Ultrasensitive Negative Feedback Control: A Natural Approach for the Design of Synthetic Controllers

Many of the most important potential applications of Synthetic Biology will require the ability to design and implement high performance feedback control systems that can accurately regulate the dynamics of multiple molecular species within the cell. Here, we argue that the use of design strategies based on combining ultrasensitive response dynamics with negative feedback represents a natural approach to this problem that fully exploits the strongly nonlinear nature of cellular information processing. We propose that such feedback mechanisms can explain the adaptive responses observed in one of the most widely studied biomolecular feedback systems—the yeast osmoregulatory response network. Based on our analysis of such system, we identify strong links with a well-known branch of mathematical systems theory from the field of Control Engineering, known as Sliding Mode Control. These insights allow us to develop design guidelines that can inform the construction of feedback controllers for synthetic biological systems.

Let us assume the closed-loop feedback control system of Fig. 5A of the main text, where the controller modifies the state of the process to obtain the desired output response. The input of the controller is the error signal, e, defined as the difference between the desired output, r (called the reference signal), and the actual output of the system, y. Based on the error signal, the controller manipulates the input to the process, u, to reduce the effect of the disturbance, u d , on the output y and therefore obtain the desired response.
To simplify the analysis, the process is a first order linear system described with the following differential equation modelv where Now, the output of the closed loop system is given by where u d (t), the additive disturbance on the output of the process, is a step signal at time t d (u d (t) = 0 for t < t d and u d (t) = a u d for t ≥ t d , with a u d the disturbance amplitude).

Proportional control (P N F )
Let us assume that the controller C produces an output value, u, proportional to the current value of the error e u(t) = k p e(t) = k p (r(t) − y(t)) , where k p is a constant called the gain of the proportional controller.
The dynamics of the closed loop system (taking into account Eqs. (A1) and (A2)) are then described by the following differential equation: To solve this differential equation the Laplace operator is used, since it allows differential equations to be transformed into algebraic equations that are much easier to solve (see (1); in particular note that the derivative operator with respect to time corresponds to a multiplication by s in the s-domain, where s is the complex Laplace variable). Taking the Laplace transform of Eq. (A3) and assuming a zero initial condition gives where R(s), Y (s) and U d (s) are the Laplace transforms of the reference, output and disturbance signals, respectively.
By applying the superposition principle (for a linear system the output response to two or more inputs is the sum of the responses which would have been caused by each input individually: see (1)), the transfer function W yr , defined as the ratio of the Laplace transform of the output y and the desired output r with u d (t) = 0, is given as and the transfer function W yu d , defined as the ratio of the Laplace transform of the output y and the disturbance u d with r(t) = 0, is given as Assuming that the desired signal is a constant, r(t) = r, and the disturbance is a step signal at time where y ss is the steady state value of the output, the input-output steady states are related via Note that the last term of Eq. (A8) is obtained by dividing numerator and denominator by a. Therefore, if Gk p >> 1, the output y is able to track the desired reference signal r (i.e. y ss ∼ r) and the controller is able to attenuate the effects of the disturbance u d on y. In terms of the error, Eq. (A8) can be rewritten as The dynamics of the closed-  As expected from Eqs. (A8)-(A10), increasing the gain value k p allows the system to attenuate the effects of the disturbance and achieve a faster response.

Integral control with infinite (I N F ) vs. finite integration period (F I N F )
In this section we consider the closed loop system of Fig. 5A of the main text with a constant reference signal r, a step disturbance u d of amplitude a u d and a controller implementing an integrator with an infinite or finite integration period.
It is well known that integral control with infinite integration period (I N F ) can be used to completely attenuate the effects of the disturbance u d on the desired output y of the closed loop system of Fig. 5A of the main text: in fact, y is able to perfectly track any step reference signal, r, for any step disturbance, u d (1). In this case the system is able to achieve an adaptive response. In general, in order to achieve an adaptive response to a reference signal corresponding to the n-th integral of a step function, where n is a positive integer, the feedback loop has to contain at least n + 1 integrators connected in series (2). Adaptive responses to step signals, where n = 0, require at least one integrator.
For such a system, the controller C produces an action which takes into account completely the history of the process with an output value, u, proportional to the integral of the error where k i is the gain of the integrator. The response of the closed loop system is obtained by solving the following integro-differential equation: Taking the Laplace transform of Eq. (A11) and noting that the integral operator with respect to time corresponds to division by s in the s-domain (1) gives where R(s), Y (s), U d (s) are the Laplace transforms of the reference, output, input and disturbance signals, respectively. It follows that the transfer functions, W yr (s) and W yu d (s), are given by By applying the final-value theorem, the input-output and error steady-states are related via e ss =r − y ss = 0 .
Therefore the steady state output y ss is equal to r for any value of the gain k i . In other words, the integral feedback control allows the system to perfectly return to its steady state value (i.e. achieve perfect adaptation) following any value of perturbation.
The gain k i determines the time dynamics of the system, i.e. the time taken for the system to reach steady state following a perturbation. In particular, the poles of the two transfer functions W yr (s) and W yu d (s) (the values of s that nullify the denominator of Eqs. (A12) and (A13)) determine the evolution of the system. The denominator of the two functions, s 2 + as + bk i , can be rewritten in the standard form where ω n = b a ak i = Gk i /τ is called the natural frequency of the system and ζ = is called the damping factor (1). The natural frequency gives information on the speed of the system response; the damping factor combines the initial speed of the response and its accuracy defined in terms of overshoot and settling time. For ζ < 1 the system is underdamped (presence of overshoot in the response) and as ζ decreases the system exhibits a faster response, but more oscillatory behaviour. So far we have assumed a controller implementing I N F , an ideal integrator that is able to store the full history of the process. We now consider a form of integral feedback that implements a finite window for error integration (F I N F ). For F I N F , the controller output signal u is given by where T m is the time window of the integral. The following integro-differential equation then describes the closed loop system: Taking the Laplace transform of Eq. (A17) and remembering that time-shifting corresponds to expo-nential multiplication in the Laplace domain gives It follows that the transfer functions, W yr (s) and W yu d (s), are given by .
By applying the final-value theorem, the input-output and error steady-states are related via (Note that as s → 0, the term e −sTm can be approximated by the first two terms of its Taylor series

Ultrasensitive control (U N F ) as a quasi sliding mode control
Ultrasensitivity describes a particular form of sensitivity in biological systems, where the system does not respond to incoming signals outside of a certain regime, but responds in a highly sensitive manner within this regime. Ultrasensitivity is characterised by a sigmoidal input-output relationship and is shown to be a ubiquitous feature in several biological systems. Ultrasensitivity can be biochemically implemented through a variety of mechanisms such as phosphorylation cycles and cooperative binding (see (3,4)) and can be described by a Hill-type function. It has been shown previously that ultrasensitivity, when implemented in a negative feedback loop, can lead to adaptive response dynamics (5).
Biochemical reactions that can implement ultrasensitivity in biological systems can take part in particular signalling and response pathways. To represent these systems using the control theory framework developed above, we imagine an ultrasensitive controller (U N F ). In this controller, the error is processed through an ultrasensitive (sigmoidal dose-response) dynamics, before being fed into the process. Thus, the input from the controller back to the process can be represented by (A22) Note that we use the sign and absolute value of the error to allow the controller to work in a symmetrical way for positive and negative values of the error.
The dynamics of the closed loop system are therefore described by the following differential equation:ẏ (t) = −a(y(t) − u d (t)) + sgn(e(t))bk p · |e(t)| n |e(t)| n + K n +u d (t)) . (A23) Rewriting Eq. (A23) in terms of e and assuming a constant reference signal, r(t) = r (note thaṫ The steady state error, e ss , is the solution of the equationė(t) = 0 which implies a(r − e ss − a u d ) = sgn(e ss )bk p · |e ss | n |e ss | n + K n . (A25) The solutions of Eq. (A25) can be visualised by plotting both sides of the equation, as shown in Fig. 5B of the main text: the intersections of the straight line a(r − e − a u d ) with sgn(e)bk p · |e| n |e| n +K n correspond to the steady states of the system. We show that U N F can achieve a steady-state error value very close to zero for any value of n, given a small value of K and without increasing the gain k p (for more details see the Results section of the main text).
From the control point of view, the ultrasensitive controller defined by Eq. (A22) is an approximation of a well-known class of controllers, based on sliding mode control (SMC), a nonlinear technique for robust control (see (6)(7)(8)). Indeed if K goes to zero, Eq. (A22) assumes the following formula (see also Fig. 6D of the main text): u(t) = k p sgn(e(t)) .  (1,7,8) to the σ-dynamics, defined aṡ For the σ-dynamics, a candidate Lyapunov function is In order for σ = 0 to be asymptotically stable, the following conditions need to be satisfied: The condition (A29) is satisfied by Eq. (A28). The derivative of Eq. (A28) is computed aṡ To attain the sliding manifold,V < 0, σ andσ must always have opposite signs. Assuming that u d and e(0) are bounded as below |u d (t)| < M and |e(0)| = |r − y(0)| < L , the product σσ will be negative if the following inequality is satisfied: Therefore, the control gain k p of the SMC controller is computed taking into account the relation (A33); the first term, a b M , is designed to compensate for the bounded disturbance u d , while the second term a b L is responsible for determining the reaching phase, when the trajectory is forced towards the sliding manifold.
Note that, if the controller can also make an action proportional to the output, then the inequality (A33) for the control gain k p is only determined by the disturbance u d , i.e. k p ≥ a b M . In particular, if the input at the process given by the controller is (A35) Inequality (A33) thus becomes k p ≥ a b M ; the term u eq = ay/b is called equivalent control and is chosen to cancel the known term, ay, on the right-hand side of Eq. (A27). In the absence of the disturbance, taking u = u eq leads toė = 0, allowing the sliding manifold σ = e = 0 to be maintained for all future time.
As shown in Fig 6 of the main text, the output response of the closed loop system with a SMC exhibits a zigzag motion of small amplitude and high frequency in the sliding mode (see the plot in the box of Fig. 6A). SMC is a high frequency switching control with a switching frequency inversely proportional to the time increment used in the simulation. Increasing the accuracy of the solution computed by the ODE solver reduces the zigzag motion, but due to the discrete-time nature of the computer simulation the output response continues to exhibit this effect, which is called chattering.
For an ideal SMC, the switching frequency goes to infinity and the amplitude of the zigzag motion goes to zero.
In addition to the practical issue of chattering, theoretical issues like the existence and uniqueness of solutions and validity of the Lyapunov analysis have to be considered due to the discontinuous nonlinearity sgn(e) in the ideal SMC (see Eq. (A26)). These issues can be avoid by using continuous/smooth approximations of the discontinuous SMC and the U N F controller is an example of a smooth control function, used to approximate the nonlinearity sgn(e). The designed U N F control is technically not a sliding mode control and there is no ideal sliding mode in the closed-loop system (A23), since the sliding variable cannot be driven to zero in a finite time. However, for small values of K, the results of U N F are close to those achieved by an ideal SMC (see Fig S8C). As shown in the main text, Eq. (A22) can be approximated by the following saturation nonlinearity with high where m is the slope for the linear regime.
For n = 2, the piecewise approximation u pw (right-hand side of Eq. (A38)) is equal to the saturation function with m = n/(4K) = 1/(2K). In this case, the U N F controller is thus well-approximated by its saturation function: u = sgn(e) · |e| 2 |e| 2 + K 2 ≈ u pw = sat(e/(2K)) =        e/(2K) , |e| ≤ 2K, sgn(e/(2K)) , |e| > 2K. (A39) As shown in Fig 6C of the main text, the U N F controller and its approximated saturation function are very similar and the effect of the disturbance on the output becomes negligible by decreasing the K value of the U N F (i.e increasing the slope m = 1/(2K) of the saturation function for n = 2) and the error lies inside the boundary layer |e| < 1/m. Therefore, the ideal discontinuous SMC can be approximated by a smooth/continuous controller, such that the error is not confined to the manifold e = 0, but lies inside the boundary layer (7), that for the case of the U N F controller, as reported in the main text, is given by In the limit K → 0 (m → ∞), the U N F controller (the saturation function sat(ne/(4K))) approaches the signum nonlinearity sgn(e) of the ideal SMC. This, finally, allows us to say that the U N F controller of Eq. (A22) is an example of quasi sliding mode control (8).