Daisyworld & Diabetes

1. Daisyworld & Diabetes Peter Saunders Department of Mathematics King’s College, London www.mth.kcl.ac.uk

2. DAISYWORLD

4. The Daisies • Two species, one with black flowers, one with white • Both grow best at 22.5C and not at all below 5 or above 40 • They grow more slowly if they are crowded • They grow in patches; they do not intermingle

5. The Temperature • Depends on the solar luminosity and the planet’s albedo. • A plant with black daisies will be warmer than one that is bare or has white daisies. • If both species are present, areas with black daisies are warmer, those with white daisies are cooler.

6. The key variables

7. An equation for the black daisies αb ( 1 – αb – αw) β(Tb) - γαb dαb/dt = = αb (αg β(Tb) – γ) b(T) is a function that is zero at 5C, rises to a maximum of one at 22.5C and then falls to zero again at 40C A simple and convenient choice is

8. An equation for the white daisies We use a similar equation for the white daisies: dαw/dt = αw (αg β(Tw) – γ) We don’t have to use the same b(T) and g but it keeps things simple. We can use different ones later if we want to.

9. Energy balance Energy arrives on Daisyworld at a rate SL(1-A) where L is the solar luminosity, S is a constant and A is the mean reflectivity Daisyworld radiates energy into space at a rate s: Stephan’s constant T: the ‘effective’ temperature. Energy in must equal energy out, and so we have

10. Heat Flow Because different regions of Daisyworld are at different temperatures, there will be heat flow. We include this in the model using the equations Note that if q=0 the whole planet is at the same temperature, i.e., the heat flow is very rapid indeed. As q increases, so do the temperature differences. Don’t worry about the 4th powers; they’re only there to make the calculations easier and don’t make any real difference.

11. The Daisyworld Equations

12. No daisies

13. Black daisies only

14. Black daisies only

15. How can an equilibrium just disappear? Plot of y=x4+2x2+vx: v= -0.125 (red), 0.8 (green), 1.5 (blue), 2.5 (black).

16. Black daisies only

17. White daisies only

18. Both kinds

20. The Glucose Control Equations (Work with Johan Koeslag, Stellenbosch University, SA) dG/dt = I + aA - bB - RG G: glucose, A: glucagon, B: insulin, I: input, R: demand dA/dt = A(f(G)h(A,B) - D) dB/dt = B(y(G)h(A,B) - D) f(G) is a decreasing and y(G) an increasing function of G h(A,B) represents inactive cells, and D switching-off of cells

21. To find the steady state value of G As usual we set all the time derivatives equal to zero. We ignore the equation for dG/dt and consider the other two: A(f(G)h(A,B) - g) = 0 B(y(G)h(A,B) - g) = 0 If neither A nor B is zero then these two equations imply f (G) = y(G)

22. Why the control is so precise We don’t know what either f or y is, but ... We know that f is a decreasing function of G while y is an increasing function so they meet (if at all) in a unique point ...

23. G … and the value of G where the curves cross is the value to which the system will always settle down, no matter what the input I and demand R may be (within reason!). Hence the “Zero Steady State Error”.

24. Why is Type 1 diabetes so hard to manage? dG/dt = I + aA - bB - RG dA/dt = A(f(G)h(A,B) - D) dB/dt = B(y(G)h(A,B) - D) In Type 1 diabetes, B=0, so the third equation doesn’t apply and we have to solve the first two for Geq and Aeq. Geq is no longer independent of I and R. It is also higher than before.

25. What about Type 2? dG/dt = I + aA - bB - RG dA/dt = A(f(G)h(A,B) - D) dB/dt = B(y(G)h(A,B) - D) We can represent “insulin resistance” by using a smaller value of beta. That leaves the set point alone; it’s just that it takes longer to get back to it.

26. But … • The explanation still depends on this ill-defined “insulin resistance” • And it doesn’t explain some of the other phenomena, such as the typically high glucagon levels diabetics have even when the blood sugar is high • So we leave the equations and go back to the biology

27. An important clue • Insulin secretion is normally pulsatile • It is typically not pulsatile in Type 2 diabetics • This indicates the beta cells are not communicating efficiently • In which case they are not able to tell the alpha cells to switch off.

28. What is Type 2 diabetes? • The common form is caused by a build up of amyloid protein. Some alpha cells do not receive the “off” signal from the beta cells. They continue to secrete glucagon even when the blood sugar is high. In the early stages (Syndrome X) beta cells can secrete sufficient extra insulin to cope. There is still a fixed point at 5 mmol/l but the return to it after a rise in glucose is slower. • As the deposits increase, more alpha cells become autonomous and eventually the beta cells cannot keep up.

29. Answers How can blood sugar regulation be so precise? Because it is done by integral rein control. Why is there diabetes but no ‘antidiabetes’? Because hGH and glucagon can back each other up. Why is Type 1 diabetes so hard to control? The set point at 5 mmol/l has disappeared. The patient must keep G away from what is now the natural equilibrium, at a much higher level of G.

30. What exactly is ‘insulin resistance’? At least part of it is that extra insulin is needed to counter the glucagon from uncontrolled alpha cells. What is Syndrome X? Just an early stage of Type 2 diabetes. Why doesn’t hGH interfere with the regulation? Somatostatin allows it to participate without destabilising the system. Why do diabetics typically have high glucagon levels even when their blood glucose is high? Amyloid protein deposits prevent some alpha cells from receiving the “off” signal from the beta cells.

31. In a glucose tolerance test, why does insulin peak so much later than glucose? Because the control is integral rather than proportional. Why are amyloid protein deposits found in advanced Type 2 diabetes? They are a prime cause of most Type 2 diabetes and would be found in all stages if we looked for them Why are there somatostatin producing D-cells in the pancreas? To allow hGH to participate and to reduce the total activity of the pancreas when not under stress.

32. What are the islet gap junctions for? So the β-cells can tell the α- and D-cells what to do. Why are obesity and cardiovascular disease associated with Type 2 diabetes? Extra insulin is being produced and therefore more somatostatin and less hGH than usual.