Center for Computational Medicine and Bioinformatics, University of Michigan Medical School, Ann Arbor, MI, USA

Laboratorio de Fisiología y Biología Molecular, Departamento de Fisiología, Biología Molecular y Celular, IFIBYNE-CONICET, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón 2, Buenos Aires, Argentina

Institut Non Linéaire de Nice, Université de Nice Sophia-Antipolis, UMR CNRS 6618, Valbonne, France

Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA

Department of Internal Medicine, Division of Hematology and Oncology and Comprehensive Cancer Center, University of Michigan Medical School, Ann Arbor, MI, USA

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 anti-cancer agents, kinase inhibitors may produce undesirable off-target effects by non-specific interactions or pathway cross-talk. We hypothesize that targeted therapies such as kinase inhibitors can produce off-target 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 off-target effect via retroactivity. The kinetics governing covalent modification cycles in a signaling network were more important for propagating an upstream off-target 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 off-target 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

Targeted therapies are used to modulate disease progression by inhibiting a specific protein within a dysregulated signaling pathway

Recent theoretical and experimental studies have demonstrated that covalently modified cascades also exhibit bidirectional signal propagation via a phenomenon termed retroactivity

Retroactivity arises due to enzyme sequestration in covalently modified cascades

**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 _{i}_{1}* serves as the activating enzyme of _{2 }and _{2}* serves as the activating enzyme of _{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 _{3}* to _{3}, more _{3 }will be available to react with and sequester _{2}*, resulting in less _{2 }substrate availability for the reaction with _{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

Ventura, Sepulchre, and Merajver

Off-target effects associated with targeted therapies are often attributed to crosstalk, which refers to inter-pathway 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 cross-talk 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

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 off-target effects under physiological conditions. Specifically, the model simulated the targeted inhibition of a specific kinase in a series of multi-cycle 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 off-target 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 off-target effect in one cascade can occur when a protein in another cascade is selectively inhibited. In each network studied, cycle _{i}

Protein _{1}* served as the activating kinase for all cascades. Cycle 2 and cycle _{n}_{n}

Topology of signaling networks studied

**Topology of signaling networks studied**. Two general types of network motifs consisting of covalently modified cycles were studied: **(A) **the vertical case where the **(B) **the lateral case where the **(C) **The **(D) **An _{2}* was added to the left-most cascade. In all networks, _{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. Off-target effects in cycle 2 were monitored by measuring the steady state concentrations of _{2 }and _{2}* as the concentration of an inhibitory drug that specifically targeted _{n}

Two general network types were considered: a vertical and a lateral case (Figure _{1}* (Figure

The general reaction scheme used for the vertical, lateral, and

Where

^{th }cycle

_{i }^{th }cycle

^{th }cycle

^{th }cycle

_{i }_{i }^{th }cycle

^{th }cycle

_{D }^{th }cycle

Parameter definitions

In order to reduce the complexity of each network studied, parameters were non-dimensionalized 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

The parameter space of each network consisted of a set of non-dimensional parameters, each with a minimum and maximum allowed value.

**
default range
**

**
parameter
**

**minimum**

**maximum**

**
description
**

**
E**

0.01

100

total kinase to total substrate ratio

**
E'**

0.01

100

total phosphatase to total substrate ratio

**
K**

0.01

100

normalized K_{m }of kinase reaction

**
K'**

0.01

100

normalized K_{m }of phosphatase reaction

**
P**

0.1

10

ratio of the kinase reaction V_{max }to the phosphatase reaction V_{max}

**
K**

0.01

100

normalized drug disassociation constant

Each cycle ** E**,

Subscripts containing _{max }and K_{m }are the standard Michaelis-Menten constants representing, respectively, the maximum velocity of a reaction (at a given enzyme concentration) and the substrate concentration necessary to achieve

(1) total enzyme to substrate ratio of the kinase and phosphatase reaction, respectively, in cycle

(2) normalized K_{m }of the kinase and phosphataste reaction, respectively, in cycle

where

(3) V_{max }ratio of the kinase and phosphatase reactions in cycle

where

(4) normalized disassociation constant of the inhibitor binding to _{n}*

** E**and

Determination of off-target effects

The concentrations of species _{i}

To determine if a detectable off-target effect occurred for a specific set of parameters, changes in the steady values of _{2 }and ** I **was varied from 10

When we tested the ** I **= 0.0000 or

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 non-dimensionalization 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 non-dimensional parameters, the selected values were mapped to corresponding dimensional parameter values (Additional File _{i }_{i}*

**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.

Click here for file

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

**Parameter space sampling to estimate the probability of off-target effects**. This file describes how the parameter space of a network was sampled to provide an estimate of the probability of off-target effects due to retroactivity alone.

Click here for file

Numeric perturbation analysis

A modified perturbation method was used to probe which model parameters were most important for producing an off-target effect as a result of the inhibition of _{n}

The reduced ranges used to perturb each parameter were arrived at by partitioning the default range established for each parameter type in Table ** E**,

A complete numeric perturbation analysis of a parameter space consisted of determining the percentage of off-target effects in 5000 randomly selected parameter sets for each parameter's sub-ranges. In the ** K**). Three of the parameters (

Results

The question we wanted to answer with our models was whether a targeted inhibitor is likely to induce a measurable off-target effect due to retroactivity in a non-targeted cascade under physiological conditions. In each network, cycle ** I **(the normalized inhibitor concentration) from 10

Specific parameter ranges promote off-target effects in cycle 2

First, we investigated the _{3}* is targeted by the inhibitor. When the full parameter space (defined in Table ** K _{3 }**(the normalized K

A numeric perturbation analysis revealed parameter value ranges that promote off-target effects in the

**A numeric perturbation analysis revealed parameter value ranges that promote off-target effects in the n = 3 network**. A perturbation analysis of the

** E _{3 }**and

The only parameter associated with cycle 2 that affected the percentage of off-target effects in this network was ** K _{2 }**(the normalized K

The cycle 1 parameters with the greatest impact on the percentage of off-target effects were ** K _{1 }**and

The value of ** K**, the normalized drug disassociation constant, had a very slight effect on the percentage of off-target effects. In general,

The results of the above analysis indicate that certain parameter value ranges are more likely to induce an off-target effect in cycle 2 as the drug concentration is increased. When we restricted the

A second numerical perturbation analysis was performed using this new restricted ** K _{3 }**values remained important for producing off-target 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,

**Additional analysis of the n = 3 and extended n = 3 networks**. This file provides additional results from the numeric perturbation analyses of the

Click here for file

While some parameters associated with cycle 2 were able to effect the percentage of off-target effects, the parameters associated with cycle 3 continued to have the greatest effect on off-target effects in the restricted ** E _{3}**,

Varying a single parameter can produce a large change in the size of the off-target effect

The magnitude of off-target effects produced by parameter sets randomly sampled from the original _{2 }protein pool (Figure

Distribution of the size of off-target effects in the

**Distribution of the size of off-target effects in the n = 3 network**. Histograms of the size of off-target effects in

We used stimulus response curves to examine how a change in a single parameter value may affect the size of an off-target effect in _{2}* as a function of the normalized inhibitor concentration (Figure ** E _{2 }**or

Varying a single parameter value can produce a large change in the off-target response

**Varying a single parameter value can produce a large change in the off-target response**. Stimulus response curves were plotted for the ** E _{2 }**and

Click here for file

Parameter sets used in stimulus response curves.

**Random set**

**
E
_{1}
**

4.87

0.1

**
E
_{2}
**

**32.56**

**0.0025**

**
E
_{3}
**

0.28

0.0025

**
E'
_{1}
**

0.05

0.1

**
E'
_{2}
**

1.26

0.00025

**
E'
_{3}
**

0.29

0.1

**
K
_{1}
**

5.07

100

**
K
_{2}
**

28.18

0.25

**
K
_{3}
**

**0.04**

**0.25**

**
K'
_{1}
**

66.34

100

**
K'
_{2}
**

9.33

0.25

**
K'
_{3}
**

0.59

0.25

**
P
_{1}
**

0.21

1

**
P
_{2}
**

3.43

1

**
P
_{3}
**

0.42

0.025

**
K**

0.05

0.0833

The two parameters sets used in Figure 5 are summarized in the table. The Random set refers to a randomly selected parameter set and the

The randomly selected parameter set produced a baseline off-target response of 0.19 in _{2}* (Figure _{2 }(data not shown). In this parameter set the original ** E _{2 }**value was 32.56 and the original

The parameter set derived from the MAPK _{2}* (Figure _{2 }(data not shown). In this parameter set the original **E**** _{2 }**value was 0.0025 and the original

A few of the enzyme to substrate ratios in the ** E _{2 }**= 0.0025,

The percentage of off-target 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 ** E _{3 }**=

In the vertical case, the percentage of off-target effects in the

The percentage of off-target effects decreased as the network size increased.

**
n
**

**
Off Target Effects
**

3

73.9

5

27.9

6.0

7

13.5

0.0

The

In the lateral case, the drop in the percentage of off-target effects was more dramatic than in the vertical case, with the

Off-target effects from retroactivity can propagate down a non-targeted 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 off-target effect near the top of another cascade. Because signal amplification is an important cellular sensory mechanism

To test for downstream propagation of off-target effects from cycle 2, we created an ^{th }cycle activated by _{2}* (Figure _{4 }and/or _{4}* occurred that was at least 0.10 of the total _{4 }protein pool, then an off-target effect was considered to have occurred in cycle 4. If an off-target 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 off-target effect with amplification was considered to have occurred in cycle 4.

When the default parameter ranges defined in Table

Off-target effects can amplify downstream of cycle 2.

**
Cycle 2
**

**
Cycle 4
**

Cycles 1-3 with restricted ranges and cycle 4 with default ranges

75.3

35.5

--

23.4

--

12.1

Cycles 1- 4 with restricted ranges

45.3

67.4

--

61.9

--

5.5

To test for downstream propagation of off-target effects from cycle 2, the

To identify the parameters that were most important for amplifying an off-target effect from cycle 2 to cycle 4 in the

Discussion

We developed a computational model to test whether targeted therapies, such as kinase inhibitors, can produce off-target effects in upstream pathways as a consequence of retroactivity alone. Using a numeric perturbation method, we identified specific conditions (Figure

A summary of conditions that favor off-target in the

**A summary of conditions that favor off-target effects in the n = 3 and the extended n = 3 networks**. The conditions that promoted off-target effects in our model are summarized for two network types. Off-target effects in

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 off-target 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

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

This work has led to very interesting and somewhat surprising results. A major importance of this work is that it did not investigate off-target 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

The binding affinity of the inhibitor for its target did not play a substantial role in the promotion of off-target effects in our model. Instead, the kinetics of the component cycles in the network were more important for increasing the likelihood of off-target effects (Figure _{max }of the deactivation reaction was larger than the V_{max }of the activation reaction and/or both enzymatic reactions in cycle

If cycle 2's cascade was extended to include cycle 4 (Figure _{2}*, off-target 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, off-target effects were promoted when cycle 2 was "off" and not consuming significant amounts of the shared upstream activator, _{1}*

The results also indicate that off-target effects were more likely when the total kinase to substrate and the total phosphatase to substrate ratios in the inhibited cycle (** E**and

The immediate experimental implications of this result is that, in the absence of kinetic information, the likelihood of off-target effects may potentially be estimated for a network configuration of this type (Figure

In agreement with the work of other groups

Conclusions

Off-target drug effects

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 off-target 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 off-target effect due to retroactive signaling is more likely when the first cycle in a non-inhibited 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

While our approach does not definitively establish that the predicted responses will occur

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 - Non-dimensionalization of the

In order to reduce the complexity of the networks studied, model parameters were non-dimensionalized. The following explains the non-dimensionalization of the

The ODEs and conservation laws governing the

- Dimensionless Parameters

- Dimensionless Variables

Algebraic rearrangement and substitution yield the following dimensionless steady state equations:

(1)

(2)

(3)

(4)

(5)

(6)

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.