Emerging Trends in Applications and Infrastructures for ...lab-sticc.univ-brest.fr/~rodin/Publis/PDF/Sarr16b.pdf · their morphology. Artificial regulatory networks have also been

Emerging Trends inApplications and

Infrastructures forComputational Biology,

Bioinformatics, andSystems Biology

Systems and Applications

CHAPTER

A DIRECTIONAL CELLULARDYNAMIC UNDER THE CONTROLOF A DIFFUSING ENERGY FORTISSUE MORPHOGENESIS:PHENOTYPE AND GENOTYPE

2

A. Sarr, A. Fronville, V. RodinComputer Science Department, University of Western Brittany, Brest, France

2.1 INTRODUCTIONFacing the complexity of in vivo experiments, understanding of the living is more and more focused onin silico models. Nowadays, the relative ease of implementation of a virtual lab has made possible theformulation and testing of many hypotheses in biology, particularly in morphogenesis.Morphogenesisis an important research field within developmental biology. It can be defined as the set of processesthat causes an organism to develop its shape. Many models exist in the area depending on the mainfactors considered in biological form creation and also on the studied organisms (prokaryotes, animals,plant, etc.). Tensegrity model is for example interested in cells’ shape changing. This model considersbiomechanical forces between cells and the extracellular matrix. The stretching of cells adhering to theextracellular matrix may result from local reshuffle in this latter [1]. However, the question of cell di-versity arises even before the acquisition of shape [2]. Indeed, when the embryo has only a few pairs ofcells, we can already see a diversification in the biochemical content of the embryonic cells or even intheir morphology. Artificial regulatory networks have also been used in morphogenesis modeling.They define a series of regulatory genes and structural genes. The first consists of a network of rulesdetermining the evolution of the system and the latter are intended to each generate a simple specificpattern [3]. However, even if the detailed knowledge of genomic sequences allow us to determinewhere and when different genes are expressed in the embryo, it is insufficient to understand howthe organism emerges [4]. The mechanisms of cell morphogenesis also include the reaction-diffusionmodels. A.M. Turing presented in 1952 his well-known model where it is suggested that a system ofchemical substances, called morphogens, reacting together and diffusing through a tissue, is adequateto account for the main phenomena of morphogenesis [5]. In this model, patterns or structures emergedue to an instability of the homogeneous equilibrium, which is triggered by random disturbances. Thismodel can surely produce structures, like spots and stripes, for a range of organisms but it does not givethe process by which the global shape emerges. Besides, because the model relies on random distur-bances to produce spotted patterns, we cannot follow in a deterministic way cell division and

Emerging Trends in Applications and Infrastructures for Computational Biology, Bioinformatics, and Systems Biology.

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# 2016 Elsevier Inc. All rights reserved.17

differentiation from the first cell of the organism and along with development of its shape. Anotherapproach commonly used for modeling multicellular systems is to make an integration of the dynamicinteractions between different spatial and/or temporal scales. This approach introduces some complex-ities in models that limit their understanding and effectiveness with respect to their purpose. Thus, weconsider that the cell must be the focus, which determines both causalities and downgrades. In otherwords, as a first step to a better understanding, observations must be restricted to a single spatial andtemporal scale of the biological organization.

Thanks to advances in microscopy and imaging, very detailed data on components and structuresof living organisms are now available. Melani et al. achieved a tracking of cell nuclei and the identi-fication of cell divisions in live zebra fish embryos using 3D+time images acquired by confocal laserscanning microscopy [6]. While the zebrafish embryo contains a few pairs of cells, we noticed (seeFig. 2.1):

1. A geometrical segmentation during cellular proliferation.2. A specific arrangement of cells at each stage guided by a deterministic process.

These observations allow us to formulate a set of principles so as to propose a model. First, the notedgeometrical segmentation allows us to adopt a discrete model in time and space to study the forms

FIGURE 2.1

The four images are a collection of screenshots from the video produced with the framework designed in [7].It was used for imaging and reconstructing the whole zebrafish embryo at its first 10 cell division cycles. Theseimages depict the first three division cycles when the embryo includes 2, 4, and 8 cells.

18 CHAPTER 2 A DIRECTIONAL CELLULAR DYNAMIC UNDER THE CONTROL OFA DIFFUSING ENERGY FOR TISSUE MORPHOGENESIS

appearing in the early stages of morphogenesis. For now, we restrict to a 2Dmodel. Besides, wemake thehypothesis that, based on spatial choices of cells, we can define amorphological dynamic of tissues emer-gence and growth at the early stage of the embryo during which the mechanical influences are insignif-icant. Furthermore, we assume that this dynamic is the fundamental principle of morphogenesis and istherefore able to describe all evolutions of a tissue, both those that modify it and those that maintain it.

Our main goal in this chapter is to develop, implement, and simulate a mathematical model thatembodies the crucial features of our hypothesis. At the same time, we show how such a model is ac-curate to describe behaviors of cells and aggregation of cells. This chapter is organized as follows,Section 2.2 presents the mathematical morphological dynamic of a multicellular system. It relies onthe viability theory [8], which offers an approach especially well suited for the biological systemswhose complexity would render a purely analytic approach unrealistic. Using computer simulations,by mimicking the tissue growth, the dynamic model allows us to generate in Section 2.3 all possiblephenotypes at the early stage of the embryo and their underlying genotype. Building on the back of thephenotypes and their associated genotype, we present in Section 2.4 a prediction tool to simulate tissueevolution at the later stages of the embryo. Because we are focusing on animal growth, cellular ex-change has been taken into account through a coupling of an energy diffusion model with the morpho-genesis model. Thereby, cells are endowed with the ability to consume, lose, and transfer energy fromand to their environment. The main contribution of this chapter is the coupling of the two dynamicmodels: differential equations manage the energy of the system and morphological equations governits growth. That coupling implies a coevolution in the system. On one hand, the global energy fed in thesystem depends on the size of the tissue, and on the other hand, the local morphological dynamics ofcells that locally transforms the tissue depends on their available energy reserve. Finally, Section 2.5concludes this chapter by giving some applications achieved with this work and then open some per-spectives to explore in coming ones in terms of simulation.

2.2 MATHEMATICAL MORPHOLOGICAL DYNAMICSIn mathematics, the viability theory offers concepts and methods to control a dynamical system in agiven fixed environment, to maintain it in a set of constraints of viability. Applied to morphogenesis,this means that we should have at least one coviable evolution of the cells’ state and their environmentamong all the available pairs of state-environment of each developmental step. This formalizationallows us to establish rules in terms of cell plus action as spatial choices of cells under logicaland temporal conditions. The application of such rules causes some biological effects on the cellsindividually but they also impact the overall tissue. However, relying on that theory to tackle issuesin morphogenesis requires first to properly define some of its concepts in the case of a multicellularsystem. In [9], we described mathematically the state, controls, and both local and global morpholog-ical dynamics of tissues. Some points of that formalization are highlighted in this section.

K!P"X#1 denotes the morphological environment2 (X$!2 denotes the set of containment cellscontained in the complement of vitellus3 ). Cells x2X[Ø are either characterized by their position

1Supplied with the structure of max-plus algebra for the operation [ and + (where K + Ø :$ Ø with K a cell tissue).2For instance, K :$ fK!Mg is the family of subsets contained in a given subset M.3In biology, the vitellus is the energy reserve used by the embryo during its development.

192.2 MATHEMATICAL MORPHOLOGICAL DYNAMICS

(living cells) or by their death made of tissues L, which are subsets of cells (L2P"X#). The subset ofeight genetic actions d of cells is

A :$ f"1,0,0#,"%1,0,0#,"0,1,0#,"0,%1,0#,"0,0,1#,"0,0, %1#,"0,0,0#,Øg

A is made of the six geometric directions, the origin, and the empty set. Here, we restrict morphogen-esis in the plan:

A :$ f"0,1#,"0, %1#,"1,0#,"%1,0#,"0,0#,Øg

For convenience, we replace (0,1), (0,%1), (1,0), (%1,0), (0,0) andØ, respectively, by 1, 2, 3, 4, 5, and 6.

A :$ f1,2,3,4,5,6g

These genetic actions allow to describe cells’ behaviors:

1. Transitions x ! x + d, where d 2 {1,2,3,4} (action)2. Quiescence x ! x + 5 $ x (no action)3. Apoptosis x ! x + 6 $ 6 (programmed cell death)

This injunction "d ¢,d “# is described by the genetic inclusion

x! x+ d ¢,x+ d “! "

where the mother cell x

• first migrates from x to x+ d ¢ using the migration action d ¢ 2A at a new position (including x orØ),• then divides, giving birth to a cell at position x + d “ using the division action d “2Anf5g.

The composition of these actions produce a mother-daughter cell pair fx+ d ¢,x+ d “g.Hence, the basic behaviors of the mother cell are described by

1. Sterile migration by taking d ¢2A and d “$ 6.2. Stationary division by taking d ¢$ 5 and d “2A.3. Migrating division by taking d ¢2Anf5g and d “2Anf5g.

In this implementation, we do not take into account migration and death and all cell transitions are allstationary divisions. Hence, the injunction "d ¢,d “# can be likened to the genetic action singletond “ :$ d.

2.2.1 GENE AND STATUS EXPRESSIONA genetic process, G, is a possible combination of genetic actions G :$ fd1,…,dig2Ai. Operating agenetic process under a given criterion, either for migration or for division, means that the processscans successively x + d1,…, x + di until the first time when the criterion is satisfied. For any tissue(phenotype), there exists a set of genetic processes (genotype) that allows to achieve its creation. Start-ing with a single cell, at each step, we compute from the previous tissues all the possible configurationsof tissue we can reach through cell division. In computer science, this issue is a case of polyominoesenumeration, except that we have added some biological constraints. The configurations are saved onthe edges of the graph and all the events involved have also been saved in the vertices: division,


quiescence, and differentiation. Thereby, from any edge in the graph, we can reconstruct the way backto the single cell. This characterization results in the determination of the lineage of any phenotype.Then, using that lineage, we construct the underlying genotype with respect to our model. A geneticprocess is identified by its color. All cells, whose last division is achieved with this genetic process,carry its color, and thus, define a pattern called a gene expression. The color levels during this divisioncycle highlights the status expression. Light colored cells are already divided, and dark cells are newlycreated in the cycle; they are both quiescent. The third category represents the proliferating cells, whichare awaiting division, which are medium colored. Differentiations can occur while generating tissues.Indeed, if a cell has to change a genetic process in order to be able to divide, its color, and that of itsdaughter, are set to the color of the new genetic process. A tissue appearing with one color would meanthat it is made of just one genetic process with no genetic differentiation occurring, ie, all cells wereable to divide using the same ordered sequence of genetic actions.

The action of one cell, x, involves a local morphological dynamic which locally transforms the tis-sue, L, at a local process time. The global morphological dynamic transforms the morphological en-vironment, K, at the end of every cell division cycle after the processing of all cells. The onlyconstraints this growth can face are spatial ones that arise from the morphological environment,and from the cells themselves.

2.3 ATTAINABLE SETS OF PHENOTYPESGiving a mathematical formalization of cells’ actions and behaviors, defining the gene and status ex-pressions, and describing the morphological dynamics, we now use this mathematical foundation toexplore a case study stated as follows. From a single cell,

1. What are the all possible tissues that can be obtained after a given number of division cycles(phenotypes)?

2. For each attained tissue, what are the minimal underlying genetic processes that have governedcell division and differentiation (genotypes)?

2.3.1 IMPLEMENTATIONOur 2D model consists of a grid of automaton elements, which represent our biological cells. The stateof each element is defined by a state vector, including three components that correspond to the featuresof interest in this case study: (i) occupation, ie, an element is either occupied by a cell or is an emptyspace; (ii) cell status, ie, the cell is either in a proliferative state (allowed to divide), quiescent (thenewly created cell is prevented from dividing), or divided (the cell is already processed in the currentdivision cycle); and (iii) cell color that identifies the genetic process that the cell is associated with.The program is developed in C++, sets are represented with the library Boost Graph and the algorithmoperates a redundancy control in sets to ensure that either a phenotype (shape plus gene and statusexpressions) or its different geometrical transformations (symmetries, rotations, and translations)are stored only once (reduced a set by 87%). As the sets’ size increase asymptomatically, we have also

212.3 ATTAINABLE SETS OF PHENOTYPES

developed a parallel implementation with Boost Thread that allowed us a significant gain in executiontime (more that 30x faster). The program offers a view in details on each generated phenotype:

• at each time, which cell was created (tagging system),• which one created it and by which genetic action (symbolized by arrows),• and what is its current cell type (color) and status (color level).

Besides, while generating a phenotype, the underlying genotype is being built gradually. The outputresults are displayed with Scilab where all tissues and their genetic processes generated after thegiven number of division cycles can be visualized. Results are depicted in Figs. 2.2–2.11.

On this basis, we are able to reconstruct the early stages of zebra fish morphogenesis depicted inFig. 2.1 and, most importantly, to shed light on the potential processes that could differentiate cells andhow they could be applied to them at different states (see Fig. 2.5). The program offers the possibility tokeep only tissues passing through some defined paths during their generation (see Fig. 2.6). These pathsare defined in a filter catalog that has to be set at the beginning of the program. The number of divisioncycles after which to apply it also has to be defined.

Turquoise[1]

2

1

FIGURE 2.2

Attainable set of tissues after one division (size 1).

4

44

422 2 22

1 1

3

3 33

11

Turquoise[1 2]

Turquoise[1 3]Purple[4] Turquoise[1 3]Purple[3]Turquoise[1 3]

FIGURE 2.3

Attainable set of tissues after two divisions (size 4). The second tissue shows two colors representing two geneticprocesses: turquoise and purple. Each is an ordered sequence of the four genetic actions (1 is north, 2 is south, 3is east, and 4 is west). These genetic processes are the minimum required to design this tissue. The arrowsdistinguish cell lineage as the creation of the tissue goes on.


FIGURE2.4

Attainab

lesetof

tissues

afterthreedivisions(size61).

1

4

4 4

31

22

1

7

6

6 8

2 4

1 3

7 5

2

6

4 8

17 3

5

2 4

3

5

1

2 4

6 8

3

5

178

32

4

1 3

2

1

3

Turquoise[12]Turquoise[1 3]Purple[4] Turquoise[1 3]Purple[3]

Turquoise[1 3]

Turquoise[1 3 4]Mauve[3 4]Marron[2]

Turquoise[1 3 2]Mauve[3]Marron[2] Turquoise[1 3 4]Mauve[3]Marron[2]

Turquoise[1 3 4]Mauve[3]Marron[2]

FIGURE 2.6

Attainable set of tissues after three divisions with application of a filter catalog after two divisions (size 4).Here, the filter catalog includes just one tissue (tissue stamped with a tick). The paths that go through the tissuesthat are not defined in the filter catalog are discarded (tissues stamped with a red cross).

11

2 2

3 3

4 4

5

6

7

8

2

1 1

FIGURE 2.5

Simulation of the early stages of zebra fish morphogenesis. To obtain the same cell arrangement at the firstthree division cycles as depicted in Fig. 2.1, it requires necessarily that gene and status expressions beunderlying by the following genetic processes: Turquoise[1 3 2], Purple[3], Brown[2].


FIGURE2.7

Attainab

lesetof

tissues

afterfourdivisions(size1029).Weshow

that

with

just

16cells,wehavesomeinterestingpatternsem

ergingin

term

sof

symmetry,bioinspiration,an

drobustness.

Turquoise[1 2 3 4]Purple[3] Turquoise[1 3 4 2]Purple[3 4]Brown[2]

14 12

5913

8

6 2 4

15 10 7

16

14

1 3 11

3 8 12

115 5 9

211 7 13

4

10

6 16

FIGURE 2.8

An example that highlights the link between phenotype and genotype. We can observe how the variations onthe genotypes affect the phenotypes. The two tissues do not have the same sequences of cells division andthen do not have either the same gene expression or the same status expression despite the fact that their overallshape is the same. We have previously addressed this issue in [10]. In that paper, we had explored andsimulated the entire base of all possible genotypes. We showed in the results that two different genotypes inthe base could achieve the same phenotype, which was The French Flag.

Turquoise[1 3 4 2]Purple[4 2 3]Brown[3 2]Green[2]

99 cells15

4

925

8

16 14 6

10

1 3 12

11

7 13

FIGURE 2.9

The result of the tissue growth after six division cycles.


2.4 PREDICTION TOOL BASED ON A COEVOLUTION OF A DYNAMIC TISSUEWITH AN ENERGY DIFFUSION2.4.1 PREDICTION OF TISSUE GROWTHHaving generated all possible phenotypes at the early stages of the embryo and highlighted the under-lying genotype for each, we can now select some of them (by bioinspiration for example) to see howthey could evolve after a given number of division cycles. During the division cycles, cells areprocessed sequentially and their status is updated asynchronously. In the case study of prediction,the second and the third components of the state vector define each element of the automaton changedefined in Section 2.3.1. Indeed, cell status and cell color of the automaton have a newmeaning: (ii) cell

6 cells72 cells

FIGURE 2.10

A scenario of irradiation, which stunts the normal growth of the tumor presented in Fig. 2.9. Indeed, after sixdivision cycles, we have 72 cells instead of 99.

FIGURE 2.11

Tree and leaf phenotypes (made of 1000 cells at least) achieved with directional division instructions definedby the genetic processes.

272.4 EVOLUTION OF A TISSUE UNDER AN ENERGY DIFFUSION

status, ie, the cell is either in a proliferative state (allowed to divide), quiescent (preventing newly cre-ated cells or already processed cells in the current division cycle from dividing), or locked due to a lackof space for division in the four possible directions by setting the evolution lock factor (ELF) of the cellto “on”; and (iii) the cell color that identifies the genetic processwith which it is associated (a dark levelmeans that ELF is “on” and a light level means that ELF is “off”). We stop the growth when wemeet therequired number of division cycles (see Fig. 2.9). This can be used to simulate responses to severalscenarios of radiotherapy in terms of effectiveness to choose the best one. To simulate an irradiationscenario on a tissue, we remove a group of cells before predicting its evolution. Then, we observe theobtained number of cells after a given number of division cycles to compare it with that obtained with-out any cell deletion at the same number of division cycles. The best scenario will be the one where thedeleted cells have been allowed to obtain the best growth slowdown, meaning the smallest number ofcells (see Fig. 2.10).

However, even though plant morphogenesis can be analyzed and understood almost entirely interms of directional cell division and expansion, in growing animals, these mechanisms are by nomeans solely responsible. Cell migration, neighbor exchange, and elective cell death are critically im-portant [11]. Up to now, tissue grows in our model by a mere reading and application of directions ofdivision. This model of growth indeed results mainly in leaf or plant phenotypes (see Fig. 2.11).Therefore, we aim now to link the morphological dynamic with an energy diffusion model to ensurea cellular exchange and death.

2.4.2 ENERGY DIFFUSION MODELTo model growing animals, we introduce energy managing and exchange between cells and their en-vironment. This allows us to define conditions under which cellular mechanisms such as quiescence,mitosis, and elective cell death are held (see Fig. 2.12).

These conditions refer to three available energy thresholds that have to be initialized: minimumEmi,medium Eme, and maximum Ema. At the beginning, we fairly distribute energy to cells’ location. Eachcell has access to an amount equal to Ema. We note En(x), the available energy of cell x at the beginningof a given division cycle n. Prior to reading its genetic process Gx, x compares its current availableenergy E(x) to these thresholds, then behaves appropriately by choosing the corresponding mechanism.The correspondences between threshold and mechanism are described in what follows.

Apoptose Quiescence

E E E E = Ema

EmaEmeEmi

Reading of GR for cell mitosis

FIGURE 2.12

Cellular mechanisms execution according to the available energy.


2.4.2.1 MitosisE(x) cannot go beyond Ema. If it is the case, we set E(x)$ Ema. Otherwise, the cell tries a mitosis with itsgenetic process.

G$Gx and Kn"x# :$!"n,Kn,x;G#

For E(x) use at reproduction, Kooijman proposed the k-rule in his study of dynamic energy and massbudgets in biological systems [12]. This rule assumes that a fixed proportion k2]0,1[ of energymobilizedfrom the reserves is spent on somatic maintenance plus growth while the remaining portion 1 % k onmaturity maintenance plus maturation plus reproduction. Given " and # 2]0,1[, the energy use at mitosisis (as depicted in Fig. 2.13):

• "k for somatic maintenance• #(1 % k) for maturity maintenance

•"1%"#k

1 + $"1%"#kfor cell growth. The parameter $ indicates that each cell type has its own energy

consumption rate for growth. To have a fair consumption rate between cell types, we can set $$ 0.Thus, $ has to be defined for each genetic process or not.

• As the reserve density at birth equals that of the mother at egg formation [12], the remaining

"1%##"1% k# + 2$"1%##2"1% k#2

1 + $"1%##"1% k#is fairly shared after reproduction between x and its daughter.

While x is dividing, each of its neighboring cells y is in a proliferating state. They have hence used apart k0 of their available energy for somatic maintenance and another part #0(1% k0) for maturity main-tenance. The parameters of energy distribution and consumption vary depending on processes that

Structure Reproduction

Reserve E

Assimilation

Growth

Somaticmaintenance

Maturitymaintenance

Reproduction

Cell

(1 ! b ) (1 ! k) + 2d (1 ! b )2 (1 ! k)2

2 [1+ d (1 ! b )(1 ! k)]

b (1 ! k)

1+ d (1 ! a)k(1 ! a)k

ak

FecesFood

FIGURE 2.13

Energy reserve use at mitosis.


the cell achieves according to its current behavior (mitosis, proliferating or quiescent). Thereby, foreach neighbor y, (1 % #0)(1 % k0) is available and x can retrieve an amount inversely proportional

to the number of neighbors of y. For instance, if y has Vy neighbors, x can receive "1%#0 #"1%k

0 #Vy

E"y#.Which is to say, the more x is subject to spatial constraints (great number of neighbors), the moreit gathers energy after reproduction. The rate change of E(x) from one cycle to the other is describedas follows:

_En"x#$"1%##"1% k#+ 2$"1%##2"1% k#2

2 1 + $"1%##"1% k#& '_En"i#"x#+

XVx

j$1

"1%#0#"1% k

0#Vjy

_En"i#"y j#

_En"i#"x# is the rate change of En(x) throughout the cycle n. These changes occurred during the localprocessing of the cells preceding x and include losses due to transfers to its divided neighbors. IfVxd is the number of divided neighbors prior to x and Vx its total number of neighbors:

_En"i#"x#$En"x#%"1%#0#"1% k

0#Vxd

VxEn"x#

_En"i#"y j# is the energy rate change of a given neighbor y j of x throughout the cycle n. These changesmight occur by two behaviors of y j:

1. Mitosis _En"i#"yj#$En"yj#% "k + "1% k## + "1%"#k1 + $"1%"#k

# $En"yj#

%"1%#0#"1% k

0#Vyjd

Vjy

En"yj#+XVyj

l$1

"1%#0#"1% k

0#Vlz

En"zl#

The terms represent, respectively, the available energy for yj at the beginning of the cycle n, thelosses during mitosis, the transfers to divided neighbors, and the gains during mitosis.

2. Quiescence _En"i#"yj#$En"yj#% k00En"yj#%"1%#

0#"1% k0#Vyjd

Vjy

En"yj#

The terms represent, respectively, the available energy for yj at the beginning of the cycle n, thelosses during quiescence, and the transfers to divided neighbors.

2.4.2.2 QuiescenceWhen E(x) is below or equal to Eme, x stays quiescent. We assume that in this state, cells have only toensure somatic maintenance and growth. Indeed, when conditions are poor, allocation to reproductioncan be blocked [12]. Therefore, the use of E(x) at quiescence is

• "0k00 for somatic maintenance• (1 % "0)k00 for structure growth

Which is to say that during this cycle where x stays quiescent, it consumes k00 of its available energy tosatisfy these two processes and 1 % k00 remains. Its local morphological dynamic is given by

G$ f5,…g)Kn"x# :$Kn

_En"x#$ "1% k00 # _En"i#"x#


2.4.2.3 ApoptosisWhen En(x) is as low as Emi or below to it, x is brought to death and its available energy is released.

G$ f6,…g)Kn"x# :$Knnfxg

E"x#$ 0

We have shown that local transitions on the tissue Kn depend on the available local energy. Likewise,the amount of the global energy supplied to cells environment at the beginning of each cycle n, is pro-portional to the cardinal of %"n%1,Kn%1# :$!"Kn%1"x1,…,xpKn%1

#;G#. Indeed, Kooijman asserts in hisbook that reserve and structure grow in harmony, and the specific somatic maintenance costs is somefunction of the structural volume. Thereby, assuming that card %(n % 1, Kn%1) $ N, the energy supplyamount at the location of each cell xi is

Ema%1=NXN

i$1

En"xi#

It may happen to stop the growth before reaching the targeted number of division cycles. This decisionis made when the global energy of the system is below or equal to N

2Emi, meaning that if the globalenergy was fairly distributed to the cells, at least half would be brought to death. In such an energylevel condition, we consider that growth is not safe for the tissue.

2.4.3 RESULTSThe tissues are displayed in a way that emphasizes the available energy for cells. A new color levelscode highlights the status that the cell is ready to get (quiescence, mitosis, or death) according to theinterval where its available energy is situated.

• dark: going to die• intermediate: going to stay quiescent• light: ready to divide

In the deb model, it is taken into account the intraspecies and interspecies variations of parametervalues, which is to say that the energy devoted to the processes of feeding, digestion, maintenance,growth, and reproduction change between individuals of a specie and also from a specie to another.For the implementation of our model, we make the choice to set the parameter values in a way thatallows a fair allocation between processes: k $ 1/2; " $ 1/2; # $ 1/2; k0 $ 2/3; "0 $ 1/2; #0 $ 1/2;and k00 $ 1/2.

In Fig. 2.14, we present the simulations results of several phenotypes among those generated inSection 2.3. We set Ema too high (2000) for these cases.

Then to obtain more specific results, we set the thresholds as follows:

• Emi $ 0.01• Eme $ 0.03• Ema $ 0.08• We set $ 2{%2,%1,1,2} for the different energy consumption rates during growth.


Simulations of the mixed model with some phenotypes have yielded some noteworthy results. Indeed,we have observed that

1. There is no more high proliferation with regular shapes and energy uptake is well observed fromthe outside to the inside of the tissue (see Fig. 2.15).

2. There is a kind of cell sort by the end of the growth when the energy level becomes too low(see Fig. 2.16).

3. Every tissue has a certain number of cycles after which it no longer has enough energy and stopsgrowing.

4. There is a symmetry acquisition during development and the tissue can stay in a homeostasisa certain number of cycles (see Fig. 2.17).

FIGURE 2.14

Simulation of 19 tissues with a high initial energy level. Cell quiescence is enabled.

Turquoise[1 3 4 2]

A

14

16 8 1 3 6

11 2 5

13 4

7

15

9

10

12

FIGURE 2.15

Phenotype A after 18 division cycles, without coupling the directional division dynamic with an energydiffusion, appears regular and reaches 448 cells (in the middle). By the coupling, apoptosis is now possible andafter 18 cycles, the tissue is made of just 54 cells and looks different.


15B

4 7 13

211 5

18

16

3 12

14 6

10

9

Turquoise[1 3 4 2] Purple[4 2 3] Brown[3 2] Green[2]

FIGURE 2.16

Phenotype B after 23 division cycles is made of 10 cells and does not have enough energy to pursue growth.We observe that the cell are separated into top, middle, and down groups (or head, body, and legs).

(A)

(C)

(B)

(D)FIGURE 2.17

Phenotype A acquires a symmetry at the 20th (A) and the 21st (B) cycle with 27 and 39 cells, respectively.Then at the 26th (C) and the 27th (D) cycle, the same scenario recurs with the same number of cells. However,energy is not spread in the same way in (D) as in (B).


2.5 DISCUSSIONTo study and understand the developmental process at the early stage of the embryo when it containsjust a few pairs of cells, spatial choices play a key role. In support of this hypothesis, we have presenteda mathematical model based on viability theory where the morphological dynamics depend on choicesof division directions through time and under some logical constraints. Major cells’ behaviors wereable to be formalized mathematically in the model and implemented using a cellular automaton-oriented multiagent system. Indeed, we use this model to generate the spaces of possible phenotypeat the early stages of embryo and their associated genotype. Due to this systematic simulation, despitethe simplicity of the model, the dedicated algorithms pose significant challenges on computing. First,representation of the sets of evolutions over time requires huge memory. In addition, running these setsis often time-consuming. However, the existing computing powers, particularly with a parallel imple-mentation on multicores processor, have opened some avenues. Thanks to this work, we highlightthe link between phenotype (cells placement and cells differentiation) and genotype (cells program).In previous work [9], we also pointed out the importance of constraints in cell development. Indeed, byrestricting the morphological environment or by imposing a commutative evolution to the tissues, thepossibilities of growth had been significantly reduced.

We have shown that the application could be, on the one hand, to determine the genotype of anyform of tissue (pathological tissue mainly), and on the other hand, the use of this genotype to predicttheir evolution in later times (pattern prediction and growth controlling). For this purpose, we havedesigned an original prediction tool in which the morphological dynamic of tissue coevolves withan energy diffusion model. The main result is the enrichment of the patterns. An interesting applicationof this coupling could be caloric management to reduce tumor growth. Indeed, this latter depends on thenutritional status of the host [13]. Hence, this would be helpful for evaluating the energy reserve thatbetter stunts the tumor’s growth. Then, according to that, the diet of the host could be controlledconsequently.

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model. In: Proceedings of the 14th annual conference on genetic and evolutionary computation, GECCO ’12.Philadelphia, PA: ACM; 2012. p. 553–60.

[2] Peyrieras N. Morphogen!ese animale; 2006; p. 179–201.[3] Chavoya A, Duthen Y. An artificial development model for cell pattern generation. In: Randall M, Abbass H,

Wiles J, editors. Progress in artificial life. Lecture notes in computer science, 4828. Berlin/Heidelberg:Springer; 2007. p. 61–71.

[4] Muller G, Newman S. Origination of organismal form: beyond the gene in developmental and evolutionarybiology. Cambridge, MA: MIT Press; 2003.

[5] TuringA. The chemical basis ofmorphogenesis. Philos Trans R Soc Lond Ser BBiol Sci 1952;237(641):37–72.[6] Melani C, Peyrieras N, Mikula K, Zanella C, Campana M, Rizzi B, et al. Cells tracking in the live zebrafish

embryo. In: Engineering in Medicine and Biology Society, 29th Annual International Conference of theIEEE, 1; 2007. p. 1631–4.

[7] Olivier N, Luengo-Oroz MA, Duloquin L, Faure E, Savy T, Veilleux I, et al. Cell lineage reconstruction ofearly zebrafish embryos using label-free nonlinear microscopy. Science 2010;329(5994):967–71.


[8] Aubin JP. Viability theory. Basel: Birkhauser; 1991.[9] Sarr A, Fronville A, Rodin V. Morphogenesis model for systematic simulation of forms co-evolution with

constraints: application to mitosis. In: TPNC 2014, 3rd international conference on the theory and practice ofnatural computing, 8890. Berlin/Heidelberg: Springer; 2014. p. 230–41.

[10] Sarr A, Fronville A, Ballet P, Rodin V. French flag tracking by morphogenetic simulation under develop-mental constraints. In: Formenti E, Tagliaferri R, Wit E, editors. Computational intelligence methodsfor bioinformatics and biostatistics. Lecture notes in computer science, 8452. Berlin/Heidelberg: SpringerInternational Publishing; 2014. p. 90–106.

[11] Davies JA. Chapter 22 — growth, proliferation and death: a brief overview. In: Davies JA, editor. Mecha-nisms of morphogenesis. 2nd. Boston: Academic Press; 2013. p. 283–305.

[12] Kooijman S. Dynamic energy and mass budgets in biological systems. Cambridge: Cambridge UniversityPress; 2008.

[13] van Leeuwen IMM, Zonneveld C, Kooijman S. The embedded tumour: host physiology is important for theevaluation of tumour growth. Br J Cancer 2003;89(12):2254–63.

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Emerging Trends in Applications and Infrastructures for ...lab-sticc.univ-brest.fr/~rodin/Publis/PDF/Sarr16b.pdf · their morphology. Artificial regulatory networks have also been