Accurate Encoding and Decoding by Single Cells: Amplitude Versus Frequency Modulation

Cells sense external concentrations and, via biochemical signaling, respond by regulating the expression of target proteins. Both in signaling networks and gene regulation there are two main mechanisms by which the concentration can be encoded internally: amplitude modulation (AM), where the absolute concentration of an internal signaling molecule encodes the stimulus, and frequency modulation (FM), where the period between successive bursts represents the stimulus. Although both mechanisms have been observed in biological systems, the question of when it is beneficial for cells to use either AM or FM is largely unanswered. Here, we first consider a simple model for a single receptor (or ion channel), which can either signal continuously whenever a ligand is bound, or produce a burst in signaling molecule upon receptor binding. We find that bursty signaling is more accurate than continuous signaling only for sufficiently fast dynamics. This suggests that modulation based on bursts may be more common in signaling networks than in gene regulation. We then extend our model to multiple receptors, where continuous and bursty signaling are equivalent to AM and FM respectively, finding that AM is always more accurate. This implies that the reason some cells use FM is related to factors other than accuracy, such as the ability to coordinate expression of multiple genes or to implement threshold crossing mechanisms.

Introduction Cells are exposed to changing environmental conditions and need to respond to external stimuli with high accuracy, e.g. to utilize nutrients and to avoid lethal stresses [1,2]. To represent (encode) chemicals in the environment, either ligand-bound receptors trigger chemical signals or ion channels allow entry of secondary messengers. These in turn activate transcription factors (TFs), which then regulate targetprotein production (decoding). In eukaryotic cells, the conventional view is that the level of signaling within the cell directly encodes the external stimuli, with consequent gradual changes in the nuclear TF concentrations. This is effectively an amplitude modulation (AM) mechanism [3][4][5][6][7][8][9][10]. However, recent single-cell experiments also show pulsating signals [3,[11][12][13][14] and bursty entry of TFs into the Here, we aim to investigate the advantages and disadvantages of CM and BM (AM and FM) for encoding and decoding of constant concentrations and ramps. To build intuition, we start with a single receptor/ion channel (CM and BM). We consider concentration sensing by a linear pathway, allowing us to gain exact results for different temporal regimes (as suitable for fast signaling and slow gene regulation). To provide analytical results, we extend the single-receptor model for ramp sensing by In FM, the nuclear TF concentration is always the same during a burst, only the frequency of occurrence changes. As a consequence, the protein ratio stays constant. Mora and Wingreen. First, we introduce an alternative mechanism to integral feedback, the incoherent feedforward loop (another common pathway motif for ramp sensing and precise adaptation [40][41][42]). This allows us to generalize the model to more than one pathway. Second, by explicitly including the time-dependence of signaling noise, we are able to provide first-order analytical results for the accuracy of ramp sensing. Taken together, a general principle emerges, favoring BM for fast signaling and CM for slow gene regulation. Finally, we generalize to many receptors and ion channels, a far more realistic situation for biological systems, allowing us to make connection with AM and FM. While we found that

Results
Cells sense external stimuli with cell-surface receptors and/or ion channels, which ultimately lead to changes in the concentration and dynamics of active transcription factors (TFs) inside the nucleus. Cells control the response at two different levels. Firstly, cell-surface receptors signal to regulate the activity of TFs in the cytoplasm. Secondly, inportin and exportin regulate the entry of active TFs into the nucleus, thereby regulating transcription (Fig. 3A). Here, we build a theoretical model that encodes information from an extra-cellular environment in an intra-cellular representation. We distinguish two ways of encoding this information: continuous modulation (CM) and bursty modulation (BM). Once the information is encoded, various proteins can act together to implement a response (decoding), involving regulatory networks. To provide a general analysis for arbitrary noise we first address concentration sensing in a simple linear pathway using the master equation. However, to derive analytical results for ramp sensing and pathways with feedback we apply the small-noise approximation. We finally extend these models to implement amplitude (AM) and frequency modulation (FM) for many receptors or ion channels. Accuracy is assessed by comparing the protein output noise for the different modulation schemes, assuming that the signal is decoded by the average concentration.

Following Mora and Wingreen [23] we build a single-receptor model that implements CM and BM.
We call the extra-cellular species c, which is encoded intra-cellularly by the signaling rate u. Assuming we are in the fast diffusion regime in which each ligand molecule can bind the receptor only once, the receptor can be in either of two uncorrelated states: on when bound and off when unbound. This allows the receptor activity, r(t), to be written mathematically as a binary response, which takes value 1 in the on state and 0 in the off state. The extra-cellular concentration c affects the unbound time intervals τ u , such that the binding rate is given by τ u −1 = k + c(t), where k + is the binding rate constant. In contrast, the bound time intervals, τ b , are exponentially distributed random numbers with average τ b −1 = k − , where k − is the unbinding rate constant, which is independent of the extra-cellular stimulus concentration (inset in Fig. 3A). As for ion channels, some are ligand-gated or regulated by receptors, while others are voltage-gated and hence dependent on action potentials [43]. In all these cases the stimulus affects the opening or closing times. In CM downstream proteins are produced with a constant rate α during each on time interval, which leads to a signaling rate u CM = αr(t), while in BM ζ = αk −1 − molecules are produced instantly at the moment of binding with rate u BM = ζ δ(t − t + i ), where t + i are the binding times (Fig. 3B). This choice for ζ allows a meaningful comparison of CM and BM as both produce, on average, the same amount of intracellular species.

General approach to concentration sensing exhibits two regimes of accuracy
In order to provide a general result for arbitrary input fluctuations, we write down the chemical master equation. For simplicity, we only consider concentration sensing with c(t) = c 0 , but the model can also be applied to ramps. Furthermore, we assume a linear pathway in which the receptor/ion channel activity r directly regulates an output species with copy number n (with production rate u and degradation rate γ) (Fig. 3C, left). Since the receptor/ion channel activity is a two-state system (on/off), there are two resulting master equations for CM (one for each state) describing the probability of being in the on and  Figure 3. Schematic view of signaling and gene regulation. (A) Cartoon of S. cerevisiae in presence of extracellular calcium, considered a paradigm of bursty frequency modulation. Calcium enters through plasma-membrane ion channels and can be stored (released) in (from) vacuoles. Intracellular calcium activates calcineurin, which dephosphorylates Crz1p. Once dephosphorylated, Crz1 binds inporting Nmd5p and enters the nucleus. Exportin Msn5p subsequently removes Crz1 from the nucleus. Cytoplasmic calcium pulses may correspond to Crz1 bursts in the nucleus [15]. Red arrows indicate movement while blue arrows stand for chemical signaling. (B) Single receptor/ion channel activity, r(t) (blue line), depends on the concentration of extra-cellular stimulus c. The signaling rate u differs between continuous (CM) and bursty modulation (BM). In CM, u is constant rate α during bound intervals, with p b the probability of being bound. In BM, ζ molecules are realized at the time of binding with τ bursts the duration between consecutive bursts (binding events). (C) Different regulatory networks. Linear pathway used for concentration sensing. Incoherent feedforward loop and integral feedback control allow chemical ramps to be sensed.
off states, i.e. p on (n, t) and p off (n, t): dp on (n, t) dt = γ(n + 1)p on (n + 1, t) + αp on (n − 1, t) + k + cp off (n, t) − (γn + α + k − )p on (n, t), Note that α ≥ k − , so molecules are generally produced in the on state. In BM, instead, the master equations which describe the probabilities p on (n, t) and p off (n, t) of having n proteins at time t, are given respectively by dp on (n, t) dt = γ(n + 1)p on (n + 1, with burst size ζ a positive integer. We solve Eqs. (1a) and (1b) with generating functions and simulate Eqs. (2a) and (2b) with the Gillespie algorithm (see Materials and Methods).
Simulations via the Gillespie algorithm show different outcomes for fast (small-noise approximation limit, Fig. 4A,B) and slow (Fig. 4C,D) dynamics of the receptor. For fast switching (k + c 0 , k − ≫ γ), for both CM and BM, the probability has an unimodal distribution (Fig. 4B). On the other hand, in the slow switching regime (k + c, k − ≪ γ), the probability distribution becomes bimodal for CM and unimodal with a long tail for BM, leading to drastically increased noise (Fig. 4D). The unimodal distribution for BM, which is simply due to the use of infinitely short pulses, would become bimodal for finite width pulses.
In order to classify the different dynamics and to compare CM and BM for arbitrary noise, we require information on the probability distribution of n output proteins. In particular, we study the average, variance and skewness (the latter is encoded in the third moment) of the distribution for both CM and BM. Constraining the average output of CM and BM to be the same (Fig. 5A,B), we identify two regimes for fast dynamics: k + c 0 < k − (Fig. 5C) and k + c 0 > k − (Fig. 5D). Specifically, for k + c 0 < k − , BM is more accurate (Fig. 5C, inset), while CM is generally more accurate when k + c 0 > k − (Fig. 5D, inset), except for minimal burst size (ζ = 1). However, for slow dynamics (and hence large noise), CM is always more accurate than BM. The study of the third moment shows that, for slow switching and hence bimodality, BM has large asymmetry (Fig. 5C,F).
These observations can be explained as follows, using the fact that the receptor/ion channel can only detect information from the extra-cellular environment during unbound (off ) time intervals, as the extra-cellular stimulus only affects the binding rate (Fig. 3). For fast dynamics, the two regimes can be understood by comparison with maximum-likelihood estimation (MLE), the most accurate strategy for encoding [44]. MLE estimates the ligand concentration c ML = k −1 + τ u −1 from the average unbound time interval τ u . The bound time intervals are discarded as they only contribute noise [44]. BM, which produces fixed-size bursts at the times of binding, approaches MLE when the bound intervals are shorter than the unbound intervals. In this case, the times of the bursts effectively estimate the unbound time intervals (Fig. 3B, bottom) and BM is more accurate than CM. However, when the bound intervals are longer than the unbound intervals, BM cannot estimate the unbound time intervals anymore and becomes less accurate than CM. Since CM produces protein during the bound intervals, it signals according to the average receptor activity (Fig. 3B, top). Hence, CM effectively contains information on both bound and unbound intervals, and thus can still provide a reasonable estimate of unbound time intervals. An interesting exception is αk −1 − = ζ = 1, for which BM becomes slightly more accurate than CM. In the latter case, since the rate of protein production during a bound interval in CM is very low, there is uncertainty as to whether CM actually produces protein or not, which reduces its accuracy. In contrast, for slow switching the burst size needs to increase since BM produces the same level of protein as CM. Hence, BM is always less accurate than CM, independent of whether bound or unbound time intervals are longer. While we analytically demonstrate the connection with MLE for fast dynamics in the next section, an extended discussion without comparison to MLE can be found in S1 Text and S1-S3 Figs. (A-B) fast (k + c 0 = 20s −1 , k − = 100s −1 , γ = 0.1s −1 , α = 100s −1 , ζ = 1) and (C-D) slow (k + c 0 = 0.01s −1 , k − = 0.05s −1 , γ = 1s −1 , α = 25s −1 , ζ = 500) switching. (A,C) Protein number as a function of time from Gillespie simulations for CM (blue lines) and BM (red lines). (B) The probability distribution for n target proteins is unimodal for both AM (blue) and FM (red). (D) The probability distribution is bimodal for AM (blue) and remains unimodal for BM (red) but with a long tail in the slow switching regime.
Small-noise approximation to ramp sensing confirms two regimes for fast dynamics To further investigate fast dynamics, we extend an analytical model for ramp sensing in the small-noise approximation [23]. Considering the single-receptor described in Fig. 3A,B, we linearize the system by averaging over a time much larger than the binding and unbinding times. We further assume exponential distributions for τ b and τ u so that ( where c(t) increases only very slowly with time (see below). Hence, signaling noise arises in CM due to variable bound time intervals (ignoring stochastic production of protein during bound intervals), while in BM the binding times (bursting times) vary. Without loss of generality, we set α = k − , which is equivalent to (Insets) Magnification of small-noise approximation region (fast switching). Analytical results for CM (blue) and numerical results for BM (red) as function of the frequency of binding events (logarithmic scale). Two regimes are shown: k − = 10 k + c 0 (α = 100s −1 , γ = 1s −1 , ζ from 1000 to 1) (left column) and k − = 0.1 k + c 0 (α = 10s −1 , γ = 1s −1 , ζ from 1000 to 1) (right column). Averages from CM and BM are constrained to be equal, i.e. ζ = αk −1 − . Variances of CM and BM exhibit two different regimes for fast switching: for k + c 0 < k − BM is more accurate than CM (inset in C), while for k + c 0 > k − CM is generally more accurate (inset in D), except for ζ = 1. Third moments show that, for large noise, the probability distributions become asymmetric. ζ = 1. Hence, as we show in S1 Text, for averaging time much longer than k −1 − and (k + c(t)) −1 , the average and autocorrelation (variance) of u(t) are given by [23] with δu(t) = 0 and Note that only the variance differs between CM and BM. In particular, in Eq. (5) the ratio k + c(t)/k − determines whether g is larger in BM or CM, which ultimately determines which scheme leads to the least noise. BM has the lower noise only when k + c(t) < k − , i.e. when τ b < τ u . In particular, in the limit of fast unbinding (k + c(t) ≪ k − ), the signaling noise for CM is twice as large as for BM. Sensing temporal ramps, i.e. the change of concentration with time, is crucial for locating nutrients and avoiding toxins. We start by considering a stimulus whose concentration is constant for t < 0 and increases linearly and slowly in time after t = 0: for constants c 0 and c 1 with c 1 t ≪ c 0 . By applying Eq. (6) to Eqs. (3)(4)(5), the signaling rate can be rewritten to first order as where u 0 , u 1 are functions of c 0 and c 1 , and δu is the noise described by δu(t)δu(t ′ ) (given in S1 Text). The condition c 1 t ≪ c 0 is necessary so that u behaves linearly in time with u 1 t ≪ u 0 . Under this condition, the factor g BM of Eq. (5) becomes where g BM is given by g * BM for a constant external concentration. We now assume that the extra-cellular stimulus is encoded in the signaling rate u which affects the production of two output proteins with concentrations x and y. Specifically, we compare the output noise of x and y between CM and BM using the incoherent feedforward (Fig. 3C, middle) and integral feedback (Fig. 3C, right) loops.

Incoherent feedforward loop
The incoherent feedforward loop is a network motif in which u directly affects two outputs x and y, while y inhibits x (Fig. 3C, middle). The loop provides precise adaptation to a step-change in stimulus and can also be used for ramp sensing. Mathematically, we use the following two coupled stochastic differential equations, where k x is the rate constant for production and degradation of x, while k y is the rate constant for degradation of y, and f (u) and g(y) are specified functions. In order to have adaptation the variable y needs to evolve slower than x, which requires k x > k y . Here we choose f (u) = e bu and g(y) = e bkyy , where constant b has units of time. This allows us to obtain an analytic solution (see S1 Text for details).

Integral feedback loop
The integral feedback loop [23] is another network motif for precise adaptation and ramp sensing. Here, u affects x only (the main output), while x activates y and y inhibits x (Fig. 3C, right). The general equations for this model are given by where k x is the rate constant for degradation of x, k y is the rate constant for production and degradation of y satisfying k x > k y , and f (y) is a monotonically decreasing function of y. Specifically, we choose f (y) = e −by , where b is a dimensionless constant. This again produces an analytic solution (see S1 Text for details).

Small-noise approximation
To analytically solve Eqs. (9) and (10) for the incoherent feedforward loop, and Eqs. (11) and (12) for the integral feedback loop, we linearize these equations within the small-noise approximation, and assume that we are in the fast-switching regime. This allows us to find analytic solutions in a particular time window and under certain conditions which we define in S1 Text. Specifically, for the incoherent feedforward loop in the small-ramp regime, the average values of u(t) , x(t) and y(t) are determined by the differential equations Eqs. (9,10). Although there are no steady states for ramps, x(t) and y(t) show time-dependent stable solutions Introducing x = x + δx and y = y + δy into Eqs. (9) and (10) with subsequent linearization the variance of the target-protein copy numbers can be derived (see Materials and Methods). To first order in small-ramp parameters the variances of x for both types of modulation are 1 ky 2(kx+ky)(1+k+c0/k−) 2 , and g CM and g * BM are parameters discussed in Eqs. (5) and (8). The corresponding results for species y are provided in Eqs. (S56) and (S58), and plots for species x and y are shown in Fig. 6B,D.
Consistent with the master equation, these results show again two regimes: ramp sensing is more accurate for BM if k + c 0 < k − , while CM is more accurate otherwise. For a constant environment (zeroth-order with c 1 = u 1 = 0) the regime is largely determined by the factor g. If k + c 0 < k − , Figure 6. Two regimes in incoherent feedforward loop based on the small-noise approximation. Output noise, i.e. relative variance of x (top) and y (bottom), as function of the non-dimensional ramp time u 1 t/u 0 for k + c 0 < k − i.e. τ b < τ u (left) and k + c 0 > k − i.e. τ b > τ u (right). CM and BM are shown by blue and red lines respectively. (A,B) BM is more accurate than AM for k + c 0 = 10 7 s −1 and k − = 6.7 × 10 7 s −1 . (C,D) CM is more accurate then BM for k + c 0 = 10 7 s −1 and k − = 6.7 × 10 6 s −1 . Remaining parameters: k + c 1 = 10 5 s −2 , k x = 5s −1 and k y = 10s −1 .
, and BM is more accurate than CM with Fig. 6A,B). This is because the variability of the bound intervals (δτ b ) 2 can be eliminated in BM (but not in CM), and the unbound intervals are well approximated by the duration between bursts (τ bursts in Fig. 3). For k + c 0 ≪ k − , BM effectively implements MLE. In contrast, CM is more accurate for k + c 0 > k − , where g CM = 2 and g BM > 2 (Fig. 6C,D). This is because BM contains no information on unbound time intervals, while CM still contains some information through the probability of being bound (p b in Fig. 3B). These results also apply to ramp sensing since the accuracy of the downstream proteins (decoding) relates again to the factor g and hence to the ratio between the bound and unbound time intervals. The integral feedback loop in Eqs. (11) and (12) shows very similar behavior (provided in S1 Text). The validity of our analytical results are confirmed by simulations of the stochastic differential equation for both pathways in S4 Fig. and S5 Fig. AM is more accurate than FM for multiple receptors/ion channels To address the question of whether AM or FM is more accurate in encoding and decoding, we consider a straightforward generalization to multiple receptors (or ion channels) (see S1 Text and S6 Fig.  for details). AM can be obtained by considering unsynchronized CM receptors. In contrast, the experimentally observed sporadic bursts of nuclear translocation [10,23] and hence FM might be explained by synchronized receptors that individually operate with BM.
For N unsynchronized (us) receptors, the resulting average and variance of the signaling rate are 1 in terms of the single-receptor quantities. Consequently, the relative variance, given by the variance divided by the average-squared, scales with the inverse of the number of receptors (N ). On the other hand, for N synchronized (s) receptors, the average and variance of the signaling rate are given respectively by The relative variance is now independent of N . Hence, unsynchronized receptors (AM) have a reduction of noise by a factor N compared to synchronized receptors (FM).
For slow dynamics, or fast dynamics with k + c > k − , CM is generally more accurate than BM (at least for ζ > 1), and with N receptors, AM is more accurate than FM by an even larger margin. In contrast, for fast dynamics with k + c < k − , BM is more accurate than CM by at most a factor of 2 (Eq. (5)). But since AM is N times more accurate than CM, AM becomes more accurate for encoding than FM for more than two receptors. Since our results from the previous sections show that larger signaling noise leads to larger output noise, the same rule emerges for decoding.
From a physical point of view, how can receptors act in a synchronized fashion? Receptors may be coupled by adaptor proteins or elastic membrane deformations, allowing them to act cooperatively [45,46]. In conclusion, while for fast dynamics (small-noise approximation) BM can be more accurate than CM up to a factor of two, two receptors/ion channels are sufficient for AM to become more accurate than FM. Since cells have thousands of receptors and ion channels, AM becomes the most accurate modulation scheme.

Discussion
Cellular responses to extra-cellular stimuli involve both encoding the external stimuli by internal signals (which is normally fast) and subsequently decoding via the regulation of protein levels (which is normally much slower). The internal representation of the external signal falls into two broad categories: continuous/amplitude modulation (CM/AM), where bound receptors continually signal and the internal concentration itself encodes the external signal, and bursty/frequency modulation (BM/FM), where receptors only signal when first bound and the signal is encoded in the frequency of peaks. Here, we compared the output noise for both types of modulation in the presence of a constant and a linearly increasing (in time) external concentration. Besides considering a linear pathway, we compared two nonlinear network motifs: the incoherent feedforward loop and the integral feedback loop. These loops are ubiquitous in biological systems. For example, the incoherent feedforward loop is found in chemotactic adaptation of eukaryotes [40] and transcription networks in bacteria [41], and the integral feedback loop is found in chemotactic adaptation of bacteria [25, 47] and in eukaryotic olfactory and phototransduction pathways [27].
We found that, for a single receptor or ion channel, BM can be more accurate than CM for fast dynamics. This situation can occur when the average duration of the active on state is shorter than the average duration of the inactive off state (Figs. 5 and 6). In this case, BM effectively implements maximum-likelihood estimation, the most accurate mechanism of sensing [44]. If instead more time is spent in the on state, then CM is generally more accurate (except when the burst size is minimal, i.e one). The reason behind this effect, which we analytically prove within the small-noise approximation, is that CM has information about both the on and off states, whereas BM only knows when a switch from off to on occurs. As such, CM effectively implements Berg and Purcell's classic result of estimating ligand concentration by time averaging [48] (see also Discussion in [44]). In addition, we found that for slow dynamics CM is always more accurate than BM, independent of whether more time is spent in the on or off states, due to increased burst sizes (Fig. 5). Taken together our results suggest that BM should be more common in signaling pathways than in gene regulation.
The generalization to multiple receptors/ion channels allows AM and FM to be compared. AM, which arises from unsynchronized CM receptors, has a reduced relative noise due to spatial averaging, while the relative noise in FM from synchronized BM receptors remains identical to the single-receptor result. (Note the observed nuclear bursts of approximately constant amplitude and duration support our FM mechanism [10,15].) As a result, AM is always more accurate than FM for more than two receptors (S6 Fig.). Since cells have tens of thousands of receptors and ion channels, this implies that the reason that FM is sometimes observed in real systems must have a different origin. At least three possibilities present themselves. Firstly, FM can help to coordinate gene expression [15,19], which is particularly useful when hundreds of genes are controlled by a single transcription factor, such as during stress response [49][50][51]. Secondly, FM can enhance co-localization of proteins inside the nucleus, providing another way to improve coordination of gene expression [52]. Thirdly, as with oscillatory signals, bursts can be used to activate transcription by threshold crossing [32] while avoiding desensitization [28]. This may then push the cell to differentiate into a new state (such as under starvation to initiate competence) [53,54]. It is also worth noting that by using seemingly redundant isoforms (such as NFAT1 and NFAT4 during an immune response), AM and FM can be combined to enhance temporal information processing [21].
While providing intuitive insights, it is clear that our models are highly oversimplified versions of signaling and gene regulation in actual cells. One of the main reasons for this is that we used idealized delta-functions as pulses in BM (and hence in FM). However, for example, in the calcium stress-response pathway in Saccharomyces cerevisiae (Fig. 3A) nuclear bursts of Crz1p are on average two minutes long (Fig. 1D, inset). Most likely cytoplasmic calcium spikes determine the nuclear bursts (Elowitz, personal communication), but since the mechanism of calcium spiking remains poorly understood, such bursts are difficult to model. A further limitation of our models is that bursts only relate to translocation, whereas additional bursts may occur further downstream during transcription [55] (e.g. due to promoter switching [24]) and translation [56]. Future models may need to include these details.
Our models suggest further experimental investigation in multiple areas. Firstly, the distribution of burst duration affects factor g (Eq. (5)), so that g = 2 in equilibrium for a single-step process and potentially g < 2 for an irreversible binding cycle dominated by energy dissipation [23,57]. These irreversible cycles are present in some ligand-gated ion channels, such as the cystic fibrosis transmembrane conductance regulator (CFTR) channels and N-Methyl-D-aspartate (NMDA) receptors. These exhibit peaked opening distributions, which can be interpreted as evidence of broken reversibility and energy consumption [58,59]. Such cases and their possible connection with accuracy need further investigation. In fact, most cellular processes rely heavily on energy consumption, including nuclear shuttling and chromosome remodeling, limiting the applicability of our equilibrium CM-receptor model. Secondly, coordination of gene expression during stress or cell-fate decisions might be another reason for implementing FM rather than AM. More quantitative experiments are needed to better understand this mechanism. Thirdly, closer inspection of Ca 2+ -independent transcription factors (as well as Ca 2+ -dependent co-regulated genes) are warranted in order to verify coordination of multiple genes [15]. Finally, to see if bursts help jump start new cellular programs (i.e. transition into a new "attractor"), global changes in gene regulation can be monitored.
A general understanding of FM may help prevent developmental defects and human diseases. Indeed, several biomedically relevant transcription factors, such as NF-κB, p53, NFAT and ERK, show oscillatory pulsing or random bursting [16,17,[33][34][35][36]54]. In fact, the destabilization of regulatory circuits can underlie human diseases: studies suggest that the coordination of gene expression could be critical in maintaining the proper functioning of key nodes in such circuits. For example, the NFATc circuit is cooperatively destabilized by a 1.5-fold increase in the DSCR1 and DYRK1A genes, which reduce NFATc activity leading to characteristics of Down's syndrome [16,60]. However, ERK pulses are regulated by both AM and FM with the same dose dependence, and it remains unclear how they affect cell proliferation and the relevance to cancer [36].
Broadly speaking, temporal ordering (regularity or periodicity) serves at least two roles in living systems [61]: extraction of energy from the environment and handling of information. While the first role is well studied in terms of molecular motors at the single-molecule level, the second role is intellectually more difficult to understand as it requires a broader, more global understanding of cells. We believe that future work that combines single-cell experiments with ideas of collective behavior and engineering principles is most likely to be successful.

Master-equation model for concentration sensing
The master equations for continuous modulation (CM), Eqs. (1a) and (1b), can be solved at steady state using generating functions. In particular, we derive the first three moments of the probability distribution using the general model in [62]. When the system is in the on/off state, the production rate of species x is α on/off . The degradation rate γ is independent of the state of the system. The probability distribution of n target proteins at time t is then described by dp s (n, t) dt = γ(n + 1)p s (n + 1, t) + α s p s (n − 1, t) + ksps(n, t) − (γn + α s + k s )p s (n, t), wheres = off (on) when s = on (off). By defining the generating functions and using Eq. (15), a solution for G s (z) can be found, which then readily gives the moments of p(n, t).
In particular, the variance and skewness are given by Full details are given in S1 Text. In order to solve the master equation for bursty modulation (BM), Eqs. (2a) and (2b), we use the Gillespie algorithm [63]. If the system is in the on state with n proteins at time t, it can either switch to the off state with transition rate given by k − /(k − + γn) or else remain in the on state and lose a protein by degradation. If instead the system is in the off state with n proteins at time t, it can either switch to the on state with switching rate k + c 0 /(k + c 0 + γn) and, via a burst, increase its number of proteins to n + ζ, or again remain in the same state and loose a protein by degradation. The time step between reactions, δt, is chosen from an exponential probability distribution λe −λδt , with λ equal to the total rate that at least one reaction occurs.

ODE models for ramp sensing
The following method applies to both the incoherent feedforward and the integral feedback loop. To solve the ordinary differential equations (9-12) we linearize around stable solutions, x(t) = x(t) + δx and y(t) = y(t) + δy, and assume that small δu leads to small δx and δy. Note that when sensing a gradually changing ramp, x(t) and y(t) are not steady states. Defining X = [x(t) y(t)] T we can rewrite these equations as where the matrix M and the constants w and z are defined in S1 Text. Analytic solutions are only available when M is time-independent. As shown in S1 Text, Eq. (19) can be solved and written as an integral, which can then be evaluated with, for example, Wolfram Mathematica 8. Supporting Information Legends S1 Text. Details of analytical calculations. . Variances of CM, BM and IM exhibit two different regimes for fast switching: for k + c 0 < k − BM is the most accurate mechanism and CM the worst (inset in C) while for k + c 0 > k − CM is generally the most accurate (except for ζ = 1) and IM the worst (inset in D). Third moments show that, for large noise, the probability distributions become asymmetric.    In this section we calculate the analytic solution for the master equation (Eq. ??,b in the main text) for continuous modulation (CM). For clarity, we repeat here the master equations for the on and off states: dp on (n, t) dt = γ(n + 1)p on (n + 1, t) + αp on (n − 1, t) + k + c p off (n, t) − (γn + α + k − )p on (n, t), dp off (n, t) dt = γ(n + 1)p off (n + 1, t) + k − p on (n, t) − (γn + k + c)p off (n, t), where p on/off (n, t) is the probability that the receptor/ion channel is in the on/off state with n output proteins at time t, α and γ are the production and degradation rates respectively, and k + c and k − are the binding and unbinding rates respectively. Note that the concentration of the input species (c) is now constant. Eqs. (??,b) can be rewritten as Eq. (??) in the main text dp s (n, t) dt = γ(n + 1)p s (n + 1, t) + α s p s (n − 1, t) + ksps(n, t) − (γn + α s + k s )p s (n, t), wheres is the on/off state when s is the off /on state, and α on = α, α off = 0, k on = k − and k off = k + c. To find the solution for the first two moments of the distribution p(n, t), we now follow Mehta and Schwab [1]. At steady state Eq. (??) becomes Ksps(n) = −(n + 1)p s (n + 1) − A s p s (n − 1) + (n + A s + K s )p s (n), where K s = k s /γ and A s = α s /γ. Using the generating function in Eq. (??) which implies which, when combined, gives To proceed further, it is useful to define the quantity H s (z) related to the generating function G s (z) by Eq. (S6) becomes which links the expressions for H on and H off . At this point the initial equation for the steady state (Eq. (S3)) becomes Multiplying by e As , taking the derivative with respect to z, substituting Eq. (S9), and defining ∆A s = As − A s , gives Finally, changing variables to u = ∆A s (z − 1) provides This is the confluent hypergeometric equation, for which the solution in terms of confluent hypergeometric functions of the first kind is given by with c s a constant of integration. Thus, through Eq. (S7), G s (z) = c s e Asz 1 F 1 (K s , 1 + K s + Ks; ∆A s (z − 1)) . (S13) To determine the constants, notice that 1 F 1 (a, b, 0) = 1 leading to G s (1) = c s e As = p s = Ks where p s is the average probability of being in state s. Rearranging terms, we obtain Finally, the probability distribution at steady state is given by [1] G s (z) = Kse As(z−1) Having an analytic expression for the steady-state probability distribution (Eq. S15), we can now calculate the first, second and third moments, which are related to the mean, variance and skewness, respectively. The mean production of the output protein is given by the mean production in the on state multiplied by the probability to be in the on state, averaged over the whole time period. For such a two-state system p on = K off K off +Kon and p off = 1 − p on . Therefore, the mean number of proteins is given by To calculate the variance, we use the following property of the generating function: Proof.
(δn) 2 = s n Using common properties of hypergeometric functions, the analytical solution for the variance is [1] (δn) For details, see the full calculation in the SI of [1].

Third moment
In order to understand more about the symmetry of the probability distribution, we calculate the third moment at steady state. As in Eq. (S17) the third moment can be found via generating functions as Thus, only s z 2 ∂ 3 z G s (z) z=1 needs to be calculated. The result is By combining Eqs. (S57)-(S59) as indicated in Eq. (S56), we obtain the analytic expression for the skewness of our system.

Input noise
In the Model section of the main text, we built a model for a single receptor/ion channel that encodes information from an cell-external environment in some cell-internal degrees of freedom. Similarly to [2], we assume that the receptor/ion channel activity (r(t)) is a two state system: on with r = 1 when the receptor is bound or the channel open, and off with r = 0 when the receptor is not bound or the channel is closed. The external concentration (c(t)) is assumed to affect the unbound/closed time interval τ u = [k + c(t)] −1 but not the bound/open time internal τ b = k −1 − , where k + and k − are both constants. Both interval durations are assumed to be independent, exponentially distributed random variables. The independence of binding and unbinding (or equivalently of opening and closing) means that the probability of a molecule binding the receptor a second time is negligible. We therefore assume the system to be in the fast diffusion regime.
The signaling rate, called u, implements two different mechanisms of encoding, either continuous (CM) or bursty (BM) modulation. CM and BM ultimately correspond to amplitude (AM) and frequency (FM) modulation, respectively, when generalized to multiple receptors/ion channels as explained in the Results section of the main text. In CM the proteins are produced with a constant rate α during the binding time. On the other hand, for BM a burst of ζ proteins is realized at the time of binding, so where αk −1 − = ζ and t + i the binding times. By taking the average of the rate u(t) over a timet much longer than both the average bound time, τ b = k −1 − , and the average unbound time, τ u = [k + c(t)] −1 , but shorter than the time during which the external concentration changes, we obtain  [2] for further details). Importantly, By considering an external concentration given by Eq. (??) in the main text, where we assume δu(t) ≪ u 0 + u 1 t and Here g CM = 2 and g * BM = 1 + (k + c 0 /k − ) 2 (cf. Eq. (??) in the main text). Again for ζ = 1, this becomes Eq. (??) in the main text. From Eqs. (S32)-(S36), the constant (t < 0) and ramp (t ≥ 0) regimes for the external species c are encoded in the rate u in the corresponding regimes since the condition c 1 t ≪ c 0 ensures u 1 t ≪ u 0 . However, to satisfy condition δu ≪ u for both CM and BM, a new condition is needed: which implies Here, we have introduced the correlation time of white noise, τ c , corresponding to the δ-function used in Eqs.(S35) and (S36). Note that condition in (S38) restricts our study to the fast switching regime. Finally, the signaling rate u in the constant regime has one small term δu/u 0 of order δ which is defined by Eqs. (S32) and (S36). Instead, small-ramp regime u contains two small terms: the small-ramp term u 1 t/u 0 of order ǫ and the small noise term δu/u 0 which now has a correction to order δ coming from the small ramp (order ǫ). With these definitions, from Eqs. (S32)-(S35) the rate u in the small ramp regime has two small corrections to the constant rate u 0 In order to linearize around linear solutions, we further assume that the small-noise amplitude is smaller than the small ramp. As a result, o(δ) ∼ o(ǫ x ) with x > 1, which means that Note that for simplicity, both in the following sections and in the main text, we set ζ = 1.

Output noise in incoherent feedforward loop
Here, b is a constant introduced to maintain the exponent unitless, and k x and k y are rate constants for x and y. This system of equations performs exact adaptation. The steady-state solution in the constant regime (t < 0 in Eq. (S32)) is which sets the initial conditions x(0) = 1 and y(0) = u 0 /k y for Eqs. (S41) and (S42) in the ramp regime (t ≥ 0 in Eq. ??). With these initial conditions, the solutions for t ≥ 0 can be written as where the integral in the expression for x(t) cannot be solved analytically. However, by assuming that the integral starts from time ǫ ≫ k −1 y , Eq. (S44) becomes Finally, by considering t such that t − ǫ ≫ k −1 x and without exceeding the small-ramp regime (e.g. k x,y ≫ 1 and ǫ small), the solution becomes Eqs. (??-b) in the main text, ky , These solutions match numerical results shown in S4A Fig. Note that the time interval over which these solutions are valid extends from a time larger than the transient time to around a time that does not exceed the small-ramp regime. These criteria also set the regime of validity for our next results.

Output variances in ramp sensing
Above we gave the average solutions for the incoherent feedforward loop, both in the constant regime (t < 0, Eq. (S43)) and in the ramp regime (t ≥ 0, Eqs. (S44) and (S45)). Now we want to linearize the equations around these solutions in order to obtain information about the noise. We assume that the input noise (δu) is smaller than the ramp, Eqs. (S38) and (S40), which translates into small output noise (δx and δy). In addition to these assumptions, we also assume b ∼ u −1 0 in order to ensure b(δu − k y δy) ≪ 1. Hence, in the ramp regime, the differential equations for δx and δy become By defining X(t) = δx δy , Eqs. (S47) and (S48) can be rewritten in a compact way for both the constant and small-ramp regimes as where and Note that M (t) (for t > 0) is independent of time, which allows Eq. (S49) to be solved analytically. This is due to our choice of f (u) and g(u) in Eqs. (??) and (??) in the main text. For the constant input regime, t < 0, the solutions for CM and BM are (δy(t) where g CM = 2 and g BM = g * BM = 1 + (k + c 0 /k − ) 2 . Hence, in the constant regime the output noise for BM is lower than the output noise for CM since k + c 0 < k − .
For the small-ramp regime, t ≥ 0, Eq. (S49) is analytically solvable for t ≫ k −1 y by evaluating the integral from time ǫ ≫ k −1 y to some time t that does not exceed the small-ramp approximation (as discussed for the average solutions). With these assumptions and by using an appropriate integrating factor, the solution for X(t) is However, for t − ǫ ≫ k −1 x,y (within the limit for t and ǫ as discussed for Eqs. (??,b) the solution is By using matrix diagonalization, expressing the noise in u by a delta function in time (Eq. S35), and integrating Eq. (S54) for X(t) 2 , we find analytical solutions for the variances. The results for CM are where g CM = 2. For BM, the g BM parameter (see Eq. (??)) affects the integration, and the results are (δy(t)) where g * BM = 1 + (k + c 0 /k − ) 2 . Note that for c 1 = u 1 = 0, the solutions coincide with the solutions for the constant regime. Furthermore, by comparing the time-dependent terms, CM is noisier than BM when k + c 0 < 1 3 k − . In our model this is due to the input noise (cf. Eq. S35). These two regimes in which BM is less noisy and hence more accurate than CM depend on the ratio of binding and unbinding rates as shown in S4B,C Fig. Clearly the analytic solutions match numerical simulations with noise. All these calculations were done using Wolfram Mathematica 8, while all the simulations of the stochastic differential equations (Eqs. S41 and S42) were done using the Euler method in MATLAB.

Output noise for integral feedback loop
A similar approach can be applied to the integral feedback loop given by Eqs. (??) and (??) in the main text, shown here for clarity with f (y) = e −by : Assuming that u is given by Eq. (S32), this system does not have analytic solutions in the ramp regime. However, in the small-ramp regime it is possible to linearize around the solutions of the constant regime. Hence, x(t) = x 0 +ǫ x (t) and y(t) = y 0 + ǫ y (t), where x 0 = 1 and y 0 = ln (u0/kx) b are the solutions for the constant regime with u = u 0 . Note that the condition k x < u 0 is required. By linearization, Eqs. (??) and (??) become Combining both equations and neglecting the second-order term u 1 t/u 0 ǫ y (t), it possible to find a second-order differential equation for ǫ x (t), given by The solution is with ǫ x (t) → u1 bkyu0 after a transient time defined by the exponential terms for any k 2 x /4 − bk x k y . Furthermore, there are two integration constants C 1 and C 2 . From Eq. (S62) we obtain ǫ y = u1 bu0 t − u1 b 2 kyu0 . Finally, the solutions of linearized Eqs. (??) and (??) in the small-ramp regime after the transient time are [2] x(t) = 1 + u 1 bk y u 0 , (S65) Within the small-noise approximation (Eq. S38), we want to find expressions for the variances. In the constant regime, u(t) = u 0 + δu(t) (t < 0 in Eq. 5) implies x(t) = 1 + δx(t) and y(t) = y 0 + δy(t). Therefore, the equations for the noise terms become Proceeding similarly to the incoherent feedforward loop, in the constant regime the solution for the variances are [2] (δx with g CM = 2 and g BM = g * BM = 1 + (k + c 0 /k − ) 2 . Hence, in the constant regime, the output noise for BM is lower than the output noise for CM (since k + c 0 < k − ).
To study the system in the small-ramp regime (t > 0 in Eq. 5), we assume that the input noise (δu) is smaller than the ramp (Eqs. S38 and S40), which translates into small output noise (δx and δy), and linearize around solutions (S65) and (S66). As a result, Eq. (??) becomes which, by using Eq. (S61), becomes where we neglect third-order terms in the small ramp (o(ǫ 3 )), second-order terms in the small noise (δ 2 ) and mixedorder terms (o(ǫ 2 δ)) due to the assumption that the noise is smaller than the ramp (cf. discussion that leads to Eq. (S40)).
Defining X(t) = δx δy , Eqs. (S71) and (S60) become where from Eqs. (S71) and (S60), using definitions of ǫ x and ǫ y , where the term e −M(t−ǫ) X(ǫ) is negligible for (t − ǫ) ≫ k −1 x,y . To calculate the variances we square Eq. (S73). Using Eqs. (S5)-(S9), the results for (δx(t)) 2 to first-order in the small-ramp parameters are Similarly the results for (δy) 2 are (δy(t)) 2 Note that for c 1 = u 1 = 0, the solutions coincide with the solutions for the constant regime. Although it is clear that BM is less noisy than CM for k + c 0 < k − , by comparing the time-dependent terms we find that, in fact, BM is always less noisy than CM. The analytical solutions are plotted in S5 Fig. and match the numerical simulations with noise. Again, all these calculations were done using Wolfram Mathematica 8, while all the simulations of the stochastic differential equations (Eqs. S59 and S60) were done using the Euler method in MATLAB.

FURTHER INVESTIGATIONS INTO THE ACCURACY
In this section we provide further explanations for the accuracy of concentration sensing by a single receptor without comparing with the maxmimum-likelihood estimation [3]. In Fig. ?? we showed results from the master equation for the two regimes k + c 0 < k − and k + c 0 > k − for slow and fast switching of the receptor. Despite its burstiness, the BM receptor turned out more accurate than the CM receptor in the k + c 0 < k − regime for fast switching.
Additional results from the master-equation model. To understand this result better we also implemented an intermediate-modulation (IM) receptor, which has features of both the CM and BM receptors. Like the CM receptor, the IM receptor signals while in the bound (on) state, but instead of a constant rate α of production it produces protein with a rate α ′ so that in each bound interval the same number of molecules are produced irrespective of the interval length, i.e. α ′ τ b = ζ, with ζ the constant burst size of BM. For this to work, the IM receptor would have to know at the time of binding when it will unbind again, in order to choose the correct rate of production. Since the rate of unbinding is a random variable this is generally not possible. Nevertheless, the IM receptor may help to further elucidate our observed trends in accuracy. In practice, we implemented this IM receptor by first simulating a time trace of bound and unbound time intervals with a Gillespie algorithm, allowing us to determine the rate of production as a function of time. Afterwards, the actual protein production and degradation were simulated.
In analogy to Fig. ?? the results for the IM receptor are shown in S1 Fig. (green lines), which also shows the results for the CM and BM receptors for comparison in blue and red, respectively. As expected, for slow switching the IM receptor has intermediate accuracy between CM and BM. CM is most accurate as continuous production during the bound intervals is balanced by degradation so the output protein level does not fluctuate excessively. BM is least accurate due to the increased burst size for slow switching. Since signaling by the IM receptor is only burst-like for the short bound intervals but not for the long bound intervals, it is somewhat more accurate than BM. Due to the non-constant rate of production, IM also fluctuates more than CM. This intermediate accuracy is clearly demonstrated by the time traces in the left panels of S2 Fig. In the k + c 0 < k − regime for fast switching, the inset of S1C Fig. shows that BM is now most accurate and that IM has again intermediate accuracy. While BM steadily produces the same amount of protein at the times of binding, IM produces this amount only during short bound intervals as its rate of production is then high, while during long bound intervals its slow production is buffered by degradation, so its protein level fluctuates more strongly. CM is even worse than IM since, due to its constant rate of production during bound time intervals, it hardly produces any protein during short bound intervals, which leads to drastic drops in protein level, while it produces a lot during long bound intervals due to its constant rate of production.
In contrast, in the k + c 0 > k − regime for fast switching, CM is generally most accurate due to its approximately constant rate of production throughout time, i.e. the receptor is almost always bound and active. IM is less accurate than CM because its rate of protein production is variable due to the variable length in bound intervals, despite the fact that the receptor is mostly bound. Interestingly, IM is even less accurate than BM under these conditions.
Inspecting the examples of time trace in the bottom right panel of S2 Fig., the burst sizes of IM can exceed the burst sizes of BM for unusually short bound intervals since production is very high and stochastic, and only on average the same amount of protein is produced during bound intervals than during a burst in BM. During long bound intervals the rate of production is very low. Hence, compared to BM, degradation prevents a net increase in protein level during a bound interval, leading to further variability. A special case is when the burst size ζ is 1. As shown in the inset of Fig. ??D, BM can be more accurate than CM. This is because the burst size of BM is minimal and in the master equation the production with minimal rate α in CM is highly stochastic.
As we now discuss, to provide further intuition for the differences in accuracy between the k + c 0 < k − and k + c 0 > k − regimes, we also simulated the variance of the signaling output (and hence the accuracy-determining factor g) directly (see Eq. 5).
Signaling output from ODE model without protein production and degradation. Factor g in Eq. 5 (and Eq. S31) determines the variance of the signaling rate u(t) without invoking any downstream protein production and degradation. For a given time interval ∆t, we can hence simulate u(t) directly. We assess the accuracy of CM, IM, and BM by plotting the histograms of the integrated signaling rate u I (∆t) := ∆t 0 u(t)dt and by determing their variances (cf. derivation of g in [2]). As slow protein production and degradation strongly affect the accuracy of the final protein output for slow switching, this approach mainly helps understand the interesting fast switching case.
We initially assume signaling during bound intervals is deterministic, leading to a linear increase of u with slope α (α ′ ) during a bound time interval for CM (IM) and a step increase by ζ for BM. At each unbound time interval, IM and BM have the same level of signaling output as IM produces the same number of proteins deterministically during each bound interval (ζ). In contrast, the signaling output from CM is generally different since the rate of signaling is always the same for each bound interval but their durations vary. Resulting time traces and variances are shown in S3A and B Figs. left panels, respectively. Specifically, S3A Fig., left panels shows clearly that for k + c 0 < k − BM and IM are most accurate with u I (t) increasing almost linearly in time. Since signaling is deterministic, BM and IM are essentially identical, and their variance may only differ due to small differences in signaling during the final bound interval (S3A Fig., bottom left panel). This last bound time interval may be interrupted in IM, but for long ∆t this difference is negligible. In contrast, S3B Fig., left panels show clearly that for k + c 0 > k − CM is most accurate, as u I (t) is now almost linear in time.
Signaling output from master-equation model without protein production and degradation. Allowing signaling to be stochastic does not change the results for the accuracy significantly. S3A Fig. right panels show that for k + c 0 < k − BM is now most accurate and that IM has intermediate accuracy (between BM and CM) due to its variability in signaling in line with S1C Fig. Additionally, S3B Fig., right panels show that CM is still most accurate but also that IM is worse than BM in line with S1D Fig. Taken together, these additional simulation results confirm our findings of the main text that BM is most accurate for k + c 0 < k − and CM is generally most accurate for k + c 0 > k − .

AM IS MORE ACCURATE THAN FM FOR MULTIPLE RECEPTORS/ION CHANNELS
Here, we provide a more detailed discussion of the accuracy of encoding by multiple receptors, i.e. using AM and FM. To determine whether AM or FM is more accurate in encoding and decoding, we generalize to multiple receptors (or ion channels) (S6 Fig.). We assume that AM is obtained by unsynchronized CM receptors (S6A Fig.), while FM is obtained by synchronized receptors that individually operate with BM (S6D Fig.). Other types of synchronization are also possible with synchronized CM receptors shown in S6B Fig. and Fig.) or variable amplitude (S6C Fig.) in contrast to the data (Fig. ??) [4,5].
To estimate the accuracy, we first consider perfect synchronization and unsynchronization in either modulation scheme. For N unsynchronized (us) receptors, we can express the resulting average and variance of the encoded input by the single-receptor quantities, i.e. u(t) us N = N u(t) 1 and δu(t)δu(t ′ ) us N = N δu(t)δu(t ′ ) 1 . As a result, the relative variance (variance divided by the average-squared) scales with N −1 . In contrast, for N synchronized (s) receptors, the average and variance of the encoded input can be written as u(t) s N = N u(t) 1 and δu(t)δu(t ′ ) s N = N 2 δu(t)δu(t ′ ) 1 , respectively. Hence, the relative variance is now independent of N , so unsynchronized receptors have an N times smaller noise than synchronized receptors. Since N unsynchronized CM receptors lead to AM, we obtain for its relative variance Conversely since N synchronized BM receptors lead to FM, the relative variance of FM is For slow dynamics, or fast dynamics with k + c > k − , CM is more accurate than BM. Hence, for N receptors, AM is even more accurate than FM. In contrast, for fast dynamics with k + c < k − , BM is up to twice as accurate as CM (Eq. (??)), and AM is N times more accurate than CM. Consequently, AM becomes more accurate for encoding than FM for more than two receptors (S6E Fig.). An exception are two receptors, for which AM and FM can be equally accurate (S6E Fig., inset). Since we generally show that larger signaling noise leads to larger output noise, the same rule emerges for decoding. To extend our results to intermediate levels of synchronization for N > 2 receptors we consider a fraction ρ of synchronized receptors while the remaining fraction (1 − ρ) are unsynchronized, with signaling either by CM or BM (S6E Fig.). When comparing CM and BM receptors for the same levels of synchronization ρ, BM receptors can remain more accurate than CM receptors (S6E Fig.). However, intermediate levels of synchronization do not strictly represent AM and FM. As shown in S6B,C Figs. synchronized CM receptors lead to pulses of variable duration, while unsynchronized BM receptors lead to highly frequent pulses with potentially variable amplitude.
Taken together, since single cells have thousands of receptors and ion channels, AM is the most accurate modulation scheme. Note that this figure is similar to Fig. ?? in main text with the addition of IM. Two regimes are shown: k− = 10 k+c0 (α = 100s −1 , γ = 1s −1 , ζ from 1000 to 1) (left column) and k− = 0.1 k+c0 (α = 10s −1 , γ = 1s −1 , ζ from 1000 to 1) (right column). Averages from CM, BM and IM are constrained to be equal, i.e. ζ (BM) = αk −1 − (CM) = α ′ τ b (IM). Variances of CM, BM and IM exhibit two different regimes for fast switching: for k+c0 < k− BM is the most accurate mechanism and CM the worst (inset in C) while for k+c0 > k− CM is generally the most accurate (except for ζ = 1) and IM the worst (inset in D). Third moments show that, for large noise, the probability distributions become asymmetric.