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Taylor series and identifying generic properties

Now lets use the Taylor series and an arbitrary ordinary differential equation and explore some potentially interesting behavior. First there are two classes (at least) of model builders. Class one is interested in building a full model of some process that is as realistic as possible. Class two is interested in building a minimal model, one that captures essential behavior and upon which, one can add more and more realism and ask: How does this altered the behavior of the minimal model?

We start with the simplest ordinary differential equation:

$\displaystyle \frac{\textrm{d}u}{\textrm{d}t} = u' = f(u).$ (2.9.1)

and ask the question - what is the behavior of this equation as we increase the complexity of f(u)? The Taylor series is a way to methodically add complexity (by adding successive terms) in order to more realisticly represent a characterization of some unknown function, f(u). Starting with the constant term, we can analyze the properties of the ODE and get some ideas about how adequately it represents some process of interest.

So we start with

$\displaystyle f(u) = f(u_0) + f'(u_0)(u-u_0) + \frac{f''}{2}(u_0)(u-u_0)^2 + \ldots$    

The values of $ f(u_0)$ and its derivatives are simply constants so that the Taylor series is simply a power series in $ (u - u_0)$. So lets rewrite the series as

$\displaystyle f(u) = a_0 + a_1(u-u_0) + a_2(u-u_0)^2 + a_3(u-u_0)^3 + \ldots$    

and start our analysis. For convenience, we will set $ u_0$ to zero.

The properties of $ u' = a_0$ are not interesting. The solutions are lines of varying slope, where the slope is determined by the value of $ a_0$.

Including the first two terms makes the solution space a bit more interesting:

$\displaystyle \frac{\textrm{d}u}{\textrm{d}t} = u' = a_0 + a_1u$ (2.9.2)

This has a single equilibrium where $ u' = 0$ and the equilibrium is located at $ u = -\frac{a_0}{a_1}$. Moreover, the equilibrium is unstable2.5 if $ a_1 < 0$ as shown in Figure 2.9.1. When there is a disturbance that moves the phase point, $ (f(u), u)$ to the left, then we see that $ u' < 0$ which pushes the phase point away from the equilibrium. Similarly when the disturbance moves the phase point to the right, $ u' > 0$ which pushes the phase point to the right, again away from the equilibrium.
Figure: Linear nullcline for Equation 2.9.2. Here the slope = 1/2 and the root is unstable.

Now, we add the quadratic term and have

$\displaystyle \frac{\textrm{d}u}{\textrm{d}t} = u' = a_0 + a_1u + a_2u^2$ (2.9.3)

We assume that all the constants are such that there are 2 intersections with the $ u' = 0$ line in the $ (u',u)$ plane. If there were no intersections, then again, the behavior is not interesting. So these two intersections represent 2 equilibria, one stable and one unstable. See Figure 2.9.2.
Figure: Quadratic nullcline for Equation 2.9.3. The left root is unstable, the right root is stable

Next we add the cubic term and have

$\displaystyle \frac{\textrm{d}u}{\textrm{d}t} = u' = a_0 + a_1u + a_2u^2 + a_3u^3$ (2.9.4)

Now, in the $ (u',u)$ plane, we have constants, $ a_i$ such that there are 3 intersections with $ u' = 0$. and depending on the values of the $ a_i$, we either have two stable and one unstable equilibrium or we have two unstable and one stable equilibrium. Figure 2.9.3 displays the cubic where we have two stable and one unstable equilibrium.

Figure: Cubic nullcline for Equation 2.9.4. The left root is unstable, the right root is stable

Now what is interesting about this from a biological modeling perspective? Many biological processes behave like switches. A neuron is either in the rest state or the excited state. A cardiac cell is either in the rest state or the excited state - and translated to muscle, the muscle is either resting or contracted. We can even look at transcription. Either the gene is being expressed or not.

All of these process have in common, switch-like behavior. From a modelling perspective, it means that the minimal complex model for describing a switch requires a cubic function on the right hand side of the ODE which means that only nonlinear systems can represent switching behavior. Also, the middle, unstable equilibrium, represent a threshold. So all switches must have a threshold, and we should be able to design experiments to reveal the threshold. Now, a distraction. If there is diffusive coupling between switches and all are initially in the same state, then as one switch is forced to change states, the switches to the left and right can potentially be induced to switch (if the diffusive element forces the local value of u to exceed the switching threshold) and the result will be a propagating wave.

It is exciting to see the verification of a theoretical argument (above taylor expansion of an arbitrary function) in real biological systems. In figure 2.9.4 we see the current voltage relationship measured in an isolated rabbit cardiac atrial cell. Using the voltage clamp procedure, the potential was gradually increased from negative to positive and the current associated with each potential was recorded. The resultant i/v is a quasi-steady state and does not accurately reflect the dynamics of a cardiac (or nerve) cell. Nevertheless, the cubic nature is clearly seen (due to calcium channels).

Figure: Current voltage relationship obtained from voltage clamp studies of cultured rabbit cardiac atrial cells. Note the cubic-like behavior

Now the fun part of modeling is to link the cubic function to some real mechanism. In the case of cardiac and neuronal cells, the cubic function represent the instantaneous current-voltage relationship of the cell. We are unsure what the cubic function represents in a gene expression system.

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Next: Methods for Solving ODEs Up: How to create a Previous: A Single Cell   Index

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Frank Starmer 2004-05-19