Limits...
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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The clines for ptc and CN.The green curve shows the ptc-cline. In the high Hill coefficient limit, ptc value drops sharply from one to zero as CN passes the threshold of min(1−κCIptc, κCNptc). Blue and red curves show the CN-clines for relatively higher and lower values of HH signaling levels, respectively. Intersection points 1 and 2 determine CI, CN and ptc in cell 1 and 2/4, respectively. Here it is assumed that the negative feedback on ptc coming from repression by CN is active. Therefore, ptc and CN are not maximally expressed. Dashed blue line shows the CN-cline for a fine-tuned set of parameters.
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pcbi-1000256-g004: The clines for ptc and CN.The green curve shows the ptc-cline. In the high Hill coefficient limit, ptc value drops sharply from one to zero as CN passes the threshold of min(1−κCIptc, κCNptc). Blue and red curves show the CN-clines for relatively higher and lower values of HH signaling levels, respectively. Intersection points 1 and 2 determine CI, CN and ptc in cell 1 and 2/4, respectively. Here it is assumed that the negative feedback on ptc coming from repression by CN is active. Therefore, ptc and CN are not maximally expressed. Dashed blue line shows the CN-cline for a fine-tuned set of parameters.

Mentions: How could we ever get such an intermediate values in our approach? First, from Equations 13 and 14, in the cells where en is not expressed and therefore ci is not repressed, namely in cells 1, 2 and 4, we have CI+CN = 1⇒CI = 1−CN (this does not depend on the high Hill coefficient approximation). Since ptc is regulated by CI-CN, we could draw one cline expressing ptc concentration as a function of CN. This curve is represented by the green graph in Figure 4. We will call it the ptc-cline. Here it is assumed that the negative feedback on ptc coming from repression by CN is active. This means that CN and ptc are not expressed maximally. For ptc to be expressed, the activation by CI requires 1−CN>κCIptc⇒CN<1−κCIptc. In addition, we need CN to be smaller than κCNptc to avoid repression of ptc by CN. Thus, for values of CN smaller than the threshold of min(1−κCIptc, κCNptc), ptc is fully expressed. As CN passes this point, the value of ptc will drop sharply. In the high Hill coefficient limit, ptc will abruptly fall to zero.


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)

The clines for ptc and CN.The green curve shows the ptc-cline. In the high Hill coefficient limit, ptc value drops sharply from one to zero as CN passes the threshold of min(1−κCIptc, κCNptc). Blue and red curves show the CN-clines for relatively higher and lower values of HH signaling levels, respectively. Intersection points 1 and 2 determine CI, CN and ptc in cell 1 and 2/4, respectively. Here it is assumed that the negative feedback on ptc coming from repression by CN is active. Therefore, ptc and CN are not maximally expressed. Dashed blue line shows the CN-cline for a fine-tuned set of parameters.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1000256-g004: The clines for ptc and CN.The green curve shows the ptc-cline. In the high Hill coefficient limit, ptc value drops sharply from one to zero as CN passes the threshold of min(1−κCIptc, κCNptc). Blue and red curves show the CN-clines for relatively higher and lower values of HH signaling levels, respectively. Intersection points 1 and 2 determine CI, CN and ptc in cell 1 and 2/4, respectively. Here it is assumed that the negative feedback on ptc coming from repression by CN is active. Therefore, ptc and CN are not maximally expressed. Dashed blue line shows the CN-cline for a fine-tuned set of parameters.
Mentions: How could we ever get such an intermediate values in our approach? First, from Equations 13 and 14, in the cells where en is not expressed and therefore ci is not repressed, namely in cells 1, 2 and 4, we have CI+CN = 1⇒CI = 1−CN (this does not depend on the high Hill coefficient approximation). Since ptc is regulated by CI-CN, we could draw one cline expressing ptc concentration as a function of CN. This curve is represented by the green graph in Figure 4. We will call it the ptc-cline. Here it is assumed that the negative feedback on ptc coming from repression by CN is active. This means that CN and ptc are not expressed maximally. For ptc to be expressed, the activation by CI requires 1−CN>κCIptc⇒CN<1−κCIptc. In addition, we need CN to be smaller than κCNptc to avoid repression of ptc by CN. Thus, for values of CN smaller than the threshold of min(1−κCIptc, κCNptc), ptc is fully expressed. As CN passes this point, the value of ptc will drop sharply. In the high Hill coefficient limit, ptc will abruptly fall to zero.

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