Four different mechanisms for switching cell polarity.

The mechanisms and design principles of regulatory systems establishing stable polarized protein patterns within cells are well studied. However, cells can also dynamically control their cell polarity. Here, we ask how an upstream signaling system can switch the orientation of a polarized pattern. We use a mathematical model of a core polarity system based on three proteins as the basis to study different mechanisms of signal-induced polarity switching. The analysis of this model reveals four general classes of switching mechanisms with qualitatively distinct behaviors: the transient oscillator switch, the reset switch, the prime-release switch, and the push switch. Each of these regulatory mechanisms effectively implements the function of a spatial toggle switch, however with different characteristics in their nonlinear and stochastic dynamics. We identify these characteristics and also discuss experimental signatures of each type of switching mechanism.

Protein-protein interaction network controlling establishment and maintenance of switchable cell polarity.

Cell polarity underlies key processes in all cells, including growth, differentiation and division. In the bacterium Myxococcus xanthus, front-rear polarity is crucial for motility. Notably, this polarity can be inverted, independent of the cell-cycle, by chemotactic signaling. However, a precise understanding of the protein network that establishes polarity and allows for its inversion has remained elusive. Here, we use a combination of quantitative experiments and data-driven theory to unravel the complex interplay between the three key components of the M. xanthus polarity module. By studying each of these components in isolation and their effects as we systematically reconstruct the system, we deduce the network of effective interactions between the polarity proteins. RomR lies at the root of this network, promoting polar localization of the other components, while polarity arises from interconnected negative and positive feedbacks mediated by the small GTPase MglA and its cognate GAP MglB, respectively. We rationalize this network topology as operating as a spatial toggle switch, providing stable polarity for persistent cell movement whilst remaining responsive to chemotactic signaling and thus capable of polarity inversions. Our results have implications not only for the understanding of polarity and motility in M. xanthus but also, more broadly, for dynamic cell polarity.

Regulation of reaction fluxes via enzyme sequestration and co-clustering.

Experimental observations suggest that cells change the intracellular localization of key enzymes to regulate the reaction fluxes in enzymatic networks. In particular, cells appear to use sequestration and co-clustering of enzymes as spatial regulation strategies. These strategies should be equally useful to achieve rapid flux regulation in synthetic biomolecular systems. Here, we leverage a theoretical model to analyse the capacity of enzyme sequestration and co-clustering to control the reaction flux in a branch of a reaction-diffusion network. We find that in both cases, the response of the system is determined by two dimensionless parameters, the ratio of total activities of the competing enzymes and the ratio of diffusion to reaction timescales. Using these dependencies, we determine the parameter range for which sequestration and co-clustering can yield a biologically significant regulatory effect. Based on the known kinetic parameters of enzymes, we conclude that sequestration and co-clustering represent a viable regulation strategy for a large fraction of metabolic enzymes, and suggest design principles for reaction flux regulation in natural or synthetic systems.

Optimal Compartmentalization Strategies for Metabolic Microcompartments.

Intracellular compartmentalization of cooperating enzymes is a strategy that is frequently used by cells. Segregation of enzymes that catalyze sequential reactions can alleviate challenges such as toxic pathway intermediates, competing metabolic reactions, and slow reaction rates. Inspired by nature, synthetic biologists also seek to encapsulate engineered metabolic pathways within vesicles or proteinaceous shells to enhance the yield of industrially and pharmaceutically useful products. Although enzymatic compartments have been extensively studied experimentally, a quantitative understanding of the underlying design principles is still lacking. Here, we study theoretically how the size and enzymatic composition of compartments should be chosen so as to maximize the productivity of a model metabolic pathway. We find that maximizing productivity requires compartments larger than a certain critical size. The enzyme density within each compartment should be tuned according to a power-law scaling in the compartment size. We explain these observations using an analytically solvable, well-mixed approximation. We also investigate the qualitatively different compartmentalization strategies that emerge in parameter regimes where this approximation breaks down. Our results suggest that the different sizes and enzyme packings of α- and ß-carboxysomes each constitute an optimal compartmentalization strategy given the properties of their respective protein shells.

A Dual-Sensing Receptor Confers Robust Cellular Homeostasis.

Cells have evolved diverse mechanisms that maintain intracellular homeostasis in fluctuating environments. In bacteria, control is often exerted by bifunctional receptors acting as both kinase and phosphatase to regulate gene expression, a design known to provide robustness against noise. Yet how such antagonistic enzymatic activities are balanced as a function of environmental change remains poorly understood. We find that the bifunctional receptor that regulates K(+) uptake in Escherichia coli is a dual sensor, which modulates its autokinase and phosphatase activities in response to both extracellular and intracellular K(+) concentration. Using mathematical modeling, we show that dual sensing is a superior strategy for ensuring homeostasis when both the supply of and demand for a limiting resource fluctuate. By engineering standards, this molecular control system displays a strikingly high degree of functional integration, providing a reference for the vast numbers of receptors for which the sensing strategy remains elusive.

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.

Optimization of collective enzyme activity via spatial localization.

The spatial organization of enzymes often plays a crucial role in the functionality and efficiency of enzymatic pathways. To fully understand the design and operation of enzymatic pathways, it is therefore crucial to understand how the relative arrangement of enzymes affects pathway function. Here we investigate the effect of enzyme localization on the flux of a minimal two-enzyme pathway within a reaction-diffusion model. We consider different reaction kinetics, spatial dimensions, and loss mechanisms for intermediate substrate molecules. Our systematic analysis of the different regimes of this model reveals both universal features and distinct characteristics in the phenomenology of these different systems. In particular, the distribution of the second pathway enzyme that maximizes the reaction flux undergoes a generic transition from co-localization with the first enzyme when the catalytic efficiency of the second enzyme is low, to an extended profile when the catalytic efficiency is high. However, the critical transition point and the shape of the extended optimal profile is significantly affected by specific features of the model. We explain the behavior of these different systems in terms of the underlying stochastic reaction and diffusion processes of single substrate molecules.

Single-cell dynamics reveals sustained growth during diauxic shifts.

Stochasticity in gene regulation has been characterized extensively, but how it affects cellular growth and fitness is less clear. We study the growth of E. coli cells as they shift from glucose to lactose metabolism, which is characterized by an obligatory growth arrest in bulk experiments that is termed the lag phase. Here, we follow the growth dynamics of individual cells at minute-resolution using a single-cell assay in a microfluidic device during this shift, while also monitoring lac expression. Mirroring the bulk results, the majority of cells displays a growth arrest upon glucose exhaustion, and resume when triggered by stochastic lac expression events. However, a significant fraction of cells maintains a high rate of elongation and displays no detectable growth lag during the shift. This ability to suppress the growth lag should provide important selective advantages when nutrients are scarce. Trajectories of individual cells display a highly non-linear relation between lac expression and growth, with only a fraction of fully induced levels being sufficient for achieving near maximal growth. A stochastic molecular model together with measured dependencies between nutrient concentration, lac expression level, and growth accurately reproduces the observed switching distributions. The results show that a growth arrest is not obligatory in the classic diauxic shift, and underscore that regulatory stochasticity ought to be considered in terms of its impact on growth and survival.

Spatial partitioning improves the reliability of biochemical signaling.

Spatial heterogeneity is a hallmark of living systems, even at the molecular scale in individual cells. A key example is the partitioning of membrane-bound proteins via lipid domain formation or cytoskeleton-induced corralling. However, the impact of this spatial heterogeneity on biochemical signaling processes is poorly understood. Here, we demonstrate that partitioning improves the reliability of biochemical signaling. We exactly solve a stochastic model describing a ubiquitous motif in membrane signaling. The solution reveals that partitioning improves signaling reliability via two effects: it moderates the nonlinearity of the switching response, and it reduces noise in the response by suppressing correlations between molecules. An optimal partition size arises from a trade-off between minimizing the number of proteins per partition to improve signaling reliability and ensuring sufficient proteins per partition to maintain signal propagation. The predicted optimal partition size agrees quantitatively with experimentally observed systems. These results persist in spatial simulations with explicit diffusion barriers. Our findings suggest that molecular partitioning is not merely a consequence of the complexity of cellular substructures, but also plays an important functional role in cell signaling.

Clustering and optimal arrangement of enzymes in reaction-diffusion systems.

Enzymes within biochemical pathways are often colocalized, yet the consequences of specific spatial enzyme arrangements remain poorly understood. We study the impact of enzyme arrangement on reaction efficiency within a reaction-diffusion model. The optimal arrangement transitions from a cluster to a distributed profile as a single parameter, which controls the probability of reaction versus diffusive loss of pathway intermediates, is varied. We introduce the concept of enzyme exposure to explain how this transition arises from the stochastic nature of molecular reactions and diffusion.

Signaling noise enhances chemotactic drift of E. coli.

Noise in the transduction of chemotactic stimuli to the flagellar motor of E. coli will affect the random run-and-tumble motion of the cell and the ability to perform chemotaxis. Here we use numerical simulations to show that an intermediate level of noise in the slow methylation dynamics enhances drift while not compromising localization near concentration peaks. A minimal model shows how such an optimal noise level arises from the interplay of noise and the dependence of the motor response on the network output. Our results suggest that cells can exploit noise to improve chemotactic performance.

Reliability of frequency and amplitude decoding in gene regulation.

In biochemical signaling, information is often encoded in oscillatory signals. However, the advantages of such a coding strategy over an amplitude-encoding scheme of constant signals remain unclear. Here we study the dynamics of a simple model gene promoter in response to oscillating and constant transcription factor signals. We find that in biologically relevant parameter regimes an oscillating input can produce a more constant protein level than a constant input. Our results suggest that oscillating signals may be used to minimize noise in gene regulation.

Multiplexing biochemical signals.

In this Letter we show that living cells can multiplex biochemical signals, i.e., transmit multiple signals through the same signaling pathway simultaneously, and yet respond to them very specifically. We demonstrate how two binary input signals can be encoded in the concentration of a common signaling protein, which is then decoded such that each of the two output signals provides reliable information about one corresponding input. Under biologically relevant conditions the network can reach the maximum amount of information that can be transmitted, which is 2 bits.

Precision of sensing cell length via concentration gradients.

Unicellular organisms are typically found to have a characteristic cell size. To achieve a homeostatic distribution of cell sizes over many generations requires that cell length is actively sensed and regulated. However, the mechanisms by which cell size is controlled remain poorly understood. Recent experiments in fission yeast have shown that cell length is controlled in part by polar gradients of the protein Pom1 together with localized measurement of concentration at midcell. Dilution as the cell grows leads to a reduction in the midcell protein concentration, which lifts a block on mitosis. Here we analyze the precision of this mechanism for length sensing in the presence of inevitable intrinsic noise in the processes leading to formation and measurement of this gradient. We find that the use of concentration gradients allows for more robust length sensing than a comparable spatially uniform system, and allows for reliable length determination even if the average protein concentration throughout the cell remains constant as the cell grows. Optimal values for the gradient decay length and receptor dissociation constant emerge from maximizing sensitivity while minimizing the impact of density fluctuations.

Mutual information in time-varying biochemical systems.

Cells must continuously sense and respond to time-varying environmental stimuli. These signals are transmitted and processed by biochemical signaling networks. However, the biochemical reactions making up these networks are intrinsically noisy, which limits the reliability of intracellular signaling. Here we use information theory to characterize the reliability of transmission of time-varying signals through elementary biochemical reactions in the presence of noise. We calculate the mutual information for both instantaneous measurements and trajectories of biochemical systems for a Gaussian model. Our results indicate that the same network can have radically different characteristics for the transmission of instantaneous signals and trajectories. For trajectories, the ability of a network to respond to changes in the input signal is determined by the timing of reaction events, and is independent of the correlation time of the output of the network. We also study how reliably signals on different time scales can be transmitted by considering the frequency-dependent coherence and gain-to-noise ratio. We find that a detector that does not consume the ligand molecule upon detection can more reliably transmit slowly varying signals, while an absorbing detector can more reliably transmit rapidly varying signals. Furthermore, we find that while one reaction may more reliably transmit information than another when considered in isolation, when placed within a signaling cascade the relative performance of the two reactions can be reversed. This means that optimizing signal transmission at a single level of a signaling cascade can reduce signaling performance for the cascade as a whole.

Effect of feedback on the fidelity of information transmission of time-varying signals.

Living cells are continually exposed to environmental signals that vary in time. These signals are detected and processed by biochemical networks, which are often highly stochastic. To understand how cells cope with a fluctuating environment, we therefore have to understand how reliably biochemical networks can transmit time-varying signals. To this end, we must understand both the noise characteristics and the amplification properties of networks. In this paper, we use information theory to study how reliably signaling cascades employing autoregulation and feedback can transmit time-varying signals. We calculate the frequency dependence of the gain-to-noise ratio, which reflects how reliably a network transmits signals at different frequencies. We find that the gain-to-noise ratio may differ qualitatively from the power spectrum of the output, showing that the latter does not directly reflect signaling performance. Moreover, we find that autoactivation and autorepression increase and decrease the gain-to-noise ratio for all of frequencies, respectively. Positive feedback specifically enhances information transmission at low frequencies, while negative feedback increases signal fidelity at high frequencies. Our analysis not only elucidates the role of autoregulation and feedback in naturally occurring biological networks, but also reveals design principles that can be used for the reliable transmission of time-varying signals in synthetic gene circuits.

Mutual information between input and output trajectories of biochemical networks.

Biochemical networks can respond to temporal characteristics of time-varying signals. To understand how reliably biochemical networks can transmit information we must consider how an input signal as a function of time--the input trajectory--can be mapped onto an output trajectory. Here we estimate the mutual information between input and output trajectories using a Gaussian model. We study how reliably the chemotaxis network of E. coli can transmit information on the ligand concentration to the flagellar motor, and find the input power spectrum that maximizes the information transmission rate.

Modeling the establishment of PAR protein polarity in the one-cell C. elegans embryo.

At the one-cell stage, the C. elegans embryo becomes polarized along the anterior-posterior axis. The PAR proteins form complementary anterior and posterior domains in a dynamic process driven by cytoskeletal rearrangement. Initially, the PAR proteins are uniformly distributed throughout the embryo. After a cue from fertilization, cortical actomyosin contracts toward the anterior pole. PAR-3/PAR-6/PKC-3 (the anterior PAR proteins) become restricted to the anterior cortex. PAR-1 and PAR-2 (the posterior PAR proteins) become enriched in the posterior cortical region. We present a mathematical model of this polarity establishment process, in which we take a novel approach to combine reaction-diffusion dynamics of the PAR proteins coupled to a simple model of actomyosin contraction. We show that known interactions between the PAR proteins are sufficient to explain many aspects of the observed cortical PAR dynamics in both wild-type and mutant embryos. However, cytoplasmic PAR protein polarity, which is vital for generating daughter cells with distinct molecular components, cannot be properly explained within such a framework. We therefore consider additional mechanisms that can reproduce the proper cytoplasmic polarity. In particular we predict that cytoskeletal asymmetry in the cytoplasm, in addition to the cortical actomyosin asymmetry, is a critical determinant of PAR protein localization.

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.

A stochastic model of Min oscillations in Escherichia coli and Min protein segregation during cell division.

The Min system in Escherichia coli directs division to the centre of the cell through pole-to-pole oscillations of the MinCDE proteins. We present a one-dimensional stochastic model of these oscillations which incorporates membrane polymerization of MinD into linear chains. This model reproduces much of the observed phenomenology of the Min system, including pole-to-pole oscillations of the Min proteins. We then apply this model to investigate the Min system during cell division. Oscillations continue initially unaffected by the closing septum, before cutting off rapidly. The fractions of Min proteins in the daughter cells vary widely, from 50%-50% up to 85%-15% of the total from the parent cell, suggesting that there may be another mechanism for regulating these levels in vivo.