4. (Hematopoietic cell replication and control) Periodic hematopoiesis is a disease characterized by large oscillations in the red blood cell count. Red blood cells are generated by a feedback mechanism that approximately obeys the difference equation 1 = *4+1 F(x,)+ -X, 1+ yt 1+ yt The function F(xt) describes production as a function of number of cells and takes the form an F(x)=Film +x" The terms here are: Xt n Parameter Meaning Normal Value Number of cells at timet About 330 (in billions) Time for cell development 5.7 days Y Fraction of cells that die each day 0.0231 F. Maximum production of cells 76.2 billion o Value where cell production 247 billion is halved Shape parameter 7.6 a) Graph and explain F(x) with these parameter values. Does this sort of feedback system make sense? b) Use a computer to find the equilibrium points with the normal parameters. c) Graph and cobweb the difference equation. If you start near the equilibrium point, do values remain nearby? What might a solution look like? d) Certain autoimmune diseases increase y, the death rate of cells. Study what happens to the equilibrium and the solution as y increases. Can you explain your results in biological terms? e) Explore what would happen if the value of n were decreased. Does the equilibrium become more sensitive to small changes in y? 4. (Hematopoietic cell replication and control) Periodic hematopoiesis is a disease characterized by large oscillations in the red blood cell count. Red blood cells are generated by a feedback mechanism that approximately obeys the difference equation 1 = *4+1 F(x,)+ -X, 1+ yt 1+ yt The function F(xt) describes production as a function of number of cells and takes the form an F(x)=Film +x" The terms here are: Xt n Parameter Meaning Normal Value Number of cells at timet About 330 (in billions) Time for cell development 5.7 days Y Fraction of cells that die each day 0.0231 F. Maximum production of cells 76.2 billion o Value where cell production 247 billion is halved Shape parameter 7.6 a) Graph and explain F(x) with these parameter values. Does this sort of feedback system make sense? b) Use a computer to find the equilibrium points with the normal parameters. c) Graph and cobweb the difference equation. If you start near the equilibrium point, do values remain nearby? What might a solution look like? d) Certain autoimmune diseases increase y, the death rate of cells. Study what happens to the equilibrium and the solution as y increases. Can you explain your results in biological terms? e) Explore what would happen if the value of n were decreased. Does the equilibrium become more sensitive to small changes in y
