Abstract
Background
It has been shown in experimental and theoretical work that covalently modified signaling cascades naturally exhibit bidirectional signal propagation via a phenomenon known as retroactivity. An important consequence of retroactivity, which arises due to enzyme sequestration in covalently modified signaling cascades, is that a downstream perturbation can produce a response in a component upstream of the perturbation without the need for explicit feedback connections. Retroactivity may, therefore, play an important role in the cellular response to a targeted therapy. Kinase inhibitors are a class of targeted therapies designed to interfere with a specific kinase molecule in a dysregulated signaling pathway. While extremely promising as anticancer agents, kinase inhibitors may produce undesirable offtarget effects by nonspecific interactions or pathway crosstalk. We hypothesize that targeted therapies such as kinase inhibitors can produce offtarget effects as a consequence of retroactivity alone.
Results
We used a computational model and a series of simple signaling motifs to test the hypothesis. Our results indicate that within physiologically and therapeutically relevant ranges for all parameters, a targeted inhibitor can naturally induce an offtarget effect via retroactivity. The kinetics governing covalent modification cycles in a signaling network were more important for propagating an upstream offtarget effect in our models than the kinetics governing the targeted therapy itself. Our results also reveal the surprising and crucial result that kinase inhibitors have the capacity to turn "on" an otherwise "off" parallel cascade when two cascades share an upstream activator.
Conclusions
A proper and detailed characterization of a pathway's structure is important for identifying the optimal protein to target as well as what concentration of the targeted therapy is required to modulate the pathway in a safe and effective manner. We believe our results support the position that such characterizations should consider retroactivity as a robust potential source of offtarget effects induced by kinase inhibitors and other targeted therapies.
Background
Cells propagate information through protein signaling pathways that are part of complex signal transduction networks [1]. The simplest view of cellular signaling entails a cascade of molecular events initiated by the recognition of a stimulus and culminating in the chemical alteration of an effector molecule. In the case of covalent modification by the addition or removal of a phosphate group, a reaction commonly found in signaling cascades, each phosphorylated protein serves as the kinase that activates the next cycle's unphosphorylated protein.
Targeted therapies are used to modulate disease progression by inhibiting a specific protein within a dysregulated signaling pathway [2]. Kinase inhibitors are a class of targeted therapies designed to interfere with a specific kinase molecule. While extremely promising as anticancer agents, kinase inhibitors can produce offtarget effects by inducing changes in molecules other than the one specifically targeted. Such offtarget effects are generally attributed to nonspecific binding or to crosstalk [3].
Recent theoretical and experimental studies have demonstrated that covalently modified cascades also exhibit bidirectional signal propagation via a phenomenon termed retroactivity [49]. This phenomenon arises because cycles in a cascade are coupled, not only to the next cycle, but also to the previous cycle (Figure 1). The cycles can be thought of as modules where each module's substrate sequesters a key component of the previous module, limiting the component's ability to participate in the previous module and inducing a natural change in the preceding module. This change may then propagate upstream through one or more preceding modules.
Figure 1. Retroactivity arises due to enzyme sequestration in covalently modified cascades. A simple signaling cascade is depicted where each sequential cycle represents the activation (denoted by *) and inactivation of protein Y_{i}. Y_{1}* serves as the activating enzyme of Y_{2 }and Y_{2}* serves as the activating enzyme of Y_{3}. The cycles can be thought of as modules where each module's substrate sequesters a key component of the previous module, limiting the component's ability to participate in the previous module. This sequestration induces a natural change in the preceding module which may propagate upstream through one or more preceding modules. In this example, a perturbation in the deactivation reaction of cycle 3 induces an effect in cycle 2. If the perturbation takes the form of an increase in the concentration or activity of the enzyme catalyzing the conversion of Y_{3}* to Y_{3}, more Y_{3 }will be available to react with and sequester Y_{2}*, resulting in less Y_{2 }substrate availability for the reaction with Y_{1}*. Thus, a reverse response can propagate upstream to a preceding cycle or cycles. In the schematic, black arrows represent the cell surface to nucleus direction of cellular signaling and red arrows represent the direction of retroactive signaling.
While retroactivity is naturally present in covalently modified cascades, signaling pathways likely have evolved to propagate signals in an optimized downstream manner. An important consequence of retroactivity, however, is that a downstream perturbation in a signaling cascade can produce an upstream effect without the need for explicit negative feedback connections [4]. Retroactivity may, therefore, play important roles in the dysregulated signaling networks of diseased cells as well as the cellular response to targeted therapies applied to dysregulated signaling networks.
Ventura, Sepulchre, and Merajver [4] demonstrated that increasing the concentration of the inactivating enzyme (e.g., a phosphatase) in the terminal cycle of a cascade can decrease the concentration of the activated protein in the previous cycle [4]. This finding led us to hypothesize that a targeted inhibitor can produce upstream offtarget effects via retroactivity that can propagate elsewhere in the signaling network.
Offtarget effects associated with targeted therapies are often attributed to crosstalk, which refers to interpathway molecular interactions arising because of explicit regulatory feedback connections between two pathways or because two pathways share one or more molecular components. It is well accepted that two pathways sharing one or more components can exhibit crosstalk with respect to a stimulation or perturbation above the shared component(s). If an upstream perturbation occurs in one of the pathways, the perturbation may affect the other pathway via the shared downstream component(s). Such a scenario could lead to specificity problems [10]. Here we propose that perturbations (e.g., from an inhibitor) that occur downstream of a shared component can also induce crosstalk effects without any explicit feedback connections via the following mechanism: the information travels upstream from the site of the perturbation through retroactivity, reaches the common component and then is delivered to the parallel pathway.
To test our hypothesis, we created a computational model that tested the application of a kinase inhibitor in a series of simple signaling networks. The objective of the model was to probe the effect of a targeted inhibitor on retroactive signaling and to test whether retroactivity is likely to contribute to measurable offtarget effects under physiological conditions. Specifically, the model simulated the targeted inhibition of a specific kinase in a series of multicycle networks. In all networks, at least two cascades were activated by the same upstream cycle with no explicit feedback connections between them. Our results indicate that within physiologically and therapeutically relevant ranges for all parameters, a targeted inhibitor can naturally induce a steady state offtarget effect via retroactivity. Our results also reveal the surprising and crucial result that a downstream kinase inhibitor has the capacity to turn "on" an otherwise "off" parallel cascade when two cascades share an upstream activator.
Methods
Model development
We designed simple signaling networks to test whether a measurable offtarget effect in one cascade can occur when a protein in another cascade is selectively inhibited. In each network studied, cycle i contained the active (phosphorylated) and inactive (unphosphorylated) forms of protein Y_{i}, where the active form was denoted by . For simplicity, we refer to activating and inactivating enzymes in a network as kinases and phosphatases, respectively.
Protein Y_{1}* served as the activating kinase for all cascades. Cycle 2 and cycle n were always in distinct cascades (Figure 2). To determine if an offtarget effect occurred due to perturbation by the inhibitor, the steady state concentration of the protein in cycle 2 was monitored as the concentration of the drug that specifically targeted Y_{n}* was increased. A competitive inhibitor was used that directly bound to Y_{n}*, limiting its ability to participate in the phosphatase reaction of cycle n, but did not change the rate of the phosphatase reaction in cycle n.
Figure 2. Topology of signaling networks studied. Two general types of network motifs consisting of covalently modified cycles were studied: (A) the vertical case where the nth cycle in the right hand cascade is inhibited and (B) the lateral case where the nth singlecycle cascade is inhibited. (C) The n = 3 network consisted of exactly 3 cycles and was the simplest form of both the vertical and lateral case. (D) An extended n = 3 network was also studied where a fourth cycle activated by Y_{2}* was added to the leftmost cascade. In all networks, Y_{1}* served as the upstream activator and cycle 2 and cycle 3 were always in distinct cascades. No additional regulatory connections were included in any network. Offtarget effects in cycle 2 were monitored by measuring the steady state concentrations of Y_{2 }and Y_{2}* as the concentration of an inhibitory drug that specifically targeted Y_{n}* was increased.
Two general network types were considered: a vertical and a lateral case (Figure 2). The vertical case consisted of two cascades where the inhibited cascade length varied (Figure 2A). This motif is similar to the upstream activation of JUN and P53 by JNK1/2 in the mitogenactivated protein kinase (MAPK) pathway [11]. The lateral case was equivalent to a fanout network topology and consisted of n  1 single cycle cascades that were all activated by Y_{1}* (Figure 2B). This motif is similar to the activation of multiple cascades by p38 [11]. The n = 3 network consisted of exactly 3 cycles and represented the simplest form of both network types studied (Figure 2C).
The general reaction scheme used for the vertical, lateral, and n = 3 networks was:
Where
is the activated protein in the i^{th }cycle
Y_{i }is the inactivated protein in the i^{th }cycle
is the kinase enzyme in the i^{th }cycle
is the phosphatase enzyme in the i^{th }cycle
D is the inhibitory drug
C_{i }is the Y_{i }and complex in the i^{th }cycle
is the and complex in the i^{th }cycle
C_{D }is the and D complex in the n^{th }cycle
Parameter definitions
In order to reduce the complexity of each network studied, parameters were nondimensionalized into 4 parameter types as described in Appendix A. The allowed value of each parameter type was restricted to the default ranges listed in Table 1. A summary of the parameter types is provided below.
Table 1. The parameter space of each network consisted of a set of nondimensional parameters, each with a minimum and maximum allowed value.
Subscripts containing k or p indicate parameters associated with a kinase or phosphatase reaction, respectively, and subscripts containing T indicate the total concentration of a species. V_{max }and K_{m }are the standard MichaelisMenten constants representing, respectively, the maximum velocity of a reaction (at a given enzyme concentration) and the substrate concentration necessary to achieve [12].
(1) total enzyme to substrate ratio of the kinase and phosphatase reaction, respectively, in cycle i:
(2) normalized K_{m }of the kinase and phosphataste reaction, respectively, in cycle i:
(3) V_{max }ratio of the kinase and phosphatase reactions in cycle i:
(4) normalized disassociation constant of the inhibitor binding to Y_{n}*:
E_{i }and E'_{i }values less than 1 indicate that the enzyme is less abundant than the substrate. K_{i }and K'_{i }values less than 1 indicate that the total available substrate exceeds the concentration needed to reach K_{m }and, consequently, the enzymatic reaction operates close to or in the zero order regime [13]. In contrast, K_{i }and K'_{i }values greater than 1 indicate that an insufficient amount of substrate exists to reach K_{m }and the enzymatic reaction operates in the linear regime [13]. P_{i }values greater than 1 indicate that the V_{max }of the kinase reaction exceeds the V_{max }of the phosphatase reaction and, consequently, the cycle tends toward the activation reaction. Likewise, P_{i }values less than 1 indicate that the cycle tends toward the deactivation reaction.
Determination of offtarget effects
The concentrations of species Y_{i}, , and the inhibitory drug D were normalized as follows:
To determine if a detectable offtarget effect occurred for a specific set of parameters, changes in the steady values of y_{2 }and were monitored as the model parameters were held fixed but I was varied from 10^{4 }to 10^{4}. If a change in the steady state value of y_{2 }or occurred that was greater than or equal to a detection threshold of 0.10 (i.e., 10% of the total protein in cycle 2), an offtarget effect in cycle 2 was reported. For numeric reasons, the range used for I was intentionally larger than needed. For a given parameter set, it was numerically more efficient to simulate with a small (10^{4}) and a large (10^{4}) value for I and then check for a change in the steady state values of y_{2 }and than it was to simulate with many values of I. In fact, the majority of offtarget effects in our model were observed as I was varied from 0.1 to 10.
When we tested the n = 3 network, we obtained the same results when we used either I = 0.0000 or I = 0.0001 (10^{4}) as the minimum drug concentration. For this reason (and because it would be experimentally challenging to distinguish 0.0000 from 0.0001 in vivo), we effectively considered I = 10^{4 }to represent the absence of the drug in the system.
Numerical simulations
For each network tested, a system of ordinary differential equations (ODEs) was used to model the rate of change of the reactants. Because we were only interested in changes in steady state values as a function of I, we first solved the system by setting the ODEs equal to zero and generating a system of steady state equations. As described in Appendix A, the model in this form was the basis for the nondimensionalization of model parameters.
For numerical reasons, it was more efficient to solve the ODEs over a very long time period rather than solving the system of steady state equations directly. After randomly selecting a set of nondimensional parameters, the selected values were mapped to corresponding dimensional parameter values (Additional File 1) and the system of ODEs was solved using the Matlab R2009b ode15s stiff solver from 0 to a maximum of 100,000 arbitrary time units. The majority (~90%) of randomly selected parameter sets obtained steady state within 5,000 arbitrary time units. The units are arbitrary because we began with dimensionless parameters lacking an explicit timescale. Finally, to confirm the numerical steady state solution, the original dimensionless parameters and the final y_{i }and y_{i}* variable values were substituted into the analytical steady state equations listed in Appendix A. Matlab source code was compiled as a C program and run on Intel Nehalem/i7 Core processors.
Additional file 1. Mapping dimensionless parameters to dimensional parameters. This file describes how randomly sampled dimensionless parameter values were mapped to dimensional parameter values prior to numeric simulation.
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Random parameter space exploration
Random parameter selection was performed via latin hypercube sampling (LHS) to provide an efficient and even sampling distribution across the range of allowed values in the parameter space [1416]. Each parameter space exploration consisted of 5000 randomly selected parameter sets. The number of parameter sets sampled was determined by calculating the percent of offtarget effects in q randomly sampled parameter sets for the n = 3 network (Figure 2C). The variation in the percent of offtarget effects stabilized when q was greater than or equal to 5000 (Additional File 2 Figure S1). The percentage of 5000 randomly selected parameter sets that produced an offtarget effect provided a probability that offtarget effects would occur in a given network's parameter space.
Additional file 2. Parameter space sampling to estimate the probability of offtarget effects. This file describes how the parameter space of a network was sampled to provide an estimate of the probability of offtarget effects due to retroactivity alone.
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Numeric perturbation analysis
A modified perturbation method was used to probe which model parameters were most important for producing an offtarget effect as a result of the inhibition of Y_{n}*. Traditional biochemical sensitivity analysis [17] with the dimensionless parameters was not possible because these parameters applied to the steady state equations and not the time dependent differential equations (Appendix A). Instead, we developed a numerical perturbation based method that allowed us to evaluate the parametric sensitivity of offtarget effects in a network's parameter space. In the method, the value of a single parameter was randomly selected from a restricted range of values while all other parameter values were randomly selected from the full range permitted by the baseline parameter space. If offtarget effects are sensitive to a given parameter, we expect that when values for the parameter under test are randomly selected from a reduced range of values, the percentage of offtarget effects produced will differ from the percentage produced when values for the parameter are instead selected from a fixed baseline range. In both cases, all other parameter values are selected from a fixed baseline range so that the only change in the system is a perturbation in the allowed range of the parameter under test.
The reduced ranges used to perturb each parameter were arrived at by partitioning the default range established for each parameter type in Table 1. The default ranges were divided into smaller perturbation subranges such that the minimum and maximum of a subrange was an order of magnitude larger than the minimum and maximum of the previous subrange. Because the E_{i}, E'_{i}, K_{i}, K'_{i}, and K_{B }parameters had a default initial range of 0.01  100.0 (Table 1), the subranges used to perturb these parameters were: (i) 0.01  0.10, (ii) 0.10  1.0, (iii) 1.0  10.0, and (iv) 10.0  100.0. Because the P_{i }parameters had a default range of 0.10  10.0, the subranges used to perturb these parameters were: (i) 0.10  1.0 and (ii) 1.0  10.0.
A complete numeric perturbation analysis of a parameter space consisted of determining the percentage of offtarget effects in 5000 randomly selected parameter sets for each parameter's subranges. In the n = 3 network (Figure 2C) there were a total of 16 parameters (5 parameters per cycle and K_{B}). Three of the parameters (P_{1}, P_{2}, and P_{3}) had 2 perturbation subranges each and the remaining 13 parameters had 4 perturbation subranges each. In this example, the analysis consisted of a total of 59 sets of 5000 simulations (58 sets for each parameter subrange and 1 set to establish the baseline percentage of offtarget effects in the parameter space prior to perturbation).
Results
The question we wanted to answer with our models was whether a targeted inhibitor is likely to induce a measurable offtarget effect due to retroactivity in a nontargeted cascade under physiological conditions. In each network, cycle n, was perturbed by an inhibitor. An offtarget effect occurred in the model if, after increasing I (the normalized inhibitor concentration) from 10^{4 }to 10^{4}, a change in the steady state concentration of Y_{2 }and/or Y_{2}* occurred that was at least 0.10 of the total Y_{2 }protein pool. For example, a change of 0.25 in Y_{2 }and 0.08 in Y_{2}* would indicate that the steady state values of Y_{2 }and Y_{2}* changed by 25% and 8% of the total Y_{2 }protein pool, respectively, and that a detectable offtarget effect occurred in Y_{2}.
Specific parameter ranges promote offtarget effects in cycle 2
First, we investigated the n = 3 network (Figure 2C) where Y_{3}* is targeted by the inhibitor. When the full parameter space (defined in Table 1 and depicted in Figure 3H) was used, 1.6% of the 5000 randomly selected parameters sets produced an offtarget effect in cycle 2. This value was essentially unchanged (1.5%) when we randomly selected 50,000 parameter sets for comparison (Additional File 2). To identify the model parameters that were most important for producing a cycle 2 offtarget effect, a numeric perturbation analysis was performed (Figure 3AF). The results of the analysis suggest that the parameters controlling the activity of cycle 3 play a large role in inducing an offtarget effect in cycle 2. Not surprisingly, K_{3 }(the normalized K_{m }of the kinase reaction in cycle 3) had the greatest effect on offtarget effects in this network (Figure 3D). K_{3 }determines how much sequestration of Y_{1}* by Y_{3 }occurs and this is the key mechanism of retroactivity. When K_{3 }was restricted to values greater than 1, the offtarget effects in the network were essentially eliminated. In contrast, when K_{3 }was restricted to values between 0.01 and 0.10, the percentage of offtarget effects increased to 4.6%. Similarly, K'_{3 }(the normalized K_{m }of the phosphatase reaction in cycle 3) also affected the percentage of offtarget effects but to a lesser degree than K_{3 }(Figure 3E).
Figure 3. A numeric perturbation analysis revealed parameter value ranges that promote offtarget effects in the n = 3 network. A perturbation analysis of the n = 3 network (G) was performed where a single parameter's value was randomly selected from a small range of values, while all other parameters were selected from the larger ranges defined in Table 1. The baseline in each plot reflects the percent of offtarget effects in cycle 2 in 5000 sampled parameter sets when all parameter values were randomly selected from the ranges defined in Table 1 and depicted in (H). All other bars reflect the results of systematically perturbing each parameter (one at a time) using the given subranges (AF). Based on this perturbation analysis, a restricted parameter space was generated (I) from which ~74% of the sampled parameter sets produced offtarget effects in cycle 2. In contrast, only ~1.6% of sampled parameter sets from the full parameter space (H) produced offtarget effects in cycle 2.
E_{3 }and E'_{3 }(the total kinase to substrate and the total phosphatase to substrate ratio, respectively, in cycle 3) also appeared to exert a large degree of control over offtarget effects (Figure 3AB). These results indicate that offtarget effects were more likely when the kinase and phosphatase enzymes of cycle 3 were saturated. P_{3}, the ratio of the V_{max }of the kinase and phosphatase reactions in cycle 3, also affected the percentage of offtarget effects (Figure 3C). When P_{3 }was less than 1, cycle 3 tended toward the deactivation reaction and the percentage of offtarget effects increased to 2.56% from 1.6%. Similarly, when P_{3 }was greater than 1, cycle 3 tended toward the activation reaction and the percentage of offtarget effects was significantly reduced relative to the baseline (0.32%).
The only parameter associated with cycle 2 that affected the percentage of offtarget effects in this network was K_{2 }(the normalized K_{m }of the kinase reaction of cycle 2). K_{2 }values between 0.01 and 0.10 are expected to produce an efficient kinase reaction because . The results of the numerical perturbation analysis indicated that when K_{2 }was restricted to values in this subrange, a small percentage of offtarget effects was observed (Figure 3D). In contrast, when K_{2 }was restricted to values between 1.0 and 10.00, the percentage of offtarget effects increased relative to the baseline. These results suggest that an offtarget effect in cycle 2 is more likely to occur in the n = 3 network when the conversion of Y_{2 }to Y_{2}* operates in the linear regime because of substrate constraints. This result is somewhat counter intuitive given the fact that we are interested in measuring a response that propagates from cycle 3 to cycle 1 and then down to cycle 2. It is reasonable to expect that an efficient kinase reaction in cycle 2 would be important for recruiting Y_{1}* to activate Y_{2 }and generate an effect in cycle 2. If the cycle 2 kinase reaction is less efficient than the cycle 3 kinase reaction, however, more Y_{1}* will be available to convert Y_{3 }to Y_{3}*, ultimately contributing to the sequestration of more Y_{3}* into a complex with D. Such a sequestration may give rise to a detectable upstream effect as a result of the reduced substrate availability in cycle 3.
The cycle 1 parameters with the greatest impact on the percentage of offtarget effects were K_{1 }and K'_{1 }(the normalized K_{m }of the kinase and phosphatase reaction, respectively, in cycle 1) (Figure 3DE). Larger values of K_{1 }acted to suppress offtarget effects, while smaller values produced an increase in offtarget effects relative to the baseline (Figure 3D). The reverse was observed for K'_{1}, with higher values producing a higher percentage of offtarget effects than smaller values (Figure 3E). Together, the K_{1 }and K'_{1 }results suggest that offtarget effects are favored when the cycle 1 phosphatase reaction tends toward inefficiency and the cycle 1 kinase reaction tends towards efficiency. This result is not surprising given that the availability of Y_{1}* is essential for the propagation of a signal from cycle 3 to cycle 2.
The value of K_{B}, the normalized drug disassociation constant, had a very slight effect on the percentage of offtarget effects. In general, K_{B }values greater than 1 produced a slight decrease in the percentage of offtarget effects relative to the baseline (Figure 3F). This result suggests that weaker binding (and larger dissociation constants) promoted fewer offtarget effects, as would be expected given the decreased sequestration of Y_{3}* that would occur. The change in the percentage of offtarget effects induced by restricting K_{B }values was fairly small compared to the change induced when other model parameter values were restricted. This result suggests that the activity and efficiency of component cycles in the network may be more important for propagating an offtarget effect than the actual kinetics of a targeted therapy.
The results of the above analysis indicate that certain parameter value ranges are more likely to induce an offtarget effect in cycle 2 as the drug concentration is increased. When we restricted the n = 3 parameter space by reducing the ranges from which some key parameters were selected (Figure 3I), the percentage of offtarget effects in 5000 randomly sampled parameter sets increased from 1.6% to 73.9%.
A second numerical perturbation analysis was performed using this new restricted n = 3 parameter space as a baseline. In general, many of the trends observed in the analysis of the original n = 3 parameter space (depicted in Figure 3H) were observed in the analysis of the restricted n = 3 parameter space (Additional File 3 Figure S3). For example, low K_{3 }values remained important for producing offtarget effects in both parameter spaces. The effects of parameters associated with cycle 2, however, were different in the two parameter spaces. When the original parameter space was tested, K_{2 }was the only cycle 2 parameter found to substantially affect the percentage of offtarget effects (Figure 3D). In the restricted parameter space, however, some ranges of E_{2}, E'_{2}, and K'_{2 }produced offtarget effect percentages that differed substantially from the baseline. For example, E_{2 }values between 10 and 100 produced offtarget effects in 92.1% of sampled parameter sets, the largest percentage of offtarget effects observed in any of our analyses (Additional File 3 Figure S3A). Because E_{2 }is the total enzyme to substrate ratio of the kinase reaction (Y_{1T}/Y_{2T}), this result suggests when more total protein exists in cycle 1 compared to cycle 2, offtarget effects in cycle 2 are more likely in this network.
Additional file 3. Additional analysis of the n = 3 and extended n = 3 networks. This file provides additional results from the numeric perturbation analyses of the n = 3 and extended n = 3 networks.
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While some parameters associated with cycle 2 were able to effect the percentage of offtarget effects, the parameters associated with cycle 3 continued to have the greatest effect on offtarget effects in the restricted n = 3 parameter space. Only parameters in cycle 3 had the ability to effectively eliminate (or substantially reduce) the percentage of offtarget effects within specific reduced ranges. Values between 10 and 100 for E_{3}, E'_{3}, K_{3 }and K'_{3 }produced offtarget effect percentages of 0%, 3.24%, 0% and 3.20%, respectively. In addition, P_{3 }values greater than 1 produced offtarget effects in 18.24% of sampled parameter sets which, compared to the baseline of 73.9%, represents a large decrease in offtarget effects (Additional File 3 Figure S3).
Varying a single parameter can produce a large change in the size of the offtarget effect
The magnitude of offtarget effects produced by parameter sets randomly sampled from the original n = 3 parameter space (depicted in Figure 3H) generally fell between .10 and .30 of the total Y_{2 }protein pool (Figure 4A). In contrast, the magnitude of offtarget effects produced by parameter sets randomly sampled from the restricted n = 3 parameter space (depicted in Figure 3I) were more uniformly distributed across a range of values (Figure 4B). These results suggest that when conditions in a network are favorable for offtarget effects, the size of an offtarget effect is highly variable.
Figure 4. Distribution of the size of offtarget effects in the n = 3 network. Histograms of the size of offtarget effects in n = 3 network (Figure 3D) are plotted for two different parameter spaces. The yaxis on each plot represents the proportion of all parameter sets that produced offtarget effects in 5000 randomly selected parameter sets. The xaxis on each plot represents the size of an offtarget effect in cycle 2 as a proportion of Y_{2T }such that each value indicates a response that was at least as big as the given value but less than the next sequential value. For example, a value of .30 indicates that the magnitude of the response was greater than or equal to .30 but less than 0.40. (A) The majority of offtarget effects in the original n = 3 parameter space (depicted in Figure 3H) were less than 0.30. (B) In contrast, the distribution of the size of offtarget effects in the restricted n = 3 parameter space (depicted in Figure 3I) was more uniform.
We used stimulus response curves to examine how a change in a single parameter value may affect the size of an offtarget effect in Y_{2}* as a function of the normalized inhibitor concentration (Figure 5). A randomly selected parameter set and a parameter set derived from a Xenopus MAPK model [18] were used (refer to Additional File 4 for the derivation of the Xenopus parameter values). In each parameter set, either E_{2 }or K_{3 }was varied, while all other parameter values were fixed to the values listed in Table 2.
Figure 5. Varying a single parameter value can produce a large change in the offtarget response. Stimulus response curves were plotted for the n = 3 network using a randomly selected parameter set and a parameter set derived from a Xenopus model [18] (all parameters values are listed in Table 2). For each parameter set, E_{2 }and K_{3 }were increased or decreased by 1 order of magnitude and the resulting stimulus response curves were plotted. (AB) The random parameter set produced an offtarget effect in Y_{2 }of 0.40 (data not shown) and in Y_{2}* of 0.19. (A) Increasing E_{2 }from 32.56 to 325.61 substantially decreased the offtarget effect in Y_{2}*, while decreasing E_{2 }to 3.26 increased the offtarget effect in Y_{2}* to 0.27. (B) Increasing K_{3 }from 0.04 to 0.41 reduced the response in Y_{2}* below the detection threshold to 0.07, while decreasing K_{3 }to 0.004 increased the offtarget response to 0.26. (CD) A second parameter set was derived from the literature using MAPK parameters from a Xenopus model. This parameter set did not initially produce an offtarget effect because the response in both Y_{2}* and Y_{2 }was 0.08, which was below the detection threshold. (C) Increasing or decreasing E_{3 }to 0.025 or 0.00025, respectively, from 0.0025 had no effect on the response to the targeted inhibitor. (D) In contrast, increasing K_{3 }from 0.25 to 2.5 eliminated the original response completely, while decreasing K_{3 }from 0.25 to 0.025 produced a large offtarget response of 0.60. Original parameter values prior to variation are indicated by ^{‡ }on the plots (see also Table 2).
Additional file 4. Xenopus MAPK Model Parameters [18,27,28]. This file explains how the Xenopus model parameters listed in Table 2 were derived.
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Table 2. Parameter sets used in stimulus response curves.
The randomly selected parameter set produced a baseline offtarget response of 0.19 in Y_{2}* (Figure 5AB) and of 0.40 in Y_{2 }(data not shown). In this parameter set the original E_{2 }value was 32.56 and the original K_{3 }value was 0.04 (Table 2). Increasing E_{2 }to 326.61 substantially decreased the response in Y_{2}* and decreasing E_{2 }to 3.26 increased the response in Y_{2}* from 0.19 to 0.27 (Figure 5A). Similarly, increasing K_{3 }to 0.41 reduced the response in Y_{2}* to 0.07 (below the detection threshold) and decreasing K_{3 }to 0.004 increased the response in Y_{2}* to 0.26 (Figure 5B).
The parameter set derived from the MAPK Xenopus model [18] produced a baseline response of 0.08 (below the detection threshold) in both Y_{2}* (Figure 5CD) and Y_{2 }(data not shown). In this parameter set the original E_{2 }value was 0.0025 and the original K_{3 }value was 0.25 (Table 2). While changing E_{2 }did not alter the response (Figure 5C), increasing K_{3 }to 2.5 completely eliminated the response in Y_{2}* and decreasing K_{3 }to 0.025 substantially increased the response in Y_{2}* to 0.60 (Figure 5D). These results suggest that when using physiologically realistic parameter values, changing one kinetic parameter or species concentration by an order of magnitude has the capacity to dramatically alter whether a targeted inhibitor induces an offtarget effect.
A few of the enzyme to substrate ratios in the Xenopus parameter set (E_{2 }= 0.0025, E'_{2 }= 0.00025, and E_{3 }= 0.0025) were outside the limits of parameter ranges allowed in our random parameter space explorations (Table 1 and Figure 3H), suggesting that offtarget effects are possible for a larger range of parameter values than we specifically tested. While we may have been too conservative in the estimation of the ranges defined in Table 1, this finding supports the position that a targeted inhibitor can naturally induce an offtarget effect via retroactivity over a range of physiologically relevant conditions.
The percentage of offtarget effects decreased as the size of the vertical and lateral networks increased
We next investigated networks with more than 3 cycles by randomly exploring the parameter spaces of the vertical (Figure 2A) and lateral (Figure 2B) cases using n = 5 and n = 7 cycles. As before, we measured the steady state change in cycle 2 as the normalized concentration of the drug that targeted cycle n was increased. The restricted parameter space depicted in Figure 3I (from which 73.9% of sampled parameter sets produced offtarget effects in cycle 2) was used for this analysis. Networks were analyzed using homogenous parameter values in cycles 4, 5, 6 and 7 that equalled the corresponding parameter values randomly selected for cycle 3 (e.g., in the n = 5 case, E_{3 }= E_{4 }= E_{5}). This allowed us to keep the size of the parameter space fixed so that 5000 parameter sets remained a reasonable number to sample from each network's parameter space.
In the vertical case, the percentage of offtarget effects in the n = 5 and n = 7 networks were 27.92% and 13.50%, respectively (Table 3). The reduced probability of offtarget effects as the cascade lengthened suggests that applying a targeted inhibitor near the bottom of a long cascade can produce a detectable offtarget response but the signal may attenuate as it travels up the cascade. This conclusion is in agreement with a recent work that investigated retroactivity in long signaling cascades [9] and found that retroactive signals are likely to attenuate as they travel up long cascades.
Table 3. The percentage of offtarget effects decreased as the network size increased.
In the lateral case, the drop in the percentage of offtarget effects was more dramatic than in the vertical case, with the n = 5 and n = 7 networks producing 6% and 0% offtarget effects, respectively (Table 3). This result suggests applying a targeted inhibitor to a cycle that is activated by a signaling molecule involved in the simultaneous activation of many other cycles decreases the likelihood of offtarget effects. This conclusion is based on a limited exploration of the parameter space (due to the homogenous parameter selection used for cycles 3 and greater) but is in agreement with a model proposed by Kim et al. [8] that showed retroactivity (or what they referred to as subsratedependent control) is attenuated by the number of substrates available.
Offtarget effects from retroactivity can propagate down a nontargeted cascade
Our results suggest that, under appropriate conditions, it is possible for a downstream perturbation from a targeted inhibitor to transmit up a cascade resulting in a detectable offtarget effect near the top of another cascade. Because signal amplification is an important cellular sensory mechanism [19], we next investigated whether offtarget effects from targeted inhibitors are likely to amplify down a nontargeted cascade.
To test for downstream propagation of offtarget effects from cycle 2, we created an extended n = 3 network by adding a 4^{th }cycle activated by Y_{2}* (Figure 2D). If a change in the steady state concentration of Y_{4 }and/or Y_{4}* occurred that was at least 0.10 of the total Y_{4 }protein pool, then an offtarget effect was considered to have occurred in cycle 4. If an offtarget effect occurred in cycle 4 and the size of the response in cycle 4 exceeded the size of the response in cycle 2, then an offtarget effect with amplification was considered to have occurred in cycle 4.
When the default parameter ranges defined in Table 1 were used for all cycles in the extended n = 3 network, the percentage of offtargets in cycle 2 and cycle 4, respectively, was 1.78% and 0.03%. We next tested the extended n = 3 network using the restricted n = 3 parameter space (depicted in Figure 3I) for cycles 1  3 and the default parameter ranges from Table 1 for cycle 4. In this partially restricted parameter space (depicted in Additional File 3 Figure S4H), the percentage of offtarget effects in cycle 2 and cycle 4 were 75.3% and 35.5%, respectively, and amplification contributed to cycle 4 offtarget effects in 23.3% of the sampled parameter sets (representing more than half of the offtarget effects in the sampled parameter sets). The remaining offtarget effects in cycle 4 occurred in 12.2% of the sampled parameter sets and had a response size that was either attenuated relative to cycle 2 or equal to the cycle 2 response (Table 4).
Table 4. Offtarget effects can amplify downstream of cycle 2.
To identify the parameters that were most important for amplifying an offtarget effect from cycle 2 to cycle 4 in the extended n = 3 network, we performed a numeric perturbation analysis (as previously described) on the partially restricted parameter space depicted in Additional File 3 Figure S4H. From these results, we generated a new parameter space (Additional File 3 Figure S4I) which produced offtarget effects of 45.3% and 67.4% in cycle 2 and cycle 4, respectively. Amplification contributed to cycle 4 offtarget effects in 61.9% of the sampled parameter sets. The remaining offtarget effects in cycle 4 occurred in 5.5% of the sampled parameter sets and had a response size that was either attenuated relative to cycle 2 or equal to the cycle 2 response (Table 4).
Discussion
We developed a computational model to test whether targeted therapies, such as kinase inhibitors, can produce offtarget effects in upstream pathways as a consequence of retroactivity alone. Using a numeric perturbation method, we identified specific conditions (Figure 6) that favored the promotion of steady state offtarget effects via retroactivity when a targeted inhibitor was applied to cycle n in a series of simple signaling networks (Figure 2).
Figure 6. A summary of conditions that favor offtarget effects in the n = 3 and the extended n = 3 networks. The conditions that promoted offtarget effects in our model are summarized for two network types. Offtarget effects in (A) the n = 3 network and (B) the extended n = 3 network were favored when cycle 3 tended toward deactivation and cycle 1 tended toward activation. Offtarget effects were favored in cycle 4 of (B) the extended n = 3 network when cycle 2 tended toward deactivation and cycle 4 tended toward activation. Blue arrows in a cycle indicate which V_{max }is larger when offtarget effects are favored. (C) A summary of the specific conditions in each cycle found to favor offtarget effects in the n = 3 network, in the extended n = 3 network, or in both networks is also provided.
Our investigation considered only the effect of retroactivity and targeted inhibitors on the individual motifs we studied in the absence of genetic and/or other regulatory relationships. This allowed us to investigate whether such motifs have the capacity to produce offtarget effects without regulatory feedback connections. In addition, the present study only considered the steady state response to a targeted therapy. The primary reason we considered only steady state responses was because it provided us with an objective measure that could be used to compare the effect of a targeted inhibitor across many different parameter sets. It is important to note that the dynamics of a retroactive signaling process are likely to induce transient changes in the levels of key signaling molecules. These transient changes, which are not observable at steady state, may lead to important in vivo responses.
It is also well known that the dynamics of signal transduction networks can be modulated by important oscillatory behavior, for example, from the P53/MDM2 regulatory feedback loop [20,21]. Because we have not considered transient dynamics, our approach cannot be assumed to apply to all signaling networks. Nevertheless, we expect conditions that favor the induction of offtarget effects at steady state to also favor the induction of detectable transient changes associated with the steady state response. In fact, this is what we observed when we plotted the timecourse of the cycle 2 proteins with the parameter sets used in Figure 5 (data not shown).
This work has led to very interesting and somewhat surprising results. A major importance of this work is that it did not investigate offtarget effects related to a specific therapeutic intervention. There are, however, examples of targeted inhibitors of great clinical interest that are involved in signaling motifs similar to the network motifs we examined. The drug NSC 74859 [22], for example, is a selective inhibitor that targets STAT3. JAK is an upstream activator of both STAT3 and PI3K [23], thus when NSC 74859 inhibits STAT3, JAK could potentially facilitate the propagation offtarget effects due to retroactive signaling from STAT3 to PI3K. Moreover, the inhibitor GSK690693 [24] targets AKT and could potentially give rise to a retroactive signal that propagates upstream to a common activator of either the MAPK or STAT3 cascades, generating offtarget effects in these pathways.
The binding affinity of the inhibitor for its target did not play a substantial role in the promotion of offtarget effects in our model. Instead, the kinetics of the component cycles in the network were more important for increasing the likelihood of offtarget effects (Figure 3 and Additional File 3 Figure S3, S4). In general, offtarget effects were more likely to occur in the networks studied when the targeted cycle n favored the deactivation reaction because the V_{max }of the deactivation reaction was larger than the V_{max }of the activation reaction and/or both enzymatic reactions in cycle n operated in or near the zero order regime. Offtarget effects were also more likely when cycle 1 (the source of the shared activator in our models) favored the activation reaction and its kinase reaction operated in or near the zero order regime.
If cycle 2's cascade was extended to include cycle 4 (Figure 2D), which was activated by Y_{2}*, offtarget effects were more likely to propagate to cycle 4 when cycle 2 favored deactivation and cycle 4 favored activation. In cycle 2 this meant that the kinetics of the kinase reaction were generally inefficient (operating in or near the linear regime) and that the V_{max }of the deactivation reaction was generally larger than the V_{max }of the activation reaction. Thus, offtarget effects were promoted when cycle 2 was "off" and not consuming significant amounts of the shared upstream activator, Y_{1}*.
The results also indicate that offtarget effects were more likely when the total kinase to substrate and the total phosphatase to substrate ratios in the inhibited cycle (E_{n }and E'_{n}, respectively) were less than 1. In the n = 3 network, this meant that there was less total protein in cycle 1 than in cycle 3 because E_{3 }< 1 implies Y_{1T }<Y_{3T}. The reason for this is that the smaller the Y_{1T}/Y_{3T }ratio, the stronger the sequestration of Y_{1}* will be. The impact of this ratio increases if cycle 3 favors the deactivation reaction such that a large fraction of Y_{3T }is in the inactive Y_{3 }form, promoting the binding of Y_{3 }to Y_{1}*.
The immediate experimental implications of this result is that, in the absence of kinetic information, the likelihood of offtarget effects may potentially be estimated for a network configuration of this type (Figure 2A,CD) based on the ratio of the concentrations of components in the inhibited cycle and the preceding cycle (using, for example, proteomic or gene expression data). While this ratio would not be an absolute predictor, the presence of this condition would suggest an increased probability of offtarget effects.
In agreement with the work of other groups [8,9], we found the probability of offtarget effects attenuated when the targeted cycle was near the bottom of a long cascade or when there were many substrates competing for a common upstream activator (Table 3). Our results also suggest that within physiologically realistic parameter ranges, changing a single kinetic parameter or species concentration by 1 order of magnitude has the capacity to dramatically alter whether an offtarget effect occurs as a direct result of targeted inhibition. It is also worth noting that, even though we varied the normalized drug concentration over a very large range, in general, the normalized inhibitor concentration needed to change by only 2 orders of magnitude to induce an offtarget effect (see, for example, Figure 5).
Conclusions
Offtarget drug effects in vivo are typically attributed to crosstalk arising from a feedback connection in a signaling network or to nonspecific interactions with other proteins. In this work we have demonstrated that offtarget drug effects can also arise naturally from retroactivity in a covalently modified signaling network. This view of signaling challenges the widespread notion that information in signaling cascades only flows from the cell surface to the nucleus and, consequently, this work has far reaching implications for targeted cancer therapies.
A crucial finding of this work is that the kinetics governing the covalently modified cycles in a signaling network are likely to be far more important for propagating an offtarget effect due to retroactive signaling than the binding affinity of the drug for the targeted protein, which is a commonly optimized property in drug development. Another particularly paramount finding is that an offtarget effect due to retroactive signaling is more likely when the first cycle in a noninhibited cascade is "off" and essentially inactive. This suggests that, in the motifs we studied, a targeted therapy has the capacity to turn "on" an otherwise "off" tributary cascade.
To emphasize, it is entirely possible for a branch of a signaling network that is "off" to become activated or "on" due to the inhibition of another protein in the network based on retroactivity alone, suggesting an inherent opportunity for negative therapeutic effects. Our findings, therefore, have implications for somatic evolution in cancer and the onset of therapeutic resistance, which has been widely reported for many targeted cancer therapeutics [25], most notably for the targeted inhibition of BCRABL by imatinib [26]. Moreover, a single mutation could conceivably give rise to a spontaneous offtarget effect without the need for any direct regulatory connections between the targeted protein and the effected protein.
While our approach does not definitively establish that the predicted responses will occur in vivo, our results demonstrate that offtarget effects are indeed possible in the absence of direct regulatory relationships and suggest that additional (and more specific) experimental and theoretical investigations are warranted. A proper characterization of a pathway's structure is important for identifying the optimal protein to target as well as what concentration of the targeted therapy is required to modulate the pathway in a safe and effective manner. We believe our results strongly support the position that such characterizations should consider retroactivity as a potential source of offtarget effects induced by kinase inhibitors and other targeted therapies. This work has also provided an initial roadmap for how to assess the likelihood of offtarget effects in a signaling network.
Authors' contributions
ACV and SDM conceived the study. ACV and JAS created the general model. ACV and MLW designed the experiments and analyzed results. MLW, HJG, and ACV wrote codes for numerical simulations. MLW wrote the manuscript and SDM, ACV, and JAS edited the manuscript. All authors read and approved the final manuscript.
Appendix A  Nondimensionalization of the n = 3 network
In order to reduce the complexity of the networks studied, model parameters were nondimensionalized. The following explains the nondimensionalization of the n = 3 network. The dimensionless parameters of the n = 5 and n = 7 vertical and lateral networks' were obtained in a similar manner.
The ODEs and conservation laws governing the n = 3 network (Figure 2C) at steady state are:
 Dimensionless Parameters
 Dimensionless Variables
Algebraic rearrangement and substitution yield the following dimensionless steady state equations:
Acknowledgements
MLW was supported by the Rackham Merit Fellowship, NIH T32 CA140044, and the Breast Cancer Research Foundation. ACV is a member of the Carrera del Investigador Científico (CONICET) and was supported by the Department of Defense Breast Cancer Research Program, the Center for Computational Medicine and Bioinformatics (University of Michigan), and the Agencia Nacional de Promoción Científica y Tecnológica (Argentina). SDM was supported by the Burroughs Wellcome Fund, NIH CA77612, the Avon Foundation, and the Breast Cancer Research Foundation. ACV thanks Dr. Bob Ziff for helpful discussions and MLW thanks Dr. Santiago Schnell for helpful discussions and computing resources.
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