Fundamental limits to position determination by concentration gradients

Position determination in biological systems is often achieved through protein concentration gradients. Measuring the local concentration of such a protein with a spatially varying distribution allows the measurement of position within the system. For these systems to work effectively, position determination must be robust to noise. Here, we calculate fundamental limits to the precision of position determination by concentration gradients due to unavoidable biochemical noise perturbing the gradients. We focus on gradient proteins with first-order reaction kinetics. Systems of this type have been experimentally characterised in both developmental and cell biology settings. For a single gradient we show that, through time-averaging, great precision potentially can be achieved even with very low protein copy numbers. As a second example, we investigate the ability of a system with oppositely directed gradients to find its centre. With this mechanism, positional precision close to the centre improves more slowly with increasing averaging time, and so longer averaging times or higher copy numbers are required for high precision. For both single and double gradients, we demonstrate the existence of optimal length scales for the gradients for which precision is maximized, as well as analyze how precision depends on the size of the concentration-measuring apparatus. These results provide fundamental constraints on the positional precision supplied by concentration gradients in various contexts, including both in developmental biology and also within a single cell.     

Shaping a Morphogen Gradient for Positional Precision

Morphogen gradients, which provide positional information to cells in a developing tissue, could in principle adopt any nonuniform profile. To our knowledge, how the profile of a morphogen gradient affects positional precision has not been well studied experimentally. Here, we compare the positional precision provided by the Drosophila morphogenetic protein Bicoid (Bcd) in wild-type (wt) embryos with embryos lacking an interacting cofactor. The Bcd gradient in the latter case exhibits decreased positional precision around mid-embryo compared with its wt counterpart. The domain boundary of Hunchback (Hb), a target activated by Bcd, becomes more variable in mutant embryos. By considering embryo-to-embryo, internal, and measurement fluctuations, we dissect mathematically the relevant sources of fluctuations that contribute to the error in positional information. Using this approach, we show that the defect in Hb boundary positioning in mutant embryos is directly reflective of an altered Bcd gradient profile with increasing flatness toward mid-embryo. Furthermore, we find that noise in the Bcd input signal is dominated by internal fluctuations but, due to time and spatial averaging, the spatial precision of the Hb boundary is primarily affected by embryo-to-embryo variations. Our results demonstrate that the positional information provided by the wt Bcd gradient profile is highly precise and necessary for patterning precision.     

The Berg-Purcell Limit Revisited

Biological systems often have to measure extremely low concentrations of chemicals with high precision. When dealing with such small numbers of molecules, the inevitable randomness of physical transport processes and binding reactions will limit the precision with which measurements can be made. An important question is what the lower bound on the noise would be in such measurements. Using the theory of diffusion-influenced reactions, we derive an analytical expression for the precision of concentration estimates that are obtained by monitoring the state of a receptor to which a diffusing ligand can bind. The variance in the estimate consists of two terms, one resulting from the intrinsic binding kinetics and the other from the diffusive arrival of ligand at the receptor. The latter term is identical to the fundamental limit derived by Berg and Purcell (Biophys. J., 1977), but disagrees with a more recent expression by Bialek and Setayeshgar. Comparing the theoretical predictions against results from particle-based simulations confirms the accuracy of the resulting expression and reaffirms the fundamental limit established by Berg and Purcell.     

A second ‘overexpression’ allele at the <Emphasis Type="Italic">Glu-B1</Emphasis> high-molecular-weight glutenin locus of wheat: sequence characterisation and functional effects

Bread is one of the major constituents of the human diet and wheat (Triticum aestivum L.) is the most important cereal for bread making. The gluten proteins (glutenins and gliadins) are recognised as important components affecting the processing quality of wheat flour. In this research, we investigated a particular glutenin subunit allele in an Australian cultivar, H45. Based on protein and DNA assays, the Glu-B1 allele of H45 seems to be Glu-B1al, an allele that includes a functional duplication of a gene encoding an x-type high-molecular-weight glutenin subunit, and is thought to increase dough strength through overexpression of that subunit. Yet H45 does not have the dough properties that would be expected if it carries the Glu-B1al allele. After confirming that H45 overexpresses Bx subunits and that it has relatively low un-extractable polymeric protein (an indicator of weak dough), we cloned and sequenced two Bx genes from H45. The sequences of the two genes differ from each other, and they each differ by four single-nucleotide polymorphisms (SNPs) from the sequence that has been reported for the Glu-B1al x-type glutenin genes of the Canadian wheat cultivar Glenlea. One of the SNPs leads to an extra cysteine residue in one of the subunits. The presence of this additional cysteine may explain the dough properties of H45 through effects on cross-linkage within or between glutenin subunits. We propose that the Glu-B1 allele of H45 be designated Glu-B1br, and we present evidence that Glu-B1br is co-inherited with low un-extractable polymeric protein.     

Restoration of native vegetation following exclosure establishment on communal grazing lands in Tigray,Ethiopia

Questions: Is plant species richness, diversity and above‐ground standing biomass enhanced after establishing exclosures on communal grazing lands? What factors influence the effectiveness of exclosures to restore degraded native vegetation in Tigray, Ethiopia? Location: Northern Ethiopia. Methods: We used a space‐for‐time substitution approach to detect changes in plant species richness, diversity and above‐ground standing biomass after conversion of communal grazing lands to exclosures. We selected replicated (n=3) 5‐, 10‐, 15‐ and 20‐year‐old exclosures and paired each exclosure with an adjacent communal grazing land to ensure that soil and terrain conditions were as similar as possible among each pair. Results: All exclosures displayed higher plant species richness, diversity and biomass than the communal grazing lands. Differences in plant species richness and biomass between an exclosure age and adjacent communal grazing land were higher in oldest than in youngest exclosures. In exclosures, much of the variability in plant species composition and biomass was explained by a combination of edaphic (total nitrogen, phosphorus, texture and soil pH) and site (precipitation and altitude) variables (R²=0.72–0.82). Edaphic and site variables also explained much of the variability in plant species composition in communal grazing lands (R²=0.76–0.82). Our study shows that all exclosures are at an early stage of succession. The increase in economically important indigenous shrub and tree species with exclosure age suggests that, with time, a valuable afromontane forest may develop. Conclusions: Establishment of exclosures on communal grazing lands is a viable option to restore degraded native vegetation. However, before expanding exclosures, the ecological consequences of additional exclosures should be investigated as further expansion of exclosures could increase grazing pressure on remaining grazing areas. Furthermore, consideration of edaphic and site variables will help optimize selection of areas for establishment of exclosures and enhance natural regeneration in exclosures in the future.     

Regulatory control and the costs and benefits of biochemical noise

Experiments in recent years have vividly demonstrated that gene expression can be highly stochastic. How protein concentration fluctuations affect the growth rate of a population of cells is, however, a wide-open question. We present a mathematical model that makes it possible to quantify the effect of protein concentration fluctuations on the growth rate of a population of genetically identical cells. The model predicts that the population's growth rate depends on how the growth rate of a single cell varies with protein concentration, the variance of the protein concentration fluctuations, and the correlation time of these fluctuations. The model also predicts that when the average concentration of a protein is close to the value that maximizes the growth rate, fluctuations in its concentration always reduce the growth rate. However, when the average protein concentration deviates sufficiently from the optimal level, fluctuations can enhance the growth rate of the population, even when the growth rate of a cell depends linearly on the protein concentration. The model also shows that the ensemble or population average of a quantity, such as the average protein expression level or its variance, is in general not equal to its time average as obtained from tracing a single cell and its descendants. We apply our model to perform a cost-benefit analysis of gene regulatory control. Our analysis predicts that the optimal expression level of a gene regulatory protein is determined by the trade-off between the cost of synthesizing the regulatory protein and the benefit of minimizing the fluctuations in the expression of its target gene. We discuss possible experiments that could test our predictions.     

Phloem unloading of potato virus X movement proteins is regulated by virus and host factors

To determine the requirements for viral proteins exiting the phloem, transgenic plants expressing green fluorescent protein (GFP) fused to the Potato virus X (PVX) triple gene block (TGB)p1 and coat protein (CP) genes were prepared. The fused genes were transgenically expressed from the companion cell (CC)-specific Commelina yellow mottle virus (CoYMV) promoter. Transgenic plants were selected for evidence of GFP fluorescence in CC and sieve elements (SE) and proteins were determined to be phloem mobile based on their ability to translocate across a graft union into nontransgenic scions. Petioles and leaves were analyzed to determine the requirements for phloem unloading of the fluorescence proteins. In petioles, fluorescence spread throughout the photosynthetic vascular cells (chlorenchyma) but did not move into the cortex, indicating a specific barrier to proteins exiting the vasculature. In leaves, fluorescence was mainly restricted to the veins. However, in virus-infected plants or leaves treated with a cocktail of proteasome inhibitors, fluorescence spread into leaf mesophyll cells. These data indicate that PVX contributes factors which enable specific unloading of cognate viral proteins and that proteolysis may play a role in limiting proteins in the phloem and surrounding chlorenchyma.     

The switching dynamics of the bacterial flagellar motor

Many bacteria are propelled by flagellar motors that stochastically switch between the clockwise and counterclockwise rotation direction. Although the switching dynamics is one of their most important characteristics, the mechanisms that control it are poorly understood. We present a statistical–mechanical model of the bacterial flagellar motor. At its heart is the assumption that the rotor protein complex, which is connected to the flagellum, can exist in two conformational states and that switching between these states depends on the interactions with the stator proteins, which drive the rotor. This couples switching to rotation, making the switch sensitive to torque and speed. Another key element is that after a switch, it takes time for the load to build up, due to conformational transitions of the flagellum. This slow relaxation dynamics of the filament leads, in combination with the load dependence of the switching frequency, to a characteristic switching time, as recently observed. Hence, our model predicts that the switching dynamics is not only controlled by the chemotaxis‐signaling network, but also by mechanical feedback of the flagellum.     

Diffusion of transcription factors can drastically enhance the noise in gene expression

We study by Green's Function Reaction Dynamics the effect of the diffusive motion of repressor molecules on the noise in mRNA and protein levels for a gene that is under the control of a repressor. We find that spatial fluctuations due to diffusion can drastically enhance the noise in gene expression. After dissociation from the operator, a repressor can rapidly rebind to the DNA. Our results show that the rebinding trajectories are so short that, on this timescale, the RNA polymerase (RNAP) cannot effectively compete with the repressor for binding to the promoter. As a result, a dissociated repressor molecule will on average rebind many times, before it eventually diffuses away. These rebindings thus lower the effective dissociation rate, and this increases the noise in gene expression. Another consequence of the timescale separation between repressor rebinding and RNAP association is that the effect of spatial fluctuations can be described by a well-stirred, zero-dimensional, model by renormalizing the reaction rates for repressor-DNA (un) binding. Our results thus support the use of well-stirred, zero-dimensional models for describing noise in gene expression. We also show that for a fixed repressor strength, the noise due to diffusion can be minimized by increasing the number of repressors or by decreasing the rate of the open complex formation. Lastly, our results emphasize that power spectra are a highly useful tool for studying the propagation of noise through the different stages of gene expression.