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4. The gas(CO,/is produced by the cellular metabolism of the body and is removed from the body when one exhales. There is a mechanism in the brain which senses the CO, level and sends signals to the breathing mechanism to take deeper breathes if the CO, level is too high. Let V, be the volume of the nth breath and let C, be the CO, concentration at the time of the nth breath. Let M be the (assumed) constant CO, concentra- tion produced by the metabolism between each breath. Then, a simple model of this system is:

4. The gas(CO,/is produced by the cellular metabolism of the body and is removed from the body when one exhales. There is a mechanism in the brain which senses the CO, level and sends signals to the breathing mechanism to take deeper breathes if the CO, level is too high. Let V, be the volume of the nth breath and let C, be the CO, concentration at the time of the nth breath. Let M be the (assumed) constant CO, concentra- tion produced by the metabolism between each breath. Then, a simple model of this system is:

Biochemistry
Biochemistry
9th Edition
ISBN:9781319114671
Author:Lubert Stryer, Jeremy M. Berg, John L. Tymoczko, Gregory J. Gatto Jr.
Publisher:Lubert Stryer, Jeremy M. Berg, John L. Tymoczko, Gregory J. Gatto Jr.
Chapter1: Biochemistry: An Evolving Science
Section: Chapter Questions
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please solve (g)and (h)
4. The gas(CO, is produced by the cellular metabolism of the body and is removed from the body when one exhales. There is a mechanism in the brain which senses the CO, level and sends signals to the breathing mechanism to take deeper breathes if the CO, level is too high. Let V, be the volume of the nth breath and let C, be the CO, concentration at the time of the nth breath. Let M be the (assumed) constant CO2 concentra- tion produced by the metabolism between each breath. Then, a simple model of this system is: Cn+1 = C, - av, + M Va+1 = BC, (19) (20) The second equation says that the volume of then+1 breath is propor- tional to the CO, concentration at the time of the nth breath. The first equation says that CO, concentration C, is lowered by an amount pro- portional to the volume of the nth breath and raised by M. Given the initial concentration Co and breath size Vo, we would like to determine the behavior of the sequences {Cn} and {Vn}. Our discussion follows (10], where more information can be found. N Show that for appropriate choices of A, and A, Ca- d, A + da solves (23) for any choice of dj and da. V For any choice of V, and Co, show that dị and da can be chosen so that C. - dịA + dzA + M/aß V - B(d,A" + daA) + M/a solve the original problem. O Prove that if 4aß < 1, then C, + M/aß and V, + M/a as n + 00, so that whatever the initial concentration and breath size, the breathing settles down to a steady rate (g) Show that if 1 < 4aß < 4, the concentrations exhibit oscillations which die out. (h) Show that if a and 3 are large enough, the size of breaths and the CO, concentration can exhibit larger and larger oscillations. Can you explain this analytically?
Transcribed Image Text:4. The gas(CO, is produced by the cellular metabolism of the body and is removed from the body when one exhales. There is a mechanism in the brain which senses the CO, level and sends signals to the breathing mechanism to take deeper breathes if the CO, level is too high. Let V, be the volume of the nth breath and let C, be the CO, concentration at the time of the nth breath. Let M be the (assumed) constant CO2 concentra- tion produced by the metabolism between each breath. Then, a simple model of this system is: Cn+1 = C, - av, + M Va+1 = BC, (19) (20) The second equation says that the volume of then+1 breath is propor- tional to the CO, concentration at the time of the nth breath. The first equation says that CO, concentration C, is lowered by an amount pro- portional to the volume of the nth breath and raised by M. Given the initial concentration Co and breath size Vo, we would like to determine the behavior of the sequences {Cn} and {Vn}. Our discussion follows (10], where more information can be found. N Show that for appropriate choices of A, and A, Ca- d, A + da solves (23) for any choice of dj and da. V For any choice of V, and Co, show that dị and da can be chosen so that C. - dịA + dzA + M/aß V - B(d,A" + daA) + M/a solve the original problem. O Prove that if 4aß < 1, then C, + M/aß and V, + M/a as n + 00, so that whatever the initial concentration and breath size, the breathing settles down to a steady rate (g) Show that if 1 < 4aß < 4, the concentrations exhibit oscillations which die out. (h) Show that if a and 3 are large enough, the size of breaths and the CO, concentration can exhibit larger and larger oscillations. Can you explain this analytically?
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