A Viability Approach for Robustness Measurement ...

A Viability Approach for Robustness Measurement,

Organizational Autopoiesis, and Cell Turnover

in a Multicellular System

ABDOULAYE SARR,1 ANYA DESILLES,2 ALEXANDRA FRONVILLE,1 and VINCENT RODIN1

ABSTRACT

In this article, we use the potential of computational biology to highlight the key role of cellapoptosis for studying some tissue’s properties through in silico experiments of morphogen-esis. Our morphogenesis model is a new approach focusing on the deterministic programwithin cells that controls their placement and their differentiation at the beginning of theembryogenesis. Indeed, when the tissue is made by just a few pair of cells, we consider thatcellular mechanisms are related neither to the influence of mechanical forces nor to the spreadof chemicals. Dynamics are based on spatial and logical choices, the other factors beinginvolved when the tissue contains a large number of cells. We had established a mathematicalformulation of such a model and had enlightened the link between phenotype (cell placementand cell differentiation) and genotype (cell program) at the early embryogenesis. Indeed, thatwork allowed for generating any early tissue and the associated program that designs it. Wepropose now to study and assess some properties of these tissues for further selection andclassification purposes. More precisely, we present in this article novel methods to measuretissue robustness based on the backward morphogenesis of our model. We also show someimplementations of their self-maintenance properties, on the one hand to deal with environ-ment disturbances through autopoiesis and on the other hand to achieve a dynamical steadystate which ensures tissue renewal.

Key words: apoptosis, autopoiesis, computational biology, homeostasis, morphogenesis, ro-

bustness, viability theory.

1. INTRODUCTION

Apoptosis is a genetically programmed cell death that occurs during embryogenesis, and then

metamorphosis (Nagata, 1997). It is involved in the acquisition of shape for the living from the egg to

the adult. During embryogenesis, the fertilized egg produces billions of cells, by series of cell division and

differentiation, to make a body. These mechanisms continue after birth until the maturation through meta-

morphosis processes. And even in adults, senescent* cells die and are replaced by new cells allowing to

maintain the body. There are two pathways known to trigger apoptosis:

1UMR CNRS 6285, Lab-STICC, CID, IHSEV, Computer Science Department, Universite de Brest, Brest, France.2ENSTA ParisTech, Palaiseau, France.*Senescence is the biological aging process; it is the result of irreversible changes in an organism that result in death.

JOURNAL OF COMPUTATIONAL BIOLOGY

Volume 23, Number 4, 2016

# Mary Ann Liebert, Inc.

Pp. 256–269

DOI: 10.1089/cmb.2015.0187

256

� severe damage or stress of the cell initiates an intracellular pathway,� extrinsic signals induce stimulation of receptors at the cellular surface (Schleich and Lavrik, 2013).

Apoptosis is a crucial mechanism to maintain life through development, homeostasis, and autopoiesis.

The review in Suzanne and Steller (2013) gives some evidences about the influence of apoptotic cells on

embryogenesis, metamorphosis, and maintenance. Otherwise, defects in apoptosis may result in serious

diseases like cancer, which is particularly characterized by an abnormal cell growth. And apoptosis of

infected cells can limit virus production or spread.

Studies of apoptosis in systems biology are mainly focused on signaling events. Many mathematical and

computational models have been developed to account for this complex system of cell communication and

molecular interactions. For instance, Monte Carlo mathematical models have been used to simulate inherent

stochasticity of signaling reactions in complex signaling pathways (Raychaudhuri et al., 2008). Two-

dimensional (2D) cellular automata have been used to simulate two anti-cancer strategies that modulate a

transmembrane protein belonging to the family of the tumor necrosis factors (Apte et al., 2010). Agent-based

modeling is another approach where each component of the model is represented by an agent with a certain

behavior based on its biological functions (Schleich and Lavrik, 2013). We adopt such a model in this work, but

we restrict ourselves to the cellular level and we do not focus on the factors that lead to cell apoptosis. Although,

systems biology approaches like those presented above contribute significantly to the understanding of this

death signaling network. However, this article is about shape emergence rather than about biochemical events;

we try to highlight the effects and the significances of cellular apoptosis on the tissue and its properties.

To pinpoint the effects of cellular apoptosis on the viability of a multicellular system, we develop some

experiments through in silico modeling. In silico experiments overcome the hurdles related to in vitro

experiments: complexity, scales, time, etc. This allows us to understand the biological systems through a

cyclic workflow that starts by the mathematical modeling of the system. Then, experimental procedures are

defined in the model to study and understand some biological processes of the system. Finally, these pro-

cedures are implemented in computer programs and tested. The aim is to be able to reproduce some mech-

anisms of the real system. At that stage, either the model is reliable and then provides a better understanding

and control on the biological system or needs to be reset, improved, reimplemented, and tested again.

The computational experimental procedures that we present in this article are based on a mathematical

model of morphogenesis relying on the viability theory concepts (Aubin, 1999). In mathematics, the

viability theory offers concepts and methods to control a dynamical system in a given fixed environment in

order to maintain it in a set of constraints. Applied to morphogenesis, this means that by using some

controls (mitosis, quiescence, apoptosis), we should have in each step at least one possible evolution of

cells through their environment. This dynamic of cells causes some biological effects on the cells indi-

vidually (state) but also impacts the overall tissue (morphology). At early embryogenesis where the tissue is

just made of a few pairs of cells, we consider that cell division and cell differentiation instructions are in

terms of spatial and logical rules. We include neither the biomechanical forces between cells and the

extracellular matrix (Henderson and Carter, 2002), nor the intercellular reaction-diffusion of molecules

(Turing, 1952). These factors have been widely studied but have yet to give relevant insights on how

division and differentiation arise. Therefore, we made assumptions in abstracting from them so as to focus

on the epigenetic factors. Instead, we establish the mathematical formulation of these cell regulations called

genetic actions (mitosis following a direction, quiescence, apoptose) and how they control both the cell

behaviors and the morphological dynamic of tissues in the environment. This dynamic is intended either to

modify the tissue or to maintain it.

For the first type of dynamic, in Sarr et al. (2014) we gave an implementation of tissue growth based on

cellular mitosis, quiescence, and differentiation. Thereby, from a given tissue, we were able to generate all

its possible evolutions through time (phenotypes) as well as the underlying genetic processes. The genetic

processes associated with a tissue are the minimal set of ordered lists of genetic actions needed to achieve it

(more details in section 2). They are useful to restore a normal pattern in response to a pathological,

accidental, or artificial disturbance, or even to direct a morphogenetic process. In viability, that was an

attainability problem that consists to seek, from an initial state, all the ones possible to reach by viable

evolutions after a given horizon time. By varying the spatial and logical constraints of cells, we showed

how the attainable sets were restricted from any initial tissue. In this article, besides cell division we

introduce cellular death mechanisms, senescence, and cellular regeneration. These cellular mechanisms are

implemented through in silico experiments to study some properties of tissues.

A VIABILITY APPROACH FOR MULTICELLULAR SYSTEMS 257

After setting in section 2 the mathematical foundations required to study tissue viability, we present in section

3 a first experiment that measures the robustness of tissues. It is quantified as the number of paths that can lead to

a specific tissue phenotype. This work is based on a backward morphogenesis where we remove cells suc-

cessively to reconstruct all possible previous states of the tissue. The cellular death involved is the programmed

cell death or apoptosis. The second experiment shows how a tissue compensates random mutations induced by

the environment to maintain itself, a property called autopoeisis (section 4). In this experiment, we have cell

proliferation and cell death by accident but feedback controls on cell regulation are triggered to reconstruct the

tissue. Finally, we provide an implementation of a dynamical steady-state that allows tissues to keep a con-

tinuous internal dynamic that renews them without changing their shape (section 5). The tissue acquires this

property thanks to a continuous process of cell death and cell regeneration. The cellular death we are referring to

in this latter experiment is a death by aging. In section 6, we present some ideas about the relevance of such a

model to contribute to the efficiency of therapies for apoptosis diseases like cancer (section 6).

2. VIABILITY CONCEPTS FOR MORPHOGENESIS

Rely on the viability theory to tackle issues in morphogenesis requires first to properly define some

concepts of this theory in the case of multicellular systems. In Sarr et al. (2014), we described mathe-

matically the state, controls, and both local and global morphological dynamics of tissues. Some points of

that formalization are highlighted in this section.

K � P(X){ denotes the morphological environment{ (X = R2 denotes the set of containment cells,

contained in the complement of vitellusx).

Cells x˛X W; are either characterized by their position (living cells) or by their death made of tissues L,

which are subsets of cells (L 2 P(X)).The subset of eight 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)‚ ;g

A is made of the six geometric directions, the origin, and the empty set. Here, we restrict morphogenesis in

the plan:

A := f(0‚ 1)‚ (0‚ - 1)‚ (1‚ 0)‚ ( - 1‚ 0)‚ (0‚ 0)‚ ;g

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

A := f1‚ 2‚ 3‚ 4‚ 5‚ 6g

These genetic actions allow for describing cell behaviors:

1. Transitions x1x + d, where d ˛{1, 2, 3, 4} (action)

2. Quiescence x1x + 5 = x (no action)

3. Apoptosis x1x + 6 = 6 (programmed cell death)

A genetic process g is a possible combination of genetic actions g := fd1‚ . . . ‚ dig 2 Ai. Operating a

genetic process under a given criterion, either for migration or for division, means that the process scans

successively x + d1, ., x + di until the first time when the criterion is satisfied. For every tissue (phe-

notype), there is a set of specific genetic processes (genotype) that helps achieve it starting from a single

cell. We had generated all the phenotypes that can be obtained after any number of division (1, 2, 3, or 4) of

a single cell. The evolution is saved and characterized using our model for all the following mechanisms:

division, quiescence, differentiation, and lineage. This characterization results in the determination of the

underlying genotype for any phenotype (see example in Fig. 1). A genetic process is identified by its color

and all cells whose last division is achieved with that genetic process carry its color. Thus, they define a

group of cells of the same genetic process, same color. In this article, we study the viability of those

generated tissues using apoptosis.

{Supplied with the structure of max-plus algebra for the operation W and + (where K + ; : = ; with K a cell tissue).{For instance, K := fK � Mg is the family of subsets contained in a given subset M.xIn biology, the vitellus is the energy reserve used by the embryo during its development.

258 SARR ET AL.

3. MEASUREMENT OF TISSUES’ ROBUSTNESS

Robustness is an important property of a biological system. It ensures the stability of the system with regard

to disturbances. In the case of morphogenesis, we consider that a tissue is robust if it emerges through several

different paths. We can rebuild these paths to determine the set of possible states of the tissue at any earlier

time. An ordered sequence to remove successively a given number of cells from a tissue results in one of its

previous states. This process, known as backward morphogenesis, gives a picture of the developmental

pathway by which we can reconstruct all the switching points that the tissue has been passed through. The

FIG. 1. One of the 1029 possible phenotypes and its genotype obtained after four divisions of a single cell. The tissue

shows three genetic processes: turquoise, purple, and brown. They are ordered sequences of the four genetic actions

that the cells have applied to design the tissue (1: north, 2: south, 3: east and 4: west). We also represent cell distinction

depending on their status during the division cycle (proliferating, divided, or quiescent). Besides, the three genetic

processes show that two differentiations occur while generating the tissue. Indeed, if a cell changes its current sequence

of genetic action to be able to divide, it is associated to a new genetic process and its daughter too. A tissue appearing

with one genetic process would mean that no differentiation occurs; that is, all cells were able to divide using the same

ordered sequence of genetic actions. The arrows distinguish cell lineage as the creation of the tissue goes on.


forward morphogenesis is given by the inverse sequence where the cells are put back by division from that

previous state to the final tissue. As cells can be added to a system by proliferation, so can they be removed by

death (Davies, 2013). Hence, we will talk about cell death or apoptosis instead of cell removal.

For a given tissue, determining such states requires establishing a system of tissue sets’ evolutions whose

mechanisms are on the one hand a backward morphogenesis, and on the other hand a forward morpho-

genesis. Robustness of tissue L in a given time t (R(L‚ t)) is defined by the number of states that L gets at t.

A state of a tissue L at t is identified by a unique pair comprising an ordered sequence of cell death, which

defines a possible shape of L at t (Lt) and a sequence of proliferations that indicates a possible evolution

path from Lt to L. The more L has states at t, the more it is robust at t. That means if there were constraints

to disrupt a path from an Lt to L, there would be other possible alternative choices for L to emerge. Low

robustness reports a weak evolution for the tissue with little alternative paths.

The algorithm for tissue robustness computing is as follows:

Suppose that we seek the robustness of L0 at tm: R(L0‚ tm)

Initialization

Sn = ;, cn ˛[0, tm], all the sets are empty,

L0 denotes the original tissue and will compose the initial set S0,

S0 = S0 W L0 and n = 0

Iteration

For i from 0 to card(Sn) – 1

cLi ˛Sn, c pair of cells (x‚ x0) 2 Li, if there exists d 2 A so as:

� w(n, Li, x, x0, d) is related,� w(n, Li, x, x0, d) is not redundant in Sn+1,� and there is an evolution path from w(n, Li, x, x0, d) to L0

w(n, Li, x, x0, d) represents the tissue Li after apoptosis of cell x at n, where x is the result of the transition

of x0 with the genetic action d (x0 + d = x). Apoptosis of x is recorded in the current state and x0 as a

corresponding regenerator.

And Sn + 1 = Sn + 1 [ w(n‚ Li‚ x‚ x0‚ d)

n = n + 1

End of iteration

The algorithm stops if

n + 1 = tm

and then R(L0‚ tm) = card(Sn + 1) .

While the backward morphogenesis is operating, there is a status expression in the tissue to identify cells.

The possible states we define here are: senescent, dead, and regenerator cells. At the beginning of cycles

(duration time for all cells to change status), all cells are marked ‘‘senescent.’’ When an apoptotic event

occurs, the dead cell is marked ‘‘dead’’ while its regenerator is marked ‘‘regenerator.’’ Status changing for

cells is possible once in a cycle. We have defined a color for each status:

� senescent: green� dead: blue� regenerator: red

We implement this model in C++ as a grid of automaton elements that represents our biological cells.

The state of each element is defined by a state vector including two components that correspond to the

features of interest in this case study: (i) occupation, that is, an element is either occupied by a cell or is an

empty space; (ii) cell status, that is, the cell is either senescent, dead, or regenerator. After setting the initial

tissue on the grid and launching the backward morphogenesis algorithm described above, the state vector

changes due to the defined rules while the algorithm is processing. The data structure is a graph im-

plemented with the library Boost_Graph. The backward and forward morphogenesis operations, that

is, the successive rules applied by the automaton elements, are recorded in the edges, and the nodes record

260 SARR ET AL.

the tissues’ shapes, that is, the successive configurations of the automaton. Tissue is visualized in Scilab.

As we focus on the early morphogenesis, we will be mainly interested in tissues of a few pair of cells.

Likewise, phenotypes and their associated genotypes we generated in previous works (Sarr et al., 2014) are

restricted to 2, 4, 8, and 16 cells. Their size was respectively 1, 4, 61, and 1029. However, these results were

achieved by building and processing very large tissue sets that grow asymptotically. For example, generating

all the possible phenotypes of 16 cells on a 2 · 2.50 GHz computer took 38 hours. Backward morphogenesis

is applied to these same phenotypes and presents almost the same properties in terms of time consumption.

We show in Figure 2 that the robustness of a square tissue made up of four cells Q at t = 2 is 8. Aside

from time consuming, we choose basic examples with few cells for readability reasons, to show distinctly

which cell has been removed at which time during the backward processing, and which one will regenerate

during the forward processing. According to the viability theory (see Fig. 3), these eight tissues are the

capture basin of two-cell tissues from which it is possible to obtain Q by successive transformations in two

steps, knowing that each transformation remains in the tissues’ viability domain. A tissue is viable after

transformation if it remains unique and related (not split).

The program can also check if a tissue L has passed by a shape L0 throughout its evolution. If this is the

case, it generates all the states having this shape L0 with their forward morphogenesis operations allowing to

move from L0 to L. This highlights the importance of programmed cell death in the sculpture of organs.

Thus, if one aims to obtain a tissue B from a tissue A, the algorithm provides the different existing ways to

sculpt A so as to get B thanks to apoptosis (see Fig. 4). In a viability theory point of view, setting a target in

the states space is an additional constraint that reduces the size of the capture basin. Without this additional

constraint, the size of the capture basin of the eight-cell tissues from which it is possible to obtain ‘‘Origin’’

by successive transformations in eight steps while satisfying tissue viability constraints is 2004. Due to the

additional constraint in the states space, the size is drastically reduced to five. A relevant application could

be to know more about how the programmed cell death sculpts organs in vertebrates. The development of

mammalian hand and foot provides a well-studied example in which elective cell death plays a direct role

in morphogenesis (Davies, 2013). Indeed, the free digit formation results from the elimination of the

interdigital tissue by apoptosis. This mechanism was widely reviewed in Hernandez-Martınez and Cov-

arrubias (2011). As we set the principles of our model, the authors assume that separation of digits is

possible due to a promotion of tissue regression and a restriction of tissue growth. However, they take into

account the components of the apoptotic machinery found in the interdigits and which are supposed to

underlie these two processes. Since the role of these components in the initiation and the execution of cell

FIG. 2. R(Q, 2): the eight states of Q at t = 2. The red part (Q2) represents a possible shape of Q at t = 2 after

successive apoptosis by the backward morphogenesis, and the arrows indicate the necessary proliferations to recreate Q

from Qt by forward morphogenesis.


death is yet to be defined, they have studied the interactions between the factors that activate the interdigital

cell death. Another example of patterning with apoptosis is seen in leg joint formation in Drosophila, in

which localized programmed cell death in a monolayer epithelium is required to create a fold in the

epithelium (Manjon et al., 2007).

Furthermore, we found that the more cells that are interconnected in the tissue, the more robust the tissue.

Conversely, tissues having low robustness are those which have low interconnection between cells. Thus,

we defined a parameter that measures this degree of connectivity between cells for a given tissue. It is given

by :P4

i = 1 i · number of cells having i neighbors.

Then we implemented a program that computes for all eight-cell tissues, the robustness at time tm = 4,

and their cells’ degree of connectivity. Then, it displays the dependency diagram of these two parameters

on the population of the 61 tissues (see Fig. 5). To better illustrate the intensity of the relationship that

FIG. 3. J.P. Aubin has defined the capture basin of a dynamic system as depicted. The x1 belongs to the capture basin

because there is at least one evolution from it that reaches the target in finite time while respecting the viability constraints.

The x0 belongs to the viability kernel but does not belong to the capture basin; any evolution from it reaches the target. The

x2 neither belongs to the capture basin nor to the viability kernel; any evolution from it satisfies the constraints. In our case,

the studied system is an evolving tissue; its state is defined by its current shape. The set of possible shapes of tissue defines

the states space. A dynamic is observed when there is a transformation of the tissue under the actions of its cells. The

constraints are set as the subset of the admissible shapes. Relying on these basic definitions, the viability kernel of a given

tissue can be seen as the set of initial shapes, from which we can find a sequence of transformations, each resulting in a

shape that respects the constraints. For the capture basin, we have to define first a subset in the set of constraints that will be

the target. Thereby, the capture basin of a given tissue is determined by the subset of initial shapes from which there is at

least one sequence of transformations on the one hand, where each results in a shape that meets the constraints and, on the

other hand that reaches, in a finite time horizon, a shape belonging to the target.

262 SARR ET AL.

exists between the two parameters, we calculated the correlation coefficient. It is 0.78, which indicates a

strong and positive correlation between the robustness of a tissue and the degree of connectivity of its cells.

Figure 6 represents the tissue with the highest degree of connectivity and robustness in the set, then a tissue

having an intermediate robustness and degree of connectivity, and finally the tissue with the lowest

robustness and degree of connectivity.

Furthermore, for a given tissue made up of 2n cells, the program can verify its compliance. That means it

could be generated with the morphological dynamics defined in our model. To do so, we check by a

backward morphogenesis if it has at least one state at time t – 2n–1. Otherwise, the tissue is not compliant

with respect to our model. An example is depicted in Figure 7. The first tissue of 16 (24) cells, randomly

designed, has no state at t – 24–1; consequently, it is not compliant. Whereas, by modifying the position of

only one cell in the tissue, the resulting tissue appears compliant with 232 states found. That would mean at

t – 8, the tissue had 232 different ways to evolve into its current shape.

FIG. 4. Algorithm has found that the tissue ‘‘Origin’’ has passed by the shape ‘‘Target’’ through its evolution. It has

generated the five states associated with ‘‘Target.’’ Each state represents a unique way to sculpt ‘‘Origin’’ in order to

obtain ‘‘Target.’’ During the process of interdigital tissue elimination, the role of apoptosis can be compared with the

work of a stone sculptor who shapes stone by progressively chipping off small fragments of material from a crude

block, eventually creating a form (Suzanne and Steller, 2013).

FIG. 5. Dependency diagram between the robustness and the cells’ degree of connectivity on the population of eight-

cell tissues.


4. AN AUTOPOETIC SYSTEM

Francisco Varela, Humberto Maturana, and Ricardo Uribe were the first to demonstrate the use of a

computational model to introduce and explore the concept of autopoiesis in order to characterize living

organisms. They define it as a form of system organization where the system, as a whole, produces and

replaces its own components and discriminates itself from its surrounding environment (Varela et al.,

1974). According to this definition, the living cell seems to be a paradigmatic model of an autopoietic

system. Thus, we implement an autopoietic tissue where cells as agents, proliferate, die, or stay quiescent

depending to some surrounding random disturbances. Then, the autopoietic tissue has the property of

removing superfluous cells and replacing the dead cells by compensatory proliferation of their neighbors.

These processes may continuously follow one another, but the tissue always has to remain constant.

Indeed, cell death in real life can be an important process during development that serves to remove

superfluous and potentially harmful cells. Likewise, in stress conditions, apoptotic cells can produce mitotic

signals to induce compensatory proliferations from the surrounding living cells so as to reshape the tissue

(Fan and Bergmann, 2014).

Unlike an autonomous entity, an agent has the ability to perform a broad range of behaviors supple-

mental to processes of self-maintenance. In the model depicted here, each cell of the tissue performs

FIG. 6. Tissues with highest, intermediate, and lowest robustness and cells’ connectivity degree.

FIG. 7. (Left) The tissue is not compliant with regard to the morphological dynamic of our model; it has no state at

time t – 8. (Right) The tissue is compliant and one of its 232 states at time t – 8 is depicted here.

264 SARR ET AL.

apoptosis, mitosis, or quiescence in one step, but at the following step, the overall shape of the tissue has to

return to normal thanks to compensatory proliferations and anoikis**. Thus, there are mechanisms un-

derlying the behaviors of the cells on the one hand and on the other hand those assuming maintenance of

the spatial configuration of the tissue. Knowing that the overall shape of the tissue must remain the same,

find all the possible feedback controls to overcome the epigenetic variations of the cells appears as a

viability issue. In our model, solve this issue translates into establishing, during the autopoietic mechanism,

the possible regulations of each cell by its genetic process:

� The genetic process of the cells that will be brought to die differentiates to a genetic process starting

with the death genetic action (d = 6). To be ready for compensatory proliferations, all their neighbors

come out of their quiescence (d 2 Anf5‚ 6g). Indeed in living, the ability of apoptotic cells to secrete

mitogenic factors to their surroundings has a key role in wound healing and tissue regeneration.� The cells whose epigenetic variation leads to mitosis leave immediately their quiescence state

(d 2 Anf5‚ 6g), and in a context of shape maintaining, as any added cells in the tissue can be

considered superfluous, all the proliferating cells produce daughter cells differentiated in a genetic

process starting with the death genetic action (d = 6). Indeed, accumulated errors can result in a

developmental mistake, typically a cell finding itself where it ought not to be. However, cell survival

relies both on their diffusible signals such as growth factors and survival signals transduced by specific

matrix receptors. This allows cells, to detect, at high spatial resolution, whether they are placed

correctly and to kill themselves if they are not (Davies, 2013).

An example of execution on a tissue made of 1023 cells is presented in Figure 8.

5. TOWARD A DYNAMICAL STEADY STATE

In the last experimental procedure we present in this article, it is not the superfluous cells that the tissue

removes but the dying cells due to their age. Their elimination remodels and renews continuously the tissue

without modifying it. We call this process the dynamical steady state (DSS). It designates here the

processes that maintain the global shape of the tissue regardless of environment and time. It relies on an

orderly continuous mechanism of apoptosis and mitosis following certain rules.

Nevertheless, there are some studies on the dynamical mechanisms that maintain the internal state of a

system facing to external constraints (homeostasis). Study computationally the homeostasis of a system

may consist first to provide, for instance, a model of its morphological, physiological, chemical, and/or

kinetic characteristics. This model could then be implemented and ran to find suitable parameters able to

FIG. 8. (A) The original tissue made of 1023 cells. (B) The tissue after wrong proliferations and necrosis due to epigenetic

mutations that affect their genetic processes. For any disturbance occurrence, the number of mutations in the tissues, the type

of mutations, and the concerned mutated cells are completely randomized. (C) Processing of the tissue’s autopoietic

mechanism by the application of all possible feedback controls that impact genetic processes of cells so as to compensate

disturbances. Superfluous and potentially harmful cells are removed by anoikis (red) whereas the dead cells are replaced by a

compensatory proliferation of their neighbors (blue). For a given dead cell, the feedback control indicates how each

surrounding living cell could regenerate it (left, right, up, or down blue arrow). (D) Tissue has returned to normal.

**Elective cell death caused by displacement from a normal tissue environment (Davis, 2013).


keep a steady-state of these integrated characteristics. Smith et al. (2013) reviewed some existing com-

putational models of the Achilles tendon tissue within a conceptual framework designed for its homeo-

stasis. The main issue is to maintain the tendon’s structure and function, knowing that the tendon is

continually damaged during use. A mathematical model of whole-body metabolism was proposed in Kim

et al. (2006) to predict glucose homeostasis during exercise. While the body performs exercise, changes in

hormonal signals are induced to modulate, in a coordinated way, metabolic flux rates of various tissues so

as to maintain blood’s glucose constant. Unlike these works, we study the steady-state of tissues’ shape by

using only cellular mechanisms regardless of environment and time.

Scaling up to another level of the biological organization such as the organism, it is the DSS that maintains

the shape of the overall body. Without this mechanism of death/birth, life would not be possible. In Spalding

et al. (2005), it is shown that tissues have different rates of cell turnover. The epithelium of the human colon

turns over at least once per week throughout life. Therefore, by age 60, a person has been through at least

3,000 replacement cycles. Likewise, the epidermal layer of the skin turns over about every 60 days. Intestinal

epithelial cells have an average lifespan of about 5 days. Unlike epithelial cells, many nonepithelial cells in

the gut are likely to be long-lived. For example, the average age of cells in the intestine is 10.7 years and 15.1

in the muscle cells of the intercostal skeletal. Because of the fact that cell turnover would be at high risk for

accumulating mutations and progressing to cancer (Frank, 2007), mainly for epithelial tissues that are short-

lived and renew continuously throughout life, indicating a way for any tissue to renew its cells seems of

utmost importance. Indeed, about 90% of human cancers arise in epithelial tissues.

Having all tissues of n cells, we would like to maintain any one of them in a DSS to ensure continuously

their cell turnover. The viability issue consists of finding in finite time for any tissue the maximum period of

senescence required for its cells and the right cell turnover sequence in order to renew entirely the tissue

without changing its overall shape. The algorithm that solves this issue by finding both the suitable value of

this parameter and the right cell turnover sequence is as follows:

1) At time t0 (initial state), all the cells have a senescence period ore age (SP) of 0. And the maximum

senescence period authorized (MSPA) is initialized to 0. The number of renewal (NR) is also initialized to 0.

2) There is a necrosis cursor (NC) that points to the position of the last dead cell, and initially it points to

the first cell of the tissue.

3) At each time, a single turnover can take place in the tissue, thus there is one action of necrosis/

proliferation that is the only dynamic of the tissue.

4) For necrosis, we choose the first cell encountered from the NC among the oldest, those with the

greatest SP. There is a necrosis counter, number of necrosis (NN), initialized to 0, which is in-

cremented each time a necrosis occurs.

5) For the proliferation compensating the dead cell, we scan the surroundings of the dead cell in a

clockwise fashion (right, down, left, up). The first cell met with the same SP as the dead cell

performs the proliferation. There is a proliferation counter that indicates the number of renewed cells,

number of proliferations (NP), initialized to 0 and incremented each time a cell is renewed.

6) We update the ages of cells SP. Cells involved in the last turnover have their SP reset to 0. The too

old cells, with an SP equal to the MSPA, are driven to apoptosis. When a cell dies by apoptosis, it

will not be renewed. SP of any other cell, different from these, is incremented.

7) After updating SPs, we check that there is no apoptosis in the tissue. If there is, it means that the

current MSPA cannot maintain the tissue in a DSS. Thus the MSPA is incremented and steps 2 to 6

are repeated.

8) When the NN has a value equal to half the number of cells, there can be two cases:

(a) if NN = NP, meaning that all the dead cells have been renewed, thus the DSS of the tissue is

achieved with the defined MSPA and cell turnover sequence. The tissue has come to be com-

pletely renewed; NR is incremented.

(b) if NN > NP, meaning that there are some dead cells that are not regenerated, thus the DSS is not

achieved due to a wrong cell turnover sequence. Then we reverse the cell turnover sequence and

neighbors scanning sequence of dead cells. This also means to operate a rotation of 90 degrees of

the tissue if we want to keep the sequences. Then steps 2 through 6 are repeated.

The algorithm can be performed on a tissue of any number of cells. For instance, in a tissue of 1060 cells,

we observe that the MSPA was increased successively from 0 to 529 with furthermore a rotation before

obtaining a DSS. Since we do not have yet an Interactive Graphical Simulation Environment, the

266 SARR ET AL.

dynamical behavior of tissues through each of these steps cannot be visualized. However, when run, the

program displays the results—either the MSPA has to be incremented, or the tissue is renewed or rotated.

To better shed light on how the algorithm actually proceeds on tissues, we present two examples with a few

pair of cells. Figure 9 presents a case of low MSPA value, and Figure 10 presents a case in which the

MSPA is high enough but the tissue is scanned the wrong way for cell turnover.

FIG. 9. At this step, the MSPA has been increased to 2. After an initialization and successive operations of necrosis/

proliferation followed each by SP update operation, the algorithm notices in (a) that there is apoptosis in the tissue. It

thus considers that the current MSPA cannot put the tissue in a DSS and hence increases it.

FIG. 10. At this step, theMSPAhas reached 3 and (a) is the 3rd necrosis/proliferation action. In (b),NN= 4 (number of cells/2)

but NP= 3, due to a wrong cell turnover sequence. Thus, in (c) the algorithm performs a rotation of the tissue and resets the

parameters. At (d), we have the 4th necrosis/proliferation action (NN= number of cells/2 =NP). In (e), with anMSPA= 3, after

SP update, the algorithm notices that there is no apoptosis in the tissue. Thus, it considers that theDSS of the tissue is achieved

with the current MSPA and cell turnover sequence. The tissue has come to be completely renewed; NR is incremented to 1.


6. CONCLUSION AND FUTURE WORKS

Based on a backward morphogenesis, we have defined a notion of tissue robustness that is highlighted to

be intricately linked to how the cells are interconnected. We have noticed relevant applications using

backward morphogenesis such as reconstructing previous states of a tissue, patterning tissues, or even, most

importantly, putting light on the controls that could lead to shrinkage, in the case of cancer for example.

Then, we have designed an autopoietic tissue that is able to deal with random epigenetic mutations that

result in proliferation and apoptosis in a tissue. The mechanism relies on the determination and application

of all the possible feedback controls on the cells’ regulation. Thus, compensatory proliferations and anoikis

are triggered in order to fairly reconstruct the original shape of the tissue. And finally, we have im-

plemented an original system allowing tissues to renew their overall cells without affecting their shape.

Complete cell turnover of tissues is continuously guaranteed after determination of a right sequence of

death/birth and an optimal senescence period for cells. We consider that this system is noteworthy because

cell turnover would be at high risk for accumulating mutations and progressing to cancer. With these three

experiments through our computational model, we have seen how apoptosis underlies the viability of

multicellular systems by allowing them to have a robustness measure, to face disruptions, and to renew.

For validation and experimentation of models dedicated to living things, it would be interesting to design

tools that can display spatial and functional aspects as well as the dynamical behaviors of the system. This

could endow simulations with an intuitive rendering allowing an easier use and study. In addition, such tools

have to be flexible enough to let humans intervene in the definition and the execution of the simulations. That

is why we plan to develop a simulation tool that can perform experimental procedures on computational

tissues like simulation of large tissue growth. We can choose the tissues to simulate according to some of their

properties, namely robustness, symmetry, or bioinspiration, and then observe how they will be beyond their

early stage of development. At that step, factors such as energy diffusion and consumption could be taken into

account in addition to the well-guided morphogenesis process under the directional cellular dynamic. These

new factors will consist somehow of additional constraints. For instance, during a simulation if these factors

allow for a given cell to divide, the direction will not be randomly chosen. It will be a defined regulation

known by the cell because its included in its genotype. As a simulation tool, Sutterlin et al. (2009) presents a

graphical modeling system for multi-agent-based simulation of tissue homeostasis.

However, much remains to be learned about how apoptosis acts during normal development to actively

induce tissue modification. Nevertheless, the breakthrough in computational biology in synergy with

parallel biological experiments attempts to explore some of the fundamental issues of apoptosis diseases

like cancer. And in the future, these works could contribute to a better understanding of this scourge and

tend effectively to the development of novel therapies.

AUTHOR DISCLOSURE STATEMENT

The authors declare that no competing financial interests exist.

REFERENCES

Apte, A., Bonchev, D., and Fong, S. 2010. Cellular automata modeling of FASL-initiated apoptosis. Chem. Biodivers.

7, 1163–1172.

Aubin, J. 1999. Mutational and morphological analysis: Tools for shape evolution and morphogenesis. Birkhauser,

Basel.

Davies, J.A. 2013. Chapter 24—Morphogenesis by elective cell death, 325–336. In Davies, J.A., ed. Mechanisms of

Morphogenesis, 2nd edition. Academic Press, Boston.

Fan, Y., and Bergmann, A. 2014. Distinct mechanisms of apoptosis-induced compensatory proliferation in proliferating

and differentiating tissues in the drosophila eye. Dev. Cell 14, 399–410.

Frank, S. 2007. Dynamics of Cancer: Incidence, Inheritance, and Evolution. Princeton Series in Evolutionary Biology.

Princeton University Press, Princeton, NJ.

Henderson, J., and Carter, D. 2002. Mechanical induction in limb morphogenesis: The role of growth-generated strains

and pressures. Bone 31, 645–653.

268 SARR ET AL.

Hernandez-Martınez, R., and Covarrubias, L. 2011. Interdigital cell death function and regulation: New insights on an

old programmed cell death model. Dev. Growth Differ. 53, 245–258.

Kim, J., Saidel, G., Kirwan, J., et al. 2006. Computational model of glucose homeostasis during exercise, 311–314. In

28th Annual International Conference of the IEEE. Engineering in Medicine and Biology Society, 2006.

Manjon, C., Sanchez-Herrero, E., and Suzanne, M. 2007. Sharp boundaries of dpp signalling trigger local cell death

required for drosophila leg morphogenesis. Nat. Cell Biol. 9, 57–63.

Nagata, S. 1997. Apoptosis by death factor. Cell 88, 355–365.

Raychaudhuri, S., Willgohs, E., Nguyen, T., et al. 2008. Monte Carlo simulation of cell death signaling predicts large

cell-to-cell stochastic uctuations through the type 2 pathway of apoptosis. Biophys. J. 95, 3559–3562.

Sarr, A., Fronville, A., and Rodin, V. 2014. Morphogenesis model for systematic simulation of forms co-evolution with

constraints: Application to mitosis, 230–241. In 3rd International Conference on the Theory and Practice of Natural

Computing, volume 8890, TPNC 2014. Springer, New York.

Schleich, K., and Lavrik, I. 2013. Mathematical modeling of apoptosis. Cell Commun. Signal. 11, 44.

Smith, D., Rubenson, J., Lloyd, D., et al. 2013. A conceptual framework for computational models of achilles tendon

homeostasis. Wiley Interdiscipl Rev Syst Biol Med. 5, 523–538.

Spalding, K., Bhardwaj, R., Buchholz, B., et al. 2005. Retrospective birth dating of cells in humans. Cell 122, 133–143.

Sutterlin, T., Huber, S., Dickhaus, H., et al. 2009. Modeling multi-cellular behavior in epidermal tissue homeostasis via

finite state machines in multi-agent systems. Bioinformatics 25, 2057–2063.

Suzanne, M., and Steller, H. 2013. Shaping organisms with apoptosis. Cell Death Differ. 20, 269–675.

Turing, A. 1952. The chemical basis of morphogenesis. Phil. Trans. R. Soc. B Biol. Sci. 237, 37–72.

Varela, F., Maturana, H., and Uribe, R. 1974. Autopoiesis: The organization of living systems, its characterization and a

model. Biosystems 5, 187–196.

Address correspondence to:

Abdoulaye Sarr

UMR CNRS 6285, Lab-STICC, CID, IHSEV

Computer Science Department

Universite de Brest

20 avenue Le Gorgeu

29200 Brest

France

E-mail: [email protected]


A Viability Approach for Robustness Measurement ...

Documents