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Many models examining intercellular signaling do not take into account the separation distances of the signaling bodies. We use a partial differential equation (PDE) based model to gain insight into spatially heterogeneous activation-inhibition intercellular signaling.

Two types of signaling molecules exist: activating and inhibiting. Each molecule is synthesized by a unique strain of cells and affects the synthesis rate of the other strain.

Figure 1 – Activation-inhibition intercellular signaling – Activating molecule (A) synthesized and diffuses to increase synthesis of inhibiting molecule (B) in secondary strain. Inhibiting molecule also diffuses back to initial cell and decreases synthesis of activating molecule.

This is modeled using the following system of PDEs:


where Ψ1 and Ψ2 represent the concentrations of the activating and inhibiting molecules, respectively, γi the degradation constant, Di the diffusion constant, λi the maximal synthesis rate of molecule i, and δ the Dirac function. fi represents the Hill function describing the dependence on the opposing molecule:


where n, b, and θ are positive. To simplify the analysis, we chose equal parameters between the activating and inhibiting sites (Appendix A).

Numerical Simulation

The above system was solved numerically using a forward Euler scheme in time and a centered difference scheme in space. Cyclical boundary conditions were assumed; meaning the spatial dimension formed a ring. This was chosen since simulating an approximate infinite line is computationally costly. However, this also allowed us to investigate two site geometries during one simulation (explained later). The ring was given a physical length of 50 and discretized into 500 intervals. For simplicity, separation distances will be reported in terms of numerical intervals rather than physical distance.

One Oscillator

We first explored the different potential dynamics when the separation distance between an activating and inhibitory site was increased.

Figure 2 – One Oscillator – The red bar represents the inhibitory site, which remains fixed in position while the activating site, blue bar, is sequentially moved around the ring.

The following is an example of the dynamics observed when the two sites are at a distance where oscillations occur.

Figure 3 – One Oscillator – One activation and one inhibitory site separated by 11 intervals. The concentration of the activating molecule is measured at the activating site and the inhibitory molecule at the inhibitory site.

By analyzing curves similar to those in figure 3 the frequency of oscillations was calculated. Figure 4 illustrates the sudden appearance of high frequency oscillations when the sites are near each other and the decreasing frequency as they are moved apart until oscillations disappear.

Figure 4 - Frequency of Activation Molecule at Activation Site vs Separation Distance – The symmetrical appearance is a result of the activation site at first moving away from the inhibitory site, making its way around the ring, and then getting closer to the inhibitory site along the opposite side. The graph of frequency of the inhibitory molecule at the inhibitory site vs separation distance was equivalent.

The observation that the steady state becomes unstable for a range of separations has been reported previously by Shymko and Glass (1974).

Another way of looking at how oscillations depend on separation distance is by plotting the value of a curve whenever its derivative is zero vs the separation distance. This will also tell us some information on how the shape of the oscillations change with separation distance.

Figure 5 - Amplitude of Oscillations of Activation Molecule at Activation Site vs Separation Distance – The amplitude of oscillations was generated by noting the value of the curve whenever its derivative was zero and plotting this for each separation distance. So that separation distances that have only one value plotted are at steady state. Whereas separation distances with two values plotted are oscillating. The larger of the two values corresponds to the peak and the smaller the trough.

Notice the range of distances that have two values plotted correspond to the range of distances that have a nonzero frequency in figure 4.

Two Oscillators

We next looked at a system consisting of two oscillators, where each consists of an activation and inhibitory site.

Figure 5 – Two Oscillators – Each oscillator consists of one activation and one inhibitory site. There are two arrangements of the system: BA AB, termed the symmetrical system, and BA BA, the unsymmetrical system. In this document we only discuss the symmetrical system. Oscillator 2 travels around the ring.

The distance between the two oscillators was varied while the distance between two sites within an oscillator was held fixed at 5 intervals. This value was chosen for demonstration purposes, however the dynamics to be described have been observed at various separation distances.

We first look at the change in frequency as the oscillators are moved apart.

Figure 6 - Frequency of Activation Molecule at Activation Site of Oscillator 2 vs Separation Distance – There are eight separation distances that resulted in a frequency of -1: 14,15,16,17,474,475,476, and 477. This value is used as a marker to denote a curve who's frequency did not converge to a single value. As will be seen later, these distances resulted in complex dynamics.

The first striking feature of this graph is the appearance of separation distances that have negative frequencies. The value of -1 is assigned to a separation distance that results in a complex oscillation whose frequency cannot be resolved using a simple method, which assumes simple periodic motion. Before we look into these curves, let's look over the entire graph. When the two oscillators are very close together, their frequency is always less than an isolated oscillator (look at figure 4 for two sites with a separation distance of 5 intervals). Interestingly, between the separation distances of 1 and 10 there appears to be a local minimum frequency. Meaning as the oscillators are moved apart, their frequency initially decreases and then begins increasing. This trend of increasing frequency continues until the numerical method calculating frequency breaks down. The following figure illustrates the concentration of the activation molecule at the activation site of oscillator 2 for the first distance that results in an oscillation whose frequency cannot be computed.

Figure 7 - Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 13 - There appears to be a constant "resetting" of the oscillations. As if a threshold is constantly being crossed that induces a reset of the oscillatory behaviour.

It is immediately obvious why the frequency measuring numerical method failed, the oscillations are far from simple. Before investigating this curve further, let's take a look at the other separation distances that resulted in complex oscillations.

Figure 8 - Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 14
Figure 9 - Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 15
Figure 10 - Concentration of activation molecule at the activation site of oscillator 2 for a separation distance of 16

The separation distances 474,475,476, and 477 resulted in similar dynamics and are not shown for brevity.

We first wanted to know whether these were truly the only separation distances that resulted in this type of dynamics. We recalculated the frequency of oscillations at each separation distance after simulating for 10,000 time units. In this way we hoped to overcome any transient dynamics and observe the true interaction. This resulted in the same set of separation distances having complex oscillations while everywhere else had regular simple oscillations. Furthermore, we recalculated the frequency of oscillations at each separation distance by using the final time point of the immediately preceding separation distance as the initial condition. We thought this would allow a quicker convergence to the stable limit cycle. However this also resulted in the same findings as before.

Appendix A - Parameters

The standard set of parameters used to observe oscillations were taken directly from Shymko and Glass (1974).

γ = 2, D = 2, λ = 54, θ = 1, b = 0, N = 8.

Both strains were assumed to have identical parameters in order to simplify the model as well as explicitly observe the dependence of dynamics on separation distance.