SummarySecreted signals, known as morphogens, provide the positional information that organizes gene expression and cellular differentiation in many developing tissues. In the vertebrate neural tube, Sonic Hedgehog (Shh) acts as a morphogen to control the pattern of neuronal subtype specification. Using an in vivo reporter of Shh signaling, mouse genetics, and systems modeling, we show that a spatially and temporally changing gradient of Shh signaling is interpreted by the regulatory logic of a downstream transcriptional network. The design of the network, which links three transcription factors to Shh signaling, is responsible for differential spatial and temporal gene expression. In addition, the network renders cells insensitive to fluctuations in signaling and confers hysteresis—memory of the signal. Our findings reveal that morphogen interpretation is an emergent property of the architecture of a transcriptional network that provides robustness and reliability to tissue patterning.

In the Ultimatum Game, two players are offered a chance to win a certain sum of money. All they must do is divide it. The proposer suggests how to split the sum. The responder can accept or reject the deal. If the deal is rejected, neither player gets anything. The rational solution, suggested by game theory, is for the proposer to offer the smallest possible share and for the responder to accept it. If humans play the game, however, the most frequent outcome is a fair share. In this paper, we develop an evolutionary approach to the Ultimatum Game. We show that fairness will evolve if the proposer can obtain some information on what deals the responder has accepted in the past. Hence, the evolution of fairness, similarly to the evolution of cooperation, is linked to reputation.

In the vertebrate neural tube, the morphogen Sonic Hedgehog (Shh) establishes a characteristic pattern of gene expression. Here we quantify the Shh gradient in the developing mouse neural tube and show that while the amplitude of the gradient increases over time, the activity of the pathway transcriptional effectors, Gli proteins, initially increases but later decreases. Computational analysis of the pathway suggests three mechanisms that could contribute to this adaptation: transcriptional upregulation of the inhibitory receptor Ptch1, transcriptional downregulation of Gli and the differential stability of active and inactive Gli isoforms. Consistent with this, Gli2 protein expression is downregulated during neural tube patterning and adaptation continues when the pathway is stimulated downstream of Ptch1. Moreover, the Shh-induced upregulation of Gli2 transcription prevents Gli activity levels from adapting in a different cell type, NIH3T3 fibroblasts, despite the upregulation of Ptch1. Multiple mechanisms therefore contribute to the intracellular dynamics of Shh signalling, resulting in different signalling dynamics in different cell types.

In the ultimatum game, two players are asked to split a certain sum of money. The proposer has to make an offer. If the responder accepts the offer, the money will be shared accordingly. If the responder rejects the offer, both players receive nothing. The rational solution is for the proposer to offer the smallest possible share, and for the responder to accept it. Human players, in contrast, usually prefer fair splits. In this paper, we use evolutionary game theory to analyse the ultimatum game. We first show that in a non-spatial setting, natural selection chooses the unfair, rational solution. In a spatial setting, however, much fairer outcomes evolve.

Spontaneous pattern formation in reaction-diffusion systems on a spatially homogeneous domain has been well studied. However, in embryonic development and elsewhere, pattern formation often takes place on a spatially heterogeneous background. We explore the effects of spatially varying parameters on pattern formation in one and two dimensions using the Gierer-Meinhardt reaction-diffusion model. We investigate the effect of the wavelength of a pre-pattern and demonstrate a novel form of moving pattern. We find that spatially heterogeneous parameters can both increase the range and complexity of possible patterns and enhance the robustness of pattern selection.

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Darwinian evolution is based on three fundamental principles, reproduction, mutation and selection, which describe how populations change over time and how new forms evolve out of old ones. There are numerous mathematical descriptions of the resulting evolutionary dynamics. In this paper, we show that apparently very different formulations are part of a single unified framework. At the center of this framework is the equivalence between the replicator-mutator equation and the Price equation. From these equations, we obtain as special cases adaptive dynamics, evolutionary game dynamics, the Lotka-Volterra equation of ecology and the quasispecies equation of molecular evolution.

The pattern of gene expression in a developing tissue determines the spatial organization of cell type generation. We previously defined regulatory interactions between a set of transcription factors that specify the pattern of gene expression in progenitors of different neuronal subtypes of the vertebrate neural tube. These transcription factors form a circuit that acts as a multistate switch, patterning the tissue in response to a gradient of Sonic Hedgehog. Here, by simplifying aspects of the regulatory interactions, we found that the topology of the circuit allows either switch-like or oscillatory behaviour depending on parameter values. The qualitative dynamics appear to be controlled by a simpler sub-circuit, which we term the AC–DC motif. We argue that its topology provides a natural way to implement a multistate gene expression switch and we show that the circuit is readily extendable to produce more distinct stripes of gene expression. Our analysis also suggests that AC–DC motifs could be deployed in tissues patterned by oscillatory mechanisms, thus blurring the distinction between pattern-formation mechanisms relying on temporal oscillations or graded signals. Furthermore, during evolution, mechanisms of gradient interpretation might have arisen from oscillatory circuits, or vice versa.