Adaptation is referred to as the ability of a gene circuit to respond to changes in input and then to return to the pre-stimulus output level, even when the input change persists (Ma et al., 2009). More precisely, with an input signal switched from a lower value I₁ to a higher value I₂, as demonstrated in Fig. 1(b), a circuit with functional adaptation should have the following response-curve criteria: (i) precision - the final state O₂ should be close to the initial state O₁, (ii) sensitivity - there should be a relatively high |O_peak| in response to the change in the input, and (iii) the system should reach equilibrium within a reasonable relaxation time. A threenode gene circuit can achieve adaptation (Ma et al., 2009), with one node receiving the input (node A), another node realizing various regulatory roles (node B), and a third node outputting the response (node C). A representative circuit topology is shown inside the red dashed box in Fig. 1(a). We restrict our study of the class of transcriptional regulatory networks (TRNs) with the AND logic. We fix node A as the input node, and node C as the output node.

Previous research identified 425 different three-node TRN network topologies that can achieve adaptation in the absence of growth feedback (Shi et al., 2017), providing the base of our computational study. These topologies can be classified into two families based on the core topology: networks with a negative feedback loop (NFBL) and networks with an incoherent feed-forward loop (IFFL) (Shi et al., 2017). More specifically, there are 206 network topologies in the NFBL family. All of these NFBL topologies have a negative feedback loop for node B.

This negative feedback loop can be formed by the loop from node B to A and back to B (such as the circuit shown in Fig. 1 (a)), by node B to C and back to B, or by a longer route, from node B to A and then to C and back to B. There is always a self-activation link from B to B in all these 206 NFBL networks. There are 219 network topologies in the IFFL family. All of them have two feedforward pathways from the input node A to the output node C. One pathway goes from node A to C directly, while the other involves node B in the middle. One of the pathways is activating while the other one is inhibitory. We use these 425 network topologies from the study Shi et al. (2017), avoiding redundancy with established results. Due to the unique focus of our research on the effects of growth feedback and the need to evaluate quantitative ratios of robust circuits among all functional ones, we have chosen to use a 20-fold increase in the number of random parameter sets for each network topology compared to the simulations in Shi et al. (2017). This approach makes it computationally prohibitive to scan all possible 16,038 three-node circuits. We carefully follow the settings in Shi et al. (2017), which also analyzed TRNs with the AND logic as in this paper. Detailed descriptions of our simulation experiments are provided in the Methods section. To make our results more convincing, we have adopted a set of adaptation criteria that are stricter than those used in Shi et al. (2017). Consequently, the ratio of adaptive circuits is somewhat lower in our study, with 4 out of the 425 network topologies not demonstrating adaptation. The specific structures of these 425 network topologies can be found in our GitHub repository (link provided in Data availability).

In our work, we use a parameter k_g to control the strength of growth feedback, which is a parameter determining the maximal growth rate of the host cells, as mathematically explained in Model description. With all the other parameters fixed, a larger k_g implies a faster cell growth rate and a stronger impact of growth feedback. To investigate the effect of growth feedback on these circuits, we systematically simulate the response of the 425 network topologies under a switch in the input signal. A three-node gene circuit subject to growth feedback has a large number of parameters, which determine the properties of the regulation links within the circuit and the circuit dynamics. For each topology, we randomly sample 2 × 10⁵ trials of circuit parameters. Altogether, our study involves analyzing approximately 8.5 × 10⁷ different circuits. We find that among these trials, only about 1.5 × 10⁵ meet the adaptation criterion in the absence of growth feedback. For these functional trials, we vary the growth feedback parameter k_g with a series of values, and find that the majority of trials (1.3×10⁵ trials, about 87%) lose their adaptation in the interval of k_g ∈ (0, 1), while only 13% of trials remain functional at k_g = 1.0.

An essential step towards understanding the detrimental or even destructive effects of growth feedback on circuit functioning is to identify the distinct failure scenarios. Our extensive simulations have yielded a comprehensive picture of these scenarios, as shown in Fig. 2. Overall, we have identified six failure scenarios that encompass more than 99.6% of the 1.3 × 10⁵ failing cases we collect. The first level of classification distinguishes between failures that occur continuously or abruptly as the growth-feedback strength k_g increases. In a continuous failure, the response curve deforms continuously as k_g increases, as exemplified in Figs. 2(a-c). In an abrupt failure, the response curve exhibits a sudden change as k_g increases through a critical value, as illustrated in Figs. 2(d-f). At the next classification level, we further divide the failures into three types of continuous failures and three types of abrupt failures.

The three types of continuous failures, denoted as types I-III as illustrated in Figs. 2(a-c), are determined according to the specific quantitative criteria in the response curve that the circuits violate. Type-I continuous failures, as shown in Fig.2(a), are associated with the violation of the precision criterion. A circuit is deemed as precise if a change in the input signal (e.g., from I₁ to I₂) generates two opposite dynamical effects in the circuit that cancel each other out after a transient and return the final output to the original state, i.e. O₂ ≈ O₁. For example, in some networks [e.g., the network in Fig. 1(a)], an increase in the input signal I will result in an increase in the concentration of gene A and a reduction in the concentration of gene B. As both genes regulate the output gene C with the respective activation links, for proper system parameter values, the two effects will cancel each other out, resulting in O₂ ≈ O₁. Type-I continuous failures constitute the largest failure category among all possible circuit topologies, suggesting that the exact cancellation is fragile and the loss of precision is the most common dynamical mechanism behind growth-feedback-induced failures.

Our simulations reveal that an exact cancellation between the two opposite sources at k_g = 0 prevents an exact cancellation at any other values of k_g. That is, the set of circuit parameter values leading to perfect precision, in general, depends on the value of k_g (see Appendix 5 for more details). The implication is that, for fixed circuit parameter values, achieving high precision under growth feedback (k_g > 0) is difficult if the circuit is precise in the absence of growth feedback (k_g = 0).

Type-II continuous failures are characterized by a continuous change in the peak of the response curve, denoted as O_peak, as k_g increases, eventually falling below a threshold, as shown in Fig. 2(b). This type of failure can make it challenging for downstream circuits to detect the peak signal, hindering information transmission in the larger system. Type-II failures are the second most common type of failure observed in our simulations. The occurrence of a high peak in the response curve requires a significant transient deviation from the final equilibrium point. In the presence of growth feedback, the transient behavior changes, which can further alter the peak height O_peak.

Type-III and type-IV failures arise due to growth-feedback-induced oscillations, while type-V and type-VI failures are caused by bistability or multistability. To provide a more detailed understanding of these different failure scenarios, we discuss the two mechanisms, respectively, in the sections of Growth-feedback induced oscillations and Bistability or multistability below.

As demonstrated by the light green and yellow slices of the pie chart in Fig. 2, a considerable portion (17%) of the circuit failures are caused by growth-feedback-induced oscillations. Growth feedback perturbations can easily change the system from the adaptive domain to the oscillation domain in these cases. Our program classifies oscillation-mediated failures into two categories: continuous (type-III) and discontinuous failures (type-IV). Type-III failures are the results of either (i) a gradual increase in the oscillation amplitude, or (ii) a gradual increase in the transient lifetime of damped oscillations. In the first case, an isolated circuit has already exhibited oscillations with small amplitudes in its gene concentrations with relatively weak growth feedback. As the feedback is strengthened with a larger value of k_g, the oscillations are intensified with a larger amplitude, leading to a circuit failure. In the second case, there is damped oscillation for small k_g with a relatively short transient time before approaching an equilibrium. After strengthening the growth feedback, the damping weakens and the oscillation’s amplitude cannot be reduced to the threshold within the time limit, as exemplified in Fig. 2(c).

The second category of growth-feedback-induced oscillation is type-IV, where oscillations emerge suddenly as the growth-feedback strength increases through a critical point. The sudden emergence of oscillations can be caused by a bifurcation or a transition into a basin of a limit-cycle attractor. A random sampling of the type-IV failure cases reveals that most of them are caused by either a saddle-node bifurcation of cycles (Strogatz, 2018) or an infinite-period bifurcation (Strogatz, 2018). In the former case, a pair of stable and unstable limit cycles suddenly emerge together. In the latter case, when observed from the opposite direction (i.e., with a decreasing k_g crossing the threshold), the oscillation in the system spends a longer and longer time around a node on the limit cycle. This node finally becomes a stable fixed point at the bifurcation point, and the oscillation period approaches infinity. One example of type-IV oscillation-mediated failures caused by an infinite-period bifurcation is shown in Fig. 2(d). In our simulations, most of the cases where there are saddle-node bifurcations of cycles are categorized as type-III failures because, prior to the bifurcation point, the system can be oscillating near the ghost cycle (Strogatz, 2018) for a long time exceeding the criterion for relaxation time, though that ghost cycle is not an attractor but only a transient in the system. More details on these two types of bifurcations in our simulated circuits can be found in Appendix 7.

Our results indicate that for various circuit topologies, the dynamic mechanisms leading to failures can differ, resulting in significantly different distributions of failure types among different networks. For instance, the fractions of failures caused by growth-induced oscillations can vary dramatically among all the topologies, as demonstrated in Fig. 3(a), where each data point represents a specific network topology. The fraction of failures caused by growth-induced oscillations can range from approximately zero to about 80%! A particular example of two different networks is presented in Figs. 3(b1) and 3(b2), both of which share the same minimal topology required for adaptation (Shi et al., 2017) - the circuit’s core function. Despite differing by only one link, the proportions of failures with unique mechanisms are quite distinct, as illustrated in Figs. 3(c1) and 3(c2). Notably, for the network in Fig. 3(b1), almost half of the failures result from oscillations, while hardly any oscillation-mediated failures occur for the network in Fig. 3(b2). The explanation is that, although the difference lies in only a single link, this link determines whether an oscillation-correlated motif exists within the network. Previously, three classes of motifs capable of supporting persistent oscillations were discussed (Novák and Tyson, 2008), including the “delayed negative-feedback loop” featured in Fig. 3(b1).

Generally, the circuit dynamics depend sensitively on the structure, but oscillations specifically require a negative feedback loop with time delay (Novák and Tyson, 2008). Since there are no explicit time-delayed terms in the dynamical equations in our model, one of the two types of motifs - an intermediate node in the path of the negative feedback loop or an additional positive feedback loop - is necessary to induce time delay (Novák and Tyson, 2008). For network topologies with a high ratio of functional failures caused by oscillations, both motifs are observed, especially the former type. For the network in Fig. 3(b1), the three links: A → C, C ⊣ B, and B → A, together constitute a negative feedback loop, making the circuit more susceptible to oscillatory behaviors. For the circuit in Fig. 3(b2), no such negative feedback loop exists. Figure 3(a) summarizes the total number of failed trials and the ratio of oscillation-induced failures for each network topology. The network topologies that contain one of the motifs for oscillation as discussed in Novák and Tyson (2008) are marked in red, while the networks that do not consist of any of them are marked in blue. Note that all the networks with relatively high ratios of oscillation-induced failures (e.g., ratio > 0.2) consist of oscillation-correlated motifs. Details about these oscillation-correlated motifs are discussed in Appendix 6.

We conclude that, for a network with an oscillation-correlated motif, even if it is functional at some parameter values, the potential of oscillatory behaviors can be triggered by growth feedback. As a result, networks without these motifs can be safer choices to avoid too many failures cases due to oscillations. Note that this relationship is not deterministic. As shown in Fig. 3, even the networks represented by blue dots that have no oscillation-correlated motifs can still have oscillation-induced failures (with small ratios). The complexity of the scenario makes it challenging to find general and relatively simple rules that connect circuit topology to the circuit’s robustness.

In this section, we describe the dynamical mechanisms behind type-V and type-VI failures, which in total take up about 14% of all the circuit failures. These failures are abrupt, meaning that the response curve undergoes an abrupt change at a certain critical value of k_g from a desirable curve of adaptation. Type-V and type-VI failures correspond to an abrupt change in O₁ and O₂, respectively. Both types of failures are closely related to bistability or multistability.

Bistability and multistability are common phenomena in nonlinear systems. Bistability refers to the situations where two stable attractors coexist in the phase space simultaneously. Multistability describes a similar coexisting phenomenon of attractors, but with more than two attractors. With bistability or multistability in the target dynamical system, the system trajectory may end up in any one of these stable attractors, depending on the initial state of the system evolution. The entire phase space can thus be separated into two or more basins of attraction. Each basin of attraction corresponds to an attractor and consists of all the initial states that eventually lead the system to the attractor. The boundary boundaries separate two different basins of attraction. A close pair of initial states but at different sides of a basin boundary lead the system to different attractors.

In our simulations, we observe bistability in most of the circuit topologies (377 out of 425 circuit topologies). While multistability has also been observed, it is relatively rare. Considering that both bistability and multistability impact the circuits similarly, our subsequent discussions will primarily focus on bistability, which can be straightforwardly extended to multistability. It is highly unusual for both attracting basins to exhibit the desired functional behavior simultaneously. This is because they are located in different regions of the system phase space, and accommodating both would impose overly stringent constraints on the circuit. After all, having one basin functional is already rare enough with a random sampling of circuit parameters. As a result, functional adaptation is typically found in only one of the basins, with adaptation being lost in the other, and the circuit is functional only locally in its phase space, rather than on a global scale. A drifting system parameter, such as k_g, can alter the dynamics of the gene circuit. In a situation with bistability, such a change in the system dynamics can modify the shape and position of the basin of attraction and the basin boundary. Consider an initial state close to a basin boundary. With the deformation caused by a drifting parameter, the boundary may shift across the initial state, leading to a sudden switching of the system’s final attractor. If the basin before the parameter change is functional and the basin after is not, this leads to a growth-feedback-induced failure. The crossing of the basin boundary dictates that the system’s final state will abruptly change from one attractor to another. This type of failure can be classified as a switching type of failure.

An example of bistability-related failures is shown in Fig. 2(e), where in the upper panel, the circuit enters into the functional region after an initial transient. In the lower panel, the circuit enters into another region that does not have adaptability, and the circuit does not respond to the switching of the input signal. Figures 4(a) and 4(b) illustrate how the basin structure of the circuit changes significantly with different values of k_g. The functional basin is in yellow and it shrinks greatly with an increasing k_g. Note that the phase space is four-dimensional, so only a two-dimensional slice is shown. For a bistability/multistability-induced type-V failure where O₁ is switched, the boundary of the functional basin crosses the initial state. For a type-VI failure, the simultaneous movement of both O₁ and the basin boundary under input I₂ results in O₁ crossing the basin boundary of O₂ states.

One might expect bifurcations to play an important role in many type-V and type-VI failures. However, in our simulations, failures precisely at the bifurcation point are not observed. This is because the bifurcation points under consideration, such as fold bifurcations, are where one of the attraction basins diminishes to zero. For a failure to occur exactly at the bifurcation point, the initial condition would need to coincide precisely with the infinitesimally small basin just before it vanishes. More realistically, failures almost always largely precede the exact bifurcation point. They happen while the basin is still contracting and the basin boundary crosses the initial condition or O₁. An example is shown in Fig. 1(b), where bistability persists, yet the lighter orange basin with a larger O₁(C) cannot be reached as the boundary shifts away from the initial condition A₀ and B₀. As another example, in Fig. 1 (c) from a different circuit, the higher O₂(C) state disappears at k_g ≈ 0.012 and switches to a lower O₂(C), but this point is not a bifurcation. It is the point where the stable O₁ continuously crosses the basin boundary of O₂.

To quantify a circuit’s robustness against growth feedback, we introduce two metrics: Q-value and R-value. We track the number of remaining functional trials for each network for various k_g values (starting from k_g = 0), denoted as Q(k_g). This measure extends the concept of Q-values in Ma et al. (2009) by accommodating non-zero values of k_g. To characterize the circuit robustness, we define the survival ratio R(k_g) as R(k_g = k) = Q(k_g = k)/Q(k_g = 0). This ratio represents the fraction of random circuit realizations that maintain functionality under growth feedback with a strength of k_g.

Note that each Q(k_g) or R(k_g) is defined for a specific network topology in a suitable parameter space. A high value of R(k_g) indicates that a large fraction of the randomly sampled circuit parameters is functional despite cell growth with any strength no larger than k_g, indicating that the topology is more robust against growth feedback. Because of the detrimental effects of growth feedback, R(k_g) decreases monotonically with respect to k_g.

To justify the utility of R(k_g), we test the circuit topologies employed in a previous work (Zhang et al., 2020), where two relatively simple network topologies were used for a comparison study in terms of their ability to resist growth feedback and remain functional. Our evaluation of R(k_g) for the two topologies has yielded results that are consistent with those in Zhang et al. (2020), as discussed in Appendix 2. To illustrate our results in a concrete way, we set k_g = 0.6 and calculate the ratio R(k_g = 0.6) for different network topologies.

Our computations have revealed a set of eight circuit topologies with optimal performance as characterized by high values of both Q(k_g = 0) and R(k_g), as indicated by the set of orange points in Fig. 5(a). The optimal circuits form a family as their topologies exhibit a high level of similarity with one other. In particular, all eight circuits in this family share a common set of links (motif), as shown in Fig. 5(b). The combination of these common links is one of the minimal topologies with perfect adaptation in a three regulatory logic (Shi et al., 2017) and is critical for the circuit to be functionally adaptable. The only difference among the circuits in this family is the links from node C. While an inhibition link from node C can be important to achieving a value of R(k_g), as discussed below, the eight optimal circuit topologies do not contain any such inhibition link from node C. (The role of this particular link will be further studied in our analysis of the results in Fig. 6.) This also explains why the family has eight members, as follows. Each link from C has two options: either the link does not appear, or it appears as an activation link. As there are three possible links from C (C to A, C to B, and C to C), there are altogether eight (2³) topologies within this optimal family, according to the simulation results in Fig. 5.

How can we quickly determine if a three-gene regulatory network with a given topology can be robust against growth feedback? Is there any structural feature of the circuit that can be used to estimate if a high value of R(k_g) can be achieved? To gain insights, we observe that the histogram in Fig. 6(b) has three peaks about low, moderate, and relatively high values of R, respectively. Computations reveal certain “shared topological similarity” (or motif) within each peak. Thus, each peak corresponds to a group of similar network topologies that simultaneously have a similar level of R. This observation suggests a correlation between the network topology and robustness against growth feedback. For convenience, we refer to these three groups of networks by the colors presented in Fig. 6. For instance, the group with the highest R (the green triangles in Fig. 6(a)(d)) is called the green group, and the group with the lowest R (the red diamonds in Fig. 6(a)(d)) is the red group.

To better distinguish the three groups, we introduce two binary variables, B₁ and B₂. For each network, B₁ = 1 if the network contains the motif in Fig. 6(c), and B₁ = 0 otherwise. Then, for each network, an additional binary variable is set to be B₂ = 1 if there is an inhibition link from the output node C, and B₂ = 0 otherwise. We find that a linear combination of the two binary variables, B_S = B₁ − B₂, can characterize the circuit topology and robustness against growth feedback. In particular, the three possible cases B_S = 0, 1, or -1 correspond to the three peaks in Fig. 6(b). This result suggests that the motif shown in Fig. 6(c) is beneficial for robustness, while an inhibition link from the output node C is detrimental. It is the balancing act of these two factors that determines the overall circuit robustness.

The discovery of this three-peak structure and the corresponding topological similarity within each peak is facilitated with the use of machine learning. In particular, we consider a simple type of artificial neural network called multilayer perceptron (MLP), where we train it to predict the R value from the input of the network topology through a small hidden layer with only two nodes, as demonstrated in Fig. 6(e). This bottleneck structure in the hidden layer plus the l-1 regularization imposed on the input matrix W_in forces the MLP to extract low-dimensional features from the input topology to estimate R. In our tests, the MLP designed this way automatically assigns different levels of weights to the input information of different links. Over an ensemble of 50 MLPs trained with different random initial values, the ranking of average importance is shown in Fig. 6 (f). The top four links are the four links used to categorize the three peaks.

The results in Fig. 6 is for k_g = 0.6. However, we find that different values of k_g lead to essentially the same ranking of R(k_g) among the circuit topologies, as illustrated in Fig. 7.

Three remarks on our categorizing rules based on the two extracted featured motifs are in order.

First, the shared motif for the green group is strikingly similar to the optimal minimal networ in Fig.5(b) (the orange group). The sole distinction lies in the self-activation link of node B. This specific link plays a crucial role. Every network in the NFBL family depends on this link to achieve adaptation (Shi et al., 2017). However, for circuits within the IFFL family, this link is not a necessity for adaptivity. Missing this link makes the motif in Fig. 6(c) no longer a minimal network for adaptation, and a circuit containing this motif may either belong to the NFBL or the IFFL family. We have thus identified two optimal groups: the green group with optimal robustness R and the orange group with both the optimal robustness R and the largest functional volume Q(k_g = 0) in the absence of growth feedback. The orange group is a subset of the green group, with an additional requirement for Q(k_g = 0).

Second, the shared motif for the red group is also exactly the group of all circuits containing an inhibition link from node C to node B, denoted as C ⊣ B. These two different definitions are in fact equivalent: all networks with Q(k_g = 0) > 300 that contain C ⊣ A or C ⊣ C also contain the motif in Fig. 6(c), yielding B_S = 0, and belong to the blue group.

Third, the three circuit groups in Fig. 6 are not correlated with the categories used in previous research on circuit functionalities without growth feedback (Ma et al., 2009; Shi et al., 2017). These studies classified adaptive networks into NFBL and IFFL families. Each family contains a few minimal topologies with or without some additional other motifs, and the two families have distinct minimal functional topologies. The minimal topology acts as the backbone for supporting circuit functionality. We find that, when growth feedback is present, the prior classification scheme and the underlying minimal topologies become less relevant. Circuits belonging to the NFBL family are spread across all three levels of R(k_g) in Fig. 6(b), as are the circuits from the IFFL family. A robust circuit can be part of either family, just as a fragile circuit can belong to both. We give that: (i) the topological motifs determining circuit functionality robustness and (ii) the motifs deciding whether a circuit belongs to the NFBL or IFFL family are independent. To quantify this irrelevance, we calculate the point biserial correlation between R(k_g = 0.6) and a binary variable determining the family to which the circuit belongs. The resulting correlation is merely 0.1, suggesting hardly any correla-tions. A further illustration and quantification of this irrelevance can be found in Appendix 4.

What are the reason and mechanism behind the phenomenological set of circuit categories? Especially, it is desired to understand why the shared motif for the green group is beneficial for circuit robustness, and why the shared motif for the red group is harmful for robustness. It is challenging to find straightforward explanations given the complexity of the problem (see Discussion section). Certain insights are as follows. We find that the average node concentrations at the equilibrium for the network topologies in the red group are consistently smaller than those in the blue group. This difference is reflected in the value of burdens b. In particular, according to Eq. (12), the cell growth rate is proportional to the term 1/(1 + b) under the same level of growth feedback. Figure 6(d) shows the average burden ⟨b⟩ for each network topology, demonstrating that the values of the term 1/(1 + ⟨b⟩) for the circuits in the red group are larger than the values in the blue group. As a result, for the same value of k_g, the growth feedback effectively received by the circuits in the red group is stronger than that of the blue group circuits. Further support is provided by the results from the limit J → ∞ (Appendix 3). In this limit, the burden b does not affect the strength of the growth feedback. As a result, the R values of the red group significantly overlap with those of the blue group, suggesting that the distinctively low values of R for the red group be a result of the burden with finite J. We also find that the existence of the shared motif for the red group has a stronger correlation to the motif necessary for growth-feedback induced oscillations. All circuits with oscillation type of failures taking up more than 20% of failures belong to the red group. This correlation can result in further fragility of the red group circuits.

A comprehensive way to understand the effects of growth feedback on gene circuits is through scaling laws, an approach commonly employed in statistical and nonlinear physics. Does a scaling law exist that characterizes quantitatively how growth feedback affects the circuit functioning? Through a systematic computational analysis of the circuit robustness, we have uncovered a scaling law that governs how the robustness measure R(k_g) deteriorates as growth feedback is strengthened, as shown in Fig. 8, where the blue curve is the result averaging over all the 425 network topologies. The three other curves represent circuit topologies that have a relatively high, moderate, and low value R among the 425 topologies tested, to demonstrate that this scaling behavior is generic. (These three topologies are the highest Q(k_g = 0) topology in each of the three groups shown in Fig. 6 and Fig. 7.) As growth feedback is strengthened, the number of circuit topologies that can maintain functioning decreases (or, equivalently, the number of failed circuits increases). The decreasing behavior of R(k_g) with k_g tends to be slower than exponential [e.g., exp(−βk_g) with β > 0 being a constant].

A general theoretical argument for the scaling law is unavailable. However, if we simplify the system by setting the parameter J in Eq. (12) to be large so the burden b(t) is much smaller than one, we are able to argue that the scaling law is approximately given by where β > 0 and 0 < λ < 1 are two specific constants that depend on the network topology, and the typical value of λ is about 0.6. The exponential scaling is assumed, given its memorylessness. That is, there is no special zero point of k_g, for the reason that a certain level of k_g is mathematically equivalent to a larger d_x, as discussed below.

The quantity R(k_g) is a simple and straightforward measure characterizing the detrimental effect of growth feedback on gene circuits. We carry out a semi-quantitative analysis of this quantity and the effect of k_g on it. The circuit dynamical equations Eqs. (9-12) can be simplified by substituting Eq. (14) into them to cancel the dN/dt terms, leading to Compared with the equations without growth feedback Eqs. (6-8), we see that introducing growth feedback is equivalent to adding a variable k_g/(1 + b) to the degradation terms for each node. Intuitively, the value Q of a network topology measures the volume of the functional region ℳ in the parameter space, which is also a function of k_g. We thus have that R(k_g = k) is the volume of the intersection between ℳ (k_g = k) and ℳ (k_g = 0) divided by the volume of ℳ (k_g = 0): where V (ℳ) is the volume of ℳ.

The picture can be further simplified if we assume the burden b is approximately a constant within a range of k_g. Since growth feedback contributes to an additional term in degradation d_x, strengthening the feedback is equivalent to increasing all three quantities d_x together. Consequently, as k_g increases, the high dimensional region M does not deform, but simply translates in the negative direction in all dimensions of degradation d_x in the parameter space. That is, as growth feedback becomes stronger, it becomes more difficult for the circuit to maintain it functioning.

We restrict our study to the class of transcriptional regulatory networks (TRNs) with the AND logic. In order not to overwhelm readers with too many terms and parameters, we first describe a partial model (an isolated circuit without growth feedback) before introducing the complete model that we study in this work. For an isolated circuit without any growth feedback, and with the topology specified inside the red dashed box in Fig. 1(a), the dynamical equations are where the dynamical variables A, B, and C are the concentrations of each protein (node). The notations are as follows. Let x and y be two arbitrary nodes. The quantity υ_x is the maximal production rate of gene x, d_x is the degradation rate of gene x, dx/dt is its time derivative of the concentration, n_xy and K_xy are the coefficients in the Hill function for a transcriptional regulation from gene x to gene y.

When growth-mediated feedback is present, the dynamical equations of the three-node circuits are modified to where the additional dynamical variable N denotes the density of the host cells, k_g is a parameter controlling the maximal growth rate of the host cells, J is a parameter reflecting how this threenode gene circuit contributes to the burden. Equations. (9) to (13) are the dynamical equations we actually use for simulating the circuit dynamics.

The growth of N is under the regulatory action of two sources: by itself following the logistic equation with the environmental capacity N₀ and by the burden b that represents the competence from the metabolism of the gene circuit. To make the computations feasible, we focus our analysis on the exponential growth phase so that N₀ ≫ N. The equation governing the growth of the cell numbers, Eq. (12), can then be rewritten as where the dilution rate dN/dt is regulated only by the burden b(t) of the gene circuit. While cell growth is inhibited by the metabolism of the gene circuit, the circuit is also regulated by the growth of N that dilutes the concentration of circuit nodes with increasing cell volume. This dilution is reflected by the additional terms −(x/N)(dN/dt) in Eqs. (9-11).

It is useful to clarify the meaning of the degradation parameter d_x and its relationship to growth feedback. While degradation and growth feedback terms have the same sign in the regulatory equations, d_x may include a constant dilution. We assume that d_x represents the sum of all the degradation effects in cells that are distinct from growth feedback. For instance, degradation tags, especially in the ssrA tagging systems (Gottesman et al., 1998), are often used in synthetic gene circuits to increase the degradation rate and thus increase the time scale of the whole system (Elowitz and Leibler, 2000; Fung et al., 2005; Stricker et al., 2008; O’Brien et al., 2012).

We use the fourth-order Runge-Kutta method to numerically integrate the dynamical equations of the gene circuits, with the integration step as t_step = 0.05. The dynamical equations we use are similar to Eqs. (9-13) but with different topologies. All the initial states of A, B, and C are taken to be 0.1. The input signal is initially I₀ = 0.06 and then switched to I₁ = 0.6. The simulation codes can be found in our GitHub repository (link provided in Code availability). The simulation results can be found in our OSF repository (link provided in Data availability).

To achieve the desired adaptation, the circuit’s output should reach a steady state before and after the input signal is switched. The values of O₁ and O₂ can be determined as the output signal associated with the steady states. However, realistically, it is not necessary for the circuit to reach an exact equilibrium. From a computational perspective, the system’s state is almost always asymptotically approaching that exact equilibrium but not actually reaching it. As we are simulating the dynamical process of the circuit, setting a condition such as dA/dt = dB/dt = dC/dt = 0 to determine O₁ or O₂ is not meaningful. From a more biological perspective, relatively small drifting or oscillations in the circuit should not harm the circuit functionality and are acceptable. Therefore, we define a “relatively steady state” where, within a time block of t_block = 200, the standard deviation of the time series of each node x(t) satisfies: std(x) < 1 × 10⁻⁴ and std(x)/mean(x) < 0.05. To further guarantee that the circuit is actually in the “relatively steady state,” two successive time blocks satisfying the standard deviation requirements are needed. The quantities O₁ or O₂ are then defined as the respective mean values of the output signal in that last time block t_block.

We introduce four criteria to determine if a circuit has functional adaptation, and the failure types. The codes implementing these criteria are available in our GitHub repository, with the link provided in Code availability. The failure type results for all circuits tested are available in our OSF repository, with the link provided in Data availability. An additional note is provided in the README file of our GitHub repository for further guidance on generating pie charts similar to Fig. 2 for any network topology or subset of topologies.

A three-node gene circuit subject to growth feedback has a large number of parameters. Let L be the number of links among the three nodes (excluding the input link). The total number of parameters is 2 · 3 + 2(L+ 1) = (2L+ 8). The values of these parameters determine the properties of the regulation links within the circuit and, as a result, the circuit dynamics. The circuit parameters are randomly generated by the Latin hypercube sampling method (Iman et al., 1980) using the function “lhsdesign” in Matlab. The parameters are sampled uniformly either on a logarithmic or a linear scale. The sampling ranges of the parameters are: υ_x ∈ [10⁻¹, 10¹] (sampled in logarithmic scale), d_x ∈ [10⁻², 1] (sampled in logarithmic scale), n_xy ∈ [1, 4] (sampled in linear scale), and K_xy ∈ [10⁻³, 1] (sampled in logarithmic scale).