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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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The clines for ptc and CN.(A) Blue and red curves show the CN-clines for relatively higher and lower values of HH signaling levels, respectively. The green curve shows the ptc-cline when . In this case, the negative feedback on ptc coming from repression by CN is active. Therefore, ptc and CN are not maximally expressed. The dashed green curve shows the other case where . In this case, both CN and ptc are maximally expressed. This means that the negative feedback on ptc is inactive. (B) The green curve shows the ptc-cline. Blue and red curves show the CN-clines for relatively higher and lower values of HH signaling levels, respectively. The blue curve shows the situation where HH signaling is strong enough so that the ptc concentration needed to produce CN is higher than the maximal possible value for ptc, namely, one. Therefore, CN will not be produced in the corresponding cell. In the high Hill coefficient approximation, this is the only way that we can have CN level in cell 2 (intersection point 2) to be different from cell 1 (intersection point 1).
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pcbi-1000256-g005: The clines for ptc and CN.(A) Blue and red curves show the CN-clines for relatively higher and lower values of HH signaling levels, respectively. The green curve shows the ptc-cline when . In this case, the negative feedback on ptc coming from repression by CN is active. Therefore, ptc and CN are not maximally expressed. The dashed green curve shows the other case where . In this case, both CN and ptc are maximally expressed. This means that the negative feedback on ptc is inactive. (B) The green curve shows the ptc-cline. Blue and red curves show the CN-clines for relatively higher and lower values of HH signaling levels, respectively. The blue curve shows the situation where HH signaling is strong enough so that the ptc concentration needed to produce CN is higher than the maximal possible value for ptc, namely, one. Therefore, CN will not be produced in the corresponding cell. In the high Hill coefficient approximation, this is the only way that we can have CN level in cell 2 (intersection point 2) to be different from cell 1 (intersection point 1).

Mentions: What would the CN level in cells 1 and 2 be when condition 1 is satisfied? As one sees from Figure 5A, there are two possibilities depending upon whether min(1−κCIptc, κCNptc) is smaller or larger than . The case corresponding to ptc-cline in solid green has been discussed before. This is the case where ptc levels are affected by the negative feedback, and CN level is equal to min(1−κCIptc, κCNptc), which is less than its maximal possible value of . When the ptc-cline is like the dashed green line in Figure 5, CN levels in both cell 1 and cell 2 is equal to the maximal amount of , which is lower than min(1−κCIptc, κCNptc). In this case, the negative feedback is not active and ptc is maximally expressed (ptc = 1). We conclude that CN level is given by , which we call ZC. We will now discuss the conditions to be satisfied by ZC for proper expression pattern of en and wg.


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.(A) Blue and red curves show the CN-clines for relatively higher and lower values of HH signaling levels, respectively. The green curve shows the ptc-cline when . In this case, the negative feedback on ptc coming from repression by CN is active. Therefore, ptc and CN are not maximally expressed. The dashed green curve shows the other case where . In this case, both CN and ptc are maximally expressed. This means that the negative feedback on ptc is inactive. (B) The green curve shows the ptc-cline. Blue and red curves show the CN-clines for relatively higher and lower values of HH signaling levels, respectively. The blue curve shows the situation where HH signaling is strong enough so that the ptc concentration needed to produce CN is higher than the maximal possible value for ptc, namely, one. Therefore, CN will not be produced in the corresponding cell. In the high Hill coefficient approximation, this is the only way that we can have CN level in cell 2 (intersection point 2) to be different from cell 1 (intersection point 1).
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1000256-g005: The clines for ptc and CN.(A) Blue and red curves show the CN-clines for relatively higher and lower values of HH signaling levels, respectively. The green curve shows the ptc-cline when . In this case, the negative feedback on ptc coming from repression by CN is active. Therefore, ptc and CN are not maximally expressed. The dashed green curve shows the other case where . In this case, both CN and ptc are maximally expressed. This means that the negative feedback on ptc is inactive. (B) The green curve shows the ptc-cline. Blue and red curves show the CN-clines for relatively higher and lower values of HH signaling levels, respectively. The blue curve shows the situation where HH signaling is strong enough so that the ptc concentration needed to produce CN is higher than the maximal possible value for ptc, namely, one. Therefore, CN will not be produced in the corresponding cell. In the high Hill coefficient approximation, this is the only way that we can have CN level in cell 2 (intersection point 2) to be different from cell 1 (intersection point 1).
Mentions: What would the CN level in cells 1 and 2 be when condition 1 is satisfied? As one sees from Figure 5A, there are two possibilities depending upon whether min(1−κCIptc, κCNptc) is smaller or larger than . The case corresponding to ptc-cline in solid green has been discussed before. This is the case where ptc levels are affected by the negative feedback, and CN level is equal to min(1−κCIptc, κCNptc), which is less than its maximal possible value of . When the ptc-cline is like the dashed green line in Figure 5, CN levels in both cell 1 and cell 2 is equal to the maximal amount of , which is lower than min(1−κCIptc, κCNptc). In this case, the negative feedback is not active and ptc is maximally expressed (ptc = 1). We conclude that CN level is given by , which we call ZC. We will now discuss the conditions to be satisfied by ZC for proper expression pattern of en and wg.

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