Implementation Considerations, Not Topological Differences, Are the Main Determinants of Noise Suppression Properties in Feedback and Incoherent Feedforward Circuits

Biological systems use a variety of mechanisms to deal with the uncertain nature of their external and internal environments. Two of the most common motifs employed for this purpose are the incoherent feedforward (IFF) and feedback (FB) topologies. Many theoretical and experimental studies suggest that these circuits play very different roles in providing robustness to uncertainty in the cellular environment. Here, we use a control theoretic approach to analyze two common FB and IFF architectures that make use of an intermediary species to achieve regulation. We show the equivalence of both circuits topologies in suppressing static cell-to-cell variations. While both circuits can suppress variations due to input noise, they are ineffective in suppressing inherent chemical reaction stochasticity. Indeed, these circuits realize comparable improvements limited to a modest 25% variance reduction in best case scenarios. Such limitations are attributed to the use of intermediary species in regulation, and as such, they persist even for circuit architectures that combine both IFF and FB features. Intriguingly, while the FB circuits are better suited in dealing with dynamic input variability, the most significant difference between the two topologies lies not in the structural features of the circuits, but in their practical implementation considerations.


Deterministic Models -Suppression of Input Fluctuations
For the FB models, both coupled and decoupled reactions schemes (Table 1) yield the same ODE model x = f (u)g(x) − l 1 ẋ y = f (u)g(x) − l 2 y.
(S1) Similarly, both coupled and decoupled IFF schemes result in the following ODE model In order to compare the performance of the two models in a meaningful way, we normalize them using the following criteria: For a given nominal value of the input u =ū, 1. the steady state values of Eq. (S2) and Eq. (S1) should be the same, 2. the production rates (inow ) at steady state should be the same for both models.
Denote the steady state value of x and y at u =ū byx andȳ respectively. The above normalization conditions imply l 1 = k 1 , l 2 = k 2 + k 12x , g(x) = 1, k2+k12x . This normalizations will be applied to the rest of this section.

Sensitivity to small local perturbations in u
Let y ss (u) be the steady state concentration of y as a function of u. The sensitivity of y ss with respect to changes in u nearū for the FB is calculated to be f (u)g(x ss ) − l 1 x ss = 0 f (u)g(x ss ) − l 2 y ss = 0 ⇓ ∂x ss ∂u (u) = − f (u)g(x ss ) f (u)g (x ss ) − l 1 ⇓ ∂y ss ∂u (u) = l 1 l 2 ∂x ss ∂u (u) = g(x ss ) g(x ss ) − x ss g (x ss ) f (u) f (u) y ss and therefore at u =ū (x ss =x, y ss =ȳ, g(x ss ) = g(x) = 1) In the case of the IFF, the sensitivity of y ss with respect to changes in u nearū is calculated to be ∂y ss ∂u is the ratio of the mediated degradation of y by x (k 12x ) to the unmediated degradation of y (k 2 ). α f f determines how much control x has over y, with larger values of α f f (i.e., more control x has on y) improving performance (less sensitive y ss to uctuations in u). Indeed, for k 2 = 0 (α f f = ∞) model (S2), studied in detail in Sontag [2010], displays perfect adaptation which is indicative of integral-like feedback (with integrator σ = y − x, the dierence in concentrations between X and Y , and error e = (k 1 − k 12 y) x). This implies that for nite α f f (i.e., k 2 > 0) the IFF can be viewed as leaky integral feedback.
We refer toα f b as the eective feedback gain and α f f as the eective feedforward gain. The motivation comes from the fact that α f b and α f f are closely related to the gains of the controller near the steady state. Indeed, if we let ξ = [ξ 1 , ξ 2 ] T be small perturbations of [x, y] T near the steady state value [x,ȳ]. The dynamics of (S1) near the steady state are given bẏ 1,0]. This can be viewed as the closed loop dynamics Similarly, the dynamics of (S2) near the steady state are given byξ has been applied). This can be viewed as the closed loop dynamics of . Both α f b and α f f are a measure of the local strength of control of (S1) and(S2).
1.2 Steady-state dependency on u ∂yss ∂u (ū) is a local measure of how sensitive the steady state of the output of interest y is to samll changes in u nearū. This measure is a function ofū and for eachū we can get a picture of the overall sensitivity of the system to changes in u. Can we calculate directly the dependency of the steady state value of y to u? For the IFF, such calculation results in where the normalization (S3) is applied. For the FB, in the general case, we can only get an implicit formula ug( l 2 l 1 y ss ) − l 2 y ss = 0 x ss = l 2 l 1 y ss Let consider a special case when the inhibition function g is given by the ramp function g R x, 0 . (S11) In this case This dependency of y ss on u for the FB is is the same as that of the IFF for the same eective gain (eq. (S10)).

Attenuation of small time-varying disturbances in u -frequency domain analysis
So far, we have considered the dependence of the steady state value on constant perturbations of u. What happens when u is time varying? How does that aect y? For simplicity of notation, we take f (u) = u. Near the steady state, for u(t) =ū + δ(t) and small δ, the dynamics are given bẏ for FB and for FF. How the eect of δ(t) on the perturbations ζ of desired output y can more easily be expressed in the frequency domain where ζ(s), δ(s) are the Laplace transform of ζ(t), δ(t) respectively, s is the frequency (Laplace) variable, and G(s) is a trasfer function gine by for FB, and for IFF (with the normalizations (S3) applied).
Note that the sensitivity ∂yss ∂u (ū) is given by G(0) (s = 0 correspond to the response to constant δ). In general, higher gain pushes one of the poles of G (the value of s for which G(s) → ±∞) toward −∞ for the FB, but has no eect on the poles of G for the FF. On the other hand, higher gain pushes the zero of G (the value of s for which G(s) = 0) toward the origin for the FF, but it has no eect on the zero of G for the FB. The H 2 -norm of G dened as is given by .
TheH 2 -norm has a few interesting interpretations.
One interpretation is that its square is a measure of the energy of the transient to the impulse response, i.e., is the Dirac delta function (whose eect is equivalent to perturbing the system away from the steady state). So the H 2 -norm is a measure of the transient deviations from the steady state. Another interpretation is that if δ is given by white noise, then the square of the H 2 -norm is the variance of the output as t → ∞ when δ(t) is white noise.
In order for either system (FB or FF) to have good disturbance attenuation properties (i.e., minimize the eect of variations of u(t) on the value of desired output y(t)), then H 2 -norm of G needs to be made small. In this respect, the FB architecture is better suited since this norm can be made arbitrarely small by choosing arbitrarely large gain, i.e., On the other hand, this is not true for the IFF architecture since This result is not surprising since the controller implemented by the IFF does not eect the poles of the system. Since the action of the controller has no eect on the behavior of x, and the transient behavior of uncontrolled x contributes to the transient behavior of y.
with innite gain and zero gain respectively (i.e., maximum control and no control). For the FB, the controller controls both x and y, and is thus able to suppress the transient behavior of both species.
In conclusion, while the steady state behavior of both FB and IFF is the same for the same eectual gains, it is not true for the transient behavior. The FB architecture is better suited to deal with attenuating transient eects of disturbances.
1.4 IFFL circuit, non-catalytic eect of x So far we have assumed that the role of x in the mediated degradation of y in the IFF is purely catalytic, i.e., x is not degraded as part of the reaction. What happens if the role of x is not always purely catalytic?
Assume that the following reaction takes place instead of reaction with frequency a. Then the corresponding ODE is given bẏ If an isolated equilibrium point in the positive orthant exists, then we dene the load eect of the noncatalytic degradation of y as ρ := ak12ȳ k1+ak12ȳ (i.e., the ratio of the non-catalytic degradation of x, to the total degradation of x at steady state). The sensitivity of y e with respect to changes in u nearū is calculated to be ∂y e ∂u . Notice thatα f f < α f f , and larger the load ρ, the smaller the eective gainα f f of the loaded circuit.

Stochastic Models -Suppression of Chemical Reaction Stochasticity
In this section we examine how the inherent stochasticity of the chemical reactions aects the ability of FB and IFF topologies to maintain a desired number of molecules of Y . In this section x and y are random variables that refer to the number of molecules of X and Y respectively. We study how dierent eective gain α f b and α f f change the noise properties of FBL and IFFL respectively, by looking at the steady state value of the variance of y.

Feedback Inhibition
Consider the case when for a given α f b , the inhibition function g is parameterized by g R where x.
I.e., g R is a linearly decresing function of x for x < x 0 := l1 and identically 0 for x > x 0 ( Figure 3A, solid gray line). Notice that this choice of the inhibition function g is consistent with the denition of α f b in Eq. S5. Let the set Γ be the set of all feedback gains for which x 0 is an integer, and consider only α f b ∈ Γ . Such set of α f b is sucient for our purpose of studying the noise properties of the FB architecture.
The stochastic models based on the coupled FB reaction scheme and the decoupled FB reaction scheme (Table 1), yield the following moment equation for the coupled FB, and b xy = 0 for the decoupled FB.
By Lemma 1 in the appendix, we have that for any ) and therefore Pr (x > x 0 ) = 0. This allows us to replace g R in all the expectations in the moments equation (S19) by g L (since . Notice that for each ratiox y , the optimal gain is given by α opt f b = 1 +x/ȳ, with corresponding variance var opt (y) = 2 + 1 +x y (x + y)y 1 + 1 +x y x +ȳ + 1 +x yȳ . The reduction of the variance by the best feedback strategy is greater for smaller the ratiosx/ȳ, with the best case scenario (x/ȳ → 0) reducing the variance by 25% (var(y, α f b → 1,x → 0) = 3 4ȳ ).
For the decoupled FB, the mean remains the same and the steady state variance is evaluated as

Incoherent Feedforward
The moment equations of the stochastic model based on the coupled and decoupled IFF reaction scheme (Table 1) where b 4 = 1 for the coupled case and b 4 = 0 for the decoupled case. Note that x is simply a birth death process, and therefore has a Poisson stationary distribution with mean and variance given byx = f (ū) k1 .
Special case -no regulation In the case k 12 = 0, k 2 > 0 (no regulation, α f f = 0), the mean of the stationary distribution of y is given by < y >= f (ū) k2 =ȳ and the variance var(y) =ȳ (both coupled and decoupled implementations give the same values, y is simply a birth death process).
General case For k 2 > 0, k 12 > 0, using derivative matching Singh and Hespanha [2011] we approximate E x 2 y and E xy 2 by

E [xy] E [y]
2 and solve L I m XY + b I = 0 for this approximations to get expressions for < y 2 >, the second moment of the stationary distibution of y, > 1, ∀α f f > 0 and therefore var(y) <ȳ. So for the coupled production, regulation reduces the variance. <xy> xȳ > 1 implies that there is a positive correlation between x and y, i.e. uctuations in x carry information about the uctuations in y and regulation can use this information to suppress such uctuations. Aggressive action (regulation) on such imperfect information can have adverse eect and eventually loose the benet of regulation ( <xy> xȳ → 1 and var(y) →ȳ as α f f → ∞).
In the case of the decoupled IFF (b 4 = 0), <xy> xȳ < 1, ∀α f f > 0 and therefore var(y) >ȳ. Therefore for the decoupled IFF realizations, any regulation increases the variance (amplies the stochastic noise). In fact, the stronger the gain, the larger the amplication. <xy> xȳ < 1 implies that there is negative correlation between x and y, i.e., any regulation pushes x and y in opposite direction and therefore increases the variance (regulation is acting on stochastic noise, uctuations in x do not carry any useful information about uctuations in y).
Special case -innite gain In the case k 2 = 0 (innite gain α f f ) the explicit expressions for the variance and the penalty terms are calculated. The mean of the stationary distribution of y for the coupled implementation is given by < y >= k1 k12 =ȳ and the variance var(y) =ȳ. There is no improvement of the variance of y compared to the no regulation (k 12 = 0) instance. For the decoupled IFF, we get The same parameter values used in the coupled case would result in a higher mean for the decoupled case. In order for any comparison to be meaningful, we set both coupled and decoupled models to have the sameȳ. To this end, the mediated consumption rate constant for the decoupled modelk 12 should bek 12 = k 12 1 + k 12 k 1 +xk 12 The rate constantk 12 is larger than k 12 , implying that faster mediated degradation of y is required for the decoupled system to maintain the same expected value of y. Using this rate constant, the adjusted variance is var(y) =ȳ +ȳ + 1 1 +x/ȳ So there is a decoupled production penalty of P f f decouple =ȳ +1 1+x/ȳ . So the most aggressive regulation can more than double the variance in the worst case scenario (very small average population of X) and increase the variance only by a little in the best case scenario (very large average population of X).

Comparing the Dierent Architectures
Both FB and IFF circuits respond in a similar manner to chemical reaction stochasticity and achieve comparable reductions in steady state variance (no more than 25% in all but a few extreme scenarios). Direct comparison of coupled FB and IFF reductions for optimal values of gains, shows that for average molecule counts of X and Y that are no smaller than 2, the FB realizations achieve larger reductions than the IFF counterparts. The larger the average population of X, the more FB outperforms the IFF. Notice also that the optimal gain for the FB is larger than the optimal gain for the IFF.

Concurrent Suppression of Both Sources of Variability
In this section we consider both sources of noise simultaneously. We rely on the analytical results of the There seems to be a tradeo on how aggressively the circuit should regulate the production (for the FB) or the degradation (for the IFF) of Y .
For the last two realizations, the feedback inhibition function g given by (S18). Let the input u be a random variable distributed according to a Poisson distribution (such as the stationary distribution of a birth death process) with meanū, and let f (u) = au + b. For the rst realization (no feedback) where C := aū . For constant C and b (which guarantees constant average inow f (u)), smallerū implies larger total variance as the result of a larger input noise component (the second term). This is to be expected since smallerū means that Poisson variable u becomes more noisy. Note that for this realization, both coupled and decoupled implementations yield the same var(y). The second FB realization (coupled, very strong feedback) results in a smaller variance, var(y) = C+b l2 , for any niteū. This is a result of strong feedback completely suppressing the variance term due to randomness in u but not changing the variance term due to randomness of the chemical reactions. So for the coupled production, strong feedback is prefered to no feedback. We will later show that, depending on the noise characteristics of u, more moderate feedback yields a bigger reduction in the overall variance.
For the third FB realization, (coupled, very strong feedback), strong feedback again completely suppresses the variance term due to randomness in u but it also doubles the variance term due to randomness of the chemical reactions resulting in var(y) = 2 C+b l2 . So this realization is preferable to no feedback only if u < C 2 l(b+C) , i.e. the input noise level is above a specic threshhold.
In general, for both FB and IFF circuits the best regulation strategy is dependent on the relative dominance of the terms in the variance decomposition. Figure 4 shows numerical simulation results for dierent circuit realizations and dierent input noise scenarios. As the input noise level is high, the extrinsic noise term is dominant and stronger regulation is preferred. For low levels of input noise, the intrisic noise term becomes dominant and small levels of regulations are preferred especially for the decoupled realizations.

Combining FB and IFF into a Single Circuit
The reaction scheme is the same as the IFF scheme of Table 1 with that change that the rate of production reactions is given by f (u)g(x) and we follow the same normalization rules for g. This results in the following We dene the eective gain α comb := α f b + α f f + α f b α f f , and note that both equations 1 and 3 hold.
Near the steady state, for u(t) =ū + δ(t) and small δ, the dynamics are given bẏ where I is the identity matrix, B i is the stoichiometry of reaction i, and ω i is the propensity of reaction i.   Fig 4. The propensities of the reactions are shown above the reaction arrows, and g(x) =