Stationary states of an active Brownian particle in a harmonic trap.

We study the stationary states of an overdamped active Brownian particle (ABP) in a harmonic trap in two dimensions via mathematical calculations and numerical simulations. In addition to translational diffusion, the ABP self-propels with a certain velocity, whose magnitude is constant, but its direction is subject to Brownian rotation. In the limit where translational diffusion is negligible, the stationary distribution of the particle's position shows a transition between two different shapes, one with maximum and the other with minimum density at the center, as the trap stiffness is increased. We show that this nonintuitive behavior is captured by the relevant Fokker-Planck equation, which, under minimal assumptions, predicts a continuous phase transition-like change between the two different shapes. As the translational diffusion coefficient is increased, both these distributions converge into the equilibrium, Boltzmann form. Our simulations support the analytical predictions and also show that the probability distribution of the orientation angle of the self-propulsion velocity undergoes a transition from unimodal to bimodal forms in this limit. We also extend our simulations to a three-dimensional trap and find similar behavior.

Comparison of thermal and athermal dynamics of the cell membrane slope fluctuations in the presence and absence of Latrunculin-B.

Conventionally, only the normal cell membrane fluctuations have been studied and used to ascertain membrane properties like the bending rigidity. A new concept, the membrane local slope fluctuations was introduced recently (Vaippullyet al2020Soft Matter167606), which can be modelled as a gradient of the normal fluctuations. It has been found that the power spectral density (PSD) of slope fluctuations behave as (frequency)-1while the normal fluctuations yields (frequency)-5/3even on the apical cell membrane in the high frequency region. In this manuscript, we explore a different situation where the cell is applied with the drug Latrunculin-B which inhibits actin polymerization and find the effect on membrane fluctuations. We find that even as the normal fluctuations show a power law (frequency)-5/3as is the case for a free membrane, the slope fluctuations PSD remains (frequency)-1, with exactly the same coefficient as the case when the drug was not applied. Moreover, while sometimes, when the normal fluctuations at high frequency yield a power law of (frequency)-4/3, the pitch PSD still yields (frequency)-1. Thus, this presents a convenient opportunity to study membrane parameters like bending rigidity as a function of time after application of the drug, while the membrane softens. We also investigate the active athermal fluctuations of the membrane appearing in the PSD at low frequencies and find active timescales of slower than 1 s.

Force-velocity relation and load-sharing in the linear polymerization ratchet revisited: the effects of barrier diffusion.

We study the velocity-force (V-F) relation for a Brownian ratchet consisting of a linear rigid polymer growing against a diffusing barrier, acted upon by a opposing constant force (F). Using a careful mathematical analysis, we derive the V-F relations in the extreme limits of fast and slow barrier diffusion. In the first case, V depends exponentially on the load F, in agreement with the well-known formula proposed by Peskin, Odell and Oster (1993), while the relationship becomes linear in the second case. For a bundle of two filaments growing against a common barrier, equal sharing of load in the corresponding V-F relation is predicted by a mean-field argument in both limits. However, the scaling behaviour of velocity with the number of filaments is different for the two cases. In the limit of large D, the validity of the mean-field approach is tested, and partially supported by a detailed and rigorous analysis. Our principal predictions are also verified in numerical simulations.

Microtubule catastrophe under force: mathematical and computational results from a Brownian ratchet model.

In the intracellular environment, the intrinsic dynamics of microtubule filaments is often hindered by the presence of barriers of various kind, such as kinetochore complexes and cell cortex, which impact their polymerisation force and dynamical properties such as catastrophe frequency. We present a theoretical study of the effect of a forced barrier, also subjected to thermal noise, on the statistics of catastrophe events in a single microtubule as well as a 'bundle' of two parallel microtubules. For microtubule dynamics, which includes growth, detachment, hydrolysis and the consequent dynamic instability, we employ a one-dimensional discrete stochastic model. The dynamics of the barrier is captured by over-damped Langevin equation, while its interaction with a growing filament is assumed to be hard-core repulsion. A unified treatment of the continuum dynamics of the barrier and the discrete dynamics of the filament is realized using a hybrid Fokker-Planck equation. An explicit mathematical formula for the force-dependent catastrophe frequency of a single microtubule is obtained by solving the above equation, under some assumptions. The prediction agrees well with results of numerical simulations in the appropriate parameter regime. More general situations are studied via numerical simulations. To investigate the extent of 'load-sharing' in a microtubule bundle, and its impact on the frequency of catastrophes, the dynamics of a two-filament bundle is also studied. Here, two parallel, non-interacting microtubules interact with a common, forced barrier. The equations for the two-filament model, when solved using a mean-field assumption, predicts equal sharing of load between the filaments. However, numerical results indicate the existence of a wide spectrum of load-sharing behaviour, which is characterized using a dimensionless parameter.

Detection of sub-degree angular fluctuations of the local cell membrane slope using optical tweezers.

Normal thermal fluctuations of the cell membrane have been studied extensively using high resolution microscopy and focused light, particularly at the peripheral regions of a cell. We use a single probe particle attached non-specifically to the cell-membrane to determine that the power spectral density is proportional to (frequency)-5/3 in the range of 5 Hz to 1 kHz. We also use a new technique to simultaneously ascertain the slope fluctuations of the membrane by relying upon the determination of pitch motion of the birefringent probe particle trapped in linearly polarized optical tweezers. In the process, we also develop the technique to identify pitch rotation to a high resolution using optical tweezers. We find that the power spectrum of slope fluctuations is proportional to (frequency)-1, which we also explain theoretically. We find that we can extract parameters like bending rigidity directly from the coefficient of the power spectrum particularly at high frequencies, instead of being convoluted with other parameters, thereby improving the accuracy of estimation. We anticipate this technique for determination of the pitch angle in spherical particles to high resolution as a starting point for many interesting studies using the optical tweezers.

Spatio-temporal correlations between catastrophe events in a microtubule bundle: a computational study.

We explore correlations between dynamics of different microtubules in a bundle, via numerical simulations, using a one-dimensional stochastic model of a microtubule. The guanosine triphosphate (GTP)-bound tubulins undergo diffusion-limited binding to the tip. Random hydrolysis events take place along the microtubule and converts the bound GTP in tubulin to guanosine diphosphate (GDP). The microtubule starts depolymerising when the monomer at the tip becomes GDP-bound; in this case, detachment of GDP-tubulin ensues and continues until either GTP-bound tubulin is exposed or complete depolymerisation is achieved. In the latter case, the microtubule is defined to have undergone a "catastrophe". Our results show that, in general, the dynamics of growth and catastrophe in different microtubules are coupled to each other; the closer the microtubules are, the stronger the coupling. In particular, all microtubules grow slower, on average, when brought closer together. The reduction in growth velocity also leads to more frequent catastrophes. More dramatically, catastrophe events in the different microtubules forming a bundle are found to be correlated; a catastrophe event in one microtubule is more likely to be followed by a similar event in the same microtubule. This propensity of bunching disappears when the microtubules move farther apart.

Temporal cooperativity of motor proteins under constant force: insights from Kramers' escape problem.

The microtubule-bound motors kinesin and dynein differ in many respects, a striking difference being that while kinesin is known to function mostly alone, dynein operates in large groups, much like myosinV in actin. Optical tweezer experiments in vitro have shown that the mean detachment time of a bead attached to [Formula: see text] kinesins under stall conditions is a slowly decreasing function of [Formula: see text], while for dyneins, the time increases almost linearly with [Formula: see text]. This makes dynein a team worker, capable of producing and sustaining a large collective force without detaching. We characterize this phenomenon as 'temporal cooperativity' under load. In general, it is unclear which biophysical properties of a single motor determine whether it behaves cooperatively or not in a group. Our theoretical analysis shows that this is determined by two dimensionless parameters: (i) the ratio of single molecule, load-independent detachment and attachment rates and (ii) the ratio of the applied force per motor to the detachment force of a single motor. We show that the attachment-detachment dynamics of a motor assembly may be mapped to the motion of a hypothetical, overdamped Brownian particle in an effective potential, the form of which depends on the load-dependence of binding and unbinding rate of the motor. In this picture, the total number [Formula: see text] of motors is proportional to the inverse temperature and cooperative behaviour arises from the trapping of the particle in the minima of the potential, when present. In the latter case, application of results from Kramers' theory predicts that the mean time of escape of the particle, equivalent to the mean detachment time of the bead under stall, increases exponentially with the number of motors, indicating cooperative behaviour. If the potential does not have minima, the detachment time depends only weakly on [Formula: see text], which suggests non-cooperative behaviour. In the large [Formula: see text] limit, the emergence of cooperative behaviour is shown to be similar to a continuous phase transition.

Pyopericardium progressing to tamponade in a patient with immune thrombocytopenia.

Pericardial effusion can develop during any stage of pericarditis, and small effusions that appear rapidly can cause cardiac tamponade. Pyopericardium is a rare aetiology for tamponade. This is a case of an elderly diabetic lady, on steroid therapy for immune thrombocytopenia, who presented with fever and acute dyspnoea. She developed cardiac tamponade due to pyopericardium with Staphylococcus as the causative organism. Staphylococcus pyopericardium, in the absence of a primary focus of infection, progressing to tamponade is an uncommon scenario.

Polymerisation force of a rigid filament bundle: diffusive interaction leads to sublinear force-number scaling.

Polymerising filaments generate force against an obstacle, as in, e.g., microtubule-kinetochore interactions in the eukaryotic cell. Earlier studies of this problem have not included explicit three-dimensional monomer diffusion, and consequently, missed out on two important aspects: (i) the barrier, even when it is far from the polymers, affects free diffusion of monomers and reduces their adsorption at the tips, while (ii) parallel filaments could interact through the monomer density field ("diffusive coupling"), leading to negative interference between them. In our study, both these effects are included and their consequences investigated in detail. A mathematical treatment based on a set of continuum Fokker-Planck equations for combined filament-wall dynamics suggests that the barrier-induced monomer depletion reduces the growth velocity and also the stall force, while the total force produced by many filaments remains additive. However, Brownian dynamics simulations show that the linear force-number scaling holds only when the filaments are far apart; when they are arranged close together, forming a bundle, sublinear scaling of force with number appears, which could be attributed to diffusive interaction between the growing polymer tips.

Ultrasensitivity and fluctuations in the Barkai-Leibler model of chemotaxis receptors in Escherichia coli.

A stochastic version of the Barkai-Leibler model of chemotaxis receptors in Escherichia coli is studied here with the goal of elucidating the effects of intrinsic network noise in their conformational dynamics. The model was originally proposed to explain the robust and near-perfect adaptation of E. coli observed across a wide range of spatially uniform attractant/repellent (ligand) concentrations. In the model, a receptor is either active or inactive and can stochastically switch between the two states. The enzyme CheR methylates inactive receptors while CheB demethylates active receptors and the probability for a receptor to be active depends on its level of methylation and ligand occupation. In a simple version of the model with two methylation sites per receptor (M = 2), we show rigorously, under a quasi-steady state approximation, that the mean active fraction of receptors is an ultrasensitive function of [CheR]/[CheB] in the limit of saturating receptor concentration. Hence the model shows zero-order ultrasensitivity (ZOU), similar to the classical two-state model of covalent modification studied by Goldbeter and Koshland (GK). We also find that in the limits of extremely small and extremely large ligand concentrations, the system reduces to two different two-state GK modules. A quantitative measure of the spontaneous fluctuations in activity is provided by the variance [Formula: see text] in the active fraction, which is estimated mathematically under linear noise approximation (LNA). It is found that [Formula: see text] peaks near the ZOU transition. The variance is a non-monotonic, but weak function of ligand concentration and a decreasing function of receptor concentration. Gillespie simulations are also performed in models with M = 2, 3 and 4. For M = 2, simulations show excellent agreement with analytical results obtained under LNA. Numerical results for M = 3 and M = 4 are qualitatively similar to our mathematical results in M = 2; while all the models show ZOU in mean activity, the variance is found to be smaller for larger M. The magnitude of receptor noise deduced from available experimental data is consistent with our predictions. A simple analysis of the downstream signaling pathway shows that this noise is large enough to affect the motility of the organism, and may have a beneficial effect on it. The response of mean receptor activity to small time-dependent changes in the external ligand concentration is computed within linear response theory, and found to have a bilobe form, in agreement with earlier experimental observations.

Transport of organelles by elastically coupled motor proteins.

Motor-driven intracellular transport is a complex phenomenon where multiple motor proteins simultaneously attached on to a cargo engage in pulling activity, often leading to tug-of-war, displaying bidirectional motion. However, most mathematical and computational models ignore the details of the motor-cargo interaction. A few studies have focused on more realistic models of cargo transport by including elastic motor-cargo coupling, but either restrict the number of motors and/or use purely phenomenological forms for force-dependent hopping rates. Here, we study a generic model in which N motors are elastically coupled to a cargo, which itself is subjected to thermal noise in the cytoplasm and to an additional external applied force. The motor-hopping rates are chosen to satisfy detailed balance with respect to the energy of elastic stretching. With these assumptions, an (N + 1) -variable master equation is constructed for dynamics of the motor-cargo complex. By expanding the hopping rates to linear order in fluctuations in motor positions, we obtain a linear Fokker-Planck equation. The deterministic equations governing the average quantities are separated out and explicit analytical expressions are obtained for the mean velocity and diffusion coefficient of the cargo. We also study the statistical features of the force experienced by an individual motor and quantitatively characterize the load-sharing among the cargo-bound motors. The mean cargo velocity and the effective diffusion coefficient are found to be decreasing functions of the stiffness. While the increase in the number of motors N does not increase the velocity substantially, it decreases the effective diffusion coefficient which falls as 1/N asymptotically. We further show that the cargo-bound motors share the force exerted on the cargo equally only in the limit of vanishing elastic stiffness; as stiffness is increased, deviations from equal load sharing are observed. Numerical simulations agree with our analytical results where expected. Interestingly, we find in simulations that the stall force of a cargo elastically coupled to motors is independent of the stiffness of the linkers.

Effects of aging in catastrophe on the steady state and dynamics of a microtubule population.

Several independent observations have suggested that the catastrophe transition in microtubules is not a first-order process, as is usually assumed. Recent in vitro observations by Gardner et al. [M. K. Gardner et al., Cell 147, 1092 (2011)] showed that microtubule catastrophe takes place via multiple steps and the frequency increases with the age of the filament. Here we investigate, via numerical simulations and mathematical calculations, some of the consequences of the age dependence of catastrophe on the dynamics of microtubules as a function of the aging rate, for two different models of aging: exponential growth, but saturating asymptotically, and purely linear growth. The boundary demarcating the steady-state and non-steady-state regimes in the dynamics is derived analytically in both cases. Numerical simulations, supported by analytical calculations in the linear model, show that aging leads to nonexponential length distributions in steady state. More importantly, oscillations ensue in microtubule length and velocity. The regularity of oscillations, as characterized by the negative dip in the autocorrelation function, is reduced by increasing the frequency of rescue events. Our study shows that the age dependence of catastrophe could function as an intrinsic mechanism to generate oscillatory dynamics in a microtubule population, distinct from hitherto identified ones.

Zero-order ultrasensitivity: a study of criticality and fluctuations under the total quasi-steady state approximation in the linear noise regime.

Zero-order ultrasensitivity (ZOU) is a long known and interesting phenomenon in enzyme networks. Here, a substrate is reversibly modified by two antagonistic enzymes (a 'push-pull' system) and the fraction in modified state undergoes a sharp switching from near-zero to near-unity at a critical value of the ratio of the enzyme concentrations, under saturation conditions. ZOU and its extensions have been studied for several decades now, ever since the seminal paper of Goldbeter and Koshland (1981); however, a complete probabilistic treatment, important for the study of fluctuations in finite populations, is still lacking. In this paper, we study ZOU using a modular approach, akin to the total quasi-steady state approximation (tQSSA). This approach leads to a set of Fokker-Planck (drift-diffusion) equations for the probability distributions of the intermediate enzyme-bound complexes, as well as the modified/unmodified fractions of substrate molecules. We obtain explicit expressions for various average fractions and their fluctuations in the linear noise approximation (LNA). The emergence of a 'critical point' for the switching transition is rigorously established. New analytical results are derived for the average and variance of the fractional substrate concentration in various chemical states in the near-critical regime. For the total fraction in the modified state, the variance is shown to be a maximum near the critical point and decays algebraically away from it, similar to a second-order phase transition. The new analytical results are compared with existing ones as well as detailed numerical simulations using a Gillespie algorithm.

Memory, bias, and correlations in bidirectional transport of molecular-motor-driven cargoes.

Molecular motors are specialized proteins that perform active, directed transport of cellular cargoes on cytoskeletal filaments. In many cases, cargo motion powered by motor proteins is found to be bidirectional, and may be viewed as a biased random walk with fast unidirectional runs interspersed with slow tug-of-war states. The statistical properties of this walk are not known in detail, and here, we study memory and bias, as well as directional correlations between successive runs in bidirectional transport. We show, based on a study of the direction-reversal probabilities of the cargo using a purely stochastic (tug-of-war) model, that bidirectional motion of cellular cargoes is, in general, a correlated random walk. In particular, while the motion of a cargo driven by two oppositely pulling motors is a Markovian random walk, memory of direction appears when multiple motors haul the cargo in one or both directions. In the latter case, the Markovian nature of the underlying single-motor processes is hidden by internal transitions between degenerate run and pause states of the cargo. Interestingly, memory is found to be a nonmonotonic function of the number of motors. Stochastic numerical simulations of the tug-of-war model support our mathematical results and extend them to biologically relevant situations.

Microtubule catastrophe from protofilament dynamics.

The disappearance of the guanosine triphosphate- (GTP) tubulin cap is widely believed to be the forerunner event for the growth-shrinkage transition ("catastrophe") in microtubule filaments in eukaryotic cells. We study a discrete version of a stochastic model of the GTP cap dynamics, originally proposed by Flyvbjerg, Holy, and Leibler [Phys. Rev. Lett. 73, 2372 (1994)]. Our model includes both spontaneous and vectorial hydrolysis, as well as dissociation of a nonhydrolyzed dimer from the filament after incorporation. In the first part of the paper, we apply this model to a single protofilament of a microtubule. A catastrophe transition is defined for each protofilament, similarly to the earlier one-dimensional models, the frequency of occurrence of which is then calculated under various conditions but without explicit assumption of steady-state conditions. Using a perturbative approach, we show that the leading asymptotic behavior of the protofilament catastrophe in the limit of large growth velocities is remarkably similar across different models. In the second part of the paper, we extend our analysis to the entire filament by making a conjecture that a minimum number of such transitions are required to occur for the onset of microtubule catastrophe. The frequency of microtubule catastrophe is then determined using numerical simulations and compared with analytical and semianalytical estimates made under steady-state and quasi-steady-state assumptions, respectively, for the protofilament dynamics. A few relevant experimental results are analyzed in detail and compared with predictions from the model. Our results indicate that loss of GTP cap in two to three protofilaments is necessary to trigger catastrophe in a microtubule.

Effectiveness of a dynein team in a tug of war helped by reduced load sensitivity of detachment: evidence from the study of bidirectional endosome transport in D. discoideum.

Bidirectional cargo transport by molecular motors in cells is a complex phenomenon in which the cargo (usually a vesicle) alternately moves in retrograde and anterograde directions. In this case, teams of oppositely pulling motors (e.g., kinesin and dynein) bind to the cargo, simultaneously, and 'coordinate' their activity such that the motion consists of spells of positively and negatively directed segments, separated by pauses of varying duration. A set of recent experiments have analyzed the bidirectional motion of endosomes in the amoeba D. discoideum in detail. It was found that in between directional switches, a team of five to six dyneins stall a cargo against a stronger kinesin in a tug of war, which lasts for almost a second. As the mean detachment time of a kinesin under its stall load was also observed to be ∼1 s, we infer that the collective detachment time of the dynein assembly must also be similar. Here, we analyze this inference from a modeling perspective, using experimentally measured single-molecule parameters as inputs. We find that the commonly assumed exponential load-dependent detachment rate is inconsistent with observations, as it predicts that a five-dynein assembly will detach under its combined stall load in less than a hundredth of a second. A modified model where the load-dependent unbinding rate is assumed to saturate at stall-force level for super-stall loads gives results which are in agreement with experimental data. Our analysis suggests that the load-dependent detachment of a dynein in a team is qualitatively different at sub-stall and super-stall loads, a conclusion which is likely to have implications in other situations involving collective effects of many motors.

A first-passage-time theory for search and capture of chromosomes by microtubules in mitosis.

The mitotic spindle is an important intermediate structure in eukaryotic cell division, in which each of a pair of duplicated chromosomes is attached through microtubules to centrosomal bodies located close to the two poles of the dividing cell. Several mechanisms are at work toward the formation of the spindle, one of which is the 'capture' of chromosome pairs, held together by kinetochores, by randomly searching microtubules. Although the entire cell cycle can be up to 24 hours long, the mitotic phase typically takes only less than an hour. How does the cell keep the duration of mitosis within this limit? Previous theoretical studies have suggested that the chromosome search and capture is optimized by tuning the microtubule dynamic parameters to minimize the search time. In this paper, we examine this conjecture. We compute the mean search time for a single target by microtubules from a single nucleating site, using a systematic and rigorous theoretical approach, for arbitrary kinetic parameters. The result is extended to multiple targets and nucleating sites by physical arguments. Estimates of mitotic time scales are then obtained for different cells using experimental data. In yeast and mammalian cells, the observed changes in microtubule kinetics between interphase and mitosis are beneficial in reducing the search time. In Xenopus extracts, by contrast, the opposite effect is observed, in agreement with the current understanding that large cells use additional mechanisms to regulate the duration of the mitotic phase.

Theoretical results for chemotactic response and drift of E. coli in a weak attractant gradient.

The bacterium Escherichia coli (E. coli) moves in its natural environment in a series of straight runs, interrupted by tumbles which cause change of direction. It performs chemotaxis towards chemo-attractants by extending the duration of runs in the direction of the source. When there is a spatial gradient in the attractant concentration, this bias produces a drift velocity directed towards its source, whereas in a uniform concentration, E. coli adapts, almost perfectly in case of methyl aspartate. Recently, microfluidic experiments have measured the drift velocity of E. coli in precisely controlled attractant gradients, but no general theoretical expression for the same exists. With this motivation, we study an analytically soluble model here, based on the Barkai-Leibler model, originally introduced to explain the perfect adaptation. Rigorous mathematical expressions are obtained for the chemotactic response function and the drift velocity in the limit of weak gradients and under the assumption of completely random tumbles. The theoretical predictions compare favorably with experimental results, especially at high concentrations. We further show that the signal transduction network weakens the dependence of the drift on concentration, thus enhancing the range of sensitivity.

A catalytic role of heparin within the extracellular matrix.

We investigated the mechanism by which heparin enhances the binding of vascular endothelial growth factor (VEGF) to the extracellular matrix protein fibronectin. In contrast to other systems, where heparin acts as a protein scaffold, we found that heparin functions catalytically to modulate VEGF binding site availability on fibronectin. By measuring the binding of VEGF and heparin to surface-immobilized fibronectin, we show that substoichiometric amounts of heparin exposed cryptic VEGF binding sites within fibronectin that remain available after heparin removal. Measurement of association and dissociation kinetics for heparin binding to fibronectin indicated that the interaction is rapid and transient. We localized the heparin-responsive element to the C-terminal 40-kDa Hep2 domain of fibronectin. A mathematical model of this catalytic process was constructed that supports a mechanism whereby the heparin-induced conformational change in fibronectin is accompanied by release of heparin. Experiments with endothelial extracellular matrix suggest that this process may also occur within biological matrices. These results indicate a novel mechanism whereby heparin catalyzes the conversion of fibronectin to an open conformation by transiently interacting with fibronectin and progressively hopping from molecule to molecule. Catalytic activation of the extracellular matrix might be an important mechanism for heparin to regulate function during normal and disease states.

Self-consistent theory of reversible ligand binding to a spherical cell.

Ligand binding to receptors is the initial event in many signaling processes, and a quantitative understanding of this interaction is important for modeling cell behavior. In this paper, we study the kinetics of reversible ligand binding to receptors on a spherical cell surface using a self-consistent stochastic theory. Binding, dissociation, diffusion and rebinding of ligands are incorporated into the theory in a systematic manner. We derive explicitly the time evolution of the ligand-bound receptor fraction p(t) in various regimes. Contrary to the commonly accepted view, we find that the well-known Berg-Purcell scaling for the association rate is modified as a function of time. Specifically, the effective on-rate changes non-monotonically as a function of time and equals the intrinsic rate at very early as well as late times, while being approximately equal to the Berg-Purcell value at intermediate times. The effective dissociation rate, as it appears in the binding curve or measured in a dissociation experiment, is strongly modified by rebinding events and assumes the Berg-Purcell value except at very late times, where the decay is algebraic and not exponential. In equilibrium, the ligand concentration everywhere in the solution is the same and equals its spatial mean, thus ensuring that there is no depletion in the vicinity of the cell. Implications of our results for binding experiments and numerical simulations of ligand-receptor systems are also discussed.