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.

Graphical AbstractHighlights d Mechanical stress releases p120-catenin from apical junctions d Loss of p120-catenin increases endocytic E-cadherin turnover d Loss of p120 speeds stress-dependent remodeling of the junctional network d Mechanical stress dependence of E-cadherin turnover sets tissue viscoelasticity SUMMARY Studying how epithelia respond to mechanical stresses is key to understanding tissue shape changes during morphogenesis. Here, we study the viscoelastic properties of the Drosophila wing epithelium during pupal morphogenesis by quantifying mechanical stress and cell shape as a function of time. We find a delay of 8 h between maximal tissue stress and maximal cell elongation, indicating a viscoelastic deformation of the tissue. We show that this viscoelastic behavior emerges from the mechanosensitivity of endocytic E-cadherin turnover. The increase in E-cadherin turnover in response to stress is mediated by mechanosensitive relocalization of the E-cadherin binding protein p120-catenin (p120) from cell junctions to cytoplasm. Mechanosensitivity of E-cadherin turnover is lost in p120 mutant wings, where E-cadherin turnover is constitutively high. In this mutant, the relationship between mechanical stress and stress-dependent cell dynamics is altered. Cells in p120 mutant deform and undergo cell rearrangements oriented along the stress axis more rapidly in response to mechanical stress. These changes imply a lower viscosity of wing epithelium. Taken together, our findings reveal that p120-dependent mechanosensitive E-cadherin turnover regulates viscoelastic behavior of epithelial tissues.

The principal pacemaker of the circadian clock of the cyanobacterium S. elongatus is a protein phosphorylation cycle consisting of three proteins, KaiA, KaiB and KaiC. KaiC forms a homohexamer, with each monomer consisting of two domains, CI and CII. Both domains can bind and hydrolyze ATP, but only the CII domain can be phosphorylated, at two residues, in a well-defined sequence. While this system has been studied extensively, how the clock is driven thermodynamically has remained elusive. Inspired by recent experimental observations and building on ideas from previous mathematical models, we present a new, thermodynamically consistent, statistical-mechanical model of the clock. At its heart are two main ideas: i) ATP hydrolysis in the CI domain provides the thermodynamic driving force for the clock, switching KaiC between an active conformational state in which its phosphorylation level tends to rise and an inactive one in which it tends to fall; ii) phosphorylation of the CII domain provides the timer for the hydrolysis in the CI domain. The model also naturally explains how KaiA, by acting as a nucleotide exchange factor, can stimulate phosphorylation of KaiC, and how the differential affinity of KaiA for the different KaiC phosphoforms generates the characteristic temporal order of KaiC phosphorylation. As the phosphorylation level in the CII domain rises, the release of ADP from CI slows down, making the inactive conformational state of KaiC more stable. In the inactive state, KaiC binds KaiB, which not only stabilizes this state further, but also leads to the sequestration of KaiA, and hence to KaiC dephosphorylation. Using a dedicated kinetic Monte Carlo algorithm, which makes it possible to efficiently simulate this system consisting of more than a billion reactions, we show that the model can describe a wealth of experimental data.

Many organisms possess both a cell cycle to control DNA replication and a circadian clock to anticipate changes between day and night. In some cases, these two rhythmic systems are known to be coupled by specific, cross-regulatory interactions. Here, we use mathematical modeling to show that, additionally, the cell cycle generically influences circadian clocks in a nonspecific fashion: The regular, discrete jumps in gene-copy number arising from DNA replication during the cell cycle cause a periodic driving of the circadian clock, which can dramatically alter its behavior and impair its function. A clock built on negative transcriptional feedback either phase-locks to the cell cycle, so that the clock period tracks the cell division time, or exhibits erratic behavior. We argue that the cyanobacterium Synechococcus elongatus has evolved two features that protect its clock from such disturbances, both of which are needed to fully insulate it from the cell cycle and give it its observed robustness: a phosphorylation-based protein modification oscillator, together with its accompanying push-pull read-out circuit that responds primarily to the ratios of different phosphoform concentrations, makes the clock less susceptible to perturbations in protein synthesis; the presence of multiple, asynchronously replicating copies of the same chromosome diminishes the effect of replicating any single copy of a gene.Kai | circadian rhythms | cell cycle | oscillations | simulation C ircadian clocks-autonomous oscillators with a roughly 24-h period that can be entrained to daily cycles of light and dark-are thought to confer important advantages on living cells by allowing them to anticipate diurnal environmental changes. Recent decades have seen considerable progress in elucidating both the architecture and the function of these biological timekeepers. Circadian clocks, however, are not the only oscillatory systems present in living cells. Most notably, cell growth and division are governed by a cell cycle, which can in many contexts be viewed as an autonomous oscillator. Much recent attention has been directed toward the connections between these two rhythmic systems, which are relevant for processes ranging from plants' response to shade (1) to cancer susceptibility (2, 3). In particular, it is now clear that circadian clocks can exert specific regulatory influences on the cell cycle, and a number of experimental and modeling studies have sought to tease out the implications of this regulation (4-11). Here, we argue that, in addition to direct, specific regulation of one oscillator by the other, there must also be more generic connections between the circadian clock and the cell cycle (2, 10-12). In particular, we focus on the consequences of the discrete gene replication events that accompany DNA replication. We show that, as a result of the regular jumps in gene copy number caused by these events, the cell cycle must, very generally, contribute a periodic forcing to the circadian clock. This forcing can markedly change clock ...