Cooperativity Can Enhance Cellular Signal Detection
Most sensory cells use surface receptors to detect environmental stimuli and initiate downstream signaling. Cooperative interactions among sensory receptors is known to play a crucial role in enhancing the sensitivity of biochemical processes such as oxygen sensing by hemoglobin, but whether cooperativity enhances the fidelity with which a system with multiple receptors can accurately and quickly detect a signal is poorly understood. We model the kinetics of small clusters of receptors in the presence of ligand, where the receptors act independently or cooperatively. We show that the interaction strength and how it is coupled to the dynamics influences the macroscopic observables. Contrary to recent reports, our analysis shows that receptor cooperativity can increase the signal-to-noise ratio, but this increase depends on the underlying dynamics of the signaling receptor cluster.
pacs:87.16.Xa, 87.18.Cf, 87.18.Tt, 87.18.Vf
Biological systems must precisely respond to small changes in environmental cues to carry out essential functions. The best studied example of this precision is the cooperative binding of oxygen to hemoglobin, which allows for tight binding of molecular oxygen under high concentrations in the lung and near complete unbinding under low concentrations at distal locations. This switch-like response is essential for efficiently carrying out tasks, and cooperativity allows hemoglobin to tune binding and unbinding over a very narrow physiological range. Similarly, sensory cells must respond to both small changes in external ligand concentrations and shallow chemical gradients with high accuracy. For example, chemotactic bacteria such as Escherichia coli can detect and respond to extremely small changes in attractant concentrations near the fundamental Berg-Purcell limit Berg and Purcell (1977), which is set by ligand diffusion. It is interesting to imagine that cooperative interactions between receptors may also help increase signal detection, since recent experiments have demonstrated that the bacterial flagellar motor, which is involved in the biochemical pathway for chemotaxis, is highly cooperative ((3); (4)) and cell surface receptors form clusters ((5)). Indeed, it has been suggested that clustering allow receptors to interact cooperatively ((6)), but there is no theoretical basis to suggest that cooperativity helps with signal detection.
Cell signaling is initiated through ligand binding to the extracellular domains of surface receptors, leading to receptor activation (Fig. 1(a)). The increase in receptor activity is directly proportional to the amplitude of the downstream cellular signal. In recent years, there have been several theoretical studies exploring the role of cooperativity in biological signal transduction Bialek and Setayeshgar (2008); (8); Aquino et al. (2011); Skoge et al. (2011). Working within the framework of the Monod-Wyman-Changeux (MWC) model, Bialek and Setayeshgar Bialek and Setayeshgar (2008) showed that receptor cooperativity lowers the threshold required to sense a change in concentration, bringing the value closer to the Berg-Purcell limit. Moreover, Hu et al. (8) employed an Ising-type model to show that receptor cooperativity improves gradient sensing in a two dimensional model of the cell embedded in a chemical gradient. A crucial concept in signal transduction is the ability to detect the signal over the intrinsic noise in the system, which is known as the signal-to-noise (SNR) ratio. This is particularly important in the present case since cooperativity not only sensitizes the receptors’ response to a signal, but also amplifies their intrinsic noise due to stochastic fluctuations between activated and inactivated states. In this regard, Skoge et al. Skoge et al. (2011), also employing a dynamic Ising model, recently showed that cooperativity in chemoreceptor activity slows down receptor activation leading to a reduction in the signal-to-noise ratio compared to non-interacting receptors. They showed that this result was true for one and two dimensional Ising models making strong implications that cooperativity does not enhance signal detection.
The well-known advantages of cooperativity in signal detection juxtaposed with the recent claim that cooperativity fails to enhance the SNR motivated us to more closely examine the role of receptor kinetics on this phenomena. In this Letter, we formulate the equilibrium energetics of a small cluster of interacting receptors using the formalism of statistical mechanics, and the system’s dynamics are then studied using classical chemical kinetics, in which forward and reverse rate constants determine the transition rates between distinct structural states of the system (Fig. 1). The MWC (11) and Koshland-Nemethy-Filmer (KNF) (12) models make up two seminal schemes for describing the kinetic states involved in the allosteric activation of proteins. Here, we closely follow the KNF model, which assumes that receptor activation immediately follows from ligand binding, and the activation state of neighboring subunits influence the kinetics of activation and deactivation. In general, there are very few constraints placed on a system’s dynamics given its energetics and conformational state space, and because of this, the kinetics can be imposed in a number of ways. For the spin system, it is common to use Glauber dynamics (13) to determine the time evolution of a system. However, detailed balance only restrains the ratio of the forward to backward rate constants, and many biochemical systems fail to follow either of these dynamics (14); (15). We show here that the relationship between SNR and receptor cooperatively is highly dependent on the manner in which detailed balance is enforced.
We consider small clusters of interacting receptors, and each receptor can exist in two states: activated or inactivated. The interaction between receptors is based on the physical proximity of the receptors in a cluster rather than their requirement to fall on a regular lattice, and therefore, the activation/deactivation rate of any given receptor depends on the activation state of all other receptors in the cluster (Fig. 1(b)). The corresponding Hamiltonian is
where is the intrinsic coupling, or represents the active or inactive receptor states, denotes links between receptors, and is related to the energy of ligand binding. Specifically, depends on the bulk ligand concentration and the ligand dissociation constant . For , the system is cooperative and receptors are biased toward the active state, while is a model of independent receptors.
We describe the time evolution of a system of receptors as shown in Fig. 1(c) using the master equation, written here in matrix form
where is a column vector with components and is a tridiagonal matrix, whose nonzero elements are given by , and . Here is the normalized probability of observing active receptors, is the rate at which a single receptor deactivates given that receptors are active, and is the rate at which a single receptor is activated given that receptors are active. We use reflecting boundary conditions such that and for the fully inactive and fully activated states, respectively (20). The average number of active receptors at any point in time, , is given by the probability distribution function: , where are integer components of the column vector .
The ratio of the forward and reverse transition rates between any two adjacent states in Fig. 1(c) is related to the energetics of the system through the principle of detailed balance
where is the energy difference, in thermal units, between the two states and the values and account for the number of available receptors for activation or deactivation, respectively. Our energetic and kinetic formulation in Eqs. 2-3 adheres closely to the approach laid out in Ref. (16). According to Eq. 1, the energy of the system with activated receptors is leading to . Given the definition of , the ligand dependent term becomes , where and are the base activation and deactivation rates for a solitary receptor, respectively. The dissociation constant is defined as the ratio of and . Thus, a definition of the rate constants that is consistent with Eq. 3 is
where is a phenomenological parameter specifying how the energy of coupling affects the forward and backward transition rates, and we assume that the ligand concentration only influences the activation rate. When , Eq. 4 reduces to the standard kinetics for ligand binding to a cluster of non-interacting receptors. For clusters exhibiting positive cooperativity with , tuning from 0 to 1 allows us to explore a range of kinetic models. For instance, Glauber dynamics results from , while a kinetic model in which cooperativity only influences the forward rates and not the reverse rates results from .
The equilibrium probability distribution, , can be determined from Eq. 2 by setting the rate of change to zero. With this quantity, the equilibrium probability of a single receptor being activated as a function of ligand concentration is . This can be solved analytically to determine the dependence of receptor activity on ligand concentration and coupling
where with . For four receptors (), is shown in Fig. 2(a), and we see that as the coupling constant, , increases the activation curve becomes more sigmoidal as expected for highly cooperative transitions such as oxygen binding to hemoglobin. Unlike dynamic properties discussed next, the equilibrium properties depend only on the ratio of forward and backward transition rates.
To determine the kinetic properties of the system after a change in ligand concentration at , we must solve the full master equation in Eq. 2, which can be done formally as
where the exponentiated rate matrix becomes the conditional probability matrix. The average activity of the system, , follows from , and the change in the system due to the external jump in ligand concentration, , is
where is the observation time after the perturbation. Assuming that the change in ligand concentration is small, linear response theory (17) tells us that the return to equilibrium is proportional to the natural fluctuations in the system, and we can rewrite the time averaged dynamic response function Skoge et al. (2011), , as
The time auto-correlation function can be written as , which is
where is an element of the conditional probability matrix and . Using the spectral decomposition, can be diagonalized as , where is formed from the eigenvectors of and is a diagonal matrix of its eigenvalues () with and for .
As shown in Fig. 2(b), increases as cooperativity is introduced into the system. As expected from a shift in the energetics of the active state, leads to greater activity than - the non-interacting case. Additionally, the response time of the system depends on the kinetic model, with eliciting faster dynamics than , since the forward rate constants are larger. The response kinetics of a cluster can be understood more quantitatively in terms of the correlation time , which characterizes the time required for the response to decay to equilibrium
where and is the largest non-zero eigenvalue of the master equation. For a cluster of non-interacting receptors, . The addition of cooperativity slows the dynamics, as can be seen qualitatively in Fig. 2(b), but as shown in Fig. 2(d), is much smaller for dynamics than Galuber dynamics, which have . In fact, this difference becomes more pronounced as the cluster size increases. Nonetheless, regardless of the kinetic model, the static response, , is independent of , and the solid and dashed lines converge at longer times.
For a given observation time , the signal-to-noise ratio is defined as
where is the average intrinsic noise of the receptor cluster averaged over time . The response of a multiple independent receptors is always amplified relative to that of the single receptor. Thus, a cluster of non-interacting receptors has a signal-to-noise ratio which is times the SNR of a single receptor. When cooperativity is introduced into the system, this linear scaling no longer holds, and the quantity must be calculated using the master equation.
The average variance in activity of a cluster of receptors, after averaging for time , is given in ref. Berg and Purcell (1977) as
where is a column-vector with components and is defined as
is a diagonal matrix with elements , where is a time dependent function used to describe the dynamics of the variance. As the observation time, , increases, integration of the signal over multiple transitions from active to inactive leads to a reduction in the variance as shown in Fig. 2(c). However, the speed with which the variance decays is directly related to the time required for a receptor to cycle between active and inactive states, which is roughly twice the system’s correlation time , as shown previously (18); (19). For large observation times, much greater than , the variance becomes
In the case of , we recover the signal-to-noise ratio of non-interacting receptors with for long averaging time,
We calculated the SNR at time for clusters containing 4 receptors and no cooperativity, (blue curve in Fig. 3(a)). As expected, the SNR increases as the ligand concentration is increased. Next, we introduced increasing amounts of cooperativity by tuning from to . For these calculations, we assumed model kinetics with . Surprisingly, we realized that cooperativity increased the SNR, as can be most clearly seen in Fig. 3(b) where the ratio of the SNR for a cooperative cluster to a non cooperative cluster is always greater than one. This result indicates that receptors always benefit from cooperativity when . The increased SNR for a cooperative cluster has a strong dependence on the ligand concentration, and its advantage reaaches a maximum around for the case. For a large cluster containing receptors, the SNR ratio is dramatically increased and for the cooperative cluster is over 10 times more sensitive than the non interacting receptors (Fig. 3(c)). The SNR’s dependence on ligand concentration is even more pronounced, and it is shifted to lower concentration values compared to the smaller cluster. This sharp transition arises from a combination of a more sigmoidal binding curve and a decreased correlation time . Therefore, both the equilibrium and kinetic properties play a role in increasing the SNR. For small ligand concentrations, cooperativity in large clusters dramatically suppresses the SNR (Fig. 3(c)). This suppression arises from decreased receptor activation below threshold. Finally, in Fig. 3(d), we show that a cluster of 4 receptors obeying Glauber dynamics exhibit a reduction in the SNR for high ligand concentrations consistent with a recent report Skoge et al. (2011). Nonetheless, for small ligand concentrations, cooperativity modestly increases the SNR.
Here, we show that cooperativity can increase or decrease the SNR of a receptor cluster depending on the dynamics of receptor activation. For , the SNR can be dramatically increased with respect to a group of non interacting receptors, but this increase is highly dependent on ligand concentration. Moreover, for , the SNR is either reduced or only marginally increased with respect to a cluster of non interacting receptors. This later result is consistent with previous work Skoge et al. (2011), but here we have shown that the SNR depends heavily on ligand concentration as well. Increased sensitivity comes from both the equilibrium properties of the receptors and their intrinsic dynamics. The SNR can be increased dynamically by decreasing the correlation time. Cooperativity reduces the correlation time for clusters with dynamics, while the correlation time increases for clusters obeying Glauber dynamics (). Real biological systems have complex dynamics, and it will be interesting to further explore how the detailed kinetics of a particular signalling system influence their macroscopic properties.
We thank Natalia Jura (UCSF) for insightful conversations. This work was supported by NIH Grant No. R21 GM100224-01 (M.G.).
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