Discovering adaptation-capable biological network structures using control-theoretic approaches

Constructing biological networks capable of performing specific biological functionalities has been of sustained interest in synthetic biology. Adaptation is one such ubiquitous functional property, which enables every living organism to sense a change in its surroundings and return to its operating condition prior to the disturbance. In this paper, we present a generic systems theory-driven method for designing adaptive protein networks. First, we translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as ‘design requirements’ for the underlying networks. We go on to prove that a protein network with different input–output nodes (proteins) needs to be at least of third-order in order to provide adaptation. Next, we show that the necessary design principles obtained for a three-node network in adaptation consist of negative feedback or a feed-forward realization. We argue that presence of a particular class of negative feedback or feed-forward realization is necessary for a network of any size to provide adaptation. Further, we claim that the necessary structural conditions derived in this work are the strictest among the ones hitherto existed in the literature. Finally, we prove that the capability of producing adaptation is retained for the admissible motifs even when the output node is connected with a downstream system in a feedback fashion. This result explains how complex biological networks achieve robustness while keeping the core motifs unchanged in the context of a particular functionality. We corroborate our theoretical results with detailed and thorough numerical simulations. Overall, our results present a generic, systematic and robust framework for designing various kinds of biological networks.

(C) A proof that the peak response time of any network admissible for perfect adaptation is minimum across a class of other network structures.
(D) Although the negative feedback topology is touted to be capable of perfect adaptation, we, for the first time, showed that not all negative feedback motifs can do justice to this claim. Biochemical networks with negative feedback loops that contain an edge from the output to the input node can not provide perfect adaptation (refer to theorem 4 of the revised manuscript). In this sense, the necessary structural conditions for perfect adaptation put forth in this work are the strictest among the existing literature.
(E) Most importantly, although negative feedback has been shown to be capable of perfect adaptation for networks of small size (not more than three nodes) for larger networks, this assertion has been largely on the basis of intuition (refer to page 53 of the supplementary section in the paper-The topological requirements for robust perfect adaptation in networks of any size. Robyn A, Lance L. Nature Communications.2018;9(13): 1757-1769.). In this paper, we present a rigorous proof (refer to theorem 5 of the revised manuscript) of the fact that a balancer module of any size shall always be locally unstable if all the loops are positive. Therefore, it requires at least one negative feedback to produce perfect adaptation. A stronger condition for the stability of linearised system has enabled us to utilize the wealth of combinatorial matrix theory in order to accomplish this. Table 1 positions our contributions relative to the existing literature in the key aspects of methodology and results. As summarized in the table, the paper makes significant contributions in both the aspects, while making the same assumptions as as those in Araujo et al..
Finally, regarding the assumption of controller nodes, we take this opportunity to point out that (as shown in line 322 of the revised manuscript) the controller module can be conceived as the sub-network containing N −2 nodes of any N−node network. Evidently, the remaining two are the input and the output nodes. Therefore, for a three node network with different I/O nodes, there can be only one controller node possible.
2. The authors have stated several conclusions in the manuscript. Based on the abstract, the authors have: 1. The authors "translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as 'design requirements' for the underlying networks. We go on to prove that a protein network with different input-output nodes (proteins) needs to be at least of third-order in order to provide adaptation." 2. The authors "show that the necessary design principles obtained for a three-node network in adaptation consist of negative feedback or a feed-forward realization. Interestingly, the design principles obtained by the proposed method remain the same for a network of arbitrary size and connectivity." 1 3. The authors "prove that the motifs discovered for adaptation are non-retroactive for a canonical downstream connection. This result explains how complex biological networks achieve robustness while keeping the core motifs unchanged in the context of a particular functionality." Points 1 and 2 are not new and have been demonstrated in different papers. The third part is not clear.

Response:
We agree with the reviewer that a two-node network with different I/O nodes is known to be unable to meet the requirements of perfect adaptation. However, in this work, we, for the first time, using a systems theoretic approach have demonstrated that a two-node network is indeed capable of perfect adaptation provided the output is measured as the concentration of the same node that receives the disturbance input. This has been discussed at length in the section'two-node networks" of the supplementary text.
Secondly, the two-node and three-node networks serve as the building blocks of our approach that aims to discover the structural conditions on networks of any size for perfect adaptation. Araujo et al. (2018) in their seminal study, showed that feedback loops (balancer modules) or incoherent feedforward paths (opposer modules) is essential for perfect adaptation. Further, the insistence on the loops being negative was inspired from the demonstrations in small networks i.e, networks with at most five nodes.Therefore, negative feedback as the necessary element in the balancer module for perfect adaptation has been an extremely important conjectural assertion that required a rigorous proof which we presented in the current work.
Subsequently, we argue that the necessary structural requirements obtained by our approach are the strictest in the existing literature. We showed in theorem 4 of the revised manuscript that not all negative feedbacks with the characteristics of a balancer module can provide perfect adaptation. Negative feedback loops which do not contain any edge from the output to the input node can only be candidate units for adaptation.
Further, the third conclusion refers to lemma 6 of the revised manuscript. where we proved the modularity property of adaptation network in presence of a downstream system connected bidirectionally with the output node.

The manuscript did not emphasize enough what's new in their study. What I find interesting is 1) the discussion on the network's response, which is not well-studied based on my knowledge. 2) to apply the method to higher-order networks. This topic has been studies but it is still very interesting if the authors could summarize new principles based on their own work. The authors have mentioned both parts, but the conclusions are not clear enough to guide further study for either theoretical or experimental biologists. And it's not clear from the writing whether the authors need to assume a single control node for large networks.
Response: The manuscript has been modified in accordance with this remark. We have revised the abstract section where we emphasised on the novel contributions of this work. Secondly, in the introduction section, we contextualized these contributions based on the existing literature. We hope that the revised version clearly highlights the contribution of this work. Finally, the revised version of the discussion section carries an effective road map for the future works to follow.
As shown in line 322 of the revised manuscript, the controller module can be conceived as the sub-network containing N −2 nodes of any N−node network. Evidently, the remaining two are the input and the output nodes. Therefore, we do not assume a single control node.  • Adaptation characterised by Sensitivity and precision.
• Adopted computational screening approach for a three-enzyme network.
• Justified the result for a three-protein system using Jacobian analysis.
NFBLB or IFFLP are the only two three-node motifs admissible for adaptation.
• Each enzyme can have two statesactive and inactive.
Tang et al (2016) Composed a Jacobian-based approach to deduce the condition (the 'cofactor condition') for infinite precision.
All the existing admissible network structures in the literature obeys the proposed condition.
The elements of the Jacobian matrix do not change throughout the entire state space.
Araujo et al (2018) • Utilizing the Jacobian as the digraph matrix, the admissible structures were obtained via the application of graph theory.
• Condition for infinite precision with respect to step input was obtained through Jacobian analysis.
• The sign of the determinant of the digraph matrix was used to obtain a week necessary condition for linear asymptotic stability.
• Balancer module (feedback loops with buffer node) or Opposer module (same as IFFLP) are the only two necessary modules for adaptation in networks of any size.
• Conjectured that balancer module should contain at least one negative feedback for stability.
The elements of the Jacobian matrix do not change their sign throughout the entire state space.
• Condition for infinite precision with respect to bounded input was obtained through Jacobian analysis.
• Utilizing the Jacobian as the digraph matrix, the admissible structures were obtained via the application of combinatorial matrix theory.
Feedback loops with buffer node / Haldane motifs or network motifs with opposing forward paths are the only two necessary modules for adaptation in networks of any size.
Similar to Araujo et al.

Current Manuscript
• Employed the notion of controllability to deduce the condition for non-zero sensitivity.
• Showed using control theory, that perfect adaptation ipso facto minimizes the peak time among the class of stable responses.
• Obtained the conditions for infinite precision for a network of any size with respect to staircase type disturbance.
• A set of N conditions concerning the sign of each coefficient in the characteristic polynomial of the digraph matrix was derived for the local asymptotic stability of the system. This serves as the strongest necessary conditions for stability among the ones in the literature.
• Negative feedback that contains the edge from output to the input node cannot provide adaptation.
• The adaptive response remains unchanged in the presence of a downstream system.

Reviewer 2
1. The manuscript is clearly-written and describes a timely subject that is of interest in the field of understanding and designing biological feedback mechanisms. However, I do have a major concern: I am not clear on how the work distinguishes itself from several previously-published papers. I hope that the authors can address the ways in which the current work differentiates itself from the key results reported in the following: text can act as an appropriate starting point it is to be noted that the present study deals with the most generic situation of networks of any size and configuration. Furthermore, in this context, the current work uses a stronger set of conditions for stability and discovers new conditions for non zero sensitivity (controllability at each steady-state) in order to find out network structures of any size that can attain perfect adaptation. A short and comprehensive discussion on the same has been added in the revised manuscript with line numbers ranging from 51-55. Response: While we partly agree with the argument that the Jacobian treatment of the problem is not new, we argue that the Jacobian treatment hitherto adopted has been limited in its capacity to characterise adaptation with respect to both precision and sensitivity. Therefore, in addition to the Jacobian analysis, we employed a number of well-known concepts inspired by control theory to deduce precise mathematical (hence structural) conditions both for non-zero sensitivity and infinite precision. This has enabled us to contribute to this field of study in the following ways.

Ref 34 -Robyn P. Araujo and Lance A. Liotta (2018). [And Araujo and Liotta cite, in their Methods, a previous publication by
(A) The Jacobian treatment can be used in finding the conditions for infinite precision-an important quality for perfect adaptation but it does not guarantee non-zero sensitivity (which is also another important ask for adaptation). In this respect, our work demonstrates that the well-known concept of controllability is required over and above the Jacobian analysis. 3 Table 1: A comparative study between the current manuscript and the relevant literature

Reference Contribution Assumptions
Methodological Results Ma et al (2011) • Adaptation characterised by Sensitivity and precision.
• Adopted computational screening approach for a three-enzyme network.
• Justified the result for a three-protein system using Jacobian analysis.
NFBLB or IFFLP are the only two three-node motifs admissible for adaptation.
• Each enzyme can have two statesactive and inactive.
Tang et al (2016) Composed a Jacobian-based approach to deduce the condition (the 'cofactor condition') for infinite precision.
All the existing admissible network structures in the literature obeys the proposed condition.
The elements of the Jacobian matrix do not change throughout the entire state space.
Araujo et al (2018) • Utilizing the Jacobian as the digraph matrix, the admissible structures were obtained via the application of graph theory.
• Condition for infinite precision with respect to step input was obtained through Jacobian analysis.
• The sign of the determinant of the digraph matrix was used to obtain a week necessary condition for linear asymptotic stability.
• Balancer module (feedback loops with buffer node) or Opposer module (same as IFFLP) are the only two necessary modules for adaptation in networks of any size.
• Conjectured that balancer module should contain at least one negative feedback for stability.
The elements of the Jacobian matrix do not change their sign throughout the entire state space.
• Condition for infinite precision with respect to bounded input was obtained through Jacobian analysis.
• Utilizing the Jacobian as the digraph matrix, the admissible structures were obtained via the application of combinatorial matrix theory.
Feedback loops with buffer node / Haldane motifs or network motifs with opposing forward paths are the only two necessary modules for adaptation in networks of any size.
Similar to Araujo et al.

Current Manuscript
• Employed the notion of controllability to deduce the condition for non-zero sensitivity.
• Showed using control theory, that perfect adaptation ipso facto minimizes the peak time among the class of stable responses.
• Obtained the conditions for infinite precision for a network of any size with respect to staircase type disturbance.
• A set of N conditions concerning the sign of each coefficient in the characteristic polynomial of the digraph matrix was derived for the local asymptotic stability of the system. This serves as the strongest necessary conditions for stability among the ones in the literature.
• Negative feedback that contains the edge from output to the input node cannot provide adaptation.
• The adaptive response remains unchanged in the presence of a downstream system.

Similar to Araujo et al.
(B) The present work offers a systematic treatment in developing the conditions for perfect adaptation in presence of step type disturbance. Conversely, this also aids in a systematic understanding behind why certain networks such as Voltage gated Sodium channel can only perform perfect adaptation only once ( fig. 3 of the revised manuscript).
(C) A proof that the peak response time of any network admissible for perfect adaptation is minimum across a class of other network structures.
(D) Although the negative feedback topology is touted to be capable of perfect adaptation, we, for the first time, showed that not all negative feedback motifs can do justice to this claim. Biochemical networks with negative feedback loops that contain an edge from the output to the input node can not provide perfect adaptation (refer to theorem 4 of the revised manuscript). In this sense, the necessary structural conditions for perfect adaptation put forth in this work are the strictest among the existing literature.
(E) Most importantly, although negative feedback has been shown to be capable of perfect adaptation for networks of small size (not more than three nodes) for larger networks, this assertion has been largely on the basis of intuition (refer to page 53 of the supplementary section in the paper-The topological requirements for robust perfect adaptation in networks of any size. Robyn A, Lance L. Nature Communications.2018;9(13):1757-1769.). In this paper, we present a rigorous proof (refer to theorem 5 of the revised manuscript) of the fact that a balancer module of any size shall always be locally unstable if all the loops are positive. Therefore, it requires at least one negative feedback to produce perfect adaptation. A stronger condition for the stability of linearised system has enabled us to utilize the wealth of combinatorial matrix theory in order to accomplish this.  Table 1 positions our contributions relative to the existing literature in the key aspects of methodology and results. As summarized in the table, the paper makes significant contributions in both the aspects, while making the same assumptions as as those in Araujo et al..