Advanced Review
Implementation of integral feedback control in biological systems Pramod R. Somvanshi,† Anilkumar K. Patel,† Sharad Bhartiya and K. V. Venkatesh∗ Integral control design ensures that a key variable in a system is tightly maintained within acceptable levels. This approach has been widely used in engineering systems to ensure offset free operation in the presence of perturbations. Several biological systems employ such an integral control design to regulate cellular processes. An integral control design motif requires a negative feedback and an integrating process in the network loop. This review describes several biological systems, ranging from bacteria to higher organisms in which the presence of integral control principle has been hypothesized. The review highlights that in addition to the negative feedback, occurrence of zero-order kinetics in the process is a key element to realize the integral control strategy. Although the integral control motif is common to these systems, the mechanisms involved in achieving it are highly specific and can be incorporated at the level of signaling, metabolism, or at the phenotypic levels. © 2015 Wiley Periodicals, Inc. How to cite this article:
WIREs Syst Biol Med 2015, 7:301–316. doi: 10.1002/wsbm.1307
INTRODUCTION
L
iving systems interact with their environment by sensing and monitoring stimuli through signaling networks and subsequently altering their phenotypic response through regulatory mechanisms. The regulatory mechanisms consist of several molecular interactions, which may be characterized by conserved motifs, such as multiple feedbacks, feed-forward loops, and cascade among others. It has been demonstrated that these motifs endow the organism with specific functional system level properties.1,2 Some of these properties include robustness, ultrasensitive response, bistability, and others, which in turn help the organism to interact with its environment, thereby enhancing its survival potential. While a system level property is associated with a specific motif, several motifs can yield the same system level property.3,4 † Both
the authors have contributed equally to this manuscript.
∗ Correspondence
to: [email protected]
Department of Chemical Engineering, IIT Bombay, Mumbai, India Conflict of interest: The authors have declared no conflicts of interest for this article.
Volume 7, September/October 2015
A commonly encountered system level property is homeostasis. Homeostasis refers to the ability of a biological system to regulate its internal state in the face of changing external perturbations.5 In a homeostatic system, the variable of interest is maintained at a constant level by manipulating the internal reaction fluxes to compensate for these perturbations. Examples of such behavior include homeostasis of plasma calcium and glucose levels in mammals6,7 and homeostasis of cytosolic nitrate in higher plants.8 Adaptation is another commonly encountered system level property, wherein the system responds to a persistent perturbation in the environment by a transient response of a sensory or signaling molecule.9–11 However, the signaling molecule eventually returns to its pre-stimulus level despite the presence of the persistent perturbation, thereby, realizing a distinctly new phenotypic behavior. The ultimate goal of these various system level properties is to enhance the potential for survival in the presence of environmental uncertainties. From a regulatory perspective, both homeostasis and adaptation have the similar property of keeping
© 2015 Wiley Periodicals, Inc.
301
wires.wiley.com/sysbio
Advanced Review
the output signal at a desired level in the face of sustained changes in one or more of the input signals. In numerous instances, these system level properties of homeostasis and adaptation are achieved through a common design principle called integral control,12–14 which ensures that the key variable of the system is maintained within a narrow band of acceptable levels.15 In the presence of perturbations in the environment, the system responds by changing a system parameter (e.g., level of a regulatory molecule) to negate the perturbation effect and thereby reduce the deviation of the key variable from its pre-stimulus value.16 It is desirable that the change in the system parameter is not only proportional to the magnitude of the deviation in the key variable but also proportional to the duration over which the deviation prevails. A single measure accounting for both these aspects is the time integral of the deviation in the key variable. This ensures that the regulatory action persists until the key variable identically assumes its basal value and thus displays homeostasis/adaptation.17 Two requirements for integral control are: (1) Quantification of the extent of deviation of the key variable from its homeostatic target, that is, magnitude of the error; and (2) generation of a regulatory action, which is nearly proportional to the time integral of the deviation or the error. Thus the integral action depends not only on the current value but also on the history of error. In biological systems, the integral control strategy is mainly implemented through a network consisting of molecular interactions. However, as discussed later in this article, the inevitable degradation of bio-molecules makes the implementation of perfect integration difficult. See Box 1 for a brief description on integral control. Unlike other control mechanisms such as proportional and derivative actions, integral control is active as long as the deviation from the target is nonzero for a step change in the perturbation. This in turn ensures that the perfect adaptation and homeostatic behavior are achieved. See Box 2 for the connection between integral control and perfect adaptation/homeostasis. Integral control strategy to achieve homeostasis/adaptation has been implicated in several biological systems. For example, perfect adaptation in bacterial chemotaxis is achieved by manipulating the methylation state of the receptor.18 The molecular mechanism involved in achieving integral control is highly specific to the biological system. In each case, the regulatory action is related to the integral of the deviation in the key variable (i.e., error). This review describes several biological systems ranging from bacteria to higher organisms in which the principle of integral control is hypothesized. We first describe 302
the structural motif common to these systems, which results in an integral control design. Subsequently, we delineate the underlying design using various examples. In each case, the key variable that is regulated and the system parameter that is manipulated are identified.
BIOLOGICAL SYSTEMS WITH INTEGRAL NEGATIVE FEEDBACK MECHANISM As discussed above, implementation of integral control in biological systems requires a negative feedback of the measurement of the error as well as a physics-driven mechanism to integrate the error or the deviation signal. Consider a biological network consisting of a signaling molecule A, a regulatory protein P, and a metabolite M. Figure 1 shows four possible configurations that achieve a negative feedback design. If the external cue E activates the signaling molecule A, the negative feedback can be achieved either by inhibition of the signaling molecule itself (see Figure 1(a)) by the feedback signal or inhibition of the regulatory protein (see Figure 1(b)). Upon a step change in E and the concomitant increase in the signaling molecule A, the negative feedback upregulates regulatory protein P, which in turn reduces A to its pre-stimulus level (see Figure 1(a), right panel). On the other hand, in case of the inhibition of the regulatory molecule, the regulatory protein is downregulated by the negative feedback, which in turn reduces A (see Figure 1(b), right panel). Figures 1(c) and (d) show implementation of negative feedback when the external cue E inhibits the signaling molecule A. In this case, the level of the signaling molecule drops upon a step change in the external cue E and recovers to its basal level subsequently because of the presence of integral control (see right panels in Figure 1). In addition to the negative feedback of error, integral control requires that the synthesis of the regulatory protein must be proportional to the integral of the error.17 A biologically relevant implementation of integral control for the network shown in Figure 1(a) is shown in Box 3. There exist various approaches to realize error integration in biological systems. One plausible approach for integrating a signal (error) is via zero-order kinetics. The kinetic rate of a reaction is said to be zero-order if the rate is apparently independent of substrate or enzyme concentration, implying that the rate of reaction is proportional to the rate constant. Thus, a zero-order process is temporary as it cannot prevail after the substrate has been exhausted. For biological systems, zero-order kinetics can be achieved mainly through saturation of enzymes
© 2015 Wiley Periodicals, Inc.
Volume 7, September/October 2015
WIREs Systems Biology and Medicine
Implementation of integral feedback control in biological systems
BOX 1 A PRIMER ON INTEGRAL CONTROL
D XSET
+
c
Controller –
Process
X
e = XSET – XM
XM
Sensor
The diagram shows a simple control system with a negative feedback loop, where X, which is kept around a desired value (i.e., X SET ), is the key variable of interest. Perturbation (D) in the process results in a deviation in variable X from X SET . X is measured by a sensor to generate the measured signal X M . The measured variable X M is subtracted from the desired variable to generate an error signal (e). Error (e) is sensed and signaled to the controller, which generates a control signal (c), which is injected into the process to eliminate the deviation in X. The control strategy is critically linked to the generation of control signal (c) from the error signal, and this determines the performance of the system. There are mainly three types of controllers,15 namely (1) proportional, (2) integral, and (3) derivative. This review deals mainly with the integral controller. As the names suggest, the proportional controller produces the control signal, which is proportional to the error signal, and the integral control produces the control signal, which is proportional to the integral of the error signal, that is, the area under the curve of error profile. Thus mathematically, an integral control signal can be defined as: t
c (t) = KI ∗
∫0
e (𝜂) d𝜂,
(i)
where gain (KI ) is a constant multiplied by the integral of error to obtain the control signal. Following are some of the characteristics of an integral control: 1 It ensures that at steady state conditions, the error is zero. 2 If the gain of system KI is low, the response time of the system is higher. 3 On the other hand, the response can become faster and even become oscillatory if the gain is high.
as explained in You et al.19,20 Let the rate of degradation of the regulatory molecule P in Figure 1(a) follow Michaelis–Menten kinetics. P rdP = k0 ∗ . P + Km
(1)
Here the zero-order kinetics is manifested for the time period when concentration of P is very large compared to the half saturation constant Km as shown Volume 7, September/October 2015
below, For Km
Implementation of integral feedback control in biological systems Pramod R. Somvanshi,† Anilkumar K. Patel,† Sharad Bhartiya and K. V. Venkatesh∗ Integral control design ensures that a key variable in a system is tightly maintained within acceptable levels. This approach has been widely used in engineering systems to ensure offset free operation in the presence of perturbations. Several biological systems employ such an integral control design to regulate cellular processes. An integral control design motif requires a negative feedback and an integrating process in the network loop. This review describes several biological systems, ranging from bacteria to higher organisms in which the presence of integral control principle has been hypothesized. The review highlights that in addition to the negative feedback, occurrence of zero-order kinetics in the process is a key element to realize the integral control strategy. Although the integral control motif is common to these systems, the mechanisms involved in achieving it are highly specific and can be incorporated at the level of signaling, metabolism, or at the phenotypic levels. © 2015 Wiley Periodicals, Inc. How to cite this article:
WIREs Syst Biol Med 2015, 7:301–316. doi: 10.1002/wsbm.1307
INTRODUCTION
L
iving systems interact with their environment by sensing and monitoring stimuli through signaling networks and subsequently altering their phenotypic response through regulatory mechanisms. The regulatory mechanisms consist of several molecular interactions, which may be characterized by conserved motifs, such as multiple feedbacks, feed-forward loops, and cascade among others. It has been demonstrated that these motifs endow the organism with specific functional system level properties.1,2 Some of these properties include robustness, ultrasensitive response, bistability, and others, which in turn help the organism to interact with its environment, thereby enhancing its survival potential. While a system level property is associated with a specific motif, several motifs can yield the same system level property.3,4 † Both
the authors have contributed equally to this manuscript.
∗ Correspondence
to: [email protected]
Department of Chemical Engineering, IIT Bombay, Mumbai, India Conflict of interest: The authors have declared no conflicts of interest for this article.
Volume 7, September/October 2015
A commonly encountered system level property is homeostasis. Homeostasis refers to the ability of a biological system to regulate its internal state in the face of changing external perturbations.5 In a homeostatic system, the variable of interest is maintained at a constant level by manipulating the internal reaction fluxes to compensate for these perturbations. Examples of such behavior include homeostasis of plasma calcium and glucose levels in mammals6,7 and homeostasis of cytosolic nitrate in higher plants.8 Adaptation is another commonly encountered system level property, wherein the system responds to a persistent perturbation in the environment by a transient response of a sensory or signaling molecule.9–11 However, the signaling molecule eventually returns to its pre-stimulus level despite the presence of the persistent perturbation, thereby, realizing a distinctly new phenotypic behavior. The ultimate goal of these various system level properties is to enhance the potential for survival in the presence of environmental uncertainties. From a regulatory perspective, both homeostasis and adaptation have the similar property of keeping
© 2015 Wiley Periodicals, Inc.
301
wires.wiley.com/sysbio
Advanced Review
the output signal at a desired level in the face of sustained changes in one or more of the input signals. In numerous instances, these system level properties of homeostasis and adaptation are achieved through a common design principle called integral control,12–14 which ensures that the key variable of the system is maintained within a narrow band of acceptable levels.15 In the presence of perturbations in the environment, the system responds by changing a system parameter (e.g., level of a regulatory molecule) to negate the perturbation effect and thereby reduce the deviation of the key variable from its pre-stimulus value.16 It is desirable that the change in the system parameter is not only proportional to the magnitude of the deviation in the key variable but also proportional to the duration over which the deviation prevails. A single measure accounting for both these aspects is the time integral of the deviation in the key variable. This ensures that the regulatory action persists until the key variable identically assumes its basal value and thus displays homeostasis/adaptation.17 Two requirements for integral control are: (1) Quantification of the extent of deviation of the key variable from its homeostatic target, that is, magnitude of the error; and (2) generation of a regulatory action, which is nearly proportional to the time integral of the deviation or the error. Thus the integral action depends not only on the current value but also on the history of error. In biological systems, the integral control strategy is mainly implemented through a network consisting of molecular interactions. However, as discussed later in this article, the inevitable degradation of bio-molecules makes the implementation of perfect integration difficult. See Box 1 for a brief description on integral control. Unlike other control mechanisms such as proportional and derivative actions, integral control is active as long as the deviation from the target is nonzero for a step change in the perturbation. This in turn ensures that the perfect adaptation and homeostatic behavior are achieved. See Box 2 for the connection between integral control and perfect adaptation/homeostasis. Integral control strategy to achieve homeostasis/adaptation has been implicated in several biological systems. For example, perfect adaptation in bacterial chemotaxis is achieved by manipulating the methylation state of the receptor.18 The molecular mechanism involved in achieving integral control is highly specific to the biological system. In each case, the regulatory action is related to the integral of the deviation in the key variable (i.e., error). This review describes several biological systems ranging from bacteria to higher organisms in which the principle of integral control is hypothesized. We first describe 302
the structural motif common to these systems, which results in an integral control design. Subsequently, we delineate the underlying design using various examples. In each case, the key variable that is regulated and the system parameter that is manipulated are identified.
BIOLOGICAL SYSTEMS WITH INTEGRAL NEGATIVE FEEDBACK MECHANISM As discussed above, implementation of integral control in biological systems requires a negative feedback of the measurement of the error as well as a physics-driven mechanism to integrate the error or the deviation signal. Consider a biological network consisting of a signaling molecule A, a regulatory protein P, and a metabolite M. Figure 1 shows four possible configurations that achieve a negative feedback design. If the external cue E activates the signaling molecule A, the negative feedback can be achieved either by inhibition of the signaling molecule itself (see Figure 1(a)) by the feedback signal or inhibition of the regulatory protein (see Figure 1(b)). Upon a step change in E and the concomitant increase in the signaling molecule A, the negative feedback upregulates regulatory protein P, which in turn reduces A to its pre-stimulus level (see Figure 1(a), right panel). On the other hand, in case of the inhibition of the regulatory molecule, the regulatory protein is downregulated by the negative feedback, which in turn reduces A (see Figure 1(b), right panel). Figures 1(c) and (d) show implementation of negative feedback when the external cue E inhibits the signaling molecule A. In this case, the level of the signaling molecule drops upon a step change in the external cue E and recovers to its basal level subsequently because of the presence of integral control (see right panels in Figure 1). In addition to the negative feedback of error, integral control requires that the synthesis of the regulatory protein must be proportional to the integral of the error.17 A biologically relevant implementation of integral control for the network shown in Figure 1(a) is shown in Box 3. There exist various approaches to realize error integration in biological systems. One plausible approach for integrating a signal (error) is via zero-order kinetics. The kinetic rate of a reaction is said to be zero-order if the rate is apparently independent of substrate or enzyme concentration, implying that the rate of reaction is proportional to the rate constant. Thus, a zero-order process is temporary as it cannot prevail after the substrate has been exhausted. For biological systems, zero-order kinetics can be achieved mainly through saturation of enzymes
© 2015 Wiley Periodicals, Inc.
Volume 7, September/October 2015
WIREs Systems Biology and Medicine
Implementation of integral feedback control in biological systems
BOX 1 A PRIMER ON INTEGRAL CONTROL
D XSET
+
c
Controller –
Process
X
e = XSET – XM
XM
Sensor
The diagram shows a simple control system with a negative feedback loop, where X, which is kept around a desired value (i.e., X SET ), is the key variable of interest. Perturbation (D) in the process results in a deviation in variable X from X SET . X is measured by a sensor to generate the measured signal X M . The measured variable X M is subtracted from the desired variable to generate an error signal (e). Error (e) is sensed and signaled to the controller, which generates a control signal (c), which is injected into the process to eliminate the deviation in X. The control strategy is critically linked to the generation of control signal (c) from the error signal, and this determines the performance of the system. There are mainly three types of controllers,15 namely (1) proportional, (2) integral, and (3) derivative. This review deals mainly with the integral controller. As the names suggest, the proportional controller produces the control signal, which is proportional to the error signal, and the integral control produces the control signal, which is proportional to the integral of the error signal, that is, the area under the curve of error profile. Thus mathematically, an integral control signal can be defined as: t
c (t) = KI ∗
∫0
e (𝜂) d𝜂,
(i)
where gain (KI ) is a constant multiplied by the integral of error to obtain the control signal. Following are some of the characteristics of an integral control: 1 It ensures that at steady state conditions, the error is zero. 2 If the gain of system KI is low, the response time of the system is higher. 3 On the other hand, the response can become faster and even become oscillatory if the gain is high.
as explained in You et al.19,20 Let the rate of degradation of the regulatory molecule P in Figure 1(a) follow Michaelis–Menten kinetics. P rdP = k0 ∗ . P + Km
(1)
Here the zero-order kinetics is manifested for the time period when concentration of P is very large compared to the half saturation constant Km as shown Volume 7, September/October 2015
below, For Km
