Trade-offs and constraints in allosteric sensing

Abstract

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.

Figure 1. The MWC model of an allosteric sensor and illustrations of the different sensing characteristics of the system.

(A) The sensor exists in two conformational states, which have different affinities for the signal molecule being sensed. A sensor can transition from one conformation to another only when not bound by a signal molecule, but adding extra transitions does not alter our results. (B) The dynamic range, r, is the difference between the saturation and basal levels of activity. (C) The Hill number, h, is a measure of the steepness of the response curve and, indirectly, of the cooperativity of the activation of the sensors and the non-linearity of the response. (D) The intrinsic noise, , quantifies the relative magnitude of the intrinsic fluctuations in the numbers of active sensors. Inset: histogram of the levels of activity at equilibrium. (E) The capacity, I _opt, provides an upper bound on the number of states that can be sensed and distinguished despite intrinsic noise (represented by the black vertical bars). In this example, the sensing system can distinguish between 3 states: low (yellow), medium (green) and high (blue). (F) The static gain, G ₀, is the change in activity in response to a small step increment in the input signal. The frequency-dependent gain (red curve) decreases as frequency increases: the system is a low-pass filter. (G) The response time, , measures the time to reach the level of activity corresponding to half of its equilibrium level.

Figure 2. The biases K and c and the number of subunits n completely determine the value of all characteristics except the response time, and consequently each characteristic is not independent.

Contour plots of the five characteristics we can derive analytically from Eq. (1). From top to bottom: the dynamic range, the Hill number, the intrinsic noise at the threshold, the capacity, and the static gain at the threshold. From left to right, the number of subunits n is respectively 1, 2, 4 and 8. The total number of sensors in the system is 100 and is used to calculate the intrinsic noise and the capacity. The white areas in the contour plots of the capacity (fourth row) correspond to parameter sets for which the magnitude of the intrinsic noise is large enough to invalidate the approximation we use to calculate the capacity. For n  = 1, although the Hill number is always one, there is small variation in the static gain (between 0 and 0.06 units).

Figure 3. The intrinsic fluctuations follow a binomial distribution and can maximize their variance away from the threshold of the response curve.

(A) Comparison between Eq. (5) (in dark blue) and numerical simulation (red line). Total number of sensors is 20; the initial number of input molecules is 224; K = 100; c = 0.01; n = 4; f_R =  0.01 s⁻¹; f_T =  0.01 s⁻¹; b_R =  1 s⁻¹; and b_T =  100 s⁻¹. (B) The threshold of the response (the midpoint between the basal and saturation levels) need not coincide with the level of input at which 50% of the sensors are active. (C) The maximum of the variance in the system, which is always located at the 50% of activity level, need not coincide with the threshold value of the input signal. (D) The intrinsic noise decreases with increasing input because more sensors become activated. In (B, C, D), the dark blue dots represent numerical simulation, the red curve the analytical solutions from Eqs. (1) and (10), and the light blue box shows the dynamic range of activity. The total number of sensors in the system is 100; K = 2; c = 0.1; f_R = 0.01 s⁻¹; f_T = 0.01 s⁻¹; b_R = 10 s⁻¹; and b_T = 100 s⁻¹.

Figure 4. Maximising the information transferred through the sensing system determines optimal distributions for outputs and inputs.

(A) Optimal distribution of inputs (dark blue area) and activity (red curve) for a system where n = 1, K = 100, and c = 0.01. (B) Optimal distribution of outputs for the system in (A), which has a dynamic range of about 0.5. (C) Optimal distribution of inputs (dark blue area) and activity (red curve) for a system where n = 4, K = 100, and c = 0.01. (D) Optimal distribution of outputs for the system in (C), whose dynamic range is close to 1.

Figure 5. For randomly sampled parameter values, constraints exist between pairs of characteristics.

(A) Scatterplot of the Hill number and the dynamic range for our randomly sampled parameter sets. (B) Scatterplot of the normalised response time (Methods) and the Hill number. (C) Scatterplot of the capacity and the dynamic range. (D) Scatterplot of the Hill number and the intrinsic noise measured at the threshold of the response curve. For each number of subunits, n, there are 10,000 data points.

Figure 6. The mutual information between pairs of characteristics quantifies the dependency of one characteristic on another.

The four matrices show the mutual information between all pairs of characteristics for different numbers of allosteric binding sites per sensor n = 1, 2, 4 and 8. The mutual information is normalised by the entropy of the characteristics on the rows. The darker colours represent pairs that are relatively unconstrained and the brighter colours indicate pairs that are more constrained. The three scatter plots give three examples of different constraints. From top to bottom, we have scatterplots of the static gain versus the Hill number, of the capacity versus the dynamic range, and of the dynamic range versus the Hill number. When , we observe a dark row corresponding to pairs involving the Hill coefficient because the Hill coefficient is always one when . The diagonals are white because the normalised mutual information of a characteristic with itself is always maximal.