In-Vivo Real-Time Control of Protein Expression from Endogenous and Synthetic Gene Networks

We describe an innovative experimental and computational approach to control the expression of a protein in a population of yeast cells. We designed a simple control algorithm to automatically regulate the administration of inducer molecules to the cells by comparing the actual protein expression level in the cell population with the desired expression level. We then built an automated platform based on a microfluidic device, a time-lapse microscopy apparatus, and a set of motorized syringes, all controlled by a computer. We tested the platform to force yeast cells to express a desired fixed, or time-varying, amount of a reporter protein over thousands of minutes. The computer automatically switched the type of sugar administered to the cells, its concentration and its duration, according to the control algorithm. Our approach can be used to control expression of any protein, fused to a fluorescent reporter, provided that an external molecule known to (indirectly) affect its promoter activity is available.

IRMA was developed as a testbed synthetic network in yeast for the design and validation of reverse engineering and modelling approaches [1].
IRMA consists of ve genes, CBF1, ASH1, SWI5, GAL4 and GAL80 and its topology comprises both transcriptional and protein-protein regulation mechanisms ( [1]). Figure 2 in the main text shows a schematic diagram of the regulatory interactions among the ve genes. The network topology comprises a positive feedback loop from CBF1 back to itself via GAL4 and SWI5, and a negative feedback loop from CBF1 back to itself via ASH1. A further regulation is present between SWI5 and GAL80 via the GAL10 promoter bound by GAL4. Transcription of network genes can be controlled by the presence of Galactose (GAL) in the growth medium, whose presence inhibits transcription of SWI5 from the GAL10 promoter.
To capture the dynamics of the network a hybrid model (Supplementary Figure S1) approximating the dynamics in Glucose (F 1 ) and Galactose (F 2 ), has been readapted from [1]. Both the vector elds F 1 and F 2 share the same model structure as well as most of the parameters (v 3 ,k 4 andγ need a specic argumentation) as reported below: where ], x 4 = [GAL80], x 5 = [ASH1] are the system states. We used Hill functions to model transcription rates from promoters; the multiple regulation on CBF1 is modelled by the product of two Hill functions (AND regulation). A time delay τ is present in the equation for x 1 modelling the transcription of CBF1, which is aected by a 100 minute-long time delay due to the sequential recruitment of chromatin-modifying complexes to the HO promoter (which follows binding of SWI5 and other transcription factors) [2]. A list of all model parameters can be found in Supplementary Table S1 in [1]. Note that the model is hybrid as parametersv 3 ,k 4 andγ switch between two dierent sets of values depending on the carbon source (Galactose or Glucose).
Control algorithm: design and implementation GAL1 promoter In this system, the GAL1 promoter drives the expression of the Gal1-Gfp fusion protein. Thus it can be described as a single input-single output (SISO) dynamical system. The input u(t) describes the presence of galactose or glucose in the growth medium. The output y(t) is the measured average level of uorescence of the Gal1-Gfp protein in the cell population.
The control objective is a set-point regulation, where the cell population is required to express, over several generations, a xed amount of uorescence (control reference r(t)). To this end, we designed a simple control algorithm based on a Proportional-Integral regulator (PI) whose outputû(t) is a function of the control error e(t) (the mismatch between the desired and current output of the system e(t) = r(t) − y(t)) dened as:û (t) = K p · e(t) + K I · t 0 e(t)dt (6) Since cells can consume either galactose or glucose in a mutually exclusive manner, the continuous signalû(t) has to be decoded into a discrete signal denoting either galactose (u(t) = 2) or glucose (u(t) = 0).
The above constraint allows an analogy with the problems faced in the design of feedback control strategies for power electronic circuits [3]. Here, switches and SCRs (silicon controlled rectiers) can only be turned on or o, some output is typically measured or estimated and, particularly in industrial applications, compensating noise and external disturbances is of utmost importance.
The simplest and most widely used control technique in this context is to use the PI regulator coupled to a PWM (Pulse Width Modulation) control strategy with a classic limited integral anti-windup [4]. This is also the strategy we adopted to control the cell population. In the simplest feedback implementation of the PI-PWM, a sawtooth signal is compared withû(t) in order to modulate the width of a rectangular pulse train, which is then used as control input (see [5] for further details). Namely, let be the sawtooth signal; then To control the GAL1 promoter we chose the sawtooth wave parameters as follows: α = 0, β = 2 and T = 5min. The gains of the PI controller, namely K p = 6 and K i = 0.3, were tuned following the Cohen-Coon strategy ( [6]) using a linear transfer function describing the GAL1 promoter previously derived in [7]: with parameters µ = 0.28, d = 68 and Θ = 134.
IRMA network IRMA can be modeled as an input-output system where the input u models the presence/absence of Galactose and the output y is the concentration of one of its genes, namely Cbf1 (x 1 ). Note that the input acts nonlinearly on the dynamics of the network as the presence of Galactose changes the values of all the Galactose-dependent parameters (namelyv 3 ,k 4 andγ). Moreover, as soon as Glucose is administered to the cells, these stop responding to Galactose, even if it is still present in the medium. Therefore, the control input (interpreted as Galactose concentration of 2 w/v% in the total volume of uid reaching yeasts) is restricted to be either ON (u = 1) or OFF (u = 0). The system output y = x 1 cannot be measured directly as a concentration. Instead, the cells were engineered so that CBF1p is fused with a GFP, the green uorescent protein [1]. In this way, higher concentrations of Cbf1p are associated to higher levels of uorescence. Moreover, as outlined in [8], the amount of estimated uorescence can be directly related to the actual number of molecules. From a control perspective, the gene network model is, therefore, a highly nonlinear, hybrid, time-delayed dynamical system of the form: Hybrid systems are often used to model gene networks (e.g. see [911]), where it is quite common to observe threshold dependent and switch-like activation or inhibition functions governing the dynamics of protein-protein or protein-gene interactions.
From a control viewpoint, the task is to regulate the expression of CBF1 to some desired value, say r(t) by modulating the control input u(t) (the carbon source being administered to the cell). In what follows, we will ignore phenomena like Zeno behaviors (which would imply the gene network to switch on and o an unbounded number times in a nite and bounded length of time). To solve this problem, the challenge is to design an eective, yet simple, control strategy able to cope with the many unavoidable constraints, which characterize biological systems (i.e. high levels of noise, incomplete knowledge and unmodelled dynamics). Cells are living organisms and they resist external actions very eectively.
Moreover, it is important to minimize the need of knowing in great detail the biological process to be controlled, in order to make the control strategy as general as possible and applicable to any process of interest.
The main design constraint that needs to be taken into account when synthesizing a controller for a biological system such as IRMA is that the cells tend to consume Glucose as their primary carbon source, even when galactose is present, since they obtain energy from it at a lesser energetical cost [12]. Moreover GLU-genes repress GAL genes as outlined in [13]; therefore our control input is binary: either galactose or glucose, but not both.
As in the case of the GAL1 promoter, we used the PI-PWM control strategy for the set-point control task. The sawtooth wave parameters for the PWM were set to α = 0, β = 10E − 5 and T = 5 min. In this way, the amplitude of the signal is 10% of the Cbf1 level at steady state in Glucose and the period is twice the settling time of Cbf1. With this choice, the constraint is satised of the control input being binary.
In the case of the signal tracking control task, in order to compensate for the estimated delay of 100min in the CBF1 gene transcription, a strategy was implemented, which is inspired by the classical Smith Predictor scheme in [14], as shown in Supplementary Figure S2. The delay-free version of IRMA's model (block M in Supplementary Figure S2) is used to produce a numerical estimate of the anticipated system outputŷ. This is then delayed and subtracted from the quantied uorescence output, y, of the network so as to produce a signal which is ltered via a linear lter F and added back to y. The lter was designed empirically to suppress high frequency noise entering the control loop and is characterized y: The output y s of the delay compensation scheme is then compared to the desired output (y ref ) to obtain the error signal e. A Proportional-Integral controller takes e in input and computes the control signalû(t) with the gains K p = 175.6 and K I = 2.11. These gains were found with the Cohen-Coon method, as previously described for the Gal1 promoter; in the case of IRMA the parameters of the transfer function in eq. (8) were found to be µ = 0.0467, d = 146.85 and Θ = 667.62. A comparison between the step response of the linear approximation with the non linear model is shown in Supplementary Figure  S3. The computed analog control signalû is then converted into binary values by means of the PWM modulator described above that feeds both the physical plant and the model used for delay compensation with the same binary input u, thus closing the feedback loop.

Control algorithm implementation
The control strategy described in the previous section was implemented as a Finite State Automaton in MATLAB (Mathworks Inc.) for the set-point regulation of both GAL1 promoter and the IRMA network.
The Finite State Automaton implementing the PI-PWM control algorithm is very simple: at each step (k) an image is acquired by the microscope, and the normalized uorescence signal is computed thanks to an image segmentation algorithm described in the next section. The uorescence signal y s (k) is compared against the reference signal y ref (k), to obtain the error e(k). The control input u(k) is then computed using the discrete-time implementation of the PI controller discussed in [4].The control input u(k) is used to determine the duration of the pulse of Glucose or Galactose by means of the PWM strategy. The duration of each pulse corresponds to the time interval during which the syringe loaded with Galactose remains higher than the one containing Glucose (or vice-versa). At the next instant (k + 1) a new image is acquired and the feedback computation takes place. The error e(k + 1) is available for a new control iteration and each step is repeated again.
In the case of the signal tracking control task for the IRMA network, the Finite State Automaton implementing the control algorithm is more complex, since in this case there is the need of simulating the IRMA model M in the Predictor block, necessary for the delay compensation. As represented in Supplementary Figure S4, at the k-th step, e(k) is known, thus it is possible to calculate the control input u(k) to be applied to both the plant and the IRMA model M via the PI controller [4]. The control input is then used to compute the duration of the pulse of Glucose or Galactose by means of the PWM strategy.
In-silico analysis of the control algorithm for the IRMA network Given the complexity of the IRMA model, prior to the experimental implementation of the control strategy in -vivo, we decided to test and validate the performance of the control scheme in -silico.
In the numerical implementation, several alternative design options were evaluated. For example, it was found that using a nonlinear model for delay compensation (in the block labelled as 'M' in Supplementary Figure S2) gives a comparable performance to that obtained when using a linearized one (as suggested in [14]).
In addition, we performed numerical simulations of the control scheme against uncertainties in the delay estimation by considering the scenario where the delay τ is set to zero. The results are presented in Supplementary Figure S5 and Supplementary Figure S6 showing the ability of the control scheme to guarantee a reasonable performance also in this case.
Finally, we explored the eects of simplifying the control strategy further by removing the prediction scheme used to compensate the delay. Although a simpler control strategy may decrease the overall performance, the reason for taking this step is that we prefer to sacrice precision in favour of a control strategy which can easily be used to control other gene networks and proteins, without requiring a dynamical model of the system to be controlled.
In-silico results reported in Supplementary Figure S7 and Supplementary Figure S8 show the performance of such a simple PI/PWM control strategy when implemented to control gene expression levels in IRMA. The in silico experiments conrm that this strategy force eectively the Cbf1 concentration to track the desired reference signal when the delay τ = 0.

Microscopy and image analysis
The closed-loop control platform described above, employs an inverted epiuorescence Nikon-TI Eclipse microscope. The microscope is programmed to acquire two types of images: (a) a phase contrast image (phase-contrast) and (b) two uorescence images (one for the green spectrum for GFP and one in the red spectrum for Sulforhodamine B). The red dye Sulforhodamine B is added to the galactose medium and it is used to check for the proper administration of the control input. Once cells have been imaged, image analysis methods can be applied to estimate their uorescence [15]. To this end, we developed a custom image processing algorithm to fully exploit the peculiarities of our platform. [16] Our aim was to carry out both cell segmentation and cell-tracking, maximizing sensitivity as primary objective, and then rening the results by improving specicity. We based our segmentation method on global edge linking system. Yeast cells in phase contrast images occur in clustered, low intensity, convex and often quasi-circular shapes surrounded by a white halo (Supplementary Figure S9 C).
The contrast between the pixels belonging to the cells and the pixels belonging to the halos is usually so high that edge points can be detected by the evaluation of the magnitude of the gradient calculated in each point of the image (Supplementary Figure S9).
Due to the shape of yeast cell, edge points can be connected with the Circular Hough Transform (CHT) [17]. CHT can detect almost all cells within the image, even when cells edges overlap; unfortunately the CHT algorithm is computationally expensive (both in memory and time) and shows limited specicity capabilities. Therefore, in order to limit the computational cost associated to the algorithm, we introduce a preprocessing stage of the image to select the regions in the image where cells are located ( Supplementary  Figure S9 B). Such regions are identied via a mask obtained through morphological operations (aperture, closure and lling) and thresholding (see Supplementary Figure S9 A). False detections are reduced as a result of the segmentation of these regions. Moreover, since the area of the regions in the image containing cells is smaller than the area of the background, computational time is considerably reduced.
Once cells have been located in the phase contrast image, a binary lter is used to detect only pixels of the GFP uorescence eld within cells. Let us dene the uorescence eld image I as: then with x and y generic coordinates and L the number of bits used for image encoding and ω the set of intensity values the pixel in the image can assume. The mask imageM can be similarly dened as: whereM (x, y) = 1 denotes a cell belonging pixel whileM (x, y) = 0 indicates background pixels. The latter class of pixels is useful to estimate the amplitude of the background signal, which can be subtracted the raw signal to obtain a normalised uorescence intensity. In order to compute the normalised signal, we use the following equation: background signal (13) with i and j spanning the rows and columns, respectively, of the arrays. ¬M (x, y) is a transformation ofM that is simply meant to complement the binary values of the original matrix (so as to select image areas not belonging to cells). The quantity GF P avg is the quantied uorescence output y used by the control algorithm to dene the control input to the cells.
In order to evaluate single cell uorescence for each frame acquired during the experiment, an o-line analysis (not during the control) is performed by using the same principles of the algorithm described above; a maskM is built for each cell in a frame, it allows to calculate the uorescence of the selected cell only (also the background value is subtracted). This analysis is useful to compute uorescence histograms for each frame (see Supplementary Movie 1 and 2) and to calculate the standard deviation and the coecient of variability of the output for each in -vivo experiments performed (see Supplementary Figures S10, S11, S12, S13 and S14 -S15-S16-S19-S20-S22-S23).
Quantitative analysis of the experimental results for the IRMA network The experimental results presented in the main text can be further analyzed by considering the internal signals involved in the computation of the control action. Supplementary Figure S21 presents these data for the experiment presented in Fig. 8 (main text). The close matching between the output signal y s and the desired output behaviour y ref is reected in the magnitude and shape of the error signal e. It is interesting to observe that, in the rst 100 minutes, the system is reacting without any delay to the frequently switching input provided to it. We argue that this phenomenon can be explained by considering the molecular mechanisms giving rise to the time delay τ : as outlined in [2], the delay is due to the chromatin remodelling step needed to initiate the transcription under the control of the HO promoter. Therefore, we hypothesise that fast switching between Galactose and Glucose attenuates the eect of the time delay by inhibiting a complete remodeling of the chromatin.
For the in-vivo signal tracking control experiment reported in Figure 8 of the main text and in Supplementary Figure S19, we computed the control error as the dierence between the average uorescence of the cell population (output) and the control reference, and reported its mean µ, variance σ, and, coecient of variation CV = µ σ in Supplementary Figure S21. It can be appreciated that the control error is much smaller than the uorescence signal, demonstrating that the control action is able to keep the signal close to the reference.
As concerns the set point regulation experiment reported in Figure 7 of the main text and in Supplementary Figure S15, we report the control error, computed as described before, and shown in Supplementary Figure S15. In this case, we also performed an additional statistical analysis to test the control action performance in regulating the protein expression level to the desired set-point. Indeed, due to cell-to-cell variability, the uorescence level in the cell population varies among the cells. Referring to Supplementary Figure S15, we considered two classes of events: (NC) the uorescence measured in single cells during the rst 180 minutes of experiment, when No Control input is applied; (C ) the uorescence measured in single cells after the rst 180 minutes of experiment, when the Control action has began. We then compared the control error in class (NC) (dashed black line in Supplementary Figure S15) to the control error in class (C ) (solid black line in Supplementary Figure S15) using a one-tail t-test to check if we can reject the null hypothesis H 0 = e N C ≤ e C , where e . represents the control error. We obtained a signicant p − value of 1.75E-11, that demonstrates that despite the cell-to-cell variability (see standard deviation bounds in Supplementary Figure S15) the control action is really eective.

Microuidic device fabrication protocol
Replica molding technique has been used to obtain replicas of the device presented in [18] by Ferry and colleagues. Before the fabrication of the microuidic devices the master is exposed to chlorotrimethylsilane (Sigma-Aldrich Co.) vapours for 10 min so as to create an anti-sticking silane layer for PDMS. A 10:1 mixture of PDMS prepolymer and curing agent (Sylgard 184, Dow Corning) is prepared and degassed under vacuum for 1 hour. Then the mixture is poured on the patterned, and to facilitate the polymerization and the cross-linking, it is cured on a hot plate at 75 • C for 3h. After this step the PDMS layer, containing the microuidic channels, is peeled from the master and it is cut with a scalpel to separate the single devices; holes are bored through them with a 20-gauge blunt needle in order to create uidic ports for the access of cells and liquid substances. The PDMS layers obtained are rinsed in isopropyl alcohol in a sonic bath for 10 min to remove debris. For each PDMS piece containing microchannels a thin glass slide (150um) is cleaned in acetone and isopropyl alcohol in a sonic bath for 10 min for each step. Finally the PDMS layers and glass slides are exposed to oxygen plasma in a RIE (Reactive Ion Etching) machine for 10 s and brought into contact forms a strong irreversible bond between two surfaces. As last step all devices were checked for faults inside and outside the channels.

Experimental setup
The experimental setup is the same for both strains of cells used in this study. On day 0 batch cultures are inoculated in 10 mL GAL/RAF+Sulforhodamine B (Sigma-Aldrich) (2%) Synthetic Complete medium (SC). On day 1 the batch culture is diluted at intervals of 12 hours (nal OD 600 0.01). On day 2, 60mL syringes (Becton, Dickinson and Company, NJ) lled with 10 mL SC+GAL/RAF (2%) and SC+GLC (2%) media are prepared, as well as sink syringes (lled with 10 mL ddH2O); capillaries and needles are used to allow connection to the microuidic device. Temperature in the microenvironment sorrounding the moving stage of the microscope is allowed to settle at 30 • C. Before connecting media and sink syringes, the microuidic device MFD0005a wetting is carried out as described in [18]. After air bubbles are removed, media and water lled 60 mL syringes are attached to the device and correct functioning is checked by inspecting the red-uorescence emitted by Sulforhodamine B as a result of the automatic height control of syringes. This allowed us to carry out a correct calibration of the actuation strategy before the actual experiment is run. At this point cells (IC18 or yGIL337 strain) are injected in the microuidic device by pouring the batch culture in a 60 mL syringe similar to the ones used to media and sinks. Once cells are trapped in the dened area (see [18] for details) Perfect Focus System is activated to assist autofocusing during the experiment and the microscope is programmed to acquire images at every 5 minutes interval: phase contrast (40 ms, exposure time) and epiuorescence images (green uorescence, 300 ms exposure time; red uorescence, 100 ms exposure time) were acquired to allow the control algorithm to (a) locate the cells (phase contrast images) (b) quantify the synthesised GFP (green uorescence) and (c) verify the correct administration of Galactose/Glucose (red uorescence).
Each signal tracking experiment starts with an initial calibration phase, needed to establish a linear relationship between the uorescence units obtained a read out of the cellular state and the arbitrary units the mathematical model is based on. In order to obtain this conversion, during the rst 600 min we supply SC+Galactose/Ranose for 180 min and Glucose for 420 min. This input signal is meant to (a) limit the stress on cells that were mechanically loaded and (b) obtain the whole uorescence dynamical range spanning the lower and upper steady states. Imaging is carried out using a Nikon TI-Eclipse microscope equipped with a 40X objective. Fluorescence images are taken using FITC (excitation 460/40 nm, emission 510nm/ 50nm) and TRITC (excitation 530/30 nm, emission 590nm/ 60nm) lters. Images are acquired by a Peltier-cooled Andor Clara camera controlled by Nikon Instrument Software v.3.10.
A set point experiment diers from the previously described since the control scheme adopted (PI -PWM cascade without prediction block) is not based on the mathematical model of the synthetic gene network; thus the initial calibration phase is not required. To calculate the maximum amount of protein that the cell population can express, at the beginning of the experiment, the cells are fed with SC+Galactose/Ranose for 180 min thus, the set point is calculated as a percentage of the average of the uorescence values measured during these rst minutes; then the control starts (it lasts for 2000 min). Figure S1. IRMA hybrid model. A hybrid model featuring two distinct vector elds (F 1 and F 2 ) has been derived from the model presented in [1]. As long as Glucose is administered (u = 0) F 1 is activated, while the system switches to F 2 as soon as Galactose is added to the medium to reect the inner dynamics of the synthetic circuits to be controlled. Figure S2. IRMA control scheme. The upper block scheme represents the control algorithm. The lower block magnies the Predictor block referred to as P red in the previous schematic. The y ref signal sets the desired output y for the controlled system P . The prediction block (P red) uses the input u and output y related to the actual plant P to compute an anticipated version of the output obtained by simulating the responseŷ of mathematical model of P in which τ = 0. This signal is immediately used to assess the eectiveness of the control action by feeding it back to the rst comparator that computes the error e made by the system. Moreover, the actual output y of the plant, is compared with a delayed version of theŷ signal (as eect of the e −τ block contribution) to account for discrepancies between the predicted (via IRMA's model M ) and real plant behavior. A low-pass lter meant to suppress high-frequency noise is applied to the resulting signal to obtain (y s ) that is nally fed back to the comparator that will subtract it from y ref so as to obtain the control error e. Figure S3. Cohen-Coon approximation for IRMA. In order to design a suitable PI controller we estimated three parameters, namely Θ, µ and d (as referenced in [6]) from the step response prole of the IRMA nonlinear model in equation 1-5. The solid blue line represents the response of our gene network (Cbf1p being the output) to the addition of Galactose to the growth media at t = 0 s while the dashed blue line shows the same information for the time delayed linear system identied with the method in [6]. Figure S4. Finite State Automaton implementing the control algorithm in Supplementary Figure S2. In the initial state, state 0, the calibration is carried out as previously described. The system cycles on this state until the initialization is completed and then moves to state 1. At this point given the error e, the PI -PWM block is simulated to compute the control input u. In state 2 the model prediction is calculated given u; the input is then applied to the physical system by means of hydrostatic pressure modulation in step 3 (the correct amounts of Galactose/Ranose and Glucose are provided at the end of this step). In state 4 the delayed version of computed output is calculated; during state 5, the presence of a new image is veried, and the image processing algorithm is run in order to obtain the system output measure. Given this it is possible to calculate y s and the error e for the next control iteration. The algorithm then moves to state 1 for a new control iteration to start. Figure S5. In-silico prediction-based signal tracking control of IRMA. The predictor-based algorithm is applied to control the dynamical model of IRMA to a time varying reference signal (y ref , in blue); the computed control input (higher state standing for Galactose and lower state meaning Glucose providing) is represented in red (u). The good overlap between the reference signal and the simulated Cbf1 time evolution (y) provides evidence for the robustness of the designed control scheme in two cases: (top panel) with no delay (τ = 0 min) and with τ = 100 min (bottom panel). Figure S6. In-silico prediction-based set point control of IRMA. The predictor-based algorithm is applied to control the dynamical model of IRMA to a constant reference signal (y ref in blue). The set point is calculated as the 80% of the maximum value for the simulated Cbf1 time evolution evaluated until t = 0min. The control input (computed after time 0 where higher state standing for Galactose and lower state meaning Glucose providing) is represented in red (u). The simulation was performed with the dynamical model without delay (top panel) or with a delay τ = 100 min (bottom panel). In both cases, the control action is able to guarantee good dynamical performances of the system, indeed the simulated Cbf1 time evolution (y in green) tightly matches the reference signal. Figure S7. In-silico PI/PWM signal tracking control of IRMA. The PI/PWM control algorithm is applied to control the dynamical model of IRMA to a time varying reference signal (y ref , in blue); the computed control input (high level: Galactose; low level: Glucose) is shown in red (u); the Cbf1 time evolution is shown in green (y). When the control is applied to the model without the delay, the control output (y) follows the reference signal (top panel); whereas the PI -PWM is not able to achieve the control objective for the model with the delay (τ = 100 min) (bottom panel). Figure S8. In-silico PI/PWM set point control of IRMA. The PI/PWM control algorithm is applied to control the dynamical model of IRMA to a constant reference signal (y ref inblue). The set point is equal to 80% of the maximum value for the simulated Cbf1 time evolution evaluated until t = 0min. The control input, computed after time 0, is shown in red (u high level: Galactose; low level: Glucose). The simulation was performed with the dynamical model without delay (top panel) or with a delay τ = 100 min (bottom panel). When the control is applied to the model without delay, the control output (y) follows the reference signal (top panel); on the contrary, the PI -PWM is not able to achieve the control objective for the model with the delay (τ = 100 min) (bottom panel). Figure S9. Image processing. The algorithm applies Otsu thresholding to binarize the grey scale phase contrast image (A). Convex hulls (B) are then used to limit the application of the Circular Hought Transform to nd cells' centers and edges (C).      Figures S10, S11, S12 and S13, the number of cells (top) and the coecient of variation (bottom) are shown. Figure S15. In -vivo set point control experiment for the IRMA network -uorescence standard deviation. By using the o-line analysis described in the text it is possible to calculate the standard deviation of the uorescence for each frame acquired during the control. The desired amount of protein (y ref in blue), the quantied GFP (y green line), the standard deviation's upper and lower bounds (thin green lines) and the control error e in black are shown; mean µ, variance σ and coecient of variation CV of the control error are also shown; the p-value was computed as described in the Supplementary Information text (top panel). The input signal u computed by the control algorithm is shown in red (bottom panel).     Figure S21. Internal signals of the control experiment in Fig. 8 (main text). Time evolution of the most relevant signals in the control loop are shown. In particular the Galactose concentration in the medium (u) provided to the cells has been plotted in red, while the output of the delay-free model (ŷ) and its delayed version (ŷ τ ) are shown in green and violet respectively. The error signal e (black) calculated as the dierence between y ref and y s (cyan) is also depicted; mean µ, variance σ and coecient of variation CV of the control error are also shown. Figure S22. In -vivo signal tracking control experiment 2 for the IRMA networkuorescence standard deviation. By using the o-line analysis described in the text it is possible to calculate the standard deviation of the uorescence for each frame acquired during the control. The desired amount of protein (y ref in blue), the quantied GFP (y green line) and its upper and lower bound of the standard deviation (thin green lines) are plotted; the control error calculated as the dierence between the feedback signal and the control reference is shown in black (top panel). The input signal u computed by the control algorithm is shown in red (bottom panel).