UNIVERSITÉ DE GENÈVE

BACHELOR THESIS

Numerical model of biological cellmotility

Author:Clément THORENS

Supervisors:Prof. Bastien CHOPARD

Raphaël CONRADINDr. Christophe COREIXAS

Dr. Franck RAYNAUD

A thesis submitted in fulfillment of the requirementsfor the degree of Bachelor of Computer Science

September 2, 2020

iii

Contents

1 Models 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 PalaCell2D Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Adherence force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Membrane force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.3 Inner pressure force . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.4 Outer pressure force . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Minimal model for spontaneous cell polarization . . . . . . . . . . . . . 4

1.3.1 Vertices velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3.2 Switch probability . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.3 Cell centre computation . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Combination of the the two models . . . . . . . . . . . . . . . . . . . . . 6

2 Numerical experiments 7

2.1 Setting the parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Sanity checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Switch distances . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Force distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.3 Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Circular migration . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Larger scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

iv

2.4 Performances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Further discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Bibliography 19

v

List of Figures

2.1 Samples of the shape of the moving cells and the trajectory of theircentres after 4’000 iterations. Units are one size of the cell(rmax). Onthe top figures, blue vertices are in protrusion and red vertices arein retraction. On the bottom figures the starting point of the cell ispictured in red and its trajectory in blue. . . . . . . . . . . . . . . . . . . 8

2.2 Distances r at which the states changed during a run. The area underthe histogram is summed to 1. . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Percentage of the three forces contribution for points in protrusion(left)and in retraction(right).The vertical black bars mark the beginning ofthe nth iteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 The forces have a different weight depending on the general state ofthe cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.5 Sample trajectories with α = 50, rmax = 11 and N = 1. . . . . . . . . . . . 11

2.6 Illustrations of short circular migration for k1 = 0 (A and B) and nocircular migration for k1 = 0.2 (C and D) . . . . . . . . . . . . . . . . . . 14

2.7 Illustrations of two systems with 400 cells . . . . . . . . . . . . . . . . . 15

2.8 Time needed to perform 100 iterations for 1 node in blue and 2 nodesin red . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.9 Time needed to perform 500 iterations on 1 cell for different domainsizes.For 1 node in blue, 2 nodes in red. . . . . . . . . . . . . . . . . . . 17

vii

List of Tables

2.1 Parameters of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1

Chapter 1

Models

1.1 Introduction

Modelling cells or any other biological process can involve many parameters andoften requires to inject in the model some knowledges about the system we want tosimulate. Even if a complex model might not be undesirable, it is still interestingto see if we can obtain the same results using a simpler one. Here we are going tocombine two pre-existing models in order to run cell motility simulations using fewbiological principles.

The first model(PalaCell2D) is still in development and aims to replicate cell growthand signalling by applying a set of forces on the vertices that outline the cell. It isbased on Palabos(Latt et al., 2020), a lattice Boltzman Solver written in C++. Sincewe are interested in cell motility, we are going to drop the part of the model assignedto growth and signalling and only keep the one that handle collisions and cohesionbetween the vertices. The second model is from (Raynaud et al., 2016). It reproducesthe motility of a fish epidermal keratinocyte and its polarization by giving a speedto the vertices describing the cell membrane. Both models are described in moredetails in the first chapter.

Our main objective is to integrate in PalaCell2D the possibility for the cells to po-larize, in order to be motile, using the techniques from (Raynaud et al., 2016). Theadded value in the final result in comparison with the model developed in (Raynaudet al., 2016), where the cells were not interacting between each other, is that the cellswill be able to make collisions.

A fundamental difference between the two models is that one makes use of forcesand the other of speeds. To incorporate the second model into the first one, we aregoing to add a force that represents the speed applied on the vertices. The valida-tion of the model combination will be realised by comparing the output to otherexperiments on the keratinocyte trajectories.

1.2 PalaCell2D Model

The PalaCell2D model describes the cell as a list of vertices linked by edges to forma polygon. The number of vertices changes over time. The length between two

2 Chapter 1. Models

vertices determines if a vertex has to be added or removed. The decision is madewhen this length reaches an upper(adding) or lower(removing) threshold.

The model offers two states for a cell, relaxing and growing. The cell can also be setin a mode where it reacts to signalling to control if it is growing or relaxing. In ourcase, we will leave the cell in the relaxing state. For each iteration, the position ofevery vertex is updated considering the sum of 4 forces:

F = FAi + FMi + FIPi + FOPi (1.1)

Where the index i represents a single vertex.

F has two components X and Y and the position is updated as:

(X(t + ∆t), Y(t + ∆t)) = (X(t), Y(t)) + µ(FX, FY)∆t (1.2)

Where µ is used to obtain a speed from a force.

The different forces are described below.

1.2.1 Adherence force

The adherence force controls the adhesion between two cells when their verticesare closer than dmax. This force is constituted of a spring between two connectedvertices.

It is defined as :

FAi = k1 ∑j

xi − xj

‖xi − xj‖(‖xi − xj‖ − d0 −

d30

(4‖xi − xj‖)2 )

if ‖xi − xj‖ < dmax.

(1.3)

Where:

• xj denotes the connected vertices with j the index of the neighbouring cell ver-tices

• d0 is the rest length of the spring

• dmax is the maximal interaction length with d0 < dmax.

• k1 is the constant related to the adherence force

1.2.2 Membrane force

The membrane force is the force that the two neighbours, from the same cell, of thevertex i apply on it.

It is defined as :FMi = k3l0(

xi+1 − xi

‖xi+1 − xi‖+

xi−1 − xi

‖xi−1 − xi‖) (1.4)

1.2. PalaCell2D Model 3

Where:

• l0 is the characteristic distance between vertices

• k3 is the constant related to the membrane force

l0 can be modified to change the number of points. If ‖xi+1 − xi‖ < l02 , then xi is

removed. If ‖xi+1 − xi‖ > 2l0, then new vertices are added.

1.2.3 Inner pressure force

The inner pressure force represents the force coming from the inner of the cell and isapplied on each vertex.

It is defined as :FPi = k2np

Snvert

(1.5)

Where:

• n is the normal at the vertex defined as the opposite of the membrane force.

• p is the pressure inside the cell

• S is the surface of the cell

• nvert is the number of vertices of the cell

The pressure inside the cell p is computed using the following formula :

p = p0 + η ∗ (mV− ρ0) (1.6)

Where

• p0 is the rest pressure in the cell

• ρ0 is the target density of the cell

• η is the pressure sensitivity

• m is the mass of the cell

• V is the volume of the cell

In the formula, the surface of the cell is divided by the total number of verticesof the cell to ensure that adding or removing a vertex is taken into account in thecomputation.

4 Chapter 1. Models

1.2.4 Outer pressure force

The outer pressure force is a weighted sum of the inner pressure force from con-nected vertices of the neighbouring cells.

It is defined as :

FOPi = ∑j∈J

FIPj

wij(1.7)

Where:

• J is the set of the connected vertices

• FIPj is the inner pressure force of the connected vertex j

• wij is a weight such that ∑j1

wij= 1

1.3 Minimal model for spontaneous cell polarization

1.3.1 Vertices velocity

The model studied in (Raynaud et al., 2016) describes the cell by its contour repre-sented by a list of vertices. Each vertex has two states: protrusion or retraction. Theprobability for a vertex to switch from a state to another is defined by its distance tothe cell center.

In protrusion state, the vertex will move with a constant velocity Vp outward nor-mally to the membrane cell. In retraction state, the vertex will move towards the cellcentre with a velocity Vr. Vr depends of the distance between the vertex and the cellcentre(defined in 1.3.3).

Vp and Vr are computed as follow:

Vp = 10−4rmax (1.8)

Vr = Vmax − (rmax − r

rmax − rmin)2(Vmax −Vmin) (1.9)

Where

• r is the distance between the vertex and the cell centre.

• rmax is the distance where the vertex is the most likely to switch from protru-sion to retraction.

• rmin = 0.25rmax

• Vmax = 2Vp

• Vmin = 0.2Vp

1.3. Minimal model for spontaneous cell polarization 5

1.3.2 Switch probability

The switch from protrusion to retraction or from retraction to protrusion is deter-mined at each iteration by computing a probability defined as:

PP→R =

{τ(Nrmax ,σ2 + (1− Nrmax ,σ2) ni

N ) if r < rmax

τ if r > rmax(1.10)

PR→P =

{τ ni

N if r > rmin

τ if r < rmin(1.11)

Where

• τ is an overall rate of transition set to the inverse of the time step.

• Nrmax ,σ2 is a Gaussian random number with mean rmax and variance σ2 com-puted as follow :

Nrmax ,σ2 = e−(r−rmax)2

2σ2 (1.12)

• ni the number of neighbours in the opposite state.

• N is the total number of neighbours that influence the state of the vertex.

1.3.3 Cell centre computation

In (Raynaud et al., 2016) a method to find the centroid of a non-intersecting polygonis used to compute the cell centre:

xc =1

6A

n−1

∑0

(xi + xi+1)(xiyi+1 − xi+1yi) (1.13)

yc =1

6A

n−1

∑0

(yi + yi+1)(xiyi+1 − xi+1yi) (1.14)

Where

• (xi, yi) are the coordinates of a vertex.

• A is the signed area of the polygon computed as bellow.

A =12

n−1

∑0

xiyi+1 − xi+1yi (1.15)

In our case, the method used in PalaCell2D has been kept as it is, i.e. computingthe centre of mass of all the vertices. Since they have all the same weights, the twomethods output the same result. The cell centre is computed at each time step forevery cell.

6 Chapter 1. Models

1.4 Combination of the the two models

To add the idea developed in (Raynaud et al., 2016) to the PalaCell2D model, thevelocity Vr|Vp has to be transformed into a force. In PalaCell2D, the variation of ve-locity of a particle is computed by multiplying the sum of the forces by the time stepdt. In our case, we have that dt = 1 and we are assuming that µ = 1, so velocity andforce are equivalent. Thus Vr|Vp is simply converted into a force Fretraction|Fprotrusionand added to the other forces. A main implication of this choice is that a point couldbe in protrusion state and still get closer to the centres(or moves away from it inretraction state) in the next iteration because the resulting force was directed to thecenter, which is not possible with a velocity based model. The results of this dif-ference are explored in 2.2.4. The calculation of Vr|Vp is done the same way thatin the original model, except an addition of a coefficient α ie. Fretraction = αVr andFprotrusion = αVp.

The PalaCell2D model gives the possibility to parallelise the computation. It impliesthat all the vertices of a cell are not necessarily available to a single computation unit.We can only assure that a specific vertex will have access to, at least, its two closestneighbours. However, to calculate the next state of a vertex we need to access its2N neighbours. For this reason, the possibility of parallelisation has been removedwhen N 6= 1.

A slight modification has to be made to avoid loops in the cell membrane to beproduced. The original PalaCell2D code was detecting loops checking only the twoneighbours in order to allow parallelism, this was not sufficient when the forcesapplied on vertices were too strong. Since the program is no longer parallelisablewhen N 6= 1, the range in which vertices check if loops were created was extendedto allow a more efficient loop detection.

Three parameters have been added to the pre-existing ones. The coefficient α, rmaxas described above and a number of neighbours N. N gives the number of neigh-bouring vertices that is going to be checked while computing the probability of avertex to switch state. The state of the vertex itself is not taking in account to com-pute the switch probability. The value of N will be multiplied by 2 to always keep itsymmetric.

While testing the model, a lot a vertices were going much further from the centerthan rmax. It has been corrected by changing 1.3.2 to force any vertices to switch fromprotrusion to retraction when r reach rmax. The switch probability is now computedas follow:

PP→R =

{τ(Nrmax ,σ2 + (1− Nrmax ,σ2) ni

N ) if r < rmax

1 if r > rmax(1.16)

PR→P =

{τ ni

N if r > rmin

τ if r < rmin(1.17)

7

Chapter 2

Numerical experiments

2.1 Setting the parameters

The parameters of the simulation have to be set. To limit the number of runs neededto find the values that give the best results, the original parameters were left to theirprevious values(i.e. like in PalaCell2D). Only the three new parameters remained tobe adjusted. The results obtained in (Raynaud et al., 2016) were used as referencesto evaluate the output.

Multiple set of parameters have been tested and for a lot of them, the cell does notpolarize itself spontaneously, i.e. the cell enters a state where one half of the cell ismostly composed of vertices in protrusion state and the other half mostly composedof vertices in retraction state. In the case where the cell gets polarized and starts tomove, we can observe three main behaviours. Three sets of parameters have beenchosen to illustrate the different shapes and movements obtained and can be seen infigure 2.1.

It appears than the value α should be chosen according to rmax , if α is too small thevertices find an equilibrium state where all the points are in retraction and nothingmove. If α is too big, multiples loops are created within the membrane leading toan erratic behaviour. We can notice that the first situation was not possible in theoriginal model where the state of a point was giving its movement direction and thenno equilibrium could be reached. Here the 5 forces can cancel themself, resulting ina total force equals to zero.

It also comes out that the choice of N does not influence much the output, but itseems that a small N is preferable to keep the cell polarized and to have a straighttrajectory. For this reason we will fix N to 1 (2 neighbours). It also removed one of theconstraint that prevent the parallelisation of the program, since only two neighboursare needed and they will always be available (more details in 2.4).

Figure 2.1 bellow shows samples of the shape of cells when they are moving andthe trajectories of their centres after 4’000 iterations for different value of α. Even ifthe general shape is very similar, their trajectories vary a lot depending on α. 2.1Ddoes not move much.2.1C and 2.1E are more motile, but 2.1E has straight segmentsand that is what we are looking for, knowing that a keratinocyte cell has straightmovements(Allen et al., 2018). This is why we will fix the parameters (α, rmax, N)to (50, 11, 1) for the next runs. The set parameters that will be used is summarized

8 Chapter 2. Numerical experiments

in table 2.1. The videos of the three cells moving can be seen on additional videos:alpha30.avi, alpha50.avi and alpha100.avi.

(A) α = 30 rmax = 11 N = 1 (B) α = 50 rmax = 11 N = 1 (C) α = 100 rmax = 11 N = 1

(D) α = 30 rmax = 11 N = 1 (E) α = 50 rmax = 11 N = 1 (F) α = 100 rmax = 11 N = 1

FIGURE 2.1: Samples of the shape of the moving cells and the trajec-tory of their centres after 4’000 iterations. Units are one size of thecell(rmax). On the top figures, blue vertices are in protrusion and redvertices are in retraction. On the bottom figures the starting point of

the cell is pictured in red and its trajectory in blue.

TABLE 2.1: Parameters of the model

N α rmax η p0 ρ0

1 50 11 25.0 0.125 1.05

k1 l0 k3 k2 d0 dmax

0.2 0.5 0.2 0.1 0.5 1.0

2.2 Sanity checks

We ran some sanity checks to verify that they were no unexpected behaviour duringthe simulation. The parameters used are the one found in 2.1.

2.2.1 Switch distances

The first check controls the distance from the center at which the state is switched,both from retraction to protrusion and from protrusion to retraction. In Fig 2.2, mostof the switches happened between r=9 and r=11 with a small spike at 11 for figure

2.2. Sanity checks 9

2.2A. It is in accordance with the modification made in the probability switch to forceall vertices to be at most at rmax from the centre. We could also expect 2.2B to haveswitches around rmin. It does not happen because when the vertices in retractionstates get close to the centre, their also get closer to each other and they are removedand thus does not reach rmin. It is a major change compared to (Raynaud et al., 2016).

(A) Protrusion to retraction (B) Retraction to protrusion

FIGURE 2.2: Distances r at which the states changed during a run.The area under the histogram is summed to 1.

2.2.2 Force distribution

The contribution of each force has been measured for both vertices in retraction andin protrusion. The result is shown in 2.3. The different forces applied on each vertexat a same iteration are stacked on top of each other and the iterations are shown fromleft to right. This choice is forced by the fact that it is not relevant to follow the evo-lution of a specific vertex since they are created and removed multiples times duringthe run. The vertical lines on the x-axis show the beginning of the nth iteration.

During the testing phase, these distributions were useful to identify which forceswere causing the system to reach a stable state with all the particles in retraction anda total force equals to zero. Reducing the value of the inner pressure force appearedto be the main solution to a problem in the simulation of multiple cells(see section2.3.1), where all the vertices were going into a retraction state and stayed like this.

The membrane force has a more important weight for a vertex in protrusion state.It is in accordance with the fact that the points are moving away from each otherand they are hold together by the membrane force. In general, the motion force forvertex in protrusion state is less prominent than in retraction state.

On both graphs 2.3 we can see a different force distribution at the beginning. If wefocus only on the first 400th iterations without distinguishing protrusion and retrac-tion (see figure 2.4A). We observe a clear difference between the first 200 iterationsand the next ones. It corresponds to the phase where the cell is not polarized (seefigure 2.4B). The alternation between high membrane force values and low mem-brane force values on the right of the figure 2.4A is explained by points being inprotrusion (high membrane force values) and points in retraction (low membrane

10 Chapter 2. Numerical experiments

(A) In protrusion state (B) In retraction state

FIGURE 2.3: Percentage of the three forces contribution for points inprotrusion(left) and in retraction(right).The vertical black bars mark

the beginning of the nth iteration.

force values) displayed next to each other, as seen in the last paragraph and on fig-ures 2.3A and 2.3B. In figure 2.4B, there is not line of vertices with the same state andthen it induces more tension on the membrane when the points are spreading out.

(A) Forces distribution on the first 400 it-erations

(B) Sample shape of the cell before beingpolarized

FIGURE 2.4: The forces have a different weight depending on the gen-eral state of the cell

2.2.3 Trajectories

Figure 2.5 shows different trajectories for 4000 iterations with the same parameters.The spatial units are the size of one cell (when the cell goes from 1 to 2, it moved bya distance of itself once). The size of the cell is defined as rmax. The trajectories arenot always as good as the one showed in figure 2.1E, but overall it contains straightsegments, which is important to validate the model, knowing that little is known onkeratinocyte trajectories except that they are rectilinear (Allen et al., 2018).

2.2. Sanity checks 11

FIGURE 2.5: Sample trajectories with α = 50, rmax = 11 and N = 1.

12 Chapter 2. Numerical experiments

2.2.4 Miscellaneous

More information about the cell displacement has been collected. The speed of a cellis computed by considering when the centre moves from once the size of the cell,characterized here by rmax. Using the parameters above, it happened every 140-160iterations. We conclude that 140-160 iterations correspond to 1 minute where the cellmoved by 20 µm (Löber, Ziebert, and Aranson, 2015).

The number of vertices going in the opposite directions than they should (i.e. awayfrom the centre in retraction and closer from the centre in retraction) was computed.It concerns around 35% for a single cell and 27% for multiple cells in a constraintcircular space.

2.3 Collisions

2.3.1 Circular migration

To test the collisions between cells, we tried to reproduce the results presented in(Löber, Ziebert, and Aranson, 2015). In particular the appearance of a collectivecircular migration for a low density of cells constraint in a circular box while thecell-cell adherence is non-zero versus when it is set to zero.

In our model, two forces are involved in the collision process: The outer pressureforce and the adherence force. In the following results the outer pressure forcehas been kept constant, while the adherence force has been set to two differentvalues(k1=0.2 and k1=0).

To constraint the cells inside a circular space, an additional force is used. When acell particle reached a specific range away from the defined centre of a circular box,a force is applied to the particle to simulate an elastic collision with a wall.

It comes out that the cells were easily going into a stable state where all the verticeswere in retraction. Once a cell is in such a state, it stops to move. Although collisionswith others cells can allow the cell to switch to a moving state, in most cases the cellwill stay in this state until the end of the simulation. Some cells were also gettingstuck against the wall (i.e points in protrusion touching the wall and the other pointsin retraction). To prevent that to happen, the coefficient k2, related to the pressure ofthe cell, has been reduced (as seen in section 2.2.2). Setting it to 0.02 instead of 0.1reduced the probability of getting this behaviour without changing the behaviour ofa single cell. Even if this state is less likely to happen, we can still observe it duringlong runs. An example can be seen on additional videos 40000iterations.avi, wherethe system runs for 40’000 iterations.

The angular momentum of every cell about the center of the wall is computed andaveraged to estimate if a collective circular migration happened:

L =∑P

i mimathb f ri ×mathb f vi

P(2.1)

2.3. Collisions 13

Where:

• mi is the mass of the ith cell.

• ri is the normalized position vector of the center of the ith cell relative to thecenter of the wall.

• vi is the normalized velocity vector computed using the current position of theith cell and its position in the next iteration.

• P is the total number of cells in the system.

The ratio in (Löber, Ziebert, and Aranson, 2015) experiment between the box and thecells is 1

10 . This has been changed, as no circular movement was observed with suchratio. Using a cells size of 11 and a box radius of 65, we could perceive a circularmigration, however it was not persistent. The cells are rotating in a direction forapproximatively 1500-2000 iterations and then collisions between cells, rotating ornot, disrupt the movement, that will start again, clockwise or counter-clockwise, fewiterations later. The non-persistence of the circular movement could be explainedby the non-persistence of a single cell movement. It could be interesting to test infurther works if improving the persistence of a single cell movement could improvethe whole system persistence.

The measure of the angular momentum for 10’000 iterations and a snapshot of therotating cells are shown bellow in figures 2.6A and 2.6B. We can compare this resultto the same measure for a non-zero adherence force (k1 set to 0.2) on figure 2.6C and2.6D where no clear circular migration appears. The associated videos are Circular-Mvt.avi and noCircularMvtK02.avi.

2.3.2 Larger scale

The model has been tested with a larger number of cell. On figure 2.7 we can seesnapshots of two system on the 5000th iterations on the left and their momentumson the right. The top line(2.7A and 2.7B) shows a run with the adherence force setto 0 and the bottom line(2.7C and 2.7D) with an adherence force as previously(table2.1). In the two cases, the momentum is not showing a clear circular migration asabove with 21 cells. However, on the two videos, 400Cells7600IterNoAdh.avi and400CellsAdh12440Iter.avi, some cells are forming small clusters that goes in the samedirection for several iterations. This effect if more present when the adherence forceis not set to zero. More generally, the cells tend to form clusters when they adhere toeach other.

14 Chapter 2. Numerical experiments

(A) Shot circular migration at 2760 itera-tions

(B) Momentum of the system with itera-tions on the x-axis

(C) No circular migration at 2760 itera-tions

(D) Momentum of the system for a non-zero adherence force

FIGURE 2.6: Illustrations of short circular migration for k1 = 0 (A andB) and no circular migration for k1 = 0.2 (C and D)

2.3. Collisions 15

(A) 400 cells without adherence at itera-tion 500

(B) Momentum of the system withoutadherence

(C) 400 cells with adherence at iteration500

(D) Momentum of the system with ad-herence

FIGURE 2.7: Illustrations of two systems with 400 cells

16 Chapter 2. Numerical experiments

2.4 Performances

Performances of the simulation have been measured on 100 iterations and a domainsize of 600X600(more comments on that bellow) without writing any output. Evenif only the number of particles in the system really matter for the performance, theresults are given by the number of cells. Each cell was initialized with a total of 122points. This number can vary by ±5 during the run.

Fixing the parameters to α = 50 and N = 1(as previously) removes the formation ofloops within the membrane of the cell and then the utility of the function that detectsloops and deletes them. Furthermore, since N = 1, the points only need informationabout the state of its two closest neighbours. With this set of parameters, we solvedthe limitations that was causing the impossibility to parallelize the computation andthen we can split it on multiple nodes.

On the graph 2.8, the line in blue shows the result on 1 node and the line in red on2 nodes in parallel. We can observe a linear progression for both of them and theparallelization successfully allows to divide the computation time by almost 2.

The size of the domain has an major influence on the computation time. For 500 iter-ations, 1 cell on 1 node and 2 nodes. Firgure 2.9 shows a quadratic evolution of thecomputation time for the different different domain sizes. It is due to functions thatchecks every position of the domain in the original PalaCell2D code. A part of thesefunctions were not used and have been removed to improve the computation time,but since some are still needed, we have not been able to get ride of the quadraticcomplexity. Even if we have a polynomial growth for the domain size, once the sizeis fixed, increasing the number of cells induces a linear growth, as shown in figure2.8, for any domain size.

2.4. Performances 17

FIGURE 2.8: Time needed to perform 100 iterations for 1 node in blueand 2 nodes in red

FIGURE 2.9: Time needed to perform 500 iterations on 1 cell for dif-ferent domain sizes.For 1 node in blue, 2 nodes in red.

18 Chapter 2. Numerical experiments

2.5 Further discussions

The model offers 12 parameters to be tweaked, here only a small subset of valueshave been tested. Only the values of N, α, rmax, k1 and k2 have been changed dur-ing the testing phase. It would be interesting to also modify η, p0, ρ0, l0, k3, d0and dmax, especially with a more biological approach instead of fitting a model pre-viously made by trying multiple parameters. Furthermore, other behaviours wereobserved while developing the model. For example the cell was observed to bespinning around itself during some runs. These observations were not documentedbecause no set of parameters for which it always happened were found. Addition-ally, the shape of the cell is sensitive to the parameters k3, the coefficient giving itsweight to the membrane force, and l0, the characteristic distance between vertices.We could also notice that some features of the PalaCell2D model have not been used,like the possibility for a cell to growth or to react to signalling.

All of this unexplored possibilities and the fact that the PalaCell2D model was notfinished during the writing of this thesis could give the opportunity to simulateother kind of cells or cell activities in further development of the previous work.

19

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Latt, Jonas et al. (2020). “Palabos: Parallel Lattice Boltzmann Solver”. In: ISSN: 0898-1221. DOI: https://doi.org/10.1016/j.camwa.2020.03.022. URL: http://www.sciencedirect.com/science/article/pii/S0898122120301267.

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