Effects of growth feedback on gene circuits: A dynamical understanding

Joint Public Review:

Using computational modeling, this manuscript explores the effect of growth feedback on the performance of gene networks capable of adaptation. The authors selected 425 hypothetical synthetic circuits that were shown to achieve nearly perfect adaptation in two earlier computational studies (see Ma et al. 2009, and Shi et al. 2017). They examined the effects of cell growth feedback by introducing additional terms to the ordinary differential equation-based models, and performed numerical simulations to check the retainment and the loss of the adaptation responses of the circuits in the presence of growth feedback. The authors show that growth feedback can disrupt the gene network adaptation dynamics in different ways, and report some exceptional core motifs which allow for robust performance in the presence of growth feedback. They also used a metric to establish a scaling law between a circuit robustness measure and the strength of growth feedback. These results have important implications in the field of synthetic biology, where unforeseen interactions between designed gene circuits and the host often disrupt the desired behavior. The paper's conclusions are supported by their simulation results, although these are presented in their summary formats and it would be useful for the community if the detailed results for each topology were available as a supplementary file or through the authors' GitHub repository.

Strengths
- This work included a detailed investigation of the reasons for adaptation failure upon introducing cell growth to the systems. The comprehensiveness of the analysis makes the work stand out among studies of functional screening of network topologies of gene regulation.

- The authors' approaches for assessment of robustness, such as the survival ratio Q, can be useful for a wide range of topologies beyond adaptation. The scaling law obtained with those approaches is interesting.

Weaknesses
- The title suggests that the work investigates the 'effects of growth feedback on gene circuits'. However, the performance of 'nearly perfect adaptation' was chosen for the majority of the work, leaving the question of whether the authors' conclusion regarding the effects of growth feedback is applicable to other functional networks.

- This work relies extensively on an earlier study, evaluating only a selected set of 425 topologies that were shown to give adaptive responses (Shi et al., 2017). This limited selection has two potential issues. First, as the authors mentioned in the introduction, growth feedback can also induce emerging dynamics even without existing function-enabling gene circuits, as an example of the "effects of growth feedback on gene circuits". Limiting the investigation to only successful circuits for adaptation makes it unclear whether growth feedback can turn the circuits that failed to produce adaptation by themselves into adaptation-enabling circuits. Secondly, as the Shi et al. (2017) study also used numerical experiments to achieve their conclusions about successful topologies, it is unclear whether the numerical experiments in the present study are compatible with the earlier work regarding the choice of equation forms and ranges of parameter values. The authors also assumed that all readers have sufficient understanding of the 425 topologies and their derivation before reading this paper.

- The authors' model does not describe the impact of growth via a biological mechanism: they model growth as an additional dilution rate and calculate growth rate based on a phenomenological description with growth rate occurring at a maximum (k_g) scaled by the circuit 'burden' b(t). Therefore, the authors' model does not capture potential growth rate changes in parameter values (e.g., synthetic protein production falls with increasing growth rate; see Scott & Hwa, 2023).

- The authors made several claims about the bifurcations (infinite-period, saddle-node, etc) underlying the abrupt changes leading to failures of adaptations. There is a lack of evidence supporting these claims. Both local and global bifurcations can be demonstrated with semi-analytic approaches such as numerical continuation along with investigations of eigenvalues of the Jacobian matrix. The claims based on ODE solutions alone are not sound.

- The impact of biochemical noise is not evaluated in this work; the author's analysis is only carried out in a deterministic regime.

Author Response

Joint Public Review

“Using computational modeling, this manuscript explores the effect of growth feedback on the performance of gene networks capable of adaptation. The authors selected 425 hypothetical synthetic circuits that were shown to achieve nearly perfect adaptation in two earlier computational studies (see Ma et al. 2009, and Shi et al. 2017). They examined the effects of cell growth feedback by introducing additional terms to the ordinary differential equation-based models, and performed numerical simulations to check the retainment and the loss of the adaptation responses of the circuits in the presence of growth feedback. The authors show that growth feedback can disrupt the gene network adaptation dynamics in different ways, and report some exceptional core motifs which allow for robust performance in the presence of growth feedback. They also used a metric to establish a scaling law between a circuit robustness measure and the strength of growth feedback. These results have important implications in the field of synthetic biology, where unforeseen interactions between designed gene circuits and the host often disrupt the desired behavior. The paper’s conclusions are supported by their simulation results, although these are presented in their summary formats and it would be useful for the community if the detailed results for each topology were available as a supplementary file or through the authors’ GitHub repository.”

We will update our GitHub repository with detailed results for each topology, along with other simulation details and results that might be of interest to the readers.

Strengths: “This work included a detailed investigation of the reasons for adaptation failure upon introducing cell growth to the systems. The comprehensiveness of the analysis makes the work stand out among studies of functional screening of network topologies of gene regulation.” “The authors’ approaches for assessment of robustness, such as the survival ratio Q, can be useful for a wide range of topologies beyond adaptation. The scaling law obtained with those approaches is interesting.”

We are grateful to the referees and editors for their positive assessment of our work.

Weaknesses 1: “The title suggests that the work investigates the ’effects of growth feedback on gene circuits’. However, the performance of ’nearly perfect adaptation’ was chosen for the majority of the work, leaving the question of whether the authors’ conclusion regarding the effects of growth feedback is applicable to other functional networks.”

We will change the title of the paper from “Effects of growth feedback on gene circuits: A dynamical understanding” to “Effects of growth feedback on adaptive gene circuits: A dynamical understanding,” because the focus of our current work was on gene circuits with adaptation. Our work provided a framework that can be readily generalized to investigate the effects of growth feedback in other functional networks such as bistable gene circuits.

Weaknesses 2: “This work relies extensively on an earlier study, evaluating only a selected set of 425 topologies that were shown to give adaptive responses (Shi et al., 2017). This limited selection has two potential issues. First, as the authors mentioned in the introduction, growth feedback can also induce emerging dynamics even without existing function-enabling gene circuits, as an example of the ”effects of growth feedback on gene circuits”. Limiting the investigation to only successful circuits for adaptation makes it unclear whether growth feedback can turn the circuits that failed to produce adaptation by themselves into adaptation-enabling circuits. Secondly, as the Shi et al. (2017) study also used numerical experiments to achieve their conclusions about successful topologies, it is unclear whether the numerical experiments in the present study are compatible with the earlier work regarding the choice of equation forms and ranges of parameter values. The authors also assumed that all readers have sufficient understanding of the 425 topologies and their derivation before reading this paper.”

We will make the following revisions.

1. We will modify the title of the paper as discussed above. The reviewers/editors are insightful that growth feedback could turn a non-adaptive circuit into an adaptation-enabling one - an interesting possibility worth further study.

2. We will provide details of all the pertinent numerical simulations, highlighting the differences from those in the previous work (Shi et al., 2017). Briefly, our adaptation criteria are stricter than those utilized in that work. As a result, out of the 425 topologies, random sampling based on our criteria identified adaptation in 414 topologies. For the remaining 11 topologies, either our more strict criteria have eliminated the possibility for the gene circuits to be adaptive, or the adaptive region in the high-dimensional parameter space is too small to be detected by random sampling.

3. We will describe the 425 topologies utilized in our study and provide more detail in the GitHub repository, including the topological structures and the parameter sets leading to adaptation.

Weaknesses 3: “The authors’ model does not describe the impact of growth via a biological mechanism: they model growth as an additional dilution rate and calculate growth rate based on a phenomenological description with growth rate occurring at a maximum (kg) scaled by the circuit ’burden’ b(t). Therefore, the authors’ model does not capture potential growth rate changes in parameter values (e.g., synthetic protein production falls with increasing growth rate; see Scott & Hwa, 2023).”

We considered dilution due to cell growth as the dominant factor of growth feedback. In fact, we studied the adaptive circuits without growth and their ability to maintain their adaptive behaviors after dilution into a fresh medium, based on a recent work [Zhang, et al., Nature Chemical Biology 16.6 (2020): 695-701]. A higher growth rate can change synthetic protein production. However, the dynamic roles of the dilution and growth-affected production rate should be analogous, given that they both act as inhibitory factors arising from cell growth as mentioned by the reviewers/editors. Taking the growth effect on the production rate into account would require a more comprehensive study. We will elaborate on the limitation of our modeling framework and include the pertinent references (e.g., Scott & Hwa, 2023).

Weaknesses 4: “The authors made several claims about the bifurcations (infinite-period, saddle-node, etc) underlying the abrupt changes leading to failures of adaptations. There is a lack of evidence supporting these claims. Both local and global bifurcations can be demonstrated with semi-analytic approaches such as numerical continuation along with investigations of eigenvalues of the Jacobian matrix. The claims based on ODE solutions alone are not sound.”

We will add this material to our next version of the paper. A further semi-analytic analysis can better justify the numerically discovered bifurcations.

Weaknesses 5: “The impact of biochemical noise is not evaluated in this work; the author’s analysis is only carried out in a deterministic regime.”

Our work focused on uncovering the deterministic dynamical mechanisms underlying growthfeedback induced circuit failures in situations where all protein concentrations are high so that neglecting the effects of biochemical noises can be justified. Incorporating noises into our analysis will significantly complicate the study and likely prevent the dynamical origin of the failures from being unveiled. Nonetheless, the effects of biochemical noises are important and we will provide a discussion in the revised manuscript.