Regulation of reaction fluxes via enzyme sequestration and co-clustering

Abstract

Experimental observations suggest that cells change the intracellular localization of key enzymes to regulate the reaction fluxes in enzymatic networks. In particular, cells appear to use sequestration and co-clustering of enzymes as spatial regulation strategies. These strategies should be equally useful to achieve rapid flux regulation in synthetic biomolecular systems. Here, we leverage a theoretical model to analyse the capacity of enzyme sequestration and co-clustering to control the reaction flux in a branch of a reaction–diffusion network. We find that in both cases, the response of the system is determined by two dimensionless parameters, the ratio of total activities of the competing enzymes and the ratio of diffusion to reaction timescales. Using these dependencies, we determine the parameter range for which sequestration and co-clustering can yield a biologically significant regulatory effect. Based on the known kinetic parameters of enzymes, we conclude that sequestration and co-clustering represent a viable regulation strategy for a large fraction of metabolic enzymes, and suggest design principles for reaction flux regulation in natural or synthetic systems.

1. Introduction

Cells must exert rapid and precise control over the flux of molecules in their reaction networks [1], and similar tasks are pursued in synthetic biology [2]. At a branch point in a reaction network, where an intermediate product is used in two or more competing pathways, it is often necessary to regulate the reaction flux into each branch, e.g. in response to a change in the environmental conditions. One way to regulate reaction fluxes, albeit slowly, is to adjust enzyme abundances, by controlling transcription, translation or degradation rates. More rapid flux regulation can be achieved by adjusting enzyme activities, via allosteric mechanisms or post-translational modifications [3–5]. However, an increasing amount of evidence suggests that cells complement these biochemical regulation mechanisms with mechanisms based on the spatial rearrangement of enzymes [6–8]. First synthetic systems have also been created, in which a metabolic flux is regulated by controlling the clustering of enzymes [2]. Here, we focus on the capacity of such physical mechanisms to regulate reaction fluxes at a branch point in a reaction network.

Many enzymes have been found to form intracellular aggregates [7,9,10], through diverse mechanisms that include direct binding between the enzymes or with scaffolds, adsorption to specific spots on the cellular membrane or the cytoskeleton, or liquid-droplet phase separation [9,11–16]. Importantly, assembly and disassembly of enzyme clusters is often reversible upon depletion or supply of metabolites or other environmental stimuli [7,8,17,18]. This suggests that cells use enzyme localization as a strategy to regulate biochemical reaction fluxes dynamically [9,19]. In particular, co-clustering of enzymes that operate sequentially within the same reaction pathway may upregulate the pathway flux [20], whereas the sequestration of enzymes may downregulate the pathway flux, as illustrated in figure 1.

Figure 1.

Regulation of reaction fluxes via changes in the intracellular localization of enzymes. At a branch point in a reaction network, the sequestration of one downstream enzyme (here called E_P) into clusters can direct the reaction flux away from the reaction catalysed by the sequestered enzyme. By contrast, co-clustering of the upstream enzyme E_I with the downstream enzyme E_P can bias the flux towards the product P. We analyse the effectiveness of these spatial regulation strategies within a reaction–diffusion model. Geometrical parameters of this model are illustrated on the right-hand side. (Online version in colour.)

A well-known example for the co-clustering strategy is the purinosome, a multi-enzyme cluster comprising enzymes of the de novo purine biosynthesis pathway [17,21]. Co-clustering has also been observed for enzymes involved in glucose metabolism and de novo pyrimidine biosynthesis [22–25]. In some such assemblies, intermediate products are directly ‘channelled’ or transferred between enzymes [20]. However, theoretical analyses have shown that proximity of consecutive enzymes is sufficient to increase the efficiency of diffusive transfer, without the need for direct or guided transfer of intermediate products [26–28]. Furthermore, it has been demonstrated experimentally that the efficiency enhancement obtained by co-clustering can change the ratio of fluxes at a branch point in a reaction network [28].

The reversible sequestration of individual enzymes into filaments or clusters has been observed in the de novo CTP and purine biosynthesis pathways [7,19], and was often associated with downregulation of the corresponding pathway [29,30]. For example, FGAMS, one of the six enzymes of the de novo purine biosynthesis pathway, can be sequestered into a cluster, leading to downregulation of the pathway [30]. Thus, this pathway appears to have the capacity for two spatial regulation strategies in response to changing environmental conditions: the assembly of all pathway enzymes into the purinosome as a mechanism for pathway upregulation, and the sequestration of a single enzyme into a cluster as a mechanism for pathway downregulation [19]. While the mechanistic origin of this downregulation remains unclear, the spatial sequestration alone may suffice, without the need for reducing the catalytic activities of the enzymes.

Here, we analyse the reaction–diffusion dynamics associated with enzyme sequestration and co-clustering to assess the capacity of these mechanisms to regulate reaction fluxes. Prior theoretical work on the reaction–diffusion dynamics of spatially arranged enzymes [26–28,31] has primarily focused on the question of how the reaction flux can be maximized, or considered fixed parameter values to describe specific experimental situations. By contrast, our focus here is on the question of flux regulation. Furthermore, while the prior studies considered co-clustering, they did not analyse the sequestration of enzymes as a spatial regulation strategy. Here, we treat co-clustering and sequestration on an equal footing and explore their flux regulation capacity over the entire parameter space. Our analysis reveals that in both cases, the strength of the regulatory effect, i.e. the fold-change in the reaction flux, is determined by two quantitative characteristics, which are dimensionless combinations of the biochemical and physical parameters of the system: (i) the ratio of the total activities of the two competing enzymes and (ii) a ratio of diffusion to reaction timescales. This result leads us to an approximate criterion for whether a given enzyme with known biochemical parameters will permit significant flux regulation by co-clustering or sequestration. The criterion is general, based only on the assumption of diffusive substrate transfer, i.e. without invoking direct or guided transfer of intermediate products. To illustrate the use of this criterion, we apply it to the well-studied case of purine biosynthesis, and also to a large dataset of biochemically characterized enzymes.

2. Results and discussion

The spatial flux regulation strategies considered here rely on the fact that changes in the localization of enzymes can alter the concentration profile of intermediate products, which in turn determines the reaction fluxes produced by enzymes that further process the intermediate product. Clearly, if all enzymes using the intermediate product are saturated irrespective of their location, spatial regulation strategies can have no effect. For the purpose of our study, it is therefore important to verify that the relevant enzymes are not all saturated. Further below, we will consider this question in some detail for the example of the branch point at the end of the purine biosynthesis pathway. On a global level, the results from a recent study [32] indicate that in a substantial fraction of metabolic reactions (approx. 30%), the cellular metabolite concentration is smaller than the Michaelis constant of the enzyme (fig. 6a of [32]). This observation suggests that an analysis of spatial regulation strategies is broadly relevant, both for understanding natural biological systems and for the design of synthetic systems.

2.1. Reaction flux regulation at a branch point

In the following, we consider a branch point in a reaction network, as illustrated in figure 1, bottom. Intermediate products I, produced by an upstream enzyme E_I, are processed by two downstream enzymes, E_P and E_Q, into two distinct products, P and Q. We assume both E_P and E_Q to be in the regime of low substrate saturation, such that their reaction fluxes are susceptible to spatial regulation (and approximately linear in the substrate concentration). We define the branching fraction J_p as the fraction of I that is converted into P, and similarly J_q = 1 − J_p as the fraction that is converted into Q.

Our reference scenario is that of no spatial regulation, i.e. a uniform distribution of all three enzymes, illustrated in figure 1a (‘well-mixed’ enzymes). The branching fractions for this reference scenario, $J_{\hspace{0.17em}p}^0$ and $J_q^0$, take on the simple form

$$J_{\hspace{0.17em}p}^0=\frac{\alpha }{1+\alpha }\text{and}{J}_q^0=\frac{1}{1+\alpha },$$ 2.1

where α is defined as

$$\alpha =\frac{k_{\hspace{0.17em}p}[E_P]}{k_q[E_Q]},$$ 2.2

with k_p, k_q denoting the catalytic efficiencies (k_cat/K_M) of the competing downstream enzymes E_P and E_Q, and [E_P], [E_Q] their concentrations. The dimensionless quantity α therefore corresponds to the ratio of the total activities of the two enzymes, i.e. it quantifies the intrinsic bias of the branched pathway at given enzyme concentrations and catalytic efficiencies.

The α-dependence of the branching fraction $J_{\hspace{0.17em}p}^0(\alpha )$ can be regarded as a regulation function for non-spatial control of flux branching towards the product P. This function describes regulation via changes of the enzyme concentrations [E_P], [E_Q] or via changes of their catalytic efficiencies k_p, k_q by allosteric or other means, all affecting the ratio α according to equation (2.2). As shown in figure 2, black curve, the splitting of intermediates between the two branches is symmetric about α = 1, with $J_{\hspace{0.17em}p}^0<J_q^0$ when the total activity of E_Q is greater than that of E_P (α < 1) and vice versa. However, this symmetry will be broken when the spatial arrangements of the downstream enzymes differ. The pathway flux can be biased towards P and away from Q by co-clustering E_I with E_P to decrease the average distance between these sequential enzymes, as illustrated in figure 1c. Conversely, the flux can be shunted away from P by sequestering only the E_P enzymes into clusters, which increases the E_I − E_P distance (figure 1b). We now turn to the description of such effects, which amount to post-translational flux regulation by spatial localization.

Figure 2.

Branching fraction J_p as a function of the enzyme activity ratio α for the enzyme sequestration strategy (green solid curve), the enzyme co-clustering strategy (orange solid curve) and for a well-mixed system of uniformly distributed enzymes (black curve). The dashed lines show the corresponding approximative forms in the limit [E_P]/c_c,s ≪ 1, equation (2.7). The parameter values are k_p = k_q = 10⁵ M⁻¹ s⁻¹, c_c,s = 25 mM, r_c = r_s = 0.4 μm, and D = 100 μm² s⁻¹. (Inset) The behaviour of the dimensionless function γ(β), introduced in equation (2.7) and defined in equation (2.9). (Online version in colour.)

2.2. Reaction–diffusion dynamics of spatial flux regulation

To analyse the regulation capacity of both localization strategies, i.e. enzyme co-clustering and enzyme sequestration, we consider a model that describes the reaction–diffusion dynamics within a spherical domain with a single central enzyme cluster, as illustrated in figure 1. This domain represents the spatial region around a typical example of one of several such clusters within a larger cell. Since we are interested in the branching fraction (rather than the absolute magnitude) of the fluxes, we only need to consider a single, representative domain. We denote the radius of this domain by R, which corresponds to half of the typical distance between clusters. We consider the typical cluster radius, denoted by r_s for E_P sequestration and r_c for E_P − E_I co-clusters, as a fixed intrinsic property of the particular enzymes and their assembly processes. The associated fixed local concentration of E_P enzymes within each type of cluster is denoted by c_s and c_c, respectively.

As a reference for the spatial regulatory effect, we take the spatially uniform, well-mixed scenario (see previous section). Hence, we consider $J_{\hspace{0.17em}p}^0$ as the basal, unregulated level of the branching fraction, to be compared with the J_p value obtained for each localization strategy. Importantly, we always compare branching fractions obtained for the same total number of enzymes and the same catalytic efficiency, assuming that the enzymes are merely rearranged for spatial regulation. This constraint relates the domain radius R, the cluster radii r_c,s and the local E_P concentrations c_c,s within the clusters to the concentration [E_P] of the well-mixed scenario via R = r_c,s(c_c,s/[E_P])^1/3.

We assume that I molecules move by diffusion and denote their diffusion coefficient by D. The steady-state radial concentration profile of I molecules, denoted as ρ(r), must then satisfy the local diffusion–reaction balance

$$D{\mathrm{\nabla }}^2\rho (r)+F\hspace{0.17em}\hspace{0.17em}[\rho (r),r]=0,$$ 2.3

where the reaction term F [ρ(r), r] depends on the localization strategy. For sequestration of E_P, the reaction term is

$$F^{\mathrm{seq}}\hspace{0.17em}\hspace{0.17em}[\rho (r),r]=\left\{\begin{array}{cc}{j}_s-k_q\hspace{0.17em}{e}_q\hspace{0.17em}\rho (r)& r>r_s\\ -k_{\hspace{0.17em}p}\hspace{0.17em}{c}_s\rho (r)& r\le r_s,\end{array}\right.$$ 2.4

where j_s denotes the production flux of intermediates by E_I outside the E_P cluster and $e_q=[E_Q]/(1-r_s^3/R^3)$ is the concentration of E_Q outside the cluster (chosen to guarantee that the mean concentration over the whole domain is the same as in the well-mixed case). For E_P − E_I co-clusters, the reaction term is

$$F^{\mathrm{cocl}}\hspace{0.17em}\hspace{0.17em}[\rho (r),r]=\left\{\begin{array}{cc}{j}_c-k_{\hspace{0.17em}p}\hspace{0.17em}{c}_c\hspace{0.17em}\rho (r)& r\le r_c\\ -k_q\hspace{0.17em}{e}_q\hspace{0.17em}\rho (r)& r>r_c,\end{array}\right.$$ 2.5

where j_c is the production flux of I in the cluster, while $e_q=[E_Q]/(1-r_c^3/R^3)$ is the concentration of E_Q between the clusters. Local conservation of mass requires that the radial concentration profile ρ(r) is continuous and smooth at the interface of the clusters and the bulk, i.e. at r_s or r_c. At the boundaries, we have reflective (no-flux) boundary conditions, such that ρ′(0) = ρ′(R) = 0. At the outer boundary, this boundary condition represents symmetric exchange of intermediates I with the adjacent domains around neighbouring clusters.

For both localization strategies, the model can be solved analytically to determine the steady-state fluxes. In particular, the branching fraction towards the product P is obtained as the ratio between the absolute flux towards P and the production flux of I,

$$J_{\hspace{0.17em}p}^s=\frac{{\int }_0^{r_s}{r}^2{k}_{\hspace{0.17em}p}{c}_s\rho (r)\hspace{0.17em}\text{d}r}{{\int }_{r_s}^R{r}^2{j}_s\hspace{0.17em}\text{d}r}\text{and}{J}_{\hspace{0.17em}p}^c=\frac{{\int }_0^{r_c}{r}^2{k}_{\hspace{0.17em}p}{c}_c\rho (r)\hspace{0.17em}\text{d}r}{{\int }_0^{r_c}{r}^2{j}_c\hspace{0.17em}\text{d}r}.$$ 2.6

In general, these branching fractions depend separately on the mean enzyme concentrations [E_P] and [E_Q] (see appendix A for the full solutions). However, in the regime of typical biological parameter values, with [E_P] and [E_Q] (nano- to micromolar range) much smaller than the local concentrations c_c,s inside the clusters (millimolar range), the branching fractions depend on [E_P] and [E_Q] only via the total enzyme activity ratio α of equation (2.2), as we will see below (and in appendix A).

2.3. Effects of sequestration and co-clustering on the regulation function

Figure 2 compares the α-dependence of the branching fraction J_p for the three localization strategies (well-mixed, co-clustered and sequestered) for a representative parameter set (see caption). Across the full α-range, co-clustering of E_I and E_P increases J_p compared to the well-mixed system, although the enhancement becomes negligible at α > 10. By contrast, sequestration of E_P generally decreases J_p (see also equation (2.10) further below). At the same α-value, these two effects correspond to spatial up- and downregulation of J_p, respectively. For instance, for α = 0.5 co-clustering increases the flux towards P about twofold, while sequestration of E_P reduces the flux about twofold. The magnitude of these effects grows with the number of enzymes in each cluster, irrespective of whether the cluster size r_c,s or the enzyme concentration c_c,s in the cluster is increased (see appendix A). For co-clustering, the branching fraction tends to a finite value for α ≪ 1. In this parameter regime, the largest enhancement compared to a well-mixed system can be achieved.

To characterize and understand the behaviour of the branching fractions, we Taylor expand the full analytical solution of appendix A, using the ratio [E_P]/c_c,s as the small parameter. The leading-order expressions for the branching fractions take the simple forms

$$J_{\hspace{0.17em}p}^s=\frac{\alpha }{1+\alpha +{\beta }_s\gamma ({\beta }_s)}\text{and}{J}_{\hspace{0.17em}p}^c=\frac{\alpha +{\beta }_c\gamma ({\beta }_c)}{1+\alpha +{\beta }_c\gamma ({\beta }_c)},$$ 2.7

for sequestration and co-clustering, respectively, which are shown as dashed lines in figure 2. Here, the dimensionless parameters β_s and β_c are defined as

$${\beta }_c=\frac{k_{\hspace{0.17em}p}\hspace{0.17em}{c}_c\hspace{0.17em}{r}_c^2}{3D}\text{and}{\beta }_s=\frac{k_{\hspace{0.17em}p}\hspace{0.17em}{c}_s\hspace{0.17em}{r}_s^2}{3D},$$ 2.8

and correspond to the ratio of two timescales, the typical time for I molecules to diffuse through the cluster, ${\tau }_d\sim r_{c,s}^2/D$, and the timescale for I to react with E_P enzymes within the cluster, τ_r ∼ (k_pc_c,s)⁻¹. Additionally, in equation (2.7) we have introduced the dimensionless function

$$\gamma (\beta )=1-{\beta }^{-1}+{\left[\sqrt{3\beta }\coth \left(\sqrt{3\beta }\right)-1\right]}^{-1},$$ 2.9

which depends only weakly on β, with values varying between 1 and 1.2 (figure 2, inset).

We now turn to the biophysical interpretation of the above results. By comparing the form of the regulation functions (2.7) with the one for the well-mixed case, equation (2.1), we immediately see that co-clustering effectively shifts the regulation function to the left, according to α → α + β_cγ(β_c). While this does not change the shape of the regulation function, it cuts off the low α-regime of $J_{\hspace{0.17em}p}^0(\alpha )$, since the total enzyme activity ratio α is necessarily positive. In the case of sequestration, we can rewrite equation (2.7) as

$$J_{\hspace{0.17em}p}^s=\frac{\alpha /(1+{\beta }_s\gamma ({\beta }_s))}{1+\alpha /(1+{\beta }_s\gamma ({\beta }_s))},$$

which by comparison with equation (2.1) shows that sequestration effectively stretches the α-axis of the regulation function according to α → α/(1 + β_sγ(β_s)). Furthermore, we can interpret the function γ(β) of equation (2.9) physically using a coarse-grained approximation of the full reaction–diffusion dynamics. Towards this end, we consider the cluster and the bulk region as two separate domains, in each of which the intermediates are well mixed. We describe transport between the domains with an effective permeability, D/Δ, where Δ is an effective exchange length scale. By comparing the expressions for the branching fractions obtained from this approximation (see appendix A) with equation (2.7), we find that γ corresponds to the exchange length scale rescaled by the cluster radius, γ(β_c,s) = Δ/r_c,s. Therefore, γ(β) is a dimensionless measure for the efficiency of the transport of I molecules between the clusters and their surroundings.

2.4. Branch point regulation factors

For both clustering strategies, the effect on the branching fraction is greatest when reactions within the cluster are faster than exchange between domains (β ≫ 1), whereas there is little effect on the branching fraction when exchange is fast compared with reactions (β ≪ 1). We quantify the regulatory effect of the two enzyme clustering strategies by the spatial regulation factor ${\eta }_{c,s}=J_{\hspace{0.17em}p}^{c,s}/J_{\hspace{0.17em}p}^0$, which measures the fold-change of the branching fraction towards P for clustering relative to the well-mixed case. These regulation factors take the form

$$\begin{array}{rl}{\eta }_s& =1-\frac{{\beta }_s\gamma ({\beta }_s)}{1+\alpha +{\beta }_s\gamma ({\beta }_s)}\text{and}\\ {\eta }_c& =1+\frac{{\beta }_c\gamma ({\beta }_c)/\alpha }{1+\alpha +{\beta }_c\gamma ({\beta }_c)},\end{array}$$ 2.10

for enzyme sequestration and co-clustering, respectively (implying that η_s < 1 and η_c > 1). The strongest upregulation by enzyme co-clustering is achieved when two conditions are met: (i) the average enzymatic activity of E_Q is much larger than of E_P (small α) and (ii) the enzymatic activity in the cluster is large compared with the rate of diffusive exchange (large β_c), see figure 3a. On the other hand, when the average enzymatic activity of E_P is much larger than that of E_Q ($\alpha \hspace{0.17em}\gtrsim \hspace{0.17em}5$) the pathway is strongly biased towards P already with uniformly distributed enzymes, and hence the enhancement from enzyme co-clustering becomes negligible.

Figure 3.

Capability of the spatial strategies to regulate the flux distribution at the branch point. (a) Enhancement towards the P branch by E_I − E_P co-clustering. Black lines divide regimes at the enhancement threshold η_c = 2, while green lines divide regimes based on the upregulated branching fraction $J_{\hspace{0.17em}p}^c=0.2$. (b) Downregulation of the P branch by enzyme sequestration. Black lines denote η_s = 0.5, green lines indicate where the branching fraction of the well-mixed system $J_{\hspace{0.17em}p}^0=0.2$. (Online version in colour.)

The spatial regulation factors, equation (2.10), quantify the strength of the regulatory effects. In order to identify the biologically interesting parameter regimes of the spatial regulation strategies, we consider a twofold regulatory effect as the minimum for biological significance. Hence, we consider upregulation by co-clustering to be biologically relevant only when the regulation factor is at least η_c ≥ 2, while for downregulation via sequestration we require η_s ≤ 0.5. However, a large regulation factor alone does not guarantee that the regulatory effect is biologically useful. The biological relevance also depends on the absolute value of the branching fraction, which is shown in figure 4a,b for the co-clustered and sequestered enzyme arrangement, respectively, again as a function of α and β. In particular, it is important for each regulation scenario that the branching fraction in the higher state is not too small to be significant. For instance, strong downregulation would be of no significance, if the basal branching fraction was already small. As an exemplary threshold for biological significance we consider J_p ≥ 0.2, i.e. in the case of co-clustering we require the branching fraction in the upregulated state to be at least $J_{\hspace{0.17em}p}^c\ge 0.2$, and in the case of sequestration we require the basal branching fraction of uniformly distributed enzymes to be at least $J_{\hspace{0.17em}p}^0\ge 0.2$.

Figure 4.

Branching fractions J_p for the (a) co-clustering and (b) sequestration scenarios as a function of α and β_c,s. (Online version in colour.)

Together, these constraints render the parameter region delimited by the solid lines in figure 3 as biologically relevant. For the case of co-clustering, the minimum relevant value of β_c is independent of α as long as α ≲ 0.1. When β_c is above this minimum threshold, the co-cluster is sufficiently large and dense, such that the required fraction of I molecules is converted to P before diffusing out of the cluster. For the case of enzyme sequestration, the strongest downregulation of J_p is achieved for large β_s and α ≤ 1, which however lies outside the biologically relevant parameter regime (top left in figure 3b). Nevertheless, sequestration can suppress the production of P more than a 100-fold within the relevant regime, where $J_{\hspace{0.17em}p}^0\ge 0.2$.

Interestingly, for both localization strategies, and for either up- or downregulation of J_p, the criteria combine to produce a lower bound on the timescale ratio β of equation (2.8) in a similar range

$$\beta =\frac{k_{\hspace{0.17em}p}\hspace{0.17em}{c}_{c,s}\hspace{0.17em}{r}_{c,s}^2}{3D}\hspace{0.17em}\gtrsim \hspace{0.17em}1.$$

Note that β depends only on the molecular properties of the enzyme P (its catalytic efficiency k_p and diffusion constant D) and the physical properties of the clusters (their radius r_c,s and local enzyme concentration c_c,s). Hence the condition $\beta \hspace{0.17em}\gtrsim \hspace{0.17em}1$ is a molecular design constraint, i.e. only enzymes with a combination of physical and biochemical parameters that satisfy this constraint will be suitable for regulation of reaction fluxes via enzyme sequestration and co-clustering. The biologically relevant parameter regime in figure 3 also constrains the total activity ratio α. However, this constraint is less limiting, since α is more easily changed to fall within the desired regime by adjusting the concentrations of the enzymes P and Q (cf. equation (2.2)).

2.5. Application of the theoretical framework

To apply the general framework presented above, we first consider the example of the branch point at the end of the de novo purine biosynthesis pathway (figure 5). The product of the purine biosynthesis pathway, inosine monophosphate (IMP), can be processed further in two competing pathways, to produce the nucleoside monophosphates GMP or AMP. The two competing enzymes at this branch point are inosine-5’-monophosphate dehydrogenase (IMPDH) and adenylosuccinate synthase (ADSS). These competing enzymes display a high degree of spatial organization. The enzyme IMPDH, in particular, has been found to either colocalize with the purinosome or form large filamentous structures under different conditions [33]. The enzyme ADSS has also been observed to colocalize with the purinosome [33]. Thus, this branch point appears to have the capacity for two spatial regulation strategies in response to changing environmental conditions: co-clustering with the purinosome as a mechanism for upregulation, and sequestration of a single enzyme into a cluster as a mechanism for downregulation.

Figure 5.

Schematic of the branch point at the end of the de novo purine biosynthesis pathway. Experimentally observed condition-dependent enzyme localizations, featuring both co-clustering and sequestration, are indicated by the dashed arrows. (Online version in colour.)

The enzyme kinetics of the branch point enzymes is relatively well characterized. The kinetic parameters of human IMPDH are k_cat = 1.8 s⁻¹ and K_M = 14 μM for type I IMPDH and k_cat = 1.4 s⁻¹ and K_M = 9 μM for type II IMPDH [34], which leads to a catalytic efficiency of k_cat/K_M ≈ 10⁵ M⁻¹ s⁻¹ in both cases. For ADSS, the kinetic parameters are k_cat = 5.4 s⁻¹ and K_M = 45 μM [35], leading to k_cat/K_M ≈ 3.5 × 10⁵ M⁻¹ s⁻¹. The intracellular concentration of IMP has been determined for HeLa cells to 0.02 nmol/(million cells) under purine rich conditions and to 0.06 nmol/(million cells) under purine-depleted conditions [33]. With a typical volume ∼4 × 10³ μm³ of a HeLa cell [36], this converts to 5 μM and 15 μM under purine rich and depleted conditions, respectively. Thus, the concentration of IMP in both conditions is of the order of the Michaelis constant of IMPDH and ADSS, which indicates that the enzymes are not saturated under typical conditions, such that the corresponding fluxes remain sensitive to the spatial organization of the enzymes. By contrast, the cofactor NAD⁺ required by IMPDH has a cellular concentration of around 0.3 mM [37], while the Michaelis constant of IMPDH for NAD⁺ is around K_M ≈ 0.07 mM [38], such that the flux of the GMP branch of the pathway is expected to be insensitive to small variations in the concentration of this cofactor.

Taken together, the literature values on the enzyme kinetics of the branch point enzymes in the de novo purine biosynthesis pathway suggest that the key assumptions of our theoretical framework are satisfied for this system. In order to verify whether this system is also in the parameter regime where co-clustering and sequestration have a biologically significant effect, we revisit the criterion $\beta \hspace{0.17em}\gtrsim \hspace{0.17em}1$ for biological significance obtained in the previous section. Assuming a local enzyme concentration within clusters of c_c,s ∼ 25 nM [28], a cluster radius of r_c,s ∼ 0.4 μm [30] and a diffusion coefficient D ∼ 100 μm² s⁻¹ for a small metabolite [39], this lower bound translates into a threshold for the catalytic efficiency of the clustered enzyme of

$$\frac{k_{\mathrm{cat}}}{K_M}\hspace{0.17em}\gtrsim \hspace{0.17em}\frac{3D}{c_{c,s}{r}_{c,s}^2}\approx {10}^5\hspace{0.17em}\text{M}\hspace{0.17em}{\text{s}}^{-1}.$$ 2.11

According to the literature values (see above), both ADSS and IMPDH satisfy the criterion (2.11). Thus, we expect that co-clustering (with the purinosome) and sequestration of these enzymes will have a significant post-transcriptional regulation effect, purely from the spatial rearrangement of the enzymes, i.e. without requiring additional allosteric interactions. Note that based on this logic, one would expect single cells to never simultaneously sequester IMPDH into filaments and co-cluster in purinosomes, since simultaneous usage of two antagonistic regulatory effects would appear to be counterproductive. By contrast, mixtures of diffuse IMPDH with sequestered filaments, or mixtures of diffuse IMPDH with co-clustered purinosomes would be useful intermediate states, enabling gradual regulation. To the best of our knowledge, the experimental data required to verify this expectation are currently not yet available.

How broadly are the spatial regulation strategies of co-clustering and sequestration applicable in cells? In other words, what fraction of enzymes with known enzyme kinetics satisfies the criterion of equation (2.11)? To address this question, we relied on a quantitative survey of enzymes [40]. These data suggest that approximately 70% of enzymes in carbohydrate metabolism and approximately 50% of other classes of metabolic enzymes satisfy the bound of equation (2.11). Given that about 30% of all metabolic reactions fulfil the additional requirement that the in vivo metabolite concentration is smaller than the K_M of the enzyme (see above), our criterion suggests that both sequestration and co-clustering are viable flux regulation strategies for a large fraction of metabolic enzymes.

3. Conclusion

According to our model and analysis, the control that can be exerted on a reaction–diffusion system by clustering of catalysts can be understood in simple physical terms. As long as the clusters are well separated (such that [E_P]/c_c,s ≪ 1), the branching fraction of both clustering strategies (sequestration or co-clustering of catalysts) is determined by two dimensionless parameters, (i) the ratio of the total activities of the two competing enzymes and (ii) the relative timescales of diffusion through and reactions within a cluster. We have seen that both strategies can achieve significant levels of regulation for realistic parameters and for a substantial fraction of natural enzymes. Taken together, our findings help to interpret the design of biological systems and suggest design principles for synthetic systems.

The general perspective that we took here is also relevant from an evolutionary angle: in order for more specialized mechanisms such as direct channelling or guidance of intermediate products to evolve, spatial proximity must already have been established. Since we found that a functional effect of spatial proximity is expected already for generic enzymes within a broad parameter regime, this functional effect may already provide a selective force for clustering. After clustering has evolved, more specialized mechanisms that build upon spatial proximity can then be explored by evolution.

Appendix A

(a) Analytic solution for the full reaction–diffusion model

Our model is based on the assumption of a larger system, in which identical clusters are arranged with a typical spacing, such that the system can be partitioned into domains as illustrated in figure 1. Each domain contains a cluster positioned at its centre. Since in this work we are only interested in the branching fraction (rather than the absolute magnitude) of the fluxes, it suffices to consider a single, representative domain in isolation. By assuming that both the cluster and the domain are spherically symmetric, the reaction–diffusion dynamics, as described by equations (2.3)–(2.5) together with the boundary conditions, becomes analytically solvable. To determine the steady-state radial concentration profile of I molecules, ρ(r), we solve these equations inside and outside of the cluster, to obtain the general solutions ρ_c(r) (within the cluster) and ρ_b(r) (for the bulk outside of the cluster). We match these solutions by requiring that the concentration profile is continuous and differentiable at the interface, ρ_c(r_c,s) = ρ_b(r_c,s) and ρ′_c(r_c,s) = ρ′_b(r_c,s). Subsequently, the obtained solution for ρ(r) is inserted into equation (2.6) to determine the branching fraction.

The full analytic solution of the branching fractions for the co-clustering and sequestration strategies are given by

$$J_{\hspace{0.17em}p}^c=1-\frac{3/\mu \left[(\tilde{R}-1)\sqrt{\nu }\cosh \left((\tilde{R}-1)\sqrt{\nu }\right)+(\tilde{R}\nu -1)\sinh \left((\tilde{R}-1)\sqrt{\nu }\right)\right]\left[\sqrt{\mu }\cosh \left(\sqrt{\mu }\right)-\sinh \left(\sqrt{\mu }\right)\right]}{\sqrt{\nu }\cosh \left((\tilde{R}-1)\sqrt{\nu }\right)\left[\tilde{R}\sqrt{\mu }\cosh \left(\sqrt{\mu }\right)-\sinh \left(\sqrt{\mu }\right)\right]+\sinh \left((\tilde{R}-1)\sqrt{\nu }\right)\left[\tilde{R}\nu \sinh \left(\sqrt{\mu }\right)-\sqrt{\mu }\cosh \left(\sqrt{\mu }\right)\right]}$$ A 1

and

$$J_{\hspace{0.17em}p}^s=\frac{3/\nu \left[(\tilde{R}-1)\sqrt{\nu }\cosh \left((\tilde{R}-1)\sqrt{\nu }\right)+(\tilde{R}\nu -1)\sinh \left((\tilde{R}-1)\sqrt{\nu }\right)\right]\left[\sqrt{\mu }\cosh \left(\sqrt{\mu }\right)-\sinh \left(\sqrt{\mu }\right)\right]{({\tilde{R}}^3-1)}^{-1}}{\sqrt{\nu }\cosh \left((\tilde{R}-1)\sqrt{\nu }\right)\left[\tilde{R}\sqrt{\mu }\cosh \left(\sqrt{\mu }\right)-\sinh \left(\sqrt{\mu }\right)\right]+\sinh \left((\tilde{R}-1)\sqrt{\nu }\right)\left[\tilde{R}\nu \sinh \left(\sqrt{\mu }\right)-\sqrt{\mu }\cosh \left(\sqrt{\mu }\right)\right]},$$ A 2

where, for ease of notation, we expressed all parameter dependencies in terms of the three non-dimensional parameters $\mu =(k_{\hspace{0.17em}p}{c}_{c,s}{r}_{c,s}^2/D)$, $\nu =(k_q{e}_q{r}_{c,s}^2/D)$, and $\tilde{R}={(c_{c,s}/e_{\hspace{0.17em}p})}^{1/3}$.

In the limit e_p/c_c,s ≪ 1, the expressions of the branching fraction take on the simple form of equation (2.7), which only depends on α and $k_{\hspace{0.17em}p}{c}_c{r}_c^2/(3D)$. Figure 6 plots the branching fractions as a function of α with different values for the cluster sizes and enzyme concentrations within the clusters, illustrating that the spatial effects generally increase with the number of enzymes in each cluster.

Figure 6.

Regulation function J_p(α) for (a) the co-clustering and (b) the sequestration strategy, with different cluster radii r_c,s and concentrations c_c,s. The well-mixed case is shown as a reference. These graphs illustrate that the magnitude of the spatial effects grows with the number of enzymes in each cluster, irrespective of whether the cluster size or the enzyme concentration in the cluster is increased. The other parameters were fixed to k_p = k_q = 10⁶ M⁻¹ s⁻¹, and D = 100 μm² s⁻¹. (Online version in colour.)

(b) Partially well-mixed approximation of the branching fraction

Here we show that the function γ(β_c,s) that appears in equation (2.7) can be interpreted as an exchange length scale rescaled by the cluster radius r_c,s. Towards this end, we consider a coarse-grained description, where the cluster and the surrounding bulk are assumed to be two separate domains, in each of which the intermediates are well mixed. The transport of intermediates between these two domains occurs with an effective permeability, D/Δ, where Δ is the exchange length scale (and D is the diffusion coefficient of the full reaction–diffusion model).

For the sequestration strategy, the steady-state density of the intermediate in the cluster and bulk, ρ_c and ρ_b, can be calculated from the flux balance equations

$$0=4\pi r_s^2\frac{D}{\mathrm{\Delta }}({\rho }_b-{\rho }_c)-\frac{4}{3}\pi r_s^3{k}_{\hspace{0.17em}p}{c}_s{\rho }_c$$ A 3

and

$$0=4\pi r_s^2\frac{D}{\mathrm{\Delta }}({\rho }_c-{\rho }_b)+\frac{4}{3}\pi [R^3-r_s^3]\hspace{0.17em}\hspace{0.17em}[j_s-k_q{e}_q{\rho }_b].$$ A 4

Using the solution of the intermediate concentration inside the cluster, ρ_c, we then obtain the branching fraction

$$J_{\hspace{0.17em}p}^s=\frac{4/3\pi r_s^3{k}_{\hspace{0.17em}p}{c}_s{\rho }_c}{4/3\pi \hspace{0.17em}\hspace{0.17em}[R^3-r_s^3]\hspace{0.17em}\hspace{0.17em}{j}_s}=\frac{\alpha }{1+\alpha +{\beta }_s\mathrm{\Delta }/r_s}.$$ A 5

For the strategy of E_I − E_P co-clustering, the flux balance equations are

$$0=4\pi r_c^2\frac{D}{\mathrm{\Delta }}({\rho }_b-{\rho }_c)+\frac{4}{3}\pi r_c^3\hspace{0.17em}\hspace{0.17em}[j_c-k_{\hspace{0.17em}p}{c}_c{\rho }_c]$$ A 6

and

$$0=4\pi r_c^2\frac{D}{\mathrm{\Delta }}({\rho }_c-{\rho }_b)-\frac{4}{3}\pi [R^3-r_c^3]\hspace{0.17em}\hspace{0.17em}{k}_q{e}_q{\rho }_b,$$ A 7

leading to the branching fraction

$$J_{\hspace{0.17em}p}^c=\frac{4/3\pi r_c^3{k}_{\hspace{0.17em}p}{c}_c{\rho }_c}{4/3\pi r_c^3{j}_i}=\frac{\alpha +{\beta }_c\mathrm{\Delta }/r_c}{1+\alpha +{\beta }_c\mathrm{\Delta }/r_c}.$$ A 8

Comparing these expressions with equation (2.7), we see that the function γ(β_c,s) corresponds to the ratio of the exchange length scale Δ and the cluster radius r_c,s.

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Authors' contributions

All authors designed the research. F.H. and F.T. performed the research. All authors analysed the results and wrote the paper.

Competing interests

The authors declare no competing interests.

Funding

This work was supported by the German Excellence Initiative via the programme ‘Nanosystems Initiative Munich’ and the German Research Foundation via SFB 1032 ‘Nanoagents for Spatiotemporal Control of Molecular and Cellular Reactions’. F.H. was supported by a DFG Fellowship through the Graduate School of Quantitative Biosciences Munich (QBM).