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Mahan
Ghafari
^{ab} and
Alireza
Mashaghi
*^{a}
^{a}Leiden Academic Centre for Drug Research, Faculty of Mathematics and Natural Sciences, Leiden University, Leiden, The Netherlands. E-mail: a.mashaghi.tabari@lacdr.leidenuniv.nl
^{b}Department of Physics, Emory University, Atlanta, GA 30322, USA

Received
24th April 2017
, Accepted 24th August 2017

First published on 8th September 2017

We study the impact of topology on the response of a transcriptional cascade with certain circuit topologies to a constant and time-varying input signal. We systematically analyze the response of the output to activating and repressing cascades. We identify two types of responses for a linear cascade, namely the “Decaying mode”, where the input signal becomes exceedingly weaker as it propagates, and the “Bistable mode”, where the input signal can either be amplified or die out in the pathway. We examine how the transition occurs from one mode to the other as we add coherent and/or incoherent feed-forward loops in an otherwise linear cascade. We find that pathways with at least one incoherent feedforward loop can perform adaptive responses with the quality of response varying among different topologies. Furthermore, we study the origin of a (non)monotonic input–output profile for various circuit topologies over a wide range of parameter space. For a time-varying input signal, we identify some circuit topologies that are more prone to noise propagation than others that are more reliable in blocking out high-amplitude fluctuations. We discuss the effect of cell to cell variation in protein expression on the output of a linear cascade and compare the robustness of activating and repressing cascades to noise propagations. In the end, we apply our model to study an example of a transcription cascade that guides the development of Bacillus subtilis spores and discuss an example from a metabolic pathway where a transition from the decaying to bistable mode can occur by changing the topology of interactions in the pathway.

(1) |

(2) |

(3) |

From eqn (3), we find that an activating cascade exhibits a saddle-node bifurcation and has three fixed points whereas a repressing cascade has only one stable fixed point. The critical protein binding constant, K_{crit} = V/(2γ), sets the boundary between two distinct behaviors for the activating cascade (for detailed analysis see Appendix A, ESI†). (i) Decaying mode: for K > K_{crit}, there is only one stable fixed point at X_{(3)}* = 0, as given by eqn (3). In this parameter regime, the activation threshold for each element of the cascade is too high such that the input signal cannot transmit through the downstream elements (also see Fig. 3). Therefore, irrespective of the magnitude of the initial signal, the steady-state level of each element decreases as the signal transmits through the cascade until it eventually dies out as it reaches the most downstream gene. As a result, the output gene does not get activated and the cell is at a disadvantage. (ii) Bistable mode: for K < K_{crit}, there are two stable fixed points at X_{(1)}* and X_{(3)}*, and one unstable fixed point at X_{(2)}*. Depending on the strength of the input signal, the steady-state level of each element would either increase to reach a non-zero value (cell is not at a disadvantage), when S_{X} > X_{(2)}*, or decrease to zero (cell is at a disadvantage), when S_{X} < X_{(2)}*. Since in real biological systems there are always cell to cell variations in the production of each protein, for cells in which X* < X_{(2)}*, the downstream genes cannot get activated which can be disadvantageous for them.

Note that in this scenario, there is a chance for the signal not to die out as it propagates further downstream and reaches the non-trivial fixed point X_{(1)}*. (iii) Transition point: K = K_{crit} defines the boundary between the first two modes of the cascade. By changing the maximum expression level, V, or degradation/dilution rate, γ, we can tune the transition point. For instance, by increasing V and/or decreasing γ, we can shift the threshold value up such that the transition from Bistable to Decaying mode becomes less likely. Thus, we can synthetically tune an activating cascade to ensure that all genes get activated. Such points of qualitative change in the behavior of signal-response profile has also been found in other types of biological networks.^{42–46} On the other hand, as eqn (3) suggests, a repressing cascade has only one stable fixed point. Therefore, given that the duration of the constant input signal is long enough to effectively initiate the repressing chain of regulations,^{28,47} regardless of the magnitude of the input signal, the steady-state expression level of the output always approaches a non-zero value. This implies that repressing cascades are more robust to fluctuations in the input signal and, thus, can be found more frequently in linear transcriptional cascades.^{48,49}

This preliminary finding forms the basis for our analysis in the following sections regarding the role of topology in regulatory cascades. We now introduce one interlocked (in)coherent feedforward loop into the cascade by adding one extra interaction, i.e. an activating or a repressing regulation, to the linear cascade (see the curved arrows in Fig. 1a). We consider the behavior of the cascade near its transition point (K = K_{crit}). By changing the position of the feedforward loop along the cascade, we measure the steady-state response of each element and the half-life time for the activation of the last element X_{N} (i.e. output) as a function of the arrow position. We find that the kinetics and the steady-state response of the output for both cases with an extra activating and repressing interaction of length two, i.e. connecting a pair of elements through a feedforward loop, significantly depends on their positions in the cascades (see Fig. 1).

To explain the dependency of the steady-state level of the output on the curved arrow, we need to understand its effect on the steady-state value of intermediate nodes X_{j}*. The one-dimensional map to find the steady-state solution in this case is very similar to the one we found for a linear cascade, except for the fact that the regulating function for the doubly regulated node, m, where the (in)coherent feedforward loop acts, differs from others

(4) |

Note that all the interactions in the linear cascade are either activating, i.e. f(X_{j−1}*,K_{A}) = A(X_{j−1}*,K_{A}), or repressing i.e. f(X_{j−1}*,K_{A}) = R(X_{j−1}*,K_{R}), Hill functions and the extra regulation by the curved arrow, f_{c}, can create an interlocked coherent or incoherent feedforward loop depending on whether it is activating A(X_{j−1}*,K_{A}) or repressing R(X_{j−1}*,K_{R}) the production of X_{m}. These two regulations should be multiplied together (AND gate) to get the steady-state level of X_{m}*.

Therefore, for an activating cascade, the steady-state expression level of each member of the pathway, X_{j}*, can be divided into two sub-groups: before the doubly regulated node, the steady-state of each element approaches the non-zero fixed point (given by eqn (3)). After the doubly regulated node (i.e. j ≥ m), the value of X_{j}* changes depending on the nature of interaction from the curved arrow. Consequently, the cascade may either maintain its initial non-zero steady-state or jump over the unstable fixed point and approach zero, i.e. the downstream elements will no longer get activated (see the green and red dotted lines in Fig. 1b) – size of the pathway N determines the number of times we apply the mapping rule in eqn (4). Therefore, an incoherent feedforward loop in the cascade has the ability to shut down the regulatory cascade which could potentially damage the cell function. By changing the position of the curved arrow, we can manipulate the “jump” in the steady-state. While having an incoherent feedforward loop in the upstream (e.g. X_{2} repressing X_{4}) can have a dramatic impact on the fate of the output, i.e. repressing link in the upstream causes a significant decrease in the expression level of the output, a coherent feedforward loop can increase the expression of output to relatively higher amounts (given that the steady-state value of each element remains above the unstable fixed point). Fig. 1c and d show that placing the incoherent feedforward closer to downstream elements of the cascade (e.g. X_{8} regulating X_{10}) makes close to no change in the steady-state and half-life time of the output. Our simulation results also show that for an activating cascade of particular size, there is an optimum position for the interlocked coherent feedforward loop to minimize the steady-state and maximize the delay in activation of the output (i.e. X_{5} activating X_{7}).

On the other hand, a repressing cascade demonstrates a completely different behavior. Since there is only one stable fixed point for the cascade, the steady-state expression level of downstream elements would ultimately approach that fixed point. However, the extra feedforward loop introduces a “lag” in the steady-state response of each downstream element after the doubly regulated node. In other words, for a linear cascade with one coherent feedforward loop (Fig. 1b), the steady-state level of each element X_{i}* is equal to the steady-state level of X_{i−2}* of a regular linear cascade with no extra interaction (see the green and black dashed lines in Fig. 1d). Similarly, for a linear cascade with one incoherent feedforward loop, the steady-state level of X_{i}* would be the same as X_{i−4}* in a linear cascade (see the red dashed lines in Fig. 1d). This explains the ups and downs in the steady-state expression level of the output as the position of the feedforward loop changes along the cascade (see Fig. 1f). We can also see in Fig. 1f that the variations around the steady-state solution for a linear cascade (solid blue and red lines) are lower for a cascade with an incoherent feedforward loop than those with a coherent feedforward loop. The half-life activation time for the production of the output in a repressing cascade is much shorter compared to the activating cascade (see Fig. 1h). This suggests that long activating transcriptional cascades can lead to long response times for downstream elements while repressing cascades tend to respond with almost no delay. Therefore, our analysis suggests that long activating cascades can exhibit longer delays (on the order of cell generation) more reliably for biological processes such as cell differentiation which requires proteins that are produced in the mother cell to be used in the next generation of daughter cells.

As we discussed in the previous section, the curved arrows can have a considerable influence on the fate of the input, especially when they are in the upstream. We consider an example of a cross configuration with X_{2} regulating X_{4} and X_{3} regulating X_{5} as shown in Fig. 2a. We find three possible scenarios for the bistable mode: (i) in CXC configuration (see Fig. 2b), both activating curved arrows will reduce the expression level of the intermediate elements compared to a normal linear cascade with no extra interaction. However, their effect is not considerable enough to change the fixed point of the system (i.e. no “jumping” over the fixed point occurs). Thus, the pathway will restore the input signal by approaching its non-zero stable fixed point. (ii) For CXI configuration (see Fig. 2c), the first curved arrow decreases the steady-state of X_{4}, similar to case (i) (because the two pathways are identical up to this point), while the second arrow (repressing interaction) causes the steady-state level to drop significantly and switches the stable fixed point of the pathway. Consequently, the input signal will die out as it propagates downstream. (iii) For both IXC and IXI configurations (Fig. 2d and e), the scenarios look similar. The first repressing regulation changes the fixed point of the pathway, whereas the second curved arrow (either the activating or repressing interaction) only determines the magnitude by which the steady-state is going to drop. A similar analysis exploits the steady-state expression levels of the elements of the pathway with Parallel and Series topologies (see Appendix C, ESI†). Our simulations suggest that for CαI configurations, if the magnitude of the repressing interaction is not too strong, then downstream genes would still be able to get activated (see also Fig. S4b and d, ESI†). However, IαC and IαI configurations are less likely to activate downstream elements of the cascade.

(5) |

We were not able to find a closed form for the input–output profile of a repressing cascade. However, the results of our computer calculations can be found in Fig. 3. As demonstrated in Fig. 3a, the expression level of X_{1}* that is needed to activate X_{N}* gets exceedingly high for K > K_{crit}, whereas it diminishes to lower values for K < K_{crit}. This highlights the point that we made earlier about the “Decaying mode”, where it becomes exceedingly unlikely for a signal to propagate through the pathway with K > K_{crit}, and “Bistable mode”, where the input signal can get amplified as it propagates through the downstream elements. Fig. 3b shows that there is a unique cut-off value for the protein binding constant, K, given N, beyond which X_{N−1} cannot effectively repress the production of X_{N} (vertical dashed lines). It also demonstrates that the expression level of X_{1}* required for repressing the output becomes exceedingly large for linear cascades of size N = 4, 6, and 8. This happens because for these cascades, the steady-state level of X_{N−1} is below the fixed point value of a repressing cascade, X_{(1)}* (see eqn (3)). Thus, if the cut-off value of K is higher than X_{(1)}*, it will be implossible for X_{N−1} to repress the production of the output.

Now, we examine the monotonicity of the input–output profile for three realizations of cross, parallel, and series topologies (see Fig. 4). Our numerical results indicate that for the CαC configuration with two coherent feedforward loops (Fig. 4b) the input–output profile is monotonic. We observe that for K > K_{crit}, the response of the output is weaker than that for K < K_{crit} as it is harder for the signal to activate downstream elements. We also note that when K > K_{crit}, the response of the CPC pathway is strongest, whereas for K < K_{crit} CXC has the strongest response. For CαIs with the first feedforward loop being coherent and the second one incoherent, the CPI pathway has the highest peak in the non-monotonic response (see Fig. 4c). While for K_{R} > K_{crit} the repressing interaction cannot get fully activated, i.e. the expression level of the output reaches a non-zero value as the amplitude of the signal increases, for K_{R} < K_{crit} the input–output profile performs an almost adaptive response where the expression level of the output settles at a low persistence amount. As demonstrated in Fig. 4d, the IαC category of pathways demonstrates a fully adaptive response for K_{R} < K_{crit}, i.e. the system's ability to respond to a change in the input stimulus and returning to its pre-stimulated output level, with ISC having the highest peak in the adaptive response. Comparing the adaptive response in Fig. 4c and d, we find that if the incoherent feedforward loop appears in the pathway before the coherent one, it has a stronger effect in controlling the adaptive behavior of the response. For IαI, all the circuit topologies perform an adaptive response with IPI and ISI having the highest peaks. Also for K_{R} < K_{crit}, the input–output profiles demonstrate lower peak amplitudes and faster decays. It is also important to note that for the ISI topology, the steady-state level of the intermediate element X_{5}* (the first doubly regulated element) acts as an input to the second part of the pathway, hence, creating an input–output profile that has two peaks. The first one corresponds to the activation of the first incoherent feedforward loop by the input and the other corresponds to the activation of the second incoherent feedforward loop by the first one. This is one of the characteristic features of incoherent feedforward loops connected in Series which has been analyzed in regulatory networks.^{50} Therefore, we have shown that while some activating cascades with one incoherent feedforward loop can perform pulse-like responses, others can also produce sustained responses which make them more robust against cell-to-cell variations.

(6) |

Therefore, the amplitude of variations in the output around its steady-state solution decays as 1/ω^{2} for high frequency changes in the input singal. In other words, the feedforward loop acts as a low pass filter to the input signal with high frequency. For lower frequencies in the input, the variations would be larger for proteins with smaller degradation rates (i.e. smaller values of ω_{c}). This property of feedforward loops has also been highlighted in the work by Guantes and collaborators.^{53} The variations in each element also depend linearly on the amplitude of oscillations in the input α, as well as the gain from the parent element(s) which directly regulates them. Note that the amplitude of variation in the doubly regulated element, X_{3}, to the first leading term in ω, only depends on g_{3}1 which changes with the expression level of X_{1}. This indicates that the noise propagation in a feedforward loop with high frequency input only depends on the parent element with the most variation in the amplitude. In other words, when X_{3} is both regulated by X_{2} and X_{1} (parent elements), the amplitude of variation in X_{3} is most dominated by the parent element X_{1} which is closer to the source and has a higher variation amplitude. Therefore, cascades where the most upstream elements are interacting with the most downstream elements are more susceptible to noise propagation. In the case of three different circuit topologies demonstrated in Fig. 4a, the amplitude of variation for the doubly regulated elements is:

(i) Cross configuration

(7) |

(8) |

(9) |

We find that while the doubly regulated elements in cross and series configurations exhibit the same dependency on ω (eqn (7) and (9)), parallel configuration shows a relatively weaker dependency (eqn (8)). The reason why parallel configuration is a weaker low-pass filter is that there are long-range interactions between the upstream and downstream elements, i.e. the upstream element X_{2} regulates X_{6} which is almost at the end of the pathway. This topological property of the Parallel configuration makes it less robust to noise propagations. For a slowly varying input signal (ω ≪ ω_{c1}, ω_{c2}, ω_{c3}), the characteristic frequency of each element is higher than the input signal. Thus, the signal is perceived as a constant input for each member of the cascade. Thus, each element has enough time to reach its steady-state expression level before it starts to sense the oscillations of the input signal (see Fig. 5b). On the other hand, if the oscillations in the input are too fast (ω ≫ ω_{c1}, ω_{c2}, ω_{c3}), the dynamics of each element is too slow to sense the rapid oscillations in the input (see Fig. 5a). Consequently, the variations decay to zero for very large oscillations (see eqn (7)–(9)). Although our approximation is only suitable for small amplitudes and is expected to hold for small perturbations around the mean, our experience with the simulations was that it provides a good guide for describing output for all amplitudes. This analysis shows that by adjusting the pathway parameters γ and K we can make our linear pathway robust to fluctuations of high frequency and each circuit topology demonstrates a unique low-pass filtering property with parallel topologies being the most susceptible one to noise. We have shown that, in addition to all these features, they are also able to reduce signal fluctuations. This further justifies their ubiquity in biological networks, and perhaps accounts in part for the robustness of living systems.

Fig. 5 The effect of large amplitude noise propagation on the expression level of each element in a linear activating cascade with cross topology given in Fig. 4a. The expression level of each element is demonstrated as a function of time for (a.1) and (a.2) ω = 10, α = 10, s = 1 and for (b.1) and (b.2) ω = 0.1, α = 1, and s = 1. The expression level of the output X_{8} for (a) is ≈0.68, for (b) is ≈0.70, and for a cascade with constant input signal is ≈0.70. Thus, there is a minor change in the steady-state expression level of the output when the noise frequency is high (regardless of the amplitude α) compared to a constant input (i.e. low-pass filtering property of the linear cascade). |

As we discussed in Section 3.1, this activating cascade should be in the bistable mode (i.e. has a non-zero steady-state fixed point) so that all the elements of the cascade are guaranteed to get activated (as long as the input signal is persistent). We also know that there should be a well-defined delay between the activation of each element in an activating cascade so that each gene is activated at a particular time. Following the approach in Sections 3.1 and 3.2 (see eqn (3) and (4)), we can find the steady-state level of each element:

(10) |

We also know that IPC configuration in Fig. 5b is capable of producing a sustained response to input signal (see Section 3.3) which is essential for maintaining the robustness of the cascade (Fig. 6).

Fig. 6 (a) A transcriptional cascade for the development of Bacillus subtilis spores. Note that the original cascades include multiple sub-branches that represent groups of tens of hundreds of genes.^{54} We only selected the longest activating cascade to study the kinetics and steady-state response of the output and removed all the other sub-branches out of the cascade. (b–e) Are the underlying topological substructures in the cascade. |

(11) |

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7cp02671d |

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