How Accurately Can a Single Receptor Measure Ligand Concentrations

Biological sensory systems often display remarkable accuracy and sensitivity, sometimes detecting the presence of just a few molecules in the environment (1). During embryo growth, important developmental decisions are sometimes made based on the positional information in just a few molecules (2). Therefore, biophysicists have wondered what determines the fundamental precision limit of biological sensory performance. A new study by Kaizu et al. (3) revisits this important problem and demonstrates that statistical fluctuations and many-body correlations both can influence this fundamental sensory limit. Their result represents an improved estimate of the sensory precision limit based on fundamental physics. The elementary event in sensory processes is the binding of a diffusing ligand molecule to a single sensory receptor. In what is now a classic in the field, Berg and Purcell (4) considered this problem in the regime where the binding reaction is diffusion-limited and receptor switching is Markovian. In this case the receptor switches between bound and unbound states with exponentially distributed waiting times. They obtained that the precision of ligand concentration measured by the receptor is δcc=24Dσc(1−n¯)T, (1) where c is the ligand concentration in the environment and δc is the error in the concentration estimate. The value T is the time interval of measurement. The value σ is the receptor size and n¯ is average probability of finding the receptor bound with ligand. This result makes intuitive sense (5), but because it is in the diffusion-limited regime, the ligand binds immediately when it reaches the receptor and reaction rate constants do not appear explicitly. The applicability of this result outside of the diffusion-limit regime is not clear. Bialek and Setayeshgar (5) sought to generalize the result from Berg and Purcell (4) by considering the binding-unbinding dynamics together with the ligand diffusion process. They utilized the fluctuation-dissipation theorem (FDT) to solve the coupled diffusion-reaction problem. FDT allows one to estimate the time it takes for a fluctuation in a system to relax to the equilibrium average. In our case, the fluctuation in consideration is the binding occupancy of the receptor. Bialek and Setayeshgar (5) predicted that the ligand concentration measurement error is related to fluctuation of the receptor occupancy around the mean, and the result from Berg and Purcell (4) should be modified to δcc=1πDσcT+2kac(1−n¯)T, (2) where ka is the receptor associate rate, given that the receptor and ligand are at contact. This result incorporates important spatial correlation effects from diffusion. However, in the diffusion-limited regime, the second term in the square-root should disappear and only diffusion effects remain. In this limit, Eq. 2 does not agree with the result from Berg and Purcell (4), apart from a geometric factor of 2π. Indeed, when the average receptor occupancy is high (n¯≈1), Eq. 2 does not diverge as in Eq. 1. But when the receptor is constantly occupied, it cannot measure the concentration field and the error should diverge. Kaizu et al. (3) revisited this problem. Their analysis resolves the conflicting results on fundamental limits to the precision in chemical sensing. Kaizu et al. (3) argues that the concentration measurement limit should be δcc=12πDσc(1−n¯)T+2kac(1−n¯)T. (3) They then validate this result by performing rigorous simulation of the reaction-diffusion process. In this new result, while the second reaction term agrees with Bialek and Setayeshgar (5), the diffusion term agrees with Berg and Purcell (4). Kaizu et al. (3) speculates that linearization from FDT misses some ligand rebinding events and receptor-ligand correlations. These nonlinear effects may be the reason behind the apparent discrepancy between Eqs. 1 and 2. An improved estimate of concentration measurement limit has implications in all sorts of biological sensory systems, ranging from chemotaxis in eukaryotes and prokaryotes, to the sense of smell. Other applications include developmental systems where tissue patterning requires sensitive measurement of spatial morphogen concentrations. Perhaps morphological variations can be ultimately traced to slight errors in concentration measurements. The work of Kaizu et al. (3) completes a piece of the puzzle in this important problem. However, it only treats the case of single isolated receptors. Cells can employ multiple receptors and utilize energy to accurately detect signals at a wide range of ligand concentrations (6). Receptor-receptor correlations and their implication in sensory systems are also interesting. Physical effects and statistical fluctuations are sure to play roles in complex receptor arrays as well.