Mathematical modeling of eukaryotic cell migration: insights beyond experiments.

A migrating cell is a molecular machine made of tens of thousands of short-lived and interacting parts. Understanding migration means understanding the self-organization of these parts into a system of functional units. This task is one of tackling complexity: First, the system integrates numerous chemical and mechanical component processes. Second, these processes are connected in feedback interactions and over a large range of spatial and temporal scales. Third, many processes are stochastic, which leads to heterogeneous migration behaviors. Early on in the research of cell migration it became evident that this complexity exceeds human intuition. Thus, the cell migration community has led the charge to build mathematical models that could integrate the diverse experimental observations and measurements in consistent frameworks, first in conceptual and more recently in molecularly explicit models. The main goal of this review is to sift through a series of important conceptual and explicit mathematical models of cell migration and to evaluate their contribution to the field in their ability to integrate critical experimental data.

Dephosphorylation accelerates the dissociation of ZAP70 from the T cell receptor.

Protein-protein binding domains are critical in signaling networks. Src homology 2 (SH2) domains are binding domains that interact with sequences containing phosphorylated tyrosines. A subset of SH2 domain-containing proteins has tandem domains, which are thought to enhance binding affinity and specificity. However, a trade-off exists between long-lived binding and the ability to rapidly reverse signaling, which is a critical requirement of noise-filtering mechanisms such as kinetic proofreading. Here, we use modeling to show that the unbinding rate of tandem, but not single, SH2 domains can be accelerated by phosphatases. Using surface plasmon resonance, we show that the phosphatase CD45 can accelerate the unbinding rate of zeta chain-associated protein kinase 70 (ZAP70), a tandem SH2 domain-containing kinase, from biphosphorylated peptides from the T cell receptor (TCR). An important functional prediction of accelerated unbinding is that the intracellular ZAP70-TCR half-life in T cells will not be fixed but rather, dependent on the extracellular TCR-antigen half-life, and we show that this is the case in both cell lines and primary T cells. The work highlights that tandem SH2 domains can break the trade-off between signal fidelity (requiring long half-life) and signal reversibility (requiring short half-life), which is a key requirement for T cell antigen discrimination mediated by kinetic proofreading.

Molecular flexibility of DNA as a key determinant of RAD51 recruitment.

The timely activation of homologous recombination is essential for the maintenance of genome stability, in which the RAD51 recombinase plays a central role. Biochemically, human RAD51 polymerises faster on single-stranded DNA (ssDNA) compared to double-stranded DNA (dsDNA), raising a key conceptual question: how does it discriminate between them? In this study, we tackled this problem by systematically assessing RAD51 binding kinetics on ssDNA and dsDNA differing in length and flexibility using surface plasmon resonance. By directly fitting a mechanistic model to our experimental data, we demonstrate that the RAD51 polymerisation rate positively correlates with the flexibility of DNA. Once the RAD51-DNA complex is formed, however, RAD51 remains stably bound independent of DNA flexibility, but rapidly dissociates from flexible DNA when RAD51 self-association is perturbed. This model presents a new general framework suggesting that the flexibility of DNA, which may increase locally as a result of DNA damage, plays an important role in rapidly recruiting repair factors that multimerise at sites of DNA damage.

Simulation of receptor triggering by kinetic segregation shows role of oligomers and close contacts.

The activation of T cells, key players of the immune system, involves local evacuation of phosphatase CD45 from a region of the T cell's surface, segregating it from the T cell receptor. What drives this evacuation? In the presence of antigen, what ensures evacuation happens in the subsecond timescales necessary to initiate signaling? In the absence of antigen, what mechanisms ensure that evacuation does not happen spontaneously, which could cause signaling errors? Phenomena known to influence spatial organization of CD45 or similar surface molecules include diffusive motion in the lipid bilayer, oligomerization reactions, and mechanical compression against a nearby surface, such as that of the cell presenting the antigen. Computer simulations can investigate hypothesized spatiotemporal mechanisms of T cell signaling. The challenge to computational studies of evacuation is that the base process, spontaneous evacuation by simple diffusion, is in the extreme rare event limit, meaning direct stochastic simulation is unfeasible. Here, we combine particle-based spatial stochastic simulation with the weighted ensemble method for rare events to compute the mean first passage time for cell surface availability by surface reorganization of CD45. We confirm mathematical estimates that, at physiological concentrations, spontaneous evacuation is extremely rare, roughly 300 years. We find that dimerization decreases the time required for evacuation. A weak bimolecular interaction (dissociation constant estimate 460 µM) is sufficient for an order of magnitude reduction of spontaneous evacuation times, and oligomerization to hexamers reduces times to below 1 s. This introduces a mechanism whereby externally induced CD45 oligomerization could significantly modify T cell function. For large regions of close contact, such as those induced by large microvilli, molecular size and compressibility imply a nonzero reentry probability of 60%, decreasing evacuation times. Simulations show that these reduced evacuation times are still unrealistically long (even with a fourfold variation centered around previous estimates of parameters), suggesting that a yet-to-be-described mechanism, besides compressional exclusion at a close contact, drives evacuation.

Intrinsic Disorder in the T Cell Receptor Creates Cooperativity and Controls ZAP70 Binding.

Many immunoreceptors have cytoplasmic domains that are intrinsically disordered (i.e., have high configurational entropy), have multiple sites of posttranslational modification (e.g., tyrosine phosphorylation), and participate in nonlinear signaling pathways (e.g., exhibiting switch-like behavior). Several hypotheses to explain the origin of these nonlinearities fall under the broad hypothesis that modification at one site changes the immunoreceptor's entropy, which in turn changes further modification dynamics. Here, we use coarse-grain simulation to study three scenarios, all related to the chains that constitute the T cell receptor (TCR). We find that first, if phosphorylation induces local changes in the flexibility of the TCR ζ-chain, this naturally leads to rate enhancements and cooperativity. Second, we find that TCR CD3ɛ can provide a switch by modulating its residence in the plasma membrane. By constraining our model to be consistent with the previous observation that both basic residues and phosphorylation control membrane residence, we find that there is only a moderate rate enhancement of 10% between first and subsequent phosphorylation events. Third, we find that volume constraints do not limit the number of ZAP70s that can bind the TCR but that entropic penalties lead to a 200-fold decrease in binding rate by the seventh ZAP70, potentially explaining the observation that each TCR has around six ZAP70 molecules bound after receptor triggering. In all three scenarios, our results demonstrate that phenomena that change an immunoreceptor chain's entropy (stiffening, confinement to a membrane, and multiple simultaneous binding) can lead to nonlinearities (rate enhancement, switching, and negative cooperativity) in how the receptor participates in signaling. These polymer-entropy-driven nonlinearities may augment the nonlinearities that arise from, e.g., kinetic proofreading and cluster formation. They also suggest different design strategies for engineered receptors, e.g., whether or not to put signaling modules on one chain or multiple clustered chains.

Determination of the molecular reach of the protein tyrosine phosphatase SHP-1.

Immune receptors signal by recruiting (or tethering) enzymes to their cytoplasmic tails to catalyze reactions on substrates within reach. This is the case for the phosphatase SHP-1, which, upon tethering to inhibitory receptors, dephosphorylates diverse substrates to control T cell activation. Precisely how tethering regulates SHP-1 activity is incompletely understood. Here, we measure binding, catalysis, and molecular reach for tethered SHP-1 reactions. We determine the molecular reach of SHP-1 to be 13.0 nm, which is longer than the estimate from the allosterically active structure (5.3 nm), suggesting that SHP-1 can achieve a longer reach by exploring multiple active conformations. Using modeling, we show that when uniformly distributed, receptor-SHP-1 complexes can only reach 15% of substrates, but this increases to 90% when they are coclustered. When within reach, we show that membrane recruitment increases the activity of SHP-1 by a 1000-fold increase in local concentration. The work highlights how molecular reach regulates the activity of membrane-recruited SHP-1 with insights applicable to other membrane-tethered reactions.

Cargo navigation across 3D microtubule intersections.

The eukaryotic cell's microtubule cytoskeleton is a complex 3D filament network. Microtubules cross at a wide variety of separation distances and angles. Prior studies in vivo and in vitro suggest that cargo transport is affected by intersection geometry. However, geometric complexity is not yet widely appreciated as a regulatory factor in its own right, and mechanisms that underlie this mode of regulation are not well understood. We have used our recently reported 3D microtubule manipulation system to build filament crossings de novo in a purified in vitro environment and used them to assay kinesin-1-driven model cargo navigation. We found that 3D microtubule network geometry indeed significantly influences cargo routing, and in particular that it is possible to bias a cargo to pass or switch just by changing either filament spacing or angle. Furthermore, we captured our experimental results in a model which accounts for full 3D geometry, stochastic motion of the cargo and associated motors, as well as motor force production and force-dependent behavior. We used a combination of experimental and theoretical analysis to establish the detailed mechanisms underlying cargo navigation at microtubule crossings.

Hydrodynamics of transient cell-cell contact: The role of membrane permeability and active protrusion length.

In many biological settings, two or more cells come into physical contact to form a cell-cell interface. In some cases, the cell-cell contact must be transient, forming on timescales of seconds. One example is offered by the T cell, an immune cell which must attach to the surface of other cells in order to decipher information about disease. The aspect ratio of these interfaces (tens of nanometers thick and tens of micrometers in diameter) puts them into the thin-layer limit, or "lubrication limit", of fluid dynamics. A key question is how the receptors and ligands on opposing cells come into contact. What are the relative roles of thermal undulations of the plasma membrane and deterministic forces from active filopodia? We use a computational fluid dynamics algorithm capable of simulating 10-nanometer-scale fluid-structure interactions with thermal fluctuations up to seconds- and microns-scales. We use this to simulate two opposing membranes, variously including thermal fluctuations, active forces, and membrane permeability. In some regimes dominated by thermal fluctuations, proximity is a rare event, which we capture by computing mean first-passage times using a Weighted Ensemble rare-event computational method. Our results demonstrate a parameter regime in which the time it takes for an active force to drive local contact actually increases if the cells are being held closer together (e.g., by nonspecific adhesion), a phenomenon we attribute to the thin-layer effect. This leads to an optimal initial cell-cell separation for fastest receptor-ligand binding, which could have relevance for the role of cellular protrusions like microvilli. We reproduce a previous experimental observation that fluctuation spatial scales are largely unaffected, but timescales are dramatically slowed, by the thin-layer effect. We also find that membrane permeability would need to be above physiological levels to abrogate the thin-layer effect.

The Influence of Molecular Reach and Diffusivity on the Efficacy of Membrane-Confined Reactions.

Signaling by surface receptors often relies on tethered reactions whereby an enzyme bound to the cytoplasmic tail of a receptor catalyzes reactions on substrates within reach. The overall length and stiffness of the receptor tail, the enzyme, and the substrate determine a biophysical parameter termed the molecular reach of the reaction. This parameter determines the probability that the receptor-tethered enzyme will contact the substrate in the volume proximal to the membrane when separated by different distances within the membrane plane. In this work, we develop particle-based stochastic reaction-diffusion models to study the interplay between molecular reach and diffusion. We find that increasing the molecular reach can increase reaction efficacy for slowly diffusing receptors, whereas for rapidly diffusing receptors, increasing molecular reach reduces reaction efficacy. In contrast, if reactions are forced to take place within the two-dimensional plasma membrane instead of the three-dimensional volume proximal to it or if molecules diffuse in three dimensions, increasing molecular reach increases reaction efficacy for all diffusivities. We show results in the context of immune checkpoint receptors (PD-1 dephosphorylating CD28), a standard opposing kinase-phosphatase reaction, and a minimal two-particle model. The work highlights the importance of the three-dimensional nature of many two-dimensional membrane-confined interactions, illustrating a role for molecular reach in controlling biochemical reactions.

Bidirectional sliding of two parallel microtubules generated by multiple identical motors.

It is often assumed in biophysical studies that when multiple identical molecular motors interact with two parallel microtubules, the microtubules will be crosslinked and locked together. The aim of this study is to examine this assumption mathematically. We model the forces and movements generated by motors with a time-continuous Markov process and find that, counter-intuitively, a tug-of-war results from opposing actions of identical motors bound to different microtubules. The model shows that many motors bound to the same microtubule generate a great force applied to a smaller number of motors bound to another microtubule, which increases detachment rate for the motors in minority, stabilizing the directional sliding. However, stochastic effects cause occasional changes of the sliding direction, which has a profound effect on the character of the long-term microtubule motility, making it effectively diffusion-like. Here, we estimate the time between the rare events of switching direction and use them to estimate the effective diffusion coefficient for the microtubule pair. Our main result is that parallel microtubules interacting with multiple identical motors are not locked together, but rather slide bidirectionally. We find explicit formulae for the time between directional switching for various motor numbers.

Extracting multivalent detachment rates from heterogeneous nanoparticle populations.

Nanoparticles can form multiple bonds with target surfaces, thereby increasing adhesion strength and internalization rate into cells. This property has helped to drive interest in nanoparticles as delivery vehicles for drugs and imaging agents, but significant gaps in our understanding of multivalent adhesion make it difficult to control and optimize binding dynamics. In previous work, we experimentally observed that multivalent nanoparticle adhesion can exhibit a time-dependent detachment rate. However, simulations later indicated that the underlying cause was variability in the number of bonds that formed between individual nanoparticles within the population. Here, we use this insight to develop a simple model to isolate a series of constant detachment rates from such heterogeneous populations. Using simulations of experimental data to train the model, we first classified nanoparticles within a given population based on the most likely equilibrium bond number, which we termed the bond potential. We then assumed that each bond potential category would follow standard first-order kinetics with constant detachment rates. Model results matched the population binding data, but only if we further divided each bond potential category into two sub-components, the second of which did not detach. We then utilized bonding rates from the simulation to estimate detachment rates for the second, slower detaching sub-component. These results confirm our hypothesis that nanoparticle populations can be sub-divided based on bond potential, each of which could be characterized by a constant detachment rate. Finally, we established relationships between the new heterogeneous population detachment model and a time-dependent, empirical detachment model that we developed in previous work. This could make it possible to determine bond potential distributions directly from experimental data without computationally costly simulations, which will be explored in future work.

Dynamics of a multicomponent vesicle in shear flow.

We study the fully nonlinear, nonlocal dynamics of two-dimensional multicomponent vesicles in a shear flow with matched viscosity of the inner and outer fluids. Using a nonstiff, pseudo-spectral boundary integral method, we investigate dynamical patterns induced by inhomogeneous bending for a two phase system. Numerical results reveal that there exist novel phase-treading and tumbling mechanisms that cannot be observed for a homogeneous vesicle. In particular, unlike the well-known steady tank-treading dynamics characterized by a fixed inclination angle, here the phase-treading mechanism leads to unsteady periodic dynamics with an oscillatory inclination angle. When the average phase concentration is around 1/2, we observe tumbling dynamics even for very low shear rate, and the excess length required for tumbling is significantly smaller than the value for the single phase case. We summarize our results in phase diagrams in terms of the excess length, shear rate, and concentration of the soft phase. These findings go beyond the well known dynamical regimes of a homogeneous vesicle and highlight the level of complexity of vesicle dynamics in a fluid due to heterogeneous material properties.

Spatial pattern formation in microtubule post-translational modifications and the tight localization of motor-driven cargo.

Microtubule (MT) "age" can be interpreted as nucleotide state, lattice defects, or post-translational modification (PTM) such as acetylation and detyrosination. In all three cases, these have been recently shown to have functionally-important effects on the dynamics of MT arrays, and can present spatial and temporal heterogeneity. While mathematical models for MT array densities are well-established, here we present equations describing MT age, defined as the mean time since the MT's building blocks (tubulin) were polymerized from their soluble dimer state. We derive the age equations using a mean first-passage time calculation and two complementary approaches: The continuum limit of spatial discretization model, and an adjoint operator approach. These equations can recapitulate the observation that the oldest (most de-tyrosinated) tubulin in axons is near the middle of axons during neuronal development in chick embryos. Furthermore, PTMs influence motor kinetics up to approximately twofold for off-rates and velocities. Our simulations demonstrate that this relatively weak dependence of motor kinetics is sufficient to target motor cargo to a specific location along the array. This localization is tightly peaked in a way that magnifies the relatively small signal of PTM spatial heterogeneity. Thus, MT age can produce long-range spatial patterning without feedbacks or diffusing signals.

Cell Surface Mechanochemistry and the Determinants of Bleb Formation, Healing, and Travel Velocity.

Blebs are pressure-driven cell protrusions implicated in cellular functions such as cell division, apoptosis, and cell motility, including motility of protease-inhibited cancer cells. Because of their mechanical nature, blebs inform us about general cell-surface mechanics, including membrane dynamics, pressure propagation throughout the cytoplasm, and the architecture and dynamics of the actin cortex. Mathematical models including detailed fluid dynamics have previously been used to understand bleb expansion. Here, we develop mathematical models in two and three dimensions on longer timescales that recapitulate the full bleb life cycle, including both expansion and healing by cortex reformation, in terms of experimentally accessible biophysical parameters such as myosin contractility, osmotic pressure, and turnover of actin and ezrin. The model provides conditions under which blebbing occurs, and naturally gives rise to traveling blebs. The model predicts conditions under which blebs travel or remain stationary, as well as the bleb traveling velocity, a quantity that has remained elusive in previous models. As previous studies have used blebs as reporters of membrane tension and pressure dynamics within the cell, we have used our system to investigate various pressure equilibration models and dynamic, nonuniform membrane tension to account for the shape of a traveling bleb. We also find that traveling blebs tend to expand in all directions unless otherwise constrained. One possible constraint could be provided by spatial heterogeneity in, for example, adhesion density.

Multisite Phosphorylation Modulates the T Cell Receptor ζ-Chain Potency but not the Switchlike Response.

Multisite phosphorylation is ubiquitous in cellular signaling and is thought to provide signaling proteins with additional regulatory mechanisms. Indeed, mathematical models have revealed a large number of mechanisms by which multisite phosphorylation can produce switchlike responses. The T cell antigen receptor (TCR) is a multisubunit receptor on the surface of T cells that is a prototypical multisite substrate as it contains 20 sites that are distributed on 10 conserved immunoreceptor tyrosine-based activation motifs (ITAMs). The TCR ζ-chain is a homodimer subunit that contains six ITAMs (12 sites) and exhibits a number of properties that are predicted to be sufficient for a switchlike response. We have used cellular reconstitution to systematically study multisite phosphorylation of the TCR ζ-chain. We find that multisite phosphorylation proceeds by a nonsequential random mechanism, and find no evidence that multiple ITAMs modulate a switchlike response but do find that they alter receptor potency and maximum phosphorylation. Modulation of receptor potency can be explained by a reduction in molecular entropy of the disordered ζ-chain upon phosphorylation. We further find that the tyrosine kinase ZAP-70 increases receptor potency but does not modulate the switchlike response. In contrast to other multisite proteins, where phosphorylations act in strong concert to modulate protein function, we suggest that the multiple ITAMs on the TCR function mainly to amplify subsequent signaling.

First-Passage Time to Clear the Way for Receptor-Ligand Binding in a Crowded Environment.

Certain biological reactions, such as receptor-ligand binding at cell-cell interfaces and macromolecules binding to biopolymers, require many smaller molecules crowding a reaction site to be cleared. Examples include the T-cell interface, a key player in immunological information processing. Diffusion sets a limit for such cavitation to occur spontaneously, thereby defining a time scale below which active mechanisms must take over. We consider N independent diffusing particles in a closed domain, containing a subregion with N_{0} particles, on average. We investigate the time until the subregion is empty, allowing a subsequent reaction to proceed. The first-passage time is computed using an efficient exact simulation algorithm and an asymptotic approximation in the limit that cavitation is rare. In this limit, we find that the mean first-passage time is subexponential, T∝e^{N_{0}}/N_{0}^{2}. For the case of T-cell receptors, we find that stochastic cavitation is exceedingly slow, 10^{9} s at physiological densities; however, it can be accelerated to occur within 5 s with only a fourfold dilution.

Wrinkling dynamics of fluctuating vesicles in time-dependent viscous flow.

We study the fully nonlinear, nonlocal dynamics of two-dimensional vesicles in a time-dependent, incompressible viscous flow at finite temperature. We focus on a transient instability that can be observed when the direction of applied flow is suddenly reversed, which induces compressive forces on the vesicle interface, and small-scale interface perturbations known as wrinkles develop. These wrinkles are driven by regions of negative elastic tension on the membrane. Using a stochastic immersed boundary method with a biophysically motivated choice of thermal fluctuations, we investigate the wrinkling dynamics numerically. Different from deterministic wrinkling dynamics, thermal fluctuations lead to symmetry-breaking wrinkling patterns by exciting higher order modes. This leads to more rapid and more realistic wrinkling dynamics. Our results are in excellent agreement with the experimental data by Kantsler et al. [Kantsler et al., Phys. Rev. Lett., 2007, 99, 17802]. We compare the nonlinear simulation results with perturbation theory, modified to account for thermal fluctuations. The strength of the applied flow strongly influences the most unstable wavelength characterizing the wrinkles, and there are significant differences between the results from perturbation theory and the fully nonlinear simulations, which suggests that the perturbation theory misses important nonlinear interactions. Strikingly, we find that thermal fluctuations actually have the ability to attenuate variability of the characteristic wavelength of wrinkling by exciting a wider range of modes than the deterministic case, which makes the evolution less constrained and enables the most unstable wavelength to emerge more readily. We further find that thermal noise helps prevent the vesicle from rotating if it is misaligned with the direction of the applied extensional flow.

Actin-myosin spatial patterns from a simplified isotropic viscoelastic model.

F-actin networks are involved in cell mechanical processes ranging from motility to endocytosis. The mesoscale architecture of assemblies of individual F-actin polymers that gives rise to micrometer-scale rheological properties is poorly understood, despite numerous in vivo and vitro studies. In vitro networks have been shown to organize into spatial patterns when spatially confined, including dense spherical shells inside spherical emulsion droplets. Here we develop a simplified model of an isotropic, compressible, viscoelastic material continually assembling and disassembling. We demonstrate that spherical shells emerge naturally when the strain relaxation rate (corresponding to internal network reorganization) is slower than the disassembly rate (corresponding to F-actin depolymerization). These patterns are consistent with recent experiments, including a collapse of shells to a central high-density focus of F-actin when either assembly or disassembly is reduced with drugs. Our results demonstrate how complex spatio-temporal patterns can emerge without spatially distributed force generation, polar alignment of F-actin polymers, or spatially nonuniform regulation of F-actin by upstream biochemical networks.

A new method to experimentally quantify dynamics of initial protein-protein interactions.

Cells run on initiation of protein-protein interactions, which are dynamically tuned spatially and temporally to modulate cellular events. This tuning can be physical, such as attaching the protein to a cargo or protein complex, thereby altering its diffusive properties, or modulating the distance between protein pairs, or chemical, by altering the proteins' conformations (e.g., nucleotide binding state of an enzyme, post-translational modification of a protein, etc.). Because a dynamic and changing subset of proteins in the cell could be in any specific state, ensemble measurements are not ideal-to untangle which of the factors are important, and how, we need single-molecule measurements. Experimentally, until now we have not had good tools to precisely measure initiation of such protein-protein interactions at the single-molecule level. Here, we develop a new method to measure dynamics of initial protein-protein interactions, allowing measurement of how properties such as the distance between proteins, and their tethered length can modulate the rate of interactions. In addition to precise measurement distance dependent motor-MT rebinding dynamics, we demonstrate the use of a dithered optical trap to measure dynamic motor-MT interactions and further discuss the possibilities of this technique being applicable to other systems.

Mechanical modulation of receptor-ligand interactions at cell-cell interfaces.

Cell surface receptors have been extensively studied because they initiate and regulate signal transduction cascades leading to a variety of functional cellular outcomes. An important class of immune receptors (e.g., T-cell antigen receptors) whose ligands are anchored to the surfaces of other cells remain poorly understood. The mechanism by which ligand binding initiates receptor phosphorylation, a process termed "receptor triggering", remains controversial. Recently, direct measurements of the (two-dimensional) receptor-ligand complex lifetimes at cell-cell interface were found to be smaller than (three-dimensional) lifetimes in solution but the underlying mechanism is unknown. At the cell-cell interface, the receptor-ligand complex spans a short intermembrane distance (15 nm) compared to long surface molecules (LSMs) whose ectodomains span >40 nm and these LSMs include phosphatases (e.g., CD45) that dephosphorylate the receptor. It has been proposed that size-based segregation of LSMs from a receptor-ligand complex is a mechanism of receptor triggering but it is unclear whether the mechanochemistry supports such small-scale segregation. Here we present a nanometer-scale mathematical model that couples membrane elasticity with the compressional stiffness and lateral mobility of LSMs. We find robust supradiffusive segregation of LSMs from a single receptor-ligand complex. The model predicts that LSM redistribution will result in a time-dependent tension on the complex leading to a decreased two-dimensional lifetime. Interestingly, the model predicts a nonlinear relationship between the three- and two-dimensional lifetimes, which can enhance the ability of receptors to discriminate between similar ligands.