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The authors have declared that no competing interests exist.

Conceived and designed the experiments: VBK LC OSS. Performed the experiments: VBK EF CW LC OSS. Analyzed the data: VBK EF OSS. Contributed reagents/materials/analysis tools: VBK EF CW LC OSS. Wrote the paper: VBK EF CW LC OSS.

Achieving a complete understanding of cellular signal transduction requires deciphering the relation between structural and biochemical features of a signaling system and the shape of the signal-response relationship it embeds. Using explicit analytical expressions and numerical simulations, we present here this relation for four-layered phosphorelays, which are signaling systems that are ubiquitous in prokaryotes and also found in lower eukaryotes and plants. We derive an analytical expression that relates the shape of the signal-response relationship in a relay to the kinetic rates of forward, reverse phosphorylation and hydrolysis reactions. This reveals a set of mathematical conditions which, when satisfied, dictate the shape of the signal-response relationship. We find that a specific topology also observed in nature can satisfy these conditions in such a way to allow plasticity among hyperbolic and sigmoidal signal-response relationships. Particularly, the shape of the signal-response relationship of this relay topology can be tuned by altering kinetic rates and total protein levels at different parts of the relay. These findings provide an important step towards predicting response dynamics of phosphorelays, and the nature of subsequent physiological responses that they mediate, solely from topological features and few composite measurements; measuring the ratio of reverse and forward phosphorylation rate constants could be sufficient to determine the shape of the signal-response relationship the relay exhibits. Furthermore, they highlight the potential ways in which selective pressures on signal processing could have played a role in the evolution of the observed structural and biochemical characteristic in phosphorelays.

Two-component phosphorelays constitute the key signaling pathways in all prokaryotes, lower eukaryotes, and plants, where they underline diverse physiological responses such as virulence, cell-cycle progression and sporulation. Despite such prevalence, our understanding of the dynamics and function of these systems remains incomplete. In particular, it is not clear why all phosphorelays studied to date embed a four-layer architecture and how their dynamics could relate to phenotypic variability in the resulting responses. Here, we use analytical approaches and numerical simulations to analyze all possible phosphorelay topologies of length four and embedding reverse phosphorylation. We find that only two topologies can embed both hyperbolic and sigmoidal signal-response relationships, and that one of these can underlie high noise (i.e. phenotypic variability) in population responses. All of the remaining topologies are either non-functional or can embed only a hyperbolic signal-response relationship. Using analytical solutions of relay dynamics, we find that reverse phosphorylation from the third layer, a topological featured commonly observed in nature, is a necessary condition for sigmoidal signal-response relationship.

Biological signaling systems allow cells to produce appropriate physiological responses to external and internal clues. Understanding the signal-response relationships of these systems and how this is shaped by specific biochemical mechanisms is fundamental to predicting and engineering cellular behavior. Among the different signaling systems that cells use, phosphorelays are found in prokaryotes, lower eukaryotes, and plants

Cartoon representation of the general four layered phosphorelay model. Hydrolysis reactions (on aspartate residues found on REC and RR proteins only) and forward and reverse phosphorylation reactions are shown, along with the possibility of the HK being bifunctional. _{s}

What are the significances, if any, of these structural and biochemical features of phosphorelays? More broadly, what is the functional benefit of having a specific phosphorelay structure for the cell? A widely held view is that phosphorelays have evolved to allow the cell to achieve signal integration using phosphorylation at their different layers

Achieving such a broader understanding requires mathematical analysis of the signal-response relationship of phosphorelays under a range of alternative biochemical assumptions and parameter ranges. Here, we take this approach and study the role of reverse phosphorylation and hydrolysis reactions in four-layered phosphorelays. Using both numerical simulations and analytical approaches we evaluate the shape of the signal-response relationship in all possible four-layered phosphorelay topologies, arising from distributing hydrolysis and reverse phosphorylation reactions on a base structure. We find that almost half of these topologies are not capable of signal transduction, and further, among those that are, only a few allow more than one type of signal-response relationship. By solving the steady state equations of the system analytically, we find mathematical criteria that relate the total protein levels in different layers and the rate constants of hydrolysis and phosphorylation reactions in a given relay to the shape of the signal-response relationship. In particular, we show that reverse phosphorylation reactions between REC-Hpt and Hpt-RR and hydrolysis at REC and RR enable sigmoidal signal-response relationships in a four-layered phosphorelay that is otherwise confined to displaying only hyperbolic response relationships. Interestingly, these topological features are found in natural four-zlayered phosphorelays. We further show that the ratio of forward and reverse phosphorylation rate constants between REC-Hpt and between Hpt-RR allows tuning the signal-response relationship among the hyperbolic and sigmoidal regimes. In the latter regime, the response of the system to a step signal is faster and noisier compared to the hyperbolic regime, thus enabling subsequent control of the timing and population-level variability of physiological responses. The emerging picture from these analytical and numerical results is that the observed features of four-layered phosphorelays endow them with tunable functionality (i.e. tunable signal-response relationship). These results account for some of the highly conserved topological and biochemical features of phosphorelays and will facilitate experimental determination of signal-response relationship in phosphorelays. In particular, they show that measuring the ratio of reverse and forward phosphorylation rate constants could be sufficient to determine the shape of the signal-response relationship in four-layered phosphorelays.

To study the role of reverse phosphorylation on the shape of the signal-response relationship, we build a generic mathematical model of a phosphorelay with four layers, which is the observed relay length in all of the commonly studied natural systems studied to date

Using a recently developed recursive technique _{h2}_{h1}_{3r}_{4r}

For the remaining 18 responsive topologies we analyzed the signal-response relationship. For each topology, we have sampled 1000 parameter sets (rate constants and total protein concentrations) from a biologically permissible range, derived the signal-response curve for each parameter set and classified this curve as hyperbolic or sigmoidal (see

Panels

ID | Reverse Phosphorylation | Hydrolysis | Sigmoid Sets | ID | Reverse Phosphorylation | Hydrolysis | Sigmoid Sets |

1 | 1 0 0 | 0 1 | 0.00 | 16 | 1 1 1 | 1 0 | 15.47 |

2 | 0 1 0 | 0 1 | 0.00 | 25 | 1 0 0 | 1 1 | 0.00 |

3 | 0 0 1 | 0 1 | 0.00 | 26 | 0 1 0 | 1 1 | 1.53 |

4 | 1 0 1 | 0 1 | 0.00 | 27 | 0 0 1 | 1 1 | 0.00 |

5 | 1 1 0 | 0 1 | 0.00 | 28 | 1 0 1 | 1 1 | 0.00 |

6 | 0 1 1 | 0 1 | 0.00 | 29 | 1 1 0 | 1 1 | 0.33 |

7 | 0 0 0 | 0 1 | 0.00 | 30 | 0 1 1 | 1 1 | 44.53 |

8 | 1 1 1 | 0 1 | 0.00 | 31 | 0 0 0 | 1 1 | 0.00 |

14 | 0 1 1 | 1 0 | 50.53 | 32 | 1 1 1 | 1 1 | 10.47 |

For each topology, we have sampled 1000 parameter sets (rate constants and total protein concentrations) from a biologically permissible range (

To further understand the effect of reverse phosphorylation in generating sigmoidality, we compared the sampled parameter sets resulting in hyperbolic vs. sigmoidal signal-response relationships. We found that a key difference between the two parameter sets is the ratio between the forward and reverse phosphorylation rate constants, where a mean ratio below one is observed in the case of sigmoidal signal-response relationships (see _{h1}_{2}_{2r} and_{3r}_{3} or k_{3r}_{4r}_{3}_{4}_{h1}_{2}_{2r}_{3}_{3r}_{4}_{4r}_{h1}_{3r}

Parameter | Topology 14 | Topology 30 | ||

Hyperbolic Regime | Sigmoidal Regime | Hyperbolic Regime | Sigmoidal Regime | |

_{3}_{3r}> |
3.519 | 0.888 | 5.067 | 0.960 |

_{3}_{3r}) |
0.022 | 0.01 | 0.016 | 0.003 |

_{3}_{3r}) |
234.465 | 37.073 | 685.298 | 27.104 |

_{4}_{4r}> |
3.183 | 0.927 | 8.001 | 0.835 |

_{4}_{4r}) |
0.004 | 0.004 | 0.006 | 0.004 |

_{4}_{4r}) |
69.841 | 47.272 | 2265.233 | 13.153 |

The results shown are for topologies 14 and 30, assuming monofunctional HK, and sampling all parameters (with equal total protein concentrations at different layers). For additional results using alternative classification and sampling schemes (different total protein concentrations at different layers), assuming bifunctional HK, as well as for results from topologies 16 and 32, see

These results can be understood intuitively if we consider the phosphorelay as a set of connected stations, through which phosphoryl groups flow at a rate dictated by the signal strength. Without the presence of reverse phosphorylation and hydrolysis reactions in intermediate layers, phosphoryl groups accumulate at the bottom of the relay at a constant rate, while intermediate layers can remain unphosphorylated until the layers below them are saturated _{h1}_{2}_{3r}_{4r}

The analyses described so far have several assumptions with regards to modeling phosphorelay dynamics. Firstly, we have assumed bimolecular phosphotransfer reactions without complex formation. This assumption would be satisfied if phosphotransfer reactions, which are distinct from enzyme-driven reactions, happen fast and any complexes formed are short-lived. While there is indication from _{3r}_{4r}_{h1}_{3r}_{4r}_{h1}_{4r}_{4}_{h1}_{3r}

In the above treatment, we have also assumed a monofunctional HK, while it is known that several HKs can show both phosphorylation and dephosphorylation activity towards their substrate (in this case REC). We find that considering such a bifunctional HK does not alter the overall analytical conclusions regarding the necessity of fast reverse phosphotransfer and presence of hydrolysis reactions for enabling sigmoidality in the system (_{h1}_{h1}

To understand the consequences of hyperbolic vs. sigmoidal signal-response relationship in a phosphorelay, we focused on the two topologies that displayed high levels of tunability between these two response types (topology 14 and 30) and further analyzed the signal-response relationship. As explained above, both of these topologies embed reverse phosphotransfer reactions between REC-Hpt and between Hpt-RR. They differ, however, in the implementation of hydrolysis reactions; topology 30 embeds hydrolysis at the level of both REC and RR, while topology 14 embeds hydrolysis only at the level of REC. For each topology we picked 100 random parameter sets from both hyperbolic and sigmoidal regimes (i.e. parameters resulting in hyperbolic and sigmoidal signal-response relationships), and analyzed the noise properties and response time of the resulting systems (

Signal-response curve for topologies 14 (A) and 30 (B). The x-axis corresponds to the signal input to the system, which in the model is approximated by varying the HK auto-phosphorylation rate constant, _{s}_{s}

In contrast to the results from the noise analysis, the results of the response time analysis differed for the two topologies. Response time refers to time required for the phosphorylated RR levels to reach steady state following a step increase (or drop) in signal levels (

Phosphorelays are extended two-component signaling systems, which embed additional proteins (or domains) between a HK and RR pair. As such they are the result of evolution exploiting the highly modular nature of two-component proteins

These results provide mathematical proof that the way in which hydrolysis and reverse phosphorylation reactions in four-layered phosphorelays is implemented in natural systems endows functionality and could allow tuning of signal-response relationships between a hyperbolic and sigmoidal regime. Together with previous mathematical analyses of phosphorelays, which showed that the maximal level of phosphorylated RR and the signal-to-noise ratio of the response saturate at a relay length of four

These findings are in line with the observations from naturally observed phosphorelays, which are all indicated to display hydrolysis at layers 2 and 4

Considering relay dynamics in light of the findings presented here could help design future experiments to better understand the control of the sporulation decision in

The main conclusions of this study are that phosphorelays can embed hyperbolic or sigmoidal signal-response relationships, and that the latter type is not possible without reverse phosphorylation and a hydrolysis reaction at the second layer. Achieved either via dynamical tuning or through evolution of kinetic rates, the hyperbolic and sigmoidal regimes should allow appropriate physiological responses as needed by the cell. We would expect that sigmoidal dynamics would be favored for responding to signals requiring binary decision making. In contrast, hyperbolic or linear signal-response relationships would be required to produce responses that should track the incoming signals. Classifying a given phosphorelay's behavior into these regimes would be highly valuable, but is currently hampered as measuring the response of a phosphorelay at different signal levels and/or different component concentrations is highly difficult. Further, the signals feeding into phosphorelays are often unknown or not feasible for experimental manipulation. The results presented here offer an alternative, in which the shape of the signal-response relationship of the relay can be predicted from the measurement of forward and reverse phosphorylation rates. These measurements are possible in most cases through

Mutations and gene duplications provide the mechanisms by which the structure and dynamics of cellular interaction networks can be changed in evolution. Mathematical and computational approaches such as the ones presented here allow mapping the signal-response relationship of the possible systems that can be generated in this way. This understanding is essential to grasp why evolution might have resulted in the observed features of biological systems and how we might further modulate them. Thus, our findings on phosphorelays should facilitate both understanding the physiology mediated by these systems in a wide range of organisms and (re)engineering these through synthetic biology.

We develop a generic model of a four-layered phosphorelay incorporating all possible combinations of reverse-phosphorylation reactions between layers and hydrolysis reactions (i.e. encompassing all possible topologies in a four-layered relay with reverse phosphorylation and hydrolysis). The hydrolysis reactions are considered possible only on REC and RR, as these proteins are phosphorylated on an aspartate residue (while HK and Hpt are phosphorylated on a histidine residue), which has an inherent instability when phosphorylated

We initially ignore dimerization of HK and complex formation during phosphotransfer. Further, we consider that dynamics of gene expression and regulation occurs at much slower time scales compared to signaling reactions (e.g. phosphorylation) and hence the protein levels at each layer are assumed to be constant. Assuming mass action kinetics, the dynamics of the concentrations of the species in this reaction system is modeled as a system of ordinary differential equations (

We analyze the behavior of all possible four-layered phosphorelays using both numerical simulations and an analytical approach. In the latter case, we derived from the steady state equations an analytical expression that relates the level of phosphorylated RR at steady state (i.e. the system output) and the signal level (i.e. system input, taken as the rate of auto-phosphorylation _{s}_{s}_{s}_{1}_{2}_{1}_{2}_{1}_{2}_{1}_{2}_{2}_{s}

The shape of the signal-response curve carries important information about the response features of a given system

The derivation of this expression is given in the _{h1}_{2}_{2r}_{3}_{3r}_{3}k_{4}_{3r}k_{4r}_{H}_{H}

In phosphorelays containing a bifunctional HK, un-phosphorylated HK molecules bind to phosphorylated REC and catalyze their dephosphorylation. This reaction extends the system shown in

We utilized the probabilistic model checking approach implemented in PRISM (v4.0.3) _{A}_{A}^{−1}. We set

To study the differences in time taken for the relay to respond to a change in input (response time) under the two regimes (sigmoidal vs. hyperbolic), the ODE model arising from topologies 14 and 30 were numerically simulated to steady state under different signal levels. For each topology we picked 100 random parameter sets from each of the sigmoidal and hyperbolic regimes. For each parameter set, we calculated _{s}_{s}

Cartoon representations of the four topologies shown in

(TIF)

Effects of key model parameters on signal-response curves. Panels (_{5}_{h1}_{s}_{2}_{3}_{4}_{2r}_{3r}_{4r}_{h1}_{h2}_{5}_{5r}_{6}_{tot}, REC_{tot}, Hpt_{tot}, RR_{tot}):

(TIF)

Analysis of the response dynamics in topologies 14 and 30. Box plots show the distribution of response off times for topologies 14 (A) and 30 (B) as measured from hyperbolic and sigmoidal regimes. Response off time is defined as the time taken for the system to reach a new steady state after the input (_{s}

(TIF)

Plot of _{s}

(PDF)

List of all possible topologies in a four-layered phosphorelay. The topologies are indicated with a binary identification code that indicates the presence (1) or absence (0) of reverse phosphotransfer reactions along the layer, and the presence (1) or absence (0) of hydrolysis reactions at layers 2 and 4.

(XLS)

Derivation of the analytical results and proofs.

(PDF)

The results of the signal-response relationship classification for the 18 responsive topologies using different classification and sampling schemes (equal or different total protein concentrations at different layers), or assuming mono- or bi-functional HK.

(XLS)

Mean of the ratio of forward to reverse rate constants based on samples resulting in hyperbolic and sigmoidal signal-response relationship in topologies 14, 16, 30 and 32. All results are based on different classification and sampling schemes (equal or different total protein concentrations at different layers), and assuming mono- or bi-functional HK are shown.

(XLSX)

Mean values of the parameters resulting in hyperbolic and sigmoidal signal-response curves in topologies 14, 16, 30 and 32. The results are based on different classification and sampling schemes (equal or different total protein concentrations at different layers), and assuming mono- or bi-functional HK.

(XLSX)

The “biologically relevant” parameter regime for the parameters of the model and references to the experimental studies, from which this information is compiled.

(XLSX)

We thank two anonymous reviewers for insightful comments and Steve Porter for discussions on experimental study of phosphorelays.