DZIAŁ NAUKOWY
M A T E M A T Y K A S T O S O W A N A 3, 200 2
A.
De u t s c h (D resden)S.
Do r m a n n (O snabriick)Principles and m athem atical m odeling of biological pattern form ation
Abstract. An overview of principles and mathematical models of bio- logical pattern formation is presented. One can distinguish preformation, optimization, topological and self-organization principles. Combinations of such principles are responsible for shape and tissue formation, differentia- tion, regeneration, morphogenetic motion, cell division and even malignant patterns, e.g. in tumor growth. Mathematical modeling allows for a system- atic analysis of the pattern-formation potential of morphogenetic principles.
Biological patterns are the result of complex interactions between a smaller or larger number of components, particularly molecules and cells. Depend- ing on the modeling perspective microscopic (from the individual compo- nent level) and macroscopic models (from the population perspective) are distinguished. The specific question directs the choice of the appropriate perspective. A selection of microscopic and macroscopic model types is in- troduced.
1 . I N T R O D U C T I O N
Life is characteristised by an inherent onto- and phylogenetic time scale responsible for individual morphogenesis and evolutionary change. Here, we focus on pattern forming principles characterizing individual morphogenesis.
There is wide agreement that mutation and selection are dominant processes in phylogenetic dynamics, that can be analyzed in a theoretical framework based on refinements of Darwin’s selection theory. The situation is different
This article is partly based on material taken from the book: A. Deutsch, S. Dormann, Cellular Automaton Modeling of Biological Pattern Formation, Birkhauser, Boston, to appear 2003.
[16]
in morphogenesis, for which a general theory is still missing - here the mechanisms and principles seem to be much more diverse and are only partly understood.
Throughout development, all cells share a more or less identical sequence of genes. In order to explain adjustments underlying regulation and the way in which integrated pattern generation might be achieved through morpho- genesis, a variety of concepts including preformation, optimization, topo- logical and self-organization ideas have been proposed. Particularly in the last decades, mathematical modeling has been established as a theoreti- cal method to analyze principles of biological morphogenesis. Mathematical models have been addressed to central problems of developmental biology including shape and tissue formation, differentiation, regeneration, morpho- genetic motion, cell division and even malignant patterns, e.g. in tumor growth. In particular, formal models offer the possibility for simulation, i.e.
numerical experiments. Here, we describe principles of biological pattern formation and mathematical modeling concepts.
Pattern formation arises from the interplay of (active or passive) cell motion and/or short- or long-range mechano-chemical interaction of cells and/or molecules. What are appropriate mathematical structures to analyze such systems? One can distinguish microscopic and macroscopic modeling perspectives on biological pattern formation focusing on the individual com- ponent (e.g. molecules or cells) or the population level, respectively. Macro- scopic (deterministic) modeling ideas have been traditionally employed and predominantly formulated as partial differential equations [44, 50]. Interest in microscopic approaches, i.e. spatial stochastic processes, has lately grown due to the recent availability of “individual cell data” (genetic and pro- teomic) and has triggered the development'of new mathematical tools, for example cellular automaton models [13]. Such models allow one to follow and analyze spatio-temporal dynamics at the individual cell level.
2 . P R I N C I P L E S O F B I O L O G I C A L P A T T E R N F O R M A T I O N
2.1. P reform ation
U ncoiling. De-velopment means de-veiling, un-coiling, an indication that development, originally, was viewed as preformistic - in the process of uncoiling nothing new can be created, structure is simply uncovered.
Note that also the word evolution bears a preformistic meaning, namely ex-
pression of the preformed germ [ 6 ]. Preformed development only allows for
differential growth or uncoiling. Until Darwin’s discovery of evolution, i.e. a
successive change of organism types in the course of time, the preformation
hypothesis was the best conceivable rational conception of morphogenesis,
18 A . Deutsch and S. Dor mann
Fig. 1. Homunculus. Hartsoeker’s drawing dates to 1694. The microscopic figure inside a human spermatozoon was interpreted as a sitting “homunculus” . Similar “homunculi”
were expected to occupy the female egg.
particularly in accordance with the Christian understanding of a perfect static world. Preformation implies that each generation contains the com- plete information to form all subsequent generations, since no information is created de-novo. Consequently, if organisms are allowed to evolve to new va- rieties, preformation is only possible if the number of generations is limited.
Ultimately, preformation is a theological concept since there is the need for an arbitrary beginning, a creational act, introducing an obvious dichotomy of creator and the created (preformed) beings: Within each animalcule is a smaller animalcule and within that a smaller one and so on (emboitement principle) [2]. Thus, the ovaries of Eve (or the testicles of Adam) contained the prefiguration of every successive human. In a naive interpretation hu- man sperms host a coiled homunculus that simply needs to uncoil like plant seeds in a flower bed (Fig. 1).
Self-assem bly and substitution. The suggestion of C onrad H.
W addin gton (1905-1975) to focus on generation of forms by self-assembly reduces pattern generation to a geometrical problem induced by preformed and disjunct elements - consistent with a geometry-founded Platonian un- derstanding of the world [75]. It is questionable if self-assembly can account for more than the shape of a virus. Nevertheless, there are still models with strong preformistic/static elements, for example L-systems, in which an initial structure develops by subsequent replacement of certain pattern elements defined by appropriately chosen rewriting rules [41] (Fig. 2 ).
P repatterns: on gradients and m orphogens. The problem how
randomly arranged discrete particles in the nucleus can be selected in differ-
ent cells in order to produce the patterned differentiation in adult morphol-
ogy was first addressed by A ugust W eism ann (1834-1914). Beginning in
the twentieth century, the role of gradients for biological pattern formation
was discussed. The early embryologists recognized the significance of epi-
genetic dynamics for pattern formation about the same time when Mendel
Fig. 2. Branching pattern formation in L-system simulation. Patterns are generated by substitution of “pattern elements” ; there is just one pattern element in the example, namely the “stalk-leaf structure” (generation 1). The rewriting rule for this element is defined pictorially by the transition from the first to the second generation. Subsequent generations are generated by successive application of this rewriting rule.
Fig. 3. Epigenetic development from a uniform distribution to a structured embryo (sketch after Aristotle)
pinpointed genetic foundations of “pattern inheritance” (Fig. 3). The notion of gradients and “morphogenetic fields” dates back to the end of the nine- teenth century and particularly to the work of Hans D riesch (1867-1941) and T h eod or B overi (1862-1915) [5]. Boveri introduced the concept of a movphogen gradient built into the egg and subdivided within daughter cells of the early embryo. Additionally, it is assumed that morphogen con- centrations can be precisely measured by cells implying corresponding cell behavioral changes, e.g. differentiation (Fig. 4).
In principle, two gradients can provide an exact positional information within a cell, e.g. a fertilized egg [79]. Meanwhile, the existence of such gradients has been demonstrated. For example, in the bicoid mutant of the fruitfly Drosophila melanogaster a gradient results from passive diffusion of proteins activated by certain developmental genes [51]. Also, it has been shown that when cells from the animal pole of an early Xenopus laevis embryo are exposed to the signaling molecule activin, they will develop as epidermis if the activin concentration is low, as muscle if it is a little higher, and as notochord if it is even larger.
Some problems go ahead with gradient theories. First of all, they assume
a rather precise measurement of concentrations by the cells. It is not clear
20 A . Deutsch and S. Dormann
Fig. 4. Cell differentiation viewed as resulting from a spatial morphogen gradient along an axis L. Differentiation into various cell types is induced by different levels of the morphogen concentration [M].
how these measurements can be achieved. Also, even in the case of one of the best studied biological systems, Drosophila melanogaster, development of polarity, i.e. the primary prepattern responsible for creation of the initial gradient and all subsequent gradients is not fully understood.
2.2. Optimization. Only in the end of the nineteenth century, se- rious research on efficient causes of biological pattern formation started.
Wilhelm His (1831-1904), an initiator of the “whole organ approach” , proposed a “physical explanation” of form formation [29]. He convincingly demonstrated that a developing gut can be modeled as a rubber tube under complex tensions. His’ model is not only a mechanical analogue assum- ing a field of forces, but also strong support for Ernst Haeckel’s (1834- 1919) student, W ilhelm Roux (1850-1924) and his intention to reduce biological phenomena to the laws of physics. Roux’s work on “developmen- tal mechanics” (1) marks the transition from teleological interpretations of embryology to search for efficient (mechanical) causes in the Aristotelian sense.
The Scottish naturalist and mathematician D ’Arcy Thompson (1860- 1948) claimed that an optimization principle, namely minimization of sur- face area under constraints regarding pressure and volume, should account for single cell shapes [71] (Fig. 5). It was Leonhard Euler (1707-1783) who argued that all phenomena in the universe are due to an optimal- ity principle. Interestingly, Euler based his argument on metaphysical rea- soning: because the God-given shape of the universe is perfect, only op- timality principles can account for processes therein. A further optimality principle for biological form is introduced by evolutionary constraints. Or- ganisms should acquire shapes that are optimally adapted to fulfill par-
f1) Roux founded the journal “Archiv fur Entwicklungsmechanik” , Engl.: “Roux’s Archives of Developmental Biology” . Recently, the name was changed to “Development, Genes, and Evolution” .
’S
Fig. 5. “Optimal shapes” : unduloid form of some Foraminifera species (Nodosaria, Rheopax, Sagrina). The unduloid is a rotationally symmetric surface of constant cur- vature (H-surface of revolution). Plateau experimentally showed that there are exactly six H-surfaces of revolution - the plane and the catenoid with mean curvature zero, the cylinder, the sphere, the unduloid and the nodoid with mean curvature nonzero. Delau- nay (1841) proved that these surfaces can be constructed by appropriate rotation of conic sections.
ticular functions, e.g. the shape of fish minimizes water resistance. How- ever, this principle does not give any cue about the formation of an actual shape.
2.3. Topology. D ’Arcy Thompson explained form and evolutionary change of form as the result of immediate, primarily mechanical forces oper- ating on the developing embryo and suggested a theory of allometric trans- formations (Fig. 6 ). Changing morphologies are explained solely as the result of coordinated differential growth during development.
e
Fig. 6. Transformation - D ’Arcy Thompson suggested allometric transformations (affine mappings) to (phenomenologically) describe body forms of several fish. Left: Diodon or porcupine-fish; right: Orthagoriscus or sun-fish (the form results from a particular defor- mation of the coordinate system in the left figure).
22 A . Deutsch and S. Dor mann
R en e T hom , a contemporary French mathematician interested in struc- tural stability in topology (stability of topological form), was convinced of the possibility of finding general laws of form evolution regardless of the underlying substance of form, a motivation that had also guided D’Arcy Thompson. Thom’s goal was to explain the “succession of form” . In his catastrophe theory every physical form is represented by a mathematical quantity called “attractor” in a space of internal variables [70]. If the at- tractor satisfies the mathematical property of being “structurally stable” , then the physical form is the stable form of an object. Changes in form, or morphogenesis, are due to the capture of the attractors of the old form by the attractors of the new form. Morphogenesis is due to the conflict between attractors. The bottom line is that dynamics and form become dual prop- erties of nonlinear systems. Catastrophe theory is a purely geometric theory of morphogenesis.
“Topological approaches” have also been applied by “neo-Darwinist”
evolutionary biologists (of the 1930s and 40s) in their attempts to integrate evolution theory within genetics [34, 42]. Changes in adult morphologies were viewed as the result of gradualistic growth changes, using concepts like allometry, in turn linked to genetics by concepts as “rate genes” or “gene balance” . But these approaches are rather limited and fail to explain, for example, changes of somite number or differentiation.
2.4. Self-organized pattern form ation
Turing (diffusive) instability. An important key to understand de- novo pattern formation in biological morphogenesis is to interpret it as a self-organization system [33]. In his pioneering paper (1952) A lan M . Tur- ing (1912-1954) analyzed a field of diffusively coupled reacting cells [73].
Counter-intuitively, starting from a slightly perturbed spatially homoge- neous configuration this system can produce periodic patterns under ap- propriately chosen conditions. The Turing instability is one of the first demonstrations of emergence with respect to chemical pattern formation:
The macroscopic periodic pattern is not present at the level of the micro-
scopic system components, the chemical molecules, but arises as a result
of component interaction. In morphogenetic applications it is assumed that
cells can perceive “morphogen” concentrations and react correspondingly,
i.e. the Turing idea can explain the formation of prepattems. This intro-
duces a dichotomy which poses conceptual difficulties since a distinction
is made between “the organization” (of the prepattern) - assumed as a
self-organization process - and “the organized” - the final pattern which is
viewed as the “hand of the prepattern” . Typically, in such models no feed-
back of the hand to the chemical pattern is considered which is biologically
unrealistic.
Noteworthy, only recently experimental manifestations of the Turing in- stability, the CIMA and the PAMBO reactions, were discovered [9, 59, 76]. It is today questionable if the Turing instability is important in biological mor- phogenesis. Note that a Turing-like instability has recently been proposed that generates oscillations of certain proteins which control the accurate placement of the midcell division site of the bacterium Escherichia coli [32].
Nevertheless, the instability principle is essential and a number of (macro- scopic) morphogenetic instabilities have been identified, e.g. chemotactical, mechanochemical and hydrodynamical instabilities [28, 30, 50]. It has been demonstrated that self-organization can account for differentiation, aggrega- tion, taxis, cell division or shape formation. Appropriately chosen reaction-
Table 1. A classification of organization principles and their morphogenetic competences
O rgan ization
principle M o rp h o g en e tica l com p eten ce (Examples)
preform ation uncoiling —> Homunculus,
self-assem b ly : key-lock principle —> macromolecular struc- ture, virus form [75],
su b stitu tio n : L -sy stem s —> branching [41]
p rep attern : positional inform ation, gradient m o d el —►
differentiation [79],
o p tim ization m in im iza tio n o f surface area —> cell form [71], differential adhesion (minimization of free energy)
—► sorting out, aggregation [22],
top olo gy allo m etric tran sform ation —* fish shapes [71], cata strop h e th eory —> organ forms etc. [70]
self-
organization
diffusive inst. —> diffusion-limited aggregation [78]
T u rin g inst. (reaction-diffusive inst.) [44, 73] —► phyllo- taxis [80]
m echanical inst. —► plant patterns, e.g. whorls [24], blastula- tion, gastrulation [19]
m ech an o -geo m etrica l inst. —> cell shape [60]
m ech an o-ch em ical inst. —> segmentation [55, 56]
h yd ro d y n am ic-ch em ica l inst. —> cell deformation, cell divi- sion, active motion [28, 38]
orientational inst. —> swarming, collective motion [4, 8, 46, 47], cytoskeleton [11]
ex cita b le m edia —> Dictyostelium discoideum taxis, aggrega- tion [12]
24 A. Deutsch and S. Dormann
diffusion models (activator-inhibitor models) have been further investigated in particular by Gierer and Meinhardt [21, 44] (see the monographs [57, 62]
for reviews).
Cell-cell interactions. Already in the nineteenth century, a cell theory was proposed by Theodor Schwann (1810-1882) and Matthias Jacob Schleiden (1804-1881), according to which animal and plant tissues solely consist of cells. Cell formation should be the common developmental prin-
ciple of organisms. Cells have an ambiguous character - they are closed functional units and, simultaneously, are capable to exchange information with other cells. Since the fifties the influence of cell-cell interactions on pat- tern formation has also been studied with mathematical models. Johannes Holtfreter (1901-1992) was among the pioneers to realize the importance of cell-cell interactions, especially differential adhesivity for cellular pattern formation [31, 67]. In particular, Holtfreter’s pioneering study combining observations on dissociated and reassociated embryonic tissue in order to analyze tissue movements and specifity apparently led directly to Stein- berg’s differential adhesion hypothesis of cell sorting based on a free energy minimization argument [49].
It has turned out that many developmental signals govern local, cell-to- cell inductive interactions rather than position-dependent or long distance gradients. In the last years more and more experimental data (genetic, pro- teomic etc.) have become available at the single cell level triggering the devel- opment of microscopic models in which individual cells can be represented.
Special emphasis has been laid on the analysis of direct cell-interaction- based self-organization, i.e. patterns arising by purely local interactions be- tween cells with discrete states. Such analysis is possible with the help of individual-based models, particularly Monte Carlo simulations and cellular automata (see below).
A classification of morphogenetic organization principles.
Table 1 gives an overview of organization principles and their prospective morphogenetic potential. Details are discussed throughout the text.
3 . H O W T O C H O O S E T H E A P P R O P R I A T E M O D E L
Being faced with the problem of constructing a mathematical model of biological pattern formation, the first modeling step is to clarify the level of organization one is primarily interested in - with regard to space, time, states and interactions.
One possibility to classify approaches to modeling spatially extended
dynamical systems is to distinguish between continuous and discrete state,
time and space variables.
The answer to the question if a process is viewed as state-, time- or space-discrete or continuous is essential, since the particular choice may influence the results significantly. For example, while the time-continuous logistic ordinary differential equation describes a simple growth process, the time-discrete logistic map leads to “complex” dynamical behavior, inclu- ding chaotic motion [36]. In order to give appropriate answers the scales of all involved processes and their relation to each other have to be spec- ified. For instance, a variable (e.g. temperature) can be regarded as con- stant if the chosen time-scale is short. On the other hand, if the time- scale is large, the detailed dynamics of the variable might become impor- tant.
In particular, the appropriate choice of space and time scales is not a mathematical problem, but a preliminary decision of causes since one has to determine and distinguish the phenomena to be considered and those to be ignored. The modeling strategy reminds of what D’Arcy Thompson called Aristotle’s parable (cited after [23]): “we must consider the various factors, in the absence of which a particular house could not have been built: the stones that compose it (material cause), the mason who laid them (efficient cause), the blueprint that he followed (formal cause), and the purpose for which the house was built (final cause)” . Applied to biologi- cal morphogenesis the house corresponds to the adult organism and the stones to the molecular components, the material cause of pattern forma- tion. While it is clear that genetic information (blueprint), natural selection and physico-chemical constraints all contribute to efficient, formal and fi- nal causes of morphogenesis, the precise role and interactions of external and internal factors cannot be easily (if at all) distinguished. For example, when trying to explain cellular shape, shall we focus on cytoskeleton archi- tecture as arising from reaction-diffusion intracellular macromolecular dy- namics; or, alternatively, shall we investigate curvature dynamics of the cell cortex, for example appearing as a minimum property of the potential energy?
In modeling the temporal evolution of spatially distributed systems, de- scribing for example the interaction of cytoskeleton molecules or the forma- tion of cellular tissue, we can think of the system as a game of formation which is defined by specifying the players (molecules and/or cells) together with the rules of interaction. In particular, the players’ internal state space representation has to be defined. Internal state may refer to position, veloc- ity, acceleration, orientation or age of cells or molecules.
In specifying the players’ interactions, namely the rules of the game, one
can choose several levels of description involving different resolutions of spa-
tial detail ranging from macroscopic to microscopic perspectives. Transitions
from one level to the other involve various approximations with regard to
26 A . Deutsch and S. Dormann
. A/V
averaging f\pj discretization over
space [s] timefT] internal state [is]
D3bO
£
%
CDcfl
o
c
o
Ph CD*-c
low representation of physical space high
Fig. 7. Model relations with respect to physical space and internal state representation.
The sketch shows the relations between various model levels. Averaging and limiting pro- cedures allow transitions between ODEs (ordinary differential equations), PDEs (partial differential equations), FDEs (finite-difference equations), IBMs (individual-based models or many particle systems). Cellular automata cap be alternatively viewed as a separate model class or as discretizations of PD E or IBM models.
the spatio-temporal nature of the underlying interactions (Fig. 7). Notably, physical space has never been explicitly defined [ 20 ], but is instead tacitly assumed in models of pattern formation.
3.1. M o d e l perspectives
V iew ing from the top : m acroscopic m odels. The simplest (macro-
scopic) approach is to assume that the (spatially extended) system is ho-
mogeneously mixing over its entire extent and to model the dynamics by
a system of ordinary differential equations (continuous time) or difference
equations (discrete time steps). This assumption is of minor value in systems in which the precise spatio-temporal patterns are of interest. In such systems, one recognizes that spatial position of interacting agents (cells, molecules) is important and allows individuals to move about in (continuous or discrete) geometrical space; the appropriate mathematical description is a system of coupled differential equations (see box) or a reaction-diffusion/advection system (partial differential equation, see box).
Model I: Coupled differential equations
An example for this approach is Turing’s cellular model [73], whose pur- pose is to explore how spatial structures (forms, patterns) can emerge from a homogeneous situation. Space is divided into discrete compartments r = 1 , . . . , T (spatially homogeneous cells) in which different components a — 1 ,..., 4 are governed by ordinary differential equations. The variables aCT(r, t) G R usually represent macroscopic quantities, e.g. densities. Spatial inter- actions are modeled by discrete coupling of components of different cells (discrete diffusion). In particular, Turing suggested the following nearest neighbor coupling of a ring of cells for a two-component one-dimensional system:
daiQt ' ^ = Fi (ai(r> *)> fl 2 (r, t))
+ Di[(ai(r - Iff) - 2ai(r, t) + affr + 1 , t) ) ] , d(L2fc ^ = F2(ai(r,t),a2{rff))
+ D 2[(a2{r - l f f ) ~ 2 a2(r, t) + a2(r + Iff)]),
with continuous time f e l and “diffusivity” Da G R , which expresses the rate of exchange of species a at cell r and the neighboring cells ( “diffusive coupling” ).
An advantage of coupled differential equations is the possible adaptation of the transport scheme (coupling) to particular system demands, e.g. to extended local neighborhood relations.
i_________________________________________________________________ ___— — i
Model II: Partial differential equations (PDEs)
The classical approach to model spatially extended dynamical systems are (deterministic) partial differential equations, which model space as a con- tinuum (2), x
GRd, where d is the space dimension. In the simplest version
(2) Note .that this implies an infinite amount of information about the state values in any arbitrarily small space-time volume.
28 A . Deutsch and S. Dormann
particle transport is assumed to be passive diffusion. Then, a two-component one-dimensional system is described by
dai(x, t) da 2 dt {x,t)
dt
= Fi{a\{x,t),a2{x,t)) + £>id ^
= F2 (ai (x, t), a2(x, t)) + D2^ ~ L^-
where t E R and D a E R are “diffusion coefficients” . A vast literature deals with a framework based on “reaction-diffusion” models (reviewed in [53]).
They describe interaction processes and demographic dynamics, commonly called reaction, which are combined with various transport processes.
i__________________________________________________________________________i Regarding the modeling perspective, the partial differential equation ap- proach (e.g. reaction-diffusion/advection) can lead to a satisfactory model if a sufficiently large number of cells or molecules allows evaluation of a local density. Quantification of densities implies a spatial average which is only meaningful if the effects of local fluctuations in the number of the considered components are sufficiently small. This modeling approach has accordingly some shortcomings - the determination of a temporal derivative implicitly involves a limit which is only justified under the assumption that the pop- ulation size gets infinitely large or, equivalently, that individuals become infinitesimally small. The following question immediately arises: How can one treat effects of local stochasticity and the fact that individuals (cells or molecules) are obviously discrete units?
Two strategies - macroscopic and microscopic, respectively, are possible:
one can start from the top (macroscopic perspective), abandon the con- tinuum of spatial scales and subsequently subdivide available space into patches. This procedure leads to finite-difference (with respect to space) patch models or coupled map lattice models (see box) which credit the indi- viduals’ discreteness but not their stochastic behavior. Note that numerical solution methods of partial differential equations typically also replace spa- tial and/or temporal derivatives by finite differences, which is equivalent to a discretization of space and/or time.
Stochastic fluctuations can also be studied after introduction of noise
terms into the reaction-diffusion equation. This so-called Landau approach
[37] offers possibilities to analyze the effect of fluctuations in corresponding
stochastic partial differential equations. The approach is somewhat para-
doxical, since one has averaged over the microscopic fluctuations to ob-
tain the macroscopic (mean-field) equation [77]. Alternatively, one can start
from the bottom, i.e. a microscopic stochastic description (Lagrangian ap-
proach) .
Viewing from the bottom: individual-based models and cellular automata
Interacting particle systems. Interacting particle systems consist of a finite population of particles (cells or molecules) moving about in discrete space and continuous time (see box). Particles are characterized by their position, velocity and internal state. There is no restriction of possible inter- actions to be studied within the modeling framework of interacting particle systems: Particle configurations may change due to direct or indirect, local or non-local interactions (some examples that have been treated mathemati- cally can be found in [40]). Particle traces governed by Newtonian dynamics can in principle (in non-quantum systems) be followed individually - an example of an individual-based description.
i i
Model III: Interacting particle systems
Interacting particle systems model directly interactions between finitely or infinitely many particles (e.g. individuals, cells, molecules). They are stochastic models consisting of a collection of spatial locations called sites and a finite set of states. Each site can be in one particular state at each time t £ R. The temporal evolution is described by specifying a rate at which each site changes its state. The rate depends upon the states of a finite number of neighboring sites. In the absence of interaction, each site would evolve according to independent finite or countable state Markov chains [40].
When an event occurs at a constant rate a then the time intervals between successive occurrences are exponentially distributed (Poisson distribution) with expectation a [16]. The ensemble of states of all lattice sites defines a configuration or a microstate of the system (3).
Model IV: Coupled map lattices
If both space (r) and time (k) are subdivided into discrete units, one refers to coupled map lattices or time- and space-dependent difference equations (4).
An example for a one-dimensional two-component system is a\{r,k + 1) = Fi(ai(r,k),a2>r(r,k))
+ D\ (ai(r - 1, k) - 2ai(r, k) + ai(r + 1, k) ) , a2(r,k + 1) = F2(ai(r, k),a2,r(r, k))
+ D 2(a2(r - 1, k) - 2a 2 (r, k) + a2(r + 1, k ) ) .
(3) A good introduction into this modeling field can be found in [17].
(4) Or finite-difference equations.
30 A. Deutsch and S. Dormann
This model approach was introduced in order to study spatio-temporal chaos, which is important in the study of turbulence [35]. Coupled map lattices are also used
elsa tool for numerical studies of partial differential equations. Lattice Boltzmann models are a particular case of coupled map lattices. They can be derived from a microscopic description (lattice-gas cel- lular automaton) of (physical) systems, which are composed of many “par- ticles” (e.g. fluid dynamics, [43]). Then, the state variables aa{r,k)
G[0,1]
are defined by averaging over an ensemble of independent copies of the lattice-gas cellular automaton, i.e. they represent the probability of the pres- ence of a particle at a cell r at time k. When spontaneous fluctuations and many-particle correlations can be ignored this approach offers an effective simulation tool in order to obtain the macroscopic behavior of the system and provides a “natural interpretation” of the numerical scheme [ 10 ].
Simulations based on principles of molecular dynamics (continuous space and time) are most effective for high particle densities since particle traces are individually evaluated by integration. In molecular dynamical algorithms integrations are performed even if the situation remains physically almost unchanged, which is typical for the low density situation. Under such cir- cumstances dynamical Monte-Carlo methods generating random sequences of events (e.g. Metropolis algorithm [45] or simulated annealing [74]) are preferable [1]. For further (statistical) analysis a large number of simulations has to be performed [26]. A “morphogenetic application” of Monte-Carlo methods is provided by simulations of tissue pattern formation based on direct cell-cell interactions [15].
Cellular autom ata. Cellular automata can be interpreted as discrete
dynamical systems - discrete in space, time and state. Spatial and temporal
discreteness are also inherent in the numerical analysis of approximate solu-
tions to e.g. partial differential equations. As long as a stable discretization
scheme is applied, the exact continuum results can be approximated more
and more closely as the number of sites and the number of time steps is
increased - the numerical scheme is convergent. The discreteness of cellular
automata with respect to the limited (discrete) number of possible states
is not typical of numerical analysis where a small number of states would
correspond to an extreme round-off error. The problem of state space lim-
its has not been addressed in a rigorous manner so far. It is possible to
devise cellular automaton rules that provide approximations to partial dif-
ferential equations and vice versa [54, 72]. For cellular automaton models
certain strategies have been developed to analyze continuous approxima-
tions [65, 66 , 68 ]. Cellular automata differ from coupled map lattices which
are characterized by a continuous state space.
I I Model V: Cellular automata
Cellular automata are discrete dynamical systems defined as follows: A cel- lular automaton is a 4-tuple {£ , £, JV, 7Z}, where
• C is a regular lattice of cells (space: in two dim. e.g. square and hexag- onal lattices).
• £ is a finite - typically small - set of elementary states. Each cell r € £ is assigned a state s(r) 6 S.
• M is a set of cells which defines the interaction neighborhood for each cell r € C.
• TZ is a function
n
:e
which assigns a new state to a cell solely depending on the neighbor states (local dynamics).
The development of a (deterministic) cellular automaton is defined by the application of function 7Z simultaneously to all cells r e £. There also ex- ist asynchronous and stochastic cellular automaton extensions (see [13] for an overview). Recently, hybrid cellular automaton models have been intro- duced which assume discrete cells on a lattice and a continuous chemical signal field. Such frameworks have found applications in chemotactic pat- tern formation and have recently been addressed to study problems of tumor growth [14].
i_________________________________________________________________________ i 3.2. From individual behavior to population dynam ics. The aim to determine how the (macroscopic) population dynamics evolves with time, in terms of the (microscopic) individual behavior that governs the motion of its constituents, is similar to the main goal of nonequilibrium statistical mechanics. The master equation, which completely specifies the change of the probability distribution over a given interacting particle system’s phase space, is far too complicated in its generality. For system analysis certain assumptions are necessary which thoroughly depend on the particular (mi- croscopic) interaction. Historically, two different directions have been devel- oped - the Brownian motion theory (initiated by Einstein, Smoluchowski and Langevin) and the kinetic theory started by Clausius, Maxwell and Boltzmann [63].
With regard to interacting particle systems as models of biological pat-
tern formation both theories can be rediscovered as modeling strategies: The
first strategy uses a (stochastic) assumption on the dynamics of individual
particle motion - this is the active walker or Langevin approach. The second
approach is based on a kinetic or Boltzmann interpretation and focuses on
the dynamics of particle distribution functions which depend on statistic
32 A . Deutsch and S. Dormann
properties, not on the details of individual particle motion. Interestingly, in many relevant situations the well-known Boltzmann equation in terms of the single particle distribution function already captures essential system characteristics [61].
Further strategies to derive macroscopic equations from individual-based models are possible. For example, alternative approximations of Eulerian and Lagrangian approaches were proposed in a model of swarming and grouping based on density-dependent individual behavior [25]. Another possibility is the method of adiabatic approximation (quasi-steady state assumption) that was recently applied to an individual-based ecological model [18]. Such strategies have however not been employed for models of morphogenesis so far.
A ctiv e walker or Langevin approach. We want to illuminate the Langevin approach with an example which has been studied particularly well within the last years. This is chemotactic aggregation as a model of bacterial pattern formation [3, 58, 68 ] where it is assumed that a chemical (diffusible or nondiffusible) substance (produced by the bacterial cells) de- termines motion of the particles insofar as these search for local maxima of the chemoattractant substance. If biological particles (e.g. cells) communi- cate by means of an external field, e.g. a chemical concentration field s(x , t) which they actively produce (or destroy), one can view them as active (or communicative) Brownian walkers [64]. Typically, the individual walker’s motion is described as linear superposition of (passive) diffusive (Brownian) and reactive (active) parts. The underlying dynamics is governed by New- ton’s law: force = mass • acceleration [27]. The equation of motion for the i-th walker (i = 1 ,. . . , N ) can be formulated as (5)
v%(t) = = - 7 •«»(*) + Vs(ft(*),*) + \/ 2 e 7 &(*), (1) where the mass of the walker has been normalized to 1 , t is time, qi(r) is the position of the walker, Vi(t) is its velocity, 7 • Vi(t) is the viscous force which slows down the particle’s motion (7 is a friction coefficient) and £*(£) is Gaussian white noise with intensity e. The gradient Vs(x,t) introduces the taxis behavior since this form assures that particles search for local maxima of the chemoattractant s(x,t), that follows an appropriate reaction-diffusion equation. Eqn. (1) is known as Langevin equation which is an inhomogeneous linear stochastic differential equation that can be regarded as a phenomeno- logical ad hoc assumption of individual behavior.
In some cases, there are averaging strategies to proceed from a fully stochastic Langevin description of single walker motion to density equa- tions [27] - especially for spatially homogeneous situations in the mean-field limit. Particularly, in the chemotaxis model (1) it can be shown that under
(5) W e follow the notation of [69].
certain conditions (particularly regarding the interaction which has to be moderate (6)) the continuous model is a good approximation of the inter- acting particle system [ 68 ].
The relation of reaction-diffusion and interacting particle systems has been systematically analyzed with regard to the interaction limiting be- havior; in particular, hydrodynamic, McKean-Vlasov and moderate inter- actions can be distinguished [52]. Recently, an alternative model of aggre- gation (without external diffusive medium), characterized solely by density- dependent long-range weak (or McKean) and short-range moderate interac- tions has been analyzed by a Langevin strategy [48].
K inetic or B oltzm ann perspective. While the active walker ap- proach starts with a description of individual particle dynamics, the kinetic interpretation neglects details of individual motion which had to be assumed in an ad hoc manner anyway. The kinetic approach focuses on bulk behavior, i.e. analysis of statistical properties of particle interactions fully contained in the single particle distribution function which represents the probability to find a particle with a given velocity at a given position. It turns out that both in continuous and discrete interacting cell systems, a particularly sim- ple description at a kinetic level can be gained by describing the state of the system in terms of the single particle distribution functions and to discard the effect of correlations (Boltzmann approximation). The corresponding equation is known as Boltzmann equation which arises as an approximate description of the temporal development in terms of the single particle dis- tribution functions if all pair, triplet, and higher order correlations between particles are completely neglected. Boltzmann showed that this assumption is approximately true in dilute gases. Nevertheless, it has turned out that the Boltzmann (or mean field) approximation is a reasonable assumption in many other, particularly biological applications [61].
With regard to biological pattern formation the Boltzmann strategy has not been utilized in a fully continuous interacting cell system so far, but in cellular automata, i.e. models discrete in space, time and state space (com- pare [ 8 ]). However, the kinetic approach is not restricted to discrete models.
A systematic comparison of the Langevin and the Boltzmann approach with respect to biological pattern formation applications is still waiting.
4. D IS C U S S I O N
We have presented principles and mathematical modeling approaches to biological pattern formation. While reaction-diffusion models are appro-
(6) Moderate interaction means that the main interaction range of each particle shrinks as N —> oo, but the number of particles in this range tends to oo.
34 A. Deutsch and S. Dormann
priate to describe the spatio-temporal dynamics of (morphogenetic) signal- ing molecules or large cell populations, microscopic models at the cellular or subcellular level have to be chosen if one is interested in the dynam- ics of small populations. Interest in such “individual-based” approaches has recently grown significantly since more and more (genetic and proteomic) data are available. Important questions arise with respect to the mathemat- ical analysis of microscopic individual-based models and the precise links to macroscopic approaches. While in physical processes typically the macro- scopic equation is already known, the master equations in biological pattern formation are far from clear. Many important questions are open for future research. One issue is the systematic study of further collective phenomena in cellular systems, in particular cellular differentiation arising from specific spatial cellular distributions. It is a huge open field to systematically link the pattern formation models presented in this article to genetic networks governing intracellular genetic and signaling networks. This will hopefully be possible in the future since experimentally the data needed for the math- ematical model become now available.
Models for spatio-temporal pattern formation in interacting cell systems and the development of genetic traits in genetic populations are interestingly of similar mathematical structure - in both situations the dynamics is due to local interactions and migration. For example, cellular interaction and inter- actions of genes on the DNA as well as diffusive cellular motion and random genetic mutations are described by the same mathematical models, respec- tively (e.g. reaction-diffusion systems). Further clarification of the relations between morpho- and phylogenetic mathematical models is a challenging future research field.
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Center for High Performance Computing (ZHR) Dresden University of Technology
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