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Comparing theoretical DT precision to SDC precision for a single cell. (A) DT precision shows a maximum as a function of shape parameter f for any value of the profile lengthscale. (B) Ratio j of DT to SDC precision shows a crossover ( j ¼ 1) as a function of profile lengthscale l=a for 1D, 2D, and 3D geometries. Here j ¼ 50 is the central cell of N ¼ 100 target cells. For each value of ^ l the value of f which maximizes precision in the DT model (f à ) as seen in A is used.
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Morphogen profiles allow cells to determine their position within a developing organism, but not all morphogen profiles form by the same mechanism. Here, we derive fundamental limits to the precision of morphogen concentration sensing for two canonical mechanisms: the diffusion of morphogen through extracellular space and the direct transport of mo...
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Context 1
... method can at best reduce the variance by a factor of 2 as the noise from degradation is eliminated but the noise from arrival remains. This would cause negligible change to our results presented in Figure 3 and Figure 4 given the order of magnitude difference in precision seen between our two models. ...
Context 2
... transport process influences the precision only via m j , not t . For a given N, j, and ^ l, we find that the precision is maximized at a particular f à >0 ( Figure 3A). The reason is that an exponential profile (f ) 1) has constant steepness but small amplitude, whereas a power-law profile (f ( 1) has low steepness but large amplitude due to its long tail; the optimum is in between. ...
Context 3
... now observe how the precision of the DT and SDC models compare in a representative system. Figure 3B shows j ¼ P 2 DT =P 2 SDC as a function of profile length ^ l for a cell in the center (j ¼ N=2) of a line of N ¼ 100 target cells, where for each ^ l we use the f à that maximizes P 2 DT as seen in Figure 3A. Since b and T have been equated between the two models, j is independent of both. ...
Context 4
... now observe how the precision of the DT and SDC models compare in a representative system. Figure 3B shows j ¼ P 2 DT =P 2 SDC as a function of profile length ^ l for a cell in the center (j ¼ N=2) of a line of N ¼ 100 target cells, where for each ^ l we use the f à that maximizes P 2 DT as seen in Figure 3A. Since b and T have been equated between the two models, j is independent of both. ...
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Citations
... [1] To provide positional information, spatial concentration patterns such as morphogen gradients are often employed. [2][3][4][5][6][7] Generally, in the presence of a spatially heterogeneous concentration distribution of a certain molecule such as a morphogen, cells can infer their positions based on the readout of the local concentration of such a molecule. [1,2,[8][9][10][11] Various mechanisms for establishing nonuniform distribution of molecules have been extensively studied, e.g., the Turing mechanism, [12,13] the wave-pinning (WP) model, [14] the active transport (AT) model, [15][16][17] and the synthesis-degradationdiffusion (SDD) model. ...
... [1,2,[8][9][10][11] Various mechanisms for establishing nonuniform distribution of molecules have been extensively studied, e.g., the Turing mechanism, [12,13] the wave-pinning (WP) model, [14] the active transport (AT) model, [15][16][17] and the synthesis-degradationdiffusion (SDD) model. [4,7,18] The Turing mechanism usually involves two mutually interacting molecular species with strong nonlinearity and distinct diffusion constants, leading to the Turing instability, i.e., the destabilization of uniform concentration distribution. [12,13] As a result, the system spontaneously resides into a stable and usually spatially periodic concentration distribution. ...
... The SDD model is one of the most studied mechanisms for morphogen gradients. [4,7,18] It involves the synthesis, degradation, and passive diffusion of one molecular species. The synthesis term is usually considered as a spatially fixed source. ...
Positional information encoded in spatial concentration patterns is crucial for the development of multicellular organisms. However, it is still unclear how such information is affected by the physically dissipative diffusion process. Here we study one-dimensional patterning systems with analytical derivation and numerical simulations. We find that the diffusion constant of the patterning molecules exhibits a nonmonotonic effect on the readout of the positional information from the concentration patterns. Specifically, there exists an optimal diffusion constant that maximizes the positional information. Moreover, we find that the energy dissipation due to the physical diffusion imposes a fundamental upper limit on the positional information.
... i is the variance of the corresponding noise term, and the proportionality constants in Eqs. 4 and 5 are set by the normalization condition A 0 = P 0 = 1. Because Eqs. 4 and 5 are not single exponential decays, we define a characteristic timescale as [30] σ η ≈ 10% [12]), this gives ...
Cells maintain a stable size as they grow and divide. Inspired by the available experimental data, most proposed models for size homeostasis assume size-control mechanisms that act on a timescale of one generation. Such mechanisms lead to short-lived autocorrelations in size fluctuations that decay within less than two generations. However, recent evidence from comparing sister lineages suggests that correlations in size fluctuations can persist for many generations. Here we develop a minimal model that explains these seemingly contradictory results. Our model proposes that different environments result in different control parameters, leading to distinct inheritance patterns. Multigenerational memory is revealed in constant environments but obscured when averaging over many different environments. Inferring the parameters of our model from Escherichia coli size data in microfluidic experiments, we recapitulate the observed statistics. Our paper elucidates the impact of the environment on cell homeostasis and growth and division dynamics.
... More recently this model was applied at the level of a single cytoneme, which can pause and reverse, in a two-dimensional array of target cells [6]. A similar model was used to compare the steady-state gradient of morphogen resulting from cytoneme transport with that resulting from diffusion [12], and a recent computational model was used to analyze morphogen gradient establishment using cytonemes [1]. ...
Spatial distributions of morphogens provide positional information in developing systems, but how the distributions are established and maintained remains an open problem. Transport by diffusion has been the traditional mechanism, but recent experimental work has shown that cells can also communicate by filopodia-like structures called cytonemes that make direct cell-to-cell contacts. Here we investigate the roles each may play individually in a complex tissue and how they can jointly establish a reliable spatial distribution of a morphogen.
... Another attempt to compare both models, distribution through simple diffusion or through cytonemes, has been developed by modeling signaling in a simple monolayer of cells. Fancher and Mugler (2020) have found that modeling graded distribution through cytonemes fits better for short-range gradients, significantly diminishing noise, while large-range gradient shapes are better attained by simple diffusion. Thus, the authors propose that some signaling contexts might utilize cytonemes while others could be based on free diffusion, or a combination of both (Fancher & Mugler, 2020). ...
... Fancher and Mugler (2020) have found that modeling graded distribution through cytonemes fits better for short-range gradients, significantly diminishing noise, while large-range gradient shapes are better attained by simple diffusion. Thus, the authors propose that some signaling contexts might utilize cytonemes while others could be based on free diffusion, or a combination of both (Fancher & Mugler, 2020). However, these models are not taking into account potential tissue context-dependent obstacles either for diffusion or for cytonemes. ...
The function of Hedgehog (Hh) as a morphogen results from its long-distance distribution from producing to neighboring receiving cells within the developing tissue. This signal distribution enables, for example, the formation of a concentration gradient eliciting distinct cellular responses that will give rise to spatial patterning. Hh is a lipid modified protein and its dispersion is better guaranteed through cytonemes, cell protrusions that allow direct cell membrane contact and signal transfer at a distance. Hh and its receptor Patched (Ptc) meet at cytoneme contacts in a way that reminds synapses. Both Hh and Ptc require a recycling process prior to presentation in cytonemes. Increasing research on the role of cytonemes in Hh signaling is revealing cellular mechanisms that link signal transport through dynamic cytonemes with concurrent regulation of cell adhesion. The equilibrium between these two processes is being unveiled as crucial to both patterned morphogen distribution and signal transfer. In addition, these discoveries are pushing forward our understanding of the role of extracellular elements involved in the Hh pathway, such as the Hh coreceptors Ihog and Boi and the glypicans Dally and Dally-like protein (Dlp).
... This also can be achieved by resetting and repeating the search, which limits the search perimeter [10][11][12]. At a larger spatial scale with multiple cells, a concentration gradient of signaling molecules can be established in a short time [13][14][15][16], which is robust to parameter variation [13,14] and internal noise [15], and even precise under a noisy environment [17]. However, there are only a few direct theoretic analyses comparing these two fundamentally different mechanisms of signaling (direct transport and diffusion), and most such studies focus on the formation of a concentration gradient [16,17]. ...
... At a larger spatial scale with multiple cells, a concentration gradient of signaling molecules can be established in a short time [13][14][15][16], which is robust to parameter variation [13,14] and internal noise [15], and even precise under a noisy environment [17]. However, there are only a few direct theoretic analyses comparing these two fundamentally different mechanisms of signaling (direct transport and diffusion), and most such studies focus on the formation of a concentration gradient [16,17]. ...
Intercellular signaling has an important role in organism development, but not all communication occurs using the same mechanism. Here, we analyze the energy efficiency of intercellular signaling by two canonical mechanisms: diffusion of signaling molecules and direct transport mediated by signaling cellular protrusions. We show that efficient contact formation for direct transport can be established by an optimal rate of projecting protrusions, which depends on the availability of information about the location of the target cell. The optimal projection rate also depends on how signaling molecules are transported along the protrusion, in particular the ratio of the energy cost for contact formation and molecule synthesis. Also, we compare the efficiency of the two signaling mechanisms, under various model parameters. We find that the direct transport is favored over the diffusion when transporting a large amount of signaling molecules. There is a critical number of signaling molecules at which the efficiency of the two mechanisms are the same. The critical number is small when the distance between cells is far, which helps explain why protrusion-based mechanisms are observed in long-range cellular communications.
... Related studies are needed to better determine robustness and fragility of BMP systems with hindered feedback. Intriguingly, recent modeling work in the Drosophila wing imaginal disc indicates that cytonemes may allow for gradient formation without the addition of extrinsic noise, suggesting a potential division of labor between cytoneme-and diffusion-based mechanisms, depending on the noise sensitivity of a given patterning niche (Fancher and Mugler, 2020). Perhaps the prevalence and complexity of feedback loops in a given patterning niche may be indicative of the relative role of diffusion and cytonemes in gradient formation. ...
Pattern formation by bone morphogenetic proteins (BMPs) demonstrates remarkable plasticity and utility in several contexts, such as early embryonic development, tissue patterning and the maintenance of stem cell niches. BMPs pattern tissues over many temporal and spatial scales: BMP gradients as short as 1-2 cell diameters maintain the stem cell niche of the Drosophila germarium over a 24-h cycle, and BMP gradients of several hundred microns establish dorsal-ventral tissue specification in Drosophila , zebrafish and Xenopus embryos in timescales between 30 min and several hours. The mechanisms that shape BMP signaling gradients are also incredibly diverse. Although ligand diffusion plays a dominant role in forming the gradient, a cast of diffusible and non-diffusible regulators modulate gradient formation and confer robustness, including scale invariance and adaptability to perturbations in gene expression and growth. In this Review, we document the diverse ways that BMP gradients are formed and refined, and we identify the core principles that they share to achieve reliable performance.
Spatial distributions of morphogens provide positional information in developing systems, but how the distributions are established and maintained remains an open problem. Transport by diffusion has been the traditional mechanism, but recent experimental work has shown that cells can also communicate by filopodia-like structures called cytonemes that make direct cell-to-cell contacts. Here we investigate the roles each may play individually in a complex tissue and how they can jointly establish a reliable spatial distribution of a morphogen. To this end, we formulate models that capture fundamental aspects of various cytoneme-based transport mechanisms. In simple cases, exact solutions are attainable, and in more complex cases, we discuss results of numerical simulations.
Intercellular signaling has an important role in organism development, but not all communication occurs using the same mechanism. Here, we analyze the energy efficiency of intercellular signaling by two canonical mechanisms: Diffusion of signaling molecules and direct transport mediated by signaling cellular protrusions. We show that efficient contact formation for direct transport can be established by an optimal rate of projecting protrusions, which depends on the availability of information about the location of the target cell. The optimal projection rate also depends on how signaling molecules are transported along the protrusion, in particular the ratio of the energy cost for contact formation and molecule synthesis. Also, we compare the efficiency of the two signaling mechanisms, under various model parameters. We find that direct transport is favored over diffusion when transporting a large amount of signaling molecules. There is a critical number of signaling molecules at which the efficiencies of the two mechanisms are the same. The critical number is small when the distance between cells is far, which helps explain why protrusion-based mechanisms are observed in long-range cellular communications.
Tissue patterning is a critical part of animal development. Here we review the role that length- and timescales play in shaping patterns during development, focusing on the mechanisms by which Notch-mediated lateral inhibition signaling generates periodic tissue patterns. Because Notch ligands and receptors are membrane bound, the signaling that underlies lateral inhibition depends on direct cell-cell contacts. Nevertheless, there are many biological examples where effective Notch signaling occurs over distances larger than adjacent cells. Here, we summarize the theoretical and experimental evidence for mechanisms that modify the scale of Notch-mediated lateral inhibition. We focus on how cell protrusions, in addition to other cell behaviors like proliferation and neighbor exchange, allow for Notch signaling to both extend lateral inhibition beyond nearest neighbors and impact the timescale of patterning. Using recent examples, we examine how dynamic cell behaviors like the formation of protrusions affect the timing of Notch-mediated lateral inhibition as well as the density of the final tissue pattern. We suggest that mechanisms that affect the length and timescale of Notch signaling may have key implications for the evolution of patterns. This review highlights the role of cell behaviors in controlling the temporal and spatial dynamics of pattern formation across scales.
