Duncan Kirby,Anton Zilman
Proofreading does not result in more reliable ligand discrimination in receptor signaling due to its inherent stochasticity
kinetic proofreading,cell signaling,estimation theory
Significance Cellular signaling requires cells to reliably discriminate between different types of molecules to convey information about the cell environment. The kinetic proofreading (KPR) model is one of the main paradigms used to explain observations of highly precise molecular discrimination by cells. However, the effect of molecular noise is typically not accounted for when quantifying discrimination performance in these systems. We show that when noise is accounted for, KPR generally does not improve signal discrimination relative to a nonproofreading receptor and can substantially degrade the performance. We compare multiple KPR models to show that increased variance in receptor activity is intrinsic to the proofreading mechanism itself. The results raise questions about the role of noise in the precision of molecular signaling.
Proofreading does not result in more reliable ligand discrimination in receptor signaling due to its inherent stochasticity

Cellular signaling requires cells to reliably discriminate between different types of molecules to convey information about the cell environment. The kinetic proofreading (KPR) model is one of the main paradigms used to explain observations of highly precise molecular discrimination by cells. However, the effect of molecular noise is typically not accounted for when quantifying discrimination performance in these systems. We show that when noise is accounted for, KPR generally does not improve signal discrimination relative to a nonproofreading receptor and can substantially degrade the performance. We compare multiple KPR models to show that increased variance in receptor activity is intrinsic to the proofreading mechanism itself. The results raise questions about the role of noise in the precision of molecular signaling.

Cells send and receive signals in the form of molecular ligands that bind corresponding cell surface receptors. Ligand–receptor binding activates intracellular cascades of molecular modifications that eventually lead to cellular response. The ability of the cell to respond distinctly to different ligands depends on the ability of the receptor to discriminate between different ligands. In the simplest—and very common—scenario, discrimination relies on the differences in the receptor-binding affinities between different ligands: cognate ligands bind the receptor more strongly (and for a longer time) than noncognate ligands. At equilibrium, the ratio of the receptor occupancies by different ligands present at the same concentration is expected to be proportional to , where ΔG is the difference in binding free energy between the ligands (1–3). However, the fidelity of many important biological processes has been observed to be higher than what can be expected from the measured differences in the equilibrium binding free energies between ligands (1, 4–9). In one prominent example, T cells appear to be able to distinguish between self-peptides and foreign antigens which may differ by as little as a single amino acid mutation (as little as a two-fold difference in affinity) (3, 4, 10–14). Such extreme specificity of the cellular response to cognate ligands is frequently explained in terms of a mechanism known as kinetic proofreading (KPR) (3, 7, 13–16). The KPR mechanism enhances occupancy differences between ligands beyond the equilibrium factor via introduction of one or more intermediate bound states between the initial binding and the final state where the output signal is produced (Fig. 1A). The delay introduced by these additional states increases the probability of the ligand to dissociate from the receptor before the signal is produced, so that the weakly binding ligands rarely stay bound long enough to reach the signaling state and produce an output. Magnifying the differences between the ligands comes at the expense of energy consumption (typically in the form of ATP hydrolysis) to ensure irreversible unbinding of the ligand from the proofreading states of the receptor (3, 16–18). KPR was originally proposed by Hopfield and Ninio as a mechanism to explain the fidelity of DNA replication and protein biosynthesis and is believed to describe the proofreading step of the endonuclease and tRNA proofreading by the ribosome (1, 2, 19). McKeithan subsequently adapted multistep KPR as an explanatory mechanism behind the remarkable specificity of the T cell receptor (TCR) in recognizing foreign versus self-antigens (3). Subsequently, proofreading models and their modifications have been applied to explain additional features of the T cell response such as the speed and antagonism, while other works explored energetic and time constraints of generalized proofreading schemes (15–17, 20–23). KPR models have also been applied to other receptor signaling systems including the FcϵRI-IgE system, RecA assembly, and various intracellular signal cascades (7, 8, 13, 14, 16, 18, 24–31). This body of work established the common view of the generalized sensing problem: discrimination of two different ligand types (such as self- vs. non-self-peptides for T cells) based on a signal produced by the ligand binding to the receptor. In the context of cellular sensing, receptor resolution for different ligands is commonly measured by the ratio of the corresponding average occupancy probabilities of the signaling state of the receptor or by the ratio of physically related quantities such as the average time spent in the signaling state or the average amount of the produced downstream signaling output (1–3, 13, 21). In this paper, we will refer to this metric as the Hopfield–Ninio (HN) metric. Classical analysis of KPR models using the HN metric suggests an unbounded geometric increase in resolution power as the number of proofreading steps increases. The HN metric characterizes the resolution in terms of ensemble average properties and thus does not take into account the stochasticity in the molecular processes of the ligand binding and unbinding, proofreading steps, and the subsequent signaling reactions. However, as known in the context of concentration sensing, this stochasticity can have substantial effects on the sensing accuracy (32–39). As illustrated in Fig. 1B, the ability of a cell to resolve two different ligands depends not only on the separation of the mean outputs of the receptor, ⟨ni⟩, but also on the overlap of the probability distributions of their outputs, P(n|koff,i), where ligand types are indexed by i, and koff,i is the ligand–receptor unbinding rate which uniquely determines the ligand affinity. The comparison of average receptor inputs, as performed by the classical HN metric, is therefore insufficient for quantifying the ability to resolve two ligands (Discussion). As we show below, the effects of noise for KPR receptors can be especially acute because enhancement of the differences in mean signal between the ligands is accompanied by attenuated receptor output, which amplifies the role of noise relative to a nonproofread receptor (3, 37, 40). This signal attenuation is an inherent property of the KPR mechanism, and KPR is most commonly invoked as a mechanism of discrimination between a cognate ligand (weakly attenuated) and a noncognate ligand (strongly attenuated) (41). In addition to signal attenuation, proofreading introduces an additional source of intrinsic noise (compared to a nonproofread receptor) arising from the stochasticity of the transitions between proofreading states. We show in this paper that the combination of signal attenuation and stochasticity of the state transitions severely limits the rationale for using proofreading schemes for ligand discrimination. It is important to distinguish the effect of the intrinsic molecular noise on signal discrimination from the role of extrinsic constraints such as the energy consumption rate or the speed of the cellular decision-making. Extrinsic constraints represent trade-offs, with better resolution achievable if only the costs can be met (39, 42–44). Quantifying discrimination performance as a balance of costs and benefits does not account for the reliability of ligand discrimination in the presence of stochastic noise (SI Appendix, section H). By contrast, as we show in this paper, the intrinsic and unavoidable molecular noise of proofreading directly affects the ligand resolution capability of the proofreading receptor relative to a nonproofread receptor. In general, the maximal information about the nature of a ligand is encoded in the whole sequence of binding and unbinding events during the measurement time. However, information processing in cellular signaling is constrained by molecularly realizable mechanisms which by necessity reduce the information available to the cell. In this paper, we specifically focus on the effect of proofreading on specificity and ligand resolution and compare the resolution of a proofreading receptor to a nonproofreading receptor using mathematical models informed by biologically realistic signaling modalities. The role of molecular noise in receptor sensing has been investigated in a variety of contexts. Extending the pioneering work by Berg and Purcell, refs. 32, 34–36, 45–51 adopted the Fisher information and the associated framework of statistical estimation theory to quantify how receptor occupancy fluctuations fundamentally limit the precision of estimating ligand concentration. Other works have investigated the specificity of response to a particular ligand using information theoretic tools (18, 22, 36, 46, 52–57). Selecting between these various approaches to account for noise in ligand processing performance depends on the question at hand. In this paper, we use the mathematical framework of statistical estimation theory to account for the role of molecular noise in quantifying ligand specificity and resolution (46, 51, 58). We view specificity as the precision of estimating the ligand binding affinity to the receptor. Accordingly, we treat the ligand discrimination task as a problem of separating the affinity estimates for different ligands, and we construct the appropriate quantitative resolution metric. The metric goes beyond a simple comparison of the mean receptor output and describes precisely the effect of intrinsic proofreading noise on the ligand discrimination task. Our results reveal that once the intrinsic molecular noise is accounted for, the proofreading receptor’s resolution ability is lower than that of a nonproofreading receptor in most of the parameter space, including the regime of low ligand concentration and strong proofreading where KPR is commonly invoked. We show that proofreading improves resolution only in a narrow nonclassical area of the parameter space with a low number of proofreading steps and weak proofreading strength and that any improvement is much lower than what would be indicated by the HN metric. At best, the resolution saturates to a small improvement relative to a nonproofread receptor. At worst, the resolution decreases relative to a nonproofreading receptor as proofreading steps are added because KPR adds to and amplifies the molecular noise more strongly than it separates the means of the receptor outputs for different ligands. This effect raises new questions about the value of KPR as a mechanism for unconstrained enhancement of ligand discrimination performance and suggests that molecular noise limits the feasibility of KPR applications which rely on multiple proofreading steps to achieve the desired specificity (18, 21, 27, 59). Despite the limited number of quantitative experimental measurements of the KPR parameters, recent results showing that the effective number of the proofreading steps for the TCR is relatively small (13, 14) and the proofreading strength is moderate (14) are consistent with our results. The paper is structured as follows. In Section 1A, we briefly review the KPR model. In Section 1B, we show how the intrinsic molecular noise affects the output distribution in the KPR model and examine the signal-to-noise ratio (SNR) of the model. In Section 1C, we formulate the problem of specificity as the statistical estimation of ligand affinity and examine how receptor occupancy fluctuations affect the ability to identify ligands, an important precursor for considering the problem of ligand discrimination. In Section 1D, we introduce a metric for ligand resolution derived from statistical estimation theory. In Section 1E, we show that resolution enhancement by KPR—relative to a nonproofread receptor—is severely restricted due to the amplification of the relative noise by the KPR scheme. In Section 1F, we investigate alternative KPR schemes and metrics to show that our results are inherent to the proofreading mechanism itself rather than a particular choice of the noise model of a discrimination performance metric. We conclude with a discussion of our results in the context of previous work and their consequences for the understanding of the roles of KPR in cellular signaling.

We model a KPR receptor as a system with N + 2 states. These states represent the unbound receptor (state 0), the bound receptor in each of the N proofreading states, and the final signaling state of the receptor (numbered as N + 1, see Fig. 1A). Thus, N denotes the number of proofreading states, and N = 0 corresponds to a nonproofreading receptor. The ligand, present at concentration c, can bind an unoccupied receptor with the overall rate konc. Once bound by the ligand, the receptor can sequentially transition from the initial bound state to each subsequent proofreading state at a rate kf. The ligand can unbind from the receptor in any state at a rate koff. We assume that the ligand–receptor association rate kon is diffusion limited and is independent of the ligand identity so that only the unbinding rate is related to the ligand-specific binding free energy (i.e., koff ∝ e−Gbind/kBT). Note that the equilibrium dissociation constant, also known as the affinity, is Kd = koff/kon and in our case is uniquely determined by koff. In the terminal bound state, the receptor produces downstream signaling molecules at a rate kp, accumulating n output molecules over time t. In the following, we measure time in units of kp−1 throughout this paper. This model of receptor output is a common representation of a signaling mechanism by phosphorylation of intracellular molecules by a receptor-bound kinase (51, 60–63). Other choices of receptor output models are investigated in Section 1F. Ligand binding and unbinding events and the forward transitions between the sequential proofreading steps possess inherent randomness due to intrinsic thermal and chemical noise. Thus, the transitions between the states of the KPR model are inherently probabilistic with the average rates described above. Together with variability of the transition times and the Poisson noise in the chemical rate of production of the output molecules, this intrinsic noise leads to the distribution P(n|koff) of possible outputs, n, from the receptor in response to a ligand with the unbinding rate koff (Section 1F for models with minimal chemical noise in output production and transition time variability). At long times—defined as min(kofft, kft)≫1—or fast phosphorylation (kpt ≫ 1), the output distribution for an N-step receptor is well approximated by the normal distribution with the mean output from the receptor ⟨n⟩N and its variance (51). In some situations, the cell needs to make a ligand discrimination decision in a short time, before the output distribution has reached the normal distribution. This regime is analyzed in Section 1F, and the main results of the paper are recapitulated in this regime as well. The expressions for the mean and the variance can be derived from the Master equation of the KPR model (Materials and Methods): where N is the number of proofreading steps (N = 0 is a nonproofreading receptor), g = koff/kf is the proofreading strength, and x = c/Kd is the nondimensionalized ligand concentration. Eq. 1 expresses the fact that at long times, the number of output molecules produced by time t is simply the product of kpt and the probability to be in the signaling state PN + 1 = x/((1 + x)(1 + g)N). Eq. 2 reflects the fact that the variance in the number of output molecules is the sum of the variance of the Poisson process producing output molecules at rate kp and the intrinsic variance of the signaling state occupancy distribution due to fluctuations in binding and unbinding events (51).

As illustrated in Fig. 1 B and C, the width of the distribution P(n|koff) is an important factor controlling the ability to resolve two ligands. In this section, we explore how proofreading affects the width of the receptor output distribution. The SNR, , is commonly used to quantify the effect of noise in any signaling system and is higher when the signal distribution is more tightly centered about its mean. In Fig. 2A, we show the SNR for a KPR receptor as a function of the number of proofreading steps, N, and the proofreading strength, g = koff/kf. Critically, the SNR decreases with increasing proofreading strength (larger koff or smaller kf). This trend can be understood from Eqs. 1 and 2, which show that in the high g limit, the mean signal, ⟨n⟩N, scales like 𝒪(1/gN), while the noise, , scales like 𝒪(1/gN/2). Thus, the SNR scales as 𝒪(g−N/2), and strong proofreading yields a lower SNR. Physically, the decrease in the SNR with increasing proofreading strength stems from the fact that a higher proofreading strength increases the variance in both the receptor occupancy distribution and the Poissonian process producing output molecules (Eq. 2), amplifying the relative noise strength. We show in Section 1F that the increase in the receptor occupancy variance is inherent to KPR and sufficient to decrease the receptor’s signal processing performance. The SNR also reveals that the output distributions from different ligands increasingly overlap as the number of proofreading steps increases, despite the increased separation of their means, as shown in Fig. 2B. The difference in the mean outputs in response to each ligand (blue and orange lines) increases with N, illustrating the classical proofreading effect. However, despite this increasing separation between the means, the output distributions become progressively more overlapped [colored envelopes represent ]. The SNR therefore already provides an indication that the noise associated with proofreading makes ligand resolution harder. In the next section, we formulate receptor specificity in a precise mathematical manner and show that the intrinsic noise of the KPR scheme poses a challenge for KPR as a mechanism for enhanced specificity.

To give a precise meaning to the problem of ligand resolution, and thereby quantify the effects of noise on ligand resolution, we frame it as a problem of determining the ligand identity based on the observed receptor output, n (46, 51). This entails statistical estimation of koff using an estimator function . For normally distributed outputs considered here, choosing to be the maximum likelihood estimator (MLE)—that is, the estimate for which P(n|koff) is maximized—guarantees an unbiased estimate (i.e., 𝔼P(n|koff)[]=koff) with a minimum variance over different estimators (35, 48, 58). At long times, the MLE is well approximated by the inverse function ⟨n⟩−1(koff) obtained by solving Eq. 1 for koff (51, 58). As illustrated in Fig. 1C, the distribution of receptor outputs, P(n|koff), gives rise to the distribution of the estimates of koff, P(|koff) because the (deterministic) estimate depends on n sampled from the distribution P(n|koff) (51, 58). Since the estimator is unbiased, P(|koff) is centered at the correct estimate, = koff. The width of the distribution of estimates depends on both the width of the output distribution, P(n|koff), and the estimator function, , as illustrated in Fig. 1C. The Cramér–Rao lower bound provides an expression for the minimum expected variance of P(|koff) as 𝔼[Var()] ≥ −I−1, where I = −𝔼[∂2lnP(n|koff)/∂koff2] is the Fisher information, and all expectations are taken over P(n|koff) (58). Intuitively, the more information about koff that is contained in P(n|koff), the lower is the variance in the estimate . For a normally distributed receptor output, n, the Fisher information can be calculated as (51, 58) follows: For sharply peaked output distributions, the distribution of estimates, P(|koff), is well approximated by , where is the estimate variance. The estimate distribution for general N has variance: For strong proofreading, the variance of the estimate scales as 𝒪(gN). We and others have recently quantified receptor specificity by the ratio /σkoff, which is similar to the SNR but measures the relative width of the estimation distribution, P(|koff) (46, 51). The specificity ratio is inspired by the measure of accuracy of ligand concentration sensing first calculated by Berg and Purcell (34). Specificity is a critical prerequisite for good ligand resolution because greater specificity allows the ligand affinities to be closer together without their estimates being statistically indistinguishable. We demonstrate in Fig. 2C that, except for low N in the weak proofreading regime, g ≲ 1, the specificity (calculated from Eq. 4) decreases with an increasing number of proofreading steps, N. Notably, this detrimental effect of proofreading—compared to the nonproofread receptor (N = 0)—persists throughout the regime where weakly binding ligands are strongly proofread but strongly binding ligands are not (corresponding to 1 < g < 10 for the weakly binding ligand in Fig. 2C), which is the regime where KPR is commonly invoked—such as in TCR signaling. The decrease in specificity at large N is a consequence of the increased relative noise in the receptor output as proofreading steps are added (Fig. 2B). Note that we interpolate the results for noninteger N, which can be thought of as the effective number of proofreading steps for imperfect proofreading such as when the proofreading steps are not perfectly irreversible (14). Specificity is maximized for low but nonzero values of N in the weak proofreading regime because of a balance between the detrimental effect of molecular noise and the beneficial effect of increased sensitivity to changes in koff arising from the term |∂⟨n⟩N/∂koff| in Eq. 3. At very small g, the receptor spends effectively no time in the proofreading states, and the specificity is decreased again. The balance of these effects results in the maximization of the specificity for low but nonzero values of N in the weak proofreading regime; the location of this maximum in (g, N)-space also depends on the ligand concentration (SI Appendix, Fig. S1C). However, the enhancement of specificity compared to a nonproofread receptor is very moderate (less than 10-fold for any N with ζ = 0.1, SI Appendix, Fig. S1G) and is substantially lower than the classical and commonly invoked exponentially strong enhancement in the strong proofreading regime. Thus, the fact that specificity is only enhanced in this regime casts light on the performance and the purpose of KPR in such applications. Another approach to quantify the effects of noise on ligand discrimination is through the mutual information, ℐ(n, koff), between the input variable (ligand affinity), koff, and the receptor output, n (18, 22, 57). Although not the main thrust of this paper, it has been shown that mutual information is closely related to the Fisher information which we considered in this section (64, 65). As shown in SI Appendix, section A, proofreading also decreases the discrimination compared to nonproofread receptor when quantified through the mutual information. We have shown that both strong proofreading and a large number of proofreading steps generally lead to very noisy estimates for a single ligand affinity. The greater the spread in the estimate distribution for a single ligand (i.e., the specificity), the less well-resolved two different ligands are expected to be. In the next section, we give this notion a precise mathematical meaning by constructing a resolution metric for two ligands.

While the SNR and the specificity are useful for quantifying noise in KPR performance, both metrics describe the signaling statistics for only a single ligand. We now extend the idea of specificity to the problem of discriminating between two ligands that are distinguished by their respective ligand–receptor dissociation rates koff, 1 and koff, 2. The differences in the affinity between the ligands are characterized by the ratio ζ = koff, 2/koff, 1 (we define ζ ∈ (0, 1] so that koff, 1 > koff, 2). An appropriate resolution metric should encapsulate how well the difference in affinity between the two ligands, ζ, can be resolved in the presence of noise in the output distribution of each ligand. The standard metric for ligand resolution in the KPR model, first used by Hopfield and Ninio, uses the ratio of the mean outputs for the two ligands (1–3): where ⟨n⟩(koff, i, c, t) is the mean signal produced by time t by the receptor in response to a ligand with dissociation rate koff, i at fixed concentration c. In our convention, ζ < 1 and ηHN ≥ 1 since the weaker binding ligand forms less long-lived ligand–receptor complexes and therefore produces less output. The larger ηHN, the better these ligands can be discriminated. For concreteness, we choose ζ = 0.1 here and throughout this paper as informed by the typical value for self-peptides versus weak antigens to be separated by the TCR (3, 12, 14, 18, 66). As discussed above, the metric ηHN relies only on the average receptor outputs and does not account for the noise in the output distributions which we have shown to be so important for ligand discrimination. To take the noise into account, a resolution metric must quantify the distance between the two estimate distributions P(|koff,1) and P(|koff,2). Intuitively, two distributions are well separated when their means are separated by more than the sum of their standard deviations (e.g., Fig. 1B). This intuition is encapsulated in Fisher’s linear discriminant (67, 68), based on which we define our ligand resolution metric: where ⟨⟩ is the mean estimate for koff generated by ligand with koff, i, and is the variance about this mean. The averages in Eq. 6 are taken over P(n|koff, i), and we assume an unbiased estimator so that ⟨⟩ = koff (58). Note that in the limit of similar ligands, we have koff,2 ≈ koff,1 + δkoff and ≈ , so that where we have used the Fisher information of (Eq. 3), and in the last line, we have interpreted dkoff = koff,1 − koff,2. Thus, the Fisher linear discriminant formulated in koff-space is very closely related to the equivalent formulation in n-space that conveys the intuitive meaning about the ability to separate the two output distributions. Note that this metric is very different from the HN metric with a weighted cost term for receptor output attenuation (SI Appendix, section H). For both ηHN and ηFLD, a higher value indicates higher resolution. However, we show in the following section that KPR has severely restricted benefits for ligand discrimination, relative to a nonproofreading receptor, when noise is accounted for in quantifying the resolution.

For a KPR receptor with N proofreading steps, the HN resolution (Eq. 5) is using x = konc/koff, 1, g = koff, 1/kf is the proofreading strength for the weaker binding ligand, and ζ = koff, 2/koff, 1. In the limit of strong proofreading (1/g → 0) and low concentration (x ≪ 1), the resolution described by Eq. 7 scales as ∼ 𝒪(1/ζN + 1). At high concentrations (x ≫ 1) and strong proofreading, the resolution scales as ∼ 𝒪(1/ζN) due to the saturation of the receptor by the ligand (Section 1C). Thus, the enhancement of the resolution by N proofreading steps—relative to a nonproofreading receptor—is given by In the strong proofreading regime, this simplifies to , while in the weak proofreading regime, this simplifies to 𝒪(1). The behavior of ηHN expresses the well-known result that in the strong proofreading regime, additional proofreading steps exponentially enhance mean resolution relative to a nonproofreading receptor. However, when noise is included in the resolution metric, as in ηFLD, increasing the number of steps does not necessarily increase the discrimination strength because the separation of the means ⟨n⟩(koff,i) grows slower with N compared to the variances , obscuring the distinctions between ligands. A closed-form expression for the resolution of the KPR receptor using the metric is cumbersome (provided for completeness in SI Appendix) but can easily be evaluated numerically. In Fig. 3A, we show how ηFLD depends on the number of proofreading steps, N, and proofreading strength, g (fixing kf = 1 so that g = koff, konc = 0.1, and ζ = 0.1). Unlike ηHN, here an intermediate proofreading strength for the weaker binding ligand [g ∼ 𝒪(1)] and a few proofreading steps yield the greatest resolution, similar to the specificity in Fig. 2C, reflecting the mathematical similarity of the two metrics (discussed in Section D). A natural threshold for “good” resolution is around ηFLD ≈ 1, where the mean responses to each ligand differ by approximately the sum of the variances of each receptor output distribution. It is important to note that the maximum resolution in Fig. 3A is 𝒪(1), much lower than the exponentially separated means of the output distributions which the HN metric classically suggests. Although the location of the maximum of the resolution, ηFLD, depends on the ligand concentration c (SI Appendix, Fig. S1 D and E), the maximum is attained at weak proofreading strengths (g < 1 for both ligands) for all concentrations (see inset of SI Appendix, Fig. S1D). The commonly considered context for KPR is where weakly binding nonagonists are suppressed by proofreading (1 < gweak < 10), while strongly binding agonists are hardly suppressed (0.1 < gstrong < 1) (12, 14). In this biologically important regime, denoted with a dashed line in Fig. 3A, the resolution monotonically decreases as proofreading steps are added. We visualize the resolution enhancement relative to the nonproofread receptor by the ratio / in Fig. 3B, which shows that the resolution does not improve with additional proofreading steps. Instead, the relative resolution is a nonincreasing function of N as N becomes large (green lines in Fig. 3B). The relative resolution scales in the large N limit as which tends to zero for N → ∞—opposite to the scaling of / shown in Fig. 3B (black line). While there can be moderate resolution enhancement for a few proofreading steps in the regime of weak proofreading, the maximum resolution is of the same order as that of a nonproofread receptor, and any enhancement is ultimately abrogated as N → ∞ (SI Appendix, Fig. S1E). The “phase diagram” of resolution for a broad range of proofreading strengths is provided in SI Appendix, Fig. S1 E and G, which also shows that the resolution relative to a nonproofread receptor (given by /) does not improve exponentially with N; this is in stark contrast to the improvement indicated by the classical analysis with . The resolution at large N decreases because the relative noise of the receptor output increases with an increasing number of proofreading steps (Fig. 2A). Although the level of molecular noise of the receptor can be reduced in different variants of the KPR mechanism, in the next section, we demonstrate that even in models with minimal noise, the inherent stochasticity of the KPR mechanism still results in diminished resolution capacity at high N compared to a nonproofread receptor (SI Appendix, section E).

The KPR model considered so far has several sources of noise. These include the intrinsic noise inherent to the probabilistic nature of the sequential state transitions during proofreading, the variability of the residence times in different states, and the noise in the production of the receptor output n. Furthermore, for mathematical tractability, all rate processes have so far been assumed to be Poisson processes with exponentially distributed reaction times. In this section, we investigate alternative KPR schemes with diminished noise to show that our results are not an artifact of a particular choice of the (high noise) model. In the first scheme (SI Appendix, section D), to understand the role of the noise in the production of the output variable, we consider a KPR receptor which produces bursts of β output molecules immediately upon arriving in the active (N + 1th) state. This model can serve as a representation of a fast signaling output burst by a G-protein coupled receptor (51, 69, 70). Unlike a Poisson rate process with rate kp, this burst process does not add extra noise associated with the output generation. The mean number of the output molecules, ⟨m⟩, produced by this burst model is The variance in the number of output molecules as a function of N is derived in SI Appendix, section D. In the second alternative scheme, we investigate the role of variance in the residence time in the active state of the receptor. To this end, in addition to the burst output, the residence time in the active state, rather than obey an exponential distribution, is a constant τ = 1/koff for each ligand. Furthermore, in SI, we present another version of this model where all transition times between states are δ- rather than exponentially distributed. This hybrid model has been investigated using stochastic simulations (Methods). The results of these minimal noise models are shown in Fig. 4A. The burst model shows a significant decrease in the resolution for the addition of a single proofreading step and a steady decline as N increases. The model with a fixed residence time is less noisy and consequently shows better resolution than either burst or Poisson types of output with exponentially distributed times. Yet, both minimal noise models exhibit decreasing discriminating power (relative to a nonproofread receptor) with an increasing number of proofreading steps, similar to the original KPR model (Eq. 9). As above, the relative resolution of these models is worse in the strong proofreading regime (g = 10, solid line) compared to the weak proofreading regime (g = 3, dashed line).

In the previous sections, we have assumed that the measurement time is sufficiently long so that the distribution of the output, n, is well approximated by a normal distribution. However, in some situations—notably, the problem of antigen detection by TCR—it is possible that a cell must determine a ligand identity very quickly, ruling out the possibility for many binding-unbinding events to occur during the measurement time. We investigated this regime using stochastic simulation (SI Appendix, section G), and the results are shown in Fig. 4B, which shows that even at very short times, ηFLD decreases with N, as seen for long measurement times. Furthermore, in some cases—also including the problem of TCR discrimination—it can be argued that the primary purpose of KPR is the maximal suppression of false-positive errors from the weakly binding ligand (even at the expense of the somewhat increased false-negative rate for the strongly binding ligand). Spurious activation of the immune system by weak binders—which carries the potential of substantial damage to the host—may be more dangerous to the host compared to false-negative errors of occasional nondetection of the strongly binding ligand because the latter can be compensated for by other clonal cells that identify the non-self-antigen (71, 72). We quantify these notions through a metric defined as the ratio of the probability of false activation to the probability of all activation events during the measurement time window. We assume that activation occurs when the output variable n crosses an activation threshold n*, with probability P(n(koff, weak, c)> n*). The threshold is set to a value such that the probability of activation by the strongly binding ligand is 10%: P(n(koff, strong, c)> n*)=0.1 (the results are robust with respect to the specific choice of this value). In Fig. 4C, we show the probability ratio of false activation to the total activation as a function of the number of proofreading steps. This plot reveals that the ratio is minimized for a small number of proofreading steps and increases as N is increased further. The location of this minimum shifts to larger N as the proofreading strength decreases, and the minimum shifts to smaller N as the proofreading strength increases. We show the ratio for gweak = 10 and gweak = 3, which represent a typical range of proofreading strengths for the TCR according to recent experimental measurements (12–14). Overall, our results demonstrate that KPR has severely limited benefits for ligand discrimination, only providing moderate enhancement in ligand resolution relative to a nonproofread receptor in the nonclassical regime of fast proofreading and a few proofreading steps.

In this paper, we have shown that intrinsic molecular noise greatly affects the ability of a KPR-based receptor to discriminate between ligands and that resolution metrics must account for this noise—beyond the comparison of the mean outputs. To capture the role of the noise, we used the metric ηFLD (Eq. 6), which, unlike the classical metric ηHN (Eq. 5), incorporates information not only about the means but also about the variances of the distributions of the estimated ligand identity, P(|koff). Taking the noise into account and comparing the discrimination performance to that of a nonproofread receptor reveals that the benefits of proofreading for ligand discrimination are severely restricted to a moderate effect at a low number of proofreading steps and low proofreading strength. The central result of the classical analysis of KPR (based on ηHN) posits that the discrimination between two ligands improves with the number of proofreading steps exponentially as 𝒪(ζ−(N + 1)) in the strong proofreading regime, where ζ = koff, 2/koff, 1 is the ratio of the affinities of the two ligands to be distinguished. By contrast, in Sections 1B–E, we demonstrated that proofreading adds to and amplifies the effect of intrinsic molecular noise in both the signaling state occupancy and the downstream output, thereby making ligand discrimination more difficult. In all regimes, for large N, the resolution, , scales inversely with N and tends to zero as N → ∞. Even in the best-case scenario, the resolution enhancement relative to a nonproofread receptor is maximized at a low number of proofreading steps and provides only a modest enhancement over the nonproofread receptor, a far cry from the geometric improvement in resolution suggested by the classical KPR analysis. One of the principal biological signaling systems where the KPR model has been commonly invoked is T cell discrimination between foreign antigens and self-peptides (41). These applications of the KPR model commonly focus on the scenario where the number of proofreading steps is large (18, 21, 27, 59) and the weakly binding self-peptides are strongly proofread and attenuated because this is where the proofreading is most effective according to the classical, noiseless, HN metric. However, the large g, large N regime is also the regime where we showed the receptor output distribution to be most noisy, leading to the degradation in discrimination performance (Section 2B). While experimental measurements of KPR parameters in biological systems have remained difficult, recent work has started to provide experimental estimates of the effective number of proofreading steps for the TCR, estimated to be N ≈ 3 (13, 14), and estimates of the proofreading rate have been found to be kf ≈ 0.3s−1 (14). Thus, for a typical agonist binding time of 1–10 s, a typical nonagonist binding time of 0.3–3 s, and taking koff = τ−1 for the binding time τ (12), the TCR operates with moderate proofreading strength: 0.3 < g < 3 for the strongly binding ligand and 1 < g < 10 for the weakly binding ligand. Weakly binding ligands may be proofread even more strongly according to recent remeasurement of ligand affinities for the TCR (14). These experimental measurements of N and g for the TCR are thus consistent with the conclusions of our theoretical framework, although full comparison of the theoretical predictions with detailed experimental data is outside the scope of the present work and will be investigated in the future. The results of this paper are not an artifact of a particular choice of the noise model. The variability of the signaling output arises from several different components. The first two are the Poisson noise of the output production in the signaling state and the variability of the residence and arrival times to the signaling state. The third is the probabilistic nature of the binding–unbinding events and state transition steps, which are intrinsic to the proofreading mechanism. As shown in Section 1F, the results of this paper—the decrease of the resolution capacity of the KPR receptor relative to a nonproofread receptor as proofreading strength or the number of proofreading steps increases—hold true even in the minimal noise models where the signaling output is produced in discrete bursts and the receptor residence times are not exponentially but rather δ-distributed. Our results also hold true for fast decision-making processes which occur at short times when multiple binding-unbinding events are unlikely. The value of kinetic proofreading in biosynthesis processes, as in the original proposal for KPR, is not invalidated by our results because in this case either correct or incorrect moieties are added to a peptide or nucleic acid chain after a single binding event which reaches the final state (1, 2). The limited utility of KPR in ligand discrimination and identifiability stems from its two inherent properties: attenuation of the occupancy of the signaling state and the accompanying amplification of the effects of noise with each additional proofreading step. The problems posed by signal attenuation were already noted by McKeithan (3), who proposed a variant KPR scheme in which the receptor is stabilized in the final signaling state. Such stabilization can be introduced into our model by reducing the unbinding rate from the final proofreading state by a factor α < 1 (Methods and Eq. 11). The mean signal for a stabilized KPR receptor is While stabilizing the signaling state does increase the mean signal, the standard deviation still scales like 𝒪(g−N/2) so that the increased noise in the signal distribution still outpaces the increased difference in signals as proofreading steps are added (SI Appendix, Fig. S2C). Furthermore, improved KPR schemes such as adaptive sorting may provide solutions for KPR signaling in addition to those identified here and will be studied in further work (22, 40, 73). In this paper, we have constrained ourselves the comparison of KPR to nonproofread receptors with output modalities motivated by biologically common kinase phosphorylation and G-protein coupled receptor (GPCR)-type signaling. While maximal information on the ligand identity may be contained in the full series of ligand–receptor binding and unbinding events (35), not all of this information may be accessible in biological systems because of the constraints of the physically realizable signaling output modalities. Future work will investigate potential mechanisms of specificity enhancement based on fuller extraction of information from the time series of binding-unbinding events (50). Our results rest on the appropriate choice of resolution metric ηFLD that is widely used in statistical inference as a measure of the estimate quality and discriminability. However, in different biological situations, a different metric can be more appropriate—such as the fraction of the false-positive vs. total activations. As shown in Section 1F, the results of this paper are not an artifact of a particular metric choice and hold also for different measures of the discrimination quality; the fraction of false positives generally decreases with an increase in the number of proofreading steps, except for a moderate minimum at low N at low proofreading strength. It is important to distinguish these results from other works that have sought to balance ligand discrimination performance with the extrinsically imposed costs of proofreading such as input energy and decision speed, finding that different choices of the extrinsic costs optimize the proofreading at different numbers of proofreading steps (22, 39, 42–44, 74). These approaches typically posit that larger N will enhance ligand discrimination so long as the extrinsic costs can be met. By contrast, our results indicate that the discrimination capacity itself decreases with the proofreading strength due to the intrinsically stochastic nature of KPR. Significant stochastic fluctuations are an unavoidable consequence of the proofreading mechanism itself, and they dominate KPR signaling dynamics so that a mean-field metric like ηHN is insufficient for quantifying the specificity of the signaling process (SI Appendix, section H). This inadequacy of the mean-field metric echoes effects observed in other biological processes that are dominated by fluctuations, such as population extinction and first passage events (75, 76). Our results naturally lead to a question: If KPR commonly does not enhance the resolution of a sensory system, what other purpose might it serve in receptor systems? One possibility is that KPR might enable multiplexing of signals in receptor mixtures by producing multiple statistically independent outputs from different proofreading states (51, 77, 78). These outputs can be used to infer the presence of multiple ligands in a mixture (51), convey information about ligand concentration or other environmental conditions in addition to ligand identity (78), or perform other complex signaling processing tasks (79). Mounting experimental evidence indicates that the TCR (as a case study for KPR) may indeed produce multiple intracellular second messengers which may enable T cells to detect antigen quantity alongside antigen quality (79–83). Our results lay the ground for further theoretical work aimed at the understanding the effects of noise in multioutput receptor signaling pathways (79, 84–86).

The KPR receptor states are enumerated by index i, with i = 0 representing the unbound state and i = N the signaling state. The system is fully characterized by the pair of state variables (i, n) representing the current receptor state and the number of copies of the downstream signal molecule, respectively. The probability of the system to be in the state i at time t is (c, koff, t) which depends on the ligand concentration c and the unbinding rate koff. The dynamics of this probability distribution is described by the Master equation where kon is the association rate of the free ligands with an unoccupied receptor, kf is the rate at which the receptor transitions from state i to state i + 1, and kp is the rate of signal production from the terminal receptor state; we assume that the ligands are in a large extracellular volume so that c remains constant. A stabilized KPR receptor mentioned in Discussion has koff reduced by a factor α in the last line of Eq. 11. For the burst model of Section 2F, the terms involving kp are not present, and the term kf becomes (SI Appendix). The mean and variance for the burst model are solved in a similar manner to that of the phosphorylation model, described below. The probability distribution of the output signal is P(n|kkoff)=∑i(c, koff). Accordingly, the mean downstream output signal n is Taking the derivative with respect to time and using the master equation above, the summation over Pi reduces this expression to Reindexing the term leads to where PN is the steady-state occupancy probability for the Nth receptor state. The steady-state receptor occupancy probabilities Pi(c, koff) can be obtained by algebraically solving the system of equations resulting from setting all time derivatives and kp to zero in Eq. 11 and summing over the state variable n in . Finally, the mean signal downstream of an N-state KPR receptor is Using the expression for PN for a general N-step proofreading receptor that can be obtained from the Master equation Eq. 11 yields Eq. 1 for the mean signal (3): The case of N = 0 is a nonproofreading receptor. The same procedure can be used to compute the average output from the G-protein type KPR receptor, with the main difference being that the Master equation yields the relation ∂⟨m⟩/∂t = kfPN − 1 for N > 0 and ∂⟨m⟩/∂t = koncP0 for N = 0. The variance in the downstream signal, , can be obtained from the second moment of the signal using the identity = ⟨n2⟩−⟨n⟩2. The second moment is defined as Once again, taking the derivative with respect to time and substituting the Master equation yields a closed-form expression: Reindexing the term leads to A closed-form expression for the term 2kp∑nn can be obtained by solving the associated set of inhomogeneous first-order differential equations: for i ∈ [0, N]. Substituting the resulting expression for ⟨n⟩i into the differential equation for and integrating over time yields an expression for the second moment, which can then be used to find the variance . Using closed-form expressions for PN, the variance of the downstream signal is, in the long time limit min(kofft, kft)→∞, Eq. 2. For the G-protein type KPR receptor, the Master equation yields ∂⟨m2⟩/∂t = kf∑n(2n + 1) , but the procedure is otherwise the same. We view the T cell sensing problem as fundamentally a classification task based on estimating the ligand dissociation rate koff. A MLE for koff can be constructed from the distribution P(n|koff, c) = 𝒩(⟨n⟩, ). At long times, the limit of large numbers applies, making the distribution P(n|koff) narrowly peaked about its mean so that this MLE is well approximated by refs. 35 and 51: The inverse function ⟨n⟩−1 is easily obtained by simple algebraic manipulation of Eq. 1 for all choices of N considered in this paper. Variation in the observed signal n results in variation of the estimate for koff. We define a particular trajectory of the random variable n, which depends on the ligand dissociation rate, as n(koff, t). An expression for the estimation variance can be obtained by taking a first-order Taylor expansion of n about koff: where in the last line, we have defined δn = n(koff + δkoff, t)−n(koff, t) and squared the expression. Averaging this expression over the (time-dependent) signal distribution 𝒩(⟨n⟩, ) and interpreting quantities like ⟨δ⋅2⟩ as variances yields a simple form for the variance of the estimate distribution: This expression can be more rigorously derived by using the Cramér–Rao lower bound, which states that the variance in an unbiased estimator is bounded below by where ℓ(x; θ) is the log-likelihood of observation x given parameter θ (58). In the long-time limit of the KPR model, the likelihood for n is well approximated by a Gaussian, which allows one to write the right-hand side of the Cramér–Rao lower bound as the inverse Fisher information (51, 58). Means and variances for the semideterministic KPR models and the stabilized KPR receptor model were computed by Gillespie simulation (87). The fixed residency time in the final proofreading state was simulated by replacing the random time step in the Gillespie algorithm with a deterministic time step τ = 1/koff. For the minimal noise model treated in SI Appendix, the receptor either transitions to the next proofreading state or the ligand unbinds the receptor. The probability of either event is kf/(kf + koff) and koff/(kf + koff), respectively. Once the next reaction is selected in the simulation, the time for that reaction to happen is determined as either 1/kf or 1/koff, respectively.

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