Mathematical description of the model of spatial organisation of gravitropic reaction

The basic scheme of the model as derived from those of Rawitscher (1932) and Merkys et al. (1972), was that changes of an apex angle occurred as a result of four consecutive stages - the physical change which occurs when the subject is disoriented, conversion of the physical change into a physiological change, transmission of the physiological signal and, finally, the growth response in which differential regulation of growth generates the change in an apex angle.

As in Barlow et al. (1991; Stočkus 1994) 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. To simulate the gravitropic reaction under different gravity Ks can be replaced by Ks g , where g is gravity vector in relative units (1.0 on the Earth).

After perception the signal is transmitted in a basipetal direction. In the previous model (Meškauskas, Moore & Novak Frazer, 1998) it was shown, that the "wave with decrement", which is most commonly used to describe the signal transmission (Johnson, 1971; Brown & Chapman, 1977; Stočkus 1994), cannot alone explain the development of curvature in a mushroom stem. The main argument is that the point of bending of a C. cinereus stem moves with decreasing speed, not characteristic for the "wave with decrement" equation. However, if an autotropism in the realisation point is also included, the combination explains this phenomenon. Hence, in this model, the classic equation of signal transmission was used:

(2),

here l is the relative distance from the base of the axial organ (0 = base, 1 = tip), and v is the signal transmission speed, which is constant for a wave equation. Hence the signal reaches the certain realisation point delayed by time (1-l)/v and the signal level exponentially decreases during transmission (constant S_DE determines this decrement).

If now we would suppose that the local bending velocity is simply proportional to this local signal level S(l,t), we would obtain a model very similar to the model 4 in Stočkus & Moore (1996). However, for exact simulation of the spatial development of gravitropic reaction in C. cinereus the description of the realisation must be a more complicated function not from only S(l,t), but also involving two additional signals.

The compensation signal arises in the same point where the bending process develops, that means it is not being transmitted through the stem. This signal was introduced to explain the straightening of the apical part before reaching the vertical position. The level of this signal was supposed to be proportional to local curvature, C_L It is known that the compensation process is much more strongly expressed closer to the apex than in lower (but still apical) subsections (Meškauskas, Moore & Novak Frazer, 1998). To produce the working model, it was necessary to take this fact into account. Hence, it was supposed, that the distribution of ability for the straightening reaction is not uniform and exponentially decreases in the basipetal direction. The level of the straightening signal at position l at given time t is:

(3).

here parameters A_U and A_D determine the distribution of autotropism along the stem.

The proposition that perception of the gravitropic signal takes place in the stem apex is enough to create the mathematical model that can reproduce all C. cinereus stem shapes that occur during the living gravitropic reaction. However, it can’t simulate the process as it develops in time (the goodness of fit test indicates significant differences). The bending process, simulated by this simpler model, is too slow in the beginning of the gravitropic reaction (in approximately the first three hours) and too fast in the later stages. Hence we included the local perception of the signal, which is confirmed both experimentally (Greening, Holden & Moore, 1993) and by mathematical modelling (Meškauskas, Moore & Novak Frazer, 1998). It was supposed, that the hyphal system has a sufficient level of autonomy not only respond to the signal from the tip, but also to perceive the gravitropic "irritation" in the site of realisation. After adding this parameter, we achieved exact simulations. However, it was necessary to suppose that the local perception function differs significantly from the perception function in the apex and can be approximated as

(4).

Similarly to sin (a _L), this function is maximal when a _L = 0 (horizontal position) , but it can reduce much faster when the parameter a _L starts to increase. Hence it was supposed, that local perception plays the most important role in the first stages of the bending process, when the a_L < 45 degrees.

The essential condition to satisfy goodness of fit test was also to suppose, that the distribution of ability for local perception is not uniform, but exponentially decreases from the apex in the basipetal direction. Hence the level of the local perception at the time t in the position l is:

(5)

. Hence the realisation of the gravitropic response, the local bending speed in point l at time t is

(6),

here K_W is the realisation constant. By summarising the equations shown above and expressing local angle through local curvature, we obtain the final equation below (t>0; 0 <= l <= 1, initial condition C_L( l,0)=0 (straight stem), boundary conditions C_L(t,0)=C_L(t,1)=0):

(7).

Using this equation, the program in Java was written to obtain numeric solutions.