Daisyworld & Diabetes Peter Saunders Department of Mathematics Kings College, London.

Daisyworld & Diabetes

Peter Saunders

Department of Mathematics

King’s College, London

DAISYWORLD

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

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.

The key variables

b: Fraction of planet covered in black daisies

w: Fraction covered in white daisies

Tb: Temperature where the black daisies areTw: Temperature where the white daisies are

L: Solar luminosity

An equation for the black daisies

dαb/dt = αb ( 1 – αb – αw) β(Tb) - γαb

= αb (αg β(Tb) – γ)

(T) is a function that is zero at 5C, rises to a maximum ofone at 22.5C and then falls to zero again at 40C

A simple and convenient choice is ( )( . )

.T

T

1

22 5

17 5

2

2

An equation for the white daisies

We use a similar equation for the white daisies:

We don’t have to use the same and but itkeeps things simple. We can use different oneslater if we want to.

dαw/dt = αw (αg β(Tw) – γ)

Energy balance

A A A Ab b g g w w

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

( )T 273 4

: Stephan’s constant T: the ‘effective’ temperature.

Energy in must equal energy out, and so we have

( ) ( )T SL A 273 14

Heat Flow

Because different regions of Daisyworld are at differenttemperatures, there will be heat flow. We include this in the model using the equations

Tb4 = T4 + q(A-Ab) Tw

4=T4 + q(A-Aw)

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 dothe temperature differences. Don’t worry about the 4th powers; they’re only there to makethe calculations easier and don’t make any real difference.

The Daisyworld Equations

db/dt = b(g

(Tb) - ) d

w/dt=w(g

(Tw) - ) (T+273)4 = SL(1-A) A = gAg + bAb + wAw

Tb4 = T4 + q(A-Ab) Tw

4 = T4 + q(A-Aw)

No daisies

TSL

2273

1 4

/

Black daisies only

0 11

b b b b

b

T a T( ( )( ) ) ( )

( )( . )

.T

Tb

b

122 5

17 5

2

2

Tbb

b

22 5 17 5

1

1. .

Black daisies only

Plot of y=x4+2x2+vx: v= -0.125 (red), 0.8 (green), 1.5 (blue), 2.5 (black).

How can an equilibrium just disappear?

Black daisies only

White daisies only

Both kinds

The Glucose Control Equations

dG/dt = I + - - RG

G: glucose, A: glucagon, B: insulin, I: input, R: demand

dA/dt = A(G)h(A,B) - D)

dB/dt = B(G)h(A,B) - D)

G) is a decreasing and G) an increasing function of G

h(A,B) represents inactive cells, and D switching-off of cells

(Work with Johan Koeslag, Stellenbosch University, SA)

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(G)h(A,B) - ) = 0 B((G)h(A,B) - ) = 0

If neither A nor B is zero then these two equations imply

(G) = (G)

Why the control is so precise

We don’t know what either or is, but ...

We know that is a decreasing function of G

while is an increasing function

so they meet (if at all) in a unique point ...

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”.

dA/dt = A(G)h(A,B) - D)

dB/dt = B(G)h(A,B) - D)

Why is Type 1 diabetes so hard to manage?

dG/dt = I + - - RG

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.

What about Type 2?

dG/dt = I + - - RG

dA/dt = A(G)h(A,B) - D)

dB/dt = B(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.

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

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.

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.

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 patientmust keep G away from what is now the naturalequilibrium, at a much higher level of G.

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 destabilisingthe system.

Why do diabetics typically have high glucagon levelseven when their blood glucose is high? Amyloid protein deposits prevent some alpha cells from receiving the “off” signal from the beta cells.

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.

In a glucose tolerance test, why does insulin peak so much later than glucose? Because the control is integral rather than proportional.

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 associatedwith Type 2 diabetes? Extra insulin is being produced and therefore more somatostatin and less hGH than usual.