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 phosphorylation cycle shapes gradients of the DYRK family kinase Pom1 at the plasma membrane

Concentration gradients regulate many cell biological and developmental processes. In rod-shaped fission yeast cells, polar cortical gradients of the DYRK family kinase Pom1 couple cell length with mitotic commitment by inhibiting a mitotic inducer positioned at midcell. However, how Pom1 gradients are established is unknown. Here, we show that Tea4, which is normally deposited at cell tips by microtubules, is both necessary and, upon ectopic cortical localization, sufficient to recruit Pom1 to the cell cortex. Pom1 then moves laterally at the plasma membrane, which it binds through a basic region exhibiting direct lipid interaction. Pom1 autophosphorylates in this region to lower lipid affinity and promote membrane release. Tea4 triggers Pom1 plasma membrane association by promoting its dephosphorylation through the protein phosphatase 1 Dis2. We propose that local dephosphorylation induces Pom1 membrane association and nucleates a gradient shaped by the opposing actions of lateral diffusion and autophosphorylation-dependent membrane detachment.     

Mathematical model of a cell size checkpoint

How cells regulate their size from one generation to the next has remained an enigma for decades. Recently, a molecular mechanism that links cell size and cell cycle was proposed in fission yeast. This mechanism involves changes in the spatial cellular distribution of two proteins, Pom1 and Cdr2, as the cell grows. Pom1 inhibits Cdr2 while Cdr2 promotes the G2 → M transition. Cdr2 is localized in the middle cell region (midcell) whereas the concentration of Pom1 is highest at the cell tips and declines towards the midcell. In short cells, Pom1 efficiently inhibits Cdr2. However, as cells grow, the Pom1 concentration at midcell decreases such that Cdr2 becomes activated at some critical size. In this study, the chemistry of Pom1 and Cdr2 was modeled using a deterministic reaction-diffusion-convection system interacting with a deterministic model describing microtubule dynamics. Simulations mimicked experimental data from wild-type (WT) fission yeast growing at normal and reduced rates; they also mimicked the behavior of a Pom1 overexpression mutant and WT yeast exposed to a microtubule depolymerizing drug. A mechanism linking cell size and cell cycle, involving the downstream action of Cdr2 on Wee1 phosphorylation, is proposed.     

A microfluidic device with passive air-bubble valves for real-time measurement of dose-dependent drug cytotoxicity through impedance sensing

The monitoring and evaluation of cell behaviors under various concentrations of diffusible molecules or drugs are important in drug screening and in many other types of biological studies. In the current study, a novel polydimethylsiloxane (PDMS)-based microfluidic device was established for the real-time monitoring of drug-induced cytotoxicity using electric cell-substrate impedance sensing (ECIS). This device consists of the following three components: a drug gradient generator, planar air-bubble valves, and parallel cell culture cavities that are combined with impedance-sensing electrodes. The gradient generator allows for the simultaneous administration of multiple drug doses to test the functional cytotoxicity, and the incorporated impedance sensing enables the dynamic, automatic and quantitative measurement of in vitro dose-dependent drug responses. The air-bubble valve presented here allows the automatic closure of the valve without the need for any external valve-control instrument. As a proof-of-concept demonstration, this device was applied to dynamically monitor the effects of the anticancer drug cisplatin on apoptosis in four cancer cell lines, which may be useful for drug discovery and other biological studies that require automated analysis combined with concentration gradients.     

Noise reduction in the intracellular pom1p gradient by a dynamic clustering mechanism

Chemical gradients can generate pattern formation in biological systems. In the fission yeast Schizosaccharomyces pombe, a cortical gradient of pom1p (a DYRK-type protein kinase) functions to position sites of cytokinesis and cell polarity and to control cell length. Here, using quantitative imaging, fluorescence correlation spectroscopy, and mathematical modeling, we study how its gradient distribution is formed. Pom1p gradients exhibit large cell-to-cell variability, as well as dynamic fluctuations in each individual gradient. Our data lead to a two-state model for gradient formation in which pom1p molecules associate with the plasma membrane at cell tips and then diffuse on the membrane while aggregating into and fragmenting from clusters, before disassociating from the membrane. In contrast to a classical one-component gradient, this two-state gradient buffers against cell-to-cell variations in protein concentration. This buffering mechanism, together with time averaging to reduce intrinsic noise, allows the pom1p gradient to specify positional information in a robust manner.     

Direct and indirect regulation of Pom1 cell size pathway by the protein phosphatase 2C Ptc1

The fission yeast cells Schizosaccharomyces pombe divide at constant cell size regulated by environmental stimuli. An important pathway of cell size control involves the membrane-associated DYRK-family kinase Pom1, which forms decreasing concentration gradients from cell poles and inhibits mitotic inducers at midcell. Here, we identify the phosphatase 2C Ptc1 as negative regulator of Pom1. Ptc1 localizes to cell poles in a manner dependent on polarity and cell-wall integrity factors. We show that Ptc1 directly binds Pom1 and can dephosphorylate it in vitro but modulates Pom1 localization indirectly upon growth in low-glucose conditions by influencing microtubule stability. Thus, Ptc1 phosphatase plays both direct and indirect roles in the Pom1 cell size control pathway.     

Ability of polymorphonuclear leukocytes to orient in gradients of chemotactic factors

Polymorphonuclear leukocyte (PMN) chemotaxis has been examined under conditions which allow phase microscope observations of cells responding to controlled gradients of chemotactic factors. With this visual assay, PMNs can be seen to orient rapidly and reversibly to gradients of N-formylmethionyl peptides. The level of orientation depends upon the mean concentration of peptide present as well as the concentration gradient. The response allows an estimation of the binding constant of the peptide to the cell. In optimal gradients, PMNs can detect a 1% difference in the concentration of peptide. At high cell densities, PMNs incubated with active peptides orient their locomotion away from the center of the cell population. This orientation appears to be due to inactivation of the peptides by the cells. Such inactivation in vivo could help to limit an inflammatory response.     

A theory of measurement error and its implications for spatial and temporal gradient sensing during chemotaxis

In order that cells respond to environmental cues, they must be able to measure ambient ligand concentration. Concentrations fluctuate, however, because of thermal noise, and one can readily show that estimates based on concentration values at a particular moment will be subject to substantial error. Cells are therefore expected to average their estimates over some limited time period. In this paper we assume that a cell uses fractional receptor occupancy as a measure of ambient ligand concentration and develop general expressions for the error a cell makes because the length of the averaging period is necessarily limited. Our analysis is general, relieving many of the assumptions underlying the seminal work of Berg and Purcell. The most important formal difference is our inclusion of occupancy-dependent dissociation--a phenomenon that has been well-documented for many systems. In addition, our formulation permits signal averaging to begin before chemical equilibrium has been established and it allows binding kinetics to be nonlinear (i.e., biomolecular rather than pseudo-first-order). The results are applied to spatial and temporal concentration gradients. In particular we estimate the minimum averaging times required for cells to detect such gradients under typical in vitro conditions. These estimates involve assigning numerical values to receptor ligand rate constants. If the rate constants are at their maximum possible values (limited only by center of mass diffusion), then either temporal or spatial gradients can be detected in minutes or less. If, however, as suggested by experiments, the rate constants are several orders of magnitude below their diffusion-limited values, then under typical constant gradient conditions the time required to detect a spatial gradient is prohibitively long, whereas temporal gradients can still be detected in reasonable lengths of time. This result was obtained for large cells such as lymphocytes, as well as for the smaller, bacterial cells. The ratio of averaging times for the two mechanisms--amounting to several orders of magnitude--is well beyond what could be reconciled by limitations of the calculation, and strongly suggests heavy reliance on temporal sensing mechanisms under typical in vitro conditions with constant spatial gradients.     

Dictyostelium Chemotactic Response to Spatial and Temporal Gradients. Theories of the Limits of Chemotactic Sensitivity and of Pseudochemotaxis

Four aspects of ameboid cell chemotaxis are discussed: 1) Ameboid cells (Dictyostelium discoideum, leukocytes) might orient to chemotaxin gradients by sensing a spatial gradient or a temporal change in the concentration. Using a moving micropipette source of cAMP, we show the D discoideum cells can orient toward a gradient in which the concentration is everywhere decreasing with time–implying a spatial mechanism. 2) The number of molecules N that must be released by a source to orient a cell is limited by the natural concentration “noise” due to diffusion. N is shown to be simply related to the cell size and the distance from the source. 3) We show that previous diffusion equations for cell population movement have not taken the speed variations (klinokinesis) into account properly, and we present a new result that does. 4) We briefly discuss reaction-diffusion models of cell orientation.     

Theoretical analysis of gradient detection by growth cones.

Gradients of diffusible and substrate-bound molecules play an important role in guiding axons to appropriate targets in the developing nervous system. Although some of the molecules involved have recently been identified, little is known about the physical mechanisms by which growth cones sense gradients. This article applies the seminal Berg and Purcell (1977) model of gradient sensing to this problem. The model provides estimates for the statistical fluctuations in the measurement of concentration by a small sensing device. By assuming that gradient detection consists of the comparison of concentrations at two spatially or temporally separated points, the model therefore provides an estimate for the steepness of gradient that can be detected as a function of physiological parameters. The model makes the following specific predictions. (a) It is more likely that growth cones use a spatial rather than temporal sensing strategy. (b) Growth cone sensitivity increases with the concentration of ligand, the speed of ligand diffusion, the size of the growth cone, and the time over which it averages the gradient signal. (c) The minimum detectable gradient steepness for growth cones is roughly in the range 1-10%. (d) This value varies depending on whether a bound or freely diffusing ligand is being sensed, and on whether the sensing occurs in three or two dimensions. The model also makes predictions concerning the role of filopodia in gradient detection.     

Mechanisms of Gradient Sensing and Chemotaxis: Conserved Pathways,Diverse Regulation

Directed cell migration is critical for normal development, immune responses, and wound healing and plays a prominent role in tumor metastasis. In eukaryotes, cell orientation is biased by an external chemoattractant gradient through a spatial contrast in chemoattractant receptor-mediated signal transduction processes that differentially affect cytoskeletal dynamics at the cell front and rear. Mechanisms of spatial gradient sensing and chemotaxis have been studied extensively in the social amoeba Dictyostelium discoideum and mammalian leukocytes (neutrophils), which are similar in their remarkable sensitivity to shallow gradients and robustness of response over a broad range of chemoattractant concentration. Recently, we have quantitatively characterized a different gradient sensing system, that of platelet-derived growth factor-stimulated fibroblasts, an important component of dermal wound healing. The marked differences between this system and the others have led us to speculate on the diversity of gradient sensing mechanisms and their biological implications.     

Theoretical analysis of gradient detection by growth cones

Gradients of diffusible and substrate‐bound molecules play an important role in guiding axons to appropriate targets in the developing nervous system. Although some of the molecules involved have recently been identified, little is known about the physical mechanisms by which growth cones sense gradients. This article applies the seminal Berg and Purcell (1977) model of gradient sensing to this problem. The model provides estimates for the statistical fluctuations in the measurement of concentration by a small sensing device. By assuming that gradient detection consists of the comparison of concentrations at two spatially or temporally separated points, the model therefore provides an estimate for the steepness of gradient that can be detected as a function of physiological parameters. The model makes the following specific predictions. (a) It is more likely that growth cones use a spatial rather than temporal sensing strategy. (b) Growth cone sensitivity increases with the concentration of ligand, the speed of ligand diffusion, the size of the growth cone, and the time over which it averages the gradient signal. (c) The minimum detectable gradient steepness for growth cones is roughly in the range 1–10%. (d) This value varies depending on whether a bound or freely diffusing ligand is being sensed, and on whether the sensing occurs in three or two dimensions. The model also makes predictions concerning the role of filopodia in gradient detection. © 1999 John Wiley & Sons, Inc. J Neurobiol 41: 230–241, 1999     

Colocalization and interaction between elongasome and divisome during a preparative cell division phase in Escherichia coli

The rod‐shaped bacterium Escherichia coli grows by insertion of peptidoglycan into the lateral wall during cell elongation and synthesis of new poles during cell division. The monofunctional transpeptidases PBP2 and PBP3 are part of specialized protein complexes called elongasome and divisome, respectively, which catalyse peptidoglycan extension and maturation. Endogenous immunolabelled PBP2 localized in the cylindrical part of the cell as well as transiently at midcell. Using the novel image analysis tool Coli‐Inspector to analyse protein localization as function of the bacterial cell age, we compared PBP2 localization with that of other E. coli cell elongation and division proteins including PBP3. Interestingly, the midcell localization of the two transpeptidases overlaps in time during the early period of divisome maturation. Försters Resonance Energy Transfer (FRET) experiments revealed an interaction between PBP2 and PBP3 when both are present at midcell. A decrease in the midcell diameter is visible after 40% of the division cycle indicating that the onset of new cell pole synthesis starts much earlier than previously identified by visual inspection. The data support a new model of the division cycle in which the elongasome and divisome interact to prepare for cell division.     

The role of cell geometry and cell-cell communication in gradient sensing

Cells can measure shallow gradients of external signals to initiate and accomplish a migration or a morphogenetic process. Recently, starting from mathematical models like the local-excitation global-inhibition (LEGI) model and with the support of empirical evidence, it has been proposed that cellular communication improves the measurement of an external gradient. However, the mathematical models that have been used have over-simplified geometries (e.g., they are uni-dimensional) or assumptions about cellular communication, which limit the possibility to analyze the gradient sensing ability of more complex cellular systems. Here, we generalize the existing models to study the effects on gradient sensing of cell number, geometry and of long- versus short-range cellular communication in 2D systems representing epithelial tissues. We find that increasing the cell number can be detrimental for gradient sensing when the communication is weak and limited to nearest neighbour cells, while it is beneficial when there is long-range communication. We also find that, with long-range communication, the gradient sensing ability improves for tissues with more disordered geometries; on the other hand, an ordered structure with mostly hexagonal cells is advantageous with nearest neighbour communication. Our results considerably extend the current models of gradient sensing by epithelial tissues, making a step further toward predicting the mechanism of communication and its putative mediator in many biological processes.     

Analysis of the signal transduction properties of a module of spatial sensing in eukaryotic chemotaxis

The movement of cells in response to a gradient in chemical concentration—known as chemotaxis—is crucial for the proper functioning of uni-and multicellular organisms. How a cell senses the chemical concentration gradient surrounding it, and what signal is transmitted to its motion apparatus is known as gradient sensing. The ability of a cell to sense gradients persists even when the cell is immobilized (i.e., its motion apparatus is deactivated). This suggests that important features of gradient sensing can be studied in isolation, decoupling this phenomenon from the movement of the cell. A mathematical model for gradient sensing in Dictyostelium cells and neutrophils was recently proposed. This consists of an adaptation/spatial sensing module. This spatial sensing module feeds into an amplification module, magnifying the effects of the former. In this paper, we analyze the spatial sensing module in detail and examine its signal transduction properties. We examine the response of this module to several inputs of experimental and biological relevance.     

Deterministic model of dermal wound invasion incorporating receptor-mediated signal transduction and spatial gradient sensing

During dermal wound healing, platelet-derived growth factor (PDGF) serves as both a chemoattractant and mitogen for fibroblasts, potently stimulating their invasion of the fibrin clot over a period of several days. A mathematical model of this process is presented, which accurately accounts for the sensitivity of PDGF gradient sensing through PDGF receptor/phosphoinositide 3-kinase-mediated signal transduction. Analysis of the model suggests that PDGF receptor-mediated endocytosis and degradation of PDGF allows a constant PDGF concentration profile to be maintained at the leading front of the fibroblast density profile as it propagates, at a constant rate, into the clot. Thus, the constant PDGF gradient can span the optimal concentration range for asymmetric phosphoinositide 3-kinase signaling and fibroblast chemotaxis, with near-maximal invasion rates elicited over a relatively broad range of PDGF secretion rates. A somewhat surprising finding was that extremely sharp PDGF gradients do not necessarily stimulate faster progression through the clot, because maintaining such a gradient through PDGF consumption is a potentially rate-limiting process.     

Mechanics of Constriction during Cell Division: A Variational Approach

During symmetric division cells undergo large constriction deformations at a stable midcell site. Using a variational approach, we investigate the mechanical route for symmetric constriction by computing the bending energy of deformed vesicles with rotational symmetry. Forces required for constriction are explicitly computed at constant area and constant volume, and their values are found to be determined by cell size and bending modulus. For cell-sized vesicles, considering typical bending modulus of , we calculate constriction forces in the range . The instability of symmetrical constriction is shown and quantified with a characteristic coefficient of the order of , thus evidencing that cells need a robust mechanism to stabilize constriction at midcell.     

Mycobacterium tuberculosis ClpX Interacts with FtsZ and Interferes with FtsZ Assembly

FtsZ assembly at the midcell division site in the form of a Z-ring is crucial for initiation of the cell division process in eubacteria. It is largely unknown how this process is regulated in the human pathogen Mycobacterium tuberculosis. Here we show that the expression of clpX was upregulated upon macrophage infection and exposure to cephalexin antibiotic, the conditions where FtsZ-ring assembly is delayed. Independently, we show using pull-down, solid-phase binding, bacterial two-hybrid and mycobacterial protein fragment complementation assays, that M. tuberculosis FtsZ interacts with ClpX, the substrate recognition domain of the ClpXP protease. Incubation of FtsZ with ClpX increased the critical concentration of GTP-dependent polymerization of FtsZ. Immunoblotting revealed that the intracellular ratio of ClpX to FtsZ in wild type M. tuberculosis is approximately 1∶2. Overproduction of ClpX increased cell length and modulated the localization of FtsZ at midcell sites; however, intracellular FtsZ levels were unaffected. A ClpX-CFP fusion protein localized to the cell poles and midcell sites and colocalized with the FtsZ-YFP protein. ClpX also interacted with FtsZ mutant proteins defective for binding to and hydrolyzing GTP and possibly for interactions with other proteins. Taken together, our results suggest that M. tuberculosis ClpX interacts stoichiometrically with FtsZ protomers, independent of its nucleotide-bound state and negatively regulates FtsZ activities, hence cell division.     

Automatic inducer addition and harvesting of recombinant Escherichia coli cultures based on indirect on-line estimation of biomass concentration and specific growth rate

This article describes a novel bioreactor configuration for production optimization of recombinant proteins in Escherichia coli. Inducer addition and harvesting are controlled on-line based on indirect estimation of biomass concentration and specific growth rate from addition of NaOH to maintain constant pH. When either a predetermined biomass concentration is reached or the cultures have obtained, a constant specific growth rate inducer is introduced automatically. The induction period is ended by automatic harvesting of the cultures either at a predetermined biomass concentration or when substrate (in this study glucose) is depleted, detected as an increase of pH, or dissolved oxygen tension. During harvesting, metabolic activities are quenched within 3 min by cooling of the cell suspension. The system has been used to optimize expression of glutathione S-transferase (GST) fusion protein of the ligand binding domain of mouse peroxisome proliferator-activated receptor, GST-PPARalpha LBD. Total yield of GST-PPARalpha LBD was independent of the time of inducer addition as long as the length of induction period corresponded to at least 0.25 cell divisions while the yield of soluble GST-PPARalpha LBD, the only active form, increased with the length of induction period. Highest yields were obtained when the inducer was added at low cell concentration as soon as constant specific growth rate was detected, resulting in induction periods corresponding to 3.4 +/- 0.4 cell divisions. The specific growth rate remained almost constant for one cell division after inducer addition, whereafter it decreased. No decrease of specific growth rate was observed when inducer was added in the lag-phase, and no soluble protein was produced. These results suggest that solely soluble GST-PPARalpha LBD acts as a growth inhibitor and that GST-PPARalpha LBD is expressed predominantly as inclusion bodies immediately after inducer addition whereas the proportion expressed as soluble protein is increased after 1 h of induction. Compared to the procedures, which are generally used for protein expression in the laboratory, this system is less labor intensive, it automatically provides recording of biomass concentration and specific growth rate, and it allows direct comparisons between expression of different proteins and performance of different constructs since the induction period is linked to growth.     

Tumor cell haptotaxis on covalently immobilized linear and exponential gradients of a cell adhesion peptide

The movement of cells up an adhesive substratum gradient has been proposed as a mechanism for directing cell migration during development and metastasis. Critical evaluation of this hypothesis (haptotaxis) benefits from the use of quantifiable, stable substratum gradients of biologically relevant adhesion molecules. We report covalent derivatization of polyacrylamide surfaces with quantifiable gradients of a nonapeptide containing the adhesive Arg-Gly-Asp sequence. Cell migration was studied by seeding derivatized surfaces evenly with B16F10 murine melanoma cells. Within 8 hr, cells on gradients redistributed markedly; higher cell densities were found at gel positions having higher immobilized peptide densities. In contrast, cells seeded on control gels with uniform concentrations of adhesive peptide did not redistribute. Redistribution occurred on gradients in both serum-free and serum-containing media. Experiments with uniform density peptide-derivatized gels demonstrated that redistribution on gradients was not due to preferential initial cell attachment or preferential growth on the higher density of immobilized peptide, but must have been due to cell translocation. Cells on exponential gradients of immobilized peptide migrated to a position on the gel surface corresponding to the highest immobilized peptide density, while cells on linear gradients of the same peptide migrated to a position of intermediate peptide density. These data suggest that the B16F10 cells respond to proportional changes in immobilized peptide density rather than to absolute changes, implying a sensing mechanism which utilizes adaptation. These results demonstrate that (1) a gradient of a small adhesive peptide is sufficient to generate redistribution of cell populations and (2) controlled quantifiable substratum gradients can be produced and used to probe the underlying cellular mechanisms of this behavior.