Modelling body-wide synchronization of DAF-16 oscillations

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Modelling body-wide synchronization

of DAF-16 oscillations

Madelon Geurts

Studentnumber: 11288035

Report Bachelor Project Physics and Astronomy

15 EC

AMOLF (Quantitative Developmental Biology) Daily supervisor: Burak Demirbas

Supervisor: Jeroen van Zon Second examiner: Pieter Rein ten Wolde University of Amsterdam & VU University Amsterdam

FEW/VU & FNWI/UvA The Netherlands

conducted between 06 - 04 – 2020 and 05 - 07 – 2020 05-07-2020

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Summary

Scientific summary

DAF-16 is a transcription factor that controls different reactions within the cell, in particular stress regulation. An Amolf research group has found that under starvation of the C. elegans worm DAF-16 tends to randomly pulse body-wide. To understand the working of the stress regulation, the communication between cells is to be re-searched: Does synchronization of oscillations of cells occur and how does it come about? The quantitative developmental biology group has started making a simple model in an effort to describe the behaviour of the oscillations. The definition of the phase of an oscillator was critical in the quantitative description of the synchroniza-tion of the cells. When oscillators synchronize, the difference between their phases goes to zero. In this thesis the phase on the limit cycle of an oscillator was described by using the speed of an oscillator to obtain an linearly increasing phase. For the phase off (but still relatively close to) the limit cycle an intersection between the un-known point and the limit cycle was made. Using linear interpolation to continuously define the phase, the phase at the unknown point was defined by the phase at the in-tersection on the limit cycle. In the end, looking at multiple cells starting at different phases gives insight in the behaviour of the synchronizing oscillators.

Using an ILP feedback model and the phase definition, the cells in the C. elegans tend to synchronize over time. This synchronization occurs within one time period, where the period is the time the oscillator takes to move one round over the limit cycle. With a maximum number of nine, more has not yet been examined, oscillators the synchronization takes approximately the same time, independent of the phases of these oscillators. This is true for cells that have parameters that produce sustained oscillations. In the future the model can be modified to take space into account. Using diffusion of ILP over space, cells would communicate differently. If diffusion is very slow, cells will probably not even synchronize or will all synchronize with a time-delay or with for example a spatial wave.

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Populair wetenschappelijke samenvatting (Dutch, 6 VWO niveau)

In de ontwikkelingsbiologie is er de afgelopen 10 jaar veel gebeurd. Door het kijken naar specifieke wormen (C. elegans) is er veel geleerd over organismes met een celkern (eukaryoten), zo ook mensen. In deze thesis is gekeken naar oscillaties (schommelin-gen) van een concentratie van een bepaalde transcriptie factor, genaamd DAF-16, in cellen. Deze oscillaties reageren verschillend op verschillende soorten stress van een worm, zoals een te hoge temperatuur. De oscillaties van deze DAF-16 transcriptie fac-tor binnen een cel kunnen in fase geraken met andere cellen, dit heet synchronisatie. Synchronisatie van oscillaties vindt op veel plekken plaats, zo ook bij het applaud-iseren van publiek. Wanneer er genoeg mensen in hun handen klappen, zullen zij tegelijk gaan klappen. Het klappen staat hier gelijk aan de oscillatie en het tegelijk klappen is als de synchronisatie.

Nu blijkt dat met behulp van een bepaald model van de worm de oscillaties van ver-schillende cellen inderdaad synchroniseren. Door het definieren van deze fase op de juiste manier kan de hoeveelheid en snelheid van de synchronizatie vastgesteld wor-den. Zo blijkt dat door cellen te laten communiceren deze cellen daadwerkelijk gaan synchroniseren. Deze vaststelling kan gebruikt worden om uiteindelijk met het juiste model de oscillaties goed te omschrijven. Zo kan er voorspeld worden hoe de periode en frequentie veranderen bij bepaalde vormen van stress in de C. elegans. Dit zal bijdragen aan het beter begrijpen en bestuderen van organismen en daarmee mensen.

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1 Introduction

1.1 IIS pathway

The insulin/IGF-1 signal (IIS) pathway controls many biological processes in worms, for example: the FoxO transcription factor DAF-16 and therefore the development of the worm. The IIS pathway, as seen in figure 1, is also present in humans [1].

Figure 1: Shown in this figure is the IIS pathway. This pathway consists of the agonistic insulin like peptides (ILP’s) that activate the pathway. Activating the pathway will lead to phosphorylating DAF-16, which will then not move into the nucleus and stay in the cytoplasm. This figure also shows the other proteins in this pathway [1].

The IIS pathway is stimulated by ILP’s (insulin like peptides). These peptides bind to the receptors and set the whole pathway in motion. The activation of DAF-2 re-sults in activating the kinase AKT-1, which phosphorylates DAF-16. Phosphorylated DAF-16 cannot move into the nucleus and stays in the cytoplasm. Under normal circumstances the dephosphorylated DAF-16 can activate DNA in the nucleus, which will eventually make a specific protein.

DAF-16 is a Fox-O transcription factor and believed to be the primary reactor to stress resistance. A Fox-O transcription factor can bind to DNA in the nucleus and set a process in motion that will encrypt mRNA. DAF-16 therefore regulates protein production which can fight different types of stress [1].

1.2 C. elegans

Caenorhabditis elegans (C. elegans) is a worm used in a lot of different experiments in biology. Due to its short life span, transparency and self-fertilizing (hermaphroditic) character, this worm is used to study organisms in, for example, developmental bi-ology and neurobibi-ology. The C. elegans is one of the best eukaryotic organisms to research to gain an understanding of all eukaryotic organisms.

A hermaphroditic organism is useful in experiments because by mutating or changing the genetics of one worm all its descendants also have the same mutation. To look at a certain asset or protein of the worm one can mutate one ancestor and create a family of worms with the same characteristics. Fluorescent markings can be used to track different types of proteins within the worm. These are visible due to its transparent form [2]. DAF-16 can have a fluorescent marking and be made visible as well. The C. elegans can be subject to different types of stress. For instance: heat, starva-tion, osmotic and oxidative stress [1]. Due to different types of stress the worm can

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Figure 2: This figure shows the life stages of the C. elegans worm. The worm starts as an embryo and moves through different life stages as it gets older. It could arrest after different stages in its life, for example after L2, then it is called a dauer larvae. In this case the worm skips the L3 stage and becomes a relatively small L4 [2].

arrest in certain stadia of its life. When a C. elegans arrests in the so-called L2 stage it will become a dauer larve, see figure 2. In this case the worm skips the L3 stage and goes straight from the dauer stage into the L4 stage. In these types of arrest the worm can survive longer than its normal life span to survive certain kinds of stress. Looking at DAF-16, which regulates its stress response, is a good start to understand the fundamentals of stress response and therefore the IIS pathway [3].

1.3 Motivation for the research conducted

The quantitative developmental group at Amolf found that, when looking at Caenorhab-ditis elegans, the overexpressed DAF-16 shuttles between the nucleus and the cyto-plasm when under stress, see figure 3. When the worm is not under stress, the DAF-16 stays mostly in the cytoplasm, this is shown as the purple base line. Otherwise the DAF-16 tends to pulse. Different types and amounts of stress also influence the pul-sating behaviour of the transcription factor. Cells tend to synchronize their oscillating behaviour. Currently, the experiments suggest that some cells react earlier to stress than others.

1.4 Scientific questions

While studying the behaviour of the oscillations the following questions should be answered: What is the best way to describe the pulsating behaviour of DAF-16 in the C. elegans? Do cells, when they communicate, tend to synchronize over time? How does this synchronizing behaviour of different cells come about? Do more cells tend to synchronise faster or slower? Eventually the goal is to find the best model that describes this synchronization of cells and predicts the length and frequency of a pulse under different types of stress.

1.5 Approach

The first step in studying the synchronization of cells during stress, is to make a model in which one cell’s oscillations can be described. In this thesis one model was chosen to compare and quantify different types of synchronization [5]. In multiple biological (and other) systems oscillations are observed. In astrophysics, chemistry and even in

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Figure 3: Pulse dynamics changes depending on the type of stress. Three types of stress are shown, where osmotic stress is that of salt concentration in its nearest surroundings. For every type of stress two different worms are shown. On the x-axis is the time and y-axis is the nuclear concentration of DAF-16. This slide was presented in a presentation by Olga Fillina on 8/10/2019 [4].

hanging clocks synchronizing oscillators are seen [6],[7]. Studying these oscillations in a far away (limit) point can be very interesting. Multiple oscillators can ’communi-cate’ in different ways. Even applauding in a large audience results in synchronised clapping. Hanging clocks, while hanging on the same wall, can synchronise over time [8]. Certain fireflies tend to start pulsing light at the same time after a while, which is another example of synchronised pulsing [9].

There are different types of coupling oscillators. If each oscillator sees all the other oscillators and can interact with each other one, this is called global or all-to-all cou-pling [10]. One way to achieve global coucou-pling is by mean field coucou-pling. This is when single oscillators see the average of all oscillators, like fireflies seeing the aver-age amount of light or audience-members hearing the clapping of the other audience members. The stronger the coupling factor, the faster and better the audience or fireflies tend to synchronize. Another way coupling can occur is through a shared external perturbation or force [11].

Next the phase of the oscillator has to be defined. Based on this definition, the syn-chronization can be quantified. Then cells with different starting conditions can be compared to understand the speed and way the cells synchronize. The goal of look-ing at the synchronizlook-ing behaviour was to understand better if and how these cells synchronize and how this should be modelled.

After studying the synchronization in a few cells, the Amolf group will also look at the space dependent component. Cells are at different parts of the worm and therefore react differently. Cells will also not react the same if they are relatively close to each other instead of far apart. This will be the next step in studying the synchronization of DAF-16 in multiple cells.

Eventually the model can become a stochastic model, instead of one with Ordinary Differential Equations (ODE’s). The final goal for this project is creating a model which can describe the pulsating behaviour of DAF-16 within a worm. The model can predict the frequency and pulsation duration. In experiments it has been found that these vary between different types of stress and the heaviness of the stress.

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2 Results

2.1 ILP feedback model

The C. elegans is an eukaryote, which means that it has a nucleus and organelles within its cells [12]. In cells proteins, molecules mainly consisting of amino acids, are produced when a transcription factor enters the nucleus. This will then activate the DNA which in turn activates the production of the proteins. There are different types of proteins that serve a variety of functions within the cell.

A receptor is a protein which can receive signals from outside the cell and then interact with intracellular signal proteins. Another type of protein is an enzyme, which can also have different functions. An enzyme can, for example, cut other molecules into separate parts. A kinase is an enzyme that does the coupling between a phosphate group and a protein. This process is called phosphorylating. Dephosphorylating is the process in which the phosphate-group is cut off of the enzyme, due to the protein phosphatase, see figure 4.

Figure 4: This figure shows the phosphoryllation and dephosphorylation process of a pro-tein. A protein can get phosphorylated, acquire a phosphate group, with the help of a kinase. This kinase uses the energy and phosphate group of ATP to phosphorylate the protein. If the protein is phosphorylated it can get dephosphorylated with the help of a phosphatase, which reverses the process.

ATP is the source of energy and the source of the phosphate group for the phospho-rylation process. ATP has 3 phosphate groups and ADP has two, and so a kinase takes a phosphate group off ATP, which makes it ADP.

There are two different types of signal molecules that can bind to a receptor; an agonist and an antagonist. An agonist is a molecule that binds to a receptor and activates a signal pathway within the cell, whereas an antagonist blocks the receptor and deactivates the signal within the cell [12].

To look at the synchronization of multiple cells within a C. elegans, a model with 4 equations for one cell was used [5]. This model is a simplified version of the IIS pathway, shown in figure 1. This simple model can be described with four ODE’s, describing the behaviour of 4 concentrations of proteins and their signal molecule in the model. These include Pre-ILP, Nucleair DAF-16 (NucD), cytoplasmic DAF-16 (CytD) and ILP, see also figure 5.

In this model cytoplasmic DAF-16 phosphorylated and cytoplasmic DAF-16 phos-phorylated are in equilibrium, just like cytoplasmic DAF-16 dephosphos-phorylated and nu-clear DAF-16 dephosphorylated. The total amount of DAF-16 (in the three forms) is assumed to be preserved. If ILP binds to the receptor (DAF-2), cytoplasmic DAF-16 becomes phosphorylated. Nuclear DAF-16, in its turn, exits the nucleus and shuttles into the cytoplasm, which will cause the ILP production to slow down. With a lower concentration of ILP less receptors are occupied, which causes cytoplasmic DAF-16 to become dephosphorylated again. This negative feedback loop causes oscillations

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Figure 5: The ILP feedback model made by Burak Demirbas [5]. This includes the concen-trations of ILP, CytD, CytD (dephosphorylated), NucD and Pre-ILP. Two arrows in different directions between two concentrations is an equilibrium. The star close to the receptor means that it is an agonist and activated the pathway.

in the concentration of the different proteins/signal molecules. The following model of four equations describes the behaviour of the concentrations of the different sub-stances for one cell:

dpre dt = fw∗ nucnw knw w + nucnw − bw∗ pre (1) dcyt

dt = fs∗ ilp ∗ (Dt − cyt − nuc) − bs∗ cyt (2) dnuc

dt = fn∗ (Dt − cyt − nuc) − bn∗ nuc (3) dilp dt = fi∗ preni kni i + preni − bi∗ ilp (4)

Here f represents the rates of incoming molecules, b represents outgoing or degrad-ing rates, n the hill coefficients and k the dissociation constants. In this model the difference in concentration for 1 variable is based on:

d[concentration] = d[production] - d[degradation].

The preservation of the total DAF-16 is used to reduce the amount of equations. This is why the equation of cytoplasmic DAF-16 in the model, equation 2, is the one that is phosphorylated. Here the ODE’s are based on the hill equations.

The Hill equation is a mathematical equation used in different segments of biology as it can describe the saturable binding of a molecule to a receptor. The Hill equation relates the response of a system to the concentration of a pharmacological agonist. In this case the ILP (insulin like peptide) binds to a receptor and activates the rest of the system. The more ILP there is, the more receptors are activated. The production of ILP and pre-ILP in the ILP feedback model is described by the hill equation [13], see equation 1 and 4.

To see the model in action, the ODE’s were numerically calculated, using Python. The concentrations of the different variables were plotted against time, see figure 6. In this model the following parameters were used: ni = 11, nw = 6, fi = 2, fs =

0.6, fn = 0.4, fw = 0.5, bi = 0.3, bs= 0.2, bn = 0.3, bw = 0.2, ki = 0.6 and kw = 0.6.

These parameters were selected so that sustained oscillations were plotted and to fit previous biological research. Eventually the n in the two different hill equations should be around 1 to make sense biologically.

A stable or attracting limit cycle is a stable cycle to which all oscillators in a certain space move. These are typical for non-linear systems, like the ILP feedback model [14]. In the case for the ILP feedback model it is a closed loop in 3-dimensions (Pre-ILP, CytD, NucD), as shown in figure 7. The basin of attraction is the space in which

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Figure 6: Concentration of the 4 different variables in the ILP feedback model, plotted against time.

all the starting points will progress into the same limit cycle. Outside this basin the points can have different endpoints or limit cycles. A bifurcation is a point in parameter-space where the parameters used, instead of giving sustained oscillators, give damped oscillations or no oscillations at all. This means that there is no limit cycle and the oscillations die out.

Figure 7: The limit cycle for the ILP-feedback model. On the three axes cytoplasmic DAF-16, nuclear DAF-16 and pre-ILP are shown (CytD, NucD, Pre-ILP). In the limit of time infinity every oscillator in its basin will reach this limit cycle.

2.2 Synchronizations of multiple cells

After picking the ILP feedback model to describe the oscillations of the different con-centrations within a cell, the synchronization is to be evaluated. To make a system of multiple cells, each cell is assigned their own set of ODE’s, in this case described by equations 1-4, see figure 8. In an organism cells can communicate, this can have effect on the synchronization, therefore multiple ways of communication are presented.

Figure 8: This figure shows that all cells have their own set of ODE’s and the concentration of ILP of each cell interacts in some way with the other cells.

First of all, to check whether cells tend to synchronize without communicating, the model in figure 8 was used. Now the ODE’s of two cells with different starting

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conditions were plotted against time, see figure 9. This shows that no synchronization takes place if cells don’t communicate, as can bee seen by looking at the peaks of the oscillations. This remains true over longer periods of time than graphed in the figure.

Figure 9: This figure shows the solution to the ODE’s over time for two cells with different starting conditions. These cells don’t communicate and therefore don’t synchronize over time. This can bee seen by looking at the peaks of both oscillations, which are at different timepoints.

There are multiple ways cells could be able to communicate. Firstly, they could follow the leader-followers model, see figure 10. In this model one cell changes be-havior, which results in a change in behavior of other cells. In this model that would mean that cell one notices the stress, reacts to it and communicates this to the other cells. As a result one cell is a bit further in the oscillation and all the other cells are synchronized.

Figure 10: The leader-followers model. Cell 1 responds to the stress and communicates this to the other cells.

Secondly a probable model could be that all cells interact with all other cells. In this case the cells all have their own pot of ILP and use this to sent and receive concentrations of ILP.

Here, we focus instead on the shared ILP ’pool’ model, see figure 11. In this case all cells produce ILP for the shared pool and use ILP from the pool. This model assumes that all cells interact the same way with the pool, so no space-dependent component is considered. If cells would have a specific point in space in the model, neighbouring cells would communicate faster than cells that are far away. The diffusion of the ILP

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would then be an important factor in describing the synchronization.

Figure 11: The shared ILP pool model for 3 cells. All cells interact with the shared pool in the same way.

To account for the pool model, equations 1-4 have to be slightly altered. All the cells will still have their own set of ODE’s for Pre-ILP, CytD and NucD but now there is only one pool of ILP. This means that all the cells produce ILP for the pool and receive ILP from the pool. The shared ILP equation (4) will then become

dilp dt = fi∗ X j preni j kni i + pre ni j − bi∗ ilp, (5)

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2.3 The phase of an oscillator

To quantify the synchronization of oscillators the correlation of solutions of the ODE’s is not precise enough, therefore the phase of an oscillator at a certain stage in time should be defined. As oscillators become more synchronized, their phases will become synchronized and the phase difference of two cells goes to zero. The definition of the phase is therefore crucial in quantifying this synchronization. There are multiple ways to define a phase at a certain point. Firstly, the phase can be defined by the angle θ to the x-axes, see figure 12a. This angle can then be calculated with θ = tan_|x||y|= tan|nuc|_|cyt|. In defining θ between (0, 2π) an algorithm is used, see section 4.1 and figure 12b.

(a) The limit cycle for the ILP feedback model with the definition of an angle and therefore a phase. The oscillator that moves over this limit cycle moves clockwise (so against regular direc-tion). To calculate the angle in this 2D figure the mentioned formula and algorithm in section 4.1 is used.

(b) The angle of a point on the limit cycle over 1 period of time. This angle does not increase linearly. This is because the point on the limit cycle moves with different velocities during one cycle.

Figure 12: This figure shows the definition of the phase as an angle over the limit cycle. This angle is defined by an angle with respect to the x-axis, made by the center point. Moving over the limit cycle the phase increased non-linearly.

The problem with this approach is that the phase of a point on the limit cycle does not increase linearly over time. When comparing two different oscillators, this can become a problem. If a cell for example has a phase around 1 and another cell has one around 4, the cells are actually closer to each other on the limit cycle (and therefore have a more similar phase) than two cells with a phase of around 4. This will give a wrong depiction of the synchronization.

Therefore, another approach was used, named the reduction theory in which the speed is also accounted for [11]. The new anti-clockwise definition for θ on the limit cycle is

θ = 2π − ωτ = 2π(1 − τ /T ), (6)

where ω is the derivative of the angle with respect to the time, τ is the time point on which the angle is to be calculated, and T is the period of time which the oscillation takes to go through one cycle. In other words: the period is the time it takes to pass the same point on the limit cycle twice. This approach insures that the angle will increase linearly over time, see figure 13b. This will be useful when an oscillator doesn’t move with a constant speed over the limit cycle. Figure 13a illustrates 15 points with an even spacing in angle, but not in space. The faster the oscillator moves over the limit cycle, the further apart the points are. This definition ensures that, even though points are very far away from each other, when two points have a high velocity, they have a relatively similar phase.

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(a) Limit cycle with 15 points. These points have even spacing of phase, not space itself. Therefore some points are closer to each other in the graph then others. This phase is based on equation 7.

(b) Phase over time, defined by 7. This phase increases linearly over time. The time it takes for 1 round trip around the limit cycle (T) is 14.87 a.u.

Figure 13: This figure shows the definition of the phase by 7. Here the phase increases linearly over time and is therefore not evenly spaced on the limit cycle.

When cells interact, they tend to move off of the limit cycle. So the phase for points off of the limit cycle should also be defined. For all the points in the basin (all the points in a space that will move to the attracting limit cycle over time) the phase can be defined in the following way: at a certain point, the state should be evolved n ∗ T times, with n a positive integer, until the point is sufficiently close to the limit cycle. Then the phase can be defined by the closest point on the limit cycle [11]. An example for this phase definition was given in a paper by Hiroya Nakao [11], see figure 14, calculated by evaluating the FitzHugh-Nagumo model. This could be a better approximation of the phase of an oscillator than by defining the phase by the closest point on the limit cycle because this approximation utilizes the attracting character of the limit cycle.

Figure 14: This figure shows the phase (in colour) of the FitzHugh-Nagumo model. In black the limit cycle is shown. This figure used the definition of the phase described in a paper by Hiroya Nakao [11].

In the FitzHugh-Nagumo model only two variables (u, v) are needed to describe the behaviour of the oscillator. Depending on the number of variables the matrix (as shown in figure 14) with phase values increases in dimension. This approach was tested for the ILP feedback model, as shown in figure 15. Here the two dimensions of NucD and CytD are shown. The used matrix is 3 dimensional. This matrix forms the conversion table for phases. The problem with this approach was that ILP was not included in the conversion table, as it is shared amongst cells. But a different concentration of ILP does actually influence the phase. Another problem with this approach is the time it takes to calculate and the relatively low accuracy for the phase. If a model with more equations than 4 was used, it would take even longer and be

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less accurate.

Figure 15: Phase definition for a certain space of NucD and CytD for a steady concentration of Pre-ILP. For every point in this space, the phase is given by a color, as shown in the bar on the right.

A better way to define the phase off of the limit cycle is by using the intersection point of the line, made by the center and the unknown point, and the limit cycle. For this method the limit cycle is split up into different points. These points can all make a line with their next point on the limit cycle. To find the phase of an unknown (blue) point, see figure 16, the intersection between the line that the center of the limit cycle makes with the unknown point and the limit cycle itself is to be calculated. Then the distance to the point that makes the line on the limit cycle is to be calculated as well. The phase is then defined using linear interpolation.

Linear interpolation is a method used to define variables in a continuous matter. In this case, to define the phase at a certain point, it first has to be calculated for a dis-crete set of points. Now linear interpolation becomes useful to transform the disdis-crete points onto a line. Assuming in the discrete set of 2 consecutive points on the limit cycle the points are close enough to each other that the linear line that connects these two is very similar to the limit cycle itself this interpolation can be applied. To use this method the distance to the points on the limit cycle is to be calculated, where after the new continuous phase is calculated with:

phase = point1 on LC + distance * (point2 on LC - point1 on LC) (7) While implementing this technique, one has to keep in mind that the line from the

Figure 16: This figure shows a limit cycle in blue. The center (green point) makes a line with the blue point intersecting the limit cycle on the red point. This red point lays on the red line, which defines the limit cycle. The red point and line can be used to calculate the linear interpolation.

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center to the unknown point also intersects the limit cycle in the other direction, as it is not a line segment.

For this method the center needs to be defined. In this paper the center is defined as: Cx=

xmax− xmin

2 + xmin, C = (Cx, Cy, Cz), (8)

where (Cx, Cy, Cz) are the coordinates of the central point and x, y, z the variables, so

in this case Pre-ILP, CytD and NucD. xmax and xmin are respectively the maximum

and minimum values of the limit cycle of that variable. Another definition of this central point could be that of an average point. This definition would look like the following Cx= X i xi N, for xi in {x1, ..., xN}, C = (Cx, Cy, Cz), (9) where {x1, ..., xN} the set is of all values of one variable with even time spacing.

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2.4 Synchronization dynamics

To illustrate the shared ’pool’ of ILP, a basic example with cell 1 at phase = 0 on the limit cycle and cell 2 at phase = π on the limit cycle was simulated, see figure 17 and 18.

Figure 17: The concentrations of ILP, Pre-ILP, CytD and NucD for two cells. Cell 1 starting at phase = 0 and cell two starting at phase = π, thus at different starting concentrations. This illustrates the shared-ILP pool model. At first sight these oscillations tend to synchronize.

Figure 18: Two points (green and blue) beginning on different points on the limit cycle (black line). These cells share a pool of ILP and tend to move to the same phase over time. The smaller dots are even spaced in time and show the behaviour of the concentration of Cytoplasmic DAF-16 and Nuclear DAF-16 over time.

Over time the cells tend to synchronize, see also 17. To quantify this synchroniza-tion process the phase was defined in various ways. The phase was defined taking speed into account, see equation 7. The phase off of the limit cycle was first defined by finding the closest point on the limit cycle and using that phase. When two cells are around phase 2π and 0, their phases are actually very close to each other, instead of 2π apart. To account for this circular motion, the phase of cells was defined in an increasing way. Every time an oscillator moved over the 2π point, 2π was added to the original phase. This resulted in figure 19a. In figure 19b the average phase (of all cells) was subtracted from the original phase to illustrate the synchronization. If this phase difference goes to 0, cells are synchronized.

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(a) This shows the increasing phase over time.

(b) This shows the phase of both cells - the average phase of the two. This stays around 0, but not completely on 0 due to its phase definition.

Figure 19: Phase of cell 1 (beginning at 0) and cell 2 (beginning at π) sharing a pool of ILP over time. These cells tend to synchronize within one or two oscillations. This figure uses the phase definition of the closest point on the limit cycle.

In comparison, if the exact same model was used but the ILP was NOT shared, the cells do not tend to synchronize and stay apart, see 20a and 20b.

(a) This shows the increasing phase over time. (b) This shows the phase of both cells - the av-erage phase of the two.

Figure 20: Phase of cell 1 (beginning at 0) and cell 2 (beginning at π) NOT sharing a pool of ILP and does not synchronize over time. This figure uses the phase definition of the closest point on the limit cycle.

Ultimately the phase was defined, with speed taken into account, using the in-tersection method, described in the end of section 2.3. For a few different starting conditions (with sharing the ILP) the phase over time was plotted, see figure 21a, 21b, 22a, 22b.

In all these figures the phase difference goes to zero over time and therefore do all the cells synchronize over time. It is striking that all these configurations take the same amount of time to synchronize, namely around 1.5 time period. This would mean that all cells, independent of the phase, synchronize over time (when they originally have sustained oscillations).

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(a) Two cells, with cell 1 starting at phase = 0 and cell 2 at phase = π.

(b) Five cells starting at five different phases, cell 1 at 0, cell 2 at π/4, cell 3 at π/2, cell 4 at π, cell 5 at 3π/2.

Figure 21: This figure shows the phase of multiple cells - the average phase of those mod 2π over time. These cells shared a pool of ILP. As seen in both figures, the phase difference tends to go to zero and thus tend to synchronize. Both figures tend to synchronize in about the same time, this is around 1.5 times one oscillation. This figure uses the intersection method to define the phase.

(a) Six cells starting at different phases, cell 1-3 at 0 and cell 4-6 at π.

(b) Six cells starting at different phases, cell 1-5 at 0 and cell 6 at π.

Figure 22: This figure shows the phase of multiple cells - the average phase of those mod 2π over time. These cells shared a pool of ILP. As seen in both figures, the phase difference tends to go to zero and thus tend to synchronize. Both figures tend to synchronize in about the same time, this is around 1.5 times one oscillation. This figure uses the intersection method to define the phase.

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3 Conclusion and Discussion

In short, for the ILP feedback model the different cells in the C. elegans do tend to synchronize over time. The best way to describe the phase so far, when having more than two variables, is the method that uses the definition of the phase on the limit cycle by equation 7 and off of the limit cycle by the intersection method. This method was based on defining the phase on the limit cycle, taking speed into account, and then using the intersection method to define the phase on the whole basin. The cells in this model tended to synchronize over time. This synchronization occurred within one time period, where the period is the time the oscillator takes to move 1 round trip over the limit cycle. With a maximum number of 9 oscillators, more has not yet been examined, the synchronization took approximately the same time, independent of the phases of these oscillators. The quantitative developmental group will use this method to look at different simple models of the IIS pathway and incorporate space into the model as well. This will give a valid method of describing the behaviour of the oscillators and describe the found results in experiments.

In the intersection method used to define a phase off of the limit cycle, there was assumed that the limit cycle has a simple shape. If a limit cycle has a shape like e.g. figure 23, the line through the point and middle point has three intersections on the limit cycle. This would not give an unambiguous definition. It is not likely that the shape of a limit cycle for a system for the IIS pathway would look like this, but the shape should be checked when using this method. The limit cycle could also have a shape that is difficult to work with if the oscillations are just pulses.

Figure 23: A random limit cycle with a shape that will not work for the intersection method. When defining the phase at the ’x’, the intersection method will find 3 points on the limit cycle. If the x is precisely between the two intersections, the phase can not be defined.

In defining the phase, the central point in the limit cycle was used (equation 8), instead of the average point (equation 9). Using the average point will give a different point to use when intersecting for the phase definition and therefore another result for the phase. One could argue that this definition would be a better fit. The first definition was chosen (equation 8) because if the speed of the oscillator over the limit cycle has a large distribution, a huge portion of the points on the limit cycle would be relatively close to each other. This way the phase would likely be less accurate. Furthermore, the definition of synchronization assumes that the limit cycle is defined by the variables that work within the cell. That means that for the ILP feedback model used in this thesis Pre-ILP, CytD and NucD define the limit cycle, not ILP. The concentration of ILP does have influence on the behaviour of the oscillator though, so it could be argued that the shared-ILP should also be a part of the limit cycle in all cells. As the ILP is shared, not taking the ILP concentration into account for the limit cycle and therefore phase is the best solution. This would only influence the phase in a way that is not interesting for the synchronization quantification.

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basin of the limit cycle. This assumption is valid if the oscillators stay close enough to the limit cycle and are not close to a bifurcation.

The point in using the ILP feedback model, as seen in figure 5, is that it is relatively simple and easy to study. The full IIS pathway, as seen in figure 1, is a more compli-cated version of the same process. By simplifying the IIS-pathway not all variables are taken into account and therefore the ILP feedback model is not fully descriptive. If another model was used, the synchronization could be very different. It is possible that the synchronization takes much longer or that the cells do not synchronize over time at all.

A next step in researching the synchronization of this model could be finding the best simple model for the IIS-pathway first. This can then be used to look at the synchronization of the different cells as different models can give very different predic-tions for the speed of the synchronization. An Interesting question can then be: Is the synchronization of cells different in different parts of the worm? This can be studied by changing the parameters of the equations in the ILP feedback model, which could result in different limit cycles. Different cells have different properties and could, for example, produce ILP’s faster or not at all, this could influence the synchronization. Another interesting step could be to take space into account. Do the ILP’s tend to spread evenly over different cells or do they diffuse slowly? How does the diffusion ve-locity influence the synchronization of the cells? Do spatial waves arise in the model? Eventually the model can become a stochastic model, instead of one with ODE’s. This will give a more realistic image of the synchronization. With random fluctua-tions in the model, the synchronization will most likely become slower.

Questions that can be answered in further research could be: Is there a common force that influences the cells (for example an external producer of ILP)? Could this com-mon force also result in the synchronization of multiple cells? What happens with the oscillations if not all cells interact with the pool of ILP’s the same way? If cells are all a bit different (so they all have different parameters) can they still synchronize? If a group of cells have different parameters than the other cells, do they synchronize with themselves or with the whole group?

Acknowledgements

I would like to thank my daily supervisor, Burak Demirbas, for his help and inspira-tion. Also, I would like to thank Jeroen van Zon for his supervision, inspiration and feedback during this thesis. Unfortunately we were not able to meet up in person on many occasions, due to the current pandemic, but they made the most of it. Further-more I would like to thank the research group Quantitative Developmental Biology for their feedback and insights in their research. I am glad to have been given the chance to take a closer look in a research group. Lastly I would like Pieter Rein ten Wolde for being my examiner.

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4 Appendices

4.1 Algorithm to calculate angle clockwise

cytdif = conc_cyt - center_cyt nucdif = conc_nuc - center_nuc θ = arctan(|nucdif |_{|cytdif |})

if nucdif > 0 and cytdif > 0: θ = 2π − θ if nucdif > 0 and cytdif < 0: θ = θ + π if nucdif < 0 and cytdif < 0: θ = π − θ

Modelling body-wide synchronization of DAF-16 oscillations