Receptor noise limitations on chemotactic sensing

Accuracy in readout of glutamate concentrations by neuronal cells

Glutamate and glycine are important neurotransmitters in the brain. An action potential propagating in the terminal of a presynaptic neuron causes the release of glutamate and glycine in the synapse by vesicles fusing with the cell membrane, which then activate various receptors on the cell membrane of the post-synaptic neuron. Entry of Ca[Formula: see text] through the activated NMDA receptors leads to a host of cellular processes of which long-term potentiation is of crucial importance because it is widely considered to be one of the major mechanisms behind learning and memory. By analysing the readout of glutamate concentration by the post-synaptic neurons during Ca[Formula: see text] signaling, we find that the average receptor density in hippocampal neurons has evolved to allow for accurate measurement of the glutamate concentration in the synaptic cleft.

Limitations on concentration measurements and gradient discerning times in cellular systems

This work reports on two results. At first we revisit the Berg and Purcell calculation that provides a lower bound to the error in concentration measurement by cells by considering the realistic case when the cell starts measuring the moment it comes in contact with the chemoattractants, instead of measuring after equilibrating with the chemotactic concentration as done in the classic Berg and Purcell paper. We find that the error in concentration measurement is still the same as evaluated by Berg and Purcell. We next derive a lower bound on measurement time below which it is not possible for the cell to discern extracellular chemotactic gradients through spatial sensing mechanisms. This bound is independent of diffusion rate and concentration of the chemoattracts and is instead set by detachment rate of ligands from the cell receptors. The result could help explain experimental observations.

Cellular memory in eukaryotic chemotaxis depends on the background chemoattractant concentration

Cells of the social amoeba Dictyostelium discoideum migrate to a source of periodic traveling waves of chemoattractant as part of a self-organized aggregation process. An important part of this process is cellular memory, which enables cells to respond to the front of the wave and ignore the downward gradient in the back of the wave. During this aggregation, the background concentration of the chemoattractant gradually rises. In our microfluidic experiments, we exogenously applied periodic waves of chemoattractant with various background levels. We find that increasing background does not make detection of the wave more difficult, as would be naively expected. Instead, we see that the chemotactic efficiency significantly increases for intermediate values of the background concentration but decreases to almost zero for large values in a switch-like manner. These results are consistent with a computational model that contains a bistable memory module, along with a nonadaptive component. Within this model, an intermediate background level helps preserve directed migration by keeping the memory activated, but when the background level is higher, the directional stimulus from the wave is no longer sufficient to activate the bistable memory, suppressing directed migration. These results suggest that raising levels of chemoattractant background may facilitate the self-organized aggregation in Dictyostelium colonies.

A negative-feedback loop maintains optimal chemokine concentrations for directional cell migration

Chemoattractant gradients frequently guide migrating cells. To achieve the most directional signal, such gradients should be maintained with concentrations around the dissociation constant (Kd)1-6 of the chemoreceptor. Whether this actually occurs in animals is unknown. Here we investigate whether a moving tissue, the zebrafish posterior lateral line primordium, buffers its attractant in this concentration range to achieve robust migration. We find that the Cxcl12 (also known as Sdf1) attractant gradient ranges from 0 to 12 nM, values similar to the 3.4 nM Kd of its receptor Cxcr4. When we increase the Kd of Cxcl12 for Cxcr4, primordium migration is less directional. Furthermore, a negative-feedback loop between Cxcl12 and its clearance receptor Ackr3 (also known as Cxcr7) regulates the Cxcl12 concentrations. Breaking this negative feedback by blocking the phosphorylation of the cytoplasmic tail of Ackr3 also results in less directional primordium migration. Thus, directed migration of the primordium is dependent on a close match between the Cxcl12 concentration and the Kd of Cxcl12 for Cxcr4, which is maintained by buffering of the chemokine levels. Quantitative modelling confirms the plausibility of this mechanism. We anticipate that buffering of attractant concentration is a general mechanism for ensuring robust cell migration.

Accuracy of position determination in Ca^{2+} signaling

A living cell senses its environment and responds to external signals. In this paper, we study theoretically the precision at which cells can determine the position of a spatially localized transient extracellular signal. To this end, we focus on the case where the stimulus is converted into the release of a small molecule that acts as a second messenger, for example, Ca^{2+}, and activates kinases that change the activity of enzymes by phosphorylating them. We analyze the spatial distribution of phosphorylation events using stochastic simulations as well as a mean-field approach. Kinases that need to bind to the cell membrane for getting activated provide more accurate estimates than cytosolic kinases. Our results could explain why the rate of Ca^{2+} detachment from the membrane-binding conventional protein kinase Cα is larger than its phosphorylation rate.

Receptor crosstalk improves concentration sensing of multiple ligands

Cells need to reliably sense external ligand concentrations to achieve various biological functions such as chemotaxis or signaling. The molecular recognition of ligands by surface receptors is degenerate in many systems, leading to crosstalk between ligand-receptor pairs. Crosstalk is often thought of as a deviation from optimal specific recognition, as the binding of noncognate ligands can interfere with the detection of the receptor's cognate ligand, possibly leading to a false triggering of a downstream signaling pathway. Here we quantify the optimal precision of sensing the concentrations of multiple ligands by a collection of promiscuous receptors. We demonstrate that crosstalk can improve precision in concentration sensing and discrimination tasks. To achieve superior precision, the additional information about ligand concentrations contained in short binding events of the noncognate ligand should be exploited. We present a proofreading scheme to realize an approximate estimation of multiple ligand concentrations that reaches a precision close to the derived optimal bounds. Our results help rationalize the observed ubiquity of receptor crosstalk in molecular sensing.

Extracellular Processing of Molecular Gradients by Eukaryotic Cells Can Improve Gradient Detection Accuracy

Eukaryotic cells sense molecular gradients by measuring spatial concentration variation through the difference in the number of occupied receptors to which molecules can bind. They also secrete enzymes that degrade these molecules, and it is presently not well understood how this affects the local gradient perceived by cells. Numerical and analytical results show that these enzymes can substantially increase the signal-to-noise ratio of the receptor difference and allow cells to respond to a much broader range of molecular concentrations and gradients than they would without these enzymes.

Dendritic Cells Interpret Haptotactic Chemokine Gradients in a Manner Governed by Signal-to-Noise Ratio and Dependent on GRK6

Navigation of cells along gradients of guidance cues is a determining step in many developmental and immunological processes. Gradients can either be soluble or immobilized to tissues as demonstrated for the haptotactic migration of dendritic cells (DCs) toward higher concentrations of immobilized chemokine CCL21. To elucidate how gradient characteristics govern cellular response patterns, we here introduce an in vitro system allowing to track migratory responses of DCs to precisely controlled immobilized gradients of CCL21. We find that haptotactic sensing depends on the absolute CCL21 concentration and local steepness of the gradient, consistent with a scenario where DC directionality is governed by the signal-to-noise ratio of CCL21 binding to the receptor CCR7. We find that the conditions for optimal DC guidance are perfectly provided by the CCL21 gradients we measure in vivo. Furthermore, we find that CCR7 signal termination by the G-protein-coupled receptor kinase 6 (GRK6) is crucial for haptotactic but dispensable for chemotactic CCL21 gradient sensing in vitro and confirm those observations in vivo. These findings suggest that stable, tissue-bound CCL21 gradients as sustainable "roads" ensure optimal guidance in vivo.

Rac signal adaptation controls neutrophil mobilization from the bone marrow

Mobilization of neutrophils from the bone marrow determines neutrophil blood counts and thus is medically important. Balanced neutrophil mobilization from the bone marrow depends on the retention-promoting chemokine CXCL12 and its receptor CXCR4 and the egression-promoting chemokine CXCL2 and its receptor CXCR2. Both pathways activate the small guanosine triphosphatase Rac, leaving the role of this signaling event in neutrophil retention and egression ambiguous. On the assumption that active Rac determines persistent directional cell migration, we generated a mathematical model to link chemokine-mediated Rac modulation to neutrophil egression time. Our computer simulation indicated that, in the bone marrow, where the retention signal predominated, egression time strictly depended on the time it took Rac to return to its basal activity (namely, adaptation). This prediction was validated in mice lacking the Rac inhibitor ArhGAP15. Neutrophils in these mice showed prolonged Rac adaptation and cell-autonomous retention in the bone marrow. Our model thus demonstrates that mobilization in the presence of two spatially defined opposing chemotactic cues strictly depends on inhibitors shaping the time course of signal adaptation. Furthermore, our findings might help to find new modes of intervention to treat conditions characterized by excessively low or high circulating neutrophils.

Modelling of Yeast Mating Reveals Robustness Strategies for Cell-Cell Interactions

Mating of budding yeast cells is a model system for studying cell-cell interactions. Haploid yeast cells secrete mating pheromones that are sensed by the partner which responds by growing a mating projection toward the source. The two projections meet and fuse to form the diploid. Successful mating relies on precise coordination of dynamic extracellular signals, signaling pathways, and cell shape changes in a noisy background. It remains elusive how cells mate accurately and efficiently in a natural multi-cell environment. Here we present the first stochastic model of multiple mating cells whose morphologies are driven by pheromone gradients and intracellular signals. Our novel computational framework encompassed a moving boundary method for modeling both a-cells and α-cells and their cell shape changes, the extracellular diffusion of mating pheromones dynamically coupled with cell polarization, and both external and internal noise. Quantification of mating efficiency was developed and tested for different model parameters. Computer simulations revealed important robustness strategies for mating in the presence of noise. These strategies included the polarized secretion of pheromone, the presence of the α-factor protease Bar1, and the regulation of sensing sensitivity; all were consistent with data in the literature. In addition, we investigated mating discrimination, the ability of an a-cell to distinguish between α-cells either making or not making α-factor, and mating competition, in which multiple a-cells compete to mate with one α-cell. Our simulations were consistent with previous experimental results. Moreover, we performed a combination of simulations and experiments to estimate the diffusion rate of the pheromone a-factor. In summary, we constructed a framework for simulating yeast mating with multiple cells in a noisy environment, and used this framework to reproduce mating behaviors and to identify strategies for robust cell-cell interactions.

One of the riddles of Nature is how cells interact with one another to create complex cellular networks such as the neural networks in the brain. Forming precise connections between irregularly shaped cells is a challenge for biology. We developed computational methods for simulating these complex cell-cell interactions. We applied these methods to investigate yeast mating in which two yeast cells grow projections that meet and fuse guided by pheromone attractants. The simulations described molecules both inside and outside of the cell, and represented the continually changing shapes of the cells. We found that positioning the secretion and sensing of pheromones at the same location on the cell surface was important. Other key factors for robust mating included secreting a protein that removed excess pheromone from outside of the cell so that the signal would not be too strong. An important advance was being able to simulate as many as five cells in complex mating arrangements. Taken together we used our novel computational methods to describe in greater detail the yeast mating process, and more generally, interactions among cells changing their shapes in response to their neighbors.

Physical Limit to Concentration Sensing Amid Spurious Ligands

To adapt their behavior in changing environments, cells sense concentrations by binding external ligands to their receptors. However, incorrect ligands may bind nonspecifically to receptors, and when their concentration is large, this binding activity may interfere with the sensing of the ligand of interest. Here, I derive analytically the physical limit to the accuracy of concentration sensing amid a large number of interfering ligands. A scaling transition is found when the mean bound time of correct ligands is twice that of incorrect ligands. I discuss how the physical bound can be approached by a cascade of receptor states generalizing kinetic proofreading schemes.

The biophysical model for accuracy of cellular sensing spatial gradients of multiple chemoattractants

Spatial gradients of surrounding chemoattractants are the key factors in determining the directionality of eukaryotic cell movement. Thus, it is important for cells to accurately measure the spatial gradients of surrounding chemoattractants. Here, we study the precision of sensing the spatial gradients of multiple chemoattractants using cooperative receptor clusters. Cooperative receptors on cells are modeled as an Ising chain of Monod-Wyman-Changeux clusters subject to multiple chemical-gradient fields to study the physical limits of multiple chemoattractants spatial gradients sensing. We found that eukaryotic cells cannot sense each chemoattractant gradient individually. Instead, cells can only sense a weighted sum of surrounding chemical gradients. Moreover, the precision of sensing one chemical gradient is signicantly affected by coexisting chemoattractant concentrations. These findings can provide a further insight into the role of chemoattractants in immune response and help develop novel treatments for inflammatory diseases.

High fidelity information processing in folic acid chemotaxis of Dictyostelium amoebae

Living cells depend upon the detection of chemical signals for their existence. Eukaryotic cells can sense a concentration difference as low as a few per cent across their bodies. This process was previously suggested to be limited by the receptor-ligand binding fluctuations. Here, we first determine the chemotaxis response of Dictyostelium cells to static folic acid gradients and show that they can significantly exceed this sensitivity, responding to gradients as shallow as 0.2% across the cell body. Second, using a previously developed information theory framework, we compare the total information gained about the gradient (based on the cell response) to its upper limit: the information gained at the receptor-ligand binding step. We find that the model originally applied to cAMP sensing fails as demonstrated by the violation of the data processing inequality, i.e. the total information exceeds the information at the receptor-ligand binding step. We propose an extended model with multiple known receptor types and with cells allowed to perform several independent measurements of receptor occupancy. This does not violate the data processing inequality and implies the receptor-ligand binding noise dominates both for low- and high-chemoattractant concentrations. We also speculate that the interplay between exploration and exploitation is used as a strategy for accurate sensing of otherwise unmeasurable levels of a chemoattractant.

Optical control demonstrates switch-like PIP3 dynamics underlying the initiation of immune cell migration

There is a dearth of approaches to experimentally direct cell migration by continuously varying signal input to a single cell, evoking all possible migratory responses and quantitatively monitoring the cellular and molecular response dynamics. Here we used a visual blue opsin to recruit the endogenous G-protein network that mediates immune cell migration. Specific optical inputs to this optical trigger of signaling helped steer migration in all possible directions with precision. Spectrally selective imaging was used to monitor cell-wide phosphatidylinositol (3,4,5)-triphosphate (PIP3), cytoskeletal, and cellular dynamics. A switch-like PIP3 increase at the cell front and a decrease at the back were identified, underlying the decisive migratory response. Migration was initiated at the rapidly increasing switch stage of PIP3 dynamics. This result explains how a migratory cell filters background fluctuations in the intensity of an extracellular signal but responds by initiating directionally sensitive migration to a persistent signal gradient across the cell. A two-compartment computational model incorporating a localized activator that is antagonistic to a diffusible inhibitor was able to simulate the switch-like PIP3 response. It was also able simulate the slow dissipation of PIP3 on signal termination. The ability to independently apply similar signaling inputs to single cells detected two cell populations with distinct thresholds for migration initiation. Overall the optical approach here can be applied to understand G-protein-coupled receptor network control of other cell behaviors.

Directional sensing of deformed cells under faint gradients

We consider the physical limit of the directional sensing ability of living cells, as in chemotaxis, under a low concentration and shallow chemoattractant gradient. Elliptic cells sense the direction, which is a stochastic variable of a characteristic distribution with peaks at directions not necessarily to the gradient. The peak positions depend on the information of the gradient that cells use to infer the direction and also the shape and orientation of cells. Cells of different shapes may use different inference strategies to increase their directional sensing performance.

How input fluctuations reshape the dynamics of a biological switching system

An important task in quantitative biology is to understand the role of stochasticity in biochemical regulation. Here, as an extension of our recent work [Phys. Rev. Lett. 107, 148101 (2011)], we study how input fluctuations affect the stochastic dynamics of a simple biological switch. In our model, the on transition rate of the switch is directly regulated by a noisy input signal, which is described as a non-negative mean-reverting diffusion process. This continuous process can be a good approximation of the discrete birth-death process and is much more analytically tractable. Within this setup, we apply the Feynman-Kac theorem to investigate the statistical features of the output switching dynamics. Consistent with our previous findings, the input noise is found to effectively suppress the input-dependent transitions. We show analytically that this effect becomes significant when the input signal fluctuates greatly in amplitude and reverts slowly to its mean.

Fundamental limits on sensing chemical concentrations with linear biochemical networks

Living cells often need to extract information from biochemical signals that are noisy. We study how accurately cells can measure chemical concentrations with signaling networks that are linear. For stationary signals of long duration, they can reach, but not beat, the Berg-Purcell limit, which relies on uniformly averaging in time the fluctuations in the input signal. For short times or nonstationary signals, however, they can beat the Berg-Purcell limit, by nonuniformly time averaging the input. We derive the optimal weighting function for time averaging and use it to provide the fundamental limit of measuring chemical concentrations with linear signaling networks.

Control parameter description of eukaryotic chemotaxis

The chemotaxis of eukaryotic cells depends both on the average concentration of the chemoattractant and on the steepness of its gradient. For the social amoeba Dictyostelium discoideum, we test quantitatively the prediction by Ueda and Shibata [Biophys. J. 93, 11 (2007)] that the efficacy of chemotaxis depends on a single control parameter only, namely, the signal-to-noise ratio (SNR), determined by the stochastic fluctuations of (i) the binding of the chemoattractant molecule to the transmembrane receptor and (ii) the intracellular activation of the effector of the signaling cascade. For SNR < or approximately equal to 1, the theory captures the experimental findings well, while for larger SNR noise sources further downstream in the signaling pathway need to be taken into account.

Noise effects in nonlinear biochemical signaling

It has been generally recognized that stochasticity can play an important role in the information processing accomplished by reaction networks in biological cells. Most treatments of that stochasticity employ Gaussian noise even though it is a priori obvious that this approximation can violate physical constraints, such as the positivity of chemical concentrations. Here, we show that even when such nonphysical fluctuations are rare, an exact solution of the Gaussian model shows that the model can yield unphysical results. This is done in the context of a simple incoherent-feedforward model which exhibits perfect adaptation in the deterministic limit. We show how one can use the natural separation of time scales in this model to yield an approximate model, that is analytically solvable, including its dynamical response to an environmental change. Alternatively, one can employ a cutoff procedure to regularize the Gaussian result.

Theoretical aspects of cellular decision-making and information-processing

Microscopic biological processes have extraordinary complexity and variety at the sub-cellular, intra-cellular, and multi-cellular levels. In dealing with such complex phenomena, conceptual and theoretical frameworks are crucial, which enable us to understand seemingly different intra- and inter-cellular phenomena from unified viewpoints. Decision-making is one such concept that has attracted much attention recently. Since a number of cellular behavior can be regarded as processes to make specific actions in response to external stimuli, decision-making can cover and has been used to explain a broad range of different cellular phenomena [Balázsi et al. (Cell 144(6):910, 2011), Zeng et al. (Cell 141(4):682, 2010)]. Decision-making is also closely related to cellular information-processing because appropriate decisions cannot be made without exploiting the information that the external stimuli contain. Efficiency of information transduction and processing by intra-cellular networks determines the amount of information obtained, which in turn limits the efficiency of subsequent decision-making. Furthermore, information-processing itself can serve as another concept that is crucial for understanding of other biological processes than decision-making. In this work, we review recent theoretical developments on cellular decision-making and information-processing by focusing on the relation between these two concepts.

Noise filtering tradeoffs in spatial gradient sensing and cell polarization response

BACKGROUND

Cells sense chemical spatial gradients and respond by polarizing internal components. This process can be disrupted by gradient noise caused by fluctuations in chemical concentration.

RESULTS

We investigated how external gradient noise affects spatial sensing and response focusing on noise-filtering and the resultant tradeoffs. First, using a coarse-grained mathematical model of gradient-sensing and cell polarity, we characterized three negative consequences of noise: Inhibition of the extent of polarization, degradation of directional accuracy, and production of a noisy output polarization. Next, we explored filtering strategies and discovered that a combination of positive feedback, multiple signaling stages, and time-averaging produced good results. There was an important tradeoff, however, because filtering resulted in slower polarization. Simulations demonstrated that a two-stage filter-amplifier resulted in a balanced outcome. Then, we analyzed the effect of noise on a mechanistic model of yeast cell polarization in response to gradients of mating pheromone. This analysis showed that yeast cells likely also combine the above three filtering mechanisms into a filter-amplifier structure to achieve impressive spatial-noise tolerance, but with the consequence of a slow response time. Further investigation of the amplifier architecture revealed two positive feedback loops, a fast inner and a slow outer, both of which contributed to noise-tolerant polarization. This model also made specific predictions about how orientation performance depended upon the ratio between the gradient slope (signal) and the noise variance. To test these predictions, we performed microfluidics experiments measuring the ability of yeast cells to orient to shallow gradients of mating pheromone. The results of these experiments agreed well with the modeling predictions, demonstrating that yeast cells can sense gradients shallower than 0.1% μm-1, approximately a single receptor-ligand molecule difference between front and back, on par with motile eukaryotic cells.

CONCLUSIONS

Spatial noise impedes the extent, accuracy, and smoothness of cell polarization. A combined filtering strategy implemented by a filter-amplifier architecture with slow dynamics was effective. Modeling and experimental data suggest that yeast cells employ these elaborate mechanisms to filter gradient noise resulting in a slow but relatively accurate polarization response.

Quantifying information transmission in eukaryotic gradient sensing and chemotactic response

Eukaryotic cells are able to sense shallow chemical gradients by surface receptors and migrate toward chemoattractant sources. The accuracy of this chemotactic response relies on the ability of cells to infer gradients from the heterogeneous distribution of receptors bound by diffusing chemical molecules. Ultimately, the precision of gradient sensing is limited by the fluctuations of signaling components, including the stochastic receptor occupancy and noisy intracellular processing. Viewing the system as a Markovian communication channel, we apply techniques from information theory to derive upper bounds on the amount of information that can be reliably transmitted through a chemotactic cell. Specifically, we derive an expression for the mutual information between the gradient direction and the spatial distribution of bound receptors. We also compute the mutual information between the gradient direction and the motility direction using three different models for cell motion. Our results can be used to quantify the information loss during the various stages of directional sensing in eukaryotic chemotaxis.

A comparison of mathematical models for polarization of single eukaryotic cells in response to guided cues

Polarization, a primary step in the response of an individual eukaryotic cell to a spatial stimulus, has attracted numerous theoretical treatments complementing experimental studies in a variety of cell types. While the phenomenon itself is universal, details differ across cell types, and across classes of models that have been proposed. Most models address how symmetry breaking leads to polarization, some in abstract settings, others based on specific biochemistry. Here, we compare polarization in response to a stimulus (e.g., a chemoattractant) in cells typically used in experiments (yeast, amoebae, leukocytes, keratocytes, fibroblasts, and neurons), and, in parallel, responses of several prototypical models to typical stimulation protocols. We find that the diversity of cell behaviors is reflected by a diversity of models, and that some, but not all models, can account for amplification of stimulus, maintenance of polarity, adaptation, sensitivity to new signals, and robustness.

How geometry and internal bias affect the accuracy of eukaryotic gradient sensing

Many motile eukaryotic cells determine their direction by measuring external chemical gradients through the binding of ligands to membrane bound receptors. This process is limited by fluctuations arising from the binding process and from the diffusion of the ligand molecules. Here, we apply estimation-theoretic methods to determine the physical limits of gradient sensing for cells that are noncircular and for cells that have an internal bias. Specifically, we derive theoretical expressions for the accuracy of gradient sensing in elliptical cells. This accuracy for highly elliptical cells can significantly deviate from the gradient sensing limits derived for circular cells. Furthermore, we find that a cell cannot improve its sensing of the gradient steepness and direction simultaneously by elongating its cell body. Finally, we derive a lower bound on the accuracy of gradient sensing for cells that possess an internal bias and compare our analytical results with recent experimental findings.

Microfluidic platform for chemotaxis in gradients formed by CXCL12 source-sink cells

Chemokine CXCL12 promotes CXCR4-dependent chemotaxis of cancer cells to characteristic organs and tissues, leading to metastatic disease. This study was designed to investigate how cells expressing CXCR7 regulate chemotaxis of a separate population of CXCR4 cells under physiologic conditions in which cells are exposed to gradients of CXCL12. We recapitulated a cancer-stroma microenvironment by patterning CXCR4-expressing cancer cells in microchannels at spatially defined positions relative to CXCL12-producing cells and CXCR7-expressing cells. CXCR7 scavenges and degrades CXCL12, which has been proposed to facilitate CXCR4-dependent chemotaxis through a source-sink model. Using the microchannel device, we demonstrated that chemotaxis of CXCR4 cells depended critically on the presence and location of CXCR7 cells (sink) relative to chemokine secreting cells (source). Furthermore, inhibiting CXCR4 on migrating cells or CXCR7 on sink cells blocked CXCR4-dependent chemotaxis toward CXCL12, showing that the device can identify new therapeutic agents that block migration by targeting chemoattractant scavenging receptors. Our system enables efficient chemotaxis under much shallower yet more physiological chemoattractant gradients by generating an in vitro microenvironment where combinations of cellular products may be secreted along with formation of a chemoattractant gradient. In addition to elucidating mechanisms of CXCL-12 mediated chemotaxis, this simple and robust method can be broadly useful for engineering multiple microenvironments to investigate intercellular communication.

Physical limits on cellular sensing of spatial gradients

Many eukaryotic cells are able to detect chemical gradients by directly measuring spatial concentration differences. The precision of such gradient sensing is limited by fluctuations in the binding of diffusing particles to specific receptors on the cell surface. Here, we explore the physical limits of the spatial sensing mechanism by modeling the chemotactic cell as an Ising spin chain subject to a spatially varying field. Our results demonstrate that the accuracy to sense the gradient direction not only increases dramatically with the cell size but also can be improved significantly by introducing receptor cooperativity. Thus, receptor coupling may open the possibility for small bacteria to perform spatial measurements of gradients, as supported by a recent experimental finding.

Implementation of dynamic Bayesian decision making by intracellular kinetics

Decision making in a noisy and dynamically changing environment is a fundamental task for a cell. To choose appropriate decisions over time, a cell must be equipped with intracellular kinetics that can conduct dynamic and efficient decision making. By using the theory of sequential inference, I demonstrate that dynamic Bayesian decision making can be implemented by an intracellular kinetics with a dual positive feedback structure. I also show that the combination of linear instantaneous and nonlinear stationary sensitivities to the input dominantly contributes to decision making efficiency, and that the state-dependent sensitivity change further suppresses noisy response. The statistical principles underlying these two factors are further clarified to be a log-likelihood-dependent quantification of the input information and uncertainty-dependent sensitivity control.

A comparison of deterministic and stochastic simulations of neuronal vesicle release models

We study the calcium-induced vesicle release into the synaptic cleft using a deterministic algorithm and MCell, a Monte Carlo algorithm that tracks individual molecules. We compare the average vesicle release probability obtained using both algorithms and investigate the effect of the three main sources of noise: diffusion, sensor kinetics and fluctuations from the voltage-dependent calcium channels (VDCCs). We find that the stochastic opening kinetics of the VDCCs are the main contributors to differences in the release probability. Our results show that the deterministic calculations lead to reliable results, with an error of less than 20%, when the sensor is located at least 50 nm from the VDCCs, corresponding to microdomain signaling. For smaller distances, i.e. nanodomain signaling, the error becomes larger and a stochastic algorithm is necessary.

External and internal constraints on eukaryotic chemotaxis

Chemotaxis, the chemically guided movement of cells, plays an important role in several biological processes including cancer, wound healing, and embryogenesis. Chemotacting cells are able to sense shallow chemical gradients where the concentration of chemoattractant differs by only a few percent from one side of the cell to the other, over a wide range of local concentrations. Exactly what limits the chemotactic ability of these cells is presently unclear. Here we determine the chemotactic response of Dictyostelium cells to exponential gradients of varying steepness and local concentration of the chemoattractant cAMP. We find that the cells are sensitive to the steepness of the gradient as well as to the local concentration. Using information theory techniques, we derive a formula for the mutual information between the input gradient and the spatial distribution of bound receptors and also compute the mutual information between the input gradient and the motility direction in the experiments. A comparison between these quantities reveals that for shallow gradients, in which the concentration difference between the back and the front of a 10-mum-diameter cell is <5%, and for small local concentrations (<10 nM) the intracellular information loss is insignificant. Thus, external fluctuations due to the finite number of receptors dominate and limit the chemotactic response. For steeper gradients and higher local concentrations, the intracellular information processing is suboptimal and results in a smaller mutual information between the input gradient and the motility direction than would have been predicted from the ligand-receptor binding process.

Phenomenological approach to eukaryotic chemotactic efficiency

Eukaryotic cells are capable of detecting small chemical gradients for a wide range of background concentrations. Ultimately, fluctuations place a limit on gradient sensing and recent work has focused on the role of stochastic receptor occupancy as one possible limiting factor. Here, we use a phenomenological approach to add spontaneous motility fluctuations to receptor noise and predict the directional statistics of eukaryotic chemotaxis. Specifically, an Itô diffusion equation with direction-dependent multiplicative noise is developed and analytically studied. We show that our approach can naturally accommodate recent experimental data for the chemotaxis of the social amoeba Dictyostelium.

Increased accuracy of ligand sensing by receptor internalization

Many types of cells can sense external ligand concentrations with cell-surface receptors at extremely high accuracy. Interestingly, ligand-bound receptors are often internalized, a process also known as receptor-mediated endocytosis. While internalization is involved in a vast number of important functions for the life of a cell, it was recently also suggested to increase the accuracy of sensing ligand as the overcounting of the same ligand molecules is reduced. Here we show, by extending simple ligand-receptor models to out-of-equilibrium thermodynamics, that internalization increases the accuracy with which cells can measure ligand concentrations in the external environment. Comparison with experimental rates of real receptors demonstrates that our model has indeed biological significance.

Maximum likelihood and the single receptor

The accuracy by which biological cells sense chemical concentration is ultimately limited by the random arrival of particles at the receptors by diffusion. This fundamental physical limit is generally considered to be the Berg-Purcell limit [Biophys. J. 20, 193 (1977)]. Here we derive a lower limit by applying maximum likelihood to the time series of receptor occupancy. The increased accuracy stems from solely considering the unoccupied time intervals--disregarding the occupied time intervals as these do not contain any information about the external particle concentration, and only decrease the accuracy of the concentration estimate. Receptors which minimize the bound time intervals achieve the highest possible accuracy. We discuss how a cell could implement such an optimal sensing strategy by absorbing or degrading bound particles.

Bayesian model predicts the response of axons to molecular gradients

Axon guidance by molecular gradients plays a crucial role in wiring up the nervous system. However, the mechanisms axons use to detect gradients are largely unknown. We first develop a Bayesian "ideal observer" analysis of gradient detection by axons, based on the hypothesis that a principal constraint on gradient detection is intrinsic receptor binding noise. Second, from this model, we derive an equation predicting how the degree of response of an axon to a gradient should vary with gradient steepness and absolute concentration. Third, we confirm this prediction quantitatively by performing the first systematic experimental analysis of how axonal response varies with both these quantities. These experiments demonstrate a degree of sensitivity much higher than previously reported for any chemotacting system. Together, these results reveal both the quantitative constraints that must be satisfied for effective axonal guidance and the computational principles that may be used by the underlying signal transduction pathways, and allow predictions for the degree of response of axons to gradients in a wide variety of in vivo and in vitro settings.