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Shape, size, and robustness: feasible regions in the parameter space of biochemical networks.

Dayarian A, Chaves M, Sontag ED, Sengupta AM - PLoS Comput. Biol. (2009)

Bottom Line: One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional.In particular, we found that, between two alternative ways of activating Wingless, one is more robust than the other.As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey, United States of America.

ABSTRACT
The concept of robustness of regulatory networks has received much attention in the last decade. One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional. In this paper, we show that, in addition to volume, the geometry of this region has important consequences for the robustness and the fragility of a network. We develop an approximation within which we could algebraically specify the feasible region. We analyze the segment polarity gene network to illustrate our approach. The study of random walks in the parameter space and how they exit the feasible region provide us with a rich perspective on the different modes of failure of this network model. In particular, we found that, between two alternative ways of activating Wingless, one is more robust than the other. Our method provides a more complete measure of robustness to parameter variation. As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks.

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Random walk in the space of admissible parameters.We choose a random point from admissible parameter set and follow a random walk until it hits a boundary after t steps. (A) The red (and dashed) and the blue (and solid) graphs represent the probability of survival as a function of time for von Dassow et al. and SLP models, respectively. These graphs results from 30,000 runs of random walks. The results given for volume are based on the fraction of feasible parameter combinations found in 1,000,000 randomly chosen combinations. (B) Histogram of violated conditions for the random walk in (A). The number above each bin indicates the corresponding condition in Tables 1 and 2.
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pcbi-1000256-g003: Random walk in the space of admissible parameters.We choose a random point from admissible parameter set and follow a random walk until it hits a boundary after t steps. (A) The red (and dashed) and the blue (and solid) graphs represent the probability of survival as a function of time for von Dassow et al. and SLP models, respectively. These graphs results from 30,000 runs of random walks. The results given for volume are based on the fraction of feasible parameter combinations found in 1,000,000 randomly chosen combinations. (B) Histogram of violated conditions for the random walk in (A). The number above each bin indicates the corresponding condition in Tables 1 and 2.

Mentions: We explore the feasible region by following random walks starting from random points. Whenever one of the random trajectories hits a boundary and exits the feasible region, we terminate the walk and keep track of the inequality that was violated. This process can be viewed as a simulation of parameter evolution due to mutations in a fitness landscape that looks like a plateau. The points in the feasible region have a constant high fitness, and the rest of the points have zero fitness. The result of the simulation is presented in Figure 3.


Shape, size, and robustness: feasible regions in the parameter space of biochemical networks.

Dayarian A, Chaves M, Sontag ED, Sengupta AM - PLoS Comput. Biol. (2009)

Random walk in the space of admissible parameters.We choose a random point from admissible parameter set and follow a random walk until it hits a boundary after t steps. (A) The red (and dashed) and the blue (and solid) graphs represent the probability of survival as a function of time for von Dassow et al. and SLP models, respectively. These graphs results from 30,000 runs of random walks. The results given for volume are based on the fraction of feasible parameter combinations found in 1,000,000 randomly chosen combinations. (B) Histogram of violated conditions for the random walk in (A). The number above each bin indicates the corresponding condition in Tables 1 and 2.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC2599888&req=5

pcbi-1000256-g003: Random walk in the space of admissible parameters.We choose a random point from admissible parameter set and follow a random walk until it hits a boundary after t steps. (A) The red (and dashed) and the blue (and solid) graphs represent the probability of survival as a function of time for von Dassow et al. and SLP models, respectively. These graphs results from 30,000 runs of random walks. The results given for volume are based on the fraction of feasible parameter combinations found in 1,000,000 randomly chosen combinations. (B) Histogram of violated conditions for the random walk in (A). The number above each bin indicates the corresponding condition in Tables 1 and 2.
Mentions: We explore the feasible region by following random walks starting from random points. Whenever one of the random trajectories hits a boundary and exits the feasible region, we terminate the walk and keep track of the inequality that was violated. This process can be viewed as a simulation of parameter evolution due to mutations in a fitness landscape that looks like a plateau. The points in the feasible region have a constant high fitness, and the rest of the points have zero fitness. The result of the simulation is presented in Figure 3.

Bottom Line: One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional.In particular, we found that, between two alternative ways of activating Wingless, one is more robust than the other.As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey, United States of America.

ABSTRACT
The concept of robustness of regulatory networks has received much attention in the last decade. One measure of robustness has been associated with the volume of the feasible region, namely, the region in the parameter space in which the system is functional. In this paper, we show that, in addition to volume, the geometry of this region has important consequences for the robustness and the fragility of a network. We develop an approximation within which we could algebraically specify the feasible region. We analyze the segment polarity gene network to illustrate our approach. The study of random walks in the parameter space and how they exit the feasible region provide us with a rich perspective on the different modes of failure of this network model. In particular, we found that, between two alternative ways of activating Wingless, one is more robust than the other. Our method provides a more complete measure of robustness to parameter variation. As a general modeling strategy, our approach is an interesting alternative to Boolean representation of biochemical networks.

Show MeSH