Accuracy of direct gradient sensing by single cells

Role of Signal Degradation in Directional Chemosensing

Directional chemosensing is ubiquitous in cell biology, but some cells such as mating yeast paradoxically degrade the signal they aim to detect. While the data processing inequality suggests that such signal modification cannot increase the sensory information, we show using a reaction-diffusion model and an exactly solvable discrete-state reduction that it can. We identify a non-Markovian step in the information chain allowing the system to evade the data processing inequality, reflecting the nonlocal nature of diffusion. Our results apply to any sensory system in which degradation couples to diffusion. Experimental data suggest that mating yeast operate in the beneficial regime where degradation improves sensing.

Physical extraction of antigen and information

To respond and adapt, cells use surface receptors to sense environmental cues. While biochemical signal processing inside the cell is studied in depth, less is known about how physical processes during cell-cell contact impact signal acquisition. New experiments found that fast-evolving immune B cells in germinal centers (GCs) apply force to acquire antigen clusters prior to internalization, suggesting adaptive benefits of physical information extraction. We present a theory of stochastic antigen transfer and show that maximizing information gain via physical extraction can explain the dramatic phenotypic transition from naive to GC B cells-attenuated receptor signaling, enhanced force usage, and decentralized contact architecture. Our model suggests that binding-lifetime measurement and physical extraction serve as complementary modes of antigen recognition, greatly extending the dynamic range of affinity discrimination when combined. This physical-information framework further predicts that the optimal size of receptor clusters decreases as affinity improves, rationalizing the use of a multifocal synaptic pattern seen in GC B cells. By linking extraction dynamics to selection fidelity via discriminatory performance, we propose that cells may physically enhance information acquisition to sustain adaptive evolution.

Learning optimal integration of spatial and temporal information in noisy chemotaxis

We investigate the boundary between chemotaxis driven by spatial estimation of gradients and chemotaxis driven by temporal estimation. While it is well known that spatial chemotaxis becomes disadvantageous for small organisms at high noise levels, it is unclear whether there is a discontinuous switch of optimal strategies or a continuous transition exists. Here, we employ deep reinforcement learning to study the possible integration of spatial and temporal information in an a priori unconstrained manner. We parameterize such a combined chemotactic policy by a recurrent neural network and evaluate it using a minimal theoretical model of a chemotactic cell. By comparing with constrained variants of the policy, we show that it converges to purely temporal and spatial strategies at small and large cell sizes, respectively. We find that the transition between the regimes is continuous, with the combined strategy outperforming in the transition region both the constrained variants as well as models that explicitly integrate spatial and temporal information. Finally, by utilizing the attribution method of integrated gradients, we show that the policy relies on a nontrivial combination of spatially and temporally derived gradient information in a ratio that varies dynamically during the chemotactic trajectories.

Memoryless chemotaxis with discrete cues

Biological systems such as axonal growth cones perform chemotaxis at micrometre-level length scales, where chemotactic molecules are sparse. Such systems lie outside the range of validity of existing models, which assume smoothly varying chemical gradients. We investigate the effect of introducing discrete chemoattractant molecules by constructing a minimal dynamical model consisting of a chemotactic cell without internal memory. Significant differences are found in the behaviour of the cell as the chemical gradient is changed from smoothly varying to discrete, including the emergence of a homing radius beyond which chemotaxis is not reliably performed.

Active oscillations in microscale navigation

Living organisms routinely navigate their surroundings in search of better conditions, more food, or to avoid predators. Typically, animals do so by integrating sensory cues from the environment with their locomotor apparatuses. For single cells or small organisms that possess motility, fundamental physical constraints imposed by their small size have led to alternative navigation strategies that are specific to the microscopic world. Intriguingly, underlying these myriad exploratory behaviours or sensory functions is the onset of periodic activity at multiple scales, such as the undulations of cilia and flagella, the vibrations of hair cells, or the oscillatory shape modes of migrating neutrophils. Here, I explore oscillatory dynamics in basal microeukaryotes and hypothesize that these active oscillations play a critical role in enhancing the fidelity of adaptive sensorimotor integration.

Collective effects in flow-driven cell migration

Autologous chemotaxis is the process in which cells secrete and detect molecules to determine the direction of fluid flow. Experiments and theory suggest that autologous chemotaxis fails at high cell densities because molecules from other cells interfere with a given cell's signal. We investigate autologous chemotaxis using a three-dimensional Monte Carlo-based motility simulation that couples spatial and temporal gradient sensing with cell-cell repulsion. Surprisingly, we find that when temporal gradient sensing dominates, high-density clusters chemotax faster than individual cells. To explain this observation, we propose a mechanism by which temporal gradient sensing allows cells to form a collective sensory unit. We demonstrate using computational fluid mechanics that that this mechanism indeed allows a cluster of cells to outperform single cells in terms of the detected anisotropy of the signal, a finding that we demonstrate with analytic scaling arguments. Our work suggests that collective autologous chemotaxis at high cell densities is possible and requires only known, ubiquitous cell capabilities.

CCR7 acts as both a sensor and a sink for CCL19 to coordinate collective leukocyte migration

Immune responses rely on the rapid and coordinated migration of leukocytes. Whereas it is well established that single-cell migration is often guided by gradients of chemokines and other chemoattractants, it remains poorly understood how these gradients are generated, maintained, and modulated. By combining experimental data with theory on leukocyte chemotaxis guided by the G protein-coupled receptor (GPCR) CCR7, we demonstrate that in addition to its role as the sensory receptor that steers migration, CCR7 also acts as a generator and a modulator of chemotactic gradients. Upon exposure to the CCR7 ligand CCL19, dendritic cells (DCs) effectively internalize the receptor and ligand as part of the canonical GPCR desensitization response. We show that CCR7 internalization also acts as an effective sink for the chemoattractant, dynamically shaping the spatiotemporal distribution of the chemokine. This mechanism drives complex collective migration patterns, enabling DCs to create or sharpen chemotactic gradients. We further show that these self-generated gradients can sustain the long-range guidance of DCs, adapt collective migration patterns to the size and geometry of the environment, and provide a guidance cue for other comigrating cells. Such a dual role of CCR7 as a GPCR that both senses and consumes its ligand can thus provide a novel mode of cellular self-organization.

Single-cell directional sensing from just a few receptor binding events

Identifying the directionality of signaling sources from noisy input to membrane receptors is an essential task performed by many cell types. A variety of models have been proposed to explain directional sensing in cells. However, many of these require significant computational and memory capacities for the cell. We propose and analyze a simple mechanism in which a cell adopts the direction associated with the first few membrane binding events. This model yields an accurate angular estimate to the source long before steady state is reached in biologically relevant scenarios. Our proposed mechanism allows for reliable estimates of the directionality of external signals using temporal information and assumes minimal computational capacities of the cell.

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.

Short-time diffusive fluxes over membrane receptors yields the direction of a signalling source

An essential ability of many cell types is to detect stimuli in the form of shallow chemical gradients. Such cues may indicate the direction that new growth should occur, or the location of a mate. Amplification of these faint signals is due to intra-cellular mechanisms, while the cue itself is generated by the noisy arrival of signalling molecules to surface bound membrane receptors. We employ a new hybrid numerical-asymptotic technique coupling matched asymptotic analysis and numerical inverse Laplace transform to rapidly and accurately solve the parabolic exterior problem describing the dynamic diffusive fluxes to receptors. We observe that equilibration occurs on long timescales, potentially limiting the usefulness of steady-state quantities for localization at practical biological timescales. We demonstrate that directional information is encoded primarily in early arrivals to the receptors, while equilibrium quantities inform on source distance. We develop a new homogenization result showing that complex receptor configurations can be replaced by a uniform effective condition. In the extreme scenario where the cell adopts the angular direction of the first impact, we show this estimate to be surprisingly accurate.

Immune cells use active tugging forces to distinguish affinity and accelerate evolution

Cells are known to exert forces to sense their physical surroundings for guidance of motion and fate decisions. Here, we propose that cells might do mechanical work to drive their own evolution, taking inspiration from the adaptive immune system. Growing evidence indicates that immune B cells-capable of rapid Darwinian evolution-use cytoskeletal forces to actively extract antigens from other cells' surfaces. To elucidate the evolutionary significance of force usage, we develop a theory of tug-of-war antigen extraction that maps receptor binding characteristics to clonal reproductive fitness, revealing physical determinants of selection strength. This framework unifies mechanosensing and affinity-discrimination capabilities of evolving cells: Pulling against stiff antigen tethers enhances discrimination stringency at the expense of absolute extraction. As a consequence, active force usage can accelerate adaptation but may also cause extinction of cell populations, resulting in an optimal range of pulling strength that matches molecular rupture forces observed in cells. Our work suggests that nonequilibrium, physical extraction of environmental signals can make biological systems more evolvable at a moderate energy cost.

Social Search and Resource Clustering as Emergent Stable States

Social search has stably evolved across various species and is often used by humans to search for resources (such as food, information, social partners). In turn, these resources frequently come distributed in patches or clusters. In the current work, we use an ecologically inspired agent-based model to investigate whether social search and clustering are stable outcomes of the dynamical mutual interactions between the two. While previous research has studied unidirectional influences of social search on resource clustering and vice versa, the current work investigates the consequential patterns emerging from their two-way interactions over time. In our model, consumers evolved search strategies (ranging from competitive to social) as adaptations to their environmental resource structures, and resources varied in distributions (ranging from random to clustered) that were shaped by agents' consumption patterns. Across four experiments, we systematically analyzed the patterns of influence that search strategies and environment structure have on each other to identify stable attractor states of both. In Experiment 1, we fixed resource clustering at various levels and observed its influence on social search, and in Experiment 2, we observed the influence of social search on resource distribution. In both these experiments we found that increasing levels of one variable produced increases in the other; however, at very high levels of the manipulated variable, the dependent variable tended to fall. Finally in Experiments 3 and 4, we studied the dynamics that arose when resource clustering and social search could both change and mutually influence each other, finding that low levels of social search and clustering were stable attractor states. Our simple 2D model yielded results that qualitatively resemble those across a wide range of search domains (from physical search for food to abstract search for information), highlighting some stable outcomes of mutually interacting consumer/resource systems.

Computational methods and diffusion theory in triangulation sensing to model neuronal navigation

Computational methods are now recognized as powerful and complementary approaches in various applied sciences such as biology. These computing methods are used to explore the gap between scales such as the one between molecular and cellular. Here we present recent progress in the development of computational approaches involving diffusion modeling, asymptotic analysis of the model partial differential equations, hybrid methods and simulations in the generic context of cell sensing and guidance via external gradients. Specifically, we highlight the reconstruction of the location of a point source in two and three dimensions from the steady-state diffusion fluxes arriving to narrow windows located on the cell. We discuss cases in which these windows are located on the boundary of a two-dimensional plane or three-dimensional half-space, on a disk in free space or inside a two-dimensional corridor, or a ball in three dimensions. The basis of this computational approach is explicit solutions of the Neumann-Green's function for the mentioned geometry. This analysis can be used to design hybrid simulations where Brownian paths are generated only in small regions in which the local spatial organization is relevant. Particle trajectories outside of this region are only implicitly treated by generating exit points at the boundary of this domain of interest. This greatly accelerates the simulation time by avoiding the explicit computation of Brownian paths in an infinite domain and serves to generate statistics, without following all trajectories at the same time, a process that can become numerically expensive quickly. Moreover, these computational approaches are used to reconstruct a point source and estimating the uncertainty in the source reconstruction due to an additive noise perturbation present in the fluxes. We also discuss the influence of various window configurations (cluster vs uniform distributions) on recovering the source position. Finally, the applications in developmental biology are formulated into computational principles that could underly neuronal navigation in the brain.

Autologous chemotaxis at high cell density

Autologous chemotaxis, in which cells secrete and detect molecules to determine the direction of fluid flow, is thwarted at high cell density because molecules from other cells interfere with a given cell's signal. Using a minimal model of autologous chemotaxis, we determine the cell density at which sensing fails, and we find that it agrees with experimental observations of metastatic cancer cells. To understand this agreement, we derive a physical limit to autologous chemotaxis in terms of the cell density, the Péclet number, and the lengthscales of the cell and its environment. Surprisingly, in an environment that is uniformly oversaturated in the signaling molecule, we find that not only can sensing fail, but it can be reversed, causing backwards cell motion. Our results get to the heart of the competition between chemical and mechanical cellular sensing, and they shed light on a sensory strategy employed by cancer cells in dense tumor environments.

Localization of signaling receptors maximizes cellular information acquisition in spatially structured natural environments

Cells in natural environments, such as tissue or soil, sense and respond to extracellular ligands with intricately structured and non-monotonic spatial distributions, sculpted by processes such as fluid flow and substrate adhesion. In this work, we show that spatial sensing and navigation can be optimized by adapting the spatial organization of signaling pathways to the spatial structure of the environment. We develop an information-theoretic framework for computing the optimal spatial organization of a sensing system for a given signaling environment. We find that receptor localization previously observed in cells maximizes information acquisition in simulated natural contexts, including tissue and soil. Specifically, information acquisition is maximized when receptors form localized patches at regions of maximal ligand concentration. Receptor localization extends naturally to produce a dynamic protocol for continuously redistributing signaling receptors, which when implemented using simple feedback, boosts cell navigation efficiency by 30-fold.

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.

Orientation of Cell Polarity by Chemical Gradients

Accurate decoding of spatial chemical landscapes is critical for many cell functions. Eukaryotic cells decode local chemical gradients to orient growth or movement in productive directions. Recent work on yeast model systems, whose gradient sensing pathways display much less complexity than those in animal cells, has suggested new paradigms for how these very small cells successfully exploit information in noisy and dynamic pheromone gradients to identify their mates. Pheromone receptors regulate a polarity circuit centered on the conserved Rho-family GTPase, Cdc42. The polarity circuit contains both positive and negative feedback pathways, allowing spontaneous symmetry breaking and also polarity site disassembly and relocation. Cdc42 orients the actin cytoskeleton, leading to focused vesicle traffic that promotes movement of the polarity site and also reshapes the cortical distribution of receptors at the cell surface. In this article, we review the advances from work on yeasts and compare them with the excitable signaling pathways that have been revealed in chemotactic animal cells. Expected final online publication date for the Annual Review of Biophysics, Volume 51 is May 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Signal processing capacity of the cellular sensory machinery regulates the accuracy of chemotaxis under complex cues

Chemotaxis is ubiquitous in many biological processes, but it still remains elusive how cells sense and decipher multiple chemical cues. In this study, we postulate a hypothesis that the chemotactic performance of cells under complex cues is regulated by the signal processing capacity of the cellular sensory machinery. The underlying rationale is that cells in vivo should be able to sense and process multiple chemical cues, whose magnitude and compositions are entangled, to determine their migration direction. We experimentally show that the combination of transforming growth factor-β and epidermal growth factor suppresses the chemotactic performance of cancer cells using independent receptors to sense the two cues. Based on this observation, we develop a biophysical framework suggesting that the antagonism is caused by the saturation of the signal processing capacity but not by the mutual repression. Our framework suggests the significance of the signal processing capacity in the cellular sensory machinery.

Reconstructing a point source from diffusion fluxes to narrow windows in three dimensions

We develop a computational approach to locate the source of a steady-state gradient of diffusing particles from the fluxes through narrow windows distributed either on the boundary of a three-dimensional half-space or on a sphere. This approach is based on solving the mixed boundary stationary diffusion equation with Neumann–Green’s function. The method of matched asymptotic expansions enables the computation of the probability fluxes. To explore the range of validity of this expansion, we develop a fast analytical-Brownian numerical scheme. This scheme accelerates the simulation time by avoiding the explicit computation of Brownian trajectories in the infinite domain. The results obtained from our derived analytical formulae and the fast numerical simulation scheme agree on a large range of parameters. Using the analytical representation of the particle fluxes, we show how to reconstruct the location of the point source. Furthermore, we investigate the uncertainty in the source reconstruction due to additive fluctuations present in the fluxes. We also study the influence of various window configurations: clustered versus uniform distributions on recovering the source position. Finally, we discuss possible applications for cell navigation in biology.

Precision of Protein Thermometry

Temperature sensing is a ubiquitous cell behavior, but the fundamental limits to the precision of temperature sensing are poorly understood. Unlike in chemical concentration sensing, the precision of temperature sensing is not limited by extrinsic fluctuations in the temperature field itself. Instead, we find that precision is limited by the intrinsic copy number, turnover, and binding kinetics of temperature-sensitive proteins. Developing a model based on the canonical TlpA protein, we find that a cell can estimate temperature to within 2%. We compare this prediction with in vivo data on temperature sensing in bacteria.

Short-range quorum sensing controls horizontal gene transfer at micron scale in bacterial communities

In bacterial communities, cells often communicate by the release and detection of small diffusible molecules, a process termed quorum-sensing. Signal molecules are thought to broadly diffuse in space; however, they often regulate traits such as conjugative transfer that strictly depend on the local community composition. This raises the question how nearby cells within the community can be detected. Here, we compare the range of communication of different quorum-sensing systems. While some systems support long-range communication, we show that others support a form of highly localized communication. In these systems, signal molecules propagate no more than a few microns away from signaling cells, due to the irreversible uptake of the signal molecules from the environment. This enables cells to accurately detect micron scale changes in the community composition. Several mobile genetic elements, including conjugative elements and phages, employ short-range communication to assess the fraction of susceptible host cells in their vicinity and adaptively trigger horizontal gene transfer in response. Our results underscore the complex spatial biology of bacteria, which can communicate and interact at widely different spatial scales.

Connection between the Bacterial Chemotactic Network and Optimal Filtering

The chemotactic network of Escherichia coli has been studied extensively both biophysically and information theoretically. Nevertheless, connection between these two aspects is still elusive. In this work, we report such a connection. We derive an optimal filtering dynamics under the assumption that E. coli's sensory system optimally infers the binary information whether it is swimming up or down along an exponential ligand gradient from noisy sensory signals. Then we show that a standard biochemical model of the chemotactic network is mathematically equivalent to this information-theoretically optimal dynamics. Moreover, we demonstrate that an experimentally observed nonlinear response relation can be reproduced from the optimal dynamics. These results suggest that the biochemical network of E. coli chemotaxis is designed to optimally extract the binary information along an exponential gradient in a noisy condition.

Average search time bounds in cue-based searches

In this work we consider search problems that evaluate the probability distribution of finding the source at each step in the search. We start with a sample strategy where the movement at each time step is in the immediate neighborhood. The jump probability is taken to be proportional to the normalized difference between the probability of finding the source at the jump location with the probability of finding the source at the present location. We evaluate a lower bound on the average search time for a searcher using this strategy. We next consider the problem of evaluating the lower bound on the search time for a generic strategy which would utilize the source probability distribution to figure out the position of the source. We derive an expression for the lower bound on the search time. We present an analytic expression for this lower bound in a case in which the particles emitted by the source diffuse in a homogeneous manner. For a general probability distribution with entropy E, we find that the lower bound goes as e^{E/2}.

Origins of eukaryotic excitability

All living cells interact dynamically with a constantly changing world. Eukaryotes, in particular, evolved radically new ways to sense and react to their environment. These advances enabled new and more complex forms of cellular behaviour in eukaryotes, including directional movement, active feeding, mating, and responses to predation. But what are the key events and innovations during eukaryogenesis that made all of this possible? Here we describe the ancestral repertoire of eukaryotic excitability and discuss five major cellular innovations that enabled its evolutionary origin. The innovations include a vastly expanded repertoire of ion channels, the emergence of cilia and pseudopodia, endomembranes as intracellular capacitors, a flexible plasma membrane and the relocation of chemiosmotic ATP synthesis to mitochondria, which liberated the plasma membrane for more complex electrical signalling involved in sensing and reacting. We conjecture that together with an increase in cell size, these new forms of excitability greatly amplified the degrees of freedom associated with cellular responses, allowing eukaryotes to vastly outperform prokaryotes in terms of both speed and accuracy. This comprehensive new perspective on the evolution of excitability enriches our view of eukaryogenesis and emphasizes behaviour and sensing as major contributors to the success of eukaryotes. This article is part of the theme issue 'Basal cognition: conceptual tools and the view from the single cell'.

Temporally regulated cell migration is sensitive to variation in body size

Few studies have measured the robustness to perturbations of the final position of a long-range migrating cell. In the nematode Caenorhabditis elegans, the QR neuroblast migrates anteriorly, while undergoing three division rounds. We study the final position of two of its great-granddaughters, the end of migration of which was previously shown to depend on a timing mechanism. We find that the variance in their final position is similar to that of other long-range migrating neurons. As expected from the timing mechanism, the position of QR descendants depends on body size, which we varied by changing maternal age or using body size mutants. Using a mathematical model, we show that body size variation is partially compensated for. Applying environmental perturbations, we find that the variance in final position increased following starvation at hatching. The mean position is displaced upon a temperature shift. Finally, highly significant variation was found among C. elegans wild isolates. Overall, this study reveals that the final position of these neurons is quite robust to stochastic variation, shows some sensitivity to body size and to external perturbations, and varies in the species.This article has an associated 'The people behind the papers' interview.

Chemokinesis-driven accumulation of active colloids in low-mobility regions of fuel gradients

Many motile cells exhibit migratory behaviors, such as chemotaxis (motion up or down a chemical gradient) or chemokinesis (dependence of speed on chemical concentration), which enable them to carry out vital functions including immune response, egg fertilization, and predator evasion. These have inspired researchers to develop self-propelled colloidal analogues to biological microswimmers, known as active colloids, that perform similar feats. Here, we study the behavior of half-platinum half-gold (Pt/Au) self-propelled rods in antiparallel gradients of hydrogen peroxide fuel and salt, which tend to increase and decrease the rods' speed, respectively. Brownian Dynamics simulations, a Fokker-Planck theoretical model, and experiments demonstrate that, at steady state, the rods accumulate in low-speed (salt-rich, peroxide-poor) regions not because of chemotaxis, but because of chemokinesis. Chemokinesis is distinct from chemotaxis in that no directional sensing or reorientation capabilities are required. The agreement between simulations, model, and experiments bolsters the role of chemokinesis in this system. This work suggests a novel strategy of exploiting chemokinesis to effect accumulation of motile colloids in desired areas.

Multicellular sensing at a feedback-induced critical point

Feedback in sensory biochemical networks can give rise to bifurcations in cells' behavioral response. These bifurcations share many properties with thermodynamic critical points. Evidence suggests that biological systems may operate near these critical points, but the functional benefit of doing so remains poorly understood. Here we investigate a simple biochemical model with nonlinear feedback and multicellular communication to determine if criticality provides a functional benefit in terms of the ability to gain information about a stochastic chemical signal. We find that when signal fluctuations are slow, the mutual information between the signal and the intracellular readout is maximized at criticality, because the benefit of high signal susceptibility outweighs the detriment of high readout noise. When cells communicate, criticality gives rise to long-range correlations in readout molecule number among cells. Consequently, we find that communication increases the mutual information between a given cell's readout and the spatial average of the signal across the population. Finally, we find that both with and without communication, the sensory benefits of criticality compete with critical slowing down, such that the information rate, as opposed to the information itself, is minimized at the critical point. Our results reveal the costs and benefits of feedback-induced criticality for multicellular sensing.

Physical limits to sensing material properties

All materials respond heterogeneously at small scales, which limits what a sensor can learn. Although previous studies have characterized measurement noise arising from thermal fluctuations, the limits imposed by structural heterogeneity have remained unclear. In this paper, we find that the least fractional uncertainty with which a sensor can determine a material constant λ0 of an elastic medium is approximately [Formula: see text] for a ≫ d ≫ ξ, [Formula: see text], and D > 1, where a is the size of the sensor, d is its spatial resolution, ξ is the correlation length of fluctuations in λ0, Δλ is the local variability of λ0, and D is the dimension of the medium. Our results reveal how one can construct devices capable of sensing near these limits, e.g. for medical diagnostics. We use our theoretical framework to estimate the limits of mechanosensing in a biopolymer network, a sensory process involved in cellular behavior, medical diagnostics, and material fabrication.

Triangulation Sensing to Determine the Gradient Source from Diffusing Particles to Small Cell Receptors

How does a cell locate the source of molecular guidance cues from within a concentration gradient? We present a computational approach to recover the source from the absorbed fluxes at narrow receptor windows located on the surface of the cell. In the limit of fast binding, we solve the steady-state diffusion equation using an asymptotic approach and hybrid stochastic-analytical simulations. We show that the sensitivity to the gradient direction decays too rapidly to enable long-distance sensing. We illustrate how this constraint can be alleviated when triangulating the source with an increasing number of receptor windows and quantify the susceptibility of this process to flux perturbations.

A mechanism for hunchback promoters to readout morphogenetic positional information in less than a minute

Cell fate decisions in the fly embryo are rapid: hunchback genes decide in minutes whether nuclei follow the anterior/posterior developmental blueprint by reading out positional information in the Bicoid morphogen. This developmental system is a prototype of regulatory decision processes that combine speed and accuracy. Traditional arguments based on fixed-time sampling of Bicoid concentration indicate that an accurate readout is impossible within the experimental times. This raises the general issue of how speed-accuracy tradeoffs are achieved. Here, we compare fixed-time to on-the-fly decisions, based on comparing the likelihoods of anterior/posterior locations. We found that these more efficient schemes complete reliable cell fate decisions within the short embryological timescales. We discuss the influence of promoter architectures on decision times and error rates, present concrete examples that rapidly readout the morphogen, and predictions for new experiments. Lastly, we suggest a simple mechanism for RNA production and degradation that approximates the log-likelihood function.

Chemical cloaking

Hiding an object in a chemical gradient requires one to suppress the distortions it would naturally cause on it. To do so, we propose a strategy based on coating the object with a chemical reaction-diffusion network which can act as an active cloaking device. By controlling the concentration of some species in its immediate surrounding, the chemical reactions redirect the gradient as if the object was not there. We also show that a substantial fraction of the energy required to cloak can be extracted from the chemical gradient itself.

Receptor Organization Determines the Limits of Single-Cell Source Location Detection

Many types of cells require the ability to pinpoint the location of an external stimulus from the arrival of diffusing signaling molecules at cell-surface receptors. How does the organization (number and spatial configuration) of these receptors shape the limit of a cell's ability to infer the source location? In the idealized scenario of a spherical cell, we apply asymptotic analysis to compute splitting probabilities between individual receptors and formulate an information-theoretic framework to quantify the role of receptor organization. Clustered configurations of receptors provide an advantage in detecting sources aligned with the clusters, suggesting a possible multiscale mechanism for single-cell source inference.

Precision of Flow Sensing by Self-Communicating Cells

Metastatic cancer cells detect the direction of lymphatic flow by self-communication: they secrete and detect a chemical which, due to the flow, returns to the cell surface anisotropically. The secretion rate is low, meaning detection noise may play an important role, but the sensory precision of this mechanism has not been explored. Here we derive the precision of flow sensing for two ubiquitous detection methods: absorption vs reversible binding to surface receptors. We find that binding is more precise due to the fact that absorption distorts the signal that the cell aims to detect. Comparing to experiments, our results suggest that the cancer cells operate remarkably close to the physical detection limit. Our prediction that cells should bind the chemical reversibly, not absorb it, is supported by endocytosis data for this ligand-receptor pair.

Absolute proteomic quantification reveals design principles of sperm flagellar chemosensation

Cilia serve as cellular antennae that translate sensory information into physiological responses. In the sperm flagellum, a single chemoattractant molecule can trigger a Ca2+ rise that controls motility. The mechanisms underlying such ultra-sensitivity are ill-defined. Here, we determine by mass spectrometry the copy number of nineteen chemosensory signaling proteins in sperm flagella from the sea urchin Arbacia punctulata. Proteins are up to 1,000-fold more abundant than the free cellular messengers cAMP, cGMP, H+ , and Ca2+ . Opto-chemical techniques show that high protein concentrations kinetically compartmentalize the flagellum: Within milliseconds, cGMP is relayed from the receptor guanylate cyclase to a cGMP-gated channel that serves as a perfect chemo-electrical transducer. cGMP is rapidly hydrolyzed, possibly via "substrate channeling" from the channel to the phosphodiesterase PDE5. The channel/PDE5 tandem encodes cGMP turnover rates rather than concentrations. The rate-detection mechanism allows continuous stimulus sampling over a wide dynamic range. The textbook notion of signal amplification-few enzyme molecules process many messenger molecules-does not hold for sperm flagella. Instead, high protein concentrations ascertain messenger detection. Similar mechanisms may occur in other small compartments like primary cilia or dendritic spines.

Different effects of fast and slow input fluctuations on output in gene regulation

An important task in the post-gene era is to understand the role of stochasticity in gene regulation. Here, we analyze a cascade model of stochastic gene expression, where the upstream gene stochastically generates proteins that regulate, as transcription factors, stochastic synthesis of the downstream output. We find that in contrast to fast input fluctuations that do not change the behavior of the downstream system qualitatively, slow input fluctuations can induce different modes of the distribution of downstream output and even stochastic focusing or defocusing of the downstream output level, although the regulatory protein follows the same distribution in both cases. This finding is counterintuitive but can have broad biological implications, e.g., slow input rather than fast fluctuations may both increase the survival probability of cells and enhance the sensitivity of intracellular regulation. In addition, we find that input fluctuations can minimize the output noise.

Ciliate Biology: The Graceful Hunt of a Shape-Shifting Predator

Even single-celled eukaryotes are capable of highly complex behaviors. A new study reveals how one unicellular predator actively manipulates and remodels its unique cytoskeletal morphology to achieve rapid shape changes and a remarkable hunting strategy.

Leader cells in collective chemotaxis: Optimality and trade-offs

Clusters of cells can work together in order to follow a signal gradient, chemotaxing even when single cells do not. Cells in different regions of collectively migrating neural crest streams show different gene expression profiles, suggesting that cells may specialize to leader and follower roles. We use a minimal mathematical model to understand when this specialization is advantageous. In our model, leader cells sense the gradient with an accuracy that depends on the kinetics of ligand-receptor binding, while follower cells follow the cluster's direction with a finite error. Intuitively, specialization into leaders and followers should be optimal when a few cells have more information than the rest of the cluster, such as in the presence of a sharp transition in chemoattractant concentration. We do find this-but also find that high levels of specialization can be optimal in the opposite limit of very shallow gradients. We also predict that the best location for leaders may not be at the front of the cluster. In following leaders, clusters may have to choose between speed and flexibility. Clusters with only a few leaders can take orders of magnitude more time to reorient than all-leader clusters.

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.

Positional Information Readout in Ca^{2+} Signaling

Living cells respond to spatially confined signals. Intracellular signal transmission often involves the release of second messengers like Ca^{2+}. They eventually trigger a physiological response, for example, by activating kinases that in turn activate target proteins through phosphorylation. Here, we investigate theoretically how positional information can be accurately read out by protein phosphorylation in spite of rapid second messenger diffusion. We find that accuracy is increased by binding of kinases to the cell membrane prior to phosphorylation and by increasing the rate of Ca^{2+} loss from the cell interior. These findings could explain some salient features of the conventional protein kinase Cα.

Bacteria push the limits of chemotactic precision to navigate dynamic chemical gradients

Ephemeral aggregations of bacteria are ubiquitous in the environment, where they serve as hotbeds of metabolic activity, nutrient cycling, and horizontal gene transfer. In many cases, these regions of high bacterial concentration are thought to form when motile cells use chemotaxis to navigate to chemical hotspots. However, what governs the dynamics of bacterial aggregations is unclear. Here, we use an experimental platform to create realistic submillimeter-scale nutrient pulses with controlled nutrient concentrations. By combining experiments, mathematical theory, and agent-based simulations, we show that individual Vibrio ordalii bacteria begin chemotaxis toward hotspots of dissolved organic matter (DOM) when the magnitude of the chemical gradient rises sufficiently far above the sensory noise that is generated by stochastic encounters with chemoattractant molecules. Each DOM hotspot is surrounded by a dynamic ring of chemotaxing cells, which congregate in regions of high DOM concentration before dispersing as DOM diffuses and gradients become too noisy for cells to respond to. We demonstrate that V. ordalii operates close to the theoretical limits on chemotactic precision. Numerical simulations of chemotactic bacteria, in which molecule counting noise is explicitly taken into account, point at a tradeoff between nutrient acquisition and the cost of chemotactic precision. More generally, our results illustrate how limits on sensory precision can be used to understand the location, spatial extent, and lifespan of bacterial behavioral responses in ecologically relevant environments.

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.

Early Cancer Detection in Blood Vessels Using Mobile Nanosensors

In this paper, we propose using mobile nanosensors (MNSs) for early stage anomaly detection. For concreteness, we focus on the detection of cancer cells located in a particular region of a blood vessel. These cancer cells produce and emit special molecules, so-called biomarkers, which are symptomatic for the presence of anomaly, into the cardiovascular system. Detection of cancer biomarkers with conventional blood tests is difficult in the early stages of a cancer due to the very low concentration of the biomarkers in the samples taken. However, close to the cancer cells, the concentration of the cancer biomarkers is high. Hence, detection is possible if a sensor with the ability to detect these biomarkers is placed in the vicinity of the cancer cells. Therefore, in this paper, we study the use of MNSs that are injected at a suitable injection site and can move through the blood vessels of the cardiovascular system, which potentially contain cancer cells. These MNSs can be activated by the biomarkers close to the cancer cells, where the biomarker concentration is sufficiently high. Eventually, the MNSs are collected by a fusion center (FC), where their activation levels are read and exploited to declare the presence of anomaly. We analytically derive the biomarker concentration in a network of interconnected blood vessels as well as the probability mass function of the MNSs' activation levels and validate the obtained results via particle-based simulations. Then, we derive the optimal decision rule for the FC regarding the presence of anomaly assuming that the entire network is known at the FC. Finally, for the FC, we propose a simple sum detector that does not require knowledge of the network topology. Our simulations reveal that while the optimal detector achieves a higher performance than the sum detector, both proposed detectors significantly outperform a benchmark scheme that uses fixed nanosensors at the FC.

Collective gradient sensing and chemotaxis: modeling and recent developments

Cells measure a vast variety of signals, from their environment's stiffness to chemical concentrations and gradients; physical principles strongly limit how accurately they can do this. However, when many cells work together, they can cooperate to exceed the accuracy of any single cell. In this topical review, I will discuss the experimental evidence showing that cells collectively sense gradients of many signal types, and the models and physical principles involved. I also propose new routes by which experiments and theory can expand our understanding of these problems.

Decision making improves sperm chemotaxis in the presence of noise

To navigate their surroundings, cells rely on sensory input that is corrupted by noise. In cells performing chemotaxis, such noise arises from the stochastic binding of signalling molecules at low chemoattractant concentrations. We reveal a fundamental relationship between the speed of chemotactic steering and the strength of directional fluctuations that result from the amplification of noise in a chemical input signal. This relation implies a trade-off between steering that is slow and reliable, and steering that is fast but less reliable. We show that dynamic switching between these two modes of steering can substantially increase the probability to find a target, such as an egg to be found by sperm cells. This decision making confers no advantage in the absence of noise, but is beneficial when chemical signals are detectable, yet characterized by low signal-to-noise ratios. The latter applies at intermediate distances from a target, where signalling molecules are diluted, thus defining a ‘noise zone’ that cells have to cross. Our results explain decision making observed in recent experiments on sea urchin sperm chemotaxis. More generally, our theory demonstrates how decision making enables chemotactic agents to cope with high levels of noise in gradient sensing by dynamically adjusting the persistence length of a biased random walk.

Many cells can navigate upwards a concentration gradient of signalling molecules, a process called chemotaxis. Chemotaxis is used e.g. by sperm cells to find the egg. To measure and compare concentrations, cells count stochastic binding events of signalling molecules that diffuse to cellular receptors. Efficient chemotaxis strategies must be adapted to this molecular shot noise of concentration measurements. We reveal a fundamental relationship between the speed of chemotactic steering and the strength of directional fluctuations that result from the amplification of noise. This implies a trade-off between steering fast and steering reliable. Inspired by recent experiments on chemotaxis of sperm cells of marine invertebrates, we develop a theory that allows to efficiently compute optimal chemotaxis strategies. We show that dynamic switching between either fast or reliable steering can substantially increase the probability for a sperm cell to find the egg. Furthermore, the optimal strategy requires only minimal computational capacities of the chemotactic agent, a key constraint for biological cells. More generally, our work demonstrates a benefit of decision making for chemotaxis in the presence of noise, which could inspire optimal control designs for artificial microswimmers.

Behavioral Variability and Phenotypic Diversity in Bacterial Chemotaxis

Living cells detect and process external signals using signaling pathways that are affected by random fluctuations. These variations cause the behavior of individual cells to fluctuate over time (behavioral variability) and generate phenotypic differences between genetically identical individuals (phenotypic diversity). These two noise sources reduce our ability to predict biological behavior because they diversify cellular responses to identical signals. Here, we review recent experimental and theoretical advances in understanding the mechanistic origin and functional consequences of such variation in Escherichia coli chemotaxis-a well-understood model of signal transduction and behavior. After briefly summarizing the architecture and logic of the chemotaxis system, we discuss determinants of behavior and chemotactic performance of individual cells. Then, we review how cell-to-cell differences in protein abundance map onto differences in individual chemotactic abilities and how phenotypic variability affects the performance of the population. We conclude with open questions to be addressed by future research.

Mixed analytical-stochastic simulation method for the recovery of a Brownian gradient source from probability fluxes to small windows

Is it possible to recover the position of a source from the steady-state fluxes of Brownian particles to small absorbing windows located on the boundary of a domain? To address this question, we develop a numerical procedure to avoid tracking Brownian trajectories in the entire infinite space. Instead, we generate particles near the absorbing windows, computed from the analytical expression of the exit probability. When the Brownian particles are generated by a steady-state gradient at a single point, we compute asymptotically the fluxes to small absorbing holes distributed on the boundary of half-space and on a disk in two dimensions, which agree with stochastic simulations. We also derive an expression for the splitting probability between small windows using the matched asymptotic method. Finally, when there are more than two small absorbing windows, we show how to reconstruct the position of the source from the diffusion fluxes. The present approach provides a computational first principle for the mechanism of sensing a gradient of diffusing particles, a ubiquitous problem in cell biology.

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.

Cell-to-cell variation sets a tissue-rheology-dependent bound on collective gradient sensing

When a single cell senses a chemical gradient and chemotaxes, stochastic receptor-ligand binding can be a fundamental limit to the cell's accuracy. For clusters of cells responding to gradients, however, there is a critical difference: Even genetically identical cells have differing responses to chemical signals. With theory and simulation, we show collective chemotaxis is limited by cell-to-cell variation in signaling. We find that when different cells cooperate, the resulting bias can be much larger than the effects of ligand-receptor binding. Specifically, when a strongly responding cell is at one end of a cell cluster, cluster motion is biased toward that cell. These errors are mitigated if clusters average measurements over times long enough for cells to rearrange. In consequence, fluid clusters are better able to sense gradients: We derive a link between cluster accuracy, cell-to-cell variation, and the cluster rheology. Because of this connection, increasing the noisiness of individual cell motion can actually increase the collective accuracy of a cluster by improving fluidity.

Physical limits to biomechanical sensing in disordered fibre networks

Cells actively probe and respond to the stiffness of their surroundings. Since mechanosensory cells in connective tissue are surrounded by a disordered network of biopolymers, their in vivo mechanical environment can be extremely heterogeneous. Here we investigate how this heterogeneity impacts mechanosensing by modelling the cell as an idealized local stiffness sensor inside a disordered fibre network. For all types of networks we study, including experimentally-imaged collagen and fibrin architectures, we find that measurements applied at different points yield a strikingly broad range of local stiffnesses, spanning roughly two decades. We verify via simulations and scaling arguments that this broad range of local stiffnesses is a generic property of disordered fibre networks. Finally, we show that to obtain optimal, reliable estimates of global tissue stiffness, a cell must adjust its size, shape, and position to integrate multiple stiffness measurements over extended regions of space.

Cells in the connective tissue are surrounded by a heterogeneous network of biopolymers. Here, the authors investigate how such heterogeneity affects cellular mechanosensing by simulating the deformation response of experimental and modelled biopolymer networks to locally applied forces.