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Characterizing the relationship between steady state and response using analytical expressions for the steady states of mass action models.

Loriaux PM, Tesler G, Hoffmann A - PLoS Comput. Biol. (2013)

Bottom Line: This suggests that activation of caspase 8 is a critical point in the death decision process.Finally, we show that changes in the threshold at which TRAIL results in cell death is not always equivalent to changes in the time of death, as is commonly assumed.Our work demonstrates that an analytical expression is a powerful tool for identifying steady state determinants of the cellular response to perturbation.

View Article: PubMed Central - PubMed

Affiliation: Signaling Systems Laboratory, Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California, United States of America.

ABSTRACT
The steady states of cells affect their response to perturbation. Indeed, diagnostic markers for predicting the response to therapeutic perturbation are often based on steady state measurements. In spite of this, no method exists to systematically characterize the relationship between steady state and response. Mathematical models are established tools for studying cellular responses, but characterizing their relationship to the steady state requires that it have a parametric, or analytical, expression. For some models, this expression can be derived by the King-Altman method. However, King-Altman requires that no substrate act as an enzyme, and is therefore not applicable to most models of signal transduction. For this reason we developed py-substitution, a simple but general method for deriving analytical expressions for the steady states of mass action models. Where the King-Altman method is applicable, we show that py-substitution yields an equivalent expression, and at comparable efficiency. We use py-substitution to study the relationship between steady state and sensitivity to the anti-cancer drug candidate, dulanermin (recombinant human TRAIL). First, we use py-substitution to derive an analytical expression for the steady state of a published model of TRAIL-induced apoptosis. Next, we show that the amount of TRAIL required for cell death is sensitive to the steady state concentrations of procaspase 8 and its negative regulator, Bar, but not the other procaspase molecules. This suggests that activation of caspase 8 is a critical point in the death decision process. Finally, we show that changes in the threshold at which TRAIL results in cell death is not always equivalent to changes in the time of death, as is commonly assumed. Our work demonstrates that an analytical expression is a powerful tool for identifying steady state determinants of the cellular response to perturbation. All code is available at http://signalingsystems.ucsd.edu/models-and-code/ or as supplementary material accompanying this paper.

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Determinants of sensitivity of TRAIL-induced cell death.(A) The dynamics of PARP cleavage are shown for EARM (left) and xEARM (right), in response to increasing doses of the TRAIL ligand (gray to blue). The abundance of cleaved PARP for each model has been normalized to the maximum observed abundance. For each model, for a particular dose of TRAIL, the time  at which PARP is 50% cleaved is indicated by the dashed red lines. For xEARM, the minimum abundance of TRAIL required to observe MOMP, , is indicated on the color scale at right. (B) Changes in  in response to changes in the steady state abundance of 12 primary xEARM species. Species have been sorted from left to right in order of those for which an increase in abundance results in the greatest increase in TRAIL sensitivity, to those for which an increase in abundance result in the greatest decrease in sensitivity. (C) Normalized sensitivity coefficients for , calculated using the xEARM model (blue), and , calculated using both EARM (white) and xEARM (gray), for each of the 12 primary species in (B).
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pcbi-1002901-g006: Determinants of sensitivity of TRAIL-induced cell death.(A) The dynamics of PARP cleavage are shown for EARM (left) and xEARM (right), in response to increasing doses of the TRAIL ligand (gray to blue). The abundance of cleaved PARP for each model has been normalized to the maximum observed abundance. For each model, for a particular dose of TRAIL, the time at which PARP is 50% cleaved is indicated by the dashed red lines. For xEARM, the minimum abundance of TRAIL required to observe MOMP, , is indicated on the color scale at right. (B) Changes in in response to changes in the steady state abundance of 12 primary xEARM species. Species have been sorted from left to right in order of those for which an increase in abundance results in the greatest increase in TRAIL sensitivity, to those for which an increase in abundance result in the greatest decrease in sensitivity. (C) Normalized sensitivity coefficients for , calculated using the xEARM model (blue), and , calculated using both EARM (white) and xEARM (gray), for each of the 12 primary species in (B).

Mentions: Finally, we sought to use py-substitution to characterize the relationship between steady state and the response to the cancer drug, dulanermin. Dulanermin is a recombinant human form of the endogenous ligand TRAIL, whose mechanism for triggering cell death is modeled in version 1.0 of the extrinsic apoptosis reaction model, or EARM [14]. This model considers the biochemical events following engagement of the death receptors 4 and 5 (DR4/5), including receptor-induced cleavage of initiator caspases, positive-feedback by effector caspases, and feed-forward amplification by the mitochondrial pathway following outer membrane permeabilization, or MOMP (Figure 5). The EARM model was trained on data derived from HeLa cells co-treated with cyclohexamide, an inhibitor of protein synthesis that results in hypersensitivity to TRAIL [43]. Accordingly, any amount of ligand in the EARM model results in cell death. The abundance of ligand still affects the time of death, defined for example by the time at which half of the caspase 3 target protein PARP has been cleaved (Figure 6A, left) [44]. Note in this section we refer to the abundance of a species rather than its concentration, as these are the units chosen by the original authors.


Characterizing the relationship between steady state and response using analytical expressions for the steady states of mass action models.

Loriaux PM, Tesler G, Hoffmann A - PLoS Comput. Biol. (2013)

Determinants of sensitivity of TRAIL-induced cell death.(A) The dynamics of PARP cleavage are shown for EARM (left) and xEARM (right), in response to increasing doses of the TRAIL ligand (gray to blue). The abundance of cleaved PARP for each model has been normalized to the maximum observed abundance. For each model, for a particular dose of TRAIL, the time  at which PARP is 50% cleaved is indicated by the dashed red lines. For xEARM, the minimum abundance of TRAIL required to observe MOMP, , is indicated on the color scale at right. (B) Changes in  in response to changes in the steady state abundance of 12 primary xEARM species. Species have been sorted from left to right in order of those for which an increase in abundance results in the greatest increase in TRAIL sensitivity, to those for which an increase in abundance result in the greatest decrease in sensitivity. (C) Normalized sensitivity coefficients for , calculated using the xEARM model (blue), and , calculated using both EARM (white) and xEARM (gray), for each of the 12 primary species in (B).
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Related In: Results  -  Collection

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getmorefigures.php?uid=PMC3585464&req=5

pcbi-1002901-g006: Determinants of sensitivity of TRAIL-induced cell death.(A) The dynamics of PARP cleavage are shown for EARM (left) and xEARM (right), in response to increasing doses of the TRAIL ligand (gray to blue). The abundance of cleaved PARP for each model has been normalized to the maximum observed abundance. For each model, for a particular dose of TRAIL, the time at which PARP is 50% cleaved is indicated by the dashed red lines. For xEARM, the minimum abundance of TRAIL required to observe MOMP, , is indicated on the color scale at right. (B) Changes in in response to changes in the steady state abundance of 12 primary xEARM species. Species have been sorted from left to right in order of those for which an increase in abundance results in the greatest increase in TRAIL sensitivity, to those for which an increase in abundance result in the greatest decrease in sensitivity. (C) Normalized sensitivity coefficients for , calculated using the xEARM model (blue), and , calculated using both EARM (white) and xEARM (gray), for each of the 12 primary species in (B).
Mentions: Finally, we sought to use py-substitution to characterize the relationship between steady state and the response to the cancer drug, dulanermin. Dulanermin is a recombinant human form of the endogenous ligand TRAIL, whose mechanism for triggering cell death is modeled in version 1.0 of the extrinsic apoptosis reaction model, or EARM [14]. This model considers the biochemical events following engagement of the death receptors 4 and 5 (DR4/5), including receptor-induced cleavage of initiator caspases, positive-feedback by effector caspases, and feed-forward amplification by the mitochondrial pathway following outer membrane permeabilization, or MOMP (Figure 5). The EARM model was trained on data derived from HeLa cells co-treated with cyclohexamide, an inhibitor of protein synthesis that results in hypersensitivity to TRAIL [43]. Accordingly, any amount of ligand in the EARM model results in cell death. The abundance of ligand still affects the time of death, defined for example by the time at which half of the caspase 3 target protein PARP has been cleaved (Figure 6A, left) [44]. Note in this section we refer to the abundance of a species rather than its concentration, as these are the units chosen by the original authors.

Bottom Line: This suggests that activation of caspase 8 is a critical point in the death decision process.Finally, we show that changes in the threshold at which TRAIL results in cell death is not always equivalent to changes in the time of death, as is commonly assumed.Our work demonstrates that an analytical expression is a powerful tool for identifying steady state determinants of the cellular response to perturbation.

View Article: PubMed Central - PubMed

Affiliation: Signaling Systems Laboratory, Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California, United States of America.

ABSTRACT
The steady states of cells affect their response to perturbation. Indeed, diagnostic markers for predicting the response to therapeutic perturbation are often based on steady state measurements. In spite of this, no method exists to systematically characterize the relationship between steady state and response. Mathematical models are established tools for studying cellular responses, but characterizing their relationship to the steady state requires that it have a parametric, or analytical, expression. For some models, this expression can be derived by the King-Altman method. However, King-Altman requires that no substrate act as an enzyme, and is therefore not applicable to most models of signal transduction. For this reason we developed py-substitution, a simple but general method for deriving analytical expressions for the steady states of mass action models. Where the King-Altman method is applicable, we show that py-substitution yields an equivalent expression, and at comparable efficiency. We use py-substitution to study the relationship between steady state and sensitivity to the anti-cancer drug candidate, dulanermin (recombinant human TRAIL). First, we use py-substitution to derive an analytical expression for the steady state of a published model of TRAIL-induced apoptosis. Next, we show that the amount of TRAIL required for cell death is sensitive to the steady state concentrations of procaspase 8 and its negative regulator, Bar, but not the other procaspase molecules. This suggests that activation of caspase 8 is a critical point in the death decision process. Finally, we show that changes in the threshold at which TRAIL results in cell death is not always equivalent to changes in the time of death, as is commonly assumed. Our work demonstrates that an analytical expression is a powerful tool for identifying steady state determinants of the cellular response to perturbation. All code is available at http://signalingsystems.ucsd.edu/models-and-code/ or as supplementary material accompanying this paper.

Show MeSH