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10. 10 Real-world Dynamics and Control In vivo, e.g., within a single living cell, various types of proteins are interacting with each other, which are in the meantime continuously synthesized by genes through mRNA. A full description of such a system would involve thousands of signals, which is difficult to achieve. However, we can simplify our world by focusing on the relationship between one particular type of protein and one particular type of mRNA. Let x1(t) be the concentration of the mRNA, and x2(t) be the concen- tration of the protein, respectively. The interaction between the two states is described by x₁ = piu k₁x1 x2 = P2x1 - k2x2 (1) (2) where P1, P2 and k₁, k2 are parameters for the production rates and degradation rates, and u is our input. (a) Let us for the moment assume x₁ is a constant, i.e., x1(t) = a for all t. Thus, we can just look at the Eq. (2). Deter- mine the sign of k2, that is, it should be positive or negative, from the physical interpretation of the system. Show your reasoning. (b) (c) Suppose we have sensors that can measure the real-time concentration of the protein x2(t). What would be the approach you would take for the design of a feedback controller, under which x2(t) can hopefully track a reference signal r? Show the main steps of your design. Explain possible uncertainties for the equation (1)-(2) considering the real world scenario as described from the begin- ning.

10. 10 Real-world Dynamics and Control In vivo, e.g., within a single living cell, various types of proteins are interacting with each other, which are in the meantime continuously synthesized by genes through mRNA. A full description of such a system would involve thousands of signals, which is difficult to achieve. However, we can simplify our world by focusing on the relationship between one particular type of protein and one particular type of mRNA. Let x1(t) be the concentration of the mRNA, and x2(t) be the concen- tration of the protein, respectively. The interaction between the two states is described by x₁ = piu k₁x1 x2 = P2x1 - k2x2 (1) (2) where P1, P2 and k₁, k2 are parameters for the production rates and degradation rates, and u is our input. (a) Let us for the moment assume x₁ is a constant, i.e., x1(t) = a for all t. Thus, we can just look at the Eq. (2). Deter- mine the sign of k2, that is, it should be positive or negative, from the physical interpretation of the system. Show your reasoning. (b) (c) Suppose we have sensors that can measure the real-time concentration of the protein x2(t). What would be the approach you would take for the design of a feedback controller, under which x2(t) can hopefully track a reference signal r? Show the main steps of your design. Explain possible uncertainties for the equation (1)-(2) considering the real world scenario as described from the begin- ning.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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10. 10 Real-world Dynamics and Control In vivo, e.g., within a single living cell, various types of proteins are interacting with each other, which are in the meantime continuously synthesized by genes through mRNA. A full description of such a system would involve thousands of signals, which is difficult to achieve. However, we can simplify our world by focusing on the relationship between one particular type of protein and one particular type of mRNA. Let x1(t) be the concentration of the mRNA, and x2(t) be the concen- tration of the protein, respectively. The interaction between the two states is described by x₁ = piu k₁x1 x2 = P2x1 - k2x2 (1) (2) where P1, P2 and k₁, k2 are parameters for the production rates and degradation rates, and u is our input. (a) Let us for the moment assume x₁ is a constant, i.e., x1(t) = a for all t. Thus, we can just look at the Eq. (2). Deter- mine the sign of k2, that is, it should be positive or negative, from the physical interpretation of the system. Show your reasoning.
Transcribed Image Text:10. 10 Real-world Dynamics and Control In vivo, e.g., within a single living cell, various types of proteins are interacting with each other, which are in the meantime continuously synthesized by genes through mRNA. A full description of such a system would involve thousands of signals, which is difficult to achieve. However, we can simplify our world by focusing on the relationship between one particular type of protein and one particular type of mRNA. Let x1(t) be the concentration of the mRNA, and x2(t) be the concen- tration of the protein, respectively. The interaction between the two states is described by x₁ = piu k₁x1 x2 = P2x1 - k2x2 (1) (2) where P1, P2 and k₁, k2 are parameters for the production rates and degradation rates, and u is our input. (a) Let us for the moment assume x₁ is a constant, i.e., x1(t) = a for all t. Thus, we can just look at the Eq. (2). Deter- mine the sign of k2, that is, it should be positive or negative, from the physical interpretation of the system. Show your reasoning.
(b) (c) Suppose we have sensors that can measure the real-time concentration of the protein x2(t). What would be the approach you would take for the design of a feedback controller, under which x2(t) can hopefully track a reference signal r? Show the main steps of your design. Explain possible uncertainties for the equation (1)-(2) considering the real world scenario as described from the begin- ning.
Transcribed Image Text:(b) (c) Suppose we have sensors that can measure the real-time concentration of the protein x2(t). What would be the approach you would take for the design of a feedback controller, under which x2(t) can hopefully track a reference signal r? Show the main steps of your design. Explain possible uncertainties for the equation (1)-(2) considering the real world scenario as described from the begin- ning.
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