- Research article
- Open Access
Specialized or flexible feed-forward loop motifs: a question of topology
- Javier Macía†^{1}Email author,
- Stefanie Widder†^{1} and
- Ricard Solé^{1, 2}
https://doi.org/10.1186/1752-0509-3-84
© Macía et al; licensee BioMed Central Ltd. 2009
- Received: 28 January 2009
- Accepted: 31 August 2009
- Published: 31 August 2009
Abstract
Background
Network motifs are recurrent interaction patterns, which are significantly more often encountered in biological interaction graphs than expected from random nets. Their existence raises questions concerning their emergence and functional capacities. In this context, it has been shown that feed forward loops (FFL) composed of three genes are capable of processing external signals by responding in a very specific, robust manner, either accelerating or delaying responses. Early studies suggested a one-to-one mapping between topology and dynamics but such view has been repeatedly questioned. The FFL's function has been attributed to this specific response. A general response analysis is difficult, because one is dealing with the dynamical trajectory of a system towards a new regime in response to external signals.
Results
We have developed an analytical method that allows us to systematically explore the patterns and probabilities of the emergence for a specific dynamical response. The method is based on a rather simple, but powerful geometrical analysis of the system's nullclines complemented by an appropriate formalization of the response probability.
Conclusion
Our analysis allows to determine unambiguously the relationship between motif topology and the set of potentially implementable functions. The distribution probability distributions are linked to the degree of specialization or flexibility of the given network topology. The implications for the emergence of different motif topologies in complex networks are outlined.
Keywords
- Phase Space
- External Input
- Relaxation Dynamic
- Feed Forward Loop
- Geometrical Element
Background
Molecular networks in cells are highly complex and dynamic. The global behaviour of these webs and their behavioral patterns are far too complicated to intuitively understand their logic. One way to address this problem is to represent them in terms of simplified interaction graphs combining both biological data and mathematical methods [1–6].
Much effort has been devoted to extract some general features of such networks, dissect them into hierarchical levels, modules and motifs to understand their functionalities, dynamics and evolution [7–16]. Simple switches and oscillators have been shown to map to the core processes of biological decision-making, implemented by two- or three-gene network motifs and characterized by their behaviour around the systems' fixed points [17–22]. However, it is reasonable to think that not only the system's steady state is of interest, but also the way such equilibrium is achieved. Such transient behavior might be characteristic, somehow representing the function performed by the genetic circuitry. In some circumstances, such as in stress responses, a quick change might be favorable [23], whereas in other occasions, e.g. cell-cell intercommunication, it might be more adequate to filter noisy signals and respond only under absolute certainty [24].
Transcriptional networks regulating cell responses exhibit several biochemical wiring patterns, termed network motifs, which appear at frequencies much higher than expected by chance, suggesting that they may have specific functions in the information processing performed by the network. Over the last years, powerful bioinformatic tools such as FANMOD [25] have been developed to detect motif distributions in complex transcriptional networks. One of these motifs is the feed-forward loop (FFL), defined by a transcription factor X that regulates a second transcription factor Y, such that both X and Y jointly regulate a target gene Z (figure 1a-b). Many examples of FFLs can be found in complex transcriptional networks. For example, in E. Coli, FFL is present in the L-arabinose system, where protein Crp is the general transcription factor (X) and AraC is the specific transcription factor (Y). This motif regulates 40 effector operons in 22 different systems in the network database [26]. A second example can be found in Saccharomyces network, where the protein Mcm1 (X) regulates the expression of Swi4 (Y). Both proteins Mcm1 and Swi4 regulate the final expression of Clb2. In the yeast network, 39 regulators have been found that are involved in 49 feedforward loops potentially controlling 240 genes [27]. In general, FFLs are known to be associated to multiple key regulations, exhibiting different functionalities, e.g. under conditions of glucose starvation (CRP), nitrogen limitation (rpoN), and noxious drugs (rob), these regulators act as X in a C1 type FFL. On the other hand, I1 type FFLs in yeast include anaerobic metabolism (HAP1 as X) and nitrogen starvation (DAL80 or GLN3 as X) systems [28]. In this context, the question about the relation between the functional response implemented by FFLs and their topology arises. The study of the response of three-gene feed-forward loops upon external input shows that they are capable of either implementing transient pulsing (rapid) or, filtering (delayed grader) dynamics [28–35]. However, despite it seems clear that motif topology has an impact on its functionality, is the mapping between motif topology and the possible dynamics one-to-one? Some studies have demonstrated that topology does not necessarily determine function [13, 36, 37]. Most analysis focused on motif's function have been carried out considering single motif networks. However, recent studies [38, 39] have provided evidence that for complex networks, the embedding of the motif with the rest of the network needs to be taken into account.
Here we have developed a method to systematically study the different functions which can be implemented by each FFL motif and how the topology determines univocally the distribution of probabilities for these functions. A will be shown below, this distribution is correlated with the degree of specialization or flexibility of each motif, by taking into account the different likelihood to perform any function. In other words: topology determines the motif's level of functional specialization. Recently, a similar question on the context of genetic clocks has been addressed [40]. The conclusions of the study suggest that for these clocks topology does not determine dynamics univocally. Although our analysis focuses on single motifs, our results provide new insights to understand the different distribution of motifs in more complex networks, as we will discuss later.
describing how concentration of different species Y and Z change during time. Here Ẏ and Ż represent the derivatives dY/dt and dZ/dt, respectively. The FFL topology is implicitly described by functions g(X) and h(X, Y). Assuming that expression of X is unregulated, the dynamics of the system can be represented in a two-dimensional diagram displaying Y against Z, the so called phase space. In the absence of input, the system evolves towards a stable state, i.e. a specific set of values for the concentrations of Y and Z that remain constant over time. This stable state is determined by the crossing of the so-called nullclines of the system [41] described by the curves Ẏ = 0 and Ż = 0, i.e. g(X) = 0 and h(X, Y) = 0 respectively. These curves define the points of the phase space where Y and Z do not change. The nullclines capture the essence of the dynamical potential of each component and the relevant chemical, physical or biological constraints. Their shapes reflect saturation effects, forbidden ranges of variables or how fast each component responds to perturbations. In other words, a careful analysis of nullclines allows us to understand the dynamics of the underlying systems and its biological implications. The crossings between both nullclines define stable fixed points, where concentrations of both Y and Z remain constant. Upon input the shape of the nullclines change, providing a new, different stable state. Hence, the system will evolve toward this new state following a trajectory, i.e. a set of intermediate states (Y, Z) within the available phase space.
Results and Discussion
General model of the FFL
The analysis is focused on the most general FFL formed by three genes. We assume that the gene circuit acts as a functional unit responding to an external input by producing output. In figure (1) we show an schematic representation of the FFL, depicting three genes G_{ X }, G_{ Y }and G_{ Z }, with regulatory interactions among each other via their corresponding proteins X, Y and Z.
The parameter γ_{ i }represents the basal production of protein i, where i = {X, Y, Z}. In this parameter we subsume the concentration of all biochemical elements which remain constant in time. The degradation rate of protein i is denoted as d_{ i }. The binding equilibrium of the regulators j with the gene G_{ i }are denoted by , with j = {X, Y, Z}. The tunable positive parameters α^{ X }and β^{ j }describe the type of regulatory interactions, i.e. activation or inhibition, for gene G_{ Y }and G_{ Z }, respectively, without any predefined specific assumptions. They provide the regulatory rates with respect to the basal transcription. Values < 1 correspond to inhibitory regulation, whereas > 1 accounts for activation. The parameter β^{ XY }accounts for the simultaneous regulation of G_{ Z }.
Traditionally, studies on FFL dynamics have been performed under the assumption of Boolean logic [28, 42] for the control of the output regulation. The presented model includes all theses cases as specific subsets of parameters. For example, assuming a Boolean AND logic for a circuit, where the output Z is positively regulated by Y and X, is described by the following parameter configuration: β^{ x }= β^{ Y }= 1, β^{ XY }> 1 and . The same circuit displaying OR logic requires β^{ x }= β^{ Y }> 1, β^{ XY }= 0 and = 0.
Finally, n and m denote the degree of multimerization of the regulators. The presented results, however, are considering the general case independent of the size of the regulatory factors.
Nullclines' analysis: Changes of the nullclines upon input
- 1.
The regulation of Y by X, described by α^{ X }, defines the direction of the shift of the straight nullcline (5) regarding its earlier position for X = 0.
- 2.
The regulation of Z by X, described by β^{ X }, displaces the crossing of the nullcline (6) with the vertical axis to a higher (β^{ X }> 1) or a lower (β^{ X }< 1) value.
- 3.
Finally, the joint regulation β^{ XY }of Z by X and Y, strongly influences the location of the horizontal asymptote Z_{ HA }.
As we will see in the following sections the geometrical features generated by the biological interactions will form the basis for the dynamical behaviour of the FFL.
Response of the FFL
Two biological scenarios for the joint regulation of G_{ Z }by Y and X are plausible: either the complex acts as an activator β^{ XY }> 1 as shown in figure (4a) or as inhibitor β^{ XY }< 1. In (4b) we show the scenario associated with the conditions β^{ XY }< 1 and β^{ XY }> β^{ Y }. The nullcline moves down, but does not cross the original nullcline. In (c) conditions β^{ XY }< 1 and β^{ XY }<β^{ Y }lead to a single crossing. The same conditions can lead to a double crossing of the nullclines, however our numerical analysis (data not shown) indicates that the probability to find an adequate configuration of parameters has very low probability (< 0.3%). For simplicity we discard these cases (see Methods for a detailed example of our analysis applied to nullclines with double crossing). By using the relatively simple configuration of the nullclines shown in (b) we already find different possible behaviour of the two circuits C4 and I1 due to their opposing α^{ X }.
Whereas for C4 the nullcline's (5) shifts to higher values, allowing only for grader trajectories (inset 1), I1 may show instead two different functionalities. The shift direction is to lower values and hence either grader or pulser can be implemented depending on the set of parameters (see inset 2). The fact that a range of feasible functional scenarios can be intuitively deduced, demands for a method of unambiguous discrimination to resolve the problem.
Separatrix
Our analysis can be generally applied to all three-gene FFLs. We have shown that both examples show enough plasticity in the dynamics to implement different type of response. In the next section we will focus on the probability of emergence of the different feasible types of dynamics.
Probability of emergence of different FFL's dynamics
In the previous section we have shown that in both examples, C4 and I1, more than one possible dynamic behavior can be obtained depending on the specific set of parameters chosen. Generally we can list six types of dynamics, namely positive or negative graders (G^{+}, G^{-}) and positive or negative pulsers (P^{+}, P^{-}). Grader response corresponds to an increase (+) or decrease (-) of the concentration of Z characterized by a transient from the initial to the final state where the concentration of Z never is higher (G^{+}) or lower (G^{-}) than the final concentration. In general, grader responses are related with responses able to filter noise and respond only on absolute certainty [24]. On the other hand, a pulser response is characterized by a rapid increase (P^{+}) or decrease (P^{-}) of the concentration of Z reaching higher (+) or lower (-) values of Z before they reach the final state. Note that for the pulsers independently of the pulse direction (P^{+} or P^{-}) the final concentration of output protein Z can be higher (T^{+}) or lower (T^{-}) than the initial concentration. Hence we separately analyze four different pulser dynamics, namely P^{+}(T^{+}), P^{+}(T^{-}), P^{-}(T^{+}), P^{-}(T^{-}). The time courses of the different functionalities are outlined in figure (6). A specific subset of dynamics can be determined for each FFL frequently containing functions, which cannot be implemented. Based on this results the main question is: Are all the feasible dynamics equally probable? To address this question we have performed an analytical study of the parametric requirements necessary to implement a given type of dynamical response.
Backbone of requirements for the FFL response
We can deduce geometrically all possible sets of trajectories. We discriminate the necessary parametric conditions for the emergence of one specific dynamic among all the possible, due to the relative position of the initial point ϕ_{X = 0}of the trajectory in relation to the separatrix S, ξ and the specific shape of the nullclines. The parametric conditions, which determine the shape of the nullcline, can be systematically formalized. The key geometrical elements of the nullclines can be described by a set of exclusive parametric combinations defined in a string, which we call the backbone of requirements. Each position in this sequence contains the solution for two possible parametric states.
The example shown in figure (7) is meant to illustrate the procedure. The cases in (a-c) display the same geometrical shape and can be described by a single backbone of requirements shown in (d). Note that not all the elements of the parametric sequence need to be defined explicitly. In certain occasions some of them are uniquely defined by previous conditions, denoted by *, or otherwise do not have an impact on the geometrical scenario, represented by '∅'. For example position one is denoted '∅'. If condition two (β^{ XY }> β^{ Y }) is satisfied, position one is not relevant, because both solutions (β^{ XY }> 1 or β^{ XY }< 1), provide the same geometry. On the other hand position four, denoted as '*', is always solved as '>', because it can be deduced from the second condition given that β^{ X }> 1.
Quantification of the dynamical probabilities
Here, represents the number of requirements having no impact on the nullclines' geometry. For this kind of elements of the sequence, both possible solutions ('>' or '<') are valid and hence discrimination is unnecessary. Therefore, different combinations of parameters are described by the same backbone sequence providing the same geometry for the nullclines. The second term , is the number of elements predefined by other conditions of the backbone. Finally, is the number of requirement actually necessary to determine univocaly the shape of the nullcline. In other words, in order to implement a determined backbone sequence it is sufficient to properly establish the conditions .
The highest level of specialization of a given motif would correspond to a distribution of probabilities displaying a single peak, whereas the maximum level of flexibility would correspond to a flat distribution of probabilities, where all the potential dynamics would be equally probable. In order to compare the level of specialization of different FFL topologies we propose to measure the peakedness of the probability distribution, i.e. the kurtosis [43] (see Methods for details about kurtosis). The values of kurtosis are K_{C 1}= 4.6, K_{C 4}= 7.8 and K_{I 1}= 1.28, respectively indicating that C1 has an intermediate level of specialization with respect to C4 (more specialized) and I1 (more flexible). The same qualitative distributions are obtained by numerical simulation choosing random sets of parameters and counting the frequency of each dynamics (data not shown).
Conclusion
FFL motifs appear frequently in cellular regulatory networks. Despite the efforts devoted to understand how FFLs encode their functionalities, the question about the relation exact between function and topology remained open. In this work we have presented a new analytical formalism based on the geometric analysis of the system's nullclines to elucidate this question. We found that the dynamical response triggered by the external input can be analysed in terms of: i) the nullcline's geometry as described by a backbone sequence of parametric conditions and ii) the specific location of the initial stable state in the phase space with respect to the nullclines and the separatrix. This puts us into the position to unambiguously enumerate the probability of a given FFL to implement a certain function. Our results support this view topology does not define a unique functionality, as have been previously discussed, ([36]). Circuits with the same topology can implement different functions, yet not all of these possible dynamics are equally probable. However, topology defines univocally the distribution of probabilities for the emergence of the different feasible responses.
For illustrative purposes we have analyzed two interesting examples, namely circuits C4 and I1, exhibiting the same regulatory interactions except a single regulation of the gene G_{ Y }by the protein X. In these cases we found two paradigmatic scenarios: circuit C4 can implement G^{+} response with significantly higher probability than the other feasible dynamics, whereas, I1 exhibit more uniform distribution of probabilities. These results demonstrate that C4 has a specialized topology, optimal for the implementation of grader response, whereas I1 has a high degree of flexibility among different dynamics. Under an evolutionary perspective, a trade-off between these different features, flexibility and specialization, is likely to play an important role. This problem will be investigated elsewhere.
In single motif networks a given function will be implemented with high probability by the most specialized topology. However, in evolved, complex networks other aspects need to be considered. In order to obtain reliable networks, i.e. robust and with high fault tolerance, complex topologies can emerge as a result of the evolutionary process. An evolved and fit network is not necessarily the sum of its optimal sub-modules. In order to provide redundancy and degeneracy, flexible sub-modules, able to change their functionalities with minimal cost are often a good solution to reliability [44]. Future work will be devoted to analyze the implications of these two characteristics in the natural emergence of current biological networks.
Methods
Analytical estimation of the separatrix
In order to test this expression we have performed numerical simulations with random sets of parameters and multiple random initial points. This separatrix expression defines properly the frontier between A_{1} and A_{2}.
Relation between dynamics and geometrical elements
Relation between geometry of the nullclines and functionality
Sl. | Relation | Function | Relation | Function |
---|---|---|---|---|
+ | S > ϕ_{X = 0}> ξ | P ^{-} | ξ > S > ϕ_{X = 0} | G ^{+} |
- | S > ϕ_{X = 0}> ξ | G ^{-} | ξ > S > ϕ_{X = 0} | G^{+} if Z_{ f } > Z_{ i }else impossible |
+ | ϕ_{X = 0}> ξ > S | P ^{-} | ξ > ϕ_{X = 0}> S | G ^{+} |
- | ϕ_{X = 0}> ξ > S | G ^{-} | ξ > ϕ_{X = 0}> S | P ^{+} |
+ | ϕ_{X = 0}> S > ξ | G^{-} if Z_{ f }<Z_{ i }else impossible | S > ξ > ϕ_{X = 0} | G ^{+} |
- | ϕ_{X = 0}> S > ξ | G ^{-} | S > ξ > ϕ_{X = 0} | P ^{+} |
Relation between geometry of the nullclines and functionality
Sl. | Relation | Function | Relation | Function |
---|---|---|---|---|
+ | S > ϕ _{X = 0}> ξ | G ^{-} | ξ > S > ϕ_{X = 0} | G^{+} if Z_{ f } > Z_{ i }else impossible |
- | S > ϕ_{X = 0}> ξ | P ^{-} | ξ > S > ϕ_{X = 0} | G ^{+} |
+ | ϕ_{X = 0}> ξ > S | G ^{-} | ξ > ϕ_{X = 0}> S | P ^{+} |
- | ϕ_{X = 0}> ξ > S | P ^{-} | ξ > ϕ_{X = 0}> S | G ^{+} |
+ | ϕ_{X = 0}> S > ξ | G ^{-} | S > ξ > ϕ_{X = 0} | P ^{+} |
- | ϕ_{X = 0}> S > ξ | G^{-} if Z_{ f }<Z_{ i }else impossible | S > ξ > ϕ_{X = 0} | G ^{+} |
Analysis of the relaxation dynamics after induction
Analysis of the dynamics in nullclines with two crossings
Backbone sequences and the associated dynamics
Backbone sequences for C4
β^{ XY }, 1 | β ^{ XY }, β ^{ Y } | Z_{0}|_{X = 0}, Z_{ HA }|_{X > 0} | Z_{0}|_{X > 0}, Z_{ HA }|_{X = 0} | Slope | ϕ_{X = 0}, ϕ_{X > 0} | ϕ_{X = 0}, ξ | Relation | Function |
---|---|---|---|---|---|---|---|---|
> | * | * | * | + | * | * | ξ > S > ϕ_{X > 0} | G ^{+} |
> | * | < | * | - | * | * | ξ > S > ϕ_{X > 0} | G ^{+} |
> | * | < | * | - | * | * | ξ > ϕ_{X > 0}> S | G ^{+} |
> | * | < | * | - | * | * | S > ξ > ϕ_{X > 0} | G ^{+} |
∅ | > | > | * | * | * | * | ξ > S > ϕ_{X > 0} | G ^{+} |
∅ | > | > | * | * | * | * | ξ > ϕ_{X > 0}> S | G ^{+} |
∅ | > | > | * | * | * | * | S > ξ > ϕ_{X > 0} | G ^{+} |
* | < | * | * | * | * | < | ξ > S > ϕ_{X > 0} | G ^{+} |
* | < | * | * | * | * | < | ξ > ϕ_{X > 0}> S | G ^{+} |
* | < | * | * | * | * | < | S > ξ > ϕ_{X > 0} | G ^{+} |
* | < | * | * | * | > | * | ϕ_{X > 0}> S > ξ | G ^{-} |
> | * | * | * | + | * | * | ξ > ϕ_{X > 0}> S | P ^{+} T ^{+} |
> | * | * | * | + | * | * | S > ξ > ϕ_{X > 0} | P ^{+} T ^{+} |
* | < | * | * | * | > | * | S > ϕ_{X > 0}> ξ | P ^{-} T ^{-} |
* | < | * | * | * | > | * | ϕ_{X > 0}> ξ > S | P ^{-} T ^{-} |
* | < | * | * | * | < | > | ϕ_{X > 0}> S > ξ | Impossible |
* | < | * | * | ∅ | < | > | ϕ_{X > 0}> ξ > S | P ^{-} T ^{+} |
* | < | * | * | ∅ | < | > | S > ϕ_{X > 0}> ξ | P ^{-} T ^{+} |
Backbone sequences for I1
β^{ XY }, 1 | β^{ XY }, β^{ Y } | Z_{0}|_{X = 0}, Z_{ HA }|_{X > 0} | Z_{0}|_{X > 0}, Z_{ HA }|_{X = 0} | Slope | ϕ_{X = 0}, ϕ_{X > 0} | ϕ_{X = 0},ξ | Relation | Function |
---|---|---|---|---|---|---|---|---|
> | * | * | * | + | * | * | ξ > S > ϕ_{X > 0} | G ^{+} |
> | * | * | * | + | * | * | ξ > ϕ_{X > 0}> S | G ^{+} |
> | * | * | * | + | * | * | S > ξ > ϕ_{X > 0} | G ^{+} |
> | * | < | * | - | * | * | ξ > S > ϕ_{X > 0} | G ^{+} |
∅ | > | > | * | * | < | * | ξ > S > ϕ_{X > 0} | G ^{+} |
* | < | * | * | * | < | * | ξ > S > ϕ_{X > 0} | G ^{+} |
* | < | * | * | * | * | > | S > ϕ_{X > 0}> ξ | G ^{-} |
* | < | * | * | * | * | > | ϕ_{X > 0}> ξ > S | G ^{-} |
* | < | * | * | * | * | > | ϕ_{X > 0}> S > ξ | G ^{-} |
> | * | < | * | - | * | * | ξ > ϕ_{X > 0}> S | P ^{+} T ^{+} |
> | * | < | * | - | * | * | S > ξ > ϕ_{X > 0} | P ^{+} T ^{+} |
∅ | > | > | * | * | < | * | ξ > ϕ_{X > 0}> S | P ^{+} T ^{+} |
∅ | > | > | * | * | < | * | S > ξ > ϕ_{X > 0} | P ^{+} T ^{+} |
* | < | * | * | * | < | * | ξ > ϕ_{X > 0}> S | P ^{+} T ^{+} |
* | < | * | * | * | < | * | S > ξ > ϕ_{X > 0} | P ^{+} T ^{+} |
∅ | > | > | * | * | > | * | ξ > S > ϕ_{X > 0} | Impossible |
∅ | > | > | * | * | > | * | ξ > ϕ_{X > 0}> S | P ^{+} T ^{-} |
∅ | > | > | * | * | > | * | S > ξ > ϕ_{X > 0} | P ^{+} T ^{-} |
* | < | * | * | * | > | < | ξ > S > ϕ_{X > 0} | Impossible |
* | < | * | * | * | > | < | ξ > ϕ_{X > 0}> S | P ^{+} T ^{-} |
* | < | * | * | * | > | < | S > ξ > ϕ_{X > 0} | P ^{+} T ^{-} |
Kurtosis as a measure of FFL specialization
where μ_{4} if the fourth moment about the mean, and σ is the standard deviation. Here K_{0} is a reference value known as excess kurtosis. In general K_{0} = 3 in order to make the kurtosis of the normal distribution equal to zero. This allows obtaining positive kurtosis, i.e. distributions with higher peakdeness than the normal distribution, or negative kurtosis flatter that the normal distribution. Without lost of generality we can consider K_{0} = 0. In this context, kurtosis is defined positive and the kurtosis values can be directly compared: systems with higher kurtosis will have higher degree of specialization
Notes
Declarations
Acknowledgements
We thank the members of the Complex Systems Lab for fruitful discussion. This work was supported by the EU grant CELLCOMPUT, the EU 6th Framework project SYNLET (NEST 043312), the James McDonnell Foundation and by the Santa Fe Institute.
Authors’ Affiliations
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