Physical Constraints on Biological Integral Control Design for Homeostasis and Sensory Adaptation

Abstract

Synthetic biology includes an effort to use design-based approaches to create novel controllers, biological systems aimed at regulating the output of other biological processes. The design of such controllers can be guided by results from control theory, including the strategy of integral feedback control, which is central to regulation, sensory adaptation, and long-term robustness. Realization of integral control in a synthetic network is an attractive prospect, but the nature of biochemical networks can make the implementation of even basic control structures challenging. Here we present a study of the general challenges and important constraints that will arise in efforts to engineer biological integral feedback controllers or to analyze existing natural systems. Constraints arise from the need to identify target output values that the combined process-plus-controller system can reach, and to ensure that the controller implements a good approximation of integral feedback control. These constraints depend on mild assumptions about the shape of input-output relationships in the biological components, and thus will apply to a variety of biochemical systems. We summarize our results as a set of variable constraints intended to provide guidance for the design or analysis of a working biological integral feedback controller.

Introduction

Homeostasis and sensory adaptation are related phenomena found in many biological contexts. Homeostasis refers to the ability of a biological system to regulate its internal state in the face of changing external inputs (1). Sensory adaptation describes the situation wherein a biological sensor shows a transient response to changing sensory inputs—thereby providing useful information to downstream systems—but eventually reverts back toward its original, prestimuli response state if the new input levels become static and sustained (see Fig. 1). This behavior prevents the sensor from permanently responding to a static input, and from losing effectiveness when the static input is large enough to saturate the sensory response, rendering the sensor incapable of detecting additional changes in the stimulus level. (Familiar examples of sensory adaptation include visual adjustment to ambient light levels and olfactory adaptation to odors.)

Figure 1.

Sensory adaptation: a system reacts to a stimulus by responding initially with a change in its output, after which the output relaxes back toward its prestimulus value despite the persistence of the stimulus. Perfect adaptation occurs when the system’s step response (the output generated by a step input) returns to its exact prestimulus value.

From a control theory perspective, both homeostasis and sensory adaptation imply the long-term preservation, via negative feedback control, of a system’s output signal in the face of sustained changes to one or more of its input signals. In the case of step input perturbations (perturbations that, once applied, remain present and constant-valued), control-theorists and engineers have long known that for adaptation to be perfect (i.e., complete) and robust (i.e., adapting perfectly, independent of the perturbation amplitude or the operating regime), a control strategy known as integral feedback control is mandatory (2–4).

Identifying and understanding the molecular implementation of integral feedback control is the subject of ongoing research in natural biological systems (5–12), and as a control structure of fundamental importance, we anticipate that further examples of integral control likely exist in a range of biological contexts. In the field of synthetic biology, where researchers seek to design novel cellular regulatory systems to alter and control the natural dynamics of cells (13–15), the ability to implement adapting sensory or purely regulatory responses is an attractive one: such systems could be used to sense changes and send signals to downstream targets over a wide dynamic range, or to maintain fixed outputs in the face of changing conditions.

However, experimental implementation of integral feedback control in a biological context is not straightforward: instead, the nature of cellular regulatory networks (density-dependent kinetics and molecular signals that easily reach saturation) give rise to several important but nonobvious design requirements (16). In that previous work, we considered the specific case of transcriptional regulatory networks; here, we extend and generalize our results to encompass any form of biochemical network. Our results indicate that there are physical limitations on the operating regime of biological integral controllers, and we will discuss how synthetic biologists seeking to design novel integral control mechanisms may constrain their parameter choices to create functioning controllers. For discussions of constraints on perfect adaptation presented from different perspectives, see the literature (5,10,17,18).

Computational Methods

Computations for the case-study system were performed in MATLAB (Ver. 7.13; The MathWorks, www.mathworks.com). In particular, numerical root finding was performed with the fzero subroutine and rate equations were solved numerically using the ode15s subroutine.

Results and Discussion

A controller for perfect adaptation

In Fig. 2, we sketch the block diagram for a negative feedback-controlled system in its most general form. It consists of two subsystems: the process (P) and the controller (C). Here, the process receives an input signal, u, and produces an output signal, y. The output signal is then fed into the controller, which generates a control-action signal, x, that is fed back into the original process element. Negative feedback means that any change to the input signal will produce a change in the control action that influences the process in an opposite manner. This type of behavior is a basic system requirement for self-regulation and stability. The dynamics of the process element may be described generically by

$$\frac{dy}{dt}=g\left(u,x,y\right).$$ (1)

(For clarity, we begin by considering only elements that can be described by a single first-order, time-invariant ordinary differential equation; more general multidimensional dynamics including those described by higher-order ordinary differential equations are addressed in the Supporting Material.)

Figure 2.

Block diagram of a negative feedback-controlled system with an input signal, u, and an output signal, y. The u to y processing is performed by the process. The output signal is fed back into the process through a controller that produces a control-action signal, x. (Arrowheads) Types of influence: (→) Upregulation. (⊣) Downregulation. Here we show a negative feedback scheme where y affects x directly, while x affects y inversely (y → x ⊣ y), as opposed to the other way around (y ⊣ x → y).

An alternate way of defining perfect adaptation is to say that it occurs when a system’s steady-state output value is independent of its input value. Mathematically, we observe that this is true for any feedback controller that prompts the dynamics of the control action, x, to have the form

$$\frac{dx}{dt}=f\left(y\right),$$ (2)

in which the function f(y) is independent of both u and x, so that the behavior of the control action, x, depends only on the system output, y. We will restrict our discussion to the case where f(y) = 0 has a single root. This root, which is unique and independent of the input, u, then defines the only y value at which the controller itself will reach a steady state. Therefore, if the overall process-controller system is stable, this y value also represents the steady state of the full process-plus-controller system. Whatever the details of the dynamics of the process in Eq. 1, Eq. 2 shows that the full system cannot be in a steady state unless we have f(y) = 0; otherwise, at least one system variable, namely the control action, would have a nonzero rate of change.

A few comments on this situation are warranted: First, Eq. 2 represents a controller containing an integrator: the control action signal, x(t), is formed by direct time integration of a y-dependent function, and thus implements a form of integral feedback control. Second, the perfect adaptation is robust: it does not depend on parameter tuning anywhere within the full system. Third, although y is the process output, its steady-state value is independent of the form of the process itself: it is implied by Eq. 2 alone and thus set by the form of the controller. This output value is generally referred to as the setpoint, denoted here by y₀.

From the perspective of synthetic biology, the key issue therefore becomes how to design a biological controller of the specific form given by Eq. 2. Broadly, there are two classes of constraints for generating perfect adaptation: those involved in making sure that both the controller and the process will be able to reach the setpoint value, and those required to preserve Eq. 2. Although we note that these constraints can play a role in analyzing the functionality of natural biological systems, our focus for the remainder of the article will be on the synthetic biology design problem.

How saturation restricts controller functionality and effectiveness

We begin with constraints on the setpoint value itself. Most cellular networks consist of molecular signals that are positive quantities by definition, and whose levels and rates of change are often easily saturable at both the high and low ends. In Fig. 3A we present a graphical illustration of Eq. 2 using a generic, monotonically increasing sigmoidal curve for the function f(y); to preserve physicality, we consider only the y > 0 domain. (Here, we assume a $y\to x⊣y$ negative feedback scheme as depicted in Fig. 2, where → denotes upregulation and $⊣$ denotes downregulation; the discussion works equally well for a $y⊣x\to y$ negative feedback scheme.)

Figure 3.

(A) A graphical illustration of a general integral controller—specifically, the rate equation for the control action signal: dx/dt = f(y). The function f(y) is represented by a generic, monotonically increasing sigmoidal curve. The y value at which f(y) is zero represents the setpoint, y₀. (B) A typical-looking steady-state x-y dose-response curve for a process element represented in Fig. 2, for a fixed u. The $y_{\text{ss}}^{\text{P}}$ range is bounded by $y_{\text{ss,min}}^{\text{P}}$ and $y_{\text{ss,max}}^{\text{P}}$.

The y value at which the function f(y) is zero corresponds to the system setpoint. If, however, f(y) does not cross zero in the y > 0 domain, then the setpoint is not physically defined: it will not be a positive, real number. In this case, the controller itself will be nonfunctional because it will be unable to reach a steady state. Put another way, such a controller lacks the necessary ability to both increase (f > 0) and decrease (f < 0) the control action signal.

This is the first hurdle in designing an integral controller: If we wish to enforce a setpoint value of y₀, we must design the form of the controller such that f(y₀) = 0 at a point where the controller will have scope to respond to both positive and negative changes in the input. Synthetic biologists have explored methods of altering the shape and range of biological response curves in networks governed by transcriptional regulation (19–27), posttranscriptional regulation (28–30), and posttranslational regulation (31,32). Employing such methods to shift and scale the controller’s response provides control over the location of the setpoint that the controller will enforce. (We will return to the discussion of setpoint design later.)

Further restrictions on the design of the controller are imposed by the nature of the process element itself. In particular, saturation effects in the steady-state dose-response profile of the process may prevent a given setpoint from being achievable in practice, even if the initial demands within the controller—i.e., those imposed on f(y)—have been satisfied. As a result, the range of enforceable setpoints becomes constrained beyond the simple y₀ > 0 requirement.

Consider the long-term response that the process element in Fig. 2 has to its inputs, independent of any dynamic feedback effects (i.e., in an open-loop configuration). The process accepts two independent input signals, u and x, which generate an output signal, y. Assuming that the process reaches a steady state that is independent of initial conditions, we will define its dose-response profile as

$$y_{\text{ss}}^{\text{P}}\equiv G\left(u,x_{\text{ss}}\right),$$ (3)

where the function G is restricted to nonnegative quantities. (The ss subscript denotes steady-state values while the superscript P signifies that this is the dose-response curve for the process alone.) For any particular u value, we can expect that a typical process element will display a saturating x-y dose-response curve, as is generally the case in biological regulation. Because y is downregulated by x (in our example), the curve will begin at some saturated upper limit, $y_{\text{ss,max}}^{\text{P}}$ (u) = G(u, x_ss = 0), decrease monotonically, then saturate at some lower limit, $y_{\text{ss,min}}^{\text{P}}$ (u) = G(u, x_ss → ∞). This is shown in Fig. 3B.

Consequently, when an integral feedback controller is connected to the process element, the controller must be designed to target a setpoint within these saturation limits—i.e., $y_{\text{ss,min}}^{\text{P}}$ < y₀ < $y_{\text{ss,max}}^{\text{P}}$—or else the process will be unable to comply and the overall system will be unable to reach a steady state. (Otherwise stated, there are no nonnegative and real x_ss values that can satisfy a setpoint outside these saturation limits, as discussed in Ni et al. (11)). In a real biochemical network, where signals cannot diverge indefinitely, such a situation would instead result in the eventual breakdown of either one or both of Eqs. 2 and 3, and therefore result in a new steady-state output that differs from the original y₀.

Because the input parameter, u, may alter the shape and positioning of the x-y dose-response curve, we can refine the setpoint limits to guarantee perfect adaptation across a range of u values, u_low to u_high. This is done by requiring that the target setpoint fall in between the largest possible $y_{\text{ss,min}}^{\text{P}}$ and the smallest possible $y_{\text{ss,max}}^{\text{P}}$. If, in addition to being downregulated by x, y is upregulated by u, then we can write these values as

$$y_{\text{ss},\text{min}}^{\text{P}}=G\left(u,x_{\text{ss}}\right){|}_{\begin{array}{l}u=u_{\text{high}}\hfill \\ x_{\text{ss}}\to \infty \hfill \end{array}}$$ (4a)

and

$$y_{\text{ss},\text{max}}^{\text{P}}=G\left(u,x_{\text{ss}}\right){|}_{\begin{array}{l}u=u_{\text{low}}\hfill \\ x_{\text{ss}}=0\hfill \end{array}}$$ (4b)

(see Fig. 4). Recall that this set of constraints on the setpoint value is imposed by the process, and thus, if we want to force the system’s steady-state output to a value outside this range, it will require alteration of the process itself, not simply a redesign of the controller.

Figure 4.

The input parameter, u, may alter the shape and positioning of the process element’s x-y dose-response curve. This figure shows an example where y is downregulated by x, and u upregulates y by shifting the x-y dose-response curve upward. In general, u may affect the curve in many other ways (e.g., scaling). Perfect adaptation is guaranteed across the input range u_low to u_high only for setpoints between $y_{\text{ss,min}}^{\text{P}}$ and $y_{\text{ss,max}}^{\text{P}}$ in this figure.

Up to this point, we have considered what is, at least in the mathematical sense, a fairly general and ideal integral controller. In practice, a controller of the exact form in Eq. 2 may only be achievable as an approximation within a certain operating regime. In what follows, we discuss this reality and how it imposes constraints on the value of x_ss, and thus further constrains the setpoint limits given by Eq. 4.

Zeroth-order kinetics for integral control

What generates robust perfect adaptation from Eq. 2 is the fact that the right-hand side of the equality is not directly dependent on either u or x. We now address design methods directed at ensuring that this is the case.

Keeping the rate of change of x independent of its own level can be difficult to engineer in cellular systems where signal levels are molecular concentrations governed by biochemical kinetics. It requires, for example, avoiding all autocatalytic mechanisms of species production. Any mechanisms working to decrease the concentration of the control action species will likely need to operate at saturation with respect to the control action species, because the kinetics of such mechanisms are not generally expected to be inherently zeroth-order.

Enzyme-catalyzed reactions

Consider an isolated controller element whose output signal, the control action x, is a molecular concentration that is increased by some production mechanism (e.g., gene expression or protein phosphorylation) and decreased by an enzyme-catalyzed removal mechanism (e.g., proteolysis or dephosphorylation). Remaining with our y→x⊣y feedback scheme, consider the case where the controller’s input signal, y, upregulates the control action signal, x, via upregulation of its production mechanism, such that the controller rate equation can be written as

$$\frac{dx}{dt}=f_{\text{prod}}\left(y\right)-V_{\text{max}}\left(\frac{x}{K_{\text{m}}+x}\right),$$ (5)

where the function f_prod(y) represents the rate of production and monotonically increases with y. Here, we treat the enzyme-catalyzed removal as following Michaelis-Menten kinetics, with maximum removal rate V_max and Michaelis constant K_m.

To make the controller dynamics a function of y alone, the x-dependence in the enzyme-catalyzed rate of removal must be removed from the proposed controller design. This can be achieved by forcing the removal reaction to operate at saturation with respect to the substrate (i.e., the control action species). In our mathematical model, this corresponds to the asymptotic regime of the Michaelis-Menten curve where x/(K_m + x) → 1. Here, x-dependence is effectively removed from Eq. 5, making the removal rate of the control action species zeroth-order (i.e., independent of its own concentration).

Because saturation occurs when x is much larger than K_m, two practical methods can be pursued during the design of the controller to promote it:

The first method is to force the overall system to operate in a high x regime. This can be accomplished by reducing the overall effectiveness of the control action species either by reducing each molecule’s intrinsic influence on the process (e.g., by lowering binding affinities, enzymatic activity, or other properties), or by interfering with the molecules’ ability to exert this influence (e.g., through allosteric inhibition, physical separation, or other barriers). This will force the controller to produce a higher x to drive the process output to the setpoint.

The second method involves lowering K_m by increasing the substrate-enzyme binding affinity. For protein degradation occurring via proteolysis, this has been demonstrated through the selection of an alternative set of proteolytic machinery (33). By tagging the substrate protein with particular amino-acid recognition sequences, it is possible to induce specific binding to this alternative protease with various K_m values (34,35).

Because the primary use of integral control is often for long-term setpoint adherence, it will likely suffice for the designer, at least initially, to focus simply on ensuring proper integrative behavior (i.e., saturation) near the steady state. Therefore, we constrain the steady-state x such that

$$x_{\text{ss}}\gg K_{\text{m}}.$$ (6)

Note that this requirement further restricts the setpoint values that the controller can maintain while continuing to guarantee integral feedback control; for example, the upper y₀ constraint (previously defined by Eq. 4b) must now account for the fact that x_ss must stay much larger than the K_m of removal. We reserve further treatment of this fact until the section on near-perfect adaptation.

The importance of zeroth-order kinetics can be seen in natural biological implementations of integral control. The celebrated example of robust near-perfect adaptivity that occurs in bacterial chemotaxis (36–38) results from what can be interpreted as an integral control network (7). That network is based on posttranslational modification signaling (phosphorylation and methylation) and the requirement for zeroth-order kinetics that we have outlined here is mirrored by the need (in the chemotaxis network) for a specific methylation step related to the control action to occur at saturation.

Dilution in growing cells

In growing cells, cellular volume increase works to dilute molecular concentrations. If volume increase is exponential in time, this effect manifests itself as a first-order removal term in a chemical species’ concentration rate equation. Consider, for example, a controller rate equation given by

$$\frac{dx}{dt}=k_{\text{pr}}^{\text{bas}}+f_{\text{pr}}^{\text{act}}\left(y\right)-k_{\text{re}}^{\text{bas}}-k_{\text{re}}^{\text{dil}}x,$$ (7)

where the general production function f_prod(y) has been split into its basal (constant) contribution, $k_{\text{pr}}^{\text{bas}}$, and the remaining y-activated contribution, $f_{\text{pr}}^{\text{act}}$ (y) (so that $f_{\text{pr}}^{\text{act}}$ (0) = 0), and where $k_{\text{re}}^{\text{bas}}$ represents zeroth-order removal and $k_{\text{re}}^{\text{dil}}x$ is the first-order removal term from dilution. In this situation, the rate constant $k_{\text{re}}^{\text{dil}}$ is the specific growth rate and can be calculated from the cell volume doubling time (often equal to the cell division time), τ₂, by $k_{\text{re}}^{\text{dil}}$ = ln 2/τ₂. This additional first-order term violates the requirement of zeroth-order kinetics in the controller, as previously discussed. Furthermore, because dilution is an intrinsic and global effect within the cell, it cannot be resolved by means of saturation.

If, however, the kinetics of the controller behave on a much faster timescale than cell growth, it is possible for integral control to be effectively preserved. This is the case for bacterial chemotaxis, where the complete adaptation network (including both the process and the controller) is based on phosphorylation and methylation signaling. The fact that these mechanisms operate on the order of seconds plays an important role in dilution being negligible. On the other hand, because transcriptional regulation operates on slower timescales (minutes), dilution poses a significant barrier to constructing transcription-based integral control networks in fast-growing cells.

To illustrate this timescale effect, we set the time derivative of Eq. 7 to zero and isolate for the steady-state y expression,

$$y_0=f_{\text{pr}}^{{\text{act}}^{-1}}\left(k_{\text{re}}^{\text{bas}}-k_{\text{pr}}^{\text{bas}}+k_{\text{re}}^{\text{dil}}{x}_{\text{ss}}\right).$$ (8)

To recover perfect adaptation to a constant setpoint, the first-order dilution rate, $k_{\text{re}}^{\text{dil}}{x}_{\text{ss}}$, must be negligibly small compared to the aggregate zeroth-order removal rate, $k_{\text{re}}^{\text{bas}}$ − $k_{\text{pr}}^{\text{bas}}$. (Note that $k_{\text{re}}^{\text{bas}}$ − $k_{\text{pr}}^{\text{bas}}$ > 0 in order for y₀ to be positive-valued; see Designing the Setpoint). This requirement further constrains x_ss such that

$$x_{\text{ss}}\ll \frac{k_{\text{re}}^{\text{bas}}-k_{\text{pr}}^{\text{bas}}}{k_{\text{re}}^{\text{dil}}}.$$ (9)

Consequently, the setpoint values to which the system can perfectly adapt are further restricted. In this case, the lower y₀ constraint, previously defined by Eq. 4a, must now account for the fact that x_ss must stay small enough to curtail the rate of dilution. Again, we reserve further treatment of this fact until the next section.

At this time, we direct the reader to Appendix A where we introduce a case-study system to help demonstrate our constraints and design methods. There, a mathematical model of the system's dynamics is described and the x_ss constraints for perfect adaptation are identified. Numerical results for the system will be presented in subsequent Appendices.

Although not addressed in this article, integral control may also be achieved without zeroth-order kinetics if the controller is of the form dx/dt = xⁿf(y) (as this will also yield a u-independent y_ss). To generate a controller of this form, the control action species’ production reaction must be autocatalytic with a reaction order (in terms of x) equal to that of its removal reaction, as discussed in Drengstig et al. (39).

Constraints for near-perfect adaptation

We have argued that preserving integration within the controller element may require constraining the value of the control action. In practice, one might approximate these constraints as

$$x_{\text{ss},\text{min}}\le x_{\text{ss}}\le x_{\text{ss},\text{max}},$$ (10)

where x_ss,min and x_ss,max are some user-defined tolerance values that allow expressions such as Eqs. 6 and 9 to be refit with standard inequalities. This acknowledges the reality that given controller constraints that can be satisfied only asymptotically, integral control must be approximated and robust adaptation to an output setpoint will be near-perfect at best; this is especially true if there are constraints that oppose one another, such as Eqs. 6 and 9. One might assign values for x_ss,min and x_ss,max by, for example, first establishing an error tolerance for y_ss: e.g., y₀ – TOL ≤ y_ss ≤ y₀ + TOL, where y₀ is found by presuming inequalities Eqs. 6 and 9 to be satisfied and evaluating the resulting ideal integral controller equation (dx/dt = f(y)) at steady state.

These x_ss and y_ss intervals may then be related by evaluating the true controller rate equation at steady state—i.e., dx/dt = f(x,y) = 0. Doing so yields the y-x dose-response curve for the controller element in open loop, $x_{\text{ss}}^{\text{C}}$ ≡ F(y_ss), also referred to as the controller nullcline. If y upregulates x, as in the y → x ⊣ y closed-loop feedback scheme, we expect $x_{\text{ss}}^{\text{C}}$ to be a monotonically increasing curve (restricted to nonnegative quantities for physicality). Considering $x_{\text{ss}}^{\text{C}}$ to be sigmoidal, its inverse, $y_{\text{ss}}^{\text{C}}$ ≡ F⁻¹(x_ss), has been drawn generically in Fig. 5; the intersection of this curve with y₀ – TOL and y₀ + TOL determines x_ss,min and x_ss,max, respectively. For a well-designed controller, the x_ss constraints should approximate integral control by constraining controller operation to the flatter portion of the $y_{\text{ss}}^{\text{C}}$ curve. Indeed, for an ideal controller, $y_{\text{ss}}^{\text{C}}$ = y₀, a flat line; therefore, the flatter this section is, the wider the x_ss,min and x_ss,max interval will be and the more robustly integration will be approximated.

Figure 5.

(Establishing x_ss constraints) The curve $y_{\text{ss}}^{\text{C}}$ represents a controller nullcline on the x_ss–y_ss plane. It is monotonically increasing because we are considering a y→x controller element; the inverse sigmoidal shape results from assumptions made about the general sigmoidal shape of dose-response curves. Integral control is approximated when the controller operates along the flatter section of the $y_{\text{ss}}^{\text{C}}$ curve; the flatter and wider this section, the more tightly the controller will maintain y_ss to y₀ (in the closed-loop setup) as x_ss changes. Here, x_ss,min and x_ss,max are chosen to keep y_ss to within the interval y₀ −TOL to y₀+TOL, the user-established error tolerance defining near-perfect adaptation to y₀. (Examining the tolerable input range) Process nullclines (dose-response curves) also illustrate how near-perfect adaptation restricts the range of tolerable input values, u. Because the intersection of process and controller nullclines indicates the steady state of the closed-loop system, we examine the effect that u has on this intersection. Taking the example that u→y and that increasing u affects the process nullcline by shifting it upward, we see that at u values larger than u_max or smaller than u_min, the process’s steady-state output, $y_{\text{ss}}^{\text{P}}$, cannot be driven to the setpoint (within tolerance) using a steady-state control action value, x_ss, that is acceptable for preserving approximate integral control.

For the network designer, how these constraints are used depends largely on the questions being asked. Up to this point, we have primarily been addressing the question: Can the controller force the process to maintain a given output setpoint over some given range of (static) input values?

This was previously addressed by Eq. 4 and Fig. 4 for an ideal integral controller. An extension for near-perfect adaptation can be found in the Supporting Material. Depending on the circumstances, however, it may be more useful to rephrase the question as: Over what range of (static) input values can the controller keep the steady-state process output near a given setpoint?

To answer this, we examine the process’ dose-response curve to find the u limits at which the curve becomes altered so that y₀ – TOL and y₀ + TOL are at the limits of attainability under acceptable controller action (see Fig. 5). These limits, u_min and u_max, bound the range of tolerable input values and can be defined implicitly by

$$G\left(u,x_{\text{ss}}\right){|}_{\begin{array}{l}{u}_{\text{min}}\hfill \\ x_{\text{ss},\text{min}}\hfill \end{array}}=y_0-\text{TOL}$$ (11a)

and

$$G\left(u,x_{\text{ss}}\right){|}_{\begin{array}{l}{u}_{\text{max}}\hfill \\ x_{\text{ss},\text{max}}\hfill \end{array}}=y_0+\text{TOL}$$ (11b)

and explicitly by inverting the dose-response function, G, with respect to u. In Appendix B, we reproduce the analysis in this section for the case-study model.

Designing the setpoint

In addition to guaranteed setpoint convergence, ideal integral control is advantageous from a design standpoint because modification of the setpoint requires modification of the controller alone. The need for redesign of the process element would arise only when changes to the system’s nonsteady-state behavior (its transient response to perturbation) that are beyond the influence of the controller are desired, or if, as we have seen, a desired setpoint cannot be reached given the process’s current dose-response profile. This convenience is transferrable to the practical case of a well-approximated integral controller, so long as the controller can be suitably modified (for setpoint alteration) without compromising the integrity of the approximation (by markedly diminishing the width of the x_ss,min and x_ss,max interval).

In Fig. 7A (see Appendix C), we illustrate the rate equation describing an ideal integral control element (dx/dt = f(y)) using the framework of the case-study model. The production rate of the control action species is a monotonically increasing sigmoid (for a system with y⊣x→y feedback, such a curve might be decreasing instead), while the species’ removal rate is constant. For this controller, the setpoint corresponds to the y value at which the production curve intersects the rate of removal; changing the setpoint requires changing the location of this intersection. If, for example, removal is an enzyme-catalyzed event (proteolysis, (de)phosphorylation, etc.) occurring at saturation, then the removal rate may be adjusted experimentally by tuning the level of the catalyzing enzyme (e.g., by modifying its gene expression rate) or by changing the removal mechanism altogether (e.g., the alternative proteolytic machinery alluded to earlier has also been used for the purpose of altering V_max (40,41)). Further discussion regarding experimental techniques for tuning response curves can be found in the Supporting Material.

Figure 7.

Controller design considerations. (A) Designing the setpoint. (B) Controller speed.

Regulation versus sensing: fast controllers with slow processes

Earlier, we showed that the unfavorable effects of dilution could be negated if the kinetics of the controller behaved on a timescale much faster than cell growth. This is a reality for controllers based on posttranslational modifications (protein-protein signaling), for example, but not for controllers based on slower processes such as transcriptional regulation. Furthermore, employing a faster controller does not necessarily affect the overall system setpoint (as we saw in the case-study example). Increasing the internal kinetics of the controller does, however, have consequences beyond the engineering difficulties associated with adapting posttranslational signaling mechanisms into synthetic networks. In particular, the speed at which a controller operates affects a system’s transient output response.

As sketched in Fig. 1, a step perturbation to the input signal of an integral controlled system will result in an initial output response followed by adaptation of the output signal back to the setpoint. Because a faster acting controller can adjust the control action more rapidly, the resulting initial output response will be comparatively less pronounced than that resulting from the use of a slower controller. We illustrate this effect for the case-study system in Fig. 7B (see Appendix C).

For a sensory-type system this change in behavior can be problematic, because a prominent transient response is useful for conveying information about changes to the environment. If, however, the goal of the controller is strictly to promote regulation (i.e., disturbance rejection and adherence to the setpoint), a minimal initial response is favorable. In the latter case, the designer would be wise to base the controller on signaling mechanisms that operated on a much faster timescale than the mechanisms governing the process.

Summary

Synthetic biology is moving in the direction of using more formal design processes to guide the development of new components and systems, as is commonplace in other forms of engineering; our aim, here, has been to offer a contribution toward the ongoing effort to put the engineering design of intracellular devices on a more rigorous footing. Integral feedback controllers, with their property of guaranteeing convergence to a desired setpoint value, are a key ingredient in the development of robust intracellular synthetic control mechanisms. Our treatment suggests a set of design constraints that can guide the development of integral feedback controllers; the constraints are of a general form, but can be realized in specific cases by referring to experimentally populated models. By viewing natural biological feedback systems as examples of control schemes, such considerations may also help us to understand their operation and functional complexities.

Appendix A: A Case Study Controller for Perfect Adaptation

We have devised a case-study model to demonstrate the application of the general constraints presented in the main text to a specific set of system equations. The model describes a system comprising three molecular species whose concentrations, denoted u, x, and y, constitute the input, output, and control action signals, respectively. The system is wired in a u→y→x⊣y negative feedback arrangement, consistent with Fig. 2. The process is described by

$$\frac{dy}{dt}=k_y^{\text{bas}}+k_y^{\text{act}}u\left(\frac{K_x^{n_x}}{K_x^{n_x}+x^{n_x}}\right)-V_y\left(\frac{y}{K_{\text{m}y}+y}\right)-k^{\text{dil}}y.$$ (12)

Here, the output species is produced at a nonzero basal rate, $k_y^{\text{bas}}$, in addition to a u-activated production rate, $k_y^{\text{act}}$u, that is susceptible to repression by the control action x. Repression, modeled using a nonlinear decreasing Hill function, suffers from leakage as it does not act on the basal term. Removal of the output species occurs via enzyme-catalyzed degradation following Michaelis-Menten kinetics where V_y is the maximal degradation rate, and via first-order dilution resulting from exponential cell growth where k^dil is the specific growth rate constant. We have incorporated the input signal, u, linearly, so that the effect of input perturbations to the input is not intrinsically dependent on its operating range. This will allow us to more easily discern the effectiveness of feedback regulation in our later analysis. The controller is described by

$$\frac{dx}{dt}=k_x^{\text{bas}}+k_x^{\text{act}}\left(\frac{y^{n_y}}{K_y^{n_y}+x^{n_y}}\right)-V_x\left(\frac{x}{K_{\text{m}x}+x}\right)-k^{\text{dil}}x,$$ (13)

where production of the control action species is activated by y and $k_y^{\text{act}}$ represents the additional production rate at maximum activation. Activation is modeled using an increasing Hill function.

In its general form, Eq. 13 does not represent an integral feedback controller due to the x-dependence in its right-hand side. Integral control may be recovered, however, if x-dependence can be made negligible. This requires constraining the controller’s operating regime, specifically the operating regime of x at steady state.

First, enzyme-catalyzed degradation must occur at saturation, meaning that x_ss ≫ K_mx. Second, the first-order dilution term must be dominated by the aggregate zeroth-order contribution. Assuming degradation occurs at saturation (and is therefore zeroth-order), this implies that k^dilx_ss ≪ V_x – $k_x^{\text{bas}}$. Taken together, these requirements constrain x_ss to the interval

$$K_{\text{m}x}\ll x_{\text{ss}}\ll \left(V_x-k_x^{\text{bas}}\right)/k^{\text{dil}}.$$ (14)

Therefore, the process must be able to reach the setpoint under an x_ss value within this interval for the system at large to exhibit perfect adaptation. We also note, however, that the controller can fail even before any process is considered. Clearly, if

$$K_{\text{m}x}\ll \left(V_x-k_x^{\text{bas}}\right)/k^{\text{dil}}$$ (15)

is not true, then Eq. 14 is impossible and integral control is compromised. If this is not the case, then the controller can still fail outright if

$$k_x^{\text{bas}}<V_x<k_x^{\text{bas}}+k_x^{\text{act}}$$ (16)

is not true. Such a circumstance leads to two equivalent consequences: (1) The control action cannot be driven between a state of increase (dx/dt > 0) and a state of decrease (dx/dt < 0) by the system output, y, rendering the controller effectively nonfunctional; and (2) the controller setpoint, given by the expression

$$y_0=K_y{\left(\frac{V_x-k_x^{\text{bas}}}{k_x^{\text{bas}}+k_x^{\text{act}}-V_x}\right)}^{\frac{1}{n_y}},$$ (17)

(derived from Eq. 13 by assuming saturated degradation and negligible dilution of x), will not be both positive and real. Verifying these inequalities (Eqs. 15 and 16) can serve as a preliminary check that perfect adaptation is potentially realizable with the controller at hand.

Appendix B: Near-Perfect Adaptation in the Case Study System

In reality, biological integral control requiring zeroth-order kinetics will likely need to be approximated, with the aim of inducing near-perfect adaptation. For the case-study system, as in the main text, we establish a tolerance interval for y_ss (y₀ – TOL ≤ y_ss ≤ y₀ + TOL) centered about the ideal setpoint given in Eq. 17, then use the location at which the controller nullcline intersects this interval to establish x_ss constraints (x_ss,min ≤ x_ss ≤ x_ss,max) that replace Eq. 14. The controller nullcline is

$$y_{\text{ss}}^{\text{C}}=K_y{\left(\frac{-k_x^{\text{bas}}+V_x\left(\frac{x}{K_{\text{m}x}+x}\right)+k^{\text{dil}}x}{k_x^{\text{bas}}+k_x^{\text{act}}-V_x\left(\frac{x}{K_{\text{m}x}+x}\right)-k^{\text{dil}}x}\right)}^{\frac{1}{n_y}}.$$ (18)

We then use the process nullcline (dose-response function) to determine the range of u values, u_min to u_max, under which y will settle to within the established y_ss tolerance interval. The process nullcline is

$$y_{\text{ss}}^{\text{P}}\left(u,x_{\text{ss}}\right)=\left(p-V_y-k^{\text{dil}}{K}_{\text{m}y}\right)/2k^{\text{dil}}+{\left({\left(p-V_y\right)}^2+2k^{\text{dil}}{K}_{\text{m}y}\left(p+V_y\right)+k^{{\text{dil}}^2}{K}_{\text{m}y}^2\right)}^{\frac{1}{2}}/2k^{\text{dil}},$$ (19)

where p represents the rate of production:

$$p\left(u,x_{\text{ss}}\right)\equiv k_y^{\text{bas}}+k_y^{\text{act}}\left(\frac{u^{n_u}}{K_u^{n_u}+u^{n_u}}\right)\left(\frac{K_x^{n_x}}{K_x^{n_x}+x_{\text{ss}}^{n_x}}\right).$$ (20)

For illustrative purposes, we begin by populating the model with the following example parameters (all concentrations in nanomolar and rates in nanomolar/min): $k_y^{\text{bas}}$ = $k_x^{\text{bas}}$ = 10, $k_y^{\text{act}}$ = 0.1, $k_x^{\text{act}}$ = 1000 (i.e., a 100-fold activation range), V_y = V_x = 100; k^dil = 0.01 (i.e., a 70-min cell doubling time); K_u = K_x = K_y = K_my = K_mx = 1000; and n_u = n_x = n_y = 1.5. These values are of orders representative of a transcriptional regulatory gene expression network. We also set TOL to 5% of the ideal setpoint, y₀ = 214, making the y_ss tolerance interval 203 ≤ y_ss ≤ 225. Computationally generated curves for the model are presented for these parameters.

Process and controller nullclines

The ideal setpoint is shown in Fig. 6A (long dashed line) along with the y_ss tolerance interval (shaded band). The intersections of the controller nullcline (increasing curve) with the boundaries of the tolerance interval indicate that x_ss,min = 2363 and x_ss,max = 3073: a relative increase in x_ss of Δx_ss = 30%. Process nullclines (decreasing curves) that intersect (x_ss,min, y₀ − TOL) and (x_ss,max, y₀ + TOL) indicate that u_min = 413 and u_max = 676, respectively: a Δu of 64%.

Figure 6.

Near-perfect adaption in the case study system. (A) Process and controller nulllines. (B) Modifying parameters to extend the tolerable input range. (C) Temporal response curves.

Modifying parameters to extend the tolerable input range

In Fig. 6B, we invoke two adjustments to our model to increase Δx_ss and Δu. The x removal via enzyme-catalyzed degradation is responsible for the constraint that excludes low values of x_ss because it governs the shape of the controller nullcline at low x_ss. We increase the binding affinity between the control action species and its degrading enzyme by lowering K_mx from 1000 to 100, thus implying enzyme saturation at lower x. This has the effect of decreasing x_ss,min (and, to a lesser degree, x_ss,max). The x removal by dilution, on the other hand, is responsible for the constraint that excludes high values of x_ss because at high x_ss (where enzyme-catalyzed degradation has saturated) it dictates the increasing shape of the controller nullcline. We mimic a faster-operating controller by raising the values of $k_x^{\text{bas}}$, $k_x^{\text{act}}$, and V_x by a factor of γ = 10. This decreases the importance of x dilution in the controller nullcline (Eq. 18) and has the effect of increasing x_ss,max (and, to a lesser degree, x_ss,min). Combined, these adjustments decrease x_ss,min to 1258 and increase x_ss,max to 7493, therefore increasing Δx to 496%. As a result, u_min = 215, u_max = 2277, and Δu increases to 957%. (Neither of the two adjustments affect the ideal setpoint.)

Temporal response curves

The system, initially at steady state, is subject to a step input perturbation, u = 215 to 2277, at t = 0. The dark curve in Fig. 6C shows the output signal for the adjusted system (K_mx = 100, γ = 10). The pre- and post-perturbation steady states are at the lower and upper boundaries of the y_ss tolerance interval, respectively, in agreement with Fig. 6B. Therefore, for either input value, the system is near-perfectly adapting to the ideal setpoint (marked with the dashed line). The light curve shows the output response for the original system (K_mx = 1000, γ = 1) for which the pre- and post-perturbation steady states fall outside the y_ss tolerance interval.

Appendix C: Other Design Considerations for the Case Study Controller

Here, we focus on ideal (as opposed to approximated) integral control. To generate curves using the original (nonideal) case-study system, we remove all x dependency in the controller rate equation by setting K_mx and k^dil (in Eq. 13 only) to 0. All other parameters remain as originally stated in Appendix B (i.e., γ = 1, unless otherwise stated).

Designing the setpoint

Fig. 7A demonstrates that with integral control, the setpoint is dictated entirely by the form of the controller. It is determined by the controller’s steady state and a designer can modify it by adjusting where the control action species’ rate of production curve intersects its rate of removal.

Controller speed

Faster working control reduces the amplitude of the initial preadaptation output response for a system under integral control. In Fig. 7B, whole-controller kinetics are scaled by multiplying $k_x^{\text{bas}}$, $k_x^{\text{act}}$, and V_x by the factor γ. Response curves are shown for γ = 0.01 (slowest; lightest curve), 0.1, 1, and 10 (fastest; darkest curve), where u is step-perturbed at t = 0 from u = 215 to u = 2277.