The interplay of spatial organization and biochemistry in building blocks of cellular signalling pathways

Abstract

Biochemical pathways and networks are central to cellular information processing. While a broad range of studies have dissected multiple aspects of information processing in biochemical pathways, the effect of spatial organization remains much less understood. It is clear that space is central to intracellular organization, plays important roles in cellular information processing and has been exploited in evolution; additionally, it is being increasingly exploited in synthetic biology through the development of artificial compartments, in a variety of ways. In this paper, we dissect different aspects of the interplay between spatial organization and biochemical pathways, by focusing on basic building blocks of these pathways: covalent modification cycles and two-component systems, with enzymes which may be monofunctional or bifunctional. Our analysis of spatial organization is performed by examining a range of ‘spatial designs’: patterns of localization or non-localization of enzymes/substrates, theoretically and computationally. Using these well-characterized in silico systems, we analyse the following. (i) The effect of different types of spatial organization on the overall kinetics of modification, and the role of distinct modification mechanisms therein. (ii) How different information processing characteristics seen experimentally and studied from the viewpoint of kinetics are perturbed, or generated. (iii) How the activity of enzymes (bifunctional enzymes in particular) may be spatially manipulated, and the relationship between localization and activity. (iv) How transitions in spatial organization (encountered either through evolution or through the lifetime of cells, as seen in multiple model organisms) impacts the kinetic module (and pathway) behaviour, and how transitions in chemistry may be impacted by prior spatial organization. The basic insights which emerge are central to understanding the role of spatial organization in biochemical pathways in both bacteria and eukaryotes, and are of direct relevance to engineering spatial organization of pathways in bottom-up synthetic biology in cellular and cell-free systems.

1. Introduction

A hallmark of all cells is their ability to perform a range of complex information processing tasks. This is a key feature which differentiates them from non-living objects of a similar scale. This is critical for the maintenance of life and conditions in a cell and for a flexible response to the environment. This is achieved via a combination of biochemical modification pathways and networks, as well as gene regulation.

A broad range of research has focused on elucidating information processing in biochemical networks, involving a combination of approaches [1]. One basic approach involves using experiments, mathematical modelling or a combination thereof to understand the behaviour of signalling pathways in specific contexts. Associated with this is also the isolation of basic themes in signalling and information processing, which transcend the individual context. Another approach which has emerged from these studies is a broader multifaceted approach to understanding multiple aspects of the behaviour and functioning of these networks. This includes non-trivial information processing characteristics (adaptation, oscillations, multistability), the roles of ingredients such as feedback, stochasticity, complex post-translational modification (e.g. multisite phosphorylation), and the robustness of networks to achieve certain tasks (e.g. [2–9]). Taken together these approaches span an entire range from the detailed quantitative investigation of specific systems to the investigation of basic elements encountered in multiple systems to broader theoretical investigations of basic underpinning features, and have significantly advanced our understanding of these systems.

Biochemical signalling networks in both bacteria and eukaryotes consist of basic building blocks. In this regard, bacteria present key distinguishing ingredients, one feature being the presence of two-component systems (TCSs) [10]. They typically consist of a membrane-bound histidine kinase, which is an environmental sensor, and a response regulator, which serves as the output [11,12]. Their widespread presence is borne out by the fact that they (histidine kinase, response regulator) are among the largest gene families in bacteria. One of the main mechanisms of output regulation (phosphorylation) is via a phosphorelay mechanism. TCSs and phosphorelays form building blocks of a wide range of regulatory processes (such as osmotic regulation, nitrogen fixation, virulence and adaptive responses) that allow bacterial cells to function and survive [1,13,14]. One of the characteristics of many enzymes in bacteria (in particular histidine kinases) is that they are bifunctional [15], functioning as kinases or phosphatases. Interestingly, both TCSs and bifunctional enzymes are also encountered in eukaryotic systems: TCSs in fungi [16] and bifunctional enzymes in metabolic and biosynthesis pathways [17,18]. Bifunctional enzymes play important roles in multiple contexts and possess distinct information processing characteristics, and for both reasons they have been the focus of a range of experimental and theoretical investigations [19–24].

The reversible modification of substrates by kinases and phosphatases (covalent modification cycle; CMC) is a basic building block for signal transduction in eukaryotes. This basic unit is built upon in many ways; for instance, by having a cascade of modifications where the modified protein at one stage acts as an enzyme for the next stage, or by having successive modifications of substrate (e.g. multisite phosphorylation) for regulating substrate function [1,25]. Other elements which are naturally incorporated within this set-up include feedback regulation (both positive and negative). In general, there is a much more intricate and sophisticated organization of biochemical networks in eukaryotes.

One of the fundamental aspects of cellular information processing is the presence of spatial organization (e.g. [1,26–32]). Indeed, the presence of basic spatial ‘compartments’ in the form of membranes, nucleus, the cytoplasm and organelles is fundamental to eukaryotes. Furthermore, it is evident that the spatial organization is intertwined with the biochemical modification pathway, at multiple levels [33–35]. This is clearly seen in many cases and stems from the fact that signalling responses occur in locations different from where they were initiated. Additionally, spatial organization at specific levels (e.g. the membrane, through the presence of membrane regions such as lipid rafts) can serve as a basic way to route signals and organize information flow [36]. Furthermore the presence of compartmentalization of various types, and the fact that these have been actively exploited in evolution, is testament to this [37]. Spatial proteomics [38] is emerging as a key large-scale investigative tool in this regard.

Bacterial systems (previously long assumed to be essentially well mixed) themselves exhibit clear hallmarks of spatial organization, even though the more elaborate machinery associated with spatial organization in eukaryotes is absent [39]. This is seen both in localization patterns of species, which may change dynamically in the course of the life of a cell, as well as in spatiotemporal self-organized patterns, such as the Min waves in Escherichia coli. These spatial organization patterns may act at the level of enzymes and substrates, and may involve monofunctional or bifunctional enzymes. An example of this is seen in the model system Caulobacter crescentus. Spatial control mechanisms regulating fundamental processes such as the cell cycle, cellular differentiation and polarity [26,40] have been studied in this organism. Caulobacter cells use multiple spatial control mechanisms [27,41–44]: for example, the dynamic localization of proteins at cell poles and the spatial distribution of signalling proteins play an important role during its asymmetric development. Furthermore the choreographed temporal and spatial control of multiple bifunctional enzyme modules (enzymes, substrates) is at the heart of cell-cycle regulation and the transition between different phases [21,26,45]. Disrupting enzyme activity as well as the localization (or spatial activity) disrupts a cell's development (it fails to form an organelle at one of its poles). This is an example of the combination of spatial organization and biochemical building blocks serving as the basic modes of regulating a central biological process.

An additional strong impetus to investigating spatial organization comes from synthetic biology. Here bottom-up approaches have been used to engineer compartmentalization and localization, and this has been used in the context of genetic circuits, biochemical pathways, as well as hybrid communicating systems (e.g. cell population communicating with a circuit) [44,46–51]. Furthermore elaborate spatial compartment networks are being constructed, with locations of compartments capable of being dynamically regulated. Multiple aspects of the compartments, including the transport and localization of individual components within them, as well as their shape, can be manipulated chemically [52,53]. Spatial organization thus implemented is a vital component of artificial cells, with a number of potential synthetic biology applications, that also offers insights into the functioning of natural systems [54].

While some aspects of spatial effects in biochemical pathways have been studied experimentally and (to a lesser extent) theoretically [29,32,55–63], a broad understanding and synthesis of different facets of the biochemistry space interplay, for use in systems and synthetic biology, starting from basic building blocks, is largely absent.

In this paper, we investigate multiple basic aspects of the interplay between chemical modification pathways and spatial organization. To do this, we focus on basic building blocks of chemical modification—TCSs and CMCs—and examine both monofunctional and bifunctional enzymes. This encompasses a range of broadly encountered basic biochemical building blocks. We develop a systems framework to examine the role of spatial organization by focusing on multiple spatial ‘designs’ (organization, involving either diffusion or localization of species). In all cases, a basic reference point is all species localized in the same location, in which case the modification can be described in purely kinetic terms. Incorporating different spatial designs allows us to explore the impact of spatial organization as a perturbation of the co-localized case. We focus on basic information processing characteristics of the chemical modification modules (ultrasensitivity, threshold behaviour, concentration robustness, bistability) and how they are altered by spatial organization. We also examine how spatial organization may be used to directly manipulate the enzyme activity. We examine further aspects of the chemistry–space interplay by dissecting alterations in spatial organization (either during a cell life cycle or through evolution), and the effect of alterations of biochemistry, when spatial organization is present.

There are many reasons for the relevance of this study and the way it is approached. Understanding the effects of spatial organization requires a synthesis of the basic effects of space impacting different modification pathways. By focusing on the basic building blocks of chemical pathways, on one hand, and basic spatial designs, on the other, we can create a broad platform to obtain a range of insights applicable in multiple contexts. Importantly, many instances of non-trivial organization of pathways can be broken down into spatial organization of particular modification stages, which is exactly what our systems analysis addresses. This analysis is also relevant to ‘spatial mutations’ (experimentally studied) resulting from mislocalization of species in natural systems. The effect on particular information processing behaviour is relevant because these have all been postulated or demonstrated to be of relevance in cells, transcending any individual context. Many aspects of cellular information processing networks (behaviour, robustness, evolution) have been studied in temporal terms with a synthesis which neglects a basic aspect of cellular organization. Revealing the effect of the spatial dimension to cellular information processing can highlight when it may be important and change our understanding of these aspects: some of this is already seen in a study of basic building blocks. Finally, there is considerable interest in compartmentalizing pathways and manipulating the localization and transport of species in synthetic biology, and these are also vital aspects of artificial cells. The broad growth of systems and synthetic biology has spawned an area which aims to bridge the living and the non-living. Spatial organization of chemistry is one of the important themes at the junction of the living and the non-living, and our approach is relevant here as well.

2. Models

Our goal is to dissect the interplay between basic biochemical modification cycles and their spatial organization. Consequently, our approach is to employ basic and fairly generic descriptions of the biochemistry (and additionally examine relevant augmentations) as well as spatial organization to isolate key patterns which emerge from the most basic aspects of the interplay

2.1. Kinetic description of the enzymatic modification modules

Our study of biochemical modification cycles involves variations along two independent axes: CMCs and TCS biochemistry, on one hand, and monofunctional and bifunctional enzymes, on the other (figure 1). Taken together, these represent some of the most basic biochemical modification scenarios which are encountered. We study the effect of spatial organization for all these modules. The kinetic models of these modification cycles employ widely used descriptions of basic enzymatic modification steps. A single covalent modification of a substrate by an enzyme is modelled in a standard way with the enzyme reversibly binding to the substrate to create a complex leading to an irreversible substrate modification and release of enzyme. For reversible modification of substrates by kinases and phosphatases, this basic description results in a standard model of the CMC (see figure 1a; in all cases below X and X* denote the unmodified and modified forms of the substrate; kinase and phosphatase are denoted by K and P, respectively, for the case of monofunctional enzymes). A TCS on the other hand involves phosphotransfer in the forward direction (X to X* conversion). For a TCS (with monofunctional enzymes), the main difference in the model (figure 1b) is that the modification of a substrate is associated with a transfer of a phosphate group of the enzyme, which then becomes inactivated. Thus, here the kinase is present in two (reversibly interconverting) forms, inactive (K) and active (K*).

Figure 1.

Schematic of models employed in the paper. X and X* represent unphosphorylated and phosphorylated substrates, respectively. (a) Covalent modification cycle (CMC), mediated by monofunctional (kinase K and phosphatase P) and bifunctional enzymes (kinase form E* and phosphatase form E). (b) Two-component system (TCS) models with monofunctional and bifunctional enzymes (same notation as before, K and K* represent inactive and active forms of the kinase; see text). (c) A schematic of a bifunctional enzyme mediating forward and reverse modifications of a substrate. The interconversion between the enzyme forms in general may be mediated by signals (e.g. Sa and Sb in the bifunctional model in (a)). (d) A dead-end complex model of a bifunctional TCS which gives rise to bistability: this model is an augmentation of the basic bifunctional TCS, incorporating an additional phosphatase F and a dead-end complex XE (based on Igoshin et al. [66]). (e) A schematic of a sample spatial organization: bifunctional enzyme localized at two regions (substrates are non-localized and diffusing). θ denotes the spatial co-ordinate here and in subsequent figures (see text). (f) A schematic model of active localization of two enzyme forms (E, E*) at a specific location (see text for details), incorporating the interconversion between the free forms (E*, E), and the localized forms (S_L1E*, S_L1E). S_L1 denotes the localizing molecule, and its attachment to the enzyme form localizes it.

An important basic variation of biochemistry is the presence of bifunctional enzymes, which can act as both kinase and phosphatase. In all cases we assume two distinct enzymatic forms (kinase form E* and phosphatase form E), which are associated with some basal interconversion. For bifunctional enzymes as part of TCSs (figure 1b), we use the description of the TCS modification cycle, with the only differences being that there is a pair of interconversion reactions between the enzyme forms reflecting this and that there is no inactive form of kinase explicitly described. This is the model employed by Batchelor & Goulian [19]. We also study bifunctional enzymes within a CMC (figure 1a). Here again, we augment the basic CMC model to incorporate interconversion of enzyme forms (discussed further below). This model, which again incorporates most aspects of bifunctionality, with covalent modification, allows us to establish correspondences with the bifunctional TCS model (discussed in electronic supplementary material, §§2.1 and 3).

2.2. Model variations and augmentations

We examine some basic augmentations to our models, because they lead to distinct patterns of information processing of interest. In the case of bifunctional enzymes in a CMC, we examine the possibility of feedback (both linear and nonlinear) of a modified substrate (directly or indirectly) affecting the balance of enzyme forms (such feedback is consistent with the biochemistry of modification). The nonlinear feedback mechanism results in bistability, a feature which is also seen in the linear feedback case with extra enzyme–substrate interactions (see electronic supplementary material and [64]). Another variation of the basic model is the one studied in [65], which uses a signal to tune the balance of kinase and phosphatase, explicitly incorporating signal binding and uptake. This allows for ultrasensitive switch-like behaviour. The possibility of a ‘dead-end complex’ (an augmentation to the basic model; see figure 1d) generating bistability has been studied in both bifunctional and monofunctional TCSs [66]. We perform a focused analysis of these cases. The kinetic (ODE) models are presented in electronic supplementary material, §§2.1 and 2.2.

2.3. Comments on kinetic models

A few points regarding the bifunctional enzyme models we employ are worth emphasizing. Firstly, in these models, the enzyme forms are all active (models with inactive enzyme forms have also been studied but are not presented here). Further, we ignore the possibility of ternary or higher complexes, which may be associated with substrates binding to enzyme forms at different locations, or dimerization. Some of these aspects have been studied in the literature (e.g. [5]). Given our focus on spatial perturbations, we kept the models as simple as possible to ascertain a fundamental understanding (which arises from the most basic aspects of bifunctionality) without invoking such additional features. Secondly, there are models of bifunctional enzymes where multiple kinetic steps are described in great detail [67]. Finally, there could be mechanistic variations in specific contexts and these are outside the scope of the current study. The models employed (which use concise descriptions of the key steps) are sufficient for the nature of the questions we address, and the analysis and insights presented here (which transparently emerge from the underlying assumptions, sometimes seen by contrasting with a closely related model with a basic difference) can serve as a platform for examining these auxiliary factors in a subsequent study.

2.4. Spatial organization

The ODE models depict kinetic modification cycles, with enzyme and substrate in the same location and no species diffusing. Our focus in spatial organization is on the effect of localization of specific entities in certain regions (see figure 1e for a schematic) and how that impacts the overall modification chemistry. Our modelling framework to achieve this involves an explicit spatial description, using PDEs (electronic supplementary material, §2.3). At the outset, this brings a set of new ingredients alongside the kinetics: the description of localization and diffusive transport of species, in a spatial domain with specified boundaries. The models involve the localization of entities (enzymes or substrates) to certain regions of the spatial domain, which we refer to as compartments. These regions are representative of compartments which may or may not be bounded by compartment boundaries. While our model incorporates these regions of localization (and not any compartment boundaries explicitly), many essential insights carry through to the case of compartments with boundaries (discussed in the electronic supplementary material).

The basic effects of localization are studied by the imposition of different patterns of localization (discussed in detail below) of enzymes and/or substrates in one or two regions (representing poles of a cell, for instance) within the spatial domain in one dimension (figure 1e). Other species diffuse freely in the intervening space, and this is described by Fickian diffusion. In all cases, the localization is realized by a combination of the initial conditions (ensuring that a species is in a given location) and non-diffusibility. As such, our depiction of the spatial localization of entities is relevant both in natural biology (for example bacteria, through localization at specific regions such as cell poles, or eukaryotes, where the model could depict a cross section of a cell, or the membrane of a cell, with localization in compartments) and in synthetic cell-free systems, with compartmentalization of reactions.

The study in a one-dimensional spatial domain is sufficient to understand the basic effects of spatial organization (which ultimately stems from different patterns of localization/non-localization of species, coupled to chemical reactions). This serves as a basis for examining specific spatial aspects (e.g. geometry) in subsequent models in two or three dimensions, while still providing important insights relevant to those cases. We have considered both no-flux boundary conditions (to describe the cross section of a cell for instance) and periodic boundary conditions (to describe the membrane of a cell in two dimensions for instance, as depicted in figure 1e) in our PDE models. The results from these choices of boundary conditions are commensurate: periodic boundary conditions involving either a single compartment (pole) or two compartments (poles) symmetrically localized is exactly equivalent to no-flux boundary conditions in half the domain size. Further, the essential qualitative conclusions, which are the focal point throughout the paper, do not depend on this choice of boundary conditions (as can be seen by an analytical study of the models). θ denotes a (dimensionless) distance co-ordinate in all cases and is scaled from 0 to 2π to serve as a common reference, and our computational results focus on the concentration of the modified and unmodified substrates X* and X as a function of the spatial co-ordinate or other parameters.

2.5. Active localization of species

While localization of species can be described by localizing the initial condition and having it non-diffusible, we also need to consider ways in which localization is imposed on diffusing species. This is the key ingredient for cells to impose localization dynamically. To describe this, the localization is realized by having the species binding to an immobile species at a location (reversibly) but the bound species being non-diffusible (figure 1f). In the case of bifunctional enzymes, we can consider (i) equal localization (equal binding/unbinding rates of each enzyme form at the location) and (ii) a bias in localization, where one form of the enzyme is localized more strongly than the other owing to differences in binding (or unbinding) rates. These are easily described in the basic model (see electronic supplementary material, §2.3) and transparently analysed analytically. Within this model one can examine variations such as the presence/absence of (i) interconversion of localized forms and (ii) enzymatic activity of localized enzymes.

2.6. Parameters

We now discuss the role of parameters in relation to the goal of our study. The parameters belong to two classes: kinetic parameters and spatial parameters (diffusivity, domain size), which we discuss in turn. It is clear that even the simple models have a number of kinetic parameters. We aim to understand basic qualitative aspects of the effects of spatial organization on biochemical modification. This is because the qualitative characteristics are the ones most easy to discern experimentally, and in general have the greatest impact in terms of biological outcome (and are consequently the first point of interest, from a modelling viewpoint). We further note the following. (i) Many of the basic qualitative effects of spatial organization stem from basic considerations (e.g. dilution, nature of localization patterns) and are seen across multiple kinetic parameter sets. Furthermore, in many cases this is directly understood through analytical work. As an example, for the case of species localized in one location, with a single species diffusing throughout the domain, the impact of the spatial organization on the modification kinetics, at steady state, is only through the dilution owing to species spreading outside the region of localization. This can be characterized analytically and computationally, revealing a basic underlying trend. While the quantitative effect depends on both kinetic parameters and the size of the domain, the underlying qualitative insight follows from basic analysis. (ii) The kinetic parameter regime is studied inasmuch as it represents a distinct basal information processing behaviour, which is another explicit vantage point of analysis. This allows us to examine the effect of spatial organization on this type of reference behaviour, and pin down the reasons for any qualitative changes. Taken together we identify both basic effects of spatial organization as well as those on particular reference kinetic characteristics. (iii) Finally, in many cases (where relevant) we observe similar trends across different models of biochemical modification, attesting to the essential robustness of the conclusions. With regard to spatial parameters, we use a sample compartment size (about 10–20% of the entire domain) and have studied a range of diffusivities of species. With regard to kinetic parameters, we employ parameter sets drawn from the literature, along with parameter sets in specific cases corresponding to particular information processing characteristics (based on kinetic analysis in the literature). For instance, for basic analysis of both CMCs and TCSs with monofunctional and bifunctional enzymes, we use modification parameters with comparable ranges of binding, unbinding and catalytic constants (ensuring that the system is not operating in a limiting regime), similar to parameters and in ranges used in the literature (and have also examined other parameter sets underpinning the study). Special parametric regimes are analysed separately (analytically) if needed.

Many aspects of qualitative behaviour/trends introduced by spatial organization are established analytically. In other cases, computational results directly support conclusions which we draw (e.g. the possibility of a certain behaviour occurring). All in all, we employ generic kinetic models, along with focused augmentations (enabling specific behaviour), and isolate basic aspects of their interplay with spatial organization. These arise from basic ingredients of the kinetic modification (e.g. the bifunctionality of enzymes) and the most basic aspects of spatial organization and can be transparently revealed.

The dimensionless equations are analysed computationally via simulations (solver ode15s) in Matlab; the kinetic schemes in Matlab are cross-validated by comparing with simulations in COPASI (where models for the kinetic schemes are automatically generated). In the case of spatial organization, a standard finite difference discretization of the PDE is employed (and results checked by doubling discretization points), which converts the PDE into a set of ODEs. This is complemented by bifurcation analysis of the steady states of ODEs for the kinetics (and, in certain cases, bifurcation analysis of the steady state of the PDE, which can be mapped onto the steady state of the ODE with a modified conservation condition: in these cases, as discussed analytically in electronic supplementary material, §3, the steady state of the PDE satisfies the kinetic ODE equations, and the only effect of spatial organization is a dilution effect which manifests itself in the conservation condition). Further details are presented in electronic supplementary material, §2.4.

3. Results

We discuss basic aspects of spatial organization of monofunctional and bifunctional enzymes. The extent to which enzyme activity can be manipulated spatially (by localization in particular) is examined next, and finally we examine how transitions in biochemistry and spatial organization impact pathway behaviour.

3.1. Monofunctional enzymes with spatial organization

We first examine a monofunctional enzyme pair (kinase, phosphatase) reversibly modifying a substrate: this modification being described by a CMC. A structured approach to the analysis of spatial organization facilitates the dissection of the interplay between space and biochemistry.

When the enzymes and substrates are co-localized in one location (which we refer to as a compartment), the behaviour of the modification system can be described via its kinetics. Spatial organization enters through altered localization patterns of enzymes and/or substrates, which we henceforth refer to as spatial designs (e.g. figure 2). We use the co-localized scenario as a reference point and examine the effect of spatial designs relative to this. Relative to this basic co-localized design, there are multiple possible alterations of localization of enzyme and substrate. The basic substrate localization alterations we examine are as follows. (i) X being diffusible and exiting out of the compartment while X* is localized (this substrate localization pattern is denoted by S₁). (ii) X* doing the same, while X remains localized (denoted S₂). (iii) Both X and X* being able to exit (denoted S₃).

Figure 2.

Spatial designs in monofunctional CMCs. (a–c) Three different kinds of localization of enzymes, depicted schematically in the left column: (a) K and P are localized in the same location (design E₀) , (b) K is localized in one location and P is everywhere (design E₂) and (c) K and P are localized in two distinct locations (design E₁). In each subplot in the middle and right column, curves describe different substrate localization patterns: all species localized (in the same location as the kinase) (design S₀), only X diffusing (design S₁), only X* diffusing (design S₂), both X and X* diffusing (design S₃), respectively. θ is the spatial coordinate. This figure demonstrates that enzyme localization can help generate non-zero uniform (red and green curves in (a)) and graded (green curves in (b) and (c)) X* profiles; combinations of substrate and enzyme localization patterns (blue curve in (c) for X) can induce zero responses (also red curves in (b) and (c) for X*). Note that the green curves in (b) and (c) are associated with a different scale, also depicted in green.

The other important aspect is the localization of enzymes. Here we have multiple possibilities. (i) Kinase localized at one location while phosphatase is localized at another location (this enzyme localization pattern is denoted by E₁). (ii) Kinase localized at one location while phosphatase is present everywhere (denoted by E₂). (iii) Both enzymes present everywhere (denoted by E₃).

Any of the associated spatial designs can be denoted by a pair of indices depicting the enzyme and substrate localization patterns. D₀₀ denotes the original localization pattern, whereas D_i,j denotes the localization pattern corresponding to E_i and S_j, with i, j = 0, 1, 2, 3. E₀ corresponds to completely localized enzyme and S₀ corresponds to completely localized substrate.

The first point to note is that transitions in localization are associated with changes in indices; furthermore, a single alteration in localization pattern is associated with the change of a single index. Transitions in localization patterns may emerge both in evolution and during the life cycle of a cell, with dynamic alterations in localization. They may also be associated with mutations.

Figure 2 depicts a snapshot of results for each of the spatial designs associated with monofunctional enzymes. All kinetic parameters are kept fixed here, and the qualitative changes introduced by (alterations of) different spatial designs are representative. Examining all these designs together reveals a few points. (i) An alteration of substrate localization affects the steady state primarily through dilution effects. Note that the extent of dilution depends on the size of the overall domain (relative to the region where other species are localized) and also which substrate is diffusing (e.g. figure 2a). (ii) The alteration of enzyme location can introduce qualitative differences. (a) One example is the transition between uniform and graded behaviour (e.g. green curves in figure 2a and 2b,c, the latter being graded). Graded behaviour occurs when the enzymes are not in the same location and both substrates diffuse. Spatial uniform (non-zero) concentration profile of the substrates (which are not localized) is seen in all other cases, with the exception of the following scenario. (b) The possibility of either modified or unmodified substrate attaining a zero steady-state concentration (e.g. red curve in figure 2b for X* and blue curve in figure 2c for X) is another example of a distinct qualitative behaviour observed. This can happen when the overall cycle of interconversion is disrupted owing to placement of enzymes, so that a particular substrate is not (or does not remain) in contact with the enzyme which acts on it. This can happen if a substrate form is modified at a particular location, but this modified form cannot diffuse/reach a location containing its associated enzyme. An echo of this is observed in compartmentalization of multisite modification and phosphotransfer cascades [56], but can be traced back to this basic feature.

The essential qualitative insights stem from basic aspects of the interplay of biochemistry and spatial organization as seen above. Quantitative aspects depend on parameters such as the domain size. Substrate diffusivity affects the steady-state behaviour only in the case of separated enzymes with both substrates diffusing. In this case, when the diffusion coefficient of both equally diffusing species is high (in dimensionless terms), the substrate concentration profiles are essentially uniform with levels explicitly determined by the kinetics, total enzyme/substrate amounts and domain size. In other cases, a gradient is obtained, and the profiles of species can be determined analytically in certain regimes. This can be consolidated further by mathematically mapping steady-state concentrations onto a compartmental model [68], highlighting the limitations of thinking about such systems in simplified compartmentalized terms.

A complementary effect of spatial organization of dilution owing to delocalization is that of elevation by localization. To illustrate this, we consider the case where kinase and phosphatase are localized in one location, with the modified substrate X* also localized there. The presence of a non-localized unmodified substrate everywhere in the domain and diffusible now allows for an elevated X* output (relative to unmodified species also localized, at the same total local concentration), because the substrate modification system can draw on substrate from outside the region of modification (not shown). We note here that the ODE kinetic model applies at steady state both to the case where all species are co-localized in one location and also to the case of no localization of any component, with enzymes present uniformly: in the latter case there are no gradients of any species, and the diffusion plays no role at steady state. Thus the above result could be interpreted by saying that localizing the enzymes and X* at a given region can elevate X* concentration levels. The effect of this elevation is seen strikingly in a case of product inhibition (figure 3c), where X* inhibits or deactivates the active form of kinase. In this instance, we find that, by having the spatial design just discussed, the resulting elevation of output (from the red to green curve) can even completely compensate for the effect of product inhibition. This localized placement of enzymes (along with localization of X*) leads to a higher output than a uniform global placement of enzymes and substrates. Analytical results in the electronic supplementary material, §3.1, reveal conditions under which this behaviour is observed. This is another illustration of how a spatial design (which can involve localizing X* and enzymes) can achieve this counterintuitive result.

Figure 3.

Spatial retroactivity in a monofunctional TCS. With localization of substrate and enzymes diffusing/non-localized, the concentration profiles of (active) kinase and phosphatase are different in a monofunctional TCS (blue curves in (a)) and CMC (red curves in (a)): in the TCS, the concentration profile of (active) kinase is graded (K* denotes active kinase in the TCS) but that of phosphatase is uniform, while they are both uniform in the CMC (K is the active kinase in the CMC, where the enzyme is assumed to be constitutively active). This shows how spatial information of the substrate can be backpropagated to the (active) enzyme at steady state, in a TCS. Also shown in (a) for reference are the localized substrate profiles (denoted by XZ and X*Z, which involve active substrate localization in a particular region). (b) How the balance of active and inactive kinase concentrations depends on the substrate in the TCS, in the kinetic (ODE) model, which is the source of the behaviour observed. (c) An illustration of how localizing enzymes and X*, while allowing for X to diffuse (in the case of the monofunctional CMC), can cause an elevation of output, which can even compensate for the effect of product inhibition (green curve compared with red curve, with the green curve being above the blue curve).

3.1.1. Distinct features associated with two-component biochemistry

The effects of spatial design studied above are relevant for modification through both CMCs and TCSs. This is because the interplay of factors responsible for the observed behaviour is similar in both cases. We now examine distinct aspects which emerge when the biochemical modification cycle is representative of a TCS: the key difference in TCSs is that the forward modification occurs via phosphotransfer and the reverse reaction by covalent modification.

The fact that substrates can affect upstream enzymes by sequestration, in a cascade (e.g. enzymatic modification cascade), is well appreciated, and is an example of what is termed ‘retroactivity’ [69]. However, the spatial dimension to retroactivity is not well appreciated, and we examine one basic instance of this: the case of a TCS where the substrates are localized, but the enzymes are diffusible. In this case (figure 3a, blue curve), at steady state, the (free) phosphatase concentration profile is spatially uniform while that of the (active) kinase is not. The dichotomy in this behaviour is understood by noting that, owing to covalent modification, the net effect of substrate on the phosphatase at steady state is only a degree of sequestration. The free enzyme profiles at steady state are determined purely by diffusion, since the net effect of the substrate in consuming the enzyme at steady state is zero (see electronic supplementary material, §3.1). By contrast, because of the phosphotransfer mechanism, the free (active) kinase profile is not uniform, but instead is determined by the balance of kinase diffusion and the balance between interconversion of inactive and active kinase (which has a dependence on the substrate at steady state). This is further consolidated by a comparison with the CMC in the same setting (where the kinase is constitutively active): there, at steady state both enzyme profiles are spatially uniform (figure 3a, red curve). This illustrates when and how retroactivity can result in the backpropagation of spatial information, and how this depends on the nature of the chemical modification mechanism.

3.2. Bifunctional enzymes with spatial organization

We now examine bifunctional enzymes. As discussed in the Introduction, bifunctional enzymes and their substrates may be subject to both temporal and spatial regulation. As an example, in the cell cycle in C. crescentus, multiple scenarios involving the localization of enzyme and substrate are observed (this is true for each of two interacting bifunctional enzyme modules). Similar scenarios could also emerge through mutations which affect the localization of enzyme or substrate.

Having studied the basic aspects of spatial organization in monofunctional enzyme modification cycles, here we primarily focus on those aspects which are particularly related, directly or indirectly, to bifunctional enzymes: both distinct characteristic behaviours of modification through bifunctional enzymes and other interactions which may naturally be relevant here. We examine both TCSs and CMCs, identifying both similarities and subtle differences between the two.

In terms of characteristic behaviour, we focus on the fact that bifunctional enzymes are associated with some form of absolute concentration robustness (ACR), which directly stems from the bifunctional nature of the enzyme. This has been studied both experimentally and theoretically. We also note that bifunctional enzymes may be associated with monostable or bistable switch-like behaviour, as discussed in the literature, both through general considerations and in specific cellular contexts [66,70]. Finally, bifunctional enzymes may be subject to feedback from the substrate, which alters the balance between enzyme forms. These characteristic aspects serve as reference points from which to evaluate the effect of spatial organization. We note here that feedback and bistability are also observed in monofunctional enzyme systems, and many of the associated insights we draw are also relevant to monofunctional enzymes. This emerges transparently from our analysis and is due to the fact that, in these instances, key dominant factors have similar consequences in both these systems.

3.2.1. Spatial designs

We first comment on the effect of different spatial designs of bifunctional enzymes and their substrates (figure 4). In contrast to monofunctional enzymes (depicted in figure 2), we note that localization of bifunctional enzymes at two locations does not result in a graded response (figure 4b), unless the balance of enzymes is different at those locations (figure 4c). Furthermore, as seen in figure 4 (in contrast to figure 2), there is no scenario (with a non-trivial balance of kinase/phosphatase forms) where the X* concentration is zero everywhere: this is because the bifunctional nature of the enzyme guarantees local reversibility, in contrast to scenarios involving monofunctional enzymes, where that may not be guaranteed. This means that, even through spatial mutations, a scenario of zero/complete substrate modification will not be generated as long as the enzyme is not completely separated from both substrates. We now examine the effect of substrate spatial organization on different characteristic behaviours of bifunctional enzymes.

Figure 4.

Spatial designs of bifunctional enzymes. (a–c) Different combinations of the enzyme localization and the balance between kinase and phosphatase forms (E* and E) of the enzyme (depicted schematically in the left column): (a) the bifunctional enzyme is localized in one location (same as in figure 2), (b) bifunctional enzyme is localized in two locations (same as in figure 2), with the same balance of enzyme forms and (c) different balance between E and E* in the two locations. A spatially varying balance of enzyme forms is a necessary factor for producing a graded response (green curve in (c): associated green scale). Moreover, because of the local reversibility of the biochemical cycle, resulting from the nature of the bifunctional enzyme, there is no case of zero concentration of substrate everywhere. Similar results are also found in the TCS (electronic supplementary material, figure S2).

3.2.2. Absolute concentration robustness

The same enzyme performing opposite modifications is at the core of different kinds of concentration robustness (in particular a weak dependence of certain outputs on total enzyme amounts). We discuss both the covalent modification and the TCS.

3.2.3. Covalent modification

Here, for enzymatic modification of substrate in the unsaturated regime, both X* and X*/X are constant over a wide range of total enzyme concentrations (figure 5a). In this case, the total amount of enzyme can essentially be factored out of the steady-state kinetics. Note that, when enzymatic kinetics is not in the unsaturated regime, X*/X remains constant, while the X* dose–response curve is essentially flat, followed by a change (electronic supplementary material, figure S1(A)). This ratio remains constant because, at steady state, the substrate modification does not have any net impact on the balance of enzyme forms (the only impact is a sequestration of enzyme). An increased amount of enzyme, however, results in a decreased amount of available substrate X + X* (because of sequestration in complexes), which is the source of the variation of X*. A sensitivity of X*/X can be obtained by having feedback from substrate X* mediating the conversion from E to E*, but this also needs the substrate modification kinetics to be far from the unsaturated regime (electronic supplementary material, figure S1(B,C)). These aspects are all explained analytically in electronic supplementary material, §3.2.

Figure 5.

The effects of spatial organization on ACR in a bifunctional CMC and TCS. (a) A regime of concentration robustness in the CMC ODE model (unsaturated substrate modification), broadly mirroring behaviour studied in the TCS model. (b–d) When enzyme is localized in one location (same as figure 2), the outputs X* show different degrees of variation with respect to the total concentration of enzyme for different spatial designs of substrate in a bifunctional CMC (c) and TCS (b,d). Note that the kinetic regime of the CMC in (c) is different from that in (a), with no ACR in the kinetics. Panel (b) shows (depicted over a broader range) that the non-localization of substrate significantly impacts the ACR in the TCS. In the CMC (c), substrate diffusion can create a regime of ACR, arising from the interplay of spatial organization and the bifunctional nature of the enzyme (green and blue curves in (c) and electronic supplementary material, figure S3(A)). Similar behaviour can also be seen in a TCS for lower total substrate amounts (d). (e,f) In the TCS, with enzymes in multiple locations (denoted by L1 and L2), if the balance of enzyme forms is different at the two locations, the ACR (for local or average substrate concentrations) is not maintained if the total enzyme concentration is varied simultaneously in the two locations (contrast with figure 5d where the robustness behaviour is observed for the case of X, X* diffusing, for the same total substrate amount).

3.2.4. Two-component systems

The source of ACR has been studied in detail by Batchelor & Goulian [19], and others, both theoretically and experimentally. The analysis indicates that, when the total amount of substrate is (substantially) greater than that of the enzyme, a regime of concentration robustness exists. This is seen in simulations from the ODE model. This behaviour occurs when the enzyme concentration is less than or comparable to the substrate concentration.

3.2.5. Effect of spatial organization on absolute concentration robustness

Spatial organization, as before, is introduced via an altered localization pattern of enzyme or substrate, relative to the case where all species are co-localized. We discuss this in turn. The exiting of enzyme from the compartment results in a reduced amount of enzyme available for modification. For both the CMC and the TCS, this generally enhances the effect of ACR based on the discussion above. In the case of substrate diffusion, there are three different possibilities of diffusing species: only X, only X* and both X and X*. As seen in figure 5b (for TCS), the particular zone of concentration robustness seen in the kinetics is now essentially destroyed. Furthermore, when both species diffuse, the system shows no echo of ACR. This is because the condition of the substrate concentration being much larger than that of the enzyme is now violated, owing to dilution. Now if we examine the CMC (in a kinetic regime where ACR for X* did not hold), the maintenance of the X*/X ratio continues to hold good, irrespective of which species diffuses. On the other hand, if we examine the output X* (figure 5c), we find that spatial organization in fact introduces ACR of the output, depending on which species is diffusing, and furthermore this is strongest when either X* or both X and X* diffuse, in the current instance. This is also seen in the TCS for a reduced level of substrate (figure 5d; also see electronic supplementary material, figures S4 and S5 for further demonstration of the introduction of these effects and dependence on the spatial design). A detailed analysis in the electronic supplementary material, §3.2, reveals the interplay between spatial organization (and the specific pattern of spatial localization) and bifunctionality, in introducing a new regime of concentration robustness (even when it did not exist in the kinetics). This is explained analytically in the electronic supplementary material. The essential insight is that the combination of bifunctionality of the enzyme (which imposes specific kinetic restrictions) along with dominant dilution of certain combinations of substrate species (which crucially relies on the spatial design) gives rise to this behaviour. In these cases, the dominant factors in the substrate conservation arise from the substrates present everywhere in the domain (i.e. non-localized), which may completely trump any effects of substrate sequestration in enzyme complexes. This explains why this absolute robustness effect is operational even if locally total enzyme amounts are greater than total substrate amounts (which from a purely kinetic study would preclude the behaviour from occurring), and also indicates why such a robustness could be created even when the completely co-localized scenario (with the same total amount of substrate) did not show this behaviour. This also demonstrates the importance of the identities of species spreading.

We examine enzymes localized in two distinct locations with substrates shuttling between them. If the balance of enzymes is the same at both locations the steady-state substrate concentration profiles are spatially uniform (seen in figure 4b). Here, the role of spatial organization is qualitatively analogous to the single location case (with two species diffusing) and can be quantitatively mapped onto that. When the balance of enzymes is different at the two ends, then in general the ACR property is lost. Simulations in both CMCs and TCSs show this. Variation of the enzyme amount (equally in the two locations) shows an absence of concentration robustness of the local as well as the global average of X* in computational analysis (figure 5e,f, electronic supplementary material, figure S6(A–F); also see electronic supplementary material, figure S6(G,H), which shows two cases where the figure 5d setting is augmented by enzyme at another location, revealing this). Can ACR be observed? If the substrate diffusivity is high (in dimensionless terms, so that substrate profiles are uniform), then if enzymatic modification of substrates is in the unsaturated limit, we see that X*/X is independent of the total enzyme concentration (at each location, varied equally). This along with substrate conservation shows X* to be independent of total enzyme amounts. Here, even with substrate modification not in an unsaturated regime, for large enough domains, the domination of X + X* in the substrate conservation condition allows for the behaviour, similar to the cases above (see electronic supplementary material).

When enzymes are localized at two (or more) locations, it is worth asking if the system can exhibit local concentration robustness to variation of enzyme amount at a specific location (while the others are fixed). This can be seen if (i) substrate diffusion is very weak, and total substrate amounts are sufficiently high (here substrate concentrations at this location remain essentially determined by the kinetics), or (ii) the balance of enzyme forms is the same at the locations (this is independent of diffusivity).

ACR is experimentally seen in the OmpR/EnvZ TCS system. We can infer from the above analysis for such systems where this behaviour is observed experimentally that either (i) there is no localization of enzyme/substrate or enzymes and substrates are both co-localized, so that the kinetic description is valid, or (ii) there can be localization of enzyme in one location or possibly more than one location, in which case the balance of enzymes is the same in all locations (though the possibility of a different balance of enzyme forms may also be possible, if substrates are highly diffusible). Here there are multiple possibilities. (a) The origins of this behaviour can be essentially traced to the enzyme kinetics, along with the dilution effect of the substrate. (b) The spatial organization of substrates (which species is (non)localized) plays a key role, along with bifunctional enzyme kinetics in establishing this effect, which may not be seen even if all substrate were co-localized with enzyme. In both cases (a) and (b), our analysis predicts that experiments performed with cells of different sizes (e.g. at different growth stages) can reveal different quantitative characteristics and margins of concentration robustness, something not seen in case (i), if all species are co-localized. (a) and (b) can be distinguished by an experiment which co-localizes substrate with enzyme: the robustness property and margin is enhanced in (a) but can be destroyed in (b). Increasing domain size can diminish the robustness margin in (a) but enhance it in (b) (see electronic supplementary material).

3.2.6. Bistability

Bistability is of natural interest in bifunctional enzymes as it allows for opposite enzyme activities to be demonstrated under the same conditions and at the same time. We focus on two different ways in which bistability may be achieved in bifunctional enzymes. One way is through co-operative allosteric feedback from substrate affecting the balance of enzyme forms. This is shown in the CMC system, clearly demonstrating that this results in a regime of bistability. Another way in which bistability can be achieved is through the presence of additional enzyme–substrate complexes, called dead-end complexes. The role of additional enzyme–substrate complexes has been studied in [64]. The effect of dead-end complexes in generating bistability in bifunctional enzymes in TCSs has been studied in [66]. It is worth pointing out that the latter mechanism does not require an explicit feedback mechanism between enzyme and substrate, and this works within the constraints of TCS biochemistry.

3.2.7. Effect of spatial organization on bistability

When the enzyme is localized in one location, then substrate diffusion can result in a destruction of bistability (see red curve in figure 6a demonstrating this for the CMC, with X* diffusing). This is because the dilution of modified substrate weakens the feedback and hence weakens the propensity for bistability. Similar insights hold good if X exits the compartment. A very similar trend is also seen in the TCS bifunctional system as well (figure 6b–d), where bistability was realized through the presence of dead-end complexes, rather than co-operative feedback. The dilution effect weakens the implicit feedback, eventually destroying bistability (these results are supported by the analytical work in electronic supplementary material, §3.2). Can bistability be preserved in such a spatial design? We show that, by starting with a greater initial substrate, we indeed get a bistable dose–response curve though its range is altered (green curve, figure 6a—this curve was obtained by incorporating the effects of dilution modifying the conservation condition, and using bifurcation analysis). Another way of maintaining bistability is if the initial compartment is a large fraction of the domain size, simply because of minimal distortion due to dilution. Both these aspects are seen in the TCS system as well. Similar insights hold good when both substrate species diffuse. Here, when the substrate modification kinetics is in the unsaturated regime and substrate diffusivities are equal, we can explicitly account for the dilution effect essentially independent of enzyme concentrations in the CMC system (analysis in electronic supplementary material). Figure 6e,f also shows how bistability can allow for enzymes localized in different regions to realize the different steady states, but this happens only when the substrate diffusivity is low (in dimensionless terms).

Figure 6.

The effects of spatial organization on bistability in a bifunctional CMC and TCS. The perturbation of bistability arising from (a) a CMC with feedback (when X* is diffusing), showing how the associated dilution effects can distort and even destroy bistability (red curve). (b–d) Different cases of a TCS with a dead-end complex model (corresponding to X diffusing, X* diffusing or both diffusing, respectively) all show a similar result (blue curves in (b) and (d), red curve in (c)). k_ph is a kinetic parameter in the dead-end complex model (see electronic supplementary material). However, a larger initial amount of substrate ([X]total) and relative compartment size can facilitate recovery of the destroyed bistability (green curve in (a), blue curve in (c) and red curve in (d)). (e,f) Bistability can be leveraged to obtain distinct steady states at different locations both in the CMC (green, blue and red curves in (e), showing distinct steady states at the two locations: the opposite combination is not shown) and TCS cases (red and blue curves in (f) depicting concentrations at the two locations), but only when the substrate diffusion is weak. A schematic of models used is depicted.

The above insights are relevant to cases where a localized enzyme and substrate may result in bistability (as suggested by Subramanian et al. [71]), at a certain stage of cellular growth: it is suggested that bistability in the localized PleC bifunctional histidine kinase activity ensures irreversibility during the transitions between phases of the cell cycle and robustness to fluctuating nutrient levels in Caulobacter. There is a subsequent alteration to the localization pattern of substrate, as there is a progression to other phases of the cell cycle. We infer that, through a sufficiently high concentration of localized substrate, a cell can allow for the maintenance of bistability, over natural growth sizes of the cell. In such a case, the breakdown of bistability would require abnormally large cells (which may arise from mutation). If the effect of bistability is not meant to be propagated to other stages of the cellular growth, this dilution must destroy bistability, which then constrains the balance of initial substrate concentration and size of the cell. In particular, a cell not growing sufficiently large cannot remove the effect of this bistability.

3.2.8. Feedback resulting in threshold-type behaviour

Our discussion of bistability in CMC bifunctional systems stemmed from co-operative feedback. It is worth examining the effect of a simpler non-cooperative feedback in such a system, and we see that it is possible to obtain a threshold effect. In such a case (figure 7a), above a certain value of signal, the output is zero (see electronic supplementary material, §3.2). We focus on a signal level below this threshold, so that a non-zero steady state is seen in the kinetics. Here we find (figure 7b) that, if X* alone leaked out of the compartment, this resulted in a reduced (but non-zero) steady state. If both X* and X leaked out of the compartment, a zero steady state for X* was obtained. Interestingly, even when only X diffused out of the compartment a zero steady state resulted. This indicates that, depending on the extent of dilution, and the particular species leaking out of the compartment, a non-zero steady state may or may not be sustained. Analytical work (electronic supplementary material) shows this.

Figure 7.

The effects of spatial organization on threshold behaviour and ultrasensitivity. (a) In the case of a bifunctional CMC with feedback (see schematic), a threshold effect can result from the kinetics (Sb is a kinetic parameter reflecting the E* to E basal conversion: beyond a certain level zero output results). (b) Depending on which substrate species is localized/non-localized, zero or non-zero concentrations for X* may be obtained, though this also depends on the size of the region where enzymes are localized, relative to the ambient region. (c,d) The perturbation of ultrasensitive behaviour in the kinetics (no feedback) (c), through dilution, and the dependence on compartment size (d) (see text for details).

While our analysis of both bistability and feedback above has focused on bifunctional enzyme systems, we point out that similar behaviour can also be observed in monofunctional enzymes, and be traced back to the same fundamental ingredients (co-operative/linear feedback, dead-end complexes). Consequently, the trends associated with spatial organization above are relevant to these cases as well.

3.2.9. Ultrasensitive switch-like behaviour

Figure 7c shows how an ultrasensitive switching behaviour is obtained in the kinetics, similar to [65]. This behaviour can be eroded because of the diffusion of the substrate species out of the compartment (figure 7d). As the size of the compartment relative to the surrounding domain is reduced, the switch amplitude is reduced, and the sharpness of switching is altered. This has been studied in other cases of enzymatic modification [55].

3.2.10. New capabilities

In addition to placing constraints on kinetic and spatial parameters for the realization of different behaviour, spatial organization can also provide new capabilities. Kinetically, a bifunctional enzyme (for given conditions, total amounts of species and signals which affect the balance of enzyme forms) exhibits either a kinase-dominant or phosphatase-dominant activity (unless there is bistability). Spatial organization can relieve this basic constraint: by having enzymes at different locations (with different balances of enzyme forms), it is possible for the bifunctional enzyme system to exhibit both extreme activities simultaneously, acting on substrates localized with them. Furthermore, if a fixed enzyme amount is now (re)distributed between multiple locations (where it may act on substrate localized there), the effect of enzyme dilution which is associated with this distribution in many instances is minimal, precisely because of ACR. Taken together, these points suggest an attractive advantage of distributing bifunctional enzymes between multiple locations to act on localized substrate there.

3.2.11. A note on measurement

We comment, in the context of spatial organization, on the effect of spatially averaged measurements, for instance by lysing a cell. In the case of localization of enzymes in one location, with others non-localized, gradients are absent. This means that spatially averaged measurements do not introduce an essential qualitative distortion (though quantitative inferences about concentrations may be incorrect). However, as seen above, a purely kinetic model that does not account for the dilution effect can have important limitations. With localization in multiple locations, gradients may be present, and spatial averaging can introduce qualitative distortions, which has important implications for both inferences drawn therefrom and models built based on this. Another basic point is related to relative enzyme and substrate concentrations (expression levels) in cells, which implicitly ignores spatial organization. Depending on the localization pattern of enzymes and substrates (and associated sizes of localization regions), conclusions simply based on relative total protein levels could be misleading, as this may not reflect relative levels at locations of co-existence.

3.3. Localization and activity

In this section, we explore new aspects of the bidirectional interplay between localization and chemistry: how chemistry is used to impose localization and how localization impacts chemistry. We previously focused on different patterns of localization of enzymes/substrates and their consequences for enzymatic modification. This was imposed by restricting the presence of relevant species to particular regions and having them non-diffusible. An important aspect of imposing patterns of localization in cells is for this to be done in a flexible and dynamic way. Thus, a basic point of interest is on how active localization (i.e. localization of diffusing species) is imposed, and how that impacts chemical modification spatially and temporally. This is relevant to both monofunctional and bifunctional enzymes. An important difference between these enzymes is that localizing a greater amount of enzyme can increase the activity of monofunctional enzymes, while for bifunctional enzymes the effect of total enzyme concentration may be weak. For bifunctional enzymes, an additional question which emerges is whether localization can be directly used to manipulate and switch enzyme activity. We study these aspects in the context of bifunctional enzymes, noting that some basic insights which emerge are also relevant to monofunctional enzymes (discussed below). We study how the dominant enzyme activity can be switched (from phosphatase to kinase or vice versa), an aspect of interest in natural and engineered contexts.

Different ‘levers’ are available to a cell for spatially manipulating activity of bifunctional enzymes: direct activity regulation (by a factor affecting the balance of enzyme forms), localization, possible feedback (where relevant) or a combination of factors (figure 8a). The impact of localization on enzymes is through the ‘localization network’ (figure 8) and associated localization parameters (binding/unbinding rate constants for each enzyme form). The localization parameters may be the same or different for the two enzyme forms, corresponding, respectively, to equal and biased localization. A bias in localization was introduced by altering the binding parameters of the enzyme forms (see Models and electronic supplementary material, §§2.3 and 3.3). It could also be incorporated by changing unbinding parameters of the two forms, and in any case analytical work explicitly reveals the effect of parameters in this localization network. Our dissection of these factors expands our analysis of spatial organization of bifunctional enzymes studied previously, and creates a bridge to multiple contexts in natural and engineered biology.

Figure 8.

Spatial manipulation of activity of bifunctional enzymes (CMC). (a) A depiction of the key factors for spatially regulating the activity of the bifunctional enzyme. The ‘localization network’ is also depicted, along with interconversion of localized forms (figure 1f). (b) The manipulation of enzymatic activity (CMC) via equal localization with and without feedback: active localization does not by itself introduce a switching in activity (blue curve), but a feedback on both localized and free forms achieves a switch in activity indicated by an elevated X* concentration (green curve). (c) This feedback is compromised if the substrate is no longer co-localized with enzymes (see text) and consequently a higher feedback strength is needed to achieve the switch in activity. (d) The case of biased/differential localization: a switch can be achieved from localization by itself without feedback (red curve), if there is no interconversion of localized forms; if interconversion of localized forms occurs, this does not happen, in which case a feedback on localized and free forms can achieve the desired switch in activity. (e) A schematic depiction of dynamic transitions in localization patterns in Caulobacter with associated bifunctional enzymes/substrates. In the legend, DivK and CtrA labelled represent the phosphorylated forms of these species. (f–h) Dynamic localization involving a transition from a single pole to a double pole localization with opposite activities motivated by what is observed in Caulobacter (here we depict how a phosphatase-dominant activity at one pole (θ = 0) transits to one involving a kinase-dominant activity at the opposite pole, though the principle is also applicable for the reverse case). (f) A scenario showing how opposite enzymatic activities can be seen in the two poles, through a combination of selective localization at one pole, and feedback. Panels (g,h) focus on activity regulation at the second pole and localization (no feedback). Panel (g) shows that even with an activity-regulating signal at the second pole, in the absence of localization in the second pole, the desired switch in activity cannot be seen (blue lines, substrates diffusing), unless the substrate is static (red lines). Panel (h) shows how weakening the localization strength at the first pole progressively (blue-red-green curves) can achieve the desired activity pattern (alternatively strengthening localization at the second pole works).

3.3.1. Direct activity regulation

The most basic way of altering enzymatic activity through the kinetics is by means of an activity-regulating signal which affects the balance of enzyme forms. In order to investigate to what extent an activity-regulating signal can switch the dominant activity of the enzyme spatially, we investigated a range of scenarios where the relative locations of three factors—the activating signal, the enzyme and the substrate—were varied (see electronic supplementary material). This corresponds to an expansion of spatial designs studied in the previous section by incorporating (i) the activating signal and (ii) localization of diffusible entities. Different possibilities for the location of these species were considered: (i) localized at either one location in the spatial domain (e.g. representing one pole of a cell), (ii) localized at two specific locations in the spatial domain, and (iii) present/spreading everywhere in the spatial domain. We examined the possibility that the relevant species was weakly diffusive/non-diffusive or was highly diffusive.

Analysing these various cases reveals the following. (i) If the enzyme is localized, then the presence of activating factor at that location is necessary. (ii) If the substrate is localized and the enzyme is weakly diffusible, then the localization of the activating factor at that location is sufficient. (iii) If the enzyme is not localized and highly diffusible, then the activating factor should be present along with the enzyme or at least in a sufficiently large fraction of the region where the enzyme is present (having activating factor present in the location of the substrate is not sufficient). However, this restricts the possibility of enzyme having different dominant activities at different locations.

3.3.2. Localization as a regulator of activity

We turn to the roles of localization as a tuner of activity, noting its potential role, which may extend to highly diffusible enzymes, and focus on the localization network. We start by examining the case of enzyme and substrate in the same location, but with an additional (active) localization mechanism for the enzymes. In all cases, parameters are chosen to result in a strong localization (of the relevant enzyme form), and this is done by having binding to the location, much stronger than unbinding, in particular having weak unbinding. The default state of the enzyme (balance of kinase/phosphatase) is assumed to be strongly towards the phosphatase. In this context, we also examine the potential role of (linear) feedback between substrate and enzyme as an additional ingredient where relevant. We first focus on bifunctional enzymes with covalent modification biochemistry, and then discuss TCSs. Unless otherwise mentioned, both localized and free forms of enzymes are assumed active.

3.3.3. Unbiased localization

In the case of equal (unbiased) localization (figure 8b) (no feedback) localization by itself does not result in a switching of dominant activity (blue curve). This is true if the localized forms do not interconvert, and also if the localized forms interconvert, at the same rates as the free forms or even if the ‘interconversion equilibrium constant’ (ratio of forward and backward rate constants) is the same for localized and free forms of enzymes. This is shown analytically in electronic supplementary material, §3.3. Now suppose a feedback from the substrate X* resulting in a greater balance of E* (the kinase form) was incorporated. If this acts only on the free (non-localized) enzymes, again, this generally does not result in a transition (red curve) if localized forms can interconvert (and localization in strong): the essential insight is that most of the enzyme is localized, and this feedback acts only on the non-localized form. If feedback is applied to both localized and free enzymes, then a transition to a dominant kinase activity is indeed achieved (green curve), demonstrating an important requirement for feedback regulation.

This previous result assumed that both enzymes and substrate were present in the same location, and not diffusing. This result is also valid in the case where the substrate is localized, but enzyme is freely diffusible. Having the presence of a localization mechanism for a diffusible enzyme increases the local enzyme concentrations but does not affect the balance of enzyme forms. On the other hand, if enzymes were completely confined to a location/compartment (and non-diffusible) and substrate was diffusible, the dilution of the substrate in the location where the enzyme is present implies a reduced feedback. This then implies a stronger feedback constant is needed to achieve a transition (figure 8c).

3.3.4. Biased localization

We now examine the effect of biased/differential localization (figure 8d). We see here that strong biased localization can by itself result in a transition of activity if no interconversion is allowed between localized forms (red curve, figure 8d). On the other hand, if interconversion between localized forms is possible (say at rates similar to the free forms), then the effect of biased localization can be reversed (blue curve). This is established analytically in electronic supplementary material, §3.3. In such a case, in the absence of activity-regulating signals, feedback from the substrate to both localized and free forms is needed (green curve). The diffusion of substrate (as opposed to co-localization with enzymes) can weaken the feedback, which would have to overcome dilution to achieve a switch in activity.

3.3.5. Localization of bifunctional enzymes with two-component biochemistry

We revisit these results in the case of bifunctional enzymes in a TCS. The distinctive feature here, of the balance of enzyme forms being affected by the substrate, is crucial. A direct consequence of this is the fact that, as the substrate amount is increased, the balance of enzyme forms is not a constant and can shift more to the phosphatase. We now turn to this distinctive tendency of switching activity, from kinase to phosphatase, and how this is affected by localization. We restrict ourself to the case that enzymes are localized but substrates are not.

3.3.6. Equal localization

We see in figure 9a that with equal localization the switching of activity from kinase to phosphatase occurs at a well-defined concentration of substrate, which incidentally is practically the same as that with no localization. To understand the interplay of different factors, we examined scenarios which involved the disabling of different factors (figure 9b–d). These factors are the activity of localized forms and the interconversion of localized forms. In the presence of localization, but with the localized forms inactive and non-interconverting, the level of substrate at which this switching was reduced to a small degree (figure 9b). This simply reflects the fact that here localization simply serves to sequester enzyme, reducing the available enzyme for conversion, and consequently the level of substrate required to effect switching is reduced. Now if the localized forms are enzymatically active, but do not interconvert, the level of substrate at which switching happens is dramatically reduced (figure 9c). This is due to the confluence of two factors. (i) A substantial fraction of enzyme is localized. (ii) The substrate modification biochemistry of a TCS results in a unidirectional conversion reaction between localized forms (which are no longer constrained by the basal interconversion balance). On the other hand, if the localized forms do interconvert but are not catalytically active, then the switching of activity happens at a substantially higher level of substrate (figure 9d). Here, a substantial amount of enzyme is localized and subject to interconversion while the effect of the substrate is present only via its action of free enzyme forms. Overall this reveals the subtle interplay of factors at work even in the case of equal localization.

Figure 9.

Switching of activity from kinase to phosphatase in a bifunctional TCS with active localization. The plots investigate how localization by itself can impact the switching from kinase to phosphatase activity in a bifunctional TCS (seen via the crossing of concentration curves for X and X* as the total substrate amount is varied). The top row (a–d) corresponds to equal localization of enzyme forms, while the bottom row (e–h) corresponds to biased localization (stronger for E*). For each row, we examine a number of scenarios corresponding to the absence of interconversion of localized forms, enzymatic activity of localized forms, or both (see text for details, and figure 8 for a depiction of the localization network of the enzymes).

3.3.7. Biased localization

Figure 9e–h focuses on the case where the localization (binding constant) of the kinase form is clearly greater than that of the phosphatase. Many of the insights parallel the earlier case. A well-defined level of substrate determines the switching of activity from kinase to phosphatase (figure 9e). If there is localization with no activity or interconversion of the localized forms, there is a small reduction in the level of substrate needed to effect the switching (figure 9f). Notice that, in such a case, steady state implies a local equilibrium between each pair of reactions, and, as a consequence, the level of substrate at which switching occurs is practically the same as the corresponding case of unbiased localization. As before, if the localized forms are active but with no interconversion the level at which switching occurs is substantially reduced for the same reason (figure 9g). An interesting contrast (to unbiased localization) emerges when we examine the case of localized forms interconverting, while being inactive. In such a case, the level of substrate at which switching occurs is significantly reduced (figure 9h).

Analytically studying the effect of differential (strong) localization of enzymes in this case shows that the net effect is a reduction of free enzyme (the active enzyme), along with a shift towards phosphatase: this is because the free kinase form is more effectively localized (and sequestered from the substrate) than the free phosphatase form. Biased localization by itself, in such a case, can effect the switch from kinase to phosphatase. This behaviour does not depend on the TCS biochemistry in any essential way; in fact, if an analogous case of biased localization with inactive localized forms were considered in the CMC, a similar behaviour ensues and can be explained analytically (and the behaviour is not seen with equal localization).

3.3.8. Requirements for localization to tune activity

A basic insight which emerges from the above studies is that localization (without feedback) can affect/switch activity if there is a difference between localized and non-localized forms (e.g. localized forms do not interconvert or are inactive). This depends on the interplay of (i) the ‘localization network’ (figure 8), (ii) the specific differences between localized and non-localized forms, and (iii) the biochemistry of substrate modification.

3.3.9. Switching activity with dynamic spatial reorganization

We examine a case of dynamic spatial regulation of bifunctional enzymes focusing on a question broadly motivated by observations in the cell cycle in Caulobacter, where (i) the establishment of a localized enzyme with a hypothesized opposite (kinase) activity to the original activity of the enzyme occurs (enzyme PleC) and (ii) the establishment of bifunctional enzymes in two opposite poles with opposing activities (for the enzyme CckA) occurs (figure 8e). The factors responsible for establishing the opposite/new activity at the second pole are unclear and being investigated. Our analysis allows for bridging the gap between the intrinsic kinetics of a bifunctional enzyme and its behaviour in such a scenario, to understand the ways in which dynamic spatial ‘control’ may be achieved by cells (also see Discussion); this is also relevant to the dynamic control of localization in synthetic biology [72,73]. We computationally study this by examining a case where an enzyme with phosphatase activity at one pole can transit to localization at two poles with opposite activities (figure 8f–h) (this is also applicable to kinase activity at the first pole). A way of achieving the desired activity is by simply having a localized activity-regulating factor at the new pole; however, as seen in figure 8g (blue curve), if the localization of enzyme at this new pole is not strong enough, then such a factor is insufficient. In effect the strong localization of enzyme at the first pole and diffusion of enzyme swamps the effect of a localized activity-regulating factor at the second pole. Figure 8h shows that if the strength of enzyme localization in pole 1 is progressively reduced, then it is possible to use activity regulation in pole 2 (green curve). This consolidates the point that a strong disparity in localization strength of enzymes at the poles can limit the efficacy of an activity-regulating signal at a given pole. Alternatively, having a sufficiently strong localization with an activity-regulating signal at the new pole can have the desired effect. On the other hand, to employ localization by itself at the second pole to achieve a transition to an opposite activity, either (i) there should be stronger localization of the desired form at the new location with no interconversion, (ii) there should be stronger localization of the undesired form, with localized forms inactive, or (iii) if the newly created pole is required to have a phosphatase-dominant activity (i.e. the first pole is kinase dominant), then having localized forms active but not interconverting promotes this.

3.3.10. Relevance to monofunctional enzymes

The above insights on bifunctional enzymes are also relevant to monofunctional enzymes, especially when there is an enzyme activation step. In fact if we regard E and E* as the inactive and active forms of an enzyme (say kinase), many of the essential insights carry through as they are primarily dependent on the balance of E and E*, and the factors which affect it. In particular, (i) the different factors affecting E*, (ii) the conditions under which localization can tune activity (both for a CMC and TCS), and (iii) the fact that localized activity regulation may not be sufficient, and localization of enzyme may be needed, are all relevant.

3.4. Evolution and spatial organization

We now explore the interplay of biochemistry and spatial organization, through a new prism: that of evolution. Evolution, in this regard, can function on both the biochemistry as well as spatial organization, causing important alterations to each. At the outset (figure 10a), for our purposes we can identify four classes of scenarios depending on whether the original ‘design’ had a non-trivial spatial aspect or not, and whether a change introduced did or did not involve a non-trivial spatial aspect. Evolutionary changes are often studied as purely kinetic changes subsumed within kinetic descriptions, which is the one category which does not involve spatial aspects in any essential way. Here we focus on two other categories: (i) biochemical alterations built upon pre-existing spatial designs and (ii) spatial alterations built on a design which did not have a spatial aspect. We present focused insights pertaining to both these aspects.

Figure 10.

The effect of pre-existing spatial organization on a transition from monofunctional to bifunctional enzymes. (a) A broad classification table for alterations through evolution, depending on the presence or the absence of spatial aspects in the original ‘design’ and in the alterations imposed. The plot then investigates the impact of prior spatial organization on the evolution from monofunctional to bifunctional enzymes via gene fusion. Two scenarios for the monofunctional enzyme were considered (see schematic): (b) K (kinase) localized in one location and P (phosphatase) everywhere, or (c) K and P localized in two distinct locations. Both scenarios result in graded profiles of substrate (not shown: see green curves in figure 2b,c for a reference). If the resulting bifunctional enzyme can localize where either parent enzyme could, then the first scenario results in essentially complete localization of the bifunctional enzyme at one location, at a balance determined by the basal interconversion rates of the enzyme forms (and consequent absence of graded profiles of substrate) while in the second scenario both enzyme forms are localized at both locations (again at a balance determined by enzyme interconversion rates, implying that a gradient of substrate is also abolished)—also see electronic supplementary material, figure S8(A,B). EZ and E*Z denote localized forms of kinase and phosphatase forms of the enzyme, respectively, at one location (localizing species Z) and EY and E*Y denote the localized forms of these enzymes at another location (localizing species Y) here and in figure 11.

3.4.1. Biochemistry

We start with the biochemical alterations. We focus on one basic aspect of evolution which is relevant in the context of our study—that involving a transition from a monofunctional enzyme pair to a bifunctional enzyme. While, in TCSs, bifunctional enzymes have emerged from a common ancestor, other bifunctional enzymes have been experimentally demonstrated to have emerged via a process of gene fusion. This allows for a single transcript producing protein which is capable of both kinase and phosphatase activity. The evolutionary advantages of such a process have been discussed in [22]. However no aspect of spatial organization has been considered. We now examine the effect of a transition from monofunctional to bifunctional enzymes, in the face of a pre-existing spatial organization.

To do this, we examine two scenarios for the monofunctional enzymes in silico. The first scenario is where one of the enzymes is localized: without loss of generality, we assume that the kinase is localized at one location of the cell, while the phosphatase is present everywhere. In the second scenario kinase and phosphatase are localized at two separate locations (see schematic in figure 10). We use this as a basis to examine the question: what effect does this have on evolvability from monofunctional to bifunctional enzymes? To make a comparison in silico, we assume equal amounts of kinase and phosphatase enzyme to start with, though our essential insights neither stem from nor require this.

The key aspect to be examined is the overall behaviour of the system, which in turn strongly depends on the localization pattern of the bifunctional enzymes created through this process. In all cases, we assume that both modified and unmodified substrates are diffusible and present everywhere in the domain: we keep this spatial ‘design’ of substrates fixed, and focus only on the enzymes. There are multiple possibilities regarding the localization pattern of the bifunctional enzymes. We examine these in turn.

The simplest and most obvious hypothesis regarding the localization of the bifunctional enzymes involves an ‘OR’ logic with regard to the localization of the parent enzymes: the bifunctional enzyme can localize at any particular location where either the parent kinase or phosphatase is localized. This, in turn, stems from a basic consideration that a binding motif which was present in the protein structure of either of the enzymes is now present in the protein structure of the bifunctional enzyme.

Computational analysis (figure 10b) reveals the basic fact that if there was only one localization present, then the bifunctional enzyme would localize exclusively (or almost exclusively) in that location. This simply follows from the fact that, with strong localization (comparable to that of the parent enzyme), almost all the enzyme ends up localized there. In terms of substrates, this creates a significant qualitative change from gradient to uniform substrate profiles (also see figures 2 and 4 for reference).

In the case of localization of the kinase and phosphatase at two locations (figure 10c), we find that the bifunctional enzyme ends up essentially equi-distributed at the two locations. In this case, the bifunctional enzyme system will exhibit the same balance of kinase and phosphatase at both poles (determined by the bifunctional enzyme interconversion rate; see electronic supplementary material, figure S8(B)), and the substrate will exhibit a uniform profile (figure 10c), a contrast from the monofunctional case where a gradient is obtained (as in figure 2). This then means that, to recapitulate the original behaviour, the activation signal (which converts the phosphatase form to kinase form) must have different levels at the two locations.

In the above scenario, the bifunctional enzyme localizes where either of the monofunctional enzymes could localize. This decouples localization from activity. We then examine conditions under which the bifunctional enzyme may approach scenarios analogous to those of a monofunctional enzyme pair (figure 11a,b). To do this, we incorporate a basic logic which couples binding at a location to its activity. Thus, in the first localization scenario, we assume (i) E* alone can bind to the designated location (say pole of the cell) and (ii) no interconversion of enzyme forms occurs when this enzyme is bound (this is consistent with the requirement that only kinase is present in this location). With these assumptions computer simulations (figure 11a) reveal simply that (essentially) all enzyme ends up as kinase at this one location (blue curves of enzyme profiles). This is again a drastic change from the monofunctional enzyme results, and suggests that a simple mirroring of the localization logic of monofunctional enzymes does not guarantee the same behaviour in a bifunctional enzyme. We then allow for interconversion of localized forms, in which we incorporate a strong unbinding of the phosphatase. In such a case, we find that there is increased phosphatase accumulated near the designated location (but not bound). However if the diffusivity of enzymes is high, this results in the phosphatase form being present everywhere, while the kinase form is bound and primarily localized in one location (red curve of enzyme profiles). This is reminiscent of the localization pattern of the monofunctional enzymes, and results in a similar substrate behaviour.

Figure 11.

Conditions where the bifunctional enzyme exhibits a comparable localization/activity to the monofunctional enzyme, building on the scenarios for the monofunctional enzymes in figure 10b,c. (a,b) The species concentration profiles when only E* (kinase form) localizes (binds) in one location and when E (phosphatase form) and E* localize in two different locations, respectively. In (a), if there is no interconversion between localized E and E* (blue curve), the bifunctional enzyme forms end up essentially fully as E* at one pole; however, when the interconversion of localized forms with a strong unbinding of E is allowed (along with high diffusivity of enzymes), broadly similar results with monofunctional enzyme can be generated (red curve: note the difference in scales between blue and red curves, and that the total free phosphatase amount is comparable to the localized kinase amount). Furthermore, in (b), the bifunctional enzyme can also show a similar behaviour (E and E* localized in two different poles with graded substrate concentration profiles) to the monofunctional enzyme pair in the presence (red curve) or absence (blue curve) of the interconversion of localized forms. This however depends on relatively high diffusivity of free enzyme relative to substrate in the former case, while it is independent of diffusivity in the latter case. (c) A schematic summary of results in figures 10 and 11.

We revisit this when kinase and phosphatase are localized in two different locations (figure 11b). Here, employing this logic results in a situation where, even without any interconversion of localized forms, enzymes can localize at the two locations in a manner reminiscent of the monofunctional enzymes (blue curve, figure 11b). The relative concentration of kinase and phosphatase at the two poles is now affected by the basal interconversion rate constants between kinase and phosphatase forms (even if interconversion does not occur when localized). If interconversion of localized forms is allowed, similar localization patterns to those of the monofunctional enzymes can ensue, if there is immediate dissociation of the other form, and the enzymes are highly diffusible (red curve, figure 11b). This parallels the insight from the previous case.

These results (summarized in figure 11c) show how factors associated with the way in which molecules localize and the relationship between the localization of the bifunctional enzyme and that of the monofunctional enzyme can have a significant impact on the consequences of a transition in the biochemistry (also see electronic supplementary material, figures S8 and S9). This suggests that drastic changes in localization patterns can emerge, identifies conditions under which that happens and also identifies requirements under which there is no drastic qualitative change.

3.4.2. Evolution of spatial organization

We now investigate the effect of a transition in spatial organization (due to altered patterns of localization of species), for fixed biochemistry. This is relevant to transitions both in evolution and occurring during the lifetime of a cell. Our analysis of spatial designs facilitates a transparent analysis of this.

Monofunctional enzymes. Figure 2 has depicted the effect of different spatial designs on the overall substrate modification. A single change in localization pattern corresponds to a change of one index. As seen there, a change of substrate localization can be associated with dilution, while localizing substrates has the opposite effect. The identity of the substrate(s) therein is important as well. Transitions in enzyme locations can result in a spatial design where there is a purely transient response in modification or de-modification of proteins with a zero steady-state output. It remains to be seen if this has been exploited in cells. Alterations in enzyme localization may facilitate or prevent graded behaviour of substrates.

Bifunctional enzymes. An analogous set of plots are presented in figure 4 and electronic supplementary material, figure S2. Distinct aspects associated with bifunctional enzymes include the following. (i) Moving from a single location organization of enzymes to a double location organization (with no alteration in balance of enzyme forms) still continues to result in a spatially uniform profile (independent of diffusion). An alteration of balance of enzyme forms spatially is necessary for generating a robust gradient. (ii) In contrast to monofunctional enzymes, even with some substrate species non-diffusible/localized, zero concentrations are never encountered, as long as enzymes are not separated from substrates initially (discussed previously). (iii) Spatial alterations can actually facilitate ACR, even if it did not exist kinetically to start with. (iv) Spatial alterations can destroy different types of behaviour such as ACR, bistability and switch-like behaviour. Some of the changes introduced by a change of spatial organization may be compensated for kinetically.

A central theme which emerges from our studies of alterations in biochemistry and spatial organization is the possibility of fundamental qualitative alterations associated with altered enzyme localization patterns. If such a transition is deleterious to a cell (e.g. resulting in a zero steady-state concentration or abolishment of a gradient), then such transitions are not expected to occur. On the other hand, if some transitions are tolerated (say with attendant dilution effects) then (i) other subsequent kinetic alterations (e.g. increasing enzyme production) could compensate for this (and in some cases understood only in light of this), (ii) the ‘memory’ of the earlier spatial organization may be erased, and this may also facilitate new transitions, with further changes in spatial organization which may significantly diverge from the original spatial design. Finally, changes in spatial organization may also enable new desirable features.

4. Conclusion and discussion

4.1. Conclusion

Spatial organization is a central aspect of cells, while biochemical pathways are central to cellular information processing. This prompts many questions regarding their interplay which are relevant to systems and synthetic biology, and the emerging field bridging the two, with broader implications for chemical information processing. Advances in imaging and the advent of spatial proteomics [38] and substantial recent experimental work in synthetic biology provide further impetus.

Our study employed models of basic biochemical building blocks of pathways and the most basic ways of imposing spatial organization: localization or delocalization of species. Since a basic way of imposing spatial organization in pathways is through single modification stages, this serves as a basis for understanding the effects of spatial organization on pathways and complex biomolecular networks. These contain additional ingredients, such as cascades, multisite substrate modification, feedback and so on. Even in these more complex cases, the effects of (de)localization based on chemical identity remain a core feature (for instance, a key effect of multisite substrate modification is determining localization).

Our study significantly expands on earlier work on CMC-based cascades [55,56] in many essential ways: the consideration of a broad range of building blocks (CMCs and TCSs, monofunctional and bifunctional enzymes and augmentations involving feedback or dead-end complexes) and a range of information processing characteristics, the systematic consideration of spatial designs, the consideration of active localization and localization as a tool for tuning activity as well as the interplay of biochemistry and spatial organization from an evolutionary perspective.

By examining a range of spatial designs in basic modification cycles (localization scenarios for enzymes and substrates), a range of insights emerge. Static gradients are observed only in specific instances. A requirement for the steady-state modification extent not to be at either extreme (zero or complete modification) is that spatial organization must ensure that the modification cycle of events is not disrupted. Instances where such a disruption happens involves a particular form of modified substrate losing contact with the enzyme which modifies it (§3.1). An echo of this feature is seen in modification cascades too [56]. Bifunctionality of enzymes both gains from and exhibits certain advantages, with regard to spatial organization (§3.2). On one hand, distinct extreme activities of the enzyme can be realized simultaneously, while, on the other hand, the bifunctionality ensures that, even with spatial organization, the modification cycle is never disrupted. We demonstrate conditions of spatial organization which are consistent with concentration robustness in modification cycles with bifunctional enzymes, and bistability and threshold behaviour. If a particular behaviour is seen experimentally either (i) spatial organization is absent and the behaviour can be understood purely in kinetic terms, (ii) specific forms of spatial organization may be present, with the kinetic regime/total protein concentrations bypassing constraints imposed by spatial organization, or (iii) the behaviour emerges from a non-trivial combination of spatial organization and kinetics, and is not attributable to kinetics alone (as shown in the case of robustness).

A basic way of imposing spatial organization is through chemistry itself. We demonstrate (§3.3) how the activity of a bifunctional enzyme may be manipulated by activity regulation, feedback (where applicable) and localization itself. We showed how activity regulation by itself may be insufficient to tune enzyme activity locally (especially where strong localization of the enzyme elsewhere may be present), and may need to act in concert with localization. On the other hand, localization by itself (without any activity-regulating factor) can be used to manipulate the activity of bifunctional enzymes when there is a difference between localized and non-localized forms (e.g. localized forms do not interconvert, or they are inactive). Differences in the nature of substrate modification (TCSs or CMCs) can be critical here. Many of these insights are also relevant to monofunctional enzymes involving an enzyme activation step. These insights are particularly relevant in instances where localization patterns change dynamically (seen in Caulobacter for instance), where reconciling local activity and overall behaviour has led to unresolved questions. From an evolutionary perspective, we examined changes in spatial organization for a fixed chemistry, and chemical changes for an existing spatial ‘design’ (§3.4). The former case highlights significant distortions in behaviour, but also new capabilities in information processing characteristics (e.g. ACR in §3.2). In the latter case, changes in chemistry (e.g. monofunctional to bifunctional enzymes, but more broadly relevant) may lead to significant unanticipated alterations in behaviour because this results in substantially altered localization patterns. Requiring a maintenance of qualitatively similar localization patterns in such cases may require (a confluence of) additional factors which are not guaranteed (and may be infeasible). This simply highlights important limitations in thinking of evolutionary changes purely in kinetic chemical terms [22,74].

4.2. Discussion

4.2.1. Systems biology

Understanding how cellular functions are realized through biomolecular networks is one of the central goals of systems biology. Understanding the many ways in which spatial organization may impact network behaviour is fundamental to this, and our analysis and extensions to more complex pathways is foundational to this, complementing data-driven approaches.

4.2.2. Space and information processing biomolecular networks

Our analysis is relevant to determining when experimentally observed behaviour may or may not be understood in purely kinetic terms. Spatial organization can cause modulatory changes (e.g. by dilution) or distort the sequence of modifications, in a way not encapsulated in the kinetics. In the latter case employing a purely kinetic description is fraught with risk, and reconciling data with the kinetic model is likely to introduce significant errors. Even in the former case, employing a (lumped) kinetic model may introduce quantitative and in certain cases qualitative distortions when the model is used extrapolatively, for instance to study dose–response curves. These aspects are crucial in understanding the robustness of biomolecular networks. Biomolecular network robustness is usually studied through kinetic parameter perturbations (often one at a time) to make inferences about system robustness. Spatial organization introduces new parameters which are associated with size and location: these parameters vary across the life cycle of the cell, and the interplay of these factors and kinetic factors determines the actual robustness of the biomolecular network.

4.2.3. Spatial organization and evolution

Spatial organization may either be a passive (present but not manipulated) or an active ingredient in evolution (§3.4). As a passive ingredient, it can exhibit a significant impact by providing constraints (and also capabilities) for the feasibility/advantages of kinetic changes (and for realizing particular behaviour). Furthermore, the order of transitions in chemistry and spatial organization are important: in some cases, alterations in spatial organization building on kinetic alterations could create an advantageous scenario without significantly compromising system behaviour (which can explain why it is observed), while reversing the order of events may be deleterious to the cell.

4.2.4. A concrete cellular context

The cell-cycle network in Caulobacter involves two interacting modules with bifunctional enzymes: the DivK-DivJ-PleC and the CckA-CtrA modules, where PleC and CckA are bifunctional enzymes modulating the phosphorylation/dephosphorylation of DivK and CtrA (key cell-cycle drivers), respectively (figure 8). Interestingly, both modules are subject to temporal and spatial regulation, which is precisely choreographed [40,75]. What relevance is our analysis here? There exist experimental studies in different cell-cycle phases and models which aim to consolidate known knowledge, explain aspects of observed behaviour and are predictive. The models make different assumptions about the behaviour of the bifunctional enzymes. Bistability and irreversible decision-making are at the core of a model at one stage of the cell cycle [3] while other models [76] treat distributed bifunctional enzymes as separate enzymes. Yet there are important questions associated with the kinetic assumptions and consistency of different models, the implications of bistability when extrapolated to different cell-cycle stages, the factors responsible for establishing a particular enzymatic activity pattern at the new pole [21,77], and understanding where bifunctionality is central.

While addressing these questions needs a dedicated study of its own, our approach can serve as a platform, with a distinct explorative framework. (i) Our analysis allows us to directly build from the study of a single module to two interacting modules (with augmentation to account for additional factors) with spatial and temporal regulation. This creates a library of scenarios for pathway behaviour in terms of assumptions about individual modules, and localization of respective components. Using this, it is possible to systematically tease out the effects of mutations which may alter spatial organization of different components in each of the modules, making a series of new testable predictions. Experiments on mutants arising from mislocalization have already been performed. (ii) Our analysis has implications for the possibility of bistability as a basis of a transition in one cell-cycle phase. We find that, once localization is lost, the bistability may be lost, but also that it may be retained (and propagated to other phases). The former case is guaranteed for sufficiently large growing cells, while the latter is possible with sufficient total substrate at the initial pole, or insufficient changes in cell size. This could be seen (or bypassed) in natural and mutant cells. (iii) Our study provides a number of insights into how opposite activity may be realized spatially and temporally at different locations (seen in Caulobacter), using basic modes of regulation, something which is not easily experimentally established.

4.2.5. Synthetic and engineered biology

Multiple synthetic biochemical circuits have been developed; for example, oscillators based on phosphorylation, and other biochemical logic modules [51,78,79]. Considerable progress has been made recently in engineering compartmentalization through various means, including the use/creation of synthetic microcompartments and in general harnessing localization and compartmentalization, both statically and dynamically, and creating networks of compartments [47,48,52,72,73,80]. Different components ranging from entire cell populations to gene regulatory machinery to enzymes have been localized. Compartmentalization has been applied to chemical pathways: this involves localization of selected components in compartments. Notable aspects of compartmentalization implemented in the bacterial cytoplasm include the reversible formation of compartments controlled by (de)phosphorylation, creation of membraneless compartments to facilitate cascade reactions and prevent substrate inhibition, synthetic scaffolds for manipulating spatial organization of enzymes and microcompartments for controlled encapsulation of enzymes [81–84]. For bottom-up synthetic biology, the basic questions centre around the effect of tuneable dials, which affect transport within and between compartments and how different components may be localized. Our analysis demonstrates the effect of (de)localization of different combinations of components. This ‘spatial design’ approach can guide rational spatial engineering in biochemical pathways, including enzymatic cascades and multisite substrate modification and assessing multiple aspects of compartmentalization. Unique features of bifunctional enzymes in this regard include the distribution of bifunctional enzymes to multiple locations without significant loss of effect, their maintenance of local reversibility of reaction and the way robust output behaviour can be obtained. Each of these features could be exploited in synthetic biology. Another aspect is using localization as a way of tuning enzymatic activity temporally and spatially.

4.2.6. Chemical information processing

Subtle differences in chemistry (TCSs and CMCs) can determine whether spatial information is backpropagated ‘spatial retroactivity’) at steady state (§3.1).

The interplay of spatial organization and chemistry is bidirectional: spatial organization affects chemical behaviour, while chemistry can impact spatial organization. Both these aspects, and their confluence, can impact a pathway behaviour at different levels/scales, as exemplified by the following. (i) For bifunctional enzymes, details such as whether localization allows for enzymatic interconversion or enzymes being active, as well as any bias in localization of enzyme forms, can significantly impact the overall enzyme activity (§3.2). (ii) For a transition from monofunctional to bifunctional enzymes (by gene fusion), replicating the localization pattern of monofunctional enzymes is difficult if only one of the parent enzymes is localized: it can be achieved in bifunctional enzymes only if a similar correlation between enzyme activity (i.e whether the enzyme is in the conformation of a kinase or a phosphatase) and localization is maintained. This in turn imposes a significant constraint at a molecular level, requiring binding capabilities correlated with distinct enzyme conformational structures (§3.3).

A consolidated view of spatial regulation of enzymes/substrates is crucial for unravelling signalling networks in bacteria and eukaryotes, identifying design principles/features, engineering pathways in synthetic biology and chemistry, with broader implications for chemical information processing. A systems analysis focusing on basic building blocks and modes of spatial organization creates a platform providing insights into all these areas, allowing for a seamless merging of the natural and the synthetic in this context.

Data accessibility

All the relevant data are contained in the main text and electronic supplementary material.

Authors' contributions

J.K. planned the work, including the conceptual approach and analysis, performed the analytical work, analysed the data, wrote the manuscript and supervised L.L. and A.A.N. L.L. performed the computational work (including code development and data analysis) and contributed results (and plots) throughout the entire Results section and provided input in editing the manuscript. A.A.N. performed computational work (including code development and data analysis) and contributed results (and plots) in §§3.2 and 3.3, as well as preliminary work for the study; also provided input in editing the manuscript. All authors read the manuscript.

Competing interests

We declare we have no competing interest.

Funding

A.A.N. was funded by an EPSRC Prize Fellowship in the early stages of the project.