Space-time dynamics of reproducing cells
Carlos Comas1 and Jorge Mateu1
1
Department of Mathematics, Universitat Jaume I, Campus Riu Sec, E-12071
Castellón, Spain
Abstract: In this paper we develop a spatially explicit marked point process
model to generate spatial patterns of reproducing and moving cells evolving
through continuous time. In this context, marked points (i.e. cells) can divide
and move as a result of their own division motions, and the division motions of
their touching neighbours. To illustrate such dynamics we present a simulated
example. Our results suggest that by replacing the mathematically convenient
growth and interaction functions by others more suited to realistic cell reproduction dynamics, this model may provide a modelling framework to simulate cell
aggregates (i.e. tissue) dynamics.
Keywords: Cell reproduction dynamics; Marked point stochastic motions; Marked
space-time point processes.
1
Introduction
Knowledge of cell growth and division dynamics has a great importance for
the study of cancer and other degenerative diseases such as Alzheimer. Understanding the chemical and physiological reasons why a given cell starts
to display uncontrolled growth (due to a mutation or a set of mutations)
is a central issue in the study of cancer. Alberts et al. (2002) compare
an organic tissue with an ecosystem where their individual members (i.e.
cells) can grow and reproduce in a collaborative way. In this context, understanding these systems may require the use of well-known concepts in
population dynamics, such as birth, death, habitat and the maintenance of
population sizes (Solé, 2003). This suggests the use of spatially explicit cell
growth models to analyse underlying properties of tissue growth dynamics.
Geometrical models of cell growth have been extensively analysed to investigate cell aggregates or tissues (see, for instance, Honda et al. 2004).
Usually these models consider spatially explicit positions of complex polyhedral structures (cells) to investigate the relation between a tissue and its
cell components. As such an alternative way to simulate and analyse such
dynamics can be done through the development of marked point processes.
In this sense, cells can be considered to be disks or spheres of distinct sizes
(i.e. marks) located at distinct spatial positions.
Although several studies have been performed to analyse snapshot spatial structures of (cancer) cells (see, Diggle, 2003), few approaches have
2
Reproducing cell motions
been formulated to analyse and generate the spatial and temporal evolution of growing particles (Renshaw and Sı̈¿ 12 kkı̈¿ 12 2001). Renshaw and
Sı̈¿ 12 kkı̈¿ 12 (2001) formulate a continuous space-time stochastic process on
the unit torus to generate spatial patterns of immovable (i.e. sedentary)
marked points evolving over time based on a simple immigration-death process. Comas and Mateu (2006) generalise this approach to the case where
growing particles can change positions as result of particles’ growth and
point random motions. The present paper, therefore, expands this work by
permitting growing and moving particles to reproduce.
Cells are the structural units of which all the plants and animals are composed. Cell division is the process by which organic tissues grow and regenerate. In complex organisms, asexual cell reproduction (i.e. mitosis) result
in two “twin” cells with half the size of and identical genetic information
to their respective mother, resulting on exponential population size growth
through local doubling.
The aim purpose of this paper is to formulate a spatially explicit marked
point process model to generate spatial patterns of reproducing and moving
cells evolving through continuous time. To define such a spatially explicit
birth-death cell model, we consider that during the mitosis process new
born cells experience motion as a result of cell replication, while decreasing
their mark size from the initial mother size to half of this mother size.
2
A spatially explicit birth-death cell model
Let us now define a spatially explicit birth-death cell model. Consider that
the resulting spatial position for a given twin cell i located at xi with size
mi (t) at time t in the next differential time affected by its own dividing
motion and the dividing motion of its touching neighbours is defined via
(the same applies for yi (t + dt))
xi (t + dt) = xi (t)
+C
1 (i) cos(θi )h1 (mi (t), m−i (t), xi (t), x−i (t))dt
P
n
+ j=i,j6=i C1 (j) cos(kθj − β(xj (t), xi (t))k) × h1 (mj (t), m−j (t), xj (t), x−j (t))
×I(kβ(xj (t), xi (t)) − θj k < π/2 or kβ(xj (t), xi (t)) − θj k > 3π/2)
x (t)−x (t)
× kxii (t)−xjj (t)k I(kxi (t) − xj (t)k = r(mi (t) + mj (t)))dt
(1)
where k · k denotes the Euclidean distance, I(·) is the indicator function,
h1 (·) is a suitable interaction function that “controls” cell motion, C1 (i) is
a constant of cell motion dependent of each dividing cell, θi denotes the
random direction (i.e. angle) of cell division and β(xi (t), xj (t)) is the angle
between the horizontal axes and the movement vector of particle j when
touching (moving) particle i. Moreover, cos(kθj − β(xj (t), xi (t))k)C1 (j) is
the projection of the dividing motion vector of cell j over the direction
Comas and Mateu
3
vector of cell i when this cell i is moved by dividing particle j. Note that
xi is a spatial position with associated vector of coordinates (xi , yi ).
To generate cell motion of a given dividing cell i moving as a response to
their own dividing motion ensuring no cell-to-cell overlappings, we consider
an interaction mechanism that ensures cell i division when: (1) this cell
does not touch any neighbour; and, (2) all the angles between the resulting
motion vectors of its touching neighbours and the motion vectors of these
neighbours promoted by touching other particles are less than or equal to
π/2. Furthermore, given that cell division not only implies twin cell motion,
but also twin cell size reduction to half of the size of mother cell, we need
to define a reasonable mechanism to simulate such process. Consider the
deterministic differential decremental size change for a given twin cell i
mi (t+dt) = mi (t)+C2 (i)I(cell i is dividing)h1 (mi (t), m−i (t), xi (t), x−i (t))dt
(2)
where C2 (i) is a constant of mark reduction dependent on each cell. Notice
that under (2) dividing cell i only reduces size when moving. Note also that
expression (2) results in constant cell size decay.
3
Illustrating the space-time process
To illustrate the type of spatial process obtained when simulating the process (1) and (2), consider an initial cell located in the unit torus center
at time t = 0 with mark size 0.1 units, with probability α = 0.5 of local
doubling and expected lifetime 1/µ = 20. Moreover, we consider that each
cell division (mitosis) takes 0.5 time units. Here, we expect that in each
time unit 50% of cells undergo cell division. Whilst the random mortality
rate µ = 0.05 avoids that a single cell becomes “dominant”, and completely
affects the resulting space-time structure.
Figure 1 shows the spatial configuration of a realisation of the above process
at times t = 5, 10, 25 and 50, and highlights that whilst the number of cells
contained in the unit torus (initially) grows exponentially, the mark
Pn size
of these cells tends to zero. Note that the total cell “mass” (i.e. i=1 =
mi (t)2 π) contained in the unit torus decreases over time.
4
Future work
The next step of our work is to consider that cells can grow once mitosis
division is over. Although cell growth clearly represent a more realistic
assumption than assuming non-growing particles, regarding real life, it is
also expected an increase in complexity of such hard-particles dynamics.
Now particle movement will be promoted by: (1) cell’s own division; (2)
cell division of touching neighbours; (3) particle’s own growth; and (4)
the growth of touching neighbours. A combination of these four movement
0.75
0.5
0.25
0
0
0.25
0.5
(d) t=50
(b) t=10
0.75
1
Reproducing cell motions
1
4
0.75
1
0
0.25
0.5
0.75
1
0.75
0.5
0.25
0.75
1
0
0.25
0.5
0.75
1
0
0.5
0
0
0
0.25
0.5
(c) t=25
(a) t=5
0.75
1
0.25
0
0.5
1
0.25
0
FIGURE 1. Simulation of a simple spatially explicit birth-death process (see
expressions (1) and (2)) shown at times t = 5, 10, 25 and 50.
effects can also result in complex cell motions. Moreover, these deterministic
effects can be also combined with random mark movements given rise to
far more complex dynamics.
References
Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., and Walter, P.
(2002). Molecular Biology of the Cell. New York: Garland.
Comas, C. and Mateu, J. (2006). On soft- and hard-particle motions for
stochastic marked point processes. Journal of Statistical Computation
and Simulation. Under review.
Diggle, P.J. (2003). Statistical Analysis of Spatial Point Patterns (second
edition). London: Edward Arnold.
Honda, H., Tanemura M., and Nagai, T. (2004). A three-dimensional vertex dynamics cell model of space-filling polyhedra simulating behavior
in a cell aggregate. Journal of Theoretical Biology, 226, 439-453.
Renshaw, E. and Särkkä, A. (2001). Gibbs point processes for studying
the development of spatial-temporal stochastic processes. Computational Statistics and Data Analysis, 36, 85-105.
Solé, R.V. (2003). Phase transitions in unstable cancer cell populations.
The European Physical Journal B, 35, 117-123.