Presence or Absence of Ras Dimerization Shows Distinct Kinetic Signature in Ras-Raf Interaction

Abstract

Initiations of cell signaling pathways often occur through the formation of multiprotein complexes that form through protein-protein interactions. Therefore, detecting their presence is central to understanding the function of a cell signaling pathway, aberration of which often leads to fatal diseases, including cancers. However, the multiprotein complexes are often difficult to detect using microscopes due to their small sizes. Therefore, currently, their presence can be only detected through indirect means. In this article, we propose to investigate the presence or absence of protein complexes through some easily measurable kinetic parameters, such as activation rates. As a proof of concept, we investigate the Ras-Raf system, a well-characterized cell signaling system. It has been hypothesized that Ras dimerization is necessary to create activated Raf dimers. Although there are circumstantial evidences supporting the Ras dimerization hypothesis, direct proof of Ras dimerization is still inconclusive. In the absence of conclusive direct experimental proof, this hypothesis can only be examined through indirect evidences of Ras dimerization. In this article, using a multiscale simulation technique, we provide multiple criteria that distinguishes an activation mechanism involving Ras dimerization from another mechanism that does not involve Ras dimerization. The provided criteria will be useful in the investigation of not only Ras-Raf interaction but also other two-protein interactions.

Significance

Ras is a small protein that regulates many important cellular functions, including cell proliferation, growth, and differentiation. Because of its ubiquity in cell regulation, mutated Ras causes many diseases, including cancer. Despite significant efforts, there is still no universally effective drug against malfunctioning Ras. This failure partially stemmed from incomplete understanding of the interaction of Ras with other proteins, in particular the serine/threonine kinase Raf. The detailed mechanism of Ras-Raf interaction has been a matter of debate for decades. In particular, it is unclear whether Ras dimerization is necessary for Ras-Raf interaction. In our studies, we have distinguished the kinetic signatures of the presence or absence of Ras dimerization. Our results offer a well-defined and well-characterized null hypothesis through which Ras-Raf interaction can be probed to its full extent.

Introduction

Protein-protein interaction plays a central role in cell signaling, in which formation of multiprotein complexes are often necessary to initiate a signaling pathway (1, 2, 3, 4, 5, 6, 7, 8, 9). Aberrant protein-protein interaction leads to various diseases, including cancers. Therefore, detecting and understanding protein-protein interaction and the resultant complex formation is central to form comprehensive understanding of the cell signaling processes. Unfortunately, direct detection of protein complexes through optical microscopes is problematic because the size of the proteins and protein complexes (1–10 nm) are often below the resolution (10–20 nm) of the best available optical microscopes. At such resolution, colocalization of proteins can be falsely interpreted as the formation of protein complexes. Therefore, indirect methods are the best approaches to detect multiprotein complexes. In particular, we ask whether it is possible to detect the formation of specific multiprotein complexes through the measurement of coarse kinetic parameters, such as activation rates. We investigate this question in the context of Ras-Raf interaction, which is a well-characterized cell signaling system.

Ever since its discovery as an oncogenic protein, Ras has been the subject of intense scientific research (10, 11, 12, 13, 14, 15, 16, 17, 18). Part of that effort identified its interaction with other proteins, in particular Raf, and contributed to the discovery of the cell signaling networks. Since then, Ras-Raf interaction has taken a center stage in cancer research, and it has been established as a model system to study how mutated protein-protein interactions lead to cancer. In fact, mutated Ras-Raf interaction has been implicated in almost 30% of all known cancers and various other diseases (10). Despite its notoriety, no universally effective drugs have been developed against Ras, leading the community to speculate that Ras is undruggable. However, this assessment reflects the frustration and the desperation of the scientific community rather than being the scientific truth. Our inability to produce effective drugs stemmed from an incomplete understanding of the interaction of Ras with other proteins (19, 20, 21, 22). In fact, the detailed mechanism of Ras-Raf interaction is still a matter of debate.

Ras is a peripheral membrane protein that acts as a switch by binding to GDP or GTP (referred to as Ras.GDP and Ras in this manuscript, respectively). In its GTP-bound form, Ras binds to various effector proteins, including Raf (23). Through a series of yet unclear steps, the binding of Ras to Raf leads to the formation of activated Raf dimers that activate the ERK (extracellular-signal-regulated kinase) signaling pathway in eukaryotes, a pathway that controls cell proliferation, cell growth, and cell differentiation, among other important cellular processes. Although the role of Raf dimer formation on ERK activation is well established (24, 25, 26, 27, 28, 29, 30), it is unclear whether Raf dimer formation is mediated by monomeric Ras or dimeric Ras (Fig. 1 A; (8,22, 23, 24, 25, 26, 27, 28, 29, 30)). Recent experiments point in both directions (31, 32, 33; 34, 35, 36, 37, 38, 39, 40, 41). Crucially, the most compelling experiments in favor of Ras dimer formation comes from direct observation using super-resolution microscopy, which has a resolution of 10–20 nm. However, Ras is ∼2 nm in diameter. Therefore, close association of Ras proteins can be interpreted as Ras dimers or even oligomers, even in the absence of any dimers or oligomers (Fig. 1 B). Also, the low resolution of the microscope can exaggerate the number of dimers and trimers in cases in which Ras truly dimerizes (Fig. 1 C). Because of this, it is difficult to resolve the debate in favor of one hypothesis or the other using direct observation.

Figure 1.

Does Ras dimerize? (A) We consider two different hypotheses for Raf activation. In both our hypotheses, Raf activates after forming Ras-Raf heterotetramer (last panel). Ras-Raf heterotetramer is a Raf dimer, formation of which is necessary for Raf activation (29,30). The mechanisms for Ras-Raf heterotetramer formation differ in the two hypotheses. In hypothesis 1, Ras cannot dimerize, but in hypothesis 2, Ras dimerization is essential for the formation of the heterotetramer. We also consider a third hypothesis (data not shown here) in which both mechanisms are allowed. (B and C) We simulated the dynamics of Ras in the absence of Raf to detect and understand the differences in Ras clustering patterns, if any, generated by the two hypotheses. Surveying the true pattern (top left panel) clearly shows the absence and presence of Ras dimers in hypothesis 1 and 2, respectively. However, when the same point pattern is observed at lower resolution (e.g., 10 nm (top right pattern) or 20 nm), we find spurious Ras dimers, and even trimers, in the observed point patterns (bottom panel). In light of this observation, because current experimental techniques to detect Ras can, at best, reach 10-nm resolution, we cannot be certain about the presence of Ras dimers and higher order clusters. To see this figure in color, go online.

To circumvent this status quo, we propose to resolve this debate using indirect evidences. In this article, we investigate three hypotheses that investigate Raf dimerization in the presence and absence of Ras dimerization. Experimental evidences show that membrane proteins are subjected to nontrivial diffusive transport (42). Therefore, assuming that either Ras or Raf is well mixed may lead to erroneous results, and spatiotemporal simulation of their interaction is necessary. We use a recently developed spatiotemporal simulation technique, Brownian dynamics (BD)-Green’s function reaction dynamics (GFRD) (43, 44, 45, 46, 47, 48, 49), to investigate Ras-Raf kinetics in biologically relevant concentrations and timescales. Ras-Raf interaction is endowed with multiple timescales that vary depending on the concentration of Ras and Raf. The concentration dependence of these timescales provides a useful set of probes through which we investigate the indirect effect of the presence or absence of Ras dimerization in Ras-Raf interaction.

Methods

Ras-Raf interaction model

The formation of phosphorylated (activated) Raf dimers is necessary to activate the various downstream effector proteins in the ERK pathway. However, the details of the dimerization process are still poorly understood. Experimental data suggest that sometimes Raf dimers form only after forming stable complexes with membrane-bound activated (GTP bound) Ras. Unfortunately, most of these measurements are based on optical microscopy, which cannot resolve the interactions between Ras and Raf in the course to form Raf dimers. Because of this, two competing hypotheses have been proposed. In this article, we investigate the consequences of the underlying assumptions of these hypotheses. In particular, we test how these assumptions influence the activation timescale of Raf dimers and the fraction of activated Raf dimers.

Hypothesis 1: Ras does not dimerize

The first hypothesis proposes that Ras (S) does not form homodimers, and it diffuses freely on the plasma membrane. Raf (F) binds to individual Ras, and the resultant heterodimer (SF) binds with itself to form the Raf dimer (S₂F₂). The Raf dimers phosphorylate each other through colliding with each other (30,50). The activation kinetics is an assumption of the model. The activation kinetics of Raf remain unclear, and currently available experimental data do not rule out activation through interaction of two diffusing S₂F₂. The following set of chemical reactions represent this hypothesis. For clarity and brevity, we represent Ras and membrane-bound Raf through the symbols S and F, respectively. Also, we represent an activated molecule by adding an asterisk (^∗) to its symbol:

$$S+F\rightleftharpoons SF,$$

$$SF+SF\to S_2{F}_2,$$

$$2S_2{F}_2\to 2S_2{F}_2^{\ast },$$

$$S_2{F}_2+S_2{F}_2^{\ast }\to 2S_2{F}_2^{\ast }.$$

Hypothesis 2: Ras homodimers are necessary for Raf dimerization

The second hypothesis assumes that free Ras cannot bind to Raf, and they need to dimerize before they can bind to Ras. In this hypothesis, Ras forms homodimers (S₂ by binding to another Ras. Two cytosolic Rafs bind to the Ras dimer and form Raf dimers that activate using the same mechanism as in the first hypothesis. The chemical reactions governing the second hypothesis are as follows:

$$S+S\rightleftharpoons S_2,$$

$$S_2+F\to S_{2}F,$$

$$S_{2}F+F\to S_2{F}_2,$$

$$2S_2{F}_2\to 2S_2{F}_2^{\ast },$$

$$S_2{F}_2+S_2{F}_2^{\ast }\to 2S_2{F}_2^{\ast }.$$

Hypothesis 3: Both mechanisms present

In this hypothesis, we assume that both free Ras and Ras dimers can bind to Rafs. However, a Raf-bound Ras cannot form a Ras dimer (i.e., the reaction SF + S → S₂F is not allowed). The activation process is the same as the two previous hypotheses. The corresponding chemical reactions are as follows:

$$S+S\rightleftharpoons S_2,$$

$$S+F\rightleftharpoons SF,$$

$$S_2+F\to S_{2}F,$$

$$SF+SF\to S_2{F}_2,$$

$$S_{2}F+F\to S_2{F}_2,$$

$$2S_2{F}_2\to 2S_2{F}_2^{\ast },$$

$$S_2{F}_2+S_2{F}_2^{\ast }\to 2S_2{F}_2^{\ast }.$$

Simulation details

BD-GFRD

Ras-Raf interactions span multiple timescales. Once Ras and Raf are within interaction distance, the formation of Ras-Raf complexes takes picoseconds to nanoseconds, but, in biologically relevant concentrations, it usually takes milliseconds or seconds for a Ras and Raf to come within interaction distance. Therefore, any simulation of Ras-Raf interaction should account for both timescales. This requirement rules out all-atom or coarse-grained spatiotemporal simulations, which, at best, can reach milliseconds timescale for a single protein. On the other extreme, well-mixed chemical reaction models can capture all timescales but leaves out important spatial correlations. GFRD (46, 47, 48,51), a recently developed multiscale method, solves this conundrum. In BD-GFRD (43, 44, 45), an updated version of GFRD, one can perform spatiotemporal simulations that can access seconds and minutes timescale.

BD-GFRD leverages the fact that biological systems are diluted, and biomolecular interactions are short ranged. These two constraints partition the simulated particles into two groups. Particles that are within interaction radius of each other are simulated using a molecular mechanics algorithm (e.g., BD). Particles that are far apart from other particles are treated as independent particles, and they are propagated diffusively using GFRD, which is an event-driven algorithm. When a particle can be treated as an isolated (noninteracting) entity, GFRD computes the Green’s function for the diffusion equation for that particle with an appropriate boundary condition. We sample the next event time and position from the computed Green’s function. Thus, by simulating the computationally expensive free diffusion using an event-driven algorithm, BD-GFRD can achieve almost a million times speed-up compared to all-atom or coarse-grained simulations. One should note that we can calculate the Green’s functions for spherical particles only (43,44,51). Therefore, in our model, we treat every complex as a sphere with radius commensurate with their mass.

The association reactions are modeled as instantaneous processes. When two molecules come within a distance r_a of each other, they react instantaneously and form the protein complex (Fig. 2 B). We use the same criterion for the activation reactions as well.

Figure 2.

Lennard-Jones Potential. (A) Lennard-Jones potential for ε = 5 kT is shown. (B) Free energy F for the potential in (A) is shown. The free energy is nonmonotonic, with a finite barrier at r ≈ 2σ. Hence, after a dissociation reaction, the dissociated molecules are placed at a distance r_d = 2σ from each other. We chose the minimal distance for association reaction r_a ≈ 1.35σ to ensure that the associating molecules are well below the free energy barrier. To see this figure in color, go online.

For dissociation reactions, we assumed that the equilibration of the bond dissociation happens much more quickly than two consecutive dissociation events. Under this assumption, the dissociation events follow a Poisson process, and the dissociation times are exponentially distributed. Once two molecules dissociate, they are placed at a distance r_d from each other. We chose r_d in such a way to ensure that the dissociated molecules do not immediately bind to each other (Fig. 2 B).

Simulation parameters

We performed all simulations on a two-dimensional 1 μm × 1 μm simulation box with a periodic boundary condition. All molecules i and j, except S₂F₂ and $S_2{F}_2^{\ast }$, that reacted together in a reaction interacted with each other through isotropic Lennard-Jones interaction with interaction strength ε_ij = 5 kT and cutoff radius r_c = 2.5 σ_ij, where σ_ij = r_i + r_j is the sum of the radius of the two interacting particles. All other molecules, including S₂F₂ and $S_2{F}_2^{\ast },$ interacted with each other through repulsive Weeks-Chandler-Anderson (WCA) potential (52) with ε_ij = 3 kT. We list the mass and radius of all molecules in Table 1 below. This particular form of the potential was inspired by computationally measured potential of mean force (PMF) of protein-protein interaction (53,54). A PMF measures the free energy landscape of the interaction potential. For a Lennard-Jones potential, the free energy landscape, F(r), is shown in Fig. 2 B. Because of the visual similarity of the F(r) with the PMFs, we chose to use Lennard-Jones and WCA interaction for our simulations.

Table 1.

Mass of the Protein Complexes Involved in Ras-Raf Interaction and the Radius of the Spheres Representing the Complexes

Molecule	Mass (kDa)	Radius of a sphere of equal mass, a (nm)	Diffusion coefficient (μm2/s)
S	22	2.0	0.95
F	66.5	2.9	0.66
S2	44	2.5	0.76
SF	88.5	3.16	0.60
S2F	110.5	3.43	0.55
S2F2 and $S_2{F}_2^{\ast }$	177	4.0	0.48

We assumed that the densities of the proteins are 1 g/cc. We approximated the diffusion constant using Stokes-Einstein relationship: $D=kT/6\pi \eta a,$by assuming that the protein is a sphere of radius a and is embedded in an isotropic fluid of viscosity η = 120 cP.

We used a Langevin integrator at 310 K for the BD. We computed the diffusion constant using Stokes-Einstein relation:

$$D=\frac{kT}{6\pi \eta a},$$

where η is the membrane viscosity. We used a membrane viscosity of 120 cP, which is close to the membrane viscosity of eukaryotic cell membranes (55,56). Please note that we use this formula merely as a convenient way to pick the diffusion coefficients. Diffusion on the membrane is not, in general, inversely proportional to the radius of the proteins or their complexes (57). However, the diffusion coefficient computed using Eq. 6 in (57) is nearly identical to what we use here. For example, for S₂, (57) it gives 0.79, whereas we use 0.76. Similarly, for S₂F₂, we use 0.48 but get 0.50 from (57). Also, instead of using any formula, we could have used experimentally available values. Unfortunately, we could find only Ras diffusion coefficient from the experiments (D. Goswami, personal communication), which was 1 μm²s⁻¹ and which is very close to what we get from our formula.

We picked r_a ≈ 1.35 σ_ij by examining the free energy landscape of the Lennard-Jones potential. The free energy for a potential U(r) is given by the following: $F=\ln \left(r\times e^{-\left(U\left(r\right)/KT\right)}/Z\right)$, where $Z=\int re-\left(U\left(r\right)/KT\right)dr$ is the partition function in two dimensions (Fig. 2 B). Also, we picked r_d ≈ 2 σ_ij using the free energy landscape. We chose the dissociation rate for all dissociation reactions to be 10/s (D. Goswami, personal communication) (58), unless otherwise mentioned.

Results

Time evolution of protein concentrations

We start each simulation with a number n_Ras of free Ras (S) molecules and a number n_Raf of free Raf (F) molecules. Subjected to the reactions in the three hypotheses, S and F combine to form various intermediate complexes that eventually produce activated S₂F₂ (Fig. 3, A–C). Despite the variety of intermediates, a common theme emerges in their time evolution. In all three hypotheses, S and F concentrations decay to produce the first intermediates that increase in concentrations until the second intermediates form at a time depending on n_Ras and n_Raf. In hypothesis 1, for example, SF is the first intermediate and S₂F₂ is the second intermediate, and in hypothesis 2, S₂ is the first intermediate and S₂F is the second intermediate. As the second intermediates form, the concentrations of the first intermediates decrease, peaking at a concentration that also depends on the initial Ras and Raf concentrations. Interestingly, how S and F decay over time depends on the model hypothesis. Ideally, such an observation would have been useful to identify the nature of Ras-Raf interaction in experiments. However, in biologically relevant concentrations, these variations occur at timescales that are too fast for current experimental probes and too slow for all-atom simulation techniques (Fig. 3, A–C).

Figure 3.

Comparison between hypotheses. (A–C) Time trace of the concentrations of the reacting molecules is shown. The concentrations, as expected, vary differently under different hypotheses. In particular, the variation of S and F with time can be a useful probe to differentiate between the three hypotheses. Remarkably, as we had anticipated, there are multiple timescales. Two particularly important timescales are the time at which S₂F₂ first forms, ${\tau }_{S_2{F}_2}$(orange arrow in (A)), and the time at which the first activation of S₂F₂ occurs, τ_A (purple arrow in (A)). For most biologically relevant concentrations, none of these timescales, including ${\tau }_{S_2{F}_2}$ and τ_A, can be probed using either experiments or all-atom (AA) simulations. We probe how (D–F) τ_A and (G–I) ${\tau }_{S_2{F}_2}$ vary with concentrations under different hypotheses. In hypothesis 1 (Ras does not dimerize), the variation is symmetric with respect to n_Ras and n_Raf. In contrast, for hypotheses 2 and 3 (Ras dimerizes), this variation is asymmetric. Hence, the (a)symmetry in timescales with variation in concentrations can be used to probe whether Ras dimerizes or not. To see this figure in color, go online.

Fortunately, Ras-Raf interaction is endowed with multiple timescales that all are potential experimental probes. For example, in hypothesis 1, S and F combine to produce the Ras-Raf complex, SF, which first occurs at timescale τ_SF. SF can either irreversibly combine with itself to form the inactive Raf dimer S₂F₂, or it can reversibly dissociate back to free Ras. Because of this, the concentration of SF changes nonmonotonically with time and peaks at time τ_peak, which depends on n_Ras and n_Raf. Also, because of the competition between the dissociation and the combination reactions, S₂F₂ forms at timescale ${\tau }_{S_2{F}_2}$, and the first activation of Raf dimers happen at timescale τ_A. Understandably, these timescales depend on n_Ras and n_Raf. Characterizing the concentration dependence of these timescales is important to construct a dynamic picture of Ras-Raf interaction and to develop them as potential experimental probes for Ras-Raf interaction.

Two different timescales, ${\tau }_{S_2{F}_2}$ and τ_A, are common to all three hypotheses. Hence, we compare and contrast them here. The Raf dimer formation varies across the three hypotheses, whereas the activation process remains same. Therefore, using these two timescales, we can probe, respectively, the direct and the indirect consequences of our assumptions in the three hypotheses. To understand these relationships, we probe these two timescales over different Ras and Raf concentrations. Visually, these two timescales behave differently under different hypotheses (Fig. 3, D–I). ${\tau }_{S_2{F}_2}$ varies symmetrically with n_Ras and n_Raf for both hypotheses 1 and 2, but τ_A varies symmetrically only for hypothesis 1. The variation of both timescales is asymmetric in other cases. Particularly, τ_A varies asymmetrically for hypotheses 2 and 3, both of which admit Ras dimerization. In the absence of Ras dimerization, τ_A varies symmetrically. Therefore, it is tempting to declare the presence and absence of asymmetry in τ_A as an indirect probe for the presence and absence of Ras dimers. To embolden this claim, we investigate the kinetics of the Raf dimer formation reaction and the activation reaction.

Concentration dependence of timescales

To disentangle how n_Ras and n_Raf influence the timescales, we study their individual contribution on the timescales ${\tau }_{S_2{F}_2}$ and τ_A. We vary either S or F concentrations, keeping the concentration of the other molecule fixed. We refer to the former as the probe and the latter as the control molecule.

Variation of ${\tau }_{S_2{F}_2}$

${\tau }_{S_2{F}_2}$ varies similarly with n_Ras (Fig. 4, A–C) and n_Raf (Fig. 4, D–F) across the three hypotheses. In all three cases, ${\tau }_{S_2{F}_2}$ decreases with increasing probe molecule concentration, eventually saturating to a timescale dependent on the control molecule concentration. For example, for n_Ras = 8 and in hypothesis 1, ${\tau }_{S_2{F}_2}$ decays from a maximal value of ∼240 ms for n_Ras = 8 to ∼50 ms for n_Raf = 50 and stays there until n_Raf = 100, the highest concentration probed (Fig. 4 A). Although the data for Fig. 4 F are noisy, the decreasing trend is easily discernible. The decay rate of ${\tau }_{S_2{F}_2}$ with the probe concentration is the same across different control molecule concentrations and across different hypotheses. In contrast, the asymptotic timescale (i.e., the timescale at a large probe molecule concentration) varies similarly with both control molecule concentrations under a hypothesis, but the variation is different across hypotheses. For example, in Fig. 4 G, the asymptotic ${\tau }_{S_2{F}_2}$ varies as n^−1.5, independent of the chosen control molecule; n is the concentration of the control molecule. However, asymptotic ${\tau }_{S_2{F}_2}$ varies as n^−1.3 and n^−1.2 for hypotheses 2 and 3, respectively (Fig. 4, H and I). Thus, the effect of the hypothesis is directly reflected in the variation of asymptotic ${\tau }_{S_2{F}_2}$with respect to the concentration of the control molecule. It is unclear what processes determine the observed scaling. We suspect the underlying reaction kinetics is the main determinant of the timescales because if these timescales resulted from a diffusion limited process, then the timescale would decay as n⁻¹, which is the well-known Berg-Purcell scaling (59).

Figure 4.

Concentration dependence of ${\mathit{\tau }}_{{\mathit{S}}_2{\mathit{F}}_2}$. We probed Raf dimer formation timescale, ${\tau }_{S_2{F}_2}$, (A–C) by varying n_Raf and keeping n_Ras constant and (D–F) by varying n_Ras and keeping n_Raf constant. For both cases, ${\tau }_{S_2{F}_2}$ varied uniformly across the three hypotheses: ${\tau }_{S_2{F}_2}$ decreased with increasing n_Raf and n_Raf, eventually saturating to an asymptotic timescale that depends on the protein concentration that is kept constant (n_Ras for (A–C) and vice versa). Although the data are noisy for (F), a decreasing trend is not difficult to infer. (G–I) The asymptotic ${\tau }_{S_2{F}_2}$ is not diffusion limited because the timescale does not decrease linearly with concentration (black dashed line). Instead, the asymptotic timescale decreases as a power law $n_{Ras,Raf}^{-\alpha }$ (red solid line), where α depends on the hypothesis (see figure). What determines the value of α is unclear, but we hypothesize that the difference arises from the difference in the underlying kinetics. To see this figure in color, go online.

Variation of τ_A

Unlike ${\tau }_{S_2{F}_2}$, τ_A varies differently with n_Raf (Fig. 5, A–C) and n_Ras (Fig. 5, D–F) across the three hypotheses. For hypothesis 1, τ_A decays with both probe concentrations and asymptotes to a value depending on the control concentrations, in a fashion similar to ${\tau }_{S_2{F}_2}$. In contrast, for hypotheses 2 and 3, the variation with respect to probe molecules depends on the choice of the probe. If the probe molecule is F and the control molecule is S, then τ_A decays with increasing concentration and asymptotes to a value that only depends on n_Ras. However, if S is chosen as the probe and F as the control, then for low n_Raf, τ_A increases with increasing n_Ras and asymptotes to a value depending on n_Raf. Remarkably, in both hypotheses 2 and 3, Ras dimerizes, whereas in hypothesis 1, Ras does not dimerize. Does Ras dimerization cause the observed anomalous variation of τ_A?

Figure 5.

Concentration dependence of activation timescale. We probed activation timescale (A–C) by varying n_Raf and keeping n_Ras constant and (D–F) by varying n_Ras and keeping n_Raf constant. For (A–C), the activation timescale varied uniformly across the three hypotheses: τ_A decreased with increasing n_Raf, eventually saturating to an asymptotic timescale that depends on n_Ras In contrast, for (D–F), τ_A varied differently across the three hypotheses. For hypothesis 1, τ_A decreased with increased concentrations for all values of n_Raf, but for hypotheses 2 and 3, τ_A increased with increasing n_Ras at low values of n_Raf. Although at higher values of n_Raf, τ_A behaved similarly to hypothesis 1. (G–I) The asymptotic τ_A is not diffusion limited because the timescale does not decrease linearly with the concentration (black dashed line). Instead, the asymptotic timescale decreases as $n_{Ras,Raf}^{-1.5}$ (red solid line) for all three hypotheses, implying that the activation time is set by a common kinetic process. To see this figure in color, go online.

To answer this question, we investigated the time series of concentrations for hypothesis 2 and 3 at low concentrations of F (Fig. 6). We find that for hypothesis 2, the delay results from the formation of S₂F, which forms from the irreversible association of a Ras dimer S₂ and free Raf F. When n_Ras is low, S₂F forms with less propensity, and enough F remains available to form at least two S₂F₂, the minimal number of inactive Raf dimer required for the activation reaction. As n_Ras increases, S₂F forms with increasing propensity, resulting in a decreasing number of F available in the simulation box. Because of this, the propensity of the formation of S₂F₂ decreases with increasing n_Ras, and the activation timescale increases. A slightly different mechanism causes the anomalous variation in hypothesis 3. In this hypothesis, as n_Ras is increased, the propensity of SF and S₂F formation increases. Because of this, free F remains present in low numbers until SF dissociates back into free S and F. A lack of free F decreases the propensity of the formation of S₂F₂ that increases the activation time (Figs. 6 and S1). Therefore, resource limitation caused by the low concentration of F is the principle reason behind the anomalous behavior. However, the low concentration of F affects the activation timescale because Ras dimerization-mediated two-step Raf dimer formation mechanism (S₂ + F → S₂F, S₂F + F → S₂F₂) requires more freely available F. Even if the formation of S₂F were reversible in both hypotheses, we would have similar delay because, in such a case, we would have to wait until the release of F from S₂F. Hence, Ras dimerization is the underlying cause of the observed anomalous behavior.

Figure 6.

Activation events in hypotheses 2 and 3. For low concentrations of Raf (8/μm² here), in hypotheses 2 and 3, the activation time increases with increasing Ras concentration. The underlying reason for this anomaly is the limited availability of F, which happens because of different reasons in the two hypotheses. (A) In hypothesis 2, F becomes limited because of the formation of S₂F, a molecule necessary to form S₂F₂. (B) In hypothesis 3, F readily forms SF. However, because of the higher concentration of S, the preferred route to form S₂F₂ is through the formation of S₂F, which can occur only after SF dissociates back to S and F. The higher the concentration of S, the higher the propensity to form SF. Because of this, the activation time increases with increasing Ras concentration. To see this figure in color, go online.

Although the variation of τ_A with respect to probe concentration depends on the underlying hypothesis, the variation of asymptotic activation time with control molecule concentration remains uniform across the three hypotheses. In all three cases, the asymptotic timescale varies as n^−1.5, where like ${\tau }_{S_2{F}_2}$, n is the concentration of the chosen control molecule (Fig. 5, G–I). Again, the asymptotic timescale does not follow Berg-Purcell scaling, implying that the underlying reaction kinetics determine the asymptotic timescale.

Reaction kinetics

Although the molecules in our simulation box move diffusively, as we have seen, their reaction timescales differ significantly from the limit set by pure diffusion. Hence, we suspect that the underlying reaction kinetics set the observed timescale. Because the activation timescale varies uniformly across the three hypotheses, we investigate its variation with the control molecule concentration n.

In our models, a S₂F₂ molecule gets activated when it collides with another S₂F₂ or $S_2{F}_2^{\ast }$ molecule. However, because the activation time is the time at which the first activation reaction happens, its kinetics is entirely determined by the concentration of S₂F₂, [S₂F₂]. In particular, we find that the concentration of S₂F₂ right before the activation event is related to the inverse of the activation time through a simple, well-mixed, mass action-like form. That is, the activation rate r_A = 1/τ_A is proportional to the mass action flux, $\left[\Phi \right]=\left(\left[S_2{F}_2\right]\times \left(\left[S_2{F}_2\right]-1\right)/2\right)$:

$$r_A=\frac{1}{{\tau }_A}=\kappa \frac{\left[S_2{F}_2\right]\times \left(\left[S_2{F}_2\right]-1\right)}{2},$$

where κ is the rate constant. This relationship holds uniformly across the three hypotheses, as we show in Fig. 7 A. In fact, the rate constant κ is exactly equal to the inverse of the activation time if there were only two S₂F₂ (∼6 × 10⁻⁴ ms⁻¹ μm⁴). The uniformity of the activation kinetics and the uniformity of the power law exponent of the asymptotic τ_A across three hypotheses lends credence to our suspicion that the underlying reaction kinetics sets the timescale. If this suspicion is true, then the kinetics of S₂F₂ should be different across the three hypotheses. We find this to be the case for hypotheses 1 and 2 (Fig. 7 B). We could not compare hypothesis 3 with the other hypotheses because S₂F₂ forms through two different processes in hypothesis 3. Therefore, the total mass action flux is a weighted sum of the two fluxes: [Φ] = κ₁[Φ₁] + κ₂[Φ₂]. Hence, the rate constant cannot be compared in a straightforward manner, like the other two hypotheses.

Figure 7.

Comparison with mass action kinetics. (A) Independent of the hypothesis, the mean activation rate (1/τ_A) follows mass action kinetics, $\left(1/{\tau }_A\right)=\kappa \left[\Phi \right],$ for a wide range of concentrations, as shown by the insensitivity of the reaction rate constant κ = 1/(τ_A × [Φ]) on the mass action flux $\left[\Phi \right]=n_{S_2{F}_2}\times \left(n_{S_2{F}_2}-1\right)/2$. (B–E) In contrast, other timescales do not arise from well-mixed mass action kinetics because κ depends on the mass action flux [Φ]. Mass action flux for various reactions are described in the text. (F–H) The deviation from the mass action behavior is reflected in the asymptotic scaling of the timescales (solid lines). For example, the difference in the deviation from mass action behavior in (C) is reflected in the different asymptotic behavior in (F). In contrast, the similarity of deviation from mass action kinetics in (D) is reflected in the same asymptotic behavior in (G). Shown are solid markers for variations with respect to n_Ras and open markers for n_Raf. For SF, there is no difference in variation with respect to n_Ras or n_Raf. S₂F, however, shows different variation. Most interestingly, the asymptotic timescale follows Berg-Purcell scaling (dashed line) for all the cases, except variation with respect to n_Ras in hypothesis 2. To see this figure in color, go online.

We consolidated the connection between the asymptotic scaling and underlying kinetics by repeating the above analysis for other molecules (Fig. 7, C–H). We list the mass action fluxes for the corresponding reactions in Table 2. We found that the asymptotic scaling of the timescales reflects the deviation from mass action kinetics. For example, in Fig. 7 C, the deviation from mass action-like behavior is different for hypothesis 2 and 3. For hypothesis 2, κ increases monotonically with [Φ]. In contrast, for hypothesis 3, κ decreases with [Φ] initially, and then, it increases with [Φ] with the same functional form as in hypothesis 2. This difference is also reflected in the scaling of asymptotic ${\tau }_{S_2}$ with n_Ras. For hypothesis 2, we observe a single power law. However, for hypothesis 3, the scaling is Berg-Purcell-like in the beginning, which changes to the same power law we observe for hypothesis 2. We observe a similar correlation between the underlying kinetics and the scaling of the asymptotic timescale for SF and S₂F.

Table 2.

Mass Action Flux: Flux of the Reactants to Produce the Molecule Listed on the Leftmost Column as Modeled by the Well-Mixed Mass Action Kinetics

Molecule	[Φ]
	Hypothesis 1	Hypothesis 2	Hypothesis 3
SF	[S][F]	–	[S][F]
S2	–	[S] × ([S] − 1)/2	[S] × ([S] − 1)/2
S2F	–	[S2][F]	[S2][F]
S2F2	[SF] × ([SF] − 1)/2	[S2F][F]	a
$S_2{F}_2^{\ast }$	$\frac{\left[S_2{F}_2\right]\times \left(\left[S_2{F}_2\right]-1\right)}{2}$	$\frac{\left[S_2{F}_2\right]\times \left(\left[S_2{F}_2\right]-1\right)}{2}$	$\frac{\left[S_2{F}_2\right]\times \left(\left[S_2{F}_2\right]-1\right)}{2}$

Some reactions are not available in all three hypotheses. Where they are absent, we mark the flux by “–” sign.

^aS₂F₂ forms through two different mechanisms. Hence, the mass action flux does not have a simple form. So, we did not compute the mass action flux for hypothesis 3.

Discussion

Using a novel spatiotemporal simulation technique, we have investigated how Ras dimerization affects Ras-Raf interaction. We have compared and contrasted the kinetics of three hypotheses that consider Ras-Raf interaction kinetics in the absence or presence of Ras dimerization. As expected, we find that multiple timescales are a feature common to all three hypotheses. These timescales depend on the initial concentrations of Ras and Raf, and the presence or absence of Ras dimers affect this trend. We find that when Ras does not dimerize (hypothesis 1), the activation timescale decreases with increasing concentrations of both Ras and Raf. If Ras dimerizes, then this symmetry in Ras and Raf concentrations no longer hold. Instead, if Ras concentration is increased, keeping Raf concentration low and constant, then the activation timescale increases with increasing Ras concentration. The underlying reason for this reversal in the trend is resource limitation. Free Raf, F, takes part in two different reactions when Ras dimerizes. When the concentration of F is low and the concentration of S is high, all F are used to form the intermediate molecules, such as SF or S₂F, and no F is left to produce S₂F₂ from these intermediates. S₂F₂ can form if only F becomes available again through the dissociation of some of these intermediate molecules. Although these results are dependent on the particular details of our hypothesis, we find this opposite trend in the concentration dependence of the activation time whenever Ras dimerizes. Therefore, we may use the concentration dependence of the activation time as an indirect probe for Ras dimerization. Although some in vivo experiments support some of our findings (60, 61, 62, 63), these predictions should be tested in in vitro experiments (33), in which the concentrations of Ras and Raf can be independently controlled. The fastest activation timescales are of the order of 10 ms, whereas the slowest timescales are of the order of several seconds, both of which are accessible to modern optical microscopes. Because of the simplicity of our system, these predictions may not be directly tested in vivo, in which an unusually low concentration of Ras may render the cell line nonviable.

The kinetics of the Raf dimer activation follow well-mixed mass action kinetics. We find this observation to be the exception rather than the norm because the formation of the intermediate complexes do not obey mass action kinetics. The underlying reason for such strange reaction kinetics is unclear, but we suspect two probable factors combine to produce such unusual kinetic behavior: 1) interparticle interaction and 2) reaction timescales. First, every other reaction except the activation reaction is an association reaction, in which the reacting molecules interact with each other through attractive Lennard-Jones potential. In the activation reaction, the inactive Raf dimers interact through repulsive WCA interaction, which can be approximated as hard-sphere interaction. It has been suggested that mass action kinetics works only when the interacting particles interact through hard-sphere interaction (64). In the presence of attractive interaction, mass action kinetics breaks down. Second, the activation reactions occur at a timescale long enough to explore the simulation box completely. Hence, the well-mixed assumption is justified. In contrast, other reactions occur at timescales that are orders of magnitude smaller than the activation timescale. It is impossible to explore the entire simulation box in such a short time, and the well-mixed approximation breaks down. It will be interesting to disentangle the role of the timescale and the interaction in the Ras-Raf kinetics. In particular, it will be useful to understand under what conditions our results are comparable to a well-mixed reactor in which reactants interact with mass action kinetics.

Despite the simplicity of our model, it has nontrivial kinetic behavior that are both biologically and chemically interesting. It is likely that our hypotheses will be limited in scope to in vitro studies, in which Ras-Raf interaction can be studied in isolation. In the presence of other interacting proteins (for example, in in vivo studies) the kinetics is likely to be far richer than what we have observed here. In particular, the competition between Raf, PI3K, PLCε, and other effectors of Ras will lead to interesting resource-limited kinetics spanning multiple timescales, potentially similar to what we have observed here. The main challenge lies in studying these proteins in biologically relevant conditions. Proteins interact with other proteins through highly anisotropic and specific forces, which we have ignored in our current model by choosing isotropic interactions of identical strength. The presence of anisotropy and specificity can vastly change the timescales of interactions and the resultant kinetics. Also, proteins can indirectly interact with other proteins through the lipids present in the membrane. Simulating the lipid dynamics in conjunction with protein dynamics is beyond the scope of BD-GFRD. Hence, we have to incorporate the effect of lipids through some effective contribution. For example, in this work, we have incorporated their effect through the viscosity of the membrane. We envision to incorporate the anisotropy and the specificity of protein interactions in our model. However, the intended purpose of BD-GFRD is not to simulate biomolecular interactions with detailed biomolecular interactions. Specialized simulations (65) are in development to answer these questions, which can accommodate detailed biomolecular interactions but fail to reach experimentally accessible timescales. BD-GFRD takes a complementary approach by trading biological details for long timescale simulation. In this way, these very different approaches can complement each other’s findings. For this work, we found it necessary to keep our model simple to understand the role of Ras dimerization in Ras-Raf interaction. For example, we have assumed that Raf is always membrane bound. This is a simplifying assumption. How Raf cycles between the cytosol and the membrane is still poorly understood, and this is an arena that can be explored in our model. However, experiments by our collaborators show that: 1) the typical lifetime of Raf interacting with Ras is ∼500 ms to 1 s, with a significant fraction of them staying on the membrane for more than 2 s, and 2) the number of Raf interacting with Ras on the membrane remains steady for the course of the experiments (D. Goswami, personal communication). These two observations can be used to justify our model, which ignores the exchange of Raf between the cytosol and the membrane. We admit that ignoring this exchange may influence the results. However, the intention of this article is to establish the simplest possible null model of Ras-Raf interaction with spatial heterogeneity. In future articles, we will contrast the results from this null model with models with increasing biological details. Particular plans include studying Ras-Raf interaction in the presence of sources of anomalous diffusion, protein clustering (66), and cytosolic exchange of the proteins. We must point out that despite these simplifications, the predictions from our model are quite general and experimentally testable with available technologies.

To conclude, Ras dimerization affects the kinetics of Ras-Raf interaction and this influence is reflected in the long timescale behavior of the underlying molecular concentrations. Our results offer a well-defined and well-characterized null hypothesis through which Ras-Raf interaction can be probed to its full extent.

Author Contributions

A.E.G. and S.S. designed research and wrote the manuscript. S.S. performed research and analyzed data.

Editor: Joseph Falke.