Steady State Spatial Patterns in a Cell
IMA Journal of Mathematics Applied in Medicine & Biology (1989) 6, 69-79
Steady-State Spatial Patterns in a Cell-Chemotaxis Model
P. GRINDROD., J. D. MURRAY
Centre for Mathematical Biology, Mathematical Institute, Oxford, UK
S. SINHA
Centre for Cellular and Molecular Biology, Hyderabad 500 007, India
[Received 13 October 1987 and in revised form 10 January 1989]
We investigate a simple cell-chemotaxis model for the generation of spatial
patterns in cell aggregations. For simple boundary-value problems, we analyse
the local and global bifurcation of spatially heterogeneous patterns away from the
uniform equilibria as the total number of cells is varied. We also discuss the
existence of periodic spatially structured solutions for the cells and chemoattractant in the infinite domain.
Keywords: cell chemotaxis; biological patterns.
1. The model mechanism
which generate biological spatial patterns, such as cartilage
patterns in the developing vertebrate limb, are unknown. There are two main
theoretical approaches to cartilage patterning, one is the reaction-diffusion
prepattern theory, wherein a heterogeneous spatial pattern in morphogen
(chemical) concentrations is formed to which the cells react in a positional
information way (Wolpert, 1971). The other is the mechanochemical approach
proposed by Oster and Murray and their co-workers (e.g. Oster et al., 1983;
Murray & Oster, 1984a,b; Oster et al., 1985).
Recently, Oster & Murray (1988) proposed a simpler chemotaxis-model
mechanism for generating the sequence of spatial patterns of cell condensations
which become the cartilage patterns. In its simplest form, their model has the
cells secreting their own chemoattractant: we briefly derive the model below in
this section. In this paper, we study the simplest version of Oster & Murray's
(1988) model and focus, in particular, on the steady-state spatially heterogeneous
patterns which it can generate.
In the early stages of cartilage formation, patterned aggregations of mesenchymal cells are formed. These aggregations undergo various morphogenetic
transformations as the limb takes shape. In the model here, we suppose the cells
move toward a source of chemoattractant, which the cells themselves secrete.
There are several well-known examples of cell aggregation arising from the cells
secreting their own chemoattractant, such as in the aggregation phase of the slime
mold Dictyostelium discoideum, where motile cells move towards a chemoattractant that is thought to be secreted by a small group of 'pacemaker' cells (see e.g.
Segel, 1980).
THE MECHANISMS
69
© Oxford University Press 1989
Downloaded from http://imammb.oxfordjournals.org/ at Indian Institute of Science and Education Mohali on February 24, 2015
AND
70
P- GRINDROD et al.
= MV2u
-aV
random cell
motility
chemotaxis
dc/dt= DV2c + bu/(u + h)diffusion
secretion
by cells
pc ,
(1.2)
decay
where M>0 and ar>0 are the cell motility and chemotactic parameters,
respectively, and D is the diffusion coefficient of c. The secretion rate is taken to
be a typical Michaelis-Menten saturating function bu/(u + h), with b and h
positive, and a first-order kinetics degradation rate for c, namely fie, is assumed.
We make the system dimensionless by introducing a typical time scale T = 1/n
based on the chemoattractant decay time, a concentration C = b/[i that is based
on the maximum secretion rate by the cells, and a length L = (ar&)tyji. We thus
define the following dimensionless quantities:
i
* = D/ab,
(
M* = M/ab,
J
with which (1.1) and (1.2) become, on dropping the asterisks for notational
convenience,
dcldt = DV2c + u/(l + u)-c.}
K
'
J
We have thus reduced the number of parameters from six in (1.1) and (1.2) to the
two dimensionless parameter groupings D and M in (1.3). In this paper, we
Downloaded from http://imammb.oxfordjournals.org/ at Indian Institute of Science and Education Mohali on February 24, 2015
The cell-chemotaxis model is based on the following scenario in the case of the
developing limb. Cells divide at the distal end of the growing limb bud which
accordingly grows. The cells pass through a 'progress zone' and after a time
become 'competent' to aggregate. This could result, for example, from an
inhibitory chemical secreted by the tip region, or simply as a consequence of cell
maturation. At a certain distance from the distal tip, or after a certain time or
number of cell divisions, the cells become competent to aggregate. Here, we are
concerned only with the pattern generation capability of the model itself and the
reader is directed to the paper by Oster & Murray (1988) for the full biological
details.
The specific model mechanism we study is the following. When cells become
competent to aggregate and commence to move, we assume that their aggregation is guided by a chemotaxis in which the cells move up a concentration
gradient of a morphogen. We assume that at a certain point after leaving the
distal tip the cells start to secrete a chemoattractant. This will cause them to start
aggregating as each cell attempts to migrate toward nearby concentrations of
attractant.
The model system consists of two conservation equations for the concentrations
u and c of the cells and the chemoattractant, respectively. We take for u and c the
equations proposed by Oster & Murray (1988), namely
SPATIAL PATTERNS IN A CELL-CHEMOTAXIS MODEL
71
2. Local bifurcation of equilibrium solutions
We begin by considering the time-dependent problem for (1.4) in one space
dimension, namely
where g(u) = u/(l + u). From the biology, we impose zero flux or periodic
boundary conditions. To be specific, we choose here the former, that is
0 = Ux = cx atJC = O, 1 forf>0.
(2.2)
The results in this section easily generalize to problems defined on compact
domains in W, with Neumann boundary conditions. However, the analysis in
Section 3 may not be directly extended, since we rely on a time-map argument
which exploits the fact that x is a scalar variable. Also, if (2.2) were replaced by
periodic boundary conditions, all the results here and in Section 3 would remain
valid with only minor adaptations. Thus, we shall see that the existence and
bifurcation of (2.1) with (2.2) naturally infers the existence of steady-state
solutions for (2.1) defined for x e U, with period 2.
First, we consider uniform equilibrium solutions for (2.1),(2.2).
If u = u(x)^0 and c = c(x)3=0 satisfy (2.1),(2.2), then (u, c) is a
steady-state or equilibrium solution, and we define the norms
DEFINITION
h(u)= max \ux\ and N(u)=
X6[0 , 1]
u(x) dx.
Jo
Here, N(u) is the usual L r norm, and, since u is a dimensionless cell density, it
is the total number of cells present.. For the time-dependent problem,
dN(u)/dt = 0. The norm h(u) is the usual °°-norm of ux(x). Clearly, h(u) = 0
implies both u and c are spatially uniform.
Downloaded from http://imammb.oxfordjournals.org/ at Indian Institute of Science and Education Mohali on February 24, 2015
consider the nonlinear spatially heterogeneous steady-state solutions of (1.4) and
investigate the possible bifurcations.
The model generates spatial patterns as one of the parameters passes through
certain bifurcation values. It is intuitively clear that cells moving according to
(1.4) need not distribute themselves uniformly in space. Suppose a small
fluctuation produces a local rise in cell density; then this region secretes more
chemoattractant than neighbouring regions, and so recruits more cells. This has
an autocatalytic effect, since the more cells that aggregate, the stronger is the
recruiting signal. Counterbalancing the autocatalytic aggregation is diffusion and
the fact that recruitment depletes the neighbourhood of cells. This creates an
effective region of inhibition around an aggregation centre so that additional
centres can form some distance away, giving rise to a regular spatial pattern.
In Section 2, we consider the local bifurcation analysis, and, in Section 3, the
global situation. We compare the analysis with the numerical results.
72
P. GRINDROD et al.
LEMMA 2.1 For all constants uo>0, there exists a uniform steady-state solution
for (2.1),(2.2)
(u,c) = (uo,g(uo)).
(2.3)
Such a solution is locally stable if and only if
M(Dn2n2 + 1) - u o /(l + uof
(2.4)
Proof. Linearizing (2.1) about (2.3) and setting
u = u0 + a(t) cos nxx
c = g(«0) + b(t) cos nnx
for any n = 1, 2, . . . , we obtain
KVM
M
*V
-ira-i
since cos nnx is the nth eigenfunction of the Laplacian defined on [0,1] with
Neumann boundary conditions.
The result follows immediately by considering the sign of the determinant of
the matrix in (2.5).
Now, if (2.4) is positive with n = 1, then it must be positive for all n > 1. So,
henceforth, we concentrate on the change of stability and associated bifurcation
of steady-state solutions in the first eigenmode. We assume
M(Dn2+l)
Steady-State Spatial Patterns in a Cell-Chemotaxis Model
P. GRINDROD., J. D. MURRAY
Centre for Mathematical Biology, Mathematical Institute, Oxford, UK
S. SINHA
Centre for Cellular and Molecular Biology, Hyderabad 500 007, India
[Received 13 October 1987 and in revised form 10 January 1989]
We investigate a simple cell-chemotaxis model for the generation of spatial
patterns in cell aggregations. For simple boundary-value problems, we analyse
the local and global bifurcation of spatially heterogeneous patterns away from the
uniform equilibria as the total number of cells is varied. We also discuss the
existence of periodic spatially structured solutions for the cells and chemoattractant in the infinite domain.
Keywords: cell chemotaxis; biological patterns.
1. The model mechanism
which generate biological spatial patterns, such as cartilage
patterns in the developing vertebrate limb, are unknown. There are two main
theoretical approaches to cartilage patterning, one is the reaction-diffusion
prepattern theory, wherein a heterogeneous spatial pattern in morphogen
(chemical) concentrations is formed to which the cells react in a positional
information way (Wolpert, 1971). The other is the mechanochemical approach
proposed by Oster and Murray and their co-workers (e.g. Oster et al., 1983;
Murray & Oster, 1984a,b; Oster et al., 1985).
Recently, Oster & Murray (1988) proposed a simpler chemotaxis-model
mechanism for generating the sequence of spatial patterns of cell condensations
which become the cartilage patterns. In its simplest form, their model has the
cells secreting their own chemoattractant: we briefly derive the model below in
this section. In this paper, we study the simplest version of Oster & Murray's
(1988) model and focus, in particular, on the steady-state spatially heterogeneous
patterns which it can generate.
In the early stages of cartilage formation, patterned aggregations of mesenchymal cells are formed. These aggregations undergo various morphogenetic
transformations as the limb takes shape. In the model here, we suppose the cells
move toward a source of chemoattractant, which the cells themselves secrete.
There are several well-known examples of cell aggregation arising from the cells
secreting their own chemoattractant, such as in the aggregation phase of the slime
mold Dictyostelium discoideum, where motile cells move towards a chemoattractant that is thought to be secreted by a small group of 'pacemaker' cells (see e.g.
Segel, 1980).
THE MECHANISMS
69
© Oxford University Press 1989
Downloaded from http://imammb.oxfordjournals.org/ at Indian Institute of Science and Education Mohali on February 24, 2015
AND
70
P- GRINDROD et al.
= MV2u
-aV
random cell
motility
chemotaxis
dc/dt= DV2c + bu/(u + h)diffusion
secretion
by cells
pc ,
(1.2)
decay
where M>0 and ar>0 are the cell motility and chemotactic parameters,
respectively, and D is the diffusion coefficient of c. The secretion rate is taken to
be a typical Michaelis-Menten saturating function bu/(u + h), with b and h
positive, and a first-order kinetics degradation rate for c, namely fie, is assumed.
We make the system dimensionless by introducing a typical time scale T = 1/n
based on the chemoattractant decay time, a concentration C = b/[i that is based
on the maximum secretion rate by the cells, and a length L = (ar&)tyji. We thus
define the following dimensionless quantities:
i
* = D/ab,
(
M* = M/ab,
J
with which (1.1) and (1.2) become, on dropping the asterisks for notational
convenience,
dcldt = DV2c + u/(l + u)-c.}
K
'
J
We have thus reduced the number of parameters from six in (1.1) and (1.2) to the
two dimensionless parameter groupings D and M in (1.3). In this paper, we
Downloaded from http://imammb.oxfordjournals.org/ at Indian Institute of Science and Education Mohali on February 24, 2015
The cell-chemotaxis model is based on the following scenario in the case of the
developing limb. Cells divide at the distal end of the growing limb bud which
accordingly grows. The cells pass through a 'progress zone' and after a time
become 'competent' to aggregate. This could result, for example, from an
inhibitory chemical secreted by the tip region, or simply as a consequence of cell
maturation. At a certain distance from the distal tip, or after a certain time or
number of cell divisions, the cells become competent to aggregate. Here, we are
concerned only with the pattern generation capability of the model itself and the
reader is directed to the paper by Oster & Murray (1988) for the full biological
details.
The specific model mechanism we study is the following. When cells become
competent to aggregate and commence to move, we assume that their aggregation is guided by a chemotaxis in which the cells move up a concentration
gradient of a morphogen. We assume that at a certain point after leaving the
distal tip the cells start to secrete a chemoattractant. This will cause them to start
aggregating as each cell attempts to migrate toward nearby concentrations of
attractant.
The model system consists of two conservation equations for the concentrations
u and c of the cells and the chemoattractant, respectively. We take for u and c the
equations proposed by Oster & Murray (1988), namely
SPATIAL PATTERNS IN A CELL-CHEMOTAXIS MODEL
71
2. Local bifurcation of equilibrium solutions
We begin by considering the time-dependent problem for (1.4) in one space
dimension, namely
where g(u) = u/(l + u). From the biology, we impose zero flux or periodic
boundary conditions. To be specific, we choose here the former, that is
0 = Ux = cx atJC = O, 1 forf>0.
(2.2)
The results in this section easily generalize to problems defined on compact
domains in W, with Neumann boundary conditions. However, the analysis in
Section 3 may not be directly extended, since we rely on a time-map argument
which exploits the fact that x is a scalar variable. Also, if (2.2) were replaced by
periodic boundary conditions, all the results here and in Section 3 would remain
valid with only minor adaptations. Thus, we shall see that the existence and
bifurcation of (2.1) with (2.2) naturally infers the existence of steady-state
solutions for (2.1) defined for x e U, with period 2.
First, we consider uniform equilibrium solutions for (2.1),(2.2).
If u = u(x)^0 and c = c(x)3=0 satisfy (2.1),(2.2), then (u, c) is a
steady-state or equilibrium solution, and we define the norms
DEFINITION
h(u)= max \ux\ and N(u)=
X6[0 , 1]
u(x) dx.
Jo
Here, N(u) is the usual L r norm, and, since u is a dimensionless cell density, it
is the total number of cells present.. For the time-dependent problem,
dN(u)/dt = 0. The norm h(u) is the usual °°-norm of ux(x). Clearly, h(u) = 0
implies both u and c are spatially uniform.
Downloaded from http://imammb.oxfordjournals.org/ at Indian Institute of Science and Education Mohali on February 24, 2015
consider the nonlinear spatially heterogeneous steady-state solutions of (1.4) and
investigate the possible bifurcations.
The model generates spatial patterns as one of the parameters passes through
certain bifurcation values. It is intuitively clear that cells moving according to
(1.4) need not distribute themselves uniformly in space. Suppose a small
fluctuation produces a local rise in cell density; then this region secretes more
chemoattractant than neighbouring regions, and so recruits more cells. This has
an autocatalytic effect, since the more cells that aggregate, the stronger is the
recruiting signal. Counterbalancing the autocatalytic aggregation is diffusion and
the fact that recruitment depletes the neighbourhood of cells. This creates an
effective region of inhibition around an aggregation centre so that additional
centres can form some distance away, giving rise to a regular spatial pattern.
In Section 2, we consider the local bifurcation analysis, and, in Section 3, the
global situation. We compare the analysis with the numerical results.
72
P. GRINDROD et al.
LEMMA 2.1 For all constants uo>0, there exists a uniform steady-state solution
for (2.1),(2.2)
(u,c) = (uo,g(uo)).
(2.3)
Such a solution is locally stable if and only if
M(Dn2n2 + 1) - u o /(l + uof
(2.4)
Proof. Linearizing (2.1) about (2.3) and setting
u = u0 + a(t) cos nxx
c = g(«0) + b(t) cos nnx
for any n = 1, 2, . . . , we obtain
KVM
M
*V
-ira-i
since cos nnx is the nth eigenfunction of the Laplacian defined on [0,1] with
Neumann boundary conditions.
The result follows immediately by considering the sign of the determinant of
the matrix in (2.5).
Now, if (2.4) is positive with n = 1, then it must be positive for all n > 1. So,
henceforth, we concentrate on the change of stability and associated bifurcation
of steady-state solutions in the first eigenmode. We assume
M(Dn2+l)