Spatial partitioning improves the reliability of biochemical signaling

Abstract

Spatial heterogeneity is a hallmark of living systems, even at the molecular scale in individual cells. A key example is the partitioning of membrane-bound proteins via lipid domain formation or cytoskeleton-induced corralling. However, the impact of this spatial heterogeneity on biochemical signaling processes is poorly understood. Here, we demonstrate that partitioning improves the reliability of biochemical signaling. We exactly solve a stochastic model describing a ubiquitous motif in membrane signaling. The solution reveals that partitioning improves signaling reliability via two effects: it moderates the nonlinearity of the switching response, and it reduces noise in the response by suppressing correlations between molecules. An optimal partition size arises from a trade-off between minimizing the number of proteins per partition to improve signaling reliability and ensuring sufficient proteins per partition to maintain signal propagation. The predicted optimal partition size agrees quantitatively with experimentally observed systems. These results persist in spatial simulations with explicit diffusion barriers. Our findings suggest that molecular partitioning is not merely a consequence of the complexity of cellular substructures, but also plays an important functional role in cell signaling.

The cell membrane is a nexus of information processing. Once regarded as a simple barrier between a cell and its surroundings, it is now clear that the membrane is a hot spot of molecular activity, where signals are integrated and modulated even before being relayed to the inside of the cell (1). Moreover, the membrane itself is structurally complex. Regions enriched in glycosphingolipids, cholesterol, and other membrane components, often called lipid rafts, transiently assemble and float within the surrounding bilayer (2), providing platforms for molecular interaction (3). Additionally, interaction of the membrane with the underlying actin cytoskeleton forms compartments in which molecules are transiently trapped (4, 5). These membrane subdomains create a highly heterogeneous environment in which molecules are far from well mixed, and it is currently unclear what effect this heterogeneity has on cell signaling.

Membrane subdomains are thought to play a dominant role in the observed aggregation of signaling molecules into clusters (6). Interestingly, these clusters have a characteristic size of only a few molecules. For example, the GPI-anchored receptor CD59 is observed to form clusters of three to nine molecules upon interaction with the cytoskeleton and lipid rafts (7, 8). Similarly, the well-studied membrane-bound GTPase Ras forms clusters of six to eight molecules, which also depend on interactions with the cytoskeleton and rafts (9, 10). Despite the important findings that aggregation of proteins induced by subdomains can affect reaction kinetics (11), enhance oligomerization (1), modulate downstream responses (12, 13), and enhance signal fidelity (13, 14), the origin of this characteristic size remains unknown. Although it is quite possible that these domains owe their size to a thermodynamic or structural origin, we here address the question of whether this size can be optimized for signaling performance. We find that the partitioning imposed by subdomains gives rise to a trade-off in cell signaling, from which an optimal size of a few molecules emerges naturally, suggesting that reliable signaling is intimately tied to the spatial structure of the membrane.

We study via stochastic analysis and spatial simulation a model that is directly motivated by both CD59 and Ras signaling at the membrane. Stimulated CD59 receptors induce the switching of several Src family kinases from an unphosphorylated to a phosphorylated state (7, 8). Similarly, stimulated EGF receptors induce the switching of Ras proteins from an inactive GDP-loaded state to an active GTP-loaded state (13). We therefore study the simple and ubiquitous motif of coupled switching reactions, in which the activation of one species (the receptor) triggers the activation of a second species (the downstream effector).

We exactly solve this stochastic model of coupled switching reactions, and we use the solution to compare signaling reliability in a spatially partitioned system to that in a well-mixed system. We demonstrate that partitioning can improve signaling performance by generating a more graded input–output relationship and by reducing the noise in the signaling response. This latter effect comes about because partitioning reduces the correlations between the states of the different output molecules. However, the stochastic exchange of proteins between partitions can generate configurations that isolate molecules and exclude them from the signaling process, thereby reducing the dynamic range of the response and increasing the output noise. The trade-off between these two effects results in an optimal partition size that agrees well with cluster sizes of signaling proteins that are observed experimentally (7–10), suggesting that cluster sizes are tuned so as to maximize information transmission.

Results

We model two coupled molecular species at the membrane, as depicted in Fig. 1A. A membrane-bound receptor (e.g., CD59 or EGF receptor) is activated via ligand stimulation, and the active receptor in turn activates a membrane-bound effector (e.g., a Src family kinase or Ras). A reaction scheme representing these processes is shown in Fig. 1B and consists of two protein species: the receptor and the downstream effector . The switching of molecules from the X to the X∗ state is driven by an external signal of strength α. Active X∗ molecules act on inactive Y molecules and promote switching to the Y∗ state. Deactivation of both active protein species occurs spontaneously and independently.

Fig. 1.

Schematic depiction of the model system. (A) We consider a model representative of signal detection by receptors and signal transmission at the cell membrane. (B) The model consists of two molecular species ( and ), which can each exist in active (X∗, Y∗) or inactive (X, Y) states. Molecules in the X state are activated by the external signal of strength α, and active X∗ molecules subsequently activate Y molecules. (C) We consider these reactions taking place in a single domain with all components well mixed, or in a domain consisting of smaller compartments, which are each individually well mixed but between which no interaction is possible. The total system volumes in the two scenarios are equal and assumed to scale with the number of molecules.

We will be concerned with how the network response, the number of active Y∗ molecules as a function of the input signal α, is affected by the spatial structure of the system. In particular, we ask how partitioning of the reaction system into noninteracting subdomains affects the reliability of signal transmission, which is determined by two principal factors: the input–output response and the output noise; together these properties determine to what extent different input signals can be reliably resolved from the network response. We focus on two system configurations, shown in Fig. 1C. In the first case, we assume that all molecules are present in a single well-mixed reaction compartment. In the second case, we consider a system partitioned into π compartments between which no interactions are permitted; here, we take the output of the system to be the total number of active Y∗ molecules in all compartments. This choice of output corresponds to a readout of the Y∗ signal by, e.g., a cytosolic component whose diffusion is much faster than the diffusion and signaling of and on the membrane. In the partitioned system, we will for simplicity first assume that the molecules are uniformly and statically distributed among compartments. However, recognizing that this scenario will not generally be realized inside cells, we will later relax this assumption and consider exchange of molecules among partitions.

We model the dynamics of the well-mixed system, as well as each compartment within the partitioned system, using a stochastic equation of the same form. We denote the total numbers of and molecules by M and N, respectively, and the numbers of active X∗ and Y∗ molecules by m and n, respectively. To parameterize the system, we scale units of time by the deactivation rate of X∗, such that the effective deactivation rate is 1. Then α denotes the rescaled activation rate of X; γ is the rate of deactivation of Y∗ relative to that of X∗; and γβ_m is the activation rate of a given Y molecule for a particular concentration of X∗ molecules. The parameter α incorporates the effective strength of the input signal and determines the mean X∗ activity via the occupancy q ≡ 〈m〉/M = α/(α + 1). The precise m-dependence of the coupling function β_m will depend on the exact nature of the interactions between X∗ and Y molecules. We take β_m ∝ m/v, with v as the volume of the compartment in which the reactions are taking place. However, our conclusions are unaffected if we instead take a Michaelis–Menten form β_m ∝ m/(m + vK) (SI Appendix, Fig. S1). The total system volume V is assumed to scale with the total number of molecules, such that M/V is constant. The coupling function in partition i ∈ {1, …, π} is then determined by m_i, the number of X∗ molecules in partition i, according to for constant β.

The probability of having m proteins in the X∗ state and n proteins in the Y∗ state evolves according to the chemical master equation (CME),

subject to suitable boundary conditions. The nature of the particular set of reactions in our model (Fig. 1B) means that the operators ℒ_m and ℒ_n have the same form,

where defines the step operator. Despite the appearance of terms containing the product mn in the operator ℒ_n(β_m, N), which make the direct calculation of moments of p_mn from the CME impossible, an exact solution to Eq. 1 can be found for arbitrary β_m using the method of spectral expansion (15, 16) as described in SI Appendix.

Partitioning Leads to a More Graded Response.

We begin by analyzing the behavior of a minimal system with M = N = 2. In the well-mixed system, all molecules are contained within π = 1 domain of volume V. In the partitioned system, π = 2 subdomains with volume V/2 each contain one and one molecule.

We first focus on the mean response 〈n〉. In the limits of small or large α, the mean response is the same in both the partitioned and mixed systems, 〈n〉/N → 0 and 〈n〉/N → β/(β + 1), respectively. However, at all intermediate values of α, the mean response of the well-mixed system is larger than that of the partitioned system; equivalently, the partitioned system exhibits a more graded response than the well-mixed system to changes in the input signal (Fig. 2A, thick solid and dashed curves). The more graded response is due to higher fluctuations in X∗ activity. When α → 0 or α → ∞, all molecules are inactive or active, respectively; however, at intermediate values of α, the number of active X∗ molecules fluctuates. Partitioning reduces the number of molecules per reaction compartment, increasing the relative size of these fluctuations according to . These fluctuations are passed through the concave dependence of n on m, resulting in a smaller mean [via Jensen’s inequality (17)], and therefore a more linear response curve (SI Appendix, Fig. S2A).

Fig. 2.

Spatial partitioning improves signaling performance. (A) The mean response 〈n〉/N as a function of the mean X∗ activity q = 〈m〉/M = α/(α + 1), and (B) the output variance as a function of the mean response, plotted for a well-mixed system with M = N = 2 (thick solid) and a partitioned system of π = 2 compartments, each containing one and one molecule (thick dashed). Partitioning linearizes the output response and reduces noise across the full range of responses, leading to a higher transmitted information. The thin solid curves show the mean field response 〈n〉/N = βq/(βq + 1) in A and the binomial noise limit (3) in B. Allowing exchange of molecules between compartments (thick dot-dashed) compresses the output response and increases the noise compared with the perfectly partitioned system, dramatically reducing information transmission. Here, β = 20 and γ = 1.

A more graded input–output relationship can potentially enhance signaling by expanding the range of input signals which the network is able to transmit without saturating the response. However, to determine whether this larger input range can be resolved in the network it is crucial to examine how the noise in the response is affected.

Partitioning Reduces Noise.

Fig. 2B shows the variance of the output as a function of the mean response 〈n〉 for the system with M = N = 2, as the input signal strength α is varied. We see that the output noise is reduced in the partitioned system relative to the well-mixed system across the full range of response levels. The noise reduction is surprising: one might expect that the increased fluctuations in X∗ activity that come with partitioning would propagate to fluctuations in Y∗ activity. Indeed, this is the case: in a single compartment, as the number of molecules is reduced, the noise in the output increases (SI Appendix, Fig. S2B). However, this effect is overcome by a second effect: partitioning reduces correlations among output molecules.

To see the effect of partitioning on correlations, we consider the expressions for the variance. In the partitioned case, because the two molecules switch independently, the variance of n is simply that of a pair of independent binomial switches with activation probability 〈n〉/N,

In contrast, in the well-mixed case, the two molecules are not independent. Because both are driven by the same set of molecules, fluctuations in β_m lead to correlations between the states of the two molecules as their switching becomes more synchronized (Fig. 3). This in turn leads to an increase in the variance, which can be written as follows:

where Δ is a correction term accounting for the correlation between molecules, which is due to “extrinsic” fluctuations in the input m(t). The functional form of Δ for any M and N follows directly from the spectral solution of the CME (SI Appendix, Eq. 68); for M = N = 2, one finds by inspection that Δ is manifestly positive, meaning that correlations increase the noise across all values of the mean. Importantly, this effect is independent of the parameters of the switching reactions.

Fig. 3.

Partitioning reduces correlations between output modules. (A) In the partitioned system, each molecule receives an independent signal m_i(t). The variance is simply that of independent two-state switches. (B) In the well-mixed system, each molecule reacts to the same m(t), which leads to correlations between in the states of different molecules and an increase in the variance . Sample trajectories are generated using parameters as in Fig. 2, with α = 1.

The reduction of noise upon partitioning extends beyond the case of one molecule per partition. Indeed, the same phenomenon is observed if we consider larger molecule numbers M > π and N > π, and compare the well-mixed system to a system with uniform partitioning of the and molecules into the π compartments. In the well-mixed case, all molecules respond to the same signal m(t) and hence are correlated with all other molecules in the system. By contrast, in the partitioned case, the N/π > 1 molecules within each partition are correlated, and indeed because the fluctuations in m_i(t) will be larger than m(t) for the mixed system, such correlations will be stronger; yet the molecules in different partitions are uncorrelated. This latter effect is sufficient to overcome the increase in correlations within each partition, such that the total noise is reduced.

To see the noise reduction explicitly, we again consider the expression for the variance. Because the dynamics of different partitions is independent, assuming that both M and N are multiples of π, the variance can be written as follows:

where and are the numbers of and molecules per compartment, respectively. Here, as before, represents the additional fluctuations due to correlations between the states of molecules within each compartment. The N-dependence of , which reflects the number of correlated pairs of molecules, can be straightforwardly factored out as , where describes how strongly correlated are molecules within each compartment. The exact form for , although straightforward to calculate for a given , is difficult to generalize for all ; nonetheless, inspection of numerical and analytic results for specific combinations of and reveals in all cases that increasing π leads to an overall reduction in . Additionally, if the switching of molecules is much slower than that of molecules, γ ≪ 1, then takes the following form:

Inserting this expression into Eq. 5 with and , one can straightforwardly see that the variance is a decreasing function of π for π < N, indicating that the noise is reduced as the system is more finely partitioned.

Partitioning Increases Information Transmission.

We have seen that partitioning has two beneficial effects on signal propagation: the input–output response becomes more graded, and the output noise at a given response level is reduced. Together, these effects mean that a larger number of distinct input signals can be encoded in the network response. To quantify the ability of the network to transmit signals, we calculate the mutual information I [α, n] (18) between the input and the number of active Y* molecules, as described in SI Appendix, Methods. The mutual information quantifies the number of resolvable output signals given a particular distribution of input signals. We find that, indeed, in the case of M = N = 2 (Fig. 2), I [α, n] is significantly larger for the partitioned system (I = 0.463 bits) than for the well-mixed system (I = 0.332 bits), confirming that signal transmission is dramatically improved by partitioning.

Exchange Between Partitions Compromises Signaling Reliability.

Thus far, we have considered only the perfectly uniform and stationary partitioning of molecules. In reality, physical transport processes such as diffusion will also give rise to a variety of configurations with different numbers of proteins in each compartment, as depicted in Fig. 4. Each of these configurations will have different properties for the transmission of the signal from α to n. It is therefore important to consider whether the benefits of partitioning described above persist once these additional configurations are taken into account.

Fig. 4.

Exchange between partitions leads to different configurations of the system with a range of signaling performance. Multiplicities listed above each configuration are due to symmetry. Parameters are as in Fig. 2.

Single-molecule tracking experiments have revealed that the timescale of diffusive mixing within a compartment (∼100 μs) is two orders of magnitude faster than the timescale of molecular exchange between compartments (∼10 ms) (19). This observation allows us to treat each configuration as static on the timescale of mixing, and then compute the total response by averaging over all configurations. Inherent in this treatment is the assumption that the timescale of signaling is also faster than that of exchange between compartments. We later relax this assumption using spatially resolved simulations and nonetheless find similar results.

The total response is computed by first enumerating the possible configurations of M molecules and N molecules distributed among π partitions. For each such configuration c, we then solve for the distribution p_mn|c and combine these distributions, weighted by the probability p_c of each configuration occurring if molecules are randomly assigned to different partitions with uniform and independent probability, to give the overall distribution .

Fig. 2B (dot-dashed curve) shows that the exchange of molecules between compartments increases the noise relative to the perfectly partitioned system considered previously when M = N = 2. This is because many of the alternative configurations generated by exchange lead to significant correlations between the states of the different molecules. Nevertheless, we see that the noise remains lower than that of the well-mixed system, because of the existence of some configurations in which the molecules are independent. However, the appearance of alternate configurations also affects the mean response (Fig. 2A); in particular, the appearance of configurations in which and molecules do not occupy the same partitions, and hence no signal can be propagated, means that the maximal output level is reduced. Given this simultaneous change in both the input–output function and the noise, it is not immediately clear whether signaling reliability is improved relative to the well-mixed system. Computing the mutual information, we see that the information transmitted by the system with exchange (I = 0.213 bits) is significantly lower than that for the well-mixed system (I = 0.332 bits), showing that the reduction of the output range compromises signal transmission to an extent that cannot be overcome by the corresponding reduction in noise.

The decrease in information transmission upon incorporating molecule exchange in the system with M = N = 2 is the result of the appearance of suboptimal protein configurations, for which signal propagation is compromised (or even impossible). However, the number and performance of such configurations will in general depend on the relative values of M, N, and π (which need not equal M or N). Although molecule exchange may make partitioning unfavorable in the extreme case of M = N = 2, for systems with higher protein numbers it can be beneficial to partition the system into π > 1 compartments, as we will see next.

Optimal Partition Size.

To study the performance of systems with higher protein numbers and different partition sizes, we compare the information transmission, including molecule exchange, for different partition numbers π as the number of proteins in the system is varied while holding M = N. Fig. 5A shows that for M = N > 3 protein copies, systems with π > 1 partition do indeed outperform the well-mixed system. Furthermore, as M = N is increased, the optimal partition number also increases such that the optimal number of proteins per partition M/π∗ = N/π∗ ∼ 3 is roughly constant (Fig. 5B). This result is robust to variations in β and γ: changing each over several orders of magnitude results in optimal partition sizes in the range M/π∗ = N/π∗ ∼ 1–10 (SI Appendix, Fig. S3 A and B). The assumption of M = N is also not crucial for this result. In fact, we find that the value of M/π∗ has only a weak dependence on N (SI Appendix, Fig. S4).

Fig. 5.

An optimal partition size. (A) For M = N > 3 molecules, a system with π > 1 partitions achieves higher information transmission that a well-mixed system (π = 1). (B) As M = N is increased, the optimal partition number also increases such that the optimal number of proteins per partition M/π∗ = N/π∗ ∼ 3 is roughly constant. Parameters are as in Fig. 2.

The optimal partition size arises from a trade-off between the reliability and efficiency of signaling. Increasing the number of partitions decreases the typical number of proteins per partition, which leads to the beneficial effects of a more graded response and reduced noise, increasing signaling reliability. However, due to molecule exchange, reducing the number of molecules per partition also increases the probability that any partition contains proteins of only one species that are therefore excluded from the signaling process, which leads to a reduced maximal response, reducing signaling efficiency.

The optimal size revealed by our study of ∼1–10 molecules per species per partition shows good quantitative agreement with the observed aggregation of CD59 receptors [3–9 molecules (7, 8)] and Ras proteins [6–8 molecules (9, 10)], which each signal via the present motif and are known to interact with rafts and the cytoskelton. It is of further interest that a recent experiment in which T-cell receptors were artificially partitioned on supported membranes found that the minimum number of agonist-bound receptors per partition necessary for downstream signaling is approximately four (20).

Explicitly Spatial Model.

Last, we confirm that the effects observed in these minimal model systems, where the contents of each compartment are well mixed and exchange can occur between any pair of compartments, persist in a more realistic model in which the diffusion of molecules in space is included explicitly. We simulate the diffusion and reaction of and molecules on a 2D lattice, as described in SI Appendix, Methods. The system is partitioned into a number of subdomains by the introduction of diffusion barriers, which are crossed with a reduced probability p_hop relative to regular diffusion steps on the lattice. Results of such simulations are shown in Fig. 6.

Fig. 6.

The effects of partitioning persist in simulations with explicit diffusion. As the probability of crossing a diffusion barrier p_hop is decreased, (A) the mean response becomes more graded, and (B) the output noise decreases. (C) The information transmission has a maximum as a function of the partition size. Here M = N = 49, β = 20, γ = 1, the system is λ = 70 lattice spacings squared, and the ratio of diffusion to reaction propensities is p_D/p_r = 1. In A and B, π = 49; in C, p_hop = 0.001, and the partition size is varied by taking from 25 to 1.

Fig. 6 A and B reveal that as the strength of the diffusion barriers is increased, the mean response becomes more graded, and the variance of Y∗ activity is reduced, analogous to the two effects observed in the minimal model system (Fig. 2). When p_hop = 0, one molecule of each species is permanently confined to a compartment, producing the graded response predicted for the perfectly partitioned system (Fig. 6A) and the associated minimal, binomial noise (Fig. 6B). Low but finite p_hop allows exchange of molecules between neighboring compartments but preserves a separation of timescales between intracompartment and intercompartment mixing. This results in a graded mean response whose maximal level is reduced (Fig. 6A) and reduced noise (Fig. 6B), precisely the features observed in the minimal model of partitioning with exchange (Fig. 2). When p_hop = 1, there are no barriers, and the system approaches the well-mixed limit (CME). Interestingly, however, the response remains more graded and the noise remains lower than the predictions of the CME due to the finite speed of diffusion (Fig. 6 A and B), with agreement only reached when the ratio of diffusion to reaction propensities is much greater than 1. This observation reveals that finite diffusion imposes an effective partitioning even when no actual partitions exist: molecules remain correlated with reaction partners within a typical distance set by diffusion, but uncorrelated with partners beyond this distance. As such, in the context of coupled reversible modification, we find that slower diffusion can linearize the response and reduce the noise, thereby improving information transmission. Interestingly, this result is in marked contrast to the case of boundary establishment in embryonic development, where faster diffusion reduces noise within each nucleus by washing out bursts of gene expression in the input signal (21). Although in the present system faster diffusion will similarly reduce any super-Poissonian component of the noise within each partition individually, this averaging does not reduce the noise in the total output across all partitions. In fact, the latter noise is enhanced with faster diffusion by virtue of increased correlations between partitions.

While slower diffusion can lead to effective partitioning in a system without explicit diffusion barriers, the noise reduction in such a system is much smaller than that in a system with actual partitions. Fig. 6B shows that finite diffusion reduces the maximal noise by (1.25 − 1)/1.25 = 20%, whereas strong partitioning (p_hop = 0.001) reduces the maximal noise by (1.25 − 0.4)/1.25 ∼ 70%. Therefore, partitioning, which introduces not only a slower effective “hop” diffusion but also a separation of tiescales between intracompartmental and intercompartmental mixing, is far more effective at conveying an information enhancement.

Fig. 6C confirms that the transmitted information varies nonmonotonically with the number of barriers in a fixed area, indicating that an optimal partition size also appears in systems where space is modeled explicitly. Like in the minimal model, this optimum persists with changes in β and γ, spanning the range of ∼1–10 molecules per partition (SI Appendix, Fig. S3 C and D). Fig. 6C also provides a measure of the scale of information transmitted by this motif. In absolute terms, the optimal information (1.35 bits) is consistent with values recently measured for signaling via the TNF–NF-κB pathway (∼0.5–1.5 bits) (22) and for patterning in the Drosophila embryo (1.5 ± 0.15 bits) (23). In relative terms, we see that partitioning increases information over the unpartitioned system by (1.35–1.04)/1.04 ∼ 30% (Fig. 6C) and decreases the maximal noise by (1 − 0.4)/1 = 60% (Fig. 6B). Thus, in both absolute and relative terms, we see that partitioning plays a critical role in producing informative and reliable membrane signaling.

As a final test, we use simulation to confirm that the effects of partitioning persist in the presence of features that are more realistic for signaling systems at the membrane, including extrinsic noise in the input (SI Appendix, Fig. S5) and receptor dimerization (SI Appendix, Fig. S6). The fact that the effects of partitioning, including the emergence of an optimal partition size, are robust to these details further underscores the generality of our findings.

Discussion

We have seen that the partitioning of a biochemical signaling system into a number of noninteracting subsystems improves the reliability of signaling via two effects. First, the nonlinear response of the network means that a reduction in the number of input molecules translates into a more graded input–output response. Second, partitioning significantly reduces the noise in the response by eliminating correlations between the states of the different output molecules, an effect that, remarkably, overcomes the increase in noise associated with fewer input molecules in each subsystem. However, we have seen that the introduction of diffusion or exchange of molecules between partitions enhances the variance and reduces the range of the response, thereby reducing signaling performance. This result is due to the presence of configurations in which the two species are isolated from one another, compromising or even arresting signal transmission in certain partitions. The interplay between these two effects leads to a partition size that optimizes information transmission, corresponding to a few molecules per partition on average, in quantitative agreement with experiments. These effects are generic, and hence the emergence of an optimal partition size is robust to the specific parameters of the model. Notably, the underlying mechanism revealed here, namely the removal of correlations, differs fundamentally from that based on cooperativity in protein activation, which has been argued to underlie optimal cluster size in sensory systems (24, 25).

Reversible modification reactions are ubiquitous in cell signaling, and interactions with the cytoskeleton and lipids provide general mechanisms for the formation of subdomains. We therefore expect the results revealed by our study to be applicable to a wide class of signaling systems at the membrane. We have focused in this paper on coupled single-site modification reactions because this motif governs pathways specifically known to be affected by the formation of membrane subdomains. However, the effects we uncover also pertain to multisite modification reactions, which are very common in cell signaling (26–29). Moreover, we have focused on systems where the reactant species are confined by a boundary that limits diffusion. However, similar effects could be observed in systems where proteins are localized to raft domains, or even scaffolds or large macromolecular complexes. In the latter case, each complex would effectively provide an independent reaction “compartment,” and the exchange between compartments would be the result of rare dissociation events, after which proteins could diffuse rapidly through the cytoplasm to a different complex. Even if the signal within each complex was not mediated via diffusive encounters, but rather via cooperative or allosteric interactions, the fundamental mechanism that we reveal here—that partitioning into subsystems removes correlations between subsystems—remains at play. The presence of scaffolds and macromolecular complexes at early stages of signaling pathways is extremely common (30), suggesting that the effects discussed here are of wide biological relevance.

Materials and Methods

The CME (1) is solved using the method of spectral expansion (15, 16). Details of this method, the computation of mutual information, and the spatial simulations are described in SI Appendix. Source code, written in MATLAB, C++, and Mathematica, used to generate all results and figures in the main text and SI Appendix, is freely available at http://partitioning.sourceforge.net.