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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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Expression pattern for key segment polarity genes and the interaction network.(A) Four cells in a parasegment with periodic boundary conditions in both dimensions. Each cell is represented by a square. The convention for numbering cells and cell faces are shown. (B) Interaction network used in reference [3]. Two green lines indicate interactions added by authors to achieve the target pattern. Black lines indicate interactions based on experimental data. Shape of the nodes indicates the corresponding component: Ellipses represent mRNAs; rectangles, proteins.
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pcbi-1000256-g001: Expression pattern for key segment polarity genes and the interaction network.(A) Four cells in a parasegment with periodic boundary conditions in both dimensions. Each cell is represented by a square. The convention for numbering cells and cell faces are shown. (B) Interaction network used in reference [3]. Two green lines indicate interactions added by authors to achieve the target pattern. Black lines indicate interactions based on experimental data. Shape of the nodes indicates the corresponding component: Ellipses represent mRNAs; rectangles, proteins.

Mentions: In the wild type segment polarity pattern, genes are expressed periodically in 14 parasegments along the fly embryo, and each parasegment consists of four stripes of cells. Because of this periodicity, one could focus only on one parasegment or in other words only on 4 cells. Figure 1A shows the wild type gene expression pattern for three key components of the segment polarity network. For simplicity, each cell is assumed to have four faces, rather than six as in the original model [3]. When using abbreviated names for components of the network, we use uppercase letters to refer to proteins and lowercase letters for the corresponding mRNAs. Wingless (WG) is a signaling molecule known experimentally to activate Engrailed (EN) through cell-to-cell communication. EN, itself a transcription factor, in turn triggers the production of another signaling molecule, Hedgehog (HH). HH then gets secreted to the neighboring cell and maintains WG expression by stabilizing an activator of wg, called Cubitus interruptus (CI). Without HH signaling, CI gets proteolytically cleaved, leaving only its amino terminus (denoted by CN), which becomes a repressor of wg. In summary, experimentally it is known that WG and EN maintain the expression of each other through cell-to-cell communication. We represent the wild type expression pattern of these mRNA components as follows:(1)where the four entries of each of the vectors correspond to the gene expression in the four cells of a parasegment. The value “0” means the gene is turned off and the value “1” means it is maximally expressed.


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)

Expression pattern for key segment polarity genes and the interaction network.(A) Four cells in a parasegment with periodic boundary conditions in both dimensions. Each cell is represented by a square. The convention for numbering cells and cell faces are shown. (B) Interaction network used in reference [3]. Two green lines indicate interactions added by authors to achieve the target pattern. Black lines indicate interactions based on experimental data. Shape of the nodes indicates the corresponding component: Ellipses represent mRNAs; rectangles, proteins.
© Copyright Policy
Related In: Results  -  Collection

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

pcbi-1000256-g001: Expression pattern for key segment polarity genes and the interaction network.(A) Four cells in a parasegment with periodic boundary conditions in both dimensions. Each cell is represented by a square. The convention for numbering cells and cell faces are shown. (B) Interaction network used in reference [3]. Two green lines indicate interactions added by authors to achieve the target pattern. Black lines indicate interactions based on experimental data. Shape of the nodes indicates the corresponding component: Ellipses represent mRNAs; rectangles, proteins.
Mentions: In the wild type segment polarity pattern, genes are expressed periodically in 14 parasegments along the fly embryo, and each parasegment consists of four stripes of cells. Because of this periodicity, one could focus only on one parasegment or in other words only on 4 cells. Figure 1A shows the wild type gene expression pattern for three key components of the segment polarity network. For simplicity, each cell is assumed to have four faces, rather than six as in the original model [3]. When using abbreviated names for components of the network, we use uppercase letters to refer to proteins and lowercase letters for the corresponding mRNAs. Wingless (WG) is a signaling molecule known experimentally to activate Engrailed (EN) through cell-to-cell communication. EN, itself a transcription factor, in turn triggers the production of another signaling molecule, Hedgehog (HH). HH then gets secreted to the neighboring cell and maintains WG expression by stabilizing an activator of wg, called Cubitus interruptus (CI). Without HH signaling, CI gets proteolytically cleaved, leaving only its amino terminus (denoted by CN), which becomes a repressor of wg. In summary, experimentally it is known that WG and EN maintain the expression of each other through cell-to-cell communication. We represent the wild type expression pattern of these mRNA components as follows:(1)where the four entries of each of the vectors correspond to the gene expression in the four cells of a parasegment. The value “0” means the gene is turned off and the value “1” means it is maximally expressed.

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