Glossary - Cellular Potts Models: Multiscale Extensions and Biological Applications - page 210

quence of the instability due to both cell proliferation and cell haptotactic
movement.

Fisher equation: Also known as the Fisher{Kolmogorov equation and the
Fisher{KPP equation, named after R. A. Fisher and A. N. Kolmogorov,
is the partial differential equation

@t = AuBu 2 + D @ 2 u

@u

@x 2 ;

originally derived for the simulation of propagation of a gene in a popu-
lation, where u = u(x;t) is a real-valued function of the one-dimensional
space-variable x and of the time t, and D, A, and B are positive param-
eters. The Fisher equation has two stationary states: u 0 is unstable,
whereas u 1 is stable. It indeed admits traveling wave solutions of the
form

u(x;t) = v(xct) v(z);
where c depends on the parameters and v is an increasing function char-
acterized by

z!1 v(z) = 1. That is, the solution
switches from the equilibrium state u = 0 to the equilibrium state u = 1.

z!1 v(z) = 0 and

lim lim

G1 phase (Gap 1 phase, or post-mitotic phase): A period in the cell
cycle during interphase, before the S phase. For most cell lines, this phase
is the major period of cell growth during its lifespan. During this stage,
new organelles are being synthesized. Therefore, the cell requires both
structural proteins and enzymes, resulting in a great amount of protein
synthesis and a high rate of metabolism in the cell.

Glycosaminoglycan (GAGs, or mucopolysaccharides): Long unbranched
polysaccharides formed by repeated disaccharide units, which consist of a
hexose (six-carbon sugar) or a hexuronic acid, linked to a hexosamine (six-
carbon sugar containing nitrogen). Protein cores are made in the rough
endoplasmic reticulum and are posttranslationally modified by glycosyl-
transferases in the Golgi apparatus, where GAG disaccharides are added
to them to yield proteoglycans; the exception is the GAG hyaluronan,
which is uniquely synthesized without a protein core and is \spun out"
by enzymes at the cell surface directly into the extracellular space. This
family of carbohydrates is essential or important for life, as it forms an
important component of connective tissues.

Gompertz function: Gompertz curve, named after Benjamin Gompertz, is
a sigmoid function. It is a type of mathematical model for a time series,
where growth is slowest at the start and end of a time period. The right-
hand or future value asymptote of the function is approached much more
gradually by the curve than the left-hand or lower-valued asymptote, in