, we show that the response to a combination of two motion vectors can be predicted by the average of the responses to the individual components. Copyright © 1996 Elsevier Science Ltd.

Recent work has demonstrated that following the clearance of infection a stable population of memory T cells remains present in peripheral organs and contributes to the control of secondary infections. However, little is known about how tissue-resident memory T cells behave in situ and how they encounter newly infected target cells. Here we demonstrate that antigen-specific CD8 + T cells that remain in skin following herpes simplex virus infection show a steady-state crawling behavior in between keratinocytes. Spatially explicit simulations of the migration of these tissue-resident memory T cells indicate that the migratory dendritic behavior of these cells allows the detection of antigen-expressing target cells in physiologically relevant time frames of minutes to hours. Furthermore, we provide direct evidence for the identification of rare antigen-expressing epithelial cells by skin-patrolling memory T cells in vivo. These data demonstrate the existence of skin patrol by memory T cells and reveal the value of this patrol in the rapid detection of renewed infections at a previously infected site.

Cell polarization is an important part of the response of eukaryotic cells to stimuli, and forms a primary step in cell motility, differentiation, and many cellular functions. Among the important biochemical players implicated in the onset of intracellular asymmetries that constitute the early phases of polarization are the Rho GTPases, such as Cdc42, Rac, and Rho, which present high active concentration levels in a spatially localized manner. Rho GTPases exhibit positive feedback-driven interconversion between distinct active and inactive forms, the former residing on the cell membrane, and the latter predominantly in the cytosol. A deterministic model of the dynamics of a single Rho GTPase described earlier by Mori et al. exhibits sustained polarization by a wave-pinning mechanism. It remained, however, unclear how such polarization behaves at typically low cellular concentrations, as stochasticity could significantly affect the dynamics. We therefore study the low copy number dynamics of this model, using a stochastic kinetics framework based on the Gillespie algorithm, and propose statistical and analytic techniques which help us analyse the equilibrium behaviour of our stochastic system. We use local perturbation analysis to predict parameter regimes for initiation of polarity and wave-pinning in our deterministic system, and compare these predictions with deterministic and stochastic spatial simulations. Comparing the behaviour of the stochastic with the deterministic system, we determine the threshold number of molecules required for robust polarization in a given effective reaction volume. We show that when the molecule number is lowered wave-pinning behaviour is lost due to an increasingly large transition zone as well as increasing fluctuations in the pinning position, due to which a broadness can be reached that is unsustainable, causing the collapse of the wave, while the variations in the high and low equilibrium levels are much less affected.

To initiate an adaptive immune response, T cells need to interact with dendritic cells (DCs), and the duration of these interactions plays an important role. In vitro and in vivo experiments have generally tried to estimate the required period of opportunity for T cell stimulation rather than the duration of individual T cell–DC interactions. Since the application of multi-photon microscopy (MPM) to living lymphoid tissues, the interactions between immune cells, as well as the duration thereof, can directly be observed in vivo. Indeed, long-lasting interactions between T cells and DCs were shown to be important for the onset of immune responses. However, because MPM imaging is typically restricted to experiments lasting 1 h, and because T cell–DC conjugates frequently move into and out of the imaged volume, it is difficult to estimate the true duration of interactions from MPM contact data. Here, we present a method to properly make such an estimate of (the average of) the distribution of contact durations. We validate the method by applying it to spatially explicit computer simulations where the true distribution of contact duration is known. Finally, we apply our analysis to a large experimental data set of T–DC contacts, and predict an average contact time of about three hours. However, we identify a mismatch between the experimental data and the model predictions, and investigate possible causes of the mismatch, including minor tissue drift during imaging experiments. We discuss in detail how future experiments can be optimized such that MPM contact data will be minimally affected by these factors.

In this chapter we examine the properties of the Cellular Potts Model (CPM) formalism which make it preeminently suitable for modelling biological cells. The most outstanding feature in which CPM differs from other modelling formalisms, is that a cell is modelled as a deformable object, and takes its shape from a combination of internal and external forces which act upon it. The energy minimisation based CPM formalism enables us to integrate these forces acting at different scales. We map the parameters of the basic CPM formalism to physical and biological properties of cells. We show through those mappings that the modelling formalism is a powerful tool for investigating a large range of biological questions, from those concerning biophysical properties of single cells, cell motion, tissue level properties, all the way up to understanding the full morphogenesis and life-cycle of an organism.

The plant growth regulator auxin controls cell identity, cell division and cell expansion. Auxin efflux facilitators (PINs) are associated with auxin maxima in distal regions of both shoots and roots. Here we model diffusion and PIN-facilitated auxin transport in and across cells within a structured root layout. In our model, the stable accumulation of auxin in a distal maximum emerges from the auxin flux pattern. We have experimentally tested model predictions of robustness and self-organization. Our model explains pattern formation and morphogenesis at timescales from seconds to weeks, and can be understood by conceptualizing the root as an ‘auxin capacitor’. A robust auxin gradient associated with the maximum, in combination with separable roles of auxin in cell division and cell expansion, is able to explain the formation, maintenance and growth of sharply bounded meristematic and elongation zones. Directional permeability and diffusion can fully account for stable auxin maxima and gradients that can instruct morphogenesis.

Adaptive immunity is initiated by T cell recognition of foreign peptides presented on dendritic cells (DCs) by major histocompatibility molecules. These interactions take place in secondary lymphoid tissues, such as lymph nodes (LNs) and spleen, and hence the anatomical structure of these tissues plays a crucial role in the development of immune responses. Two-photon microscopy (2PM) imaging in LNs suggests that T cells walk in a consistent direction for several minutes, pause briefly with a regular period, and then take off in a new, random direction. Here, we construct a spatially explicit model of T cell and DC migration in LNs and show that all dynamical properties of T cells could be a consequence of the densely packed LN environment. By means of 2PM experiments, we confirm that the large velocity fluctuations of T cells are indeed environmentally determined rather than resulting from an intrinsic motility program. Our simulations further predict that T cells self-organize into microscopically small, highly dynamic streams. We present experimental evidence for the presence of such turbulent streams in LNs. Finally, the model allows us to estimate the scanning rates of DCs (2,000 different T cells per hour) and T cells (100 different DCs per hour).

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.

Differential movement of individual cells within tissues is an important yet poorly understood process in biological development. Here we present a computational study of cell sorting caused by a combination of cell adhesion and chemotaxis, where we assume that all cells respond equally to the chemotactic signal. To capture in our model mesoscopic properties of biological cells, such as their size and deformability, we use the Cellular Potts Model, a multiscale, cell-based Monte Carlo model. We demonstrate a rich array of cell-sorting phenomena, which depend on a combination of mescoscopic cell properties and tissue level constraints. Under the conditions studied, cell sorting is a fast process, which scales linearly with tissue size. We demonstrate the occurrence of “absolute negative mobility”, which means that cells may move in the direction opposite to the applied force (here chemotaxis). Moreover, during the sorting, cells may even reverse the direction of motion. Another interesting phenomenon is “minority sorting”, where the direction of movement does not depend on cell type, but on the frequency of the cell type in the tissue. A special case is the cAMP-wave-driven chemotaxis of Dictyostelium cells, which generates pressure waves that guide the sorting. The mechanisms we describe can easily be overlooked in studies of differential cell movement, hence certain experimental observations may be misinterpreted., Synopsis The movements of biological cells during the development of an organism depend on the physical characteristics of these cells. Genetic regulation of (developmental) cell movements, in order to have an effect, must operate through or in accordance with these physical properties. With this framework in mind, the authors use a computational model to investigate the interaction of two important phenomena, namely differential adhesion and chemotaxis. Authors Käfer, Hogeweg, and Marée show that this interaction leads to fast cell sorting, a process during which the cells actually move in a specific direction, which can even be opposite to the direction of chemotaxis. The direction of motion does not only depend on intrinsic cell properties (such as cell size or the strength of the homotypic and heterotypic bonds) but also on the pattern formation that takes place during the sorting, as well as on general tissue properties. For example, both the formation of cell clusters and the level of confinement of the tissue can completely turn around the direction of sorting. The authors demonstrate that to fully understand cell tissue dynamics, it is essential to take into account mesoscopic cell properties, such as size, shape, and deformability. Because physically driven processes can profoundly influence the movement of biological cells, this should not be neglected in explanations for observed cell movement patterns.

Antigen therapy may hold great promise for the prevention of autoimmunity; however, most clinical trials have failed, suggesting that the principles guiding the choice of treatment remain ill defined. Here, we examine the antidiabetogenic properties of altered peptide ligands of CD8+ T cells recognizing an epitope of islet-specific glucose-6-phosphatase catalytic subunit–related protein (IGRP206–214), a prevalent population of autoreactive T cells in autoimmune diabetes. We show that islet-associated CD8+ T cells in nonobese diabetic mice recognize numerous IGRP epitopes, and that these cells have a role in the outcome of protocols designed to induce IGRP206–214-specific tolerance. Ligands targeting IGRP206–214-reactive T cells prevented disease, but only at doses that spared low-avidity clonotypes. Notably, near complete depletion of the IGRP206–214-reactive T-cell pool enhanced the recruitment of subdominant specificities and did not blunt diabetogenesis. Thus, peptide therapy in autoimmunity is most effective under conditions that foster occupation of the target organ lymphocyte niche by nonpathogenic, low-avidity clonotypes.