4.4. A LAPLACIAN MODEL OF BRANCHING GROWTH
and orientation of the elements being determined by the gradients of Phi.
For many organisms, thickening growth occurs all along the branches. This
type of response is not modeled here as it is intended to represent only the
branching pattern.
At each iteration the concentration gradient is calculated in the vicinity of each tip. The gradient of Phi at a point i , j is GradPhi=
(Phi(i+l,j)-Phi(i-l,j), Phi(i,j+l)-Phi(i,j-l))/2. The
gradient in the vicinity of a branch tip is then calculated by taking the vector
average of GradPhi over all the non -zero points which are within a distance
GradRadius of the tip.
Before new skeletal elements are added a decision is made whether
to branch each tip. This is decided by calculating three concentrations at a distance GradRadi us from the tip, using a linear
interpolation of the gridded data. The first value TipPhi is in the
direction of the gradient vector; the two others LeftSidePhi and
RightSidePhi are at right-angles to this direction. The ratio Branch =
min (LeftSidePhi, RightSidePhi) /TipPhi is used to decide
whether or not branching should proceed. As a single tip grows further
awayfrom the substrate the gradients at the tip increase and the contours of
Phi become pushed down around the sides of the branch, increasing the ratio Branch. If Branch becomes larger than a value BranchThreshold
then the tip splits into two. Because Branch depends on the ratio of two
Phi values it is independent of the magnitude of Phi. Instead, it gives
geometric information on how far a given tip is poking away from its
neighbors .
If the tip has not branched then the new skeletal element lies in the
direction defined by GradPhi. If the tip has just split into two then the
direction of growth of the two tips are 60 degrees to either side of the
direction of increasing gradient. This is needed to separate the two tips so
that they grow awayfrom one another.
Two versions of the model are considered. In the first version of the
model, the growth rate is set to be the same across all tips and constant with
time. In the second, the growth rate of each tip is set to be proportional to
the magnitude of the gradient vector GradPhi. The gradient is highest near
tips which poke clear of their neighbors and they then grow faster, leading
to an escalation of the growth rate. In order to control this, the growth rates
are scaled by a factor which is the same for all tips. This factor is chosen to
keep the maximum growth rate below an upper bound.
The length of each new skeletal element is set to be proportional to the
growth rate multiplied by a random factor. This factor is chosen for each tip
from a distribution which has a mean of 1 and, in the model runs presented
here, a standard deviation of 10%. The inclusion of the small random factor is
necessary to break the determinism in the model and so to allow a branching
form to develop which varies, in detail, from simulation to simulation .
The two simulations presented have been run on a grid with n =
600. The radius of the branches is BranchRadius = 2.4 pixels; the gradient near each tip is calculated within a distance GradRadius = 4.2 pixels
of each tip; the maximum growth rate is 2.5 pixels per iteration; and the
organism begins as a single tip at x = 300, Y = o. With these values the
choice BranchThreshold = 0.5 gives a suitable branch density, allowing
resolution of Phi in between the branches .
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and orientation of the elements being determined by the gradients of Phi.
For many organisms, thickening growth occurs all along the branches. This
type of response is not modeled here as it is intended to represent only the
branching pattern.
At each iteration the concentration gradient is calculated in the vicinity of each tip. The gradient of Phi at a point i , j is GradPhi=
(Phi(i+l,j)-Phi(i-l,j), Phi(i,j+l)-Phi(i,j-l))/2. The
gradient in the vicinity of a branch tip is then calculated by taking the vector
average of GradPhi over all the non -zero points which are within a distance
GradRadius of the tip.
Before new skeletal elements are added a decision is made whether
to branch each tip. This is decided by calculating three concentrations at a distance GradRadi us from the tip, using a linear
interpolation of the gridded data. The first value TipPhi is in the
direction of the gradient vector; the two others LeftSidePhi and
RightSidePhi are at right-angles to this direction. The ratio Branch =
min (LeftSidePhi, RightSidePhi) /TipPhi is used to decide
whether or not branching should proceed. As a single tip grows further
awayfrom the substrate the gradients at the tip increase and the contours of
Phi become pushed down around the sides of the branch, increasing the ratio Branch. If Branch becomes larger than a value BranchThreshold
then the tip splits into two. Because Branch depends on the ratio of two
Phi values it is independent of the magnitude of Phi. Instead, it gives
geometric information on how far a given tip is poking away from its
neighbors .
If the tip has not branched then the new skeletal element lies in the
direction defined by GradPhi. If the tip has just split into two then the
direction of growth of the two tips are 60 degrees to either side of the
direction of increasing gradient. This is needed to separate the two tips so
that they grow awayfrom one another.
Two versions of the model are considered. In the first version of the
model, the growth rate is set to be the same across all tips and constant with
time. In the second, the growth rate of each tip is set to be proportional to
the magnitude of the gradient vector GradPhi. The gradient is highest near
tips which poke clear of their neighbors and they then grow faster, leading
to an escalation of the growth rate. In order to control this, the growth rates
are scaled by a factor which is the same for all tips. This factor is chosen to
keep the maximum growth rate below an upper bound.
The length of each new skeletal element is set to be proportional to the
growth rate multiplied by a random factor. This factor is chosen for each tip
from a distribution which has a mean of 1 and, in the model runs presented
here, a standard deviation of 10%. The inclusion of the small random factor is
necessary to break the determinism in the model and so to allow a branching
form to develop which varies, in detail, from simulation to simulation .
The two simulations presented have been run on a grid with n =
600. The radius of the branches is BranchRadius = 2.4 pixels; the gradient near each tip is calculated within a distance GradRadius = 4.2 pixels
of each tip; the maximum growth rate is 2.5 pixels per iteration; and the
organism begins as a single tip at x = 300, Y = o. With these values the
choice BranchThreshold = 0.5 gives a suitable branch density, allowing
resolution of Phi in between the branches .
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