Signaling Architectures that Transmit Unidirectional Information Despite Retroactivity.

A signaling pathway transmits information from an upstream system to downstream systems, ideally in a unidirectional fashion. A key obstacle to unidirectional transmission is retroactivity, the additional reaction flux that affects a system once its species interact with those of downstream systems. This raises the fundamental question of whether signaling pathways have developed specialized architectures that overcome retroactivity and transmit unidirectional signals. Here, we propose a general procedure based on mathematical analysis that provides an answer to this question. Using this procedure, we analyze the ability of a variety of signaling architectures to transmit one-way (from upstream to downstream) signals, as key biological parameters are tuned. We find that single stage phosphorylation and phosphotransfer systems that transmit signals from a kinase show a stringent design tradeoff that hampers their ability to overcome retroactivity. Interestingly, cascades of these architectures, which are highly represented in nature, can overcome this tradeoff and thus enable unidirectional transmission. By contrast, phosphotransfer systems, and single and double phosphorylation cycles that transmit signals from a substrate, are unable to mitigate retroactivity effects, even when cascaded, and hence are not well suited for unidirectional information transmission. These results are largely independent of the specific reaction-rate constant values, and depend on the topology of the architectures. Our results therefore identify signaling architectures that, allowing unidirectional transmission of signals, embody modular processes that conserve their input/output behavior across multiple contexts. These findings can be used to decompose natural signal transduction networks into modules, and at the same time, they establish a library of devices that can be used in synthetic biology to facilitate modular circuit design.

Introduction

Cellular signal transduction is typically viewed as a unidirectional transmission of information via biochemical reactions from an upstream system to multiple downstream systems through signaling pathways (1, 2, 3, 4, 5, 6, 7). However, without the presence of specialized mechanisms, signal transmission via chemical reactions is not in general unidirectional. In fact, the chemical reactions that allow a signal to be transmitted from an upstream system to downstream systems also affect the upstream system due to the resulting reaction flux. This flux is called retroactivity, which is one of the chief hurdles to one-way transmission of information (8, 9, 10, 11, 12, 13). Signaling pathways, typically composed of phosphorylation, dephosphorylation, and phosphotransfer reactions, are highly conserved evolutionarily, such as the MAPK cascade (14) and two-component signaling systems (15). Thus, the same pathways act between different upstream and downstream systems in different scenarios and organisms, facing different effects of retroactivity in different contexts. For signal transmission to be unidirectional in these different contexts, a signaling pathway should have evolved architectures that overcome retroactivity. Specifically, these architectures should impart a small retroactivity to their upstream system (called retroactivity to the input) and should be minimally affected by the retroactivity imparted to them by their downstream systems (retroactivity to the output).

Phosphorylation-dephosphorylation (PD) cycles, phosphotransfer reactions, and cascades of these are ubiquitous in both prokaryotic and eukaryotic signaling pathways, playing a major role in cell cycle progression, survival, growth, differentiation and apoptosis (1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19). Numerous studies have been conducted to analyze such systems, starting with milestone works by Stadtman and Chock (20, 21, 22) and Goldbeter et al. (23, 24, 25), which theoretically and experimentally analyzed phosphorylation cycles and cascades. These systems were further investigated by Kholodenko et al. (26, 27, 28) and Gomez-Uribe et al. (29, 30). However, these studies considered signaling cycles in isolation, and thus did not investigate the effect of retroactivity. The effect of retroactivity on such systems was theoretically analyzed in the work by Ventura et al. (31), where retroactivity is treated as a “hidden feedback” to the upstream system. Experimental studies then confirmed the effects of retroactivity in signaling systems through in vivo experiments on the MAPK cascade (12, 13) and in vitro experiments on reconstituted covalent modification cycles (9, 11). These studies clearly demonstrated that the effects of retroactivity on a signaling system manifest themselves in two ways. They cause a slowdown of the temporal response of the signaling system’s output to its input and lead to a change of the output’s steady state.

In 2008, Del Vecchio et al. (8) demonstrated theoretically that a single PD cycle with a slow input kinase can attenuate the effect of retroactivity to the output when the total substrate and phosphatase concentrations of the cycle are increased together. Essentially, a sufficiently large phosphatase concentration along with relatively large kinetic rates of modification adjusts the cycle’s internal dynamics very quickly with respect to a relatively slower input, making any retroactivity-induced delays negligible on the timescale of the signal being transmitted (32). A similarly large concentration of the total cycle’s substrate ensures that the output’s steady state is not significantly affected by the presence of downstream sites. These theoretical findings were later verified experimentally both in vitro (11) and in vivo (33). Although a single PD cycle can attenuate the effect of retroactivity to the output, it is unfortunately unsuitable for unidirectional signal transmission. In fact, as the substrate concentration is increased, the PD cycle applies a large retroactivity to the input, causing the input signal to slow down. This was experimentally observed in (33). The experimental results of (34) further suggest that a cascade composed of two PD cycles and a phosphotransfer reaction could overcome both retroactivity to the input and retroactivity to the output. In (35), it was theoretically found that for certain parameter conditions, a cascade of PD cycles could attenuate the upward (from downstream to upstream) propagation of disturbances applied downstream of the cascade. In (36), a parametric study was performed on a cascade of single phosphorylation cycles at steady state, and parametric regimes in which the cascade would transmit signals either upstream (using retroactivity) or downstream were numerically determined. These results suggest that specific signaling architectures may be able to counteract retroactivity. However, to the best of the authors’ knowledge, no attempt has been made to systematically characterize signaling architectures with respect to their ability to overcome the effects of retroactivity and therefore enable unidirectional signal transmission.

This work presents a procedure by which to identify and characterize signaling architectures that can transmit unidirectional signals. We first model a general signaling system based on the underlying reactions that the species of the signaling system participate in. These reactions result in an ordinary differential equation (ODE) model based on the reaction-rate equations. Based on this general model, we propose a procedure to evaluate the unidirectional signaling ability of a general signaling architecture that operates on a fast timescale relative to its input. Such a model is valid for many signaling systems that transmit relatively slower signals, such as those from slowly varying “clock” proteins that operate on the timescale of the circadian rhythm (37), from proteins signaling nutrient availability (38), or from proteins whose concentration is regulated by transcriptional networks, which operate on the slower timescale of gene expression (39). Our framework provides expressions for retroactivity to the input and to the output, as well as the input-output relationship of the signaling system. These expressions are given in terms of the reaction-rate parameters and protein concentrations. Based on these expressions, we present a procedure to analyze the ability of signaling systems to transmit unidirectional signals by tuning their total (modified + unmodified) protein concentrations. We focus on total protein concentrations as a design parameter, because these appear to be highly variable in natural systems and through the course of evolution. Thus, it is possible that natural systems themselves use protein concentrations as a design parameter, optimizing them to improve systems’ performance (40, 41). Further, protein concentrations are also an easily tunable quantity in synthetic genetic circuits. The different “dials” that can be used to tune protein concentration have been characterized in (42). Protein concentrations have been tuned in (33) and (34) to show the effect of increasing substrate and phosphatase concentrations on the retroactivity attenuation properties of a signaling cycle. Thus, we analyze a number of signaling architectures composed of PD cycles and phosphotransfer systems by tuning total protein concentrations.

Methods

Problem definition

In this work, we consider a general signaling system, S, connected between an upstream and downstream system, as shown in Fig. 1 A. Here, is the state-variable vector of S, and each component of represents the concentration of a species of system S. System S receives an input from the upstream system in the form of a protein whose concentration is U, and sends an output to the downstream system in the form of a protein whose concentration is Y. When this output protein reacts with the species of the downstream system, whose normalized concentrations are represented by state variable v, the resulting reaction flux changes the behavior of the upstream system. We represent this reaction flux as an additional input, , to the signaling system. Similarly, when the input protein from the upstream system reacts with the species of the signaling system, the resulting reaction flux changes the behavior of the upstream system. We represent this as an input, , to the upstream system. We call the retroactivity to the input of S and the retroactivity to the output of S, as in (8). For system S to transmit a unidirectional signal, the effects of on the upstream system and of  on the downstream system must be small. Retroactivity to the input changes the input from to U, where is shown in Fig. 1 B. Thus, for the effect of to be small, the difference between U and must be small. Retroactivity to the output changes the output from (where “is” stands for isolated) to Y, where is shown in Fig. 1 C, and for the effect of retroactivity to the output to be small, the difference between and Y must be small. An ideal unidirectional signaling system is therefore a system where the input is transmitted from the upstream system to the signaling system without any change imparted by the latter, and the output of the signaling system is also transmitted to the downstream system without any change imparted to it by the downstream system. Based on this concept of ideal unidirectional signaling system, we then present the following definition of a signaling system that can transmit information unidirectionally. To give this definition, we assume that the proteins (besides the input species) that compose signaling system S are constitutively produced and therefore their total concentrations (modified and unmodified) are constant. The vector of these total protein concentrations is denoted by .

Figure 1

Interconnections between signaling system S and its upstream and downstream systems, along with input, output, and retroactivity signals. (A) Full system showing all interconnection signals: is the input from the upstream system to the signaling system, with state variable vector . is the output of the signaling system, sent to the downstream system, whose state variable is v. is the retroactivity signal from the signaling system to the upstream system (retroactivity to the input of S), and is the retroactivity signal from the downstream system to the signaling system (retroactivity to the output of S). (B) Ideal input, : output of the upstream system in the absence of the signaling system . (C) Isolated output, : output of the signaling system in the absence of the downstream system ( = 0). denotes the corresponding state of S. To see this figure in color, go online.

Definition 1

We will say that system S is a signaling system that can transmit unidirectional signals for all inputs , if can be chosen such that the following properties are satisfied:

is small: this is mathematically characterized by requiring that be small for all .

System S attenuates the effect of on Y: this is mathematically characterized by requiring that be small for all .

Input-output relationship: , for some , for some , and for all .

Note that definition 1 specifies that the signaling system must impart a small retroactivity to its input (requirement 1) and attenuate retroactivity to its output (requirement 2). Requirement 3 specifies that the output must not saturate with respect to the input, so that the signal is still propagated downstream by the signaling system. In particular, definition 1 specifies that these properties should be satisfied for a full range of inputs and outputs, implying that these properties must be guaranteed by the features of the signaling system and cannot be enforced by tuning the amplitudes of inputs and/or outputs.

Example

As an illustrative example of the effects of and on a signaling architecture, we consider a signaling system, S, composed of a single PD cycle (8, 11, 33). The system is shown in Fig. 2 A. It receives a slowly varying input signal, U, in the form of kinase concentration Z generated by an upstream system, and it has as the output signal Y the concentration of X∗, which in this example is a transcription factor that binds to promoter sites in the downstream system. Kinase Z phosphorylates protein X to form X∗, which is dephosphorylated by phosphatase M back to X. The state variables of S are the concentrations of the species in the cycle, that is, , where C1 and C2 are the complexes formed by X and Z during phosphorylation, and by X∗ and M during dephosphorylation, respectively. The state variable v of the downstream system is the normalized concentration of C, the complex formed by X∗ and p (i.e., where is the total concentration of the downstream promoters). This configuration, where a signaling system has as downstream system(s) gene expression processes, is common in many organisms, as it is often the case that a transcription factor goes through some form of covalent modification before activating or repressing gene expression (43). However, the downstream system could be any other system, such as another covalent modification process, which interacts with the output through a binding-unbinding reaction. We denote the total amount of cycle substrate by and the total amount of phosphatase by .

Figure 2

Tradeoff between small retroactivity to the input and attenuation of retroactivity to the output in a single phosphorylation cycle. (A) Single phosphorylation cycle, with input Z as the kinase: X is phosphorylated by Z to X∗ and dephosphorylated by the phosphatase M. X∗ is the output and acts on sites p in the downstream system, which is depicted as a gene expression system here. (B–E) Simulation results for the ODE model shown in Supporting Material, Eq. 17. Simulation parameters are given in Table S1. The ideal system is simulated for with . The isolated system is simulated for with . To see this figure in color, go online.

According to definition 1, we vary the total protein concentrations of the cycle, , to investigate the ability of this system to transmit unidirectional signals. To this end, we consider two extreme cases: first, when the total substrate concentration, , is low (simulation results in Fig. 2, B and C), and second, when it is high (simulation results in Fig. 2, D and E). For both these cases, we change proportionally to . This is because, for large Michaelis-Menten constants, we have an input-output relationship with and (details in Supporting Material, Eq. 21, as defined in Definition 1, requirement 3). To maintain the same K for fair comparison between the two cases, we vary proportionally with . Here, and are the Michaelis-Menten constant and catalytic rate constant for the phosphorylation reaction, and and are the Michaelis-Menten constant and catalytic rate constant for the dephosphorylation reaction. These reactions are shown in Supporting Material, Eq. 16. For the simulation results, we consider a sinusoidal input to see the dynamic response of the system to a time-varying signal. Results for responses to the step input are shown in Fig. S1. For these two cases, then, we see from Fig. 2 B (and Fig. S1 B) that when (and ) is low, is small, i.e., is small (satisfying requirement 1 of definition 1). This is because kinase Z must phosphorylate very little substrate X, and thus, the reaction flux due to phosphorylation to the upstream system is small. However, as seen in Fig. 2 C (and Fig. S1 C), for low , the signaling system is unable to attenuate . The difference is large, and requirement 2 of definition 1 is not satisfied for low . This large retroactivity to the output is due to the reduction in the total substrate available for the cycle because of the sequestration of X∗ by the promoter sites in the downstream system. Since is low, this sequestration results in a large relative change in the amount of total substrate available for the cycle, and thus, interconnection to the downstream system has a large effect on the behavior of the cycle. For the case when (and ) is high, the system shows exactly the opposite behavior. From Fig. 2 D (and Fig. S1 D), we see that is high (thus not satisfying requirement 1 of definition 1), since the kinase must phosphorylate a large amount of substrate, but is attenuated (satisfying requirement 2), since there is enough total substrate available for the cycle even once X∗ is sequestered. Thus, this system shows a tradeoff: by increasing (and ), we attenuate retroactivity to the output but do so at the cost of increasing retroactivity to the input. Similarly, by decreasing (and ), we make retroactivity to the input smaller, but at the cost of being unable to attenuate retroactivity to the output. Therefore, requirements 1 and 2 cannot be independently obtained by tuning and .

We note that because the signaling reactions, i.e., phosphorylation and dephosphorylation, act on a faster timescale than the input, the signaling system operates at quasi-steady state and the output is able to quickly catch up to changes in the input. It has been demonstrated in (32, 34) that this fast timescale of operation of the signaling system attenuates the temporal effects of retroactivity to the output, which would otherwise result in the output slowing down in the presence of the downstream system. Thus, although the high substrate concentration is required to reduce the effect of retroactivity to the output due to permanent sequestration, timescale separation is necessary for attenuating the temporal effects of the binding-unbinding reaction flux (32).

Generalized model

Although the single phosphorylation cycle shows some ability to attenuate retroactivity, it is not able to transmit unidirectional signals due to the tradeoff seen above. We therefore study different architectures of signaling systems, composed of phosphorylation cycles and phosphotransfer systems that are ubiquitous in natural signal transduction (1, 2, 3, 4, 5, 6, 7, 14, 15, 16, 17, 18, 19). All reactions are modeled as two-step reactions. Phosphorylation and dephosphorylation reactions proceed by first reversibly forming an intermediate complex, which then irreversibly decomposes into the enzyme and the product. Phosphotransfer reactions are modeled as reversible two-step reactions resulting in the transfer of the phosphate group via the formation of an intermediate complex. Based on these reactions, as well as production and decay of the various species, ODE models are created for the systems using reaction-rate equations. Reactions for each system analyzed and the corresponding reaction-rate equation models are shown in Supporting Material, Sections 1.3–1.10. The following general ODE model then describes any signaling system architecture in the interconnection topology of Fig. 1 A:where, the variable t represents time, U is the input signal (the concentration of the input species), is a vector of concentrations of the species of the signaling system, Y is the output signal (the concentration of the output species) and v is the state variable of the downstream system. In the cases that follow, v is the normalized concentration of the complex formed by the output species Y and its target binding sites p in the downstream system.

The internal dynamics of the upstream system are captured by the reaction-rate vector . This vector includes the production and decay terms for the input species. The internal dynamics of the signaling system are captured by the reaction-rate vector . This vector captures the reactions that occur between different species within the signaling system. The reaction-rate vector is the reaction flux resulting from the reactions between species of the upstream system and those of the signaling system. Thus, this vector affects the rate of change of both the input species and the species of the signaling system, with corresponding stoichiometry matrices A and B. The reaction-rate vector s represents the additional reaction flux due to the binding-unbinding of the output protein with the target sites in the downstream system. This vector therefore affects the rate of change of the downstream species as well as the signaling system, with corresponding stoichiometric matrices C and D. These additional reaction fluxes, and s, affect the temporal behavior of the input and the output, often slowing them down, as demonstrated previously (11).

The parameter R accounts for decay/degradation of complexes formed by the input species with species of the signaling system, thus leading to an additional channel for removal of the input species through their interaction with the signaling system. Similarly, scalar represents decay of complexes formed by the input species with species of the downstream system. This additional decay leads to an effective increase in decay of the input, thus affecting its steady-state. As species of the signaling system are sequestered by the downstream system, their free concentrations change. This is accounted for by the vectors and .

The retroactivity to the input indicated in Fig. 1 A therefore equals , which leads to both steady-state and temporal effects on the input response. The retroactivity to the output of Fig. 1 A equals , which leads to an effect on the output response. For ideal unidirectional signal transmission, the effects of and must be small. The ideal input of Fig. 1 B, , is the input when retroactivity to the input is zero, i.e., when . The isolated output of Fig. 1 C, , is the output when retroactivity to the output is zero, i.e., when .

The positive scalar captures the timescale separation between the reactions of the signaling system and the dynamics of the input. Since we consider relatively slow inputs, we have . The positive scalar captures the timescale separation between the binding-unbinding rates between the output Y and its target sites p in the downstream system and the dynamics of the input. Since binding-unbinding reactions also operate on a fast timescale, we have . We define and thus, . This allows us to apply techniques from singular perturbation to simplify and the set of equations in (1), to arrive at the results presented in the next section. Details of this analysis are shown in Supporting Material, Section 1.1.

In Results, we outline a procedure to determine whether a given signaling system satisfies definition 1. For this, we introduce the following definitions. We assume that there exist matrices M and P, and invertible matrices T and Q such that

This assumption is usually satisfied in signaling systems (32). Further, we haveandWe note that, for system 1 (Supporting Material, Section 1.1), terms and and functions and depend on the vector of total protein concentrations, .

Simulations and validity of results

For most systems, we have assumed that the Michaelis-Menten constants for the phosphorylation and dephosphorylation reactions are larger than protein concentrations. More specific assumptions are stated in Results. Our theoretical analysis for the various systems is valid for all reaction-rate parameters as long as these assumptions are satisfied. Thus, although the simulation results are performed for specific parameters, the conclusions are robust to changes in these parameters. Simulations of the full ODE systems are run on MATLAB, using the numerical ODE solvers ode23s and ode15s. All simulation parameters are picked from the biologically relevant ranges given in (35), and are listed in Table S1.

Results

The main result of this study is twofold. First, we provide a general procedure to determine whether any given signaling system enables unidirectional signal transmission. Second, using this procedure, we analyze the unidirectional signal transmission ability of both common and less frequent signaling architectures. In particular, we found that most signaling architectures transmit information via kinases. Therefore, we have analyzed several architectures where this is the case. However, both nature and a human designer have the option of designing a system that would transmit information via substrates. Since this is not frequently encountered in natural signaling architectures, we analyzed whether these designs show a disadvantage to unidirectional signaling, as indeed we find they do.

Procedure to determine unidirectional signal transmission

We outline a procedure to determine whether any given signaling system can enable unidirectional signaling in Fig. 3. First, the reaction-rate equations of the signaling system are written in form (1) (Supporting Material, Section 1.1), allowing us to note the terms , , , and R for step 2. The remaining terms for step 2 are computed using (2), (3), (4) under Generalized model. The terms in steps 3–5 are computed using the terms in step 2. The upper bound on is proportional to the terms found in step 3, and thus, as these are made small according to test (i), the first requirement of definition 1 is satisfied. The analysis giving rise to these terms is shown in theorem 1 in Supporting Material, Section 1.1. Similarly, the upper bound on is proportional to the terms in step 4, and thus, as these are made small according to test (ii), the second requirement of definition 1 is satisfied. This is derived in theorem 2 in Supporting Material, Section 1.1. Theorem 3 in Supporting Material, Section 1.1 shows that the input-output relationship for the signaling system can be computed by step 5. If this input-output relationship satisfies test (iii), the third requirement of definition 1 is satisfied. Once tests (i)–(iii) are satisfied, test (iv) checks whether all the requirements for definition 1 can be achieved simultaneously by tuning . If this is possible, the signaling system is said to be able to transmit a unidirectional signal.

Figure 3

Procedure to determine whether a given signaling system satisfies definition 1 for unidirectional signal transmission.

Note that throughout this work, is assumed to be the design parameter, since it is relatively easier to tune in both natural and synthetic circuits. However, the procedure outlined in Fig. 3 holds even if different design parameters are chosen.

As an example of the application of the procedure, we consider once again the single PD cycle (see Example). Steps 1–5 for this system are shown in Supporting Material, Section 1.3. We find that to satisfy test (i), we must have small . Further, to satisfy test (ii), we must have large and . Finally, computing  from step 5, we find that the input-output relationship has with when . These results are consistent with those described in Example, as well as with previous theoretical and experimental work (8, 11, 33). There exists a tradeoff between requirements 1 and 2 of definition 1, i.e., between imparting a small retroactivity to the input and attenuating retroactivity to the output. Thus, cannot be chosen such that all three requirements are simultaneously satisfied. Test (iv) fails, and the single PD cycle cannot achieve unidirectional signal transmission.

This way, the above procedure can be used to identify ways to tune the total protein concentration of a signaling system such that it satisfies definition 1. Using this procedure, we analyze a number of signaling architectures, including double phosphorylation systems, phosphotransfer systems, and multistage signaling architectures composed of these. For these architectures, we consider two types of input signals: a kinase input (highly represented in natural systems), where the input regulates the rate of phosphorylation, and a substrate input (less frequent in natural systems), where the input regulates the rate of production of the substrate.

Double phosphorylation cycle with input as kinase

Here, we consider a double phosphorylation cycle with a common kinase Z for both phosphorylation cycles as the input and the doubly phosphorylated substrate X∗∗ as the output. This architecture is found in the second and third stages of the MAPK cascade, where the kinase phosphorylates both the threonine and tyrosine sites in a distributive process (44). This configuration is shown in Fig. 4 A. Referring to Fig. 1 A, the input signal U is the concentration Z of the kinase and the output signal Y is the concentration of the doubly phosphorylated substrate X.

Figure 4

Tradeoff between small retroactivity to the input and attenuation of retroactivity to the output in a double phosphorylation cycle. (A) Double phosphorylation cycle, with input Z as the kinase: X is phosphorylated by Z to X∗, and further on to X∗∗. Both of these are dephosphorylated by the phosphatase M. X∗∗ is the output and acts on sites p in the downstream system, which is depicted as a gene expression system here. (B–E) Simulation results for the ODE model shown in Supporting Material, Eq. 29. Simulation parameters are given in Table S1. The ideal system is simulated for with . The isolated system for is simulated with . To see this figure in color, go online.

The input kinase is produced at a time-varying rate, . All species dilute with a rate constant δ, and the total promoter concentration in the downstream system is . The total substrate and phosphatase concentrations are and , respectively. The Michaelis-Menten constants for the two phosphorylation and the two dephosphorylation reactions are , , , and , respectively. The catalytic reaction rate constants of these reactions are , , , and , respectively. The system’s chemical reactions are shown in Supporting Material, Eq. 28. As explained before, the parameters that we tune to investigate retroactivity effects are the total protein concentrations of the phosphorylation cycle, that is, and . Specifically, using the procedure in Fig. 3, we tune and to verify if this system can transmit a unidirectional signal, according to definition 1. Steps 1–5 are detailed in Supporting Material, Section 1.4. We therefore find the following.

Retroactivity to the input: Evaluating the terms in step 3, we find that to satisfy test (i), we must have small and small . Thus, to have small retroactivity to the input, the parameter must be small. (Evaluation of terms in step 3 is shown in Supporting Material, Section 1.4).

Retroactivity to the output: Evaluating the terms in step 4, we find that to satisfy test (ii), we must have small and . Thus, to attenuate retroactivity to the output, we must have large and . (Evaluation of terms in step 4 is shown in Supporting Material, Section 1.4).

Input-output relationship: Computing shows that when , and . Under these assumptions, this system satisfies test (iii) by tuning the ratio to achieve a desired K with . (Evaluation of step 5 is shown in Supporting Material, Section 1.4).

This system shows opposing requirements to satisfy tests (i) and (ii), similar to the single phosphorylation cycle. Thus, although each of the requirements of tests (i)-(iii) are individually satisfied, the system does not satisfy test (iv), showing a tradeoff that prevents unidirectional signal transmission. Retroactivity to the input is large when substrate concentration (and ) increases, because the input Z must phosphorylate a large amount of substrate, thus leading to a large reaction flux to Z due to the phosphorylation reaction. However, if (and ) is made small, the system cannot attenuate the retroactivity to the input, since as the output X∗∗ is sequestered by the downstream system, there is not enough substrate available for the signaling system. Therefore, tests (i) and (ii) cannot be independently satisfied.

These mathematical predictions can be appreciated from the numerical simulations of Fig. 4, B–E, with a time-varying input, and from the simulations in Fig. S2, B–E, with a step input. This result is summarized in Fig. 9 B.

Figure 9

Table summarizing the results. For each inset table, a () for column implies that the system can (cannot) be designed to minimize retroactivity to the input by varying total protein concentrations, a () for column implies that the system can (cannot) be designed to attenuate retroactivity to the output by varying total protein concentrations, and column m describes the input-output relationship of the system (i.e., ) as described in definition 1, requirement 3). Thus, as seen above, systems (D), (F), and (G) fail to satisfy at least one of the three requirements for definition 1 for unidirectional signaling. Inset tables with two rows imply that one of the two rows can be achieved for a set of values for the design parameters: thus, the two rows for systems (A)–(C) show the tradeoff between the ability to minimize retroactivity to the input (first row) and the ability to attenuate retroactivity to the output (second row). Note that this tradeoff is overcome by the cascade (E). To see this figure in color, go online.

Regulated autophosphorylation followed by phosphotransfer

We now consider a signaling system composed of a phosphotransfer system, whose phosphate donor receives the phosphate group via autophosphorylation regulated by protein Z. An instance of this architecture is found in the bacterial chemotaxis network, where the autophosphorylation of protein CheA is regulated by a transmembrane receptor (e.g., Tar). CheA then transfers the phosphate group to protein CheY in a phosphotransfer reaction. CheY further undergoes dephosphorylation catalyzed by the phosphatase CheZ (45, 46, 47). A similar mechanism is also present in the ubiquitous two-component signaling networks, where the sensor protein autophosphorylates upon binding to a stimulus (e.g., a ligand) and then transfers the phosphate group to the receptor protein (15, 48). We model this regulated autophosphorylation as a phosphorylation reaction with kinase as input, since in both cases, first an intermediate complex is formed and the protein then undergoes phosphorylation. This architecture is shown in Fig. 5 A. In this case, the input signal U of Fig. 1 A is Z, which is the concentration of the kinase/stimulus Z that regulates the phosphorylation of the phosphate donor X1, which then transfers the phosphate group to protein X2. The output signal Y in Fig. 1 A is then , which is the concentration of the phosphorylated substrate . Protein is dephosphorylated by phosphatase M. Total concentrations of proteins X1, X2, and M are , and , respectively. The Michaelis-Menten constants for the phosphorylation of X1 by Z and dephosphorylation of by M are and , and the catalytic rate constants of these are and , respectively. The association rate constant of complex formation by and X1 is . These reactions are shown in Supporting Material, Eq. 48. The total concentration of promoter sites in the downstream system is . The input Z is produced at a time-varying rate, . As before, the parameters we change to analyze the system for unidirectional signal transmission are its total protein concentrations, , and . Using the procedure in Fig. 3, we analyze the system’s ability to transmit unidirectional signals as per definition 1 as , and are varied. This is done as follows. (Steps 1–5 for this system are shown in Supporting Material, Section 1.5).

Figure 5

Tradeoff between small retroactivity to the input and attenuation of retroactivity to the output in a phosphotransfer system. (A) System with phosphorylation followed by phosphotransfer, with input Z as the kinase: Z phosphorylates X1 to . The phosphate group is transferred from to X2 by a phosphotransfer reaction, forming , which is in turn dephosphorylated by the phosphatase M. is the output and acts on sites p in the downstream system, which is depicted as a gene expression system here. (B–E) Simulation results for ODE (49) in Supporting Material, Section 1.5. Simulation parameters are given in Table S1. The ideal system is simulated for with . The isolated system is simulated for with . To see this figure in color, go online.

Retroactivity to the input: Evaluating the terms in step 3, we find that to satisfy test (i), we must have small . Thus, for small retroactivity to the input, we must have small . (Evaluation of the terms in step 3 is shown in Supporting Material, Section 1.5).

Retroactivity to the output: Evaluating the terms in step 4, we find that to satisfy test (ii), and must be small. Thus, for a small retroactivity to the output, we must have large and . (Evaluation of terms in step 4 is shown in Supporting Material, Section 1.5).

Input-output relationship: Evaluating as in step 5, we find that when and . Under these assumptions, this system satisfies test (iii), where a desired K can be achieved by tuning the ratio with . (Evaluation of step 5 is shown in Supporting Material, Section 1.5).

In light of findings 1 and 2, above, we note that tests (i) and (ii) cannot be simultaneously satisfied. Test (iv) fails, and the system shows a tradeoff in attenuating retroactivity to the input and output. Retroactivity to the input can be made small by making (and ) small, since kinase Z must phosphorylate less substrate. However, the system with low is unable to attenuate retroactivity to the output, which requires that be large. This is because, as the output is sequestered by the downstream system and undergoes decay as a complex, this acts as an additional channel of removal for the phosphate group from the system, which was received from . If (and ) is small, this removal of the phosphate group affects the amount of in the system to a larger extent than when is large. Thus, there exists a tradeoff between requirements 1 and 2 of definition 1, and the system does not allow unidirectional signal transmission.

This mathematical analysis is demonstrated in the simulation results shown in Fig. 5, B–E with a time-varying input, and in the simulation results in Fig. S3, B–E with a step input. The discussion is further summarized in Fig. 9 B.

Cascade of single phosphorylation cycles

We have now seen three systems that show a tradeoff between attenuating retroactivity to the output and imparting a small retroactivity to the input: the single phosphorylation cycle, the double phosphorylation cycle, and the phosphotransfer system, all with a kinase as input. In all three cases, the tradeoff is due to the fact that, as the total substrate concentration is increased to attenuate the effect of retroactivity on the output, the system applies a large retroactivity to the input. Thus, requirements 1 and 2 of definition 1 cannot be independently achieved. In (34), a cascade of phosphotransfer systems was found to apply a small retroactivity to the input and to attenuate retroactivity to the output. Further, cascades of single and double PD cycles are ubiquitous in cellular signaling, such as in the MAPK cascade (14, 49). The two-component signaling system (Regulated autophosphorylation followed by phosphotransfer) is also often the first stage of a cascade of signaling reactions (15, 48). Motivated by this, here we consider a cascade of PD cycles to determine how a cascaded architecture can overcome this tradeoff. We have found that single and double PD cycles, and the phosphotransfer system, show similar properties with respect to unidirectional signal transmission. Thus, our findings are applicable to all systems composed of cascades of single-stage systems, such as the single PD cycle, the double PD cycle, and the phosphotransfer system analyzed in Regulated autophosphorylation followed by phosphotransfer (simulation results for cascades of different systems are shown in Figs. S15 and S16).

We consider a cascade of two single phosphorylation cycles, shown in Fig. 6 A. The input signal is Z, the concentration of kinase Z. Z phosphorylates substrate X1 to , which acts as a kinase for substrate X2, phosphorylating it to . and are dephosphorylated by phosphatases M1 and M2, respectively. The output signal is , the concentration of .

Figure 6

Tradeoff between small retroactivity to the input and attenuation of retroactivity to the output is overcome by a cascade of single phosphorylation cycles. (A) Cascade of two phosphorylation cycles, with kinase Z as the input: Z phosphorylates X1 to , acts as the kinase for X2, phosphorylating it to , which is the output, acting on sites p in the downstream system, which is depicted as a gene expression system here. and are dephosphorylated by phosphatases M1 and M2, respectively. (B and C) Simulation results for ODEs in Supporting Material, Eqs. 72–79, with N = 2. Simulation parameters are given in Table S1. The ideal system is simulated for with . The isolated system is simulated for with . To see this figure in color, go online.

The input Z is produced at a time-varying rate, , and all species dilute with rate constant δ. The substrates of the cycles are produced at constant rates and , respectively, and the phosphatases are produced at constant rates and . We then define , , , and . The concentration of promoter sites in the downstream system is . The Michaelis-Menten constants for the phosphorylation and dephosphorylation reactions are , , , and , respectively, and catalytic rate constants are , , , and . The chemical reactions for this system are shown in Supporting Material, Eq. 58. As before, the parameters we vary to analyze this system’s ability to transmit unidirectional signals are , , and . Using the procedure in Fig. 3, we seek to tune these to satisfy the requirements of definition 1. We find the following. (Steps 1–5 are detailed in Supporting Material, Section 1.6.)

Retroactivity to the input: Evaluating the terms in step 3, we find that to satisfy test (i), must be small. Thus, to have a small retroactivity to the input, must be small. (Evaluation of terms in step 3 is shown in Supporting Material, Section 1.6.)

Retroactivity to the output: As before, we evaluate the terms in step 4, and find that to satisfy test (ii), we must have small and . Thus, to attenuate retroactivity to the output, and must be large. (Evaluation of terms in step 4 is shown in Supporting Material, Section 1.6.)

Input-output relationship: Evaluating as in step 5, we find that the input-output relationship is when and . (Details are shown in Supporting Material, Section 1.6.) The ratio can thus be tuned such that the system satisfies test (iii) with . However, if the different stages of the cycle share a common phosphatase, additional cycles may be required to maintain a linear input-output response (50). Details of this analysis are shown in step 5 of Supporting Material, Section 1.7.

Finally, we see that test (iv) is satisfied for this system, since tests (i)–(iii) can be satisfied simultaneously. We thus note that the tradeoff between attenuating retroactivity to the output and imparting small retroactivity to the input, found in single-stage systems, is broken by having a cascade of two cycles. This is because the input kinase Z only directly interacts with the first cycle, and thus, when is made small, the upstream system faces a small reaction flux due to the phosphorylation reaction, making retroactivity to the input small. The downstream system sequesters the species , and when is made high, there is enough substrate X2 available for the signaling system to be nearly unaffected, thus attenuating retroactivity to the output. This is verified in Fig. 6, B and C. The tradeoff found in the single cycle in Fig. 2, B–E, is overcome by the cascade, where we have tuned and to satisfy requirement 3 of definition 1. When the total substrate concentration for a single cycle is low, the retroactivity to the input is small (Fig. 2 B), but the retroactivity to the output is not attenuated (Fig. 2 C). When the total substrate concentration of this cycle is increased, the retroactivity to the output is attenuated (Fig. 2 D), but the input, and therefore the output, is highly changed due to an increase in the retroactivity to the input (Fig. 2, D and E). When the same two cycles are cascaded, with the low substrate concentration cycle being the first and the high substrate concentration cycle being the second (and and tuned to maintain the same gain K as the single cycles), retroactivity to the input is small and retroactivity to the output is attenuated (Fig. 6, B and C). Thus, cascading two cycles overcomes the tradeoff found in a single cycle. The same conclusions can also be appreciated from the simulation results for a step-input response in Fig. S1.

These results are summarized in Fig. 9 E. Although the system demonstrated here is a cascade of single phosphorylation cycles, the same decoupling is true for cascaded systems composed of double phosphorylation cycles and phosphorylation cycles followed by phosphotransfer, which, as we saw in the previous subsections, show a similar kind of tradeoff. Cascades of such systems, with the first system with a low substrate concentration and the last system with a high substrate concentration thus both impart a small retroactivity to the input and attenuate retroactivity to the output and are therefore able to transmit unidirectional signals. This can be seen via simulation results in Supporting Material, Section 1.7.1, where a cascade of a phosphotransfer system and a single PD cycle is seen in Fig. S5 and a cascade of a single PD cycle and a double PD cycle is seen in Fig. S6.

Phosphotransfer with the phosphate donor undergoing autophosphorylation as input

Here, we consider a signaling system composed of a protein X1 that undergoes autophosphorylation and then transfers the phosphate group to a substrate X2, shown in Fig. 7 A. In Regulated autophosphorylation followed by phosphotransfer, we considered a system with regulated autophosphorylation, where the input is a ligand/kinase. In this section, motivated by proteins that undergo autophosphorylation and then transfer the phosphate group, we consider a system where the input is the protein undergoing autophosphorylation (substrate input). Based on our literature review, we have not found instances of such systems in nature, and in this section, we investigate whether they might pose a disadvantage to unidirectional signaling. The input signal U of Fig. 1 A is , the concentration of protein X1 that undergoes autophosphorylation, and the output signal Y of Fig. 1 A is , the concentration of phosphorylated protein . The total protein concentrations of substrate X2 and phosphatase M are and , respectively. The total concentration of promoters in the downstream system is . Autophosphorylation of a protein typically follows a conformational change that either allows the protein to dimerize and phosphorylate itself or stimulates the phosphorylation of the monomer (51). Here, we model the latter mechanism for autophosphorylation as a single step with rate constant . The Michaelis-Menten constant for the dephosphorylation of by M is , and the association, dissociation, and catalytic rate constants for this reaction are , , and . The association and dissociation rate constants for the complex formed by and X2 are and , the dissociation rate constant of this complex into X1 and is , and the corresponding reverse association rate constant is . The input protein X1 is produced at a time-varying rate, . Details of the chemical reactions of this system are shown in Supporting Material, Eq. 101. We use the procedure in Fig. 3 to analyze this system as per definition 1 by varying the total protein concentrations and . This is done as follows. (Steps 1–5 are detailed in Supporting Material, Section 1.8.)

Figure 7

Attenuation of retroactivity to the output by a phosphotransfer system. (A) System with autophosphorylation followed by phosphotransfer, with input as protein X1 which autophosphorylates to . The phosphate group is transferred from to X2 by a phosphotransfer reaction, forming , which is in turn dephosphorylated by the phosphatase M. is the output and acts on sites p in the downstream system, which is depicted as a gene expression system here. (B–E) Simulation results for the ODE in Supporting Material, Eq. 102. Simulation parameters are given in Table S1. The ideal system is simulated for with . The isolated system is simulated for with . To see this figure in color, go online.

Retroactivity to input: Evaluating the terms in step 3, we find that to satisfy test (i), , , , and must be small, where . However, not all these terms can be made smaller by varying and alone. Thus, the retroactivity to the input, and whether or not test (i) is satisfied, depends on the reaction rate constants of the system, and it is not possible to tune it using total protein concentrations alone. (Evaluation of terms in step 3 is shown in Supporting Material, Section 1.8.)

Retroactivity to output: Evaluating the terms in step 4, we find that to satisfy test (ii), we must have a small and . Thus, to attenuate retroactivity to the output, and must be large. (Evaluation of terms in step 4 is shown in Supporting Material, Section 1.8.)

Input-output relationship: Evaluating as in step 5, we find that the input-output relationship is when and thus, this system can satisfy test (iii) by tuning to achieve a desired K with . (Details of step 5 are shown in Supporting Material, Section 1.8.)

Thus, we find that the retroactivity to the input cannot be made small by changing concentrations alone. The retroactivity to the output can be attenuated by having a large and , since these can compensate for the sequestration of by the downstream system. This signaling system can therefore satisfy tests (ii) and (iii) for unidirectional signal transmission. Although satisfying these requirements does not increase the retroactivity to the input, thus making it possible for it to satisfy test (i) as well, retroactivity to the input depends on the reaction-rate parameters, in particular, on the forward reaction-rate constant, , of autophosphorylation of X1. If this is large, the autophosphorylation reaction applies a large reaction flux to the upstream system, thus resulting in a large retroactivity to the input. If is small, this flux is small, and thus, retroactivity to the input is small. By the way we have defined cascades (as signals between stages transmitted through a kinase), any cascade containing this system would have it as a first stage. Therefore, even cascading this system with different architectures would not overcome the above limitation. These mathematical predictions can be appreciated in the simulation results shown in Fig. 7, B–E for a time-varying input, and in the simulation results shown in Fig. S7, B–E with a step input. The result is summarized in Fig. 9 C.

Single cycle with substrate input

Here, we consider a single phosphorylation cycle where the input signal U of Fig. 1 A is X, the concentration of the substrate X, and the output signal Y is , the concentration of the phosphorylated substrate. We consider this system motivated by the various transcription factors that undergo phosphorylation before activating or repressing their targets, such as the transcriptional activator nitrogen regulator I in the Escherichia coli nitrogen assimilation system (52). However, to the best of our knowledge, based on our literature review, signals are more commonly transmitted through kinases, as opposed to being transmitted by the substrates of phosphorylations. Since these are less represented than the others in natural systems, we ask whether they have any disadvantage for unidirectional transmission, and in fact they do. Note that the system analyzed in Phosphotransfer with the phosphate donor undergoing autophosphorylation as input is a system that takes as input a kinase that undergoes autophosphorylation before donating the phosphate group, and is not the same as the system considered here, where the input is a substrate of enzymatic phosphorylation.

The signaling system we consider, along with the upstream and downstream systems, is shown in Fig. 8 A. The input protein X is produced at a time-varying rate, . It is phosphorylated by kinase Z to the output protein X∗, which is in turn dephosphorylated by phosphatase M. X∗ then acts as a transcription factor for the promoter sites in the downstream system. All the species in the system decay with rate constant δ. The total concentration of promoters in the downstream system is . The total kinase and phosphatase concentrations are and , respectively, which are the parameters of the system we vary. The Michaelis-Menten constants of the phosphorylation and dephosphorylation reactions are and , and the catalytic rate constants are and . The chemical reactions of this system are shown in Supporting Material, Eq. 112. Using the procedure in Fig. 3, we analyze whether this system can transmit a unidirectional signal according to definition 1 by varying and . This is done as follows (steps 1–5 in Supporting Material, Section 1.9).The input-output relationship is thus linear, i.e., , and K can be tuned by varying and . The system thus satisfies test (iii). (Details of step 5 are shown in Supporting Material, Section 1.9.)

Figure 8

Inability to attenuate retroactivity to the output or impart small retroactivity to the input by a single phosphorylation cycle with substrate as input. (A) Single phosphorylation cycle, with input X as the substrate: X is phosphorylated by the kinase Z to X∗, which is dephosphorylated by the phosphatase M back to X. X∗ is the output and acts as a transcription factor for the promoter sites p in the downstream system. (B–E) Simulation results for the ODEs in Supporting Material, Eqs. 113 and 114. Simulation parameters are given in Table S1. The ideal system is simulated for with . The isolated system is simulated for with . To see this figure in color, go online.

Retroactivity to the input: Evaluating the terms in step 3, we find that they cannot be made small by changing and , and therefore, test (i) fails and retroactivity to the input cannot be made small. (Evaluation of terms in step 3 is shown in Supporting Material, Section 1.9).

Retroactivity to the output: Similarly, we evaluate the terms in step 4 and find that they cannot be made small by varying and . Thus, test (ii) fails and retroactivity to the output cannot be attenuated by tuning these parameters. (Evaluation of terms in step 4 is shown in Supporting Material, Section 1.9.)

Input-output relationship: Evaluating as in step 5, we find that the input-output relationship is linear with gain when , that is:

Thus, we find that a signaling system composed of a single phosphorylation cycle with substrate as input cannot transmit a unidirectional signal, since it can neither make retroactivity to the input small nor attenuate retroactivity to the output. This is because the same protein X is the input (when unmodified) and the output (when phosphorylated). Thus, when X undergoes phosphorylation, the concentration of input X is reduced by conversion to , thus applying a large retroactivity to the input. Now, when X∗ is sequestered by the downstream system, this results in a large flux to both X and X∗, and thus the retroactivity to the output is also large. In fact, the same is true for an architecture with the input undergoing double phosphorylation, as seen in Supporting Material, Section 1.10, where X∗∗ is the output. For this architecture, as X∗∗ is sequestered, this applies a large flux to X, X∗, and X∗∗. Cascading such systems would also not enhance their ability to transmit unidirectional signals: if the system were used as the first stage to a cascade, it would apply a large retroactivity to the input for the aforementioned reasons. The way we have defined cascades above, with noninitial stages receiving their input via a kinase, this system cannot be the second stage of a cascade, since it takes its input in the form of the substrate. These results are demonstrated in the simulation results shown in Fig. 8, B–E, for a sinusoidal input, and in Figs. S18, B–E, for a step input. Results for the double phosphorylation cycle with substrate input are seen from Figs. S19 and S20. These results are summarized in Fig. 9 F.

Discussion

Retroactivity effects have been shown to be useful in certain contexts, such as transcription-factor decoy sites that convert graded dose responses to sharper, more switch-like responses (53). However, retroactivity is one of the chief hurdles to one-way transmission of information (8, 9, 10, 11, 12, 13). The goal of this work was to identify signaling architectures that can overcome retroactivity and thus allow the transmission of unidirectional signals. To achieve this, we have provided a procedure that can be used to analyze any signaling system composed of reactions such as PD and phosphotransfer. We have then considered different signaling architectures (Fig. 9) and have used this procedure to determine whether they have the ability to minimize retroactivity to the input and attenuate retroactivity to the output.

We have found that a main discriminating factor is whether the signaling architecture transmits information from kinases or from substrates. Specifically, phosphorylation cycles (single or double) and phosphotransfer systems that transmit information from an input kinase (Fig. 9, A–C) show a tradeoff between minimizing retroactivity to the input and attenuating retroactivity to the output, consistent with prior experimental studies (33, 54). Yet cascades of such systems (see, for example Fig. 9 E) can break this tradeoff. This is achieved when the first stage has low substrate concentration, thus imparting a small retroactivity to the input, and the last stage has high substrate concentration, thus attenuating retroactivity to the output. Interestingly, this low-high substrate concentration pattern appears in the MAPK signaling cascade in the mature Xenopus oocyte, where the first stage is a phosphorylation cycle with substrate concentration in the nanomolar range and the last two stages are double phosphorylation cycle with substrate concentration in the thousands of nanomoles (25). This low-high pattern indicates an ability to overcome retroactivity and transmit unidirectional signals, and although this structure may serve other purposes as well, it is possible that the substrate concentration pattern has evolved to more efficiently transmit unidirectional signals. By contrast, architectures that transmit information from a substrate (Fig. 9, D–G) do not perform as well even when cascaded. Consistent with this finding, whereas architectures that transmit signals from an input kinase are highly represented in cellular signaling, such as in the MAPK cascade and two-component signaling (1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19), those receiving signals through substrates are not as frequent in natural systems. This was in fact the reason we chose to analyze systems with substrate as input. We wished to determine whether they show a disadvantage to unidirectional signaling, potentially explaining why they are not frequently seen. It has also been reported that kinase-to-kinase relationships are highly conserved evolutionarily (55), implying that upon evolution, signaling mechanisms where kinases phosphorylate other kinases are conserved. These facts support the notion that cellular signaling has evolved to favor one-way transmission.

For graph-based methods for analyzing cellular networks (56), such as discovering functional modules based on motif search or clustering, signaling pathway architectures that transmit unidirectional signals can then be treated as directed edges. On the contrary, analysis of signaling systems (such as those with a substrate as input) that do not demonstrate the ability to transmit unidirectional signals must take into account the effects of retroactivity. These effects could result in cross talk between different targets of the signaling system, since a change in one target would affect the others by changing the signal being transmitted through the pathway (13). Our work provides a way to identify signaling architectures that overcome such effects and that can be treated as modules whose input/output behavior is largely independent from the context. Our findings further uncover a library of systems that transmit unidirectional signals, which could be used in synthetic biology to connect genetic components, enabling modular circuit design.

Author Contributions

R.S. performed the research, developed the mathematical analysis, and wrote the article. D.D.V. designed the research, assisted with the mathematical analysis, and edited the article.