Engineering aspects of enzymatic signal transduction: photoreceptors in the retina

Nonequilibrium Effects on Information Recoverability of the Noisy Channels

We investigated the impact of nonequilibrium conditions on the transmission and recovery of information through noisy channels. By measuring the recoverability of messages from an information source, we demonstrate that the ability to recover information is connected to the nonequilibrium behavior of the information flow, particularly in terms of sequential information transfer. We discovered that the mathematical equivalence of information recoverability and entropy production characterizes the dissipative nature of information transfer. Our findings show that both entropy production (or recoverability) and mutual information increase monotonically with the nonequilibrium strength of information dynamics. These results suggest that the nonequilibrium dissipation cost can enhance the recoverability of noise messages and improve the quality of information transfer. Finally, we propose a simple model to test our conclusions and found that the numerical results support our findings.

Engineering synthetic phosphorylation signaling networks in human cells

Protein phosphorylation signaling networks play a central role in how cells sense and respond to their environment. Here, we describe the engineering of artificial phosphorylation networks in which "push-pull" motifs-reversible enzymatic phosphorylation cycles consisting of opposing kinase and phosphatase activities-are assembled from modular protein domain parts and then wired together to create synthetic phosphorylation circuits in human cells. We demonstrate that the composability of our design scheme enables model-guided tuning of circuit function and the ability to make diverse network connections; synthetic phosphorylation circuits can be coupled to upstream cell surface receptors to enable fast-timescale sensing of extracellular ligands, while downstream connections can regulate gene expression. We leverage these capabilities to engineer cell-based cytokine controllers that dynamically sense and suppress activated T cells. Our work introduces a generalizable approach for designing and building phosphorylation signaling circuits that enable user-defined sense-and-respond function for diverse biosensing and therapeutic applications.

Universal energy-accuracy tradeoffs in nonequilibrium cellular sensing

We combine stochastic thermodynamics, large deviation theory, and information theory to derive fundamental limits on the accuracy with which single cell receptors can estimate external concentrations. As expected, if the estimation is performed by an ideal observer of the entire trajectory of receptor states, then no energy consuming nonequilibrium receptor that can be divided into bound and unbound states can outperform an equilibrium two-state receptor. However, when the estimation is performed by a simple observer that measures the fraction of time the receptor is bound, we derive a fundamental limit on the accuracy of general nonequilibrium receptors as a function of energy consumption. We further derive and exploit explicit formulas to numerically estimate a Pareto-optimal tradeoff between accuracy and energy. We find this tradeoff can be achieved by nonuniform ring receptors with a number of states that necessarily increases with energy. Our results yield a thermodynamic uncertainty relation for the time a physical system spends in a pool of states and generalize the classic Berg-Purcell limit [H. C. Berg and E. M. Purcell, Biophys. J. 20, 193 (1977)0006-349510.1016/S0006-3495(77)85544-6] on cellular sensing along multiple dimensions.

Cellular Signaling beyond the Wiener-Kolmogorov Limit

Accurate propagation of signals through stochastic biochemical networks involves significant expenditure of cellular resources. The same is true for regulatory mechanisms that suppress fluctuations in biomolecular populations. Wiener-Kolmogorov (WK) optimal noise filter theory, originally developed for engineering problems, has recently emerged as a valuable tool to estimate the maximum performance achievable in such biological systems for a given metabolic cost. However, WK theory has one assumption that potentially limits its applicability: it relies on a linear, continuum description of the reaction dynamics. Despite this, up to now no explicit test of the theory in nonlinear signaling systems with discrete molecular populations has ever seen performance beyond the WK bound. Here we report the first direct evidence of the bound being broken. To accomplish this, we develop a theoretical framework for multilevel signaling cascades, including the possibility of feedback interactions between input and output. In the absence of feedback, we introduce an analytical approach that allows us to calculate exact moments of the stationary distribution for a nonlinear system. With feedback, we rely on numerical solutions of the system's master equation. The results show WK violations in two common network motifs: a two-level signaling cascade and a negative feedback loop. However, the magnitude of the violation is biologically negligible, particularly in the parameter regime where signaling is most effective. The results demonstrate that while WK theory does not provide strict bounds, its predictions for performance limits are excellent approximations, even for nonlinear systems.

Phototransduction in retinal cones: Analysis of parameter importance

In daylight, cone photoreceptors in the retina are responsible for the bulk of visual perception, yet compared to rods, far less is known quantitatively about their biochemistry. This is partly because it is hard to isolate and purify cone proteins. The issue is also complicated by the synergistic interaction of these parameters in producing systems biology outputs, such as photoresponse. Using a 3-D resolved, finite element model of cone outer segments, here we conducted a study of parameter significance using global sensitivity analysis, by Sobol indices, which was contextualized within the uncertainty surrounding these parameters in the available literature. The analysis showed that a subset of the parameters influencing the circulating dark current, such as the turnover rate of cGMP in the dark, may be most influential for variance with experimental flash response, while the shut-off rates of photoexcited rhodopsin and phosphodiesterase also exerted sizable effect. The activation rate of transducin by rhodopsin and the light-induced hydrolysis rate of cGMP exerted measurable effects as well but were estimated as relatively less significant. The results of this study depend on experimental ranges currently described in the literature and should be revised as these become better established. To that end, these findings may be used to prioritize parameters for measurement in future investigations.

Optimal ligand discrimination by asymmetric dimerization and turnover of interferon receptors

In multicellular organisms, antiviral defense mechanisms evoke a reliable collective immune response despite the noisy nature of biochemical communication between tissue cells. A molecular hub of this response, the interferon I receptor (IFNAR), discriminates between ligand types by their affinity regardless of concentration. To understand how ligand type can be decoded robustly by a single receptor, we frame ligand discrimination as an information-theoretic problem and systematically compare the major classes of receptor architectures: allosteric, homodimerizing, and heterodimerizing. We demonstrate that asymmetric heterodimers achieve the best discrimination power over the entire physiological range of local ligand concentrations. This design enables sensing of ligand presence and type, and it buffers against moderate concentration fluctuations. In addition, receptor turnover, which drives the receptor system out of thermodynamic equilibrium, allows alignment of activation points for ligands of different affinities and thereby makes ligand discrimination practically independent of concentration. IFNAR exhibits this optimal architecture, and our findings thus suggest that this specialized receptor can robustly decode digital messages carried by its different ligands.

Noisy signal propagation and amplification in phenotypic transition cascade of colonic cells

Like genes and proteins, cells can use biochemical networks to sense and process information. The differentiation of the cell state in colonic crypts forms a typical unidirectional phenotypic transitional cascade, in which stem cells differentiate into the transit-amplifying cells (TACs), and TACs continue to differentiate into fully differentiated cells. In order to quantitatively describe the relationship between the noise of each compartment and the amplification of signals, the gain factor is introduced, and the gain-fluctuation relation is obtained by using the linear noise approximation of the master equation. Through the simulation of these theoretical formulas, the characters of noise propagation and amplification are studied. It is found that the transmitted noise is an important part of the total noise in each downstream cell. Therefore, a small number of downstream cells can only cause its small inherent noise, but the total noise may be very large due to the transmitted noise. The influence of the transmitted noise may be the indirect cause of colon cancer. In addition, the total noise of the downstream cells always has a minimum value. As long as a reasonable value of the gain factor is selected, the number of cells in colonic crypts will be controlled within the normal range. This may be a good method to intervene the uncontrollable growth of tumor cells and effectively control the deterioration of colon cancer.

Kalman-like Self-Tuned Sensitivity in Biophysical Sensing

Living organisms need to be sensitive to a changing environment while also ignoring uninformative environmental fluctuations. Here, we argue that living cells can navigate these conflicting demands by dynamically tuning their environmental sensitivity. We analyze the circadian clock in Synechococcus elongatus, showing that clock-metabolism coupling can detect mismatch between clock predictions and the day-night light cycle, temporarily raise the clock's sensitivity to light changes, and thus re-entraining faster. We find analogous behavior in recent experiments on switching between slow and fast osmotic-stress-response pathways in yeast. In both cases, cells can raise their sensitivity to new external information in epochs of frequent challenging stress, much like a Kalman filter with adaptive gain in signal processing. Our work suggests a new class of experiments that probe the history dependence of environmental sensitivity in biophysical sensing mechanisms.

Maximal information transmission is compatible with ultrasensitive biological pathways

Cells are often considered input-output devices that maximize the transmission of information by converting extracellular stimuli (input) via signaling pathways (communication channel) to cell behavior (output). However, in biological systems outputs might feed back into inputs due to cell motility, and the biological channel can change by mutations during evolution. Here, we show that the conventional channel capacity obtained by optimizing the input distribution for a fixed channel may not reflect the global optimum. In a new approach we analytically identify both input distributions and input-output curves that optimally transmit information, given constraints from noise and the dynamic range of the channel. We find a universal optimal input distribution only depending on the input noise, and we generalize our formalism to multiple outputs (or inputs). Applying our formalism to Escherichia coli chemotaxis, we find that its pathway is compatible with optimal information transmission despite the ultrasensitive rotary motors.

Interlinked GTPase cascades provide a motif for both robust switches and oscillators

GTPases regulate a wide range of cellular processes, such as intracellular vesicular transport, signal transduction and protein translation. These hydrolase enzymes operate as biochemical switches by toggling between an active guanosine triphosphate (GTP)-bound state and an inactive guanosine diphosphate (GDP)-bound state. We compare two network motifs, a single-species switch and an interlinked cascade that consists of two species coupled through positive and negative feedback loops. We find that interlinked cascades are closer to the ideal all-or-none switch and are more robust against fluctuating signals. While the single-species switch can only achieve bistability, interlinked cascades can be converted into oscillators by tuning the cofactor concentrations, which catalyse the activity of the cascade. These regimes can only be achieved with sufficient chemical driving provided by GTP hydrolysis. In this study, we present a thermodynamically consistent model that can achieve bistability and oscillations with the same feedback motif.

Kinetic Modelling of Competition and Depletion of Shared miRNAs by Competing Endogenous RNAs

Non-coding RNAs play a key role in the post-transcriptional regulation of mRNA translation and turnover in eukaryotes. miRNAs, in particular, interact with their target RNAs through protein-mediated, sequence-specific binding, giving rise to extended and highly heterogeneous miRNA-RNA interaction networks. Within such networks, competition to bind miRNAs can generate an effective positive coupling between their targets. Competing endogenous RNAs (ceRNAs) can in turn regulate each other through miRNA-mediated crosstalk. Albeit potentially weak, ceRNA interactions can occur both dynamically, affecting, e.g., the regulatory clock, and at stationarity, in which case ceRNA networks as a whole can be implicated in the composition of the cell's proteome. Many features of ceRNA interactions, including the conditions under which they become significant, can be unraveled by mathematical and in silico models. We review the understanding of the ceRNA effect obtained within such frameworks, focusing on the methods employed to quantify it, its role in the processing of gene expression noise, and how network topology can determine its reach.

Identifying feasible operating regimes for early T-cell recognition: The speed, energy, accuracy trade-off in kinetic proofreading and adaptive sorting

In the immune system, T cells can quickly discriminate between foreign and self ligands with high accuracy. There is evidence that T-cells achieve this remarkable performance utilizing a network architecture based on a generalization of kinetic proofreading (KPR). KPR-based mechanisms actively consume energy to increase the specificity beyond what is possible in equilibrium. An important theoretical question that arises is to understand the trade-offs and fundamental limits on accuracy, speed, and dissipation (energy consumption) in KPR and its generalization. Here, we revisit this question through numerical simulations where we simultaneously measure the speed, accuracy, and energy consumption of the KPR and adaptive sorting networks for different parameter choices. Our simulations highlight the existence of a "feasible operating regime" in the speed-energy-accuracy plane where T-cells can quickly differentiate between foreign and self ligands at reasonable energy expenditure. We give general arguments for why we expect this feasible operating regime to be a generic property of all KPR-based biochemical networks and discuss implications for our understanding of the T cell receptor circuit.

Understanding Biological Regulation Through Synthetic Biology

Engineering synthetic gene regulatory circuits proceeds through iterative cycles of design, building, and testing. Initial circuit designs must rely on often-incomplete models of regulation established by fields of reductive inquiry—biochemistry and molecular and systems biology. As differences in designed and experimentally observed circuit behavior are inevitably encountered, investigated, and resolved, each turn of the engineering cycle can force a resynthesis in understanding of natural network function. Here, we outline research that uses the process of gene circuit engineering to advance biological discovery. Synthetic gene circuit engineering research has not only refined our understanding of cellular regulation but furnished biologists with a toolkit that can be directed at natural systems to exact precision manipulation of network structure. As we discuss, using circuit engineering to predictively reorganize, rewire, and reconstruct cellular regulation serves as the ultimate means of testing and understanding how cellular phenotype emerges from systems-level network function.

Describing function-based approximations of biomolecular systems

Mathematical methods provide useful framework for the analysis and design of complex systems. In newer contexts such as biology, however, there is a need to both adapt existing methods as well as to develop new ones. Using a combination of analytical and computational approaches, the authors adapt and develop the method of describing functions to represent the input–output responses of biomolecular signalling systems. They approximate representative systems exhibiting various saturating and hysteretic dynamics in a way that is better than the standard linearisation. Furthermore, they develop analytical upper bounds for the computational error estimates. Finally, they use these error estimates to augment the limit cycle analysis with a simple and quick way to bound the predicted oscillation amplitude. These results provide system approximations that can add more insight into the local behaviour of these systems than standard linearisation, compute responses to other periodic inputs and to analyse limit cycles.

Shunt peaking in neural membranes

Capacitance limits the bandwidth of engineered and biological electrical circuits because it determines the gain-bandwidth product (GBWP). With a fixed GBWP, bandwidth can only be improved by decreasing gain. In engineered circuits, an inductance reduces this limitation through shunt peaking but no equivalent mechanism has been reported for biological circuits. We show that in blowfly photoreceptors a voltage-dependent K⁺ conductance, the fast delayed rectifier (FDR), produces shunt peaking thereby increasing bandwidth without reducing gain. Furthermore, the FDR's time constant is close to the value that maximizes the photoreceptor GBWP while reducing distortion associated with the creation of a wide-band filter. Using a model of the honeybee drone photoreceptor, we also show that a voltage-dependent Na⁺ conductance can produce shunt peaking. We argue that shunt peaking may be widespread in graded neurons and dendrites.

Non-equilibrium assembly of microtubules: from molecules to autonomous chemical robots

Biological systems have evolved to harness non-equilibrium processes from the molecular to the macro scale. It is currently a grand challenge of chemistry, materials science, and engineering to understand and mimic biological systems that have the ability to autonomously sense stimuli, process these inputs, and respond by performing mechanical work. New chemical systems are responding to the challenge and form the basis for future responsive, adaptive, and active materials. In this article, we describe a particular biochemical-biomechanical network based on the microtubule cytoskeletal filament - itself a non-equilibrium chemical system. We trace the non-equilibrium aspects of the system from molecules to networks and describe how the cell uses this system to perform active work in essential processes. Finally, we discuss how microtubule-based engineered systems can serve as testbeds for autonomous chemical robots composed of biological and synthetic components.

Criticality and Adaptivity in Enzymatic Networks

The contrast between the stochasticity of biochemical networks and the regularity of cellular behavior suggests that biological networks generate robust behavior from noisy constituents. Identifying the mechanisms that confer this ability on biological networks is essential to understanding cells. Here we show that queueing for a limited shared resource in broad classes of enzymatic networks in certain conditions leads to a critical state characterized by strong and long-ranged correlations between molecular species. An enzymatic network reaches this critical state when the input flux of its substrate is balanced by the maximum processing capacity of the network. We then consider enzymatic networks with adaptation, when the limiting resource (enzyme or cofactor) is produced in proportion to the demand for it. We show that the critical state becomes an attractor for these networks, which points toward the onset of self-organized criticality. We suggest that the adaptive queueing motif that leads to significant correlations between multiple species may be widespread in biological systems.

Optimal information transfer in enzymatic networks: A field theoretic formulation

Signaling in enzymatic networks is typically triggered by environmental fluctuations, resulting in a series of stochastic chemical reactions, leading to corruption of the signal by noise. For example, information flow is initiated by binding of extracellular ligands to receptors, which is transmitted through a cascade involving kinase-phosphatase stochastic chemical reactions. For a class of such networks, we develop a general field-theoretic approach to calculate the error in signal transmission as a function of an appropriate control variable. Application of the theory to a simple push-pull network, a module in the kinase-phosphatase cascade, recovers the exact results for error in signal transmission previously obtained using umbral calculus [Hinczewski and Thirumalai, Phys. Rev. X 4, 041017 (2014)2160-330810.1103/PhysRevX.4.041017]. We illustrate the generality of the theory by studying the minimal errors in noise reduction in a reaction cascade with two connected push-pull modules. Such a cascade behaves as an effective three-species network with a pseudointermediate. In this case, optimal information transfer, resulting in the smallest square of the error between the input and output, occurs with a time delay, which is given by the inverse of the decay rate of the pseudointermediate. Surprisingly, in these examples the minimum error computed using simulations that take nonlinearities and discrete nature of molecules into account coincides with the predictions of a linear theory. In contrast, there are substantial deviations between simulations and predictions of the linear theory in error in signal propagation in an enzymatic push-pull network for a certain range of parameters. Inclusion of second-order perturbative corrections shows that differences between simulations and theoretical predictions are minimized. Our study establishes that a field theoretic formulation of stochastic biological signaling offers a systematic way to understand error propagation in networks of arbitrary complexity.

Measuring time during early embryonic development

In most metazoans, embryonic development is orchestrated by a precise series of cellular behaviors. Understanding how such events are regulated to achieve a stereotypical temporal progression is a fundamental problem in developmental biology. In this review, we argue that studying the regulation of the cell cycle in early embryonic development will reveal novel principles of how embryos accurately measure time. We will discuss the strategies that have emerged from studying early development of Drosophila embryos. By comparing the development of flies to that of other metazoans, we will highlight both conserved and alternative mechanisms to generate precision during embryonic development.

Gene switching rate determines response to extrinsic perturbations in the self-activation transcriptional network motif

Gene switching dynamics is a major source of randomness in genetic networks, also in the case of large concentrations of the transcription factors. In this work, we consider a common network motif - the positive feedback of a transcription factor on its own synthesis - and assess its response to extrinsic noises perturbing gene deactivation in a variety of settings where the network might operate. These settings are representative of distinct cellular types, abundance of transcription factors and ratio between gene switching and protein synthesis rates. By investigating noise-induced transitions among the different network operative states, our results suggest that gene switching rates are key parameters to shape network response to external perturbations, and that such response depends on the particular biological setting, i.e. the characteristic time scales and protein abundance. These results might have implications on our understanding of irreversible transitions for noise-related phenomena such as cellular differentiation. In addition these evidences suggest to adopt the appropriate mathematical model of the network in order to analyze the system consistently to the reference biological setting.

Bridging Mechanistic and Phenomenological Models of Complex Biological Systems

The inherent complexity of biological systems gives rise to complicated mechanistic models with a large number of parameters. On the other hand, the collective behavior of these systems can often be characterized by a relatively small number of phenomenological parameters. We use the Manifold Boundary Approximation Method (MBAM) as a tool for deriving simple phenomenological models from complicated mechanistic models. The resulting models are not black boxes, but remain expressed in terms of the microscopic parameters. In this way, we explicitly connect the macroscopic and microscopic descriptions, characterize the equivalence class of distinct systems exhibiting the same range of collective behavior, and identify the combinations of components that function as tunable control knobs for the behavior. We demonstrate the procedure for adaptation behavior exhibited by the EGFR pathway. From a 48 parameter mechanistic model, the system can be effectively described by a single adaptation parameter τ characterizing the ratio of time scales for the initial response and recovery time of the system which can in turn be expressed as a combination of microscopic reaction rates, Michaelis-Menten constants, and biochemical concentrations. The situation is not unlike modeling in physics in which microscopically complex processes can often be renormalized into simple phenomenological models with only a few effective parameters. The proposed method additionally provides a mechanistic explanation for non-universal features of the behavior.

Limits to the precision of gradient sensing with spatial communication and temporal integration

Gradient sensing requires at least two measurements at different points in space. These measurements must then be communicated to a common location to be compared, which is unavoidably noisy. Although much is known about the limits of measurement precision by cells, the limits placed by the communication are not understood. Motivated by recent experiments, we derive the fundamental limits to the precision of gradient sensing in a multicellular system, accounting for communication and temporal integration. The gradient is estimated by comparing a "local" and a "global" molecular reporter of the external concentration, where the global reporter is exchanged between neighboring cells. Using the fluctuation-dissipation framework, we find, in contrast to the case when communication is ignored, that precision saturates with the number of cells independently of the measurement time duration, because communication establishes a maximum length scale over which sensory information can be reliably conveyed. Surprisingly, we also find that precision is improved if the local reporter is exchanged between cells as well, albeit more slowly than the global reporter. The reason is that whereas exchange of the local reporter weakens the comparison, it decreases the measurement noise. We term such a model "regional excitation-global inhibition." Our results demonstrate that fundamental sensing limits are necessarily sharpened when the need to communicate information is taken into account.

Cell-cell communication enhances the capacity of cell ensembles to sense shallow gradients during morphogenesis

Collective cell responses to exogenous cues depend on cell-cell interactions. In principle, these can result in enhanced sensitivity to weak and noisy stimuli. However, this has not yet been shown experimentally, and little is known about how multicellular signal processing modulates single-cell sensitivity to extracellular signaling inputs, including those guiding complex changes in the tissue form and function. Here we explored whether cell-cell communication can enhance the ability of cell ensembles to sense and respond to weak gradients of chemotactic cues. Using a combination of experiments with mammary epithelial cells and mathematical modeling, we find that multicellular sensing enables detection of and response to shallow epidermal growth factor (EGF) gradients that are undetectable by single cells. However, the advantage of this type of gradient sensing is limited by the noisiness of the signaling relay, necessary to integrate spatially distributed ligand concentration information. We calculate the fundamental sensory limits imposed by this communication noise and combine them with the experimental data to estimate the effective size of multicellular sensory groups involved in gradient sensing. Functional experiments strongly implicated intercellular communication through gap junctions and calcium release from intracellular stores as mediators of collective gradient sensing. The resulting integrative analysis provides a framework for understanding the advantages and limitations of sensory information processing by relays of chemically coupled cells.

Simple biochemical pathways far from steady state can provide switchlike and integrated responses

Covalent modification cycles (systems in which the activity of a substrate is regulated by the action of two opposing enzymes) and ligand/receptor interactions are ubiquitous in signaling systems and their steady-state properties are well understood. However, the behavior of such systems far from steady state remains unclear. Here, we analyze the properties of covalent modification cycles and ligand/receptor interactions driven by the accumulation of the activating enzyme and the ligand, respectively. We show that for a large range of parameters both systems produce sharp switchlike response and yet allow for temporal integration of the signal, two desirable signaling properties. Ultrasensitivity is observed also in a region of parameters where the steady-state response is hyperbolic. The temporal integration properties are tunable by regulating the levels of the deactivating enzyme and receptor, as well as by adjusting the rate of accumulation of the activating enzyme and ligand. We propose that this tunability is used to generate precise responses in signaling systems.

RNA-based regulation: dynamics and response to perturbations of competing RNAs

The observation that, through a titration mechanism, microRNAs (miRNAs) can act as mediators of effective interactions among their common targets (competing endogenous RNAs or ceRNAs) has brought forward the idea (i.e., the ceRNA hypothesis) that RNAs can regulate each other in extended cross-talk networks. Such an ability might play a major role in posttranscriptional regulation to shape a cell's protein repertoire. Recent work focusing on the emergent properties of the cross-talk networks has emphasized the high flexibility and selectivity that may be achieved at stationarity. On the other hand, dynamical aspects, possibly crucial on the relevant timescales, are far less clear. We have carried out a dynamical study of the ceRNA hypothesis on a model of posttranscriptional regulation. Sensitivity analysis shows that ceRNA cross-talk is dynamically extended, i.e., it may take place on timescales shorter than those required to achieve stationarity even in cases where no cross-talk occurs in the steady state, and is possibly amplified. In addition, in the case of large, transfection-like perturbations, the system may develop a strongly nonlinear, threshold response. Finally, we show that the ceRNA effect provides a very efficient way for a cell to achieve fast positive shifts in the level of a ceRNA when necessary. These results indicate that competition for miRNAs may indeed provide an elementary mechanism to achieve system-level regulatory effects on the transcriptome over physiologically relevant timescales.

Predicting chemical environments of bacteria from receptor signaling

Sensory systems have evolved to respond to input stimuli of certain statistical properties, and to reliably transmit this information through biochemical pathways. Hence, for an experimentally well-characterized sensory system, one ought to be able to extract valuable information about the statistics of the stimuli. Based on dose-response curves from in vivo fluorescence resonance energy transfer (FRET) experiments of the bacterial chemotaxis sensory system, we predict the chemical gradients chemotactic Escherichia coli cells typically encounter in their natural environment. To predict average gradients cells experience, we revaluate the phenomenological Weber's law and its generalizations to the Weber-Fechner law and fold-change detection. To obtain full distributions of gradients we use information theory and simulations, considering limitations of information transmission from both cell-external and internal noise. We identify broad distributions of exponential gradients, which lead to log-normal stimuli and maximal drift velocity. Our results thus provide a first step towards deciphering the chemical nature of complex, experimentally inaccessible cellular microenvironments, such as the human intestine.

Outside the laboratory, bacteria live in complex microenvironments characterized by competition for space and available nutrients. Although often inaccessible by experiments, understanding the spatio-temporal dynamics of bacterial microenvironments is biomedically important. For instance, the chemical environment that symbiotic Escherichia coli encounter in the human gut relates to health of the gastrointestinal tract, gut metabolism, immune response, and tissue homeostasis. Other complex microenvironments include soil and biofilms. Assuming that bacterial sensory systems have evolved to optimally sense typical gradients, we treat signaling data, the signaling pathway with its architecture and reaction rates, and computer simulations of swimming bacteria in different gradients as “prior knowledge” to “reverse engineer” E. coli's habitat. Our identified gradients are exponentially shaped with wide-ranging rate values. These microenvironments most likely stem from local fluctuating nutrient sources and degradation by competing species, in which bacteria have evolved to swim with optimal performance.

Thermodynamics of statistical inference by cells

The deep connection between thermodynamics, computation, and information is now well established both theoretically and experimentally. Here, we extend these ideas to show that thermodynamics also places fundamental constraints on statistical estimation and learning. To do so, we investigate the constraints placed by (nonequilibrium) thermodynamics on the ability of biochemical signaling networks to estimate the concentration of an external signal. We show that accuracy is limited by energy consumption, suggesting that there are fundamental thermodynamic constraints on statistical inference.

Multiplexing oscillatory biochemical signals

In recent years it has been increasingly recognized that biochemical signals are not necessarily constant in time and that the temporal dynamics of a signal can be the information carrier. Moreover, it is now well established that the protein signaling network of living cells has a bow-tie structure and that components are often shared between different signaling pathways. Here we show by mathematical modeling that living cells can multiplex a constant and an oscillatory signal: they can transmit these two signals simultaneously through a common signaling pathway, and yet respond to them specifically and reliably. We find that information transmission is reduced not only by noise arising from the intrinsic stochasticity of biochemical reactions, but also by crosstalk between the different channels. Yet, under biologically relevant conditions more than 2 bits of information can be transmitted per channel, even when the two signals are transmitted simultaneously. These observations suggest that oscillatory signals are ideal for multiplexing signals.

Characterization of tradeoffs in biomolecular signaling

Systems-level tradeoffs are fundamental in engineering, and recent work has highlighted an analogous role for them in biology. However, the extent of validity of these tradeoffs, especially for biomolecular systems, is generally unclear. Here, we address this issue for signaling tradeoffs that can constrain, for a fixed concentration of the signaling protein, a simultaneous enhancement of the gain and range of an amplifier or of the gain and threshold of a switch. We find that these gain-related tradeoffs persist in mathematical models of biomolecular reaction mechanisms that are at the core of large classes of signaling systems. Further, we find that these tradeoffs are also prevalent in the parametric functional forms commonly used to describe input-output curves in experimental analyses. Finally, we find that these tradeoffs can persist even in the presence of transcriptional feedback mechanisms that can change the concentration of the signaling protein. These results present a systematic characterization of these tradeoffs in biomolecular signaling systems.

Quantification of noise in bifunctionality-induced post-translational modification

We present a generic analytical scheme for the quantification of fluctuations due to bifunctionality-induced signal transduction within the members of a bacterial two-component system. The proposed model takes into account post-translational modifications in terms of elementary phosphotransfer kinetics. Sources of fluctuations due to autophosphorylation, kinase, and phosphatase activity of the sensor kinase have been considered in the model via Langevin equations, which are then solved within the framework of linear noise approximation. The resultant analytical expression of phosphorylated response regulators are then used to quantify the noise profile of biologically motivated single and branched pathways. Enhancement and reduction of noise in terms of extra phosphate outflux and influx, respectively, have been analyzed for the branched system. Furthermore, the role of fluctuations of the network output in the regulation of a promoter with random activation-deactivation dynamics has been analyzed.

Robustness of self-organizing chemoattractant field arising from precise pulse induction of its breakdown enzyme: a single-cell level analysis of PDE expression in Dictyostelium

The oscillation of chemoattractant cyclic AMP (cAMP) in Dictyostelium discoideum is a collective phenomenon that occurs when the basal level of extracellular cAMP exceeds a threshold and invokes cooperative mutual excitation of cAMP synthesis and secretion. For pulses to be relayed from cell to cell repetitively, secreted cAMP must be cleared and brought down to the subthreshold level. One of the main determinants of the oscillatory behavior is thus how much extracellular cAMP is degraded by extracellular phosphodiesterase (PDE). To date, the exact nature of PDE gene regulation remains elusive. Here, we performed live imaging analysis of mRNA transcripts for pdsA--the gene encoding extracellular PDE. Our analysis revealed that pdsA is upregulated during the rising phase of cAMP oscillations. Furthermore, by analyzing isolated cells, we show that expression of pdsA is strictly dependent on the presence of extracellular cAMP. pdsA is induced only at ∼1 nM extracellular cAMP, which is almost identical to the threshold concentration for the cAMP relay response. The observed precise regulation of PDE expression together with degradation of extracellular cAMP by PDE form a dual positive and negative feedback circuit, and model analysis shows that this sets the cAMP level near the threshold concentration for the cAMP relay response for a wide range of adenylyl cyclase activity. The overlap of the thresholds could allow oscillations of chemoattractant cAMP to self-organize at various starving conditions, making its development robust to fluctuations in its environment.

The fidelity of dynamic signaling by noisy biomolecular networks

Cells live in changing, dynamic environments. To understand cellular decision-making, we must therefore understand how fluctuating inputs are processed by noisy biomolecular networks. Here we present a general methodology for analyzing the fidelity with which different statistics of a fluctuating input are represented, or encoded, in the output of a signaling system over time. We identify two orthogonal sources of error that corrupt perfect representation of the signal: dynamical error, which occurs when the network responds on average to other features of the input trajectory as well as to the signal of interest, and mechanistic error, which occurs because biochemical reactions comprising the signaling mechanism are stochastic. Trade-offs between these two errors can determine the system's fidelity. By developing mathematical approaches to derive dynamics conditional on input trajectories we can show, for example, that increased biochemical noise (mechanistic error) can improve fidelity and that both negative and positive feedback degrade fidelity, for standard models of genetic autoregulation. For a group of cells, the fidelity of the collective output exceeds that of an individual cell and negative feedback then typically becomes beneficial. We can also predict the dynamic signal for which a given system has highest fidelity and, conversely, how to modify the network design to maximize fidelity for a given dynamic signal. Our approach is general, has applications to both systems and synthetic biology, and will help underpin studies of cellular behavior in natural, dynamic environments.

Cells do not live in constant conditions, but in environments that change over time. To adapt to their surroundings, cells must therefore sense fluctuating concentrations and ‘interpret’ the state of their environment to see whether, for example, a change in the pattern of gene expression is needed. This task is achieved via the noisy computations of biomolecular networks. But what levels of signaling fidelity can be achieved and how are dynamic signals encoded in the network's outputs? Here we present a general technique for analyzing such questions. We identify two sources of signaling error: dynamic error, which occurs when the network responds to features of the input other than the signal of interest; and mechanistic error, which arises because of the inevitable stochasticity of biochemical reactions. We show analytically that increased biochemical noise can sometimes improve fidelity and that, for genetic autoregulation, feedback can be deleterious. Our approach also allows us to predict the dynamic signal for which a given signaling network has highest fidelity and to design networks to maximize fidelity for a given signal. We thus propose a new way to analyze the flow of information in signaling networks, particularly for the dynamic environments expected in nature.

The energy costs of insulators in biochemical networks

Complex networks of biochemical reactions, such as intracellular protein signaling pathways and genetic networks, are often conceptualized in terms of modules--semiindependent collections of components that perform a well-defined function and which may be incorporated in multiple pathways. However, due to sequestration of molecular messengers during interactions and other effects, collectively referred to as retroactivity, real biochemical systems do not exhibit perfect modularity. Biochemical signaling pathways can be insulated from impedance and competition effects, which inhibit modularity, through enzymatic futile cycles that consume energy, typically in the form of ATP. We hypothesize that better insulation necessarily requires higher energy consumption. We test this hypothesis through a combined theoretical and computational analysis of a simplified physical model of covalent cycles, using two innovative measures of insulation, as well as a possible new way to characterize optimal insulation through the balancing of these two measures in a Pareto sense. Our results indicate that indeed better insulation requires more energy. While insulation may facilitate evolution by enabling a modular plug-and-play interconnection architecture, allowing for the creation of new behaviors by adding targets to existing pathways, our work suggests that this potential benefit must be balanced against the metabolic costs of insulation necessarily incurred in not affecting the behavior of existing processes.

The interplay of intrinsic and extrinsic bounded noises in biomolecular networks

After being considered as a nuisance to be filtered out, it became recently clear that biochemical noise plays a complex role, often fully functional, for a biomolecular network. The influence of intrinsic and extrinsic noises on biomolecular networks has intensively been investigated in last ten years, though contributions on the co-presence of both are sparse. Extrinsic noise is usually modeled as an unbounded white or colored gaussian stochastic process, even though realistic stochastic perturbations are clearly bounded. In this paper we consider Gillespie-like stochastic models of nonlinear networks, i.e. the intrinsic noise, where the model jump rates are affected by colored bounded extrinsic noises synthesized by a suitable biochemical state-dependent Langevin system. These systems are described by a master equation, and a simulation algorithm to analyze them is derived. This new modeling paradigm should enlarge the class of systems amenable at modeling. We investigated the influence of both amplitude and autocorrelation time of a extrinsic Sine-Wiener noise on: (i) the Michaelis-Menten approximation of noisy enzymatic reactions, which we show to be applicable also in co-presence of both intrinsic and extrinsic noise, (ii) a model of enzymatic futile cycle and (iii) a genetic toggle switch. In (ii) and (iii) we show that the presence of a bounded extrinsic noise induces qualitative modifications in the probability densities of the involved chemicals, where new modes emerge, thus suggesting the possible functional role of bounded noises.

Energetic costs of cellular computation

Cells often perform computations in order to respond to environmental cues. A simple example is the classic problem, first considered by Berg and Purcell, of determining the concentration of a chemical ligand in the surrounding media. On general theoretical grounds, it is expected that such computations require cells to consume energy. In particular, Landauer's principle states that energy must be consumed in order to erase the memory of past observations. Here, we explicitly calculate the energetic cost of steady-state computation of ligand concentration for a simple two-component cellular network that implements a noisy version of the Berg-Purcell strategy. We show that learning about external concentrations necessitates the breaking of detailed balance and consumption of energy, with greater learning requiring more energy. Our calculations suggest that the energetic costs of cellular computation may be an important constraint on networks designed to function in resource poor environments, such as the spore germination networks of bacteria.

The energy-speed-accuracy tradeoff in sensory adaptation

Adaptation is the essential process by which an organism becomes better suited to its environment. The benefits of adaptation are well documented, but the cost it incurs remains poorly understood. Here, by analysing a stochastic model of a minimum feedback network underlying many sensory adaptation systems, we show that adaptive processes are necessarily dissipative, and continuous energy consumption is required to stabilize the adapted state. Our study reveals a general relation among energy dissipation rate, adaptation speed and the maximum adaptation accuracy. This energy-speed-accuracy relation is tested in the Escherichia coli chemosensory system, which exhibits near-perfect chemoreceptor adaptation. We identify key requirements for the underlying biochemical network to achieve accurate adaptation with a given energy budget. Moreover, direct measurements confirm the prediction that adaptation slows down as cells gradually de-energize in a nutrient-poor medium without compromising adaptation accuracy. Our work provides a general framework to study cost-performance tradeoffs for cellular regulatory functions and information processing.

Noise propagation in two-step series MAPK cascade

Series MAPK enzymatic cascades, ubiquitously found in signaling networks, act as signal amplifiers and play a key role in processing information during signal transduction in cells. In activated cascades, cell-to-cell variability or noise is bound to occur and thereby strongly affects the cellular response. Commonly used linearization method (LM) applied to Langevin type stochastic model of the MAPK cascade fails to accurately predict intrinsic noise propagation in the cascade. We prove this by using extensive stochastic simulations for various ranges of biochemical parameters. This failure is due to the fact that the LM ignores the nonlinear effects on the noise. However, LM provides a good estimate of the extrinsic noise propagation. We show that the correct estimate of intrinsic noise propagation in signaling networks that contain at least one enzymatic step can be obtained only through stochastic simulations. Noise propagation in the cascade depends on the underlying biochemical parameters which are often unavailable. Based on a combination of global sensitivity analysis (GSA) and stochastic simulations, we developed a systematic methodology to characterize noise propagation in the cascade. GSA predicts that noise propagation in MAPK cascade is sensitive to the total number of upstream enzyme molecules and the total number of molecules of the two substrates involved in the cascade. We argue that the general systematic approach proposed and demonstrated on MAPK cascade must accompany noise propagation studies in biological networks.

On the calculation of signal transduction ability of signaling transduction pathways in intracellular communication: systematic approach

MOTIVATION

The major function of signal transduction pathways in cells is to sense signals from the environment and process the information through signaling molecules in order to regulate the activity of transcription factors. On the molecular level, the information transmitted by a small number of signal molecules is amplified in the internal signaling pathway through enzyme catalysis, molecular modification and via the activation or inhibition of interactions. However, the dynamic system behavior of a signaling pathway can be complex and, despite knowledge of the pathway components and interactions, it is still a challenge to interpret the pathways behavior. Therefore, a systematic method is proposed in this study to quantify the signal transduction ability.

RESULTS

Based on the non-linear signal transduction system, signal transduction ability can be investigated by solving a Hamilton-Jacobi inequality (HJI)-constrained optimization problem. To avoid difficulties associated with solving a complex HJI-constrained optimization problem for signal transduction ability, the Takagi-Sugeno fuzzy model is introduced to approximate the non-linear signal transduction system by interpolating several local linear systems so that the HJI-constrained optimization problem can be replaced by a linear matrix inequality (LMI)-constrained optimization problem. The LMI problem can then be efficiently solved for measuring signal transduction ability. Finally, the signal transduction ability of two important signal transduction pathways was measured by the proposed method and confirmed using experimental data, which is useful for biotechnological and therapeutic application and drug design.

Short-term integration of Cdc25 dynamics controls mitotic entry during Drosophila gastrulation

Cells commit to mitosis by abruptly activating the mitotic cyclin-Cdk complexes. During Drosophila gastrulation, mitosis is associated with the transcriptional activation of cdc25(string), a phosphatase that activates Cdk1. Here, we demonstrate that the switch-like entry into mitosis observed in the Drosophila embryo during the 14(th) mitotic cycle is timed by the dynamics of Cdc25(string) accumulation. The switch operates as a short-term integrator, a property that can improve the reliable control of timing of mitosis. The switch is independent of the positive feedback between Cdk1 and Cdc25(string) and of the double negative feedback between Cdk1 and Wee1. We propose that the properties of the mitotic switch are established by the out-of-equilibrium properties of the covalent modification cycle controlling Cdk1 activity. Such covalent modification cycles, triggered by transcriptional expression of the activating enzymes, might be a widespread strategy to obtain reliable and switch-like control of cell decisions.

Gain control in molecular information processing: lessons from neuroscience

Statistical properties of environments experienced by biological signaling systems in the real world change, which necessitates adaptive responses to achieve high fidelity information transmission. One form of such adaptive response is gain control. Here, we argue that a certain simple mechanism of gain control, understood well in the context of systems neuroscience, also works for molecular signaling. The mechanism allows us to transmit more than 1 bit (on or off) of information about the signal independent of the signal variance. It does not require additional molecular circuitry beyond that already present in many molecular systems, and in particular, it does not depend on existence of feedback loops. The mechanism provides a potential explanation for abundance of ultrasensitive response curves in biological regulatory networks.

Trade-offs and constraints in allosteric sensing

Sensing extracellular changes initiates signal transduction and is the first stage of cellular decision-making. Yet relatively little is known about why one form of sensing biochemistry has been selected over another. To gain insight into this question, we studied the sensing characteristics of one of the biochemically simplest of sensors: the allosteric transcription factor. Such proteins, common in microbes, directly transduce the detection of a sensed molecule to changes in gene regulation. Using the Monod-Wyman-Changeux model, we determined six sensing characteristics – the dynamic range, the Hill number, the intrinsic noise, the information transfer capacity, the static gain, and the mean response time – as a function of the biochemical parameters of individual sensors and of the number of sensors. We found that specifying one characteristic strongly constrains others. For example, a high dynamic range implies a high Hill number and a high capacity, and vice versa. Perhaps surprisingly, these constraints are so strong that most of the space of characteristics is inaccessible given biophysically plausible ranges of parameter values. Within our approximations, we can calculate the probability distribution of the numbers of input molecules that maximizes information transfer and show that a population of one hundred allosteric transcription factors can in principle distinguish between more than four bands of input concentrations. Our results imply that allosteric sensors are unlikely to have been selected for high performance in one sensing characteristic but for a compromise in the performance of many.

Sensing environmental changes is the first step in the process of cellular decision-making, but many different biochemical sensors exist and why one sensor is selected for a particular task over another is not known. Here we study the sensing properties of a simple and generic allosteric sensor to understand the effectiveness and limitations of its “design”. We begin by defining and calculating a set of six engineering-inspired characteristics of the sensor’s response and investigate how specifying a high performance in one characteristic constrains the sensor’s performance in others. We determine many such trade-offs and, perhaps surprisingly, that much of the space of characteristics is inaccessible given biophysically plausible ranges of parameters. Our results suggest that allosteric sensors are not under selection for high performance in one sensing characteristic but for a compromise in performance between many. Our approach provides both quantitative and qualitative insights about the function and robustness of allosteric sensors and as such is applicable to both the study of endogenous systems and the design of synthetic ones.

Noise characteristics of the Escherichia coli rotary motor

Background

The chemotaxis pathway in the bacterium Escherichia coli allows cells to detect changes in external ligand concentration (e.g. nutrients). The pathway regulates the flagellated rotary motors and hence the cells' swimming behaviour, steering them towards more favourable environments. While the molecular components are well characterised, the motor behaviour measured by tethered cell experiments has been difficult to interpret.

Results

We study the effects of sensing and signalling noise on the motor behaviour. Specifically, we consider fluctuations stemming from ligand concentration, receptor switching between their signalling states, adaptation, modification of proteins by phosphorylation, and motor switching between its two rotational states. We develop a model which includes all signalling steps in the pathway, and discuss a simplified version, which captures the essential features of the full model. We find that the noise characteristics of the motor contain signatures from all these processes, albeit with varying magnitudes.

Conclusions

Our analysis allows us to address how cell-to-cell variation affects motor behaviour and the question of optimal pathway design. A similar comprehensive analysis can be applied to other two-component signalling pathways.

Protein signaling networks from single cell fluctuations and information theory profiling

Protein signaling networks among cells play critical roles in a host of pathophysiological processes, from inflammation to tumorigenesis. We report on an approach that integrates microfluidic cell handling, in situ protein secretion profiling, and information theory to determine an extracellular protein-signaling network and the role of perturbations. We assayed 12 proteins secreted from human macrophages that were subjected to lipopolysaccharide challenge, which emulates the macrophage-based innate immune responses against Gram-negative bacteria. We characterize the fluctuations in protein secretion of single cells, and of small cell colonies (n = 2, 3,···), as a function of colony size. Measuring the fluctuations permits a validation of the conditions required for the application of a quantitative version of the Le Chatelier's principle, as derived using information theory. This principle provides a quantitative prediction of the role of perturbations and allows a characterization of a protein-protein interaction network.

Noise amplification in human tumor suppression following gamma irradiation

The influence of noise on oscillatory motion is a subject of permanent interest, both for fundamental and practical reasons. Cells respond properly to external stimuli by using noisy systems. We have clarified the effect of intrinsic noise on the dynamics in the human cancer cells following gamma irradiation. It is shown that the large amplification and increasing mutual information with delay are due to coherence resonance. Furthermore, frequency domain analysis is used to study the mechanisms.

Interdependence of behavioural variability and response to small stimuli in bacteria

The chemotaxis signalling network in Escherichia coli that controls the locomotion of bacteria is a classic model system for signal transduction. This pathway modulates the behaviour of flagellar motors to propel bacteria towards sources of chemical attractants. Although this system relaxes to a steady state in response to environmental changes, the signalling events within the chemotaxis network are noisy and cause large temporal variations of the motor behaviour even in the absence of stimulus. That the same signalling network governs both behavioural variability and cellular response raises the question of whether these two traits are independent. Here, we experimentally establish a fluctuation-response relationship in the chemotaxis system of living bacteria. Using this relationship, we demonstrate the possibility of inferring the cellular response from the behavioural variability measured before stimulus. In monitoring the pre- and post-stimulus switching behaviour of individual bacterial motors, we found that variability scales linearly with the response time for different functioning states of the cell. This study highlights that the fundamental relationship between fluctuation and response is not constrained to physical systems at thermodynamic equilibrium but is extensible to living cells. Such a relationship not only implies that behavioural variability and cellular response can be coupled traits, but it also provides a general framework within which we can examine how the selection of a network design shapes this interdependence.

Decoupling of receptor and downstream signals in the Akt pathway by its low-pass filter characteristics

In cellular signal transduction, the information in an external stimulus is encoded in temporal patterns in the activities of signaling molecules; for example, pulses of a stimulus may produce an increasing response or may produce pulsatile responses in the signaling molecules. Here, we show how the Akt pathway, which is involved in cell growth, specifically transmits temporal information contained in upstream signals to downstream effectors. We modeled the epidermal growth factor (EGF)-dependent Akt pathway in PC12 cells on the basis of experimental results. We obtained counterintuitive results indicating that the sizes of the peak amplitudes of receptor and downstream effector phosphorylation were decoupled; weak, sustained EGF receptor (EGFR) phosphorylation, rather than strong, transient phosphorylation, strongly induced phosphorylation of the ribosomal protein S6, a molecule downstream of Akt. Using frequency response analysis, we found that a three-component Akt pathway exhibited the property of a low-pass filter and that this property could explain decoupling of the peak amplitudes of receptor phosphorylation and that of downstream effectors. Furthermore, we found that lapatinib, an EGFR inhibitor used as an anticancer drug, converted strong, transient Akt phosphorylation into weak, sustained Akt phosphorylation, and, because of the low-pass filter characteristics of the Akt pathway, this led to stronger S6 phosphorylation than occurred in the absence of the inhibitor. Thus, an EGFR inhibitor can potentially act as a downstream activator of some effectors.

Stochastic signaling in biochemical cascades and genetic systems in genetically engineered living cells

Living cells, either prokaryote or eukaryote, can be integrated within whole-cell biochips (WCBCs) for various applications. We investigate WCBCs where information is extracted from the cells via a cascade of biochemical reactions that involve gene expression. The overall biological signal is weak due to small sample volume, low intrinsic cell response, and extrinsic signal loss mechanisms. The low signal-to-noise ratio problem is aggravated during initial detection stages and limits the minimum detectable signal or, alternatively, the minimum detection time. Taking into account the stochastic nature of biochemical process, we find that the signal is accompanied by relatively large noise disturbances. In this work, we use genetically engineered microbe sensors as a model to study the biochips output signal stochastic behavior. In our model, the microbes are designed to express detectable reporter proteins under external induction. We present analytical approximated expressions and numerical simulations evaluating the fluctuations of the synthesized reporter proteins population based on a set of equations modeling a cascade of biochemical and genetic reactions. We assume that the reporter proteins decay more slowly than messenger RNA molecules. We calculate the relation between the noise of the input signal (extrinsic noise) and biochemical reaction statistics (intrinsic noise). We discuss in further details two cases: (1) a cascade with large decay rates of all biochemical reactions compared to the protein decay rate. We show that in this case, the noise amplitude has a positive linear correlation with the number of stages in the cascade. (2) A cascade which includes a stable enzymatic-binding reaction with slow decay rate. We show that in this case, the noise strongly depends on the protein decay rate. Finally, a general observation is presented stating that the noise in whole-cell biochip sensors is determined mainly by the first reactions in the genetic system with weak dependence on the number of stages in the cascade.

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.