Specialized or flexible feed-forward loop motifs: a question of topology.
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Network motifs are recurrent interaction patterns, which are significantly more often encountered in biological interaction graphs than expected from random nets.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.
Affiliation: Complex Systems Lab (ICREA-UPF), Barcelona Biomedical Research Park (PRBB-GRIB), 08003 Barcelona, Spain. javier.macia@upf.edu
ABSTRACT
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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 clines 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. |
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Mentions: Furthermore, expression (6) shows a single inflection point, i.e. a point where curvature changes from concave to convex or viceversa, in the biological domain defined by Y ≥ 0 and Z ≥ 0, but no extrema (local maximum or minimum). In order to understand how the input triggers the dynamical response, we study the configuration of the two clines with X = 0, i.e. no input, versus X > 0, i.e. with input. In figure (2) we show a numerical example. In absence of input, the system is governed by a single stable fixed point, denoted by ϕX = 0, located at the crossing of the clines (5) and (6). On addition of input, the fixed point moves, because the cline configuration changes. Now the system shifts from ϕx = 0 to the new stable fixed point ϕx > 0. The point ϕX > 0 is determined by the crossing of the new location of the clines for X > 0. The dynamical response of the system, which is represented by the trajectory of the FFL in phase space, is generated by this change of the stable regime. A subset of four parameters, i.e. {αX, βX, βY, βXY}, describing the interactions of the proteins, classify the type of the circuits into coherent and incoherent according to [28]. Without input, only βY is relevant for the geometry of the cline Ż = 0. If βY > 1 the cline rises, because ZHA > Z0, whereas for βY < 1 the cline decreases as shown in figure (3a). Under the presence of an external input the other parameters become relevant (figure (3b)): |
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Affiliation: Complex Systems Lab (ICREA-UPF), Barcelona Biomedical Research Park (PRBB-GRIB), 08003 Barcelona, Spain. javier.macia@upf.edu
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 clines 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.