Computational Biology and Bioinformatics Group, Pacific Northwest National Laboratory, Richland WA 99352, USA.

Biological Sciences Division, Pacific Northwest National Laboratory, Richland WA 99352, USA.

Abstract

Background

In addition to initiating signaling events, the activation of cell surface receptors also triggers regulatory processes that restrict the duration of signaling. Acute attenuation of signaling can be accomplished either via ligand-induced internalization of receptors (

Results

Using a mathematical model, we show that downregulation and desensitization mechanisms can lead to tight and efficient input-output coupling thereby ensuring synchronous processing of ligand inputs. Frequency response analysis indicates that upstream elements of the EGFR and GPCR networks behave like low-pass filters with the system being able to faithfully transduce inputs below a critical frequency. Receptor downregulation and desensitization increase the filter bandwidth thereby enabling the receptor systems to decode inputs in a wider frequency range. Further, system-theoretic analysis reveals that the receptor systems are analogous to classical mechanical over-damped systems. This analogy enables us to metaphorically describe downregulation and desensitization as phenomena that make the systems more resilient in responding to ligand perturbations thereby improving the stability of the system resting state.

Conclusion

Our findings suggest that in addition to serving as mechanisms for adaptation, receptor downregulation and desensitization can play a critical role in temporal information processing. Furthermore, engineering metaphors such as the ones described here could prove to be invaluable in understanding the design principles of biological systems.

Background

Recent systems biology efforts are starting to establish biology as a systems science wherein concepts borrowed from engineering and physical sciences are applied to describe biological systems using an input-output relationship based formalism

Cells use surface receptor systems to survey their environment. Signal transduction, where information about extracellular stimuli is converted to a biological response, is critically important both in the context of normal cell physiology as well as in pathogenesis. Signaling receptors are known to mediate diverse cellular responses such as migration, proliferation and differentiation. Constitutive and ligand-induced receptor trafficking (

GPCR

In a physiological setting, ligand concentrations may have spatio-temporal variations that could contain information vital to the cell. The ability to accurately process time varying input signals would allow cellular systems to mount appropriate responses to environmental stimuli. In this manuscript we examine the information processing ability of cell surface receptors and restrict our analysis to the early molecular elements in receptor signaling. We make the implicit assumption that the primary task of a receptor is to transduce ligand concentration information without distortion into molecular activation information, which provides the input to downstream signaling processes. The downstream signal transduction events may be complex and could be non-linearly related to, and temporally distinct from the ligand input. For instance, an instantaneous ligand addition could lead to sustained downstream responses due to the non-linearity of the downstream processes. Even under these circumstances, we believe that the temporal distortion is introduced at stages downstream of surface receptor activation and that the cell would still benefit by maintaining a faithful temporal coupling between the input ligand waveform and the upstream molecular signal that it elicits. To summarize, since they constitute the cellular sensory machinery, receptors need to be able to detect the temporal variations in their ligand availability and then convert this information to an input for the further downstream signalling events. The fidelity of this conversion requires that the changes in ligand availability are faithfully reproduced in the input to the downstream stages.

By examining the EGFR and GPCR pathways as model systems, we show that downregulation and desensitization mechanisms can lead to tight and efficient input-output coupling thereby ensuring synchronous information processing. Analysis of the frequency response of the reaction systems indicates that the EGFR and GPCR models behave like low-pass filters that can process inputs below a critical frequency. EGFR downregulation and GPCR desensitization have a qualitatively similar effect on information processing in that they enable the system to accurately process higher frequency inputs. Examination of the governing equations also reveals that the signaling networks are analogous to a classical mechanical system involving the motion of a mass connected to a spring and a damper. In the context of this analogy, downregulation and desensitization can be metaphorically described as phenomena that make the systems more resilient in responding to external perturbations thus improving the stability of the system's resting state.

Our choice of the EGFR and GPCR receptor systems was motivated by the fact that there is sufficient experimental data to model these systems

Results and discussion

Mathematical models for the EGFR and GPCR systems

Figure

EGFR and GPCR models

**EGFR and GPCR models**. Schematic descriptions of the EGFR (A) and GPCR (B) systems (see text for details).

Supplementary Methods. Detailed descriptions of numerical and analytical solution methodology and rate constants employed in the mathematical model.

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EGFR model

The model for the EGFR (Fig. _{on }and reverse rate _{off}. Free receptors and receptor-ligand complexes are internalized with rates _{t }and _{e}, respectively. Newly synthesized receptors enter the plasma membrane at rate _{R}. The input _{on}, _{off}, _{e}, _{t}, _{T}. The receptor synthesis rate _{R }can be derived based on the steady-state condition in the absence of ligand and can be expressed as _{R }= _{t}_{T}. The parameter values employed in our simulations are as follows: _{on }= 0.097/nM/min, _{off }= 0.24/min, _{e }= 0.15/min, _{t }= 0.02/min, ^{-10 }lt/cell, and _{T }= 200,000 (see Additional file _{e }alone is varied to examine the effect of downregulation on the system response.

The signal transduction network downstream of the EGFR is quite complex and involves multiple signaling pathways

Here, we use the specific example of the EGFR system to assess the role of receptor downregulation on system response. The effect of varying a single system parameter, the extent of downregulation _{e}/_{t}) on the EGFR response is examined using the mathematical model shown in Fig. _{e}_{off }and _{a}_{T}/(_{av}_{a }= _{on}/_{off }is the receptor-ligand affinity (see Eq. 1 in the Methods section for the EGFR transfer function). _{e}, the internalization rate of receptor-ligand complexes, which amounts to changing the

GPCR model

Our GPCR model (Fig. _{on }and reverse-rate _{off }to yield inactive receptor-ligand complexes, _{fr }and reverse-rate _{rr }to yield active complexes _{a}. Activated complexes are irreversibly desensitized at rate _{ds}. Activated receptor-ligand complexes are also capable of converting inactive G-protein molecules _{a }with a second-order forward rate constant _{a }and reverse rate _{i}. The parameter values employed in our simulations are as follows: _{on }= 8.4 × 10^{7 }/M/s, _{off }= 0.37/s, _{fr }= 10/s, _{rr }= 10/s, _{ds }= 0.065/s, _{a }= 10^{-7}/#/s, _{i }= 0.2/s, ^{-10 }lt/cell, _{T }= 55000, and _{T }= 100000 (see Additional file _{ds }alone is varied to examine the effect of desensitization on the system response.

It should be noted that when ligand is added, receptors are lost due to desensitization and are not replaced by receptor synthesis or recycling terms in this GPCR model. In this regard, we restrict our system to that described by Riccobene et al., and neglect receptor trafficking and synthesis terms for the sake of simplicity. While receptor depletion would have an effect on the magnitude of the response to large ligand doses, for the ligand concentrations examined in this paper receptor depletion is limited and does not affect the results. The system output in our model is the number of activated G-protein molecules, _{a}(_{a}(

Even though these models are simple, they capture the essential features of the upstream events in the EGFR and GPCR systems. For example, the spatio-temporal distribution of activated receptors obtained using the presented simple EGFR model is very similar to results (not shown) obtained using extended models that we have previously employed

Receptor downregulation/desensitization increases response speed and improves frequency processing

We hypothesized that signaling receptors are designed to allow cells to generate biological outputs that are tightly coupled to the ligand inputs for a wide range of input patterns. Here, we show that endocytic downregulation and desensitization play a critical role towards this end. They endow the EGFR and GPCR systems with an ability to accurately transduce time-varying ligand doses.

Response to ligand impulses

Under _{D}, where _{D }is the dissociation constant. In other words, the ligand entry rate _{D}. This is a small perturbation representative of physiological ligand stimuli. In our simulations we use an extracellular volume of 4 × 10^{-10 }lt/cell. For the EGFR and GPCR receptor systems this ligand concentration and extracellular volume translate to a one-time addition of 5961 and 10612 ligand molecules respectively.

Impulse response of the EGFR and GPCR systems

**Impulse response of the EGFR and GPCR systems**. (A&C) The dimensionless response to a ligand impulse was computed for EGFR (A) and GPCR (C). The response is to a ligand impulse of magnitude 0.01_{D }nM. System responses were computed either by numerically integrating the ODEs governing the system (solid lines) or by inverting the transfer function, _{e}/_{t}) for the EGFR (B) and the desensitization rate, _{ds }for the GPCR (D). Note that the relaxation time in panel D is plotted on a logarithmic scale to capture the large change in GPCR relaxation time magnitude. This plot would be qualitatively similar to panel B when a linear scale is used.

Figure _{e}/_{t }where _{e }and _{t }are the internalization rates of receptor-ligand complexes and free receptors, respectively. For our simulations _{t }was kept fixed while _{e }was varied to obtain the desired range of

In our simulations of the GPCR system (Figs. _{ds }in a four-orders of magnitude logarithmic scale to account for the range employed in _{ds }value of 0.065 s^{-1 }as the likely desensitization rate for the fMLP receptor system. Our results for the GPCR model were qualitatively similar to the ones for the EGFR system. Increasing the desensitization rate _{ds }yielded greater response speeds (Fig. ^{4 }s at _{ds }= 6.5 × 10^{-4 }s^{-1 }to 77 s at _{ds }= 6.5 s^{-1}.

A faster response would allow the cellular system to accurately decode and transduce inputs consisting of frequent pulses. As an illustrative example, consider a scenario where the EGFR system is exposed to successive spikes in ligand entry spaced an hour apart. With a relaxation time of 27 min (as in the case with

Our results also indicate that there is a trade-off between amplitude and response speed. Downregulation and desensitization increase the response speed at the cost of response amplitude. The consequences of this trade-off on the phenotypic response of the cell would require a detailed examination of the link between receptor occupancy and cell phenotype. In the case of the EGFR we have previously shown that the occupancy of only a few receptors is sufficient to trigger mitogenesis

As seen in Figs. _{D}). The analytical approximation is not valid for these concentrations, and this regime was investigated by numerically integrating the governing equations. We found that at high input ligand dosages the parameter dependence of the system response (results not shown) is qualitatively similar to that reported in Fig.

Frequency response of the receptor systems

The frequency response of a system defines how the system handles temporal information and is computed by assessing the response of the system to sinusoidal inputs. It should be noted that it is unlikely that the receptor systems studied here would be subjected to purely sinusoidal inputs

To the best of our knowledge, there are no good estimates of the physiologically relevant input frequencies that the EGFR and GPCR systems are required to process. Here, we use mathematical analysis to identify the range of frequencies that the EGFR and GPCR systems are capable of processing. In order to compute the frequency response, we assessed the effect of endocytic downregulation and desensitization on the system response to non-negative sinusoidal inputs with a peak value _{D}_{off}), _{off}, _{off}, and _{off }and _{D }respectively are the dissociation rate and the dissociation constant for the receptor-ligand binding reaction. Figure

Characteristics of the EGFR frequency response

**Characteristics of the EGFR frequency response**. The response of the EGFR system was computed for a sinusoidal input of wavelength 200 min and a dimensionless peak value _{n }* by dividing the dimensionless response

In the transient phase of the frequency response, roughly speaking, the system integrates the input and builds up to its steady-state. Large transient times are detrimental to the fidelity of a receptor system and reflect the system's inability to cope with the variations in the input, i.e. the ligand entry rate in this case. In Fig. _{99}taken for the transient to decay by 99% from its initial value at _{99 }quantifies the time taken for the response to build-up to the steady-state oscillation. Figure _{99}/_{99 }is nearly independent of the input frequency _{crit }for which the response reaches steady-state in a single input pulse, (i.e. _{99}/_{crit }the system will respond with virtually no transient. In Fig. _{crit }is plotted as function of the extent of downregulation _{crit }is less than 200 min with _{crit }= 96 min at _{ds }is increased from _{ds }= 6.5 × 10^{-4 }s^{-1 }to 6.5 s^{-1}, the normalized rise time decreases by nearly three orders of magnitude at any given input wavelength (Fig. _{crit }beyond which the system can process inputs without a significant transient, decreases from 1.51 × 10^{5 }s at _{ds }= 6.5 × 10^{-4 }s^{-1 }to 263 s at _{ds }= 6.5 s^{-1 }(Fig.

Transients in the frequency response of the EGFR and GPCR systems

**Transients in the frequency response of the EGFR and GPCR systems**. (A&C) The rise time, _{99 }taken for the transient term in frequency response to decay by 99% was normalized with the input wavelength _{crit }for which the response reaches steady-state in a single input pulse (_{99}/_{ds }for the GPCR (D). For input wavelengths _{crit }the system responds with virtually no transient.

Next, we examined the steady-state characteristics of the frequency response of the EGFR and GPCR systems (Fig. _{delay }(= _{BW }is defined as the frequency at which the output amplitude drops to 70.7% of the zero frequency amplitude (DC gain). For a low pass filter, inputs with frequency lower than _{BW }are transduced with a nearly constant amplitude ratio, while inputs with frequency greater than _{BW }are significantly attenuated. It should be noted that although the amplitude ratio drops beyond _{BW }this does not mean that the system does not generate a significant output. At steady-state the system will display a significantly attenuated oscillation about a non-zero mean value, which is independent of the input frequency but is a function of _{BW }the system ceases to generate an output that resembles the input waveform and the overall response will simply be a transient rise to a constant mean value. In other words, _{BW }quantifies the frequency range where the system responds with high fidelity to the input signal. For the EGFR system, the bandwidth frequency increases from 4.8 × 10^{-3 }min-^{1 }(^{-2 }min-^{1 }at _{BW }increases from 3 × 10^{-5 }s^{-1 }(^{5 }s) at _{ds }= 6.5 × 10^{-4 }s^{-1 }to 1.7 × 10^{-2 }s^{-1 }(_{ds }= 6.5 s^{-1 }(Fig.

Steady-state characteristics of the EGFR and GPCR frequency response

**Steady-state characteristics of the EGFR and GPCR frequency response**. The amplitude ratio _{ds }values corresponding to each of the colored lines for the respective receptor systems are the same as the ones indicated in Figs. 2 and 4.

The steady-state time delay between the input and the output can be computed from the phase lag _{ds }= 6.5 × 10^{-4 }s^{-1 }to 62 s at _{ds }= 6.5 s^{-1 }which reflects significantly increased synchrony when desensitization rates are higher.

Our impulse response results for the EGFR system (Fig.

Receptor downregulation/desensitization enhances the stability of the system resting state

We analyzed the governing equations in order to identify the physical reasons behind the improvement in signal processing elicited by downregulation and desensitization. The linearized EGFR model is a second-order system (Eq. 1 in Methods). The behaviour of such second-order systems can be analyzed in the context of a classical harmonic oscillator. On the other hand, the linearized GPCR model is a fourth-order system (Eq. 2 in Methods), which can be conceptualized as two harmonic oscillators in series. However, to simplify the system description we identified an equivalent second-order system that would mimic the input-output behaviour of the GPCR transfer function. Figure _{a}(_{n }and damping ratio

Comparison of the governing equations for our model with those describing the motion of the mass in Fig. _{n }and damping ratio

EGFR and GPCR systems behave like over-damped mechanical oscillators

**EGFR and GPCR systems behave like over-damped mechanical oscillators**. (A) Schematic of a mass-spring-damper system. The system comprises of a mass connected to a wall via a spring with spring constant _{n}^{2}) and a damper with damping constant _{n}). The mass _{n }the natural frequency and _{n }(red line) and _{ds }(C).

Conclusion

There is considerable evidence to suggest that evolution has optimized the regulation of the biomolecular networks so that they are efficient and function in a robust fashion

Establishing equivalencies between biological reaction networks and human engineered systems enabled us to uncover the functional role of the receptor downregulation/desensitization. We note that the molecular network underlying bacterial chemotaxis has been previously shown to employ a low-pass filter to ensure optimal noise filtering

In engineering systems, a low-pass filter provides the beneficial role of noise rejection and the system generates outputs where noise frequencies are suppressed. Thus, attenuation of frequencies above the bandwidth is viewed as _{mean}) to high frequency inputs. Let us consider the case of a sinusoidal input with frequency higher than the bandwidth. This input would traditionally be viewed as "noise" in the context of engineering but it would still elicit a biological response that can be viewed here as an inaccurate reconstruction of the input – i.e., a constant steady-state response. We suggest that in these biological systems frequencies beyond the bandwidth are not necessarily noise; they may contain useful temporal information that the system is simply incapable of processing. In other words, being a low-pass filter can be viewed as an

In addition to the analogy with electrical circuitry, we established the parallelism between the cellular reaction networks and a classical mechanical over-damped mass-spring-damper system. In the context of this analogy downregulation and desensitization can be metaphorically understood as phenomena that increase the "static stability" of receptor systems. They render the systems more "resilient" and thereby enable the response to bounce back in a rapid fashion following an external perturbation.

We believe that such characterizations of biomolecular networks in the context of engineering or physical systems will significantly improve our understanding of cellular networks. The mathematical analysis methodology detailed in this paper is relatively simple and we employ techniques that are well-established in control systems literature. We believe that the novelty of our work lies in our demonstration that even simple engineering-based analyses can reveal a lot about the underlying design principles of biomolecular networks. Our analysis helped us uncover non-intuitive rationales for the existence of negative regulation mechanisms in receptor signaling networks. Following the elucidation of proteomes and protein interaction maps, an increasing class of biological problems deals with answering the questions of what a specific reaction network does and how it achieves its functionality

Methods

Governing equations and numerical solution

Ordinary differential equations (ODEs) for the EGFR and GPCR models depicted in Fig.

Numerical solutions of the EGFR and GPCR ODE systems were obtained by integrating the ODEs (Additional file

Analytical solution of the EGFR and GPCR models

Investigation of the parameter dependencies in the models becomes easier when analytical solutions are available. The non-linearity of the governing equations for the EGFR and GPCR models prevents us from obtaining exact analytical solutions for these systems. However, the response of the models to relatively small perturbations can be reasonably approximated by solving the system of equations obtained by linearization of the model around the initial steady-state. We solved linearized versions of the EGFR and GPCR governing equations using the technique of Laplace transforms (Additional file

Here _{e}/_{off }and _{on}_{T}/(_{off}_{av}_{av }is Avogadro's number. _{a}*(

The six dimensionless system parameters in Eq. 2 are: _{f }= _{a}_{T}/_{off}, _{r }= _{i}/_{off}, _{f }= _{fr}/_{off}, _{r }= _{rr}/_{off}, _{on}_{T}/(_{off}_{av}_{ds}_{off}.

Impulse response

We used the transfer functions to analyze the response of the receptor models to time-varying ligand inputs. The system output to any input _{D }(dimensionless magnitude = 0.01), _{a}* for GPCR), _{i }are the roots of the characteristic polynomial in the denominator of Eqs. 1 and 2, _{i }are coefficients obtained during partial-fractions expansion of the transfer functions (see Additional file

Frequency response analysis

In order to determine the range of input frequencies the receptor systems can process, we computed the response of the linearized EGFR and GPCR models to sinusoidal inputs of the form _{D}_{off}), _{off}, _{off }and _{off }and _{D }are respectively the dissociation rate and the dissociation constant for the receptor-ligand binding reaction. The Laplace transform of the input is given by ^{2 }+ _{tr}(_{ss}(_{tr}(_{ss}(_{i}(_{i }values for the EGFR and GPCR transfer functions are real and negative, the transient term- like the name suggests- exponentially reaches a value of zero. Further, we can show that _{tr }_{ss}(_{mean }- (_{mean }is the mean value of the steady-state response, and |_{mean }is a time-invariant constant that is a function of the poles of the transfer function. As shown in the Additional file _{mean }= _{1}_{2}) for the EGFR system and it is equal to – _{f }_{f}/(2_{1}_{2}_{3}_{mean }with an amplitude ratio (output amplitude/input amplitude) given by |_{ss }trails the sinusoidal input _{mean }about which the response continues to oscillate. We characterized the frequency response of our receptor systems by examining the transient and steady-state terms individually.

In order to quantify the transient response, we computed the time _{99 }in minutes taken for the magnitude of _{tr}(_{tr}(_{99 }characterizes the time taken for the response to rise to steady-state. We computed the dimensionless normalized rise time as _{99}/

As mentioned earlier, the steady-state frequency response of the receptor systems can be characterized by computing the frequency dependent response function _{BW}, which is defined as the frequency at which the output amplitude of the system drops to 70.7% of the zero frequency amplitude. We also computed the time-lag between the input and the output waveforms as _{delay }=

Analysis of the models in the context of classical second-order dynamics

As seen above, the EGFR response model is a second-order system. Second-order systems can be understood in the context of a classical mechanical oscillator. The transfer function of a canonical second-order system can be written as ^{2 }+ 2_{n }s + _{n}^{2}) where _{n }is the natural frequency of the oscillator. Comparing this expression with the actual transfer function of the EGFR model, we find that _{n }=

The linearized GPCR model is a fourth-order system. Such systems can be conceptually described as two second-order systems (harmonic oscillators) connected in series. However, we sought to simplify the description of the GPCR system by finding a second-order transfer function that would approximate the behaviour of the fourth-order system. For a given parameter set we first computed the fourth-order transfer function _{n }values.

Authors' contributions

HS, HSW and HR jointly designed the study and wrote the paper, and HS performed the research. All authors read and approved the final manuscript.

Acknowledgements

The research described in this paper was funded by the National Institutes of Health Grant 5R01GM072821-02 to H.R. and by the Biomolecular Systems Initiative LDRD Program at the Pacific Northwest National Laboratory, a multiprogram national laboratory operated by Battelle for the U.S. Department of Energy under Contract DE-AC06-76RL01830.