An Evolutionary Approach to Pattern Formation Mechanisms on LepidopteranWings

Peter EggenbergerATR

Human Information Processing Research Laboratories2-2 Hikaridai, Seika-cho, Soraku-gun

Kyoto 619-02, JapanEmail: eggen@hip.atr.co.jp

Raja DravidAILab

Department of Computer ScienceUniversity of Zurich

Winterthurerstrasse190, 8057 Zurich, SwitzerlandEmail: dravid@ifi.unizh.ch

Abstract- In this paper an evolutionary and developmen-tal model of cell differentiation and cell induction basedon differential gene expression is introduced. By introduc-ing morphogenetic gradients on a two-dimensional, cellu-lar grid, the model is able to evolve and generate patternsresembling those found on moth and butterflies. Inves-tigation of biological models has its interest for artificialevolution, because one can study the relationship betweenthe genome and the phenotype.

1 Introduction

We would like to present a model of the pattern formationprocess found on moth and butterflies. With it we can simu-late both source localization and pattern formation with twobasic mechanisms, genetic expression and diffusion. Thecomputer simulation of the model consists of a grid of say, 30by 30 units, representing the epidermic cells in a lepidopterawing, or compartment. Transcription factors are diffused intothe grid, creating chemical gradients, to which grid cells be-have differently. We define these behaviors to be pigment andTF-induction specifically.

If the parameters, describing the implemented physicalsystem (diffusion parameters, TF-decay, affinity settings toregulator units), are properly set, sources or patterns emerge,depending on the specific cell’s chemical environment.

We see that not all the necessary information for the spe-cific patterns is contained in the genome itself, but more soin the cell’s interactions with it’s chemical environment, orwith other words physical processes like diffusion and de-cay of corresponding transcription factors1. Therefore, it isthis mechanism which maps the genome onto the phenotype.We consider the investigation of such processes crucially im-portant to the field of artificial evolution, because only suchmechanisms allow the reduction of the genetic informationwithout losing complexity of the system. Reducing the ge-netic parameters also shortens the time needed for the systemto converge.

Our model is capable of generating a large number of pat-terns (see figures 6 and 7), from the simple background pat-terns seen on moths, to the more intricate ones observed in

1Cell behaviors are also induced through physical processes such as stressand strain [3]

butterflies. We did this using an evolution strategy (ES) to”evolve” the parameters dictating the model’s physical pro-cesses, from which the different patterns emerge. The evolu-tion strategy was used specifically because of the large num-ber of parameters involved in describing the pattern formationmechanism, thus making it very difficult to find a certain pa-rameter set (or sets) corresponding to a desired pattern. Withthe ES all we had to do was to describe the desired patternwith a fitness function.

2 Method

In subsection A we give a brief biological introduction howgenes are regulated, how they can communicate with eachother and the effects an activated gene can have. In subsectionB we introduce the mathematical implementation we use todescribe the mechanisms described in subsection A.

2.1 Biological Background

2.1.1 Gene Regulation

Two different classes of genes can be found in the DNA

� structural genes

� regulatory units

Every structural gene is controlled by several regulatoryunits (see figure 2). These units determine the activity of thestructural genes by inducing or inhibiting a special enzymeto transcribe a part of the DNA to RNA [2, 8]. The activityof the structural gene thus depends on the influences of tran-scription factors (TF) on the regulatory units of a particulargene. Experiments with reporter genes have shown that theactivity of the gene depends on the concentration of the TFsat the gene, the affinity between the TFs and the regulatoryunits and of the number of regulatory units which are influ-enced by the TFs [6, 4, 5].

2.1.2 Classes of Gene products

If a structural gene is active, either a chemical substance isinduced, or a predefined procedure is called to induce a de-velopmental process. One of the following events can occur:

1. A transcription factor is produced to regulate the geneactivities.

2. A receptor is produced on the surface of the cell to reg-ulate the communication between the cells.

2.1.3 Receptors

Receptors are proteins located on the cells membrane. Cellscan be influenced by connecting substances to their receptorsand eliciting a cellular response by the inter-cellular signalingpathway. It seems that such mechanisms are used to allow forspecific responses to a general signal like a hormone. Onlythose cells which have a specific receptor for a hormone ontheir surface will respond to it.

2.1.4 Inter-Cellular Communication

By the above described gene regulatory mechanisms, cell dif-ferentiation based on positional information of diffusing TFsor uneven distribution of TF inside the cell can be stimulated.TFs produced by a cell can diffuse to nearby cells. In thiscase they may induce a change of state of some genes in cellswhich are influenced differently due to the different concen-trations of the TF (see figure 1).

2.1.5 Cell Differentiation

Although every cell has the same genome, cells are differ-ent, because different genes are active in them. The differ-ences are due to the morphogenetic sources in the environ-ment, due to asymmetric distribution of morphogens duringcell division (symmetry breaking mechanisms) and due to dif-ferent influences of other transcription factors on the cells.The mechanism of cell differentiation can be based on neigh-boring cells (receptors) or on more distance interactions bydiffusing chemicals in the organism.

a

b

c1

2

Figure 1: The communication pathways between cells.

cis regualtorsGene

trans regulators Activation of a gene by two trans regulators

Figure 2: A structural gene is under the control of transcrip-tion factors which can activate or inhibit a gene‘s activity.Depending on the affinity between transcription factor(TF)and cis-regulators and the TF’s concentration the activity of astructural gene changes.

2.1.6 Cell Induction

Cell induction is the process where a group of cells inducesnew gene activities in another group of cells and so affectstheir development [8]. Cell induction is an important processallowing for the compartmentalization process of the organ-ism. An example of cell induction is illustrated in section 2.3.

2.2 The Model

At least for some gene it seems that the affinity of the site forbinding a TF determines the threshold of a response and thatthe number of sites determines the amplitude of the response[6]. Therefore the following equations were used to simulatesuch a gene regulatory mechanism:

Ai(cTF0 ; : : : ; cTFm) =

�i

2

1 + tanh

��

RXj=0

�(ajcTFj � �j)�!

(1)

where�(x) is the stepfunction:

�(x) =

�1 : if x > 00 : otherwise

(2)

and the variables in (1) have the following meaning:

� Ai is the activity of the gene

� �i amplitude of the activity

� tanh(x) = ex�e�xex+e�x

is the tangenshyperbolicus func-tions with values between 0 and 1

� aj affinity to encode the effect between the regulatoryunit and the transcription factorj

� cTFj transcription factorj

� �j is a threshold value

� � is a constant

� R is the number of regulatory units

Figure 3: an example for cell induction

Figure 4: result of the TF gradient (see figure 3)

The regulatory units function as reading heads for the con-centrations of morphogens or transcription factors, whichwill then determine the state of a gene. As can be seen inDrosophila, usually to a structural gene belong many regula-tory units [4, 2, 1] The differential equations describing thewhole system are:

dgi(x; y; t)

dt= Ai(cTF0 ; ::; cTFm) +D1r

2gi(x; y; t) (3)

where:

� gi(x; y; t): concentration of substance i at gridpoint(x,y) at time t

� m: number of transcription factors

� D1: diffusion constant

� Ai(cTF0 ; ::; cTFm): see eq.(1)

� D1r2gi(x; y; t): diffusion term

In this model the symmetry of the system is already bro-ken through the morphgenetic gradient and thus does not haveto rely on amplification of small differences of activator andinhibitor concentrations to generate patterns [7].

Figure 5: an example for cell differentiation

2.3 An Example for Cell Induction and Cell Differentia-tion

2.3.1 Cell Induction

In the first example (figure 3) the top left image illustratesan initial TF gradient, to which a flat, ellipsoidal cell clus-ter, composed of identical cells, is submitted. Depending onthe interactions of the TF with the cells’ reading mechanisms(regulatory units in the genome), a TF producing gene is ac-tivated in a small band of the cell cluster (top right image),creating a new source from where the new TF diffuses (bot-tom center). This new gradient in turn influences other cellsin the cluster, resulting in different (color coded) cell types(figure 4). When introducing a greater number of TFs andmore genes unpredictable patterns can emerge.

2.3.2 Cell Differentiation

Again the left image shows a morphogenetic gradient, onlythis time a regulator unit is set to simulate a pigment forma-tion process (center image), resulting in the well known ringstructures often found on butterfly wings (right image).

3 Results

These results were obtained by setting sources from whichone TF diffuses. By varying the affinities of the regulatorterms of corresponding structural genes, different patternscan be generated. Furthermore the fitness function described

Figure 6: In the middle of a plane of cells a morphogeneticsource is put and the cells read the concentration of the mor-phogen. Different cells have different gene activity patterns.Note that depending on the the cis-regulators in the cells, thesame morphogen can have different effects.

the “ideal” number of genes to be available.In figure 6 the fitness was set to 3, while in figure 7 it was

set to 4. Notice how the patterns become more complicatedas more genes are added to the simulation.

4 Discussion

Our model implies that pattern formation is dictated not onlyby the information contained in the genome, but more so bythe cells’ interactions with their chemical environment. Wetherefore believe that this mechanism is responsible for map-ping the genome onto the phenotype, being a pattern in thisspecific case study. The investigation of such processes iscrucially important to the field of artificial evolution, becauseonly such mechanisms allow the reduction of the genetic in-formation without losing complexity of the system.

5 Outlook

In further papers we will demonstrate how symmetry sys-tems, dominant in butterflies, could evolve out of the moresimple source distribution patterns found in moth.

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Figure 7: By adding a second source and depending on thenumber of genes, the patterns of different cell types becomeincreasingly complex. Different colors encode different celltypes depending on the subsets of active gene in the genome.(Every cell contains the same genome).

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