A unified approach to dissecting biphasic responses in cell signaling
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This study presents a useful mathematical analysis of different signaling networks in an attempt to provide general rules that give rise to biphasic responses, a widely observed behavior in biology in which the outputs of the network depend non-monotonically on the inputs. Determining general conditions that underlie this behavior would be useful in engineering synthetic biological systems and for mechanistically understanding biphasic responses in biological systems. However, whereas the mathematical approach and methods are solid, as they stand, the analyses are inadequate to assess how these findings are applicable in nature and which are general.
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Abstract
Biphasic responses are encountered at all levels in biological systems. At the cellular level, biphasic dose-responses are widely encountered in cell signaling and post-translational modification systems and represent safeguards against overactivation or overexpression of species. In this paper, we provide a unified theoretical synthesis of biphasic responses in cell signaling systems, by assessing signaling systems ranging from basic biochemical building blocks to canonical network structures to well-characterized exemplars on one hand, and examining different types of doses on the other. By using analytical and computational approaches applied to a range of systems across levels (described by broadly employed models), we reveal (i) design principles enabling the presence of biphasic responses, including in almost all instances, an explicit characterization of the parameter space (ii) structural factors which preclude the possibility of biphasic responses (iii) different combinations of the presence or absence of enzyme-biphasic and substrate-biphasic responses, representing safeguards against overactivation and overexpression, respectively (iv) the possibility of broadly robust biphasic responses (v) the complete alteration of signaling behavior in a network due to biphasic interactions between species (biphasic regulation) (vi) the propensity of different co-existing biphasic responses in the Erk signaling network. These results both individually and in totality have a number of important consequences for systems and synthetic biology.
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Author Response
Reviewer #1 (Public Review):
This paper examines different signaling networks and attempts to give general results for when the network will exhibit biphasic behavior, which is the situation when the output of the network is a non-monotonic function of its inputs. The strength of the paper is in the approach it takes. It starts with the simplest network motifs that produce biphasic behavior and then asks too what happens when these motifs are parts of larger networks. Their approach is in contrast to the usual way in which this question is tackled, which tends to be within the confines of a specific signaling network, where general results like the ones that the authors are after, might be hard to spot.
We thank the reviewer for the careful reading of the manuscript and for the comments and appreciate the fact the …
Author Response
Reviewer #1 (Public Review):
This paper examines different signaling networks and attempts to give general results for when the network will exhibit biphasic behavior, which is the situation when the output of the network is a non-monotonic function of its inputs. The strength of the paper is in the approach it takes. It starts with the simplest network motifs that produce biphasic behavior and then asks too what happens when these motifs are parts of larger networks. Their approach is in contrast to the usual way in which this question is tackled, which tends to be within the confines of a specific signaling network, where general results like the ones that the authors are after, might be hard to spot.
We thank the reviewer for the careful reading of the manuscript and for the comments and appreciate the fact the reviewer regards the approach as the strength of the paper.
The weakness of the paper, in my opinion, is the rather formal description of the results which I am afraid will be of rather limited utility to experimental groups seeking to make use of them. The paper attempts to provide general rules for when to expect biphasic behavior and it was hard to assess to what extent such rules exist as behaviors can change depending on the context of a larger network in which the smaller biphasic one is embedded. The other thing that made assessing the generality of the results difficult is that the input-output functions shown in all the figures are computed for a specific choice of parameters and I was left wondering how different choices of parameters might change the reported behaviors. The lack of specific proposals for how their results should guide future experiments on different signaling networks is another weakness.
We address these points in a number of ways. Initially our presentation was intended to highlight unambiguously which systems (especially the substrate modification building blocks) were capable of biphasic response and which were not, and highlighting parameter dependence on intrinsic kinetic parameters. Based on both referee comments, we make a number of changes
(a) We highlight the rationale for choosing the suite of biochemical substrate modification systems: enzyme/substrate sharing is a key driver for the origins of biphasic responses and the suite of systems we employ allows us to systematically explore this (see Response to Essential Revisions). These are building blocks of many pathways,
(b) Biphasic responses emerge from a built in competing effect. In every instance of substrate modification systems, we now highlight the mechanistic underpinning which gives rise to the competing effect responsible for the biphasic response. This will help experimentalists and modellers alike obtain insights into how such behaviour may arise, and the associated ingredients which facilitate that (which may be relevant in other systems). Similarly, we highlight how altered behaviour at the network level may arise from a biphasic interaction pattern, providing the intuition therein and guide further experimental investigation (also see Response to Essential Revisions).
(c) With regard to parameters (also see Response to Essential Comments) firstly we emphasize that we completely characterize at the substrate modification level, whether biphasic responses are possible as a function of intrinsic kinetic constants. This is done for every system studied. In Fig 2, we depict this, along with sample biphasic dose responses, for pictorial depiction. However, the essential point is that the parametric dependence on intrinsic kinetic parameters is completely done. We indicate in which cases biphasic responses are impossible irrespective of intrinsic kinetic parameters, where they can be obtained for every value of the intrinsic kinetic parameters, and where there are partial restrictions in the intrinsic kinetic parameter space for obtaining this. In the revision we have performed further parametric analysis to assess the impact of species total amount providing further insights. We have also shown that in all these systems biphasic responses can be obtained in ranges of kinetic parameters similar to those found experimentally (eg Wistel et al 2018) and for reasonable species total amounts in systems and synthetic biology. This is analyzed, and depicted in Figure 2-figure supplement 3 and Figure 2-figure supplement 4.
(d) Also, in response to another comment (about behaviour changing in networks): we first emphasize that we start at the substrate modification level to uncover drivers of biphasic responses at this level. Biphasic responses arise from an inbuilt competing effect and we demonstrate different ways in which such an inbuilt competing effect arises, through sharing of enzymes or substrates. While it is true that the behaviour can change as part of a network (a) It still remains that there are these in-built competing effects which can generate biphasic responses (both substrate and enzyme) and this can manifest at a pathway or network level under suitable conditions (b) the fact that behaviour at a network level may be altered is exactly why we consider studies at the network level showing both biphasic patterns in interaction (the overall behaviour is determined by the motif and the biphasic pattern of interaction and studies involving interaction of biphasic responses at both the network and substrate modification level!! (subsection: The network level)
(e) We have also expanded on a paragraph on testable predictions in the conclusions (p10).
Taken together, we believe that these results should interest both experimentalists and modellers and have intrinsic value as well.
While I appreciate that the authors adopted a style of presenting their results such that all the mathematics is buried in the figures, I found that it made reading the paper quite difficult, and contributed to my confusion about which results are general and insensitive to parameter choices and which are not. I believe a narrative that integrated the math with some simple intuition might have been more effective. For example, when the authors say in the text that model M0 is incapable of displaying biphasic response, how general is that result? Later on, when discussing model M2, they provide a criterion for biphasic response in terms of products of rate constants satisfying an inequality, but the meaning of this condition is not described. Such things make it hard to learn from the authors' work.
This has indeed been incorporated, and we agree that presenting the intuition and mechanistic underpinning for the behaviour aids readability. In addition to the points about parameters which are now explained at length in the paper , there are a number of paragraphs providing the mechanistic underpinning and intuition for why the behaviour is obtained. Both these are discussed at length in Response to Essential Revisions. Thus, both the mechanistic intuition and the role of parameters are addressed in detail in the revision.
When M0 is mentioned to be incapable of yielding biphasic responses we mean just that: irrespective of any parameter choice in the model. The meaning of the criterion in Model M2 is now discussed. We take the point about not being able to learn from the work seriously and have made various changes both on the intuition and clarifying the impact of parameters.
The text is sprinkled with statements like "this reveals the plurality of information processing behaviors..." where the meaning is quite opaque (for this example, there is no description of "information processing" and what it might mean in this context) and therefore it makes it hard to understand what are the lessons learned from these calculations. Another example is found in the description of Erk regulation where the authors speak of "significant robustness" but what is meant by "significant" is also unclear.
Yes, we agree that these phrases are distracting and not adding much and so we have removed them.
Overall, I think this is an interesting attempt to provide a general mathematical framework for analyzing biphasic response of signaling networks, but the authors fall short for the reasons described above. I think a lot can be fixed by improving the way the results are presented.
We have indeed taken these comments on board and aimed to improve the presentation
Reviewer #2 (Public Review):
Biphasic responses are widely observed in biological systems and the determination of general design principles underlying biphasic responses is an important problem. The authors attempt to study this problem using a range of biochemical signaling models ranging from simple enzymatic modification and de-modification of a single substrate to systems with multiple enzymes and substrates. The authors used analytical and computational calculations to determine conditions such as network topology, range of concentrations, and rate parameters that could give rise to biphasic responses. I think the approach and the result of their investigation are interesting and can be potentially useful. However, the conditions for biphasic responses are described in terms of parameter ranges or relationships in particular biochemical models, and these parameters have not been connected to the values of concentrations or rates in real biological systems. This makes it difficult to evaluate how these findings would be applicable in nature or in experiments. It might also help if some general mechanisms in terms of competition/cooperation of time scales/processes are gleaned which potentially can be used to analyze biphasic responses in real biological systems.
We thank the reviewer for a careful reading of the manuscript and for the various comments and are happy to see the reviewer find the approach interesting. We address these comments in more detail below.
Reading these comments, we recognized how various analysis and algebraic equations could appear opaque to a reader both in terms of what it conveys and its import. To address this, we made a number of changes.
First and foremost, we provide the mechanistic underpinning and intuition for why a competing effect emerges in the first place. We do this for every substrate modification system we analyze and make further comments in the subsection focussing on the network level as well as ERK This intuition should help a reader where the result is coming from and be then able to see if it might apply in a quite different system. This is discussed in detail in Response to Essential Revisions.
Secondly, we have discussed many aspects of the parameters in more detail. Our goal, especially in substrate modification systems was to be able to completely characterize the role of intrinsic kinetic parameters: whether biphasic responses was impossible irrespective of parameters, whether they were possible for every value of intrinsic kinetic parameters or whether they were possible in a subset of kinetic parameter space. This has been done for every substrate modification system, and has been discussed more explicitly in the revision. Furthermore, when biphasic responses were possible, we aimed to determine the impact of species total amounts which facilitated the response. Here we performed additional analytical and semi-analytical work. Additionally with the semi-analytical work and parameters chosen in ranges very similar to those found experimentally (eg Wistel et al 2018), we are able to show that biphasic responses can indeed be obtained in experimentally feasible ranges. Further aspects of the parameters are discussed in detail in the Response to Essential Revisions. In particular, a number of new paragraphs (p2-3, p6) and plots Figure 2-figure supplement 3 and Figure 2-figure supplement 4 specifically deal with this.
Taken together these address the reviewers points.
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eLife assessment
This study presents a useful mathematical analysis of different signaling networks in an attempt to provide general rules that give rise to biphasic responses, a widely observed behavior in biology in which the outputs of the network depend non-monotonically on the inputs. Determining general conditions that underlie this behavior would be useful in engineering synthetic biological systems and for mechanistically understanding biphasic responses in biological systems. However, whereas the mathematical approach and methods are solid, as they stand, the analyses are inadequate to assess how these findings are applicable in nature and which are general.
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Reviewer #1 (Public Review):
This paper examines different signaling networks and attempts to give general results for when the network will exhibit biphasic behavior, which is the situation when the output of the network is a non-monotonic function of its inputs. The strength of the paper is in the approach it takes. It starts with the simplest network motifs that produce biphasic behavior and then asks too what happens when these motifs are parts of larger networks. Their approach is in contrast to the usual way in which this question is tackled, which tends to be within the confines of a specific signaling network, where general results like the ones that the authors are after, might be hard to spot.
The weakness of the paper, in my opinion, is the rather formal description of the results which I am afraid will be of rather limited …
Reviewer #1 (Public Review):
This paper examines different signaling networks and attempts to give general results for when the network will exhibit biphasic behavior, which is the situation when the output of the network is a non-monotonic function of its inputs. The strength of the paper is in the approach it takes. It starts with the simplest network motifs that produce biphasic behavior and then asks too what happens when these motifs are parts of larger networks. Their approach is in contrast to the usual way in which this question is tackled, which tends to be within the confines of a specific signaling network, where general results like the ones that the authors are after, might be hard to spot.
The weakness of the paper, in my opinion, is the rather formal description of the results which I am afraid will be of rather limited utility to experimental groups seeking to make use of them. The paper attempts to provide general rules for when to expect biphasic behavior and it was hard to assess to what extent such rules exist as behaviors can change depending on the context of a larger network in which the smaller biphasic one is embedded. The other thing that made assessing the generality of the results difficult is that the input-output functions shown in all the figures are computed for a specific choice of parameters and I was left wondering how different choices of parameters might change the reported behaviors. The lack of specific proposals for how their results should guide future experiments on different signaling networks is another weakness.
While I appreciate that the authors adopted a style of presenting their results such that all the mathematics is buried in the figures, I found that it made reading the paper quite difficult, and contributed to my confusion about which results are general and insensitive to parameter choices and which are not. I believe a narrative that integrated the math with some simple intuition might have been more effective. For example, when the authors say in the text that model M0 is incapable of displaying biphasic response, how general is that result? Later on, when discussing model M2, they provide a criterion for biphasic response in terms of products of rate constants satisfying an inequality, but the meaning of this condition is not described. Such things make it hard to learn from the authors' work.
The text is sprinkled with statements like "this reveals the plurality of information processing behaviors..." where the meaning is quite opaque (for this example, there is no description of "information processing" and what it might mean in this context) and therefore it makes it hard to understand what are the lessons learned from these calculations. Another example is found in the description of Erk regulation where the authors speak of "significant robustness" but what is meant by "significant" is also unclear.
Overall, I think this is an interesting attempt to provide a general mathematical framework for analyzing biphasic response of signaling networks, but the authors fall short for the reasons described above. I think a lot can be fixed by improving the way the results are presented.
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Reviewer #2 (Public Review):
Biphasic responses are widely observed in biological systems and the determination of general design principles underlying biphasic responses is an important problem. The authors attempt to study this problem using a range of biochemical signaling models ranging from simple enzymatic modification and de-modification of a single substrate to systems with multiple enzymes and substrates. The authors used analytical and computational calculations to determine conditions such as network topology, range of concentrations, and rate parameters that could give rise to biphasic responses. I think the approach and the result of their investigation are interesting and can be potentially useful. However, the conditions for biphasic responses are described in terms of parameter ranges or relationships in particular …
Reviewer #2 (Public Review):
Biphasic responses are widely observed in biological systems and the determination of general design principles underlying biphasic responses is an important problem. The authors attempt to study this problem using a range of biochemical signaling models ranging from simple enzymatic modification and de-modification of a single substrate to systems with multiple enzymes and substrates. The authors used analytical and computational calculations to determine conditions such as network topology, range of concentrations, and rate parameters that could give rise to biphasic responses. I think the approach and the result of their investigation are interesting and can be potentially useful. However, the conditions for biphasic responses are described in terms of parameter ranges or relationships in particular biochemical models, and these parameters have not been connected to the values of concentrations or rates in real biological systems. This makes it difficult to evaluate how these findings would be applicable in nature or in experiments. It might also help if some general mechanisms in terms of competition/cooperation of time scales/processes are gleaned which potentially can be used to analyze biphasic responses in real biological systems.
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