Reduction to a Two-Dimensional Model

Let's rewrite equation (2) as

$$\frac{d \beta}{d t} = q \left(\sigma \phi(\alpha,\gamma) - \frac{k_t \beta}{q} \right).$$ (5)

We've been using $q=100$, a pretty large number; this means that $\beta$ is a fast variable. (You might ask, ``fast relative to what?'' We really should check that

$$\left\vert\frac{d \beta}{dt} \right\vert \gg \left\vert\frac... ...dt} \right\vert \gg \left\vert\frac{d \gamma}{d t} \right\vert$$

for the solutions of interest, namely the solutions shown in Figures 2 and 3. This is left as an exercise for the student.) Because $\beta$ is fast, we make a quasi-steady state approximation that it can be taken to be its value found from setting $d\beta/dt=0$, i.e.,

$$\beta = \frac{q \sigma \phi(\alpha,\gamma)}{k_t}.$$ (6)

Then we obtain the following two-dimensional set of equations:

$$\frac{d \alpha}{dt} = \nu - \sigma \phi(\alpha,\gamma),$$ (7)

$$\frac{d \gamma}{dt} = \lambda \phi(\alpha,\gamma) - k \gamma,$$ (8)

where $\phi$ is still given by equation 4, and

$$\lambda = \frac{q \sigma}{h}.$$ (9)

This probably seems a bit like FitzHugh's simplification of the Hodgkin/Huxley equations as described in Tutorial 2. Indeed, a very common way to understand ODE models which arise in the modeling of biological systems is to reduce them (rigorously or not) to a two-dimensional system.

To verify that this is a good approximation, we now integrate the two-dimensional system (7,8) using the following Matlab code. First, camp_2d.m:

global nu sigma k kt L q h lambda

nu = 0.1;
%nu = 0.04;
sigma = 1.2;
k = 0.4;
kt = 0.4;
L = 10^6;
q = 100;
h = 10;
lambda = q*sigma/h;

[T,Y] = ode23('func_camp_2d',[0,2500],[92.366,2]);  

for i=1:size(Y(:,1))
   bbeta(i) = q*sigma*phi(Y(i,1),Y(i,2))/kt; %construct beta according to
                                             %the quasi-steady state approx
end

figure(1);
hold on;
plot(T,Y(:,1),'b');
plot(T,bbeta,'r');
plot(T,Y(:,2),'g');
xlabel('t');
ylabel('\alpha,\gamma');

Text version of this program Next, func_camp_2d.m:

function dy = func_camp(t,y)

global nu sigma k kt L q h lambda

a = y(1);
g = y(2);

dalpha = nu - sigma*phi(a,g);
dgamma = lambda*phi(a,g) - k*g;

dy = [dalpha;dgamma];

Text version of this program Figures 4 and 5 show the results for $\nu = 0.1 s^{-1}$ and $\nu = 0.04 s^{-1}$, respectively. Here $\beta$ is reconstructed from $\alpha$ and $\gamma$ using equation (6). Comparing with Figures 2 and 3, we see that the agreement is quite good.

Figure 4: Oscillations of ATP ($\alpha$), and intracellular ($\beta$) and extracellular ($\gamma$) cAMP for $\nu = 0.1 s^{-1}$ found from the two-dimensional system (7,8).

Figure 5: A single spike of intracellular ($\beta$) and extracellular ($\gamma$) cAMP for $\nu = 0.04 s^{-1}$ found from the two-dimensional system (7,8).