Relating cellular signaling timescales to single-molecule kinetics: A first-passage time analysis of Ras activation by SOS

Significance

Timing in cellular signal transduction is generally observed as a functional property of an ensemble, but it is fundamentally governed by the reaction kinetics of individual signaling molecules. Here, we combined stochastic modeling and reconstitution experiments to show that the functional timescale of Ras activation by multiple Son of Sevenless (SOS) molecules is much shorter than the average activation time of individual SOS molecules at the membrane. These results illustrate how common mechanistic features of cellular signaling reactions can establish a complex relationship between individual molecular kinetics and the apparent functional kinetics of the system. They also underscore the importance of physiological protein copy numbers in establishing the functional output from a signaling module.

Abstract

Son of Sevenless (SOS) is a Ras guanine nucleotide exchange factor (GEF) that plays a central role in numerous cellular signaling pathways. Like many other signaling molecules, SOS is autoinhibited in the cytosol and activates only after recruitment to the membrane. The mean activation time of individual SOS molecules has recently been measured to be ∼60 s, which is unexpectedly long and seemingly contradictory with cellular signaling timescales, which have been measured to be as fast as several seconds. Here, we rectify this discrepancy using a first-passage time analysis to reconstruct the effective signaling timescale of multiple SOS molecules from their single-molecule activation kinetics. Along with corresponding experimental measurements, this analysis reveals how the functional response time, comprised of many slowly activating molecules, can become substantially faster than the average molecular kinetics. This consequence stems from the enzymatic processivity of SOS in a highly out-of-equilibrium reaction cycle during receptor triggering. Ultimately, rare, early activation events dominate the macroscopic reaction dynamics.

Son of Sevenless (SOS) is the primary Ras guanine nucleotide exchange factor (GEF) in a number of receptor tyrosine kinase signaling pathways, including epidermal growth factor receptor (EGFR), as well as T-cell receptor (TCR) signaling (1, 2). SOS is inactive in the cytosol, with its catalytic Ras-activating site blocked by autoinhibitory domains (1, 3–5). Upon receptor activation, SOS is recruited to the membrane via the adaptor protein Grb2 (6). This allows the autoinhibitory domains of SOS to interact with negatively charged membrane lipids, such as PIP₂, leading to structural rearrangements and subsequent release of autoinhibition (4, 5, 7, 8) (Fig. 1A). The final critical step toward SOS activation is the engagement of Ras at an allosteric binding site on SOS (1, 5, 9, 10). Until recently, very little was known from experiments about the kinetics of SOS activation. In earlier computational modeling studies of Ras activation (11, 12) and MAPK network signaling (13, 14), SOS activation was generally assumed to be fast and rapidly reversible.

Fig. 1.

SOS activation at the membrane. (A) Receptor-dependent SOS activation. SOS is initially inactive and autoinhibited in the cytosol. SOS is recruited to the membrane by binding to the phosphotyrosines of LAT (yellow p) via Grb2. Interactions with negatively charged lipids like PIP₂ assist in the release of autoinhibition. Next, the engagement of Ras at the allosteric site of SOS increases its membrane dwell time and enables processive turnovers of hundreds of Ras. (B) Kinetic scheme of SOS activation shown in A. (C) Activation time distribution ($p_{\tau }$) of a single SOS molecule for the case of one kinetic intermediate,$N=1$. The parameters were derived from single-molecule SOS activation assay (8): $k_N=0.02$ s⁻¹, $k_{-1}=0.016$ s⁻¹.

Single-molecule studies have now provided detailed measurements of SOS membrane binding kinetics and of the time delay between SOS recruitment to the membrane and initiation of its GEF activity (8, 9, 15–17). Once activated, SOS is highly processive and can catalyze nucleotide exchange for hundreds of Ras molecules during a single dwell period on the membrane (9). In the cellular context, SOS has been observed to follow an essentially one-way trafficking process in which the unbinding rate is so slow that many receptor-recruited SOS molecules are internalized prior to unbinding from the membrane (16). The system is far from equilibrium. The mean time to activation after SOS recruitment to the membrane is also surprisingly long (∼60 s), and the rise-and-fall shape of the activation time distribution reveals the presence of a rate-limiting kinetic intermediate. These distinctive timing features of SOS activation enable a kinetic proofreading mechanism, which may play a role in suppressing noise in the signaling pathway (8, 18, 19). However, the long timescale for individual SOS molecules to activate is seemingly at odds with cellular signaling timescales, which can be much faster (20–22).

Here, we examine the apparent discrepancy between cellular and molecular signaling timescales through detailed kinetic analyses of the SOS activation mechanism and the subsequent process of Ras activation. Fundamental to this problem is how the ensemble kinetics, which establish cellular signaling dynamics, can differ from the average of the individual molecular kinetics. In the case of SOS activation, the functional timescale is set by the time for the first few SOS molecules to activate, not the average time for each molecule to activate, and is thus intrinsically dependent on molecular copy number in the reaction system. As a result, hundreds of SOS molecules involved in a single reaction system can reduce the functional signaling timescale by more than an order of magnitude, even while there is no change to the individual molecular kinetics. This effect stems from the fact that, once activated, a single SOS molecule can processively activate hundreds of Ras molecules. These defining features of Ras activation by early activating SOS molecules resemble a class of stochastic processes known as the first-passage problems.

First-passage time problems describe stochastic processes in which the timing characteristics are determined by threshold crossing events. A classic example is the time it takes for a randomly diffusing molecule to first travel a defined distance, as opposed to how far it travels on average (23). In recent years, modeling biological processes as first-passage problems has drawn attention in areas including gene expression (accumulation of protein copy number that triggers cellular response) (24) and signal transduction (translocation of transcription factors into the nucleus) (25). In the case of signal transduction, signaling molecules appear to compete with each other in a race to transduce signals (25). These modeling studies represent conceptual advances in the understanding of timing in biomolecular processes driven by threshold crossing events. Under such mechanisms, early triggering events can disproportionately shape the system dynamics.

In this study, we argue that the signaling module of Ras activation by SOS is also effectively governed by a first-passage time mechanism. Part of the challenge in relating first-passage modeling studies to experiments is that, in most cases, only a final timing event marked with macroscopic changes is measurable (e.g., accumulation of proteins leading to cell lysis), while the detailed molecular kinetics and activation mechanism are not generally experimentally defined. Here, we reconstruct the ensemble dynamics from directly measured single-molecule kinetics, such that all parameters are experimentally defined. We then perform mesoscopic experiments with controlled numbers of SOS molecules to test model predictions on real systems. This approach is enabled by recently developed single-molecule array techniques (8, 9, 15, 16) and presents a rare opportunity to experimentally verify the existence of a first-passage mechanism controlling Ras activation by SOS. Quantitative understanding of this effect is critical for relating bulk biochemical data (an average over essentially infinite copy number) and single-molecule measurements to functional kinetics in cellular systems. Moreover, the relatively low protein copy number per cell of SOS in the MAPK pathway (26) implicates SOS as a kinetic bottleneck and suggests that timing and delays at this step may propagate downstream in the signaling cascade. The processivity of SOS and the resulting saturation behavior in the activation of Ras likely reflect common phenomena in cellular signal transduction. Such effects underscore the importance of the specific copy numbers of molecules involved in cellular signaling, establish a relation between protein copy number and signaling dynamics, and highlight challenges for both quantitative modeling of these processes and their experimental reconstitution.

Model: First-Passage Time Analysis of SOS Activation

Recent experiments of Ras activation by SOS have revealed multiple kinetic characteristics that warrant reconsideration of modeling this signaling module. Single-molecule analysis reveals that SOS activation (e.g., release of autoinhibition) involves a slow, rate-limiting kinetic intermediate (8), Ras activation by SOS (e.g., catalyzed nucleotide exchange) is highly processive (9, 15), and SOS exhibits very slow unbinding from the membrane once activated in both reconstitution studies and live cells (16). These characteristic features of SOS ensure that the SOS–Ras activation process operates far from chemical equilibrium and differs significantly from early assumptions of fast membrane binding and SOS activation kinetics.

In the following, we consider the subsequent timing of signal propagation after recruitment of SOS molecules to activated membrane receptors. We ask how the number of recruited SOS molecules, which will generally be a small fraction of cytosolic SOS and proportional to the signal strength (e.g., number of activated receptors), affects the timing of SOS and subsequent Ras activation. The key experimentally determined features of the SOS activation mechanism that frame this analysis are as follows: 1) SOS is inactive in the cytosol and activates only on the membrane (1, 3–5), 2) SOS activation requires progression through at least one rate-limiting kinetic intermediate (8, 19), 3) activated SOS is highly processive (9, 15), and 4) the activation reaction sequence is far out of equilibrium and follows an essentially one-way trafficking process (9, 15, 16). The activation process of SOS is represented by the kinetic scheme depicted in Fig. 1B. Progression to the activated state ($S^{*}$) and dissociation to the cytosol ($\varnothing$) are both treated as irreversible transitions in this model. This choice is based on the experimental observation that, once activated, SOS remains activated for an extended period of time (>100 s) without dissociating from the membrane (9, 15, 16) and that the recycling time for dissociated SOS to stably reassociate with the membrane is long compared with the timescale to relax back to the fully autoinhibited state (8). At each state prior to activation, SOS may either progress to the next state or dissociate from the membrane. These intermediate transitions are also treated as irreversible, for the sake of simplicity, and making them reversible does not change the qualitative behavior of the model as long as the system is far from equilibrium and there is a net flux toward the activated state (SI Appendix, SI Text and Fig. S1A).

The activation kinetics for a single SOS molecule can be described by the following chemical master equation (CME):

$$\{\begin{array}{l}{\dot{p}}_0=-(k_1+k_{-1})p_0,\\ {\dot{p}}_i=-(k_{i+1}+k_{-1})p_i+k_i{p}_{i-1}, i=\{1,2,\dots ,N\},\\ {\dot{p}}_{*}=k_A{p}_N\end{array}$$ [1]

where $p_i\left(t\right)≔p(S=S_i,t)$ is the probability of the SOS molecule being in state $S_i$ at time $t$, and $N$ denotes the number of kinetic intermediates; $\left\{S_i\right\}$ are the inactive, intermediate states, and $S^{*}$ is the activated state. Eq. 1 can be solved as a linear system of equations, $\dot{\mathbb{\mathbb{P}}}\left(t\right)=\mathbb{A}\mathbb{P}\left(\mathit{t}\right)$, where the solution, $\mathrm{\mathbb{P}}(t)={\left(\begin{array}{ccccc}{p}_0& p_1& \cdots & p_N& p_{*}\end{array}\right)}^{\mathrm{⊺}}$ (23), describes the probability distribution of SOS among each of its states. To gain intuition on the behavior of this system, we consider the simplifying case where the rate constants for transitions through the kinetic intermediates are identical ($k_1=k_2=\dots =k_N$). In reality, these intermediate kinetic transition rates may differ, but the qualitative system behavior does not depend on this as long as there is at least one rate-limiting kinetic intermediate (SI Appendix, Fig. S1B) (8, 19). The first-passage time for a single SOS molecule to reach activation, $\tau$, is then defined as the minimum time for a single SOS molecule to reach the activated state ($S^{*}$) starting from the initial membrane-recruited state ($S_0$):

$$\tau ≔\text{min} \{t:S=S^{*}|(p_0\left(0\right)=1,p_{i\ne 0}\left(0\right)=p_{*}\left(0\right)=0)\}.$$ [2]

Solving Eqs. 1 and 2 gives the activation time distribution for a SOS molecule:

$$p_{\tau }\left(t\right)=\frac{d{p}_{*}}{dt}=\frac{1}{N!}{k}_N^{N+1}{t}^N{e}^{-\left(k_N+k_{-1}\right)t},$$ [3]

which exhibits the characteristic rise-and-fall shape of a gamma distribution (Fig. 1C). This distribution has been experimentally measured for receptor-mediated SOS activation, and the first-passage activation times are well fit with a single rate-limiting kinetic intermediate ($N=1$) (8) (Fig. 1C). The intermediate states of SOS are related to the known biochemical transitions necessary for SOS activation: 1) membrane recruitment ($S_0$), 2) autoinhibition release ($S_1$), and 3) engagement of Ras at the allosteric pocket of SOS ($S^{*}$). Hereon we focus on the case of one dominant kinetic intermediate for SOS activation.

Next, we seek to construct the ensemble signaling timescale, $T$. In a cellular setting, activation of Ras is opposed by GTPase-activating proteins (GAPs), which catalyze the hydrolysis reaction of activated RasGTP back to the inactive RasGDP state. Net activation of Ras is only achieved when the SOS-catalyzed nucleotide exchange of RasGDP to RasGTP exceeds the hydrolysis rate. Deactivation by GAPs implicitly maps to an effective threshold copy number of SOS molecules that must be activated within a given reaction environment to achieve net Ras activation. Since SOS is highly processive and remains trapped in its activated state for extended times, the effective ensemble signaling timescale is primarily governed by the first-passage time probability to get enough SOS molecules activated.

In the model, we define T as the minimum time required for activation of $j$ molecules of SOS given initial recruitment of $n$ molecules at state $S_0$ (Fig. 2A):

$$T_{j,n}≔\min \{t: n^{*}=j\text{|}{n}_0(t=0)=n,n_{i\ne 0}(t=0)=n^{*}(t=0)=0\},$$ [4]

where the ensemble state is described by $\psi ≔\{n_0,n_1,\dots ,n_n,n_{*}\}$, and $n_i$ is the copy number at the $i$ state. The timing of multi-SOS activation is inherently stochastic, depending on the progression of each SOS molecule through its intermediates (Fig. 2B). A simple method to evaluate $T_{j,n}$ is the argument of survival probability (25): at time $t$, exactly one molecule activates while $(j-1)$ molecules have activated prior to time $t$ and $\left(n-j\right)$ molecules have not yet reached activation. This reasoning leads to the following expression for the signaling timescale distribution, $p_{T_j,n}(t)$:

$$p_{T_{j,n}}\left(t\right)=\frac{n!}{\left(n-j\right)!\left(j-1\right)!}\cdot p_{\tau }(t){\left(\underset{0}{\overset{t}{{\displaystyle \int }}}{p}_{\tau }\left(t^{\prime }\right)d{t}^{\prime }\right)}^{j-1}\cdot {\left(1-\underset{0}{\overset{t}{{\displaystyle \int }}}{p}_{\tau }\left(t^{\text{″}}\right)d{t}^{\text{″}}\right)}^{n-j}.$$ [5]

Fig. 2.

Signaling timescale of SOS activation from single-molecule kinetics. (A) Signaling timescale is defined as the minimum time for $j$ copy number of activated SOS given $n$ initial SOS recruitments. Each SOS molecule follows the activation mechanism depicted in Fig. 1. (B) Stochastic trajectories of multiple SOS activation assuming one kinetic intermediate $\left(N=1\right)$. (C) Signaling timescale distribution ($p_T$) of A from Eq. 5 (dashed line) and stochastic simulations. The simulations were performed with the Gillespie stochastic algorithm with SOS copy number $n=100$ and activation threshold $j=3$. The histogram compiled 20,000 runs. (D) Activation defined by a RasGTP threshold with negative pressure from GAPs. The signaling timescale here is defined as the minimum time to accumulate $j$ Ras molecules at the R* state. Production of RasGTP was simulated as a first-order reaction with $k_{\mathit{cat}}^{+}=1$ s⁻¹. RasGTP hydrolysis by GAPs was simulated as a first-order reaction with $k_{\mathit{cat}}^{-}=0.2$ s⁻¹. The simulations recorded the minimum time for the $R^{*}$ copy number to reach a threshold of 20 molecules. The protein copy number of S* at the time of threshold crossing was used to compare with Eq. 5 (in this example, $j=6$). (E) Signaling timescale of Ras activation without GAPs. Note that the width of timescale distributions for both SOS activation and Ras activation with GAP are well described by a first-passage model.

Note that we construct $p_T{}_{j,n}\left(t\right)$ from the single-molecule activation time distribution, $p_{\tau }(t)$, which can be experimentally determined. An underlying assumption of this approach is that each SOS molecule activates independently. This is justified when focusing on the initial activation timescale of only a handful of SOS molecules amid saturating numbers of RasGDP molecules, which is the case for the onset of Ras activation. At later timepoints, as RasGTP levels build up, RasGTP-mediated positive feedback in SOS recruitment and activation could further accelerate activation of Ras (11). Here, we focus on the initial activation timescale as this appears to be rate limiting. Detailed description of the model as well as expansions considering other variations, including those from membrane recruitment and protein expression, are discussed in SI Appendix, SI Text.

Results

Modeling SOS and Ras Activation as First-Passage Processes.

Results from stochastic simulations of the first-passage time of SOS activation for the case of one kinetic intermediate ($N=1$) are plotted in Fig. 2C (parameters summarized in SI Appendix, Table S1). The simulated timescale distribution is in good agreement with Eq. 5. The signaling timescale distribution is highly non-Gaussian with a skewed long tail toward longer times. This analysis of SOS activation can be extended to describe Ras activation utilizing the measured catalytic rate of SOS on membranes (9, 15). The first-passage time to reach a certain level of RasGTP accumulation is also effectively described by Eq. 5 as long as opposing GAP activity is also considered (Fig. 2 D and E). Without GAPs, the first-passage time distribution of Ras accumulation is narrower than Eq. 5 (Fig. 2E) because early activated SOS contributes more Ras turnover than later ones. However, under physiological conditions, steady deactivation by GAPs is essentially always present. The functional signaling timescale for initiation of Ras activation is thus set by the time to activate an effective threshold copy number, $j$, of SOS molecules, which are sufficient to overcome GAP activity.

Protein Copy Number Effects on Signaling Timescale and Consistency.

The copy number of initially recruited SOS molecules ($n$), which in a cellular setting would be proportionate with the number of activated receptors, affects both the shape and the average of the signaling timescale distribution. Increasing the copy number from 1 to 100 reduces the mean signaling timescale from 56 to 7 s (Fig. 3A). Here, the ensemble activation time is biased by early activating SOS molecules and not reflective of the average molecular activation kinetics. This mechanism relies on the individual molecular activation times being broadly distributed, with many molecules activating much more quickly than the average—as is the experimentally observed case with SOS. To add context, we consider the hypothetical case in which SOS activation involves more intermediates. Increasing the number of kinetic intermediates results in a narrower activation time distribution for SOS autoinhibition release (19). As a result, early activating molecules are more rare and play a less prominent role. This reduces the effective coupling between signaling timescale and copy number of activated molecules that the first-passage time activation mechanism establishes (SI Appendix, Fig. S2). Such first-passage time effects are most pronounced when activation times are broadly distributed.

Fig. 3.

Protein copy number effects on the signaling timescale distribution. (A) The signaling timescale distribution as a function of $n$ for $j=1$. (B–D) CV² of the timing of SOS activation as a function of protein copy number and activation threshold. The white lines are line scans plotted in C and D. The inset percentages of C and D mark the activation probability of the shaded area.

In cells, the timing of MAPK activation from EGFR triggering exhibits remarkable consistency with minimal cell-to-cell variations (27). This accuracy in timing likely contributes to the robust oscillations of Ras/Erk activities and gene expression that have been observed (28). With SOS as a key signaling intermediate between EGFR and MAPK, the timing consistency and robustness of SOS activation is of central importance to establish overall EGFR signaling time characteristics. In the analysis we present here, the timing consistency of SOS activation is associated with the shape of the signaling timescale distribution, which we quantify with the squared coefficient of variation (CV²) (Fig. 3 B–D and SI Appendix, Fig. S3). For the stochastic time variable $t$, CV² , where “” denotes the average normalized by activation probability, $P_{\mathit{act}}$ (since only threshold-crossing events are calculated in this average). This quantity of variance divided by the mean squared is also called the randomness parameter in the context of single-molecule enzyme kinetics (29). For a Poisson process, CV² is unity; lower CV² values correspond to a more clocklike timing behavior.

The timing consistency of SOS activation is modulated by both the copy number ($n$) and activation threshold ($j$) (Fig. 3B). In the following, we focus on the case of robust activation ($n\gg j$ such that $P_{\mathit{act}}\to 1$, SI Appendix, Fig. S3B). For a fixed threshold, an increase in SOS copy number generally reduces CV² (Fig. 3C and SI Appendix, Fig. S3 C–E), since sampling of early activating events becomes more frequent. In contrast, for a fixed copy number, an optimal intermediate-low threshold exists that minimizes CV² (Fig. 3D and SI Appendix, Fig. S3 F–H). For very low thresholds ($j\to 1$), activation relies on a rare, early event such that variation is large. For high thresholds ($j\to \text{max}\left\{j_{P_{\mathit{act}\sim 1}}\right\}$), threshold crossing is limited by the arrival of the last few, also rare, events. Of note, the minimum CV²—corresponding to the condition of high copy number and intermediate-low threshold ($n\sim 100, j\sim 10$)—is about 0.05, which is close to the timing precision in cells where a CV_cell²∼0.02 has been measured from EGFR stimulation to maximum Erk translocation (27).

In addition, the CV² analysis shows that any modulation of the copy number and activation threshold, such as cell-to-cell variation in protein expression or mutations in SOS, Ras, or GAPs, alters the timing consistency of Ras activation. Of particular interest are Ras mutants identified in cancers that disrupt SOS or GAP interactions. For example, G12 Ras mutants prevent hydrolysis of RasGTP by GAPs (30), which, in the first-passage time model, corresponds to a lower activation threshold that results in inconsistent Ras activation timing and possibly noisy Erk oscillations. These timing features reflect dynamical aspects of signal initiation by Ras mutants and are additional to the canonical view of their steady-state activation characteristics (30).

Experimental Measurement of SOS Reaction Times in Reconstituted Membranes.

To compare the model to experiments, we modified a recently developed single-molecule SOS activation assay on supported membranes (8). In these experiments, the membrane receptor scaffold to which SOS is recruited, phosphorylated LAT (pLAT), and the substrate, RasGDP, were tethered onto supported membranes at densities of about 800 and 500 molecules/µm², respectively (Fig. 4A). pLAT and Ras were laterally fluid with diffusion coefficients of about 1.5 and 3 µm²/s, respectively, measured by single-particle tracking (SPT) (19). The reaction was initiated by addition of 1 to 20 nM full-length SOS, 20 nM Grb2, and 120 µM GTP in the solution. The SH2 and SH3 domains of Grb2 bridge SOS to pLAT by binding to the phosphotyrosine residues of pLAT and the proline-rich domain of SOS, respectively. Under these experimental conditions, macroscopic LAT:Grb2:SOS condensates (from multivalent phosphotyrosine interactions) do not readily form (19). Supported membranes included 2% PIP₂ lipids, which facilitate autoinhibition release of SOS (4). Ras activation was monitored using a fluorescently labeled Ras binding domain (RBD) of Raf-1 that selectively binds RasGTP with fast on/off kinetics (8, 31). Membrane-localized RBD was visualized by total internal reflection fluorescence (TIRF) microscopy. This assay provides a real-time readout of the RasGTP densities with temporal resolution of about 1 s (8).

Fig. 4.

Reaction timescale of SOS activation on supported membranes. (A) Schematic of membrane reconstitution experiments. pLAT and RasGDP were tethered onto supported membranes containing PIP₂ lipids. Injection of SOS, Grb2, and GTP initiated nucleotide turnover of Ras, detected by RBD. SOS and RBD were fluorescently labeled with Alexa Fluor 555 and Alexa Fluor 647, respectively. The image shows the top view of the corralled membranes. (B) Time series of Ras activation at different SOS concentrations. The field of view shows nine corrals. (Scale bars, 12 µm.) (C) The left plot shows nine trajectories of RasGTP production, with one example emphasized in red. In the case of saturating Ras density, the slope of a RasGTP trajectory is proportional to the number of activated SOS. The dashed line shows the chosen slope for analysis. The triangles indicate approximate times when an additional SOS activated. (D) Reaction timescale distribution. The dashed line is a global fit to Eq. 5 with a fixed ratio between each SOS titration (18:6:1). The fitted values were $n_1=144$ (pink), $n_2=48$ (purple), $n_3=8$ (blue), and $j=2$. Each histogram is extracted from 120 to 200 corrals.

Reconstituted reactions were performed on supported lipid bilayer microarrays with 4 × 4 µm corral size. Each corral in the array is isolated by metal barriers, prefabricated onto the underlying substrate that confine membrane-associated components two-dimensionally but pose no impediment to exchange with the bulk solution (8, 9, 32). Variations of LAT densities across the corrals were small (SD = 3.9%) (SI Appendix, Fig. S4); the numbers of membrane-recruited SOS molecules were titrated roughly from 5 to 100 per corral. Several hundred corral reactions were monitored in parallel, enabling experimental measurement of the Ras activation timescale as a function of SOS copy number (Fig. 4 A and B and Movie S1).

The RasGTP density trace reflects the number of activated SOS molecules, and a typical trace is shown in Fig. 4C. Discrete changes in the rate of RasGTP production, corresponding to single-molecule SOS activation events, can be resolved (marked with triangles in the right panel of Fig. 4C). Since activated SOS does not desorb from the membrane on experimental timescales (dwell times $\ge 100$ s) (9, 15, 16), the slope of the RasGTP density curve at early timepoints ($<200$ s) is reflective of the number of activated SOS molecules ($n_{*}$). At high RasGDP densities (>100 molecules/µm²), previous single-molecule studies on supported membranes have shown that activated SOS is mostly saturated by Ras, and the reaction is limited by the $k_{\mathit{cat}}$ of SOS (9, 15). In this case, the net rate of Ras activation remains proportional to the number of activated SOS molecules. Nonetheless, we also examined the case where the RasGTP production is limited by enzyme-substrate binding and reached the same conclusion in the following analysis (SI Appendix, Fig. S5). The first-passage time of SOS activation from a single corral can be extracted by analyzing the minimum time for RasGTP production to reach a defined rate.

We test if the first-passage time distribution in SOS activation is the primary governor of Ras activation kinetics by quantitatively comparing the experiment and the model. For this test, we define a RasGTP production rate threshold of 7 molecules/s, corresponding to two activated SOS molecules per corral ($j=2$), to extract and compile the experimental first-passage times to reach this threshold from individual trajectories under different SOS concentrations (Fig. 4C). For this experimental test of the model, it makes no difference which threshold RasGTP production rate is chosen; here, we chose a low threshold such that any positive feedback from RasGTP-mediated SOS recruitment from the solution can be safely omitted. The experimentally measured first-passage time distributions for multiple SOS concentrations are plotted in Fig. 4D (Movie S1).

The first-passage kinetic model relates the copy number of recruited SOS molecules ($n$), which is experimentally varied by changing SOS concentration and the activation threshold ($j$) to the corresponding first-passage time distribution. Although individual experimental distributions can be fitted to obtain $\left\{n,j\right\}$, this is unreliable because multiple combinations of $\left\{n,j\right\}$ yield distributions that are experimentally difficult to distinguish. This degeneracy is broken by performing a global fit of several measured distributions spanning a range of SOS concentrations. The global fit is achieved under the following constraints: the ratio of SOS concentrations between samples was fixed to their experimentally set values, and the single-molecule activation time distribution of SOS was held constant at its experimentally measured distribution (8). These experimentally established constraints reduced the floating variables from $\{n_1,n_2,n_3,j;k_N,k_{-1}\}$ to $\{n_{\mathit{global}},j\}$ such that the fitting results directly examine the effect of SOS titration ($n$) on the experimental first-passage time distributions. The global fit results for the distributions are plotted (dashed lines) in Fig. 4D.

Notably, the fit independently retrieves $j=2$—the predefined threshold for extracting the individual first-passage times—solely from the distributions, without any explicit input of the threshold. An additional, independent set of experimental data is provided in SI Appendix, Fig. S6. Quantitative agreement between the kinetic model and experimental data suggests that the molecular mechanism of receptor-mediated SOS activation, and subsequent Ras activation, is dominated by an out of equilibrium, first-passage time process.

Discussion

This study demonstrates how the ensemble dynamics of Ras activation by SOS are dominated by early-activating SOS molecules and can be much faster than the average molecular kinetics. This dependence is not a result of cooperativity or any other effect that alters the kinetics of the individual molecules. Rather, it is a result of stochastic variation in the activation time of individual SOS molecules and their high degree of processivity once activated. The effect also stems from the fact that the functional response time is determined by a threshold-crossing process governed by competition between Ras activation and deactivation reactions. The overall signaling timescale is set by the first-passage time probability to get enough SOS molecules activated. As a consequence, ensemble signaling timescales intrinsically depend on the copy numbers of molecules involved as well as the full activation time distribution of each molecule—not just the average molecular kinetics (Fig. 5). With more molecules, the active molecules are progressively skewed toward rare, early activators in the overall distribution (Fig. 3A). A corollary of this result is that the strength of an incoming signal, which is represented by the number of activated receptors, is converted into a timescale of activation more so than an amount of Ras activation. Such effects may underlie the apparent binary signaling through the MAPK pathway that has been experimentally identified in some circumstances (11). Additionally, a first-passage mechanism, in which a small fraction of SOS molecules dominate Ras activation, reduces the timing variation of a signaling ensemble compared with the intrinsic variation of single SOS molecules (Fig. 3D, CV² $\sim 0.35\to 0.05$). Such a mechanism may partly contribute to the timing consistency of Erk activation in live cells (CV² $\sim 0.02$) (27).

Fig. 5.

Activation defined by the population average versus early events. A population averaging mechanism leads to a reaction timescale that is largely independent of molecular copy number (Left); a first-passage mechanism leads to a faster timescale with more participating molecules (Right).

Controlling Ras activation by rare, early activating SOS molecules—as opposed to threshold crossing by the population average—offers conceivable benefits for signal transduction. We specifically consider the case of Ras signaling in the context of TCR triggering at low receptor signals (by a handful of agonists), which is likely the physiological stimulation condition (33). The signaling system must exhibit a high degree of sensitivity and amplification with sufficient robustness. Processivity of SOS provides amplification, in which one molecule of SOS may activate hundreds of Ras molecules. Slightly counterintuitive for a signaling mechanism that relies on rare events, this sensitivity is relatively robust to protein expression variation since activation only depends on early activators and not the entire average. In contrast, activation based on the population average is relatively prone to crippling by variations in protein expression [the CV² of SOS copy number in cells is about 50% (26)].

These realizations of kinetic details in the SOS-Ras system expose general challenges for both computational modeling and experimental reconstitution studies. Neither bulk biochemical kinetics nor single-molecule kinetics are likely to directly reflect the actual ensemble kinetics of the small copy numbers of molecules within a mesoscopic intracellular reaction system. Quantitative descriptions of cellular signaling systems will require accurate knowledge of individual molecular properties as well as the relevant copy numbers under physiological reaction conditions (Fig. 5); stochastic variation cannot necessarily be ignored. However, we also see here that high precision quantitative description of a complex biochemical signaling reaction can be achieved with sufficiently detailed experimental determination of these parameters. Overall, the reaction topology in this example is common to numerous biological signaling systems; this general phenomenon is likely to be widespread.

Materials and Methods

Modeling SOS Activation.

Construction of the kinetic model for Ras activation by SOS is described in the Model: First-Passage Time Analysis of SOS Activation section and SI Appendix. All simulations used Gillespie stochastic simulation algorithm (34). Parameters are summarized in SI Appendix, Table S1.

Reconstitution of SOS Activation.

Grb2-mediated Ras activation by SOS was reconstituted in supported membranes using a protocol as described previously (8), with modifications of using patterned membranes of a larger grid size (4 × 4 µm) and higher SOS concentrations. Detailed procedures for purifying proteins, preparing supported membranes, functionalizing lipids with Ras molecules, triggering SOS activation, and acquiring the images are described in SI Appendix.

Data Availability

All study data are included in the article and/or supporting information.