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Fig. 1. The fixed-reference model relies on an abstract coordinate, which is constant through the whole simulation.

Fixed-reference models: INVERTEBRATES

see Raup's model

Fixed-reference models: FORAMINIFERA

Berger's model (1969)

The first formal model constructed in the fixed reference frame (Fig. 1) was presented over three decades ago by Berger (1969). It was the first theoretical morphospace of foraminifera based on three following parameters (Fig. 2):

Fig. 2 Basic parameters of Berger's model (1969). The origin of the xy-coordinate system represents the fixed-reference frame (based on Berger, 1969)
Fig. 3. Section through 3D-morphospace (Berger, 1969)
(1) [q-ratio] — the ratio between successive chamber radii (Ri);
(2) [a-angle] — ’angle of advance’ which is the angle between the lines connecting the center of the shell with two successive chamber midpoints;
(3) [o-lap] — ’amount of overlap’ between two successive chambers, which is used to calculate a distance between centers of these two chambers, i.e., d=Ri/[o-lap].

In the model, the shell is fixed to a point that defines the center of the shell. The model represents a simple step-by-step rotation of a circle with a certain amount of overlap and expansion of circle radius (Ri). This model simulates isometric growth (all three parameters are held constant through ontogeny) and is confined to planispiral shells composed of circular chambers. (from Topa & Tyszka, 2005)

The above fixed-reference model relies on an abstract coordinate, which is arbitrarily defined and has no morphogenetic or physiological meaning. Chambers (circles or spheres) are rotated and translated along these artificial axes, which are fixed and serve as a reference line for the growth process. Therefore, while these models can simulate simple planispiral, trochospiral or uniserial chamber arrangement, they cannot simulate more complex patterns found in foraminifera. For instance, they cannot model gradual or abrupt changes of growth modes that cause different chamber arrangements during ontogeny, such as planispiral and switching to biserial or streptospiral to uniserial. These constraints can be overcome by abandoning a fixed-reference frame in favor of a moving reference models. (from Topa & Tyszka, 2005, modified)

Brasier's model (1980)

Fig. 4. Morphospace presented by Brasier (1980); simplified

Brasier (1980) produced a morphospace model using four parameters, which correspond to Berger’s parameters, expanded by the degree of growth extension along the coiling axis and the degree of chamber compression. The latter parameter introduces allometry into the system and mimics the changing proportions of chambers during ontogeny of real foraminifera. Another important aspect of this model is that chambers are not rotated but translated. Although this approach was not developed further, it did stimulate the modeling of foraminifera (Brasier 1980; McGhee 1999; cited after Tyszka & Topa, 2005).