Mathematical description of the apex angle-based model of gravitropic reaction

Information on how the apex angle changes in time is often the only information, available in the reference data. Hence, despite of the availability of a model on spatial organisation of gravitropic reaction, it is still useful to present the apex-angle based model. For more detailed description and discussion on this and related models see Stočkus, (1994) and Stočkus & Moore (1996) in references.

As in Barlow et al. (1991) it was supposed, that after reorientation the physiological signal arises in the apex (signal perception), and that this signal at any time t is proportional to the cosine of the tip angle at this time:

(1),

here the tip angle is measured as the angle from the horizontal position. To simulate the gravitropic reaction under different gravitational accelerations, Ks can be replaced by Ks g , here g is the gravity vector in relative units (1.0 on the Earth). It is possible to combine two earlier models into one by writing the perception function as

(1a).

Here is the angle of gravitropic reaction (90 degrees for strictly positive or negative gravitropism; lower values for plagiogravitropism.). This function can also be written in alternative, but mathematically identical ways, for example as sum of sine and cosine.

After perception, the signal is transmitted in the basipetal direction. In this model, the classic equation of signal transmission was used:

(2),

This equation means that signal in the realisation site, S  is equal to the signal level in the apex as it was time before (NOTE that this model simplifies the bending process to a rotation around one joint, there is no attempt to simulate the natural bending process).

During realisation of the gravitropic response, the bending acceleration (not speed), , is proportional to the difference of two factors. The first factor is the level of gravitropic signal in the realisation site, S. The level of the second factor is proportional to the bending speed. It may be related to some internal systems concerned with regulation of the gravitropic response. You can easily demonstrate the necessity for such regulation by switching Kw, so-called signal weakening parameter, to zero in the simulation program. Hence,

(3).

By joining together the equations mentioned above, we obtain the final equation:

(4).

This is a second order differential equation with delay. It can be solved easily using classic numeric methods of solution, as is done in our demonstration program. To underline the biological meaning of the components (first perception, then the signal delay) it can be rewritten as

(4a).