A Mathematical Model for RomR Polarized Distribution During Reversal andDivision in M.xanthus December 2, 2016

Francesco Pancaldi1, Shant Mahserejian2, Chinedu Madukoma3, Jianxu Chen4,Joshua Shrout5, Mark Alber6.

1Department of Applied and Computational Mathematics and Statistics, University of Notre Dame2Department of Applied and Computational Mathematics and Statistics, University of Notre Dame3Department of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame4Department of Computer Science and Engineering, University of Notre Dame5Department of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame6Department of Applied and Computational Mathematics and Statistics, University of Notre Dame

1

Contents

1 Introduction 4

2 In vivo Data Collection 5

2.1 Data Collection Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Quantifying Data from Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Data Analysis & Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.1 Raw RomR Measurements during a Reversal Event . . . . . . . . . . . . . . . . . . . . 7

2.3.2 Relative Difference of RomR Measurements between Leading and Lagging Poles . . . 8

3 Mathematical Model 10

3.1 Bio-Physical assumptions for Myxococcus Xanthus Polarity . . . . . . . . . . . . . . . . . . . 10

3.2 Partial Differential Equations model 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.3 2D stochastic model for diffusion in changing domain . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.1 Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.2 Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3.3 Particle step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.4 Collision detection and Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.5 Computation of relevant measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.6 Additional: Receptors component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.7 Current and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Models extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4.1 Growth Component of 1D PDEs model . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4.2 Alternative Wave-like Initial Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.3 Larger Division Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.4 Post Division Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Model Results 19

4.1 1D Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 2D Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2.1 Idea to avoid receptors in 1D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3 Additional 1D Model Components Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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4.3.1 1D model Growth Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3.2 Bigger Central Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3.3 New IC Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3.4 Post Division Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Conclusions 28

3

1 Introduction

M.xanthus is a rod-shaped bacteria that tends to reverse its preferential direction of movement on averageevery 7-10 minutes with a certain probability distribution. This reversal is done through changes in bacteriapolarity. During this changes the motility engines of the cells switch location, from one pole of the cell tothe other, hence inverting the direction of motion. One of the chemical component of the cell believed toinfluence the polarity is the protein RomR. This idea is supported by some known interactions of RomRwith other recognized components of the motility engine, but most of all by the observable RomR patternin the body of the bacteria. In fact, fluorescent microscopy pictures show that RomR tends to form twomajor concentrations at the two pole of the cells, where the motility engines are located. Furthermore, whencomparing the levels of these two peaks, analyzing the fluorescence levels, the pole at the back, also calledlagging pole tend to have an higher RomR concentration than the one at the front, also called leading pole,and around the time of direction-reversal also the concentration level seem to change from one pole to theother. Therefore, suggesting that the reversal in direction and the one in concentration levels are related.In the literature usually its assumed that the change in RomR comes first, and this induces a change inthe motility engines and therefore in the direction of movement. However, the experimental data we havecollected seam to show that the direction of movement changes first and RomR follows. Additionally, in ourexperiments we have observed that during division the bacteria interrupts all movement, and a new peak inthe concentration of RomR appears at the division cite, where the cell membrane splits to form the new polesof the daughter-cells. We will first present a simple 1D PDE model inspired by the MinCDE system in E. coli.This simple model is able to reproduce most of the qualitative behaviors for the concentration of RomR alongthe cell, as seen in experiments. In this model, to reproduce the effect of division on the diffusion process,we will assume that the formation of a new boundary at the center will induce a reduction in the mobility ofthe particles around that location, hence affecting the effective diffusion coefficient. Therefore, our equationsused a non-constant diffusion coefficient as in fig 1. This may appear as a reasonable assumption when we

Figure 1: Diffusion profile for 1D PDE model.This plot shows the evolution of the diffusion coefficient along the bacteria central axis (x-coord.) during division. The initial

value of the diffusion coefficient (y-coord.) is constant along the all domain, but with the progress of division a valley is formed

around the center with minimum value at the center, and this minimum value decreases with time till it reaches 0 at the very

end of division, when the two daughter cells are completely formed and about to divide.

consider the problem from a 2D or 3D prospective, since the formation of new pole is going to create a regionclose to the center where each particle has less freedom of movement. However, is not intuitively clear whatshould be the shape of the ”diffusion profile” (function describing the diffusion coefficient along the x-axisand time) during division. In other words, even if its clear that around the center the mobility of a particleis more limited, there are a number of properties of this change in diffusibility that are not easy to predict:speed of change, minimum value, size of the region around the center with non-constant coefficient, shapeof the profile, etc.For this purpose we decided to implement a 2D stochastic model to simulate the movement of a certainnumber of particles inside a capsule-shaped domain, like the one of M. Xanthus, while the domain changesto form two symmetric ones. Our hope is that analyzing the movement of this particles we will able to

4

confirm our hypothesis deriving a plot similar to fig 1.Moreover, we will also show that this stochastic modeldoes not require receptors forming at the center to produce a new polar zone for RomR at the location ofdivision. This recently lead us to an idea that could allow us the need for receptors to justify the formationof the polar zones, and may explain the results obtained in experiments when investigating the relationbetween RomR and direction of movement in M. xanthus. We will first provide an accurate description ofthe 2D stochastic model together with all the most recent simulation results, and will conclude explainingthe idea mentioned above.

2 In vivo Data Collection

2.1 Data Collection Process

M. xanthus strain 1622psh, wild type that are gfp-tagged for RomR expression, were used in this studyon the effects of dynamic sub-cellular processes regulating motility and polarity during cell reversal events.This process is in development to incorporate division events as well, but at this stage adequate data ondivision is not available. So, the focus here is to successfully capture measurements during reversals beforemoving onward. The bacteria samples were prepared on agar chambers able to handle a cover slip suitablefor oil immersion microscopy (as done with previous work). The confocal microscopy made use of Nikonequipment and software, using a 100x 1.49NA oil immersion objective lens to deliver the clearest possibleimages using the Galvano mode. Time-lapsed images were captured in 2-channels: TD (gray-scale images,used for segmentation and tracking) and fluorescence (green images 488nm wavelength, used to collect dataon the RomR protein). The laser intensity was minimized to avoid the halting behavioral side-effects, whichallowed for the long-duration observations of undisturbed bacteria. The pinhole size was optimized to allowfor a more heightened capture of intensity while maintaining image clarity, especially in the fluorescencechannel. Finally, we made use of the digital zoom provided by the software without loss of resolution(typically 2x-3x zoom, giving a scale of 0.25 micron/pixel). Doing so enable the observation of individual andrelatively isolated cells undergoing complete reversals in their direction of motion, while recording intensitiesof gfp-tagged RomR proteins expressed by the cell. As shown in Figure 2a, these settings provided theclearest possible video image sequence at the fastest capture rate available. Images were recorded every 0.5seconds (or 2 frames/second) at a spatial resolution of 512x512. Typical total durations range from 45-60minutes, but individual events of interest tend to be much shorter. The image field captured 20-70 cells at atime, and mostly spans their complete movements throughout the region. ⇒ These microscopy proceduressuccessfully provided 2-channel time-lapse images (or videos) at the fastest time-step that we have capturedM. xanthus motion and sub-cellular dynamics, and provides potential data on many cells and their RomRpoles simultaneously, with no microscopy limitations on side-effects, clarity, and sample duration.

2.2 Quantifying Data from Images

In order to extract data about RomR from the images, information for individual cells must first be separated.This step, done by hand, identifies cells undergoing reversal (and division) events while remaining relativelyisolated throughout the duration of the video. The need for isolated cells arises from difficulties whencollecting data in the cell poles, which tend to merge with neighboring cells when they touch, thus makingthe boundaries to the region of interest too vague. A video sample is cropped to capture the entirety of theobservable behavior. For consistency, only videos of reversal events were kept if the cell traveled at leastone cell length before and after the time of reversal. Figure 2a is from one such time-lapse image sequence.Samples that met the necessary criteria were submitted for processing. The image processing algorithmutilizes the gray-scale TD channel images for segmenting the region of interest, and tracking cells betweenframes. The finer time resolution in this sample set made for little movement in the bacteria, which eased thecell tracking ensured data collection frame to frame. Even so, the compromised reduction in spatial qualityled to difficulty in the fluorescence channel, including the need for cells to remain isolated. In order to collectthe data for RomR in the cell interior, a bi-channel segmentation approach was used, where the TD channelprovides the pixels comprising the region of interest, and the corresponding pixels in the fluorescence channelare used for data collection. To the cell body identified in the gray-scale image, an extra buffer zone is added

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(a) (b)

Figure 2: Reversal event of an isolated M. xanthus cell captured and processed using 2-channel fluorescentmicroscopy time-lapse imaging, during which the cell moves forward (T=0min), pauses (T=2.5min), andthen moves in the opposite direction (T=5.0min). (a) Gray-scale TD (left) and green fluorescence (right)channels during a reversal event. (b) Images a prepared for in vivo data collection by using (left) TD imagesto segment and track the region of interest throughout the time-lapse image sequence, and processing (right)the green fluorescence images in the corresponding regions.

near the ends in order to capture the entire pole region in the green channel for each frame. Once the regionhas been segmented, some typical processing is done: de-noising to reduce the contributions from backgroundnoise, and Gaussian blur to smoothen out the rough edges in the image data. Figure 2b shows the results ofsegmentation and tracking via the TD channel, and the corresponding green fluorescence regions processedfor data collection. Once the processing for each frame is completed, the cell body is segmented and a centralskeleton line is identified for the observed cell in the gray-scale channels. The cell body is then divided into12 compartments along this skeleton line. In these corresponding compartment regions in the fluorescentchannel, the intensity of RomR expression is captured, where two compartments at each end of the cell bodyconstitutes a pole region. By doing so, the following quantitative information has been collected from thesegmented cell image.

• Cell Length

• Total Intensity

• Local Intensities at 12 sub cellular compartments

• (x,y) coordinates of each sub cellular compartment centroid.

In addition, information about the time instances when the cell is undergoing a reversal event and leadingpole orientation were also provided to aid analysis.

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2.3 Data Analysis & Experimental Results

In total 21 movies were captured, 80 reversal events were observed, of which 68 had reliable video quality, andof those only 40 traveled one cell length before and after reversal. Of the first 10 that have been processed,only 4 were error-free. Here a summary of the inventory numbers:

• 21 movies recorded;

• 40 sample videos of reversal meeting criteria;

• 24 pending processing;

• 10 Error-free processing completed - used for data analysis;

2.3.1 Raw RomR Measurements during a Reversal Event

Using those samples that met the quality standards for collecting accurate data, measurements were madeof the gfp-tagged RomR protein intensities from the processed green fluorescent images at 12 positions alongthe cell body at different points in time. The study here limits the intensities measurements to the time spenttraveling one-cell length both before and after a reversal event occurs. Normalizing these values produces theplot in Figure 3, which showcases the raw RomR levels before (green) the reversal event, and after (blue),where the ?0? location corresponds to the leading pole region. It is clear that prior to the reversing, thelagging pole has almost twice the levels of RomR than the leading pole. Immediately after the reversal event,there is a period of time where the new leading pole is brighter than the new lagging pole, a consequence ofthe continuity in RomR levels during the polarity switch. During the time after the reversal event, the newlagging pole RomR levels begin to increase, while the new leading pole RomR levels decrease, thus restoringthe orientation with more RomR at the rear of the cell body. It is worth noting that polarized RomRbrightness switch is occurring after the switch in motility direction indicating that the change in directionof motion is causing the change in RomR brightness orientation.

05

1015

−1−0.500.510.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Relative Intensity of RomR Expression One Cell−Length Before (Green) & After (Blue) Reversal

Location along Cell Body

Time

Figure 3: RomR measurements, normalized with respect to the total cell intensity, during one-cell length oftravel before (green) and after (blue) reversal event.

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2.3.2 Relative Difference of RomR Measurements between Leading and Lagging Poles

Another perspective for quantifying the behavior of RomR intensities is to calculate the relative differencebetween the poles as illustrated in Figure 4 via the following formulation:

y =Ileading − Ilagging

ITotal

where Ileading, Ilagging and ITotal are the measured RomR intensities in the leading pole region, the laggingpole region, and throughout the entire cell respectively. By the nature of the formulation, positive valuesindicate more RomR in the leading pole, and negative values indicate more RomR in the lagging pole.These values were plotted for time steps corresponding to one-cell length distance traveled before and aftera reversal event, indicated by the blue vertical line. It is worth noting that prior to reversing, the laggingpole seems to have about 20% more RomR visible. The discontinuity at the time of reversal is due to thecontinuity of RomR levels in each pole while the polarity orientation is switched. The leading pole remainsthe “brighter” pole up until the cell travels about half its body’s length, after which the two poles illuminateroughly the same amount of RomR. Unfortunately, the polar RomR being re-established back to levels priorto the reversal event are not observed due to limitations and availability of reversing cells during microscopyexperiments. However, this plot supplies further evidence that (1) the polar intensities of RomR in motilecells are typically with 20% differences, a level too small to detect by naked eye, and (2) that the change inRomR intensities surely occurs well after cell motion in the opposite direction has been regained.

Figure 4: Relative difference of RomR measurements between the leading and lagging poles during one-celllength of travel before and after reversal event. Positive values indicate that there is more RomR in theleading pole, and negative values indicate that there is more RomR in the lagging pole. Note the suddenjump during the reversal event (blue vertical line) where the leading and lagging poles are switched.

Figure 5: This plot shows the sample mean (solid lines) and +/- the sample standard deviation (dashedlines) at each time step before and after the reversal event (green and blue respectively).

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Figure 6: This plot shows the sample max and min (red dashed), the sample median (black dashed), andthe middle 50% of the sample (shaded red) at each time step.

Figure 7: The previous two plots combined

Figure 8: This plot shows all 10 plots in the sampled data set

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3 Mathematical Model

3.1 Bio-Physical assumptions for Myxococcus Xanthus Polarity

The bacteria M. xanthus is a model organism used to study collective motion of cells, is a Gram-negativebacteria, and is commonly found in top soil. This bacteria has a capsule or rod shape, with length between4 and 10 µm in length and 1 µm in diameter. The bacteria uses two motility mechanisms, A-motility andS-motility. The first, also called ”Adventurous” motility, is mainly used to explore the environment and thesecond, also called ”Social” motility, is used primarily for collective motion. One of the main characteristicsin M. xanthus movements is a complete inversion in its direction of motion that happens on average every7-9 minutes. After one of this events the bacteria moves with a preferential direction opposite to the onebefore reversal. These events are believed to be related with the distribution of a protein called RomR insidethe bacteria body. Specifically, RomR mostly accumulates at the poles (i.e. two extremities) of the bacteria.However, in our recent experiments we were able to visualize a significant portion of the protein diffusingin the middle of the cell (from one pole to the other). The concentrations in the two poles are typicallynot identical, with the leading pole (front) having less RomR than the lagging pole (back). Moreover, areversal in concentrations was observed around the reversal events mentioned above. Interesting behavior inM. xanthus motility also occurs during division events, when the cell stops all movements despite its highlysocial tendencies, and a new RomR concentration peak can be observed corresponding to the formation oftwo new poles at the division septum. Moreover, hyper-reversing mutants present additional concentrationspots along the cell body that seem to not change location during cell movement, reversals, and division.This further drives the hypothesis that the RomR molecules bind to receptors located inside the cell, ratherthan coagulate at the poles due to the geometric confines of the cell body. These empirical observationslead to several assumptions for a 1D model in which the sub-cellular dynamics of the RomR protein aresimulated:

1. RomR protein exists in two forms: bounded to the membrane, and unbounded in the cytoplasm;

2. Each RomR form has a specific diffusion coefficient (0, or very low for the bounded one);

3. RomR can change from one form to an other depending on attachment and detachment rates;

4. RomR can bind to the membrane (i.e attach) only if free active receptors are available (i.e. not occupiedor blocked);

5. Receptors are present only close to a pole, or at the cell center during division (where new future polesare formed);

6. During division, progressively more receptors are introduced near division septum at the cell center;

7. One RomR receptor can bind only a fixed number of RomR molecules (possibly more than one moleculeper receptor);

8. RomR receptors can be either active or non-active, and only active receptors can bind RomR to themembrane;

9. RomR receptors are activated (or deactivated) by a protein that diffuses in the cell body and reactswith other proteins similarly to the MinCDE system in E. Coli (see Reference);

10. During division, the two daughter cells can exchange protein for a certain amount of time (either freelyor with certain limitations);

11. When the two daughter cells are completely separated, communication between them is interrupted;

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known. M. xanthus is among many bacteria lacking a clear MinCDsystem that drives the recruitment of FtsZ for division. It is knownthat the middle of M. xanthus cells is marked by the ParA-likeprotein PomZ (7). There is support for an association of PomZwith setting of M. xanthus motility, as pomZ (originally annotatedas agmE) was originally identified as a partial A motility mutation.

We describe the resetting of M. xanthus polarity at division bycorrelating the accumulation of RomR at newly formed cell poleswith cell division (Fig. 7). Our results are consistent with an ex-planation that pausing of motility is a well-ordered step of the cellcycle. Recent evidence of the detailed orchestration of ParA/ParBimportant to chromosomal segregation suggests a distinct cycle ofapproximately 4 h where division into two cells accounts for 30 to60 min (60, 61). On the basis of our results, we link cell divisionwith the establishment of opposing motility polarity in progenycells by considering the possible RomR distribution scenarios. Weassume that sufficient phosphorylated RomR is diffused freelythroughout the cell (Fig. 7). As the motility of these predivisioncells is paused, we deduce from our experiments that RomR hasnot yet begun to accumulate via dephosphorylation at the site ofdivision (Fig. 5) but continues to diffuse freely in the phosphory-lated state. However, we propose that diffusion across the entirepredivisional cell starts to become limited at this stage (Fig. 7C)because of the constriction of cell division, limiting flow betweenthe two cell ends. This constriction also introduces a morphologychange as curvature at the predivisional cell middle is initiat-ed—we propose that RomR recognizes some component of thisdeveloping cell pole, as it must recognize existing poles. This maybe directly associated with M. xanthus ParA, which is known tolocalize to cell poles and sites of division (60, 61). While diffusionof RomR continues, the level of asymmetry in RomR distributionat the old poles of the parent cell is mirrored in the new poles at thesite of division, which we measured by using RomR-GFP (see Fig.S2 in the supplemental material). While the actual ratio of mea-

sured leading/lagging pole RomR levels of any single cell varies(from 0.43 to 0.83), the mirroring of these levels from parent toprogeny cells is very consistent (ratio of 1.06 ! 0.2 new polesinheriting old-pole RomR in both progeny cells). Upon the com-pletion of cell division, the accumulation of RomR in the laggingcell is sufficient to recruit MglB to initiate a new direction, ex-plaining why we saw progeny cells move away from each otherfollowing division.

Morphologically symmetrical M. xanthus cells inherit a clearasymmetry in the distribution of proteins that confer their motil-ity. We propose that this asymmetry is mirrored at the parent cellmidpoint because of the process of division to explain the oppos-ing polarity we observed when division was complete. This pro-posed mechanism would be sensitive at the time of motility paus-ing to the distribution of RomR, which is known to switch fromthe asymmetric pattern to a short-lived symmetric pattern to theopposite asymmetric pattern during cell directional reversals (52).Thus, disruption of the Frz system, which effects reversal timing,would be expected to disrupt the polarity pattern inherited bydaughter cells, as seen in our experiments. Because most cell typesare symmetrical, like the M. xanthus cells we examined here, gain-ing more insight into the coordination cascade that regulates thisphenotype may useful in understanding other processes that areassociated with cell division.

ACKNOWLEDGMENTSThis work was funded by a National Institutes of Health grant(R01GM100470).

Strains and helpful discussions were provided by John Kirby (DZ2,DZ4482, and DZ4483), University of Iowa; Lotte Søgaard-Andersen(pSH1208), Max Planck Institute for Terrestrial Microbiology; and DanWall (DK1240, DK7881, DK8621, and DW706), University of Wyoming.Assistance with some measurements was provided by Eric Fein and DanielAlber.

REFERENCES1. Hughes V, Jiang C, Brun Y. 2012. Caulobacter crescentus. Curr. Biol.

22:R507–R509. http://dx.doi.org/10.1016/j.cub.2012.05.036.2. Kirsebom LA, Dasgupta S, Fredrik Pettersson BM. 2012. Pleiomor-

phism in Mycobacterium. Adv. Appl. Microbiol. 80:81–112. http://dx.doi.org/10.1016/B978-0-12-394381-1.00004-0.

3. Horvitz HR, Herskowitz I. 1992. Mechanisms of asymmetric cell divi-sion: two Bs or not two Bs, that is the question. Cell 68:237–255. http://dx.doi.org/10.1016/0092-8674(92)90468-R.

4. Brown PJB, Kysela DT, Brun YV. 2011. Polarity and the diversity ofgrowth mechanisms in bacteria. Semin. Cell Dev. Biol. 22:790 –798. http://dx.doi.org/10.1016/j.semcdb.2011.06.006.

5. Reichenbach H. 1999. The ecology of the myxobacteria. Environ. Micro-biol. 1:15–21. http://dx.doi.org/10.1046/j.1462-2920.1999.00016.x.

6. Berleman JE, Scott J, Chumley T, Kirby JR. 2008. Predataxis behavior inMyxococcus xanthus. Proc. Natl. Acad. Sci. U. S. A. 105:17127–17132. http://dx.doi.org/10.1073/pnas.0804387105.

7. Treuner-Lange A, Aguiluz K, van der Does C, Gómez-Santos N, HarmsA, Schumacher D, Lenz P, Hoppert M, Kahnt J, Muñoz-Dorado J,Søgaard-Andersen L. 2013. PomZ, a ParA-like protein, regulates Z-ringformation and cell division in Myxococcus xanthus. Mol. Microbiol. 87:235–253. http://dx.doi.org/10.1111/mmi.12094.

8. Campos JM, Zusman DR. 1975. Regulation of development in Myxococ-cus xanthus: effect of 3=:5=-cyclic AMP, ADP, and nutrition. Proc. Natl.Acad. Sci. U. S. A. 72:518 –522. http://dx.doi.org/10.1073/pnas.72.2.518.

9. Xie C, Zhang H, Shimkets LJ, Igoshin OA. 2011. Statistical imageanalysis reveals features affecting fates of Myxococcus xanthus develop-mental aggregates. Proc. Natl. Acad. Sci. U. S. A. 108:5915–5920. http://dx.doi.org/10.1073/pnas.1018383108.

10. Zhang H, Angus S, Tran M, Xie C, Igoshin OA, Welch RD. 2011. Quan-

FIG 7 Model of cell division and polarity inheritance via diffusion of RomR inM. xanthus. (A) A motile predivision cell moves left to right. RomR is localizedto both poles but preferentially to the rear pole, while RomR is freely diffusedin the cytoplasm. (B) The predivisional cell pauses its motility. Key divisionproteins PomZ and FtsZ act to mark the cell division site and initiate separa-tion. (C) As FtsZ constricts the cell, this both limits the diffusion of RomRacross the entire volume and cues the accumulation of RomR at these newlydeveloping poles. RomR preferentially accumulates at the new pole of thelagging cell. (D) Upon completion of cell division, these progeny cells displaysufficiently differing polar traits to initiate motility in opposing directions.Synthesis of new TFP may coincide with these actions, but they are not re-quired to initiate motility.

Myxococcus xanthus Cell Division Resets Polarity

November 2014 Volume 196 Number 22 jb.asm.org 7

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Figure 9: Schematic of sub cellular activity during division, with depiction of known proteins distributionsbefore (A), during (B and C), and after (D) division.

3.2 Partial Differential Equations model 1D

According to the list of assumptions in Section 3.1 on the behavior of RomR, the following equations for a1D PDE model is proposed:

• Equation for cytoplasmic, unbounded RomR (note:C = C(x, t)).

∂C

∂t=

∂

∂x

(D∂C

∂x

)− rαC +Rc (1)

where D is the diffusion coefficient (could be space dependent), r and R are the binding and unbindingrates (respectively) to the RomR-receptors, and α = min(n − c, 0) with n number of active receptors(n− c number of free active receptors).

• Equation for RomR bounded to the membrane (note:c = c(x, t)).

∂c

∂t= rαC −Rc (2)

• Equation for density of active receptors.

n(x, t) =

f(N, k, h), x ∈ [0, l1] ∩ [L− l1, L] t ≤ T2

g(t)f(N, k, h), x ∈ [L/2− l2, L/2 + l2] T1 ≤ t ≤ T2

(3)

where l1 and l2 indicates respectively the region near the poles and near the center, T−1 and T2 are thestart and end times of division, N is the maximum density of active receptors at any given location, fis a function quantifying the amount of active receptors dependent on N and the quantity of chemicalsk, h, and g is a function for the growth of new receptors at the center. For example, possible forms ofEquation (3) are:

n(x, t) =

N k

kmax, x ∈ [0, 1] ∩ [9, 10] t ≤ T2

t−T1

T2−T1Nk

kmax, x ∈ [4.5, 5.5] T1 ≤ t ≤ T2

11

or

n(x, t) =

N(

1− kkmax

), x ∈ [0, 1] ∩ [9, 10] t ≤ T2

t−T1

T2−T1N(

1− kkmax

), x ∈ [4.5, 5.5] T1 ≤ t ≤ T2

• Equations for the MinCDE-like system

∂K

∂t=

∂

∂x(DK

∂K

∂x)− σ1K

1 + σ′1h+ σ2hk

∂k

∂t=

σ1K

1 + σ′1h− σ2hk (4)

∂H

∂t=

∂

∂x(DH

∂H

∂x) +

σ4h

1 + σ′4K− σ3KH

∂H

∂t= − σ4h

1 + σ′4K+ σ3KH

where the variables k,K seen in Equation (eqn:Receptors) are the densities for unbounded and boundedform of a chemical that interferes with the receptors and interacts with a second chemical h,H (theseequations are taken from the literature about the MinCDE system and have been shown to produceoscillation from on pole to the other with a period T dependent on the parameters)[Note: we couldalso assume that both chemicals interact with receptors, or only the second one (with appropriatechanges)].

To take into consideration the formation of the new ”boundary” at the center, we consider our diffusioncoefficient to be a function of space and time close to the center of the cell. Therefore, expanding thediffusion term from Equation (1) and similarly for the diffusion terms in system of equations in (4) gives thefollowing:

∂

∂x

(D∂C

∂x

)= D

∂2C

∂x2+∂D

∂x

∂C

∂x(5)

For D, a reasonable assumption will be to consider a symmetric function equal to a constant D0 away fromthe center, and with only one minimum point at the center equal to D0 for t ≤ T1, which that tends to 0 whent → T2 (see Figure 10). This way the classical diffusion term is used before division, while during divisionaround the center, there will be an advective term pointing away from the center due to the slope of thefunction for D, and the flux across the center will gradually reduce to zero (see Figure 11). Therefore, the

Figure 10: Diffusion coefficient as a function of space and time. Lower minimum correspond to longer timepast since division started. At the end of division the minimum is 0 (or a small ε)

12

Figure 11: Strength of the advective term (i.e. space derivative of D). Negative correspond to left directionand positive to right direction.

varying diffusion coefficient acts as a repulsive ”condition” at the center. This choice is somewhat justifiableby the fact that the change in the domain is producing a correspondent change in the ”effective” diffusibilityof the RomR particles, limiting it in one of the two possible directions due to the change in domain geometry.However, the manner and timing in which the diffusion coefficient changes from being constant to the finalprofile in Figure 10 is still unclear (see 2D stochastic simulation). Additionally, as mentioned in Section 3.1new receptors are added near the center (see Figure 12), thus inducing some of the RomR molecules to bindto the cell membrane near the division site.

Figure 12: Number of receptors (when not affected by other chemicals) during division. Higher peak at thecenter correspond to more time passed since division started.

13

3.3 2D stochastic model for diffusion in changing domain

Our 2D stochastic model is implemented in Matlab and each single simulation runs on a single core. However,since to obtain statistically significant measures the simulation needs to be repeated a big number of times,these can be split into an array job in which each core is doing a fraction of the total work (used arrayjob option on the CRC). Attempts to use the parallel environment in Matlab have proven not useful do toexcessive overhead communication time in the matlab parallel environment. A possible alternative to try isto write a code for the GPU environment in Matlab, before eventually passing the code to CUDA, but if wefollow this route we cannot save the position of the particles at each time step (because of the high memoryrequirement) and will instead have to calculate the quantities of interest between each time step. The codeis mainly divided in 5 parts:

• Definition of the boundary of the domain through simple equations in x and y-coord. and its changein time;

• Determining the position of each particle in the given domain as sampled from a specific distribution;

• Choice of next step for each particle, as defined by Brownian motion;

• Detection of collision between particle between time steps, either due to the movement of the particleor the movement of the domain, and implementation of reflective boundary condition when particlesreach a boundary;

• Computation of relevant measures from our simulation, such as number of particles, concentration,derivative of concentration, etc...

3.3.1 Boundary

The domain is modeled to resemble the shape of the bacteria before and during division. For this reason weare considering a total length of 10 µm and a diameter of 1 µm, with a rectangular central section 8µm long,and two semicircular sections at the two extremities each with radius 0.5µm. Moreover, the central sectionof 1µm on the right and left of the center, located at 5µm, changes during division transforming the initiallystraight membrane in two new semicircular sections, forming the new cells poles, see fig 13. This transition is

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

Figure 13: Diffusion profile for 1D PDE model.This plot shows the evolution of the domain representing the bacteria during division. The initial capsule shape slowly changes

in time to form a double capsule touching at the center.

achieved using equation for ellipsis of decreasing radius in time, for the left and right portions of the domainclose to the center. Note that at each time step the domain is closed (continuous) and that the only singularpoint (not unique tangent line) are the one exactly at the center. Also note that while it is given that theinitial and final state of the domain should be as described, the actual transition may be different, and thespeed at which the change happens may be an additional parameter to take into consideration. [If needed Ican include the equations defining the boundary]

3.3.2 Initial Condition

The distribution of particles at time step 0 is assumed to be uniform along the whole initial domain. [Iam having some problems in obtaining a completely uniform one in the actual simulation and may need to

14

discuss this point, both from a mathematical and biological point of view.] [It is also possible to change thisinitial condition with some other predetermined distribution and verify how this changes our results]

3.3.3 Particle step

Our particle are assumed to move according to Brownian motion in the prescribed domain therefore thecoordinates of each successive step are given by

xt+1 = xt +√

2D∆tξx

yt+1 = yt +√

2D∆tξy

ξx, ξy ∼ N(0, 1) (6)

V ar(xt+1 − xt) = 2D∆t⇒ D =V ar(xt+1 − xt)

2∆t(7)

with D diffusion coefficient, ∆t time between steps, and ξx, ξy independent random variables each with astandard normal distribution. [It’s also possible to add some additional component to this step (e.g. anadvective term) or assume a different type of random walk]

3.3.4 Collision detection and Reflection

Both the change in the domain and the movement of the particle can cause the current position to be outsidethe domain. This means that at some point between the two steps the particle collided with the boundary.In the case in which the particle was inside after its movement, but outside after the change in domainwe assume that the change in domain happens essentially in the y direction and therefore we change they-coordinate of the particle to be immediately inside the domain (with some tolerance) and leave the xcoordinate unchanged. In the case in which the particle is outside after its movement, we consider this tobe an actual collision and the particle is reflected with an angle symmetrical to the one of collision (respectto the normal to the boundary at the collision point) and with a distance equal to the remaining distanceto the step (i.e. the excess distance traveled outside the domain).

~v′ = ~v − 2(~v · ~n)~n (8)

It is also possible to add a damping factor 0 ≤ d ≤ 1 to represent a surface absorbing part of the energyof the movement due to the impact. [other type of boundary condition, such as adhesive and repulsivewhere implemented in previous versions of the code and can still be used, and different new ones can beimplemented if needed]

3.3.5 Computation of relevant measures

To compute relevant measures similar to the one obtainable in experiments, we subdivide the domain incompartments along the x-axis with same width ∆x and will measure these quantities as if located at thecenter of the compartment. At the moment at each time step we are counting how many particles are insideeach compartment, the volume of each compartment, the total concentration in each compartment, and thederivatives along the x-axis (as finite difference) of Number of particles and concentration.

3.3.6 Additional: Receptors component

One of the assumptions done in the 1D model was the presence of receptors that can bind to RomR particlesat the domain boundary and stop the particle movement. We have currently implemented a simple catch-and-release algorithm in which a certain number of receptors are distributed in the appropriate regions ofthe domain (poles, and developing poles) and are assigned an occupied (bounded to RomR) or free state(unbounded). If a particle is at the location of a receptor the algorithm checks the receptor state. If the

15

receptor is occupied nothing happens, if it is free the particle will bind to the receptor with a fixed probabilitypon in which case the receptor will change is state to occupied. On the other hand if a particle is binded toa receptor, when we start our time step, it will detached with a fixed probability poff , in which case it willkeep moving, otherwise will stay put until the next time step. [Note this is a very basic way of simulatingreceptors binding, and I am currently looking at some papers for receptors model with a more clear biologicalinterpretation. Also I will suggest later that possibly we don’t need the receptors at all, since we don’t haveany real prove that there are any.]

3.3.7 Current and future work

For both models, number of receptors, number of proteins, binding and unbinding rates are parameters thatneed to be calibrated with experimental data (if possible). We want to do this calibration trying to useonly data from normal cell motion (before division) and will test the results with experimental data duringand after division. Additionally, we will perform new simulation for the 2D model with different initialconditions and/or some extra component for the random walk (either fixed or stochastic) and compare theresults with the one obtained so far and that this may lead to polar peaks without the need for receptors,and the asymmetry will also emerge similarly to what seen in the experiments.

3.4 Models extensions

Both models at present lack two fundamental characteristic of the biological problem: cell growth, andrealistic initial conditions. In fact one of the most interesting features of RomR in M. xanthus is the fact thatdespite the changes occurring in the size of cell, during growth, and its splitting in two halves, during division,the dynamics of the chemicals controlling its reversal seem to be maintained almost without interruption,as observed in the fluorescent microscopy experiments. In particular, the oscillation between the two polesare maintained between two consecutive divisions event, despite the fact that the actual domain in whichthis happens effectively doubles in size; and shortly after division is completed the reversal restart in bothdaughter cells, indicating that the two daughter cells probably both inherited around half of the chemical(e.g. RomR) from their mother. To investigate these feature we can add domain growth to both model, andwe can use results from the oscillatory regime in our previous simulation, in agreement with experiment, asinitial conditions for our future simulations. This way instead of starting with a random initial condition forconcentration along the cell (1D model) or with a uniform particle distribution (2D model) we can start oursimulation with more reasonable concentration levels that reflect the polar nature of RomR distribution. Atpresent we added the growth component to the 1D model (without division), and shortly we are planningto add the same to the 2D model and to include in both more realistic initial conditions.

3.4.1 Growth Component of 1D PDEs model

To incorporate growth in our 1D PDEs model we are recurring to an approach already used in studyingpattern formations of Turing model in a growing domain (see Maini 2012 Interface Focus). Instead of dealingwith an expanding domain we will normalize our equation with respect to the length of the domain L = L(t),and we will prescribe the way this quantity changes in time.

x̄ =x

L(t),with L(t) = L0

(1 +

t

Tf

)(9)

Is easy to see that all the equation for our 1D model will remain unchanged except for the diffusion coefficientthat will change as follow.

D̄(t) =D

L2(t)(10)

where D is the old diffusion coefficient, and D̄ is the new diffusion coefficient. This way we moved ourproblem, from solving our equation in a growing domain to solve similar equations in a fixed domain inwhich diffusion becomes ”harder” in time (because the effective space particles need to cover is stretched).

16

3.4.2 Alternative Wave-like Initial Condition

Our 1D PDEs model uses a random initial condition as a starting point to develop oscillations. This approachis the same used in Howard et. al. 2001, that we used as a reference for our MinCDE-like activator-repellentsystem interacting with the RomR receptors. However, in the same paper the authors mentioned that itis also possible to obtain analogous wave solutions for the MinCDE system using an appropriate initialprofile. This profile will need to have a sufficient jump in concentration in different parts of the cell for bothattractant and repellent component. In this section we will see how, also in our model, we can avoid randominitial condition for the chemical components concentrations, and instead use a more realistic Wave-likeinitial condition. Before we illustrate the final approach that was chosen, we will explain a naive choicetaking directly from the comments of Howard et. al. about their initial condition. The authors suggest thatskewed enough I.C., where the concentrations present a higher concentration in one part of the cell comparedto the rest, will be enough to generate the initial interaction between attractant and repellent componentto initiate the temporal oscillations observed in the case with random IC. However, when this approach wasused for our model the invariable result, obtained trying several different combination of skewed profiles forthe chemicals involved, was always a RomR concentration spatio-temporal solution, in which the two poleswill quickly reset to a steady state with both having the same level of protein (see fig.29 in model resultsection). Therefore, just following the indication that worked for the simpler MinCDE system is not enoughto reproduce oscillations in our setting. We then need to find some other initial condition that could generateoscillation in our PDE model. However, the reason for us to change initial condition is that we want majoradherence to the experimental condition, in which when we start to observe a cell this is already undergoingoscillation. Moreover, our model has already proven to be able to replicate reasonable oscillations startingwith random IC. For all this reason, we decided to take as initial condition the profile obtained at a randomtime before division starts of a previously simulated cell (done with random IC). This set-up actually closelycorrespond to what is observed in microscopy movies. In fact, when in experiments we start following a cellit is usually a mobile cell (i.e. undergoing oscillations) somewhere along the period between two divisionevents. Therefore, this is an obvious choice for the IC that will closely resemble an ”experimental” one.To obtain an actual function profile the data point of the chosen solution have been converted to a splineapproximation. This allow to give approximate value for the profile at any point in the domain, even if wehave the numerical initial condition only for a finite number of points along the domain. This approach wassuccessful in reproducing the expected oscillations and essentially gave the same type of solution alreadyobtained using random IC (see Fig. 16, 28 and 30).

3.4.3 Larger Division Zone

An other improvement that was added to the PDE model was to increase the region around the center, wherethe division happens. From the above description of the model we remember that there is a region aroundthe center where the diffusion coefficient is gradually reduced through time, to simulate the formation of anew boundary, and new receptors are formed, to take into account for new pole formation. Initially the totallength of that region was considered to be 1µm, centered in the middle of the cell length, and the receptorswill gradually increase as shown in Fig. 12 while the diffusion coefficient will gradually decrease as shownin Fig. 10. This assumption considers division to have a very sharp effect at the exact center of the cell.Both diffusion coefficient and receptors number assume an extreme local value (minimum and maximumrespectively) only at the exact centered. This approach already gave some interesting results (see Fig. 17)compared to experimental data on division (see Division in Cameron’s paper), especially the asymmetry inconcentration on the two sides of the newly formed poles. However, the concentrations difference obtainedin Fig. 17 are too high, and the transition look too sharp when compared to experiments. To obviate tothese issues, we decided to introduce a larger division zone around the center. This new region is 2µm longand centered at the cell septum. Hence, on each side of the center we have a 1µm polar region exactly as atthe two cell sides, this choice will also help us with simulating the concentration in the two daughter cells(see next section). Moreover, we slightly changed the way that diffusion coefficient and receptors numberchange in this larger region. For the diffusion coefficient we only stretched the previous profile to cover thebigger area (see 14) giving a more gradual transition. For the number of receptors we added a central region(corresponding to the smaller old area) where the number of receptors assumes the current maximum localvalue (see Fig. 15, this way if we consider the number of receptors of only half of this region the profile closely

17

Figure 14: Diffusion coefficient as a function of space and time. Lower minimum correspond to longer timepast since division started. At the end of division the minimum is 0 (or a small ε)

resemble the one for the poles of the mother cell, and the profile transition at the center is totally smooth(being that all derivatives at that point will be zero). This approach resulted in smoother solutions at the

Figure 15: Diffusion coefficient as a function of space and time. Lower minimum correspond to longer timepast since division started. At the end of division the minimum is 0 (or a small ε)

center, but preserved a well visible asymmetry in the two sides at the division site (see 28). Furthermore,the obtained results have better agreement with experimental result (compare division 3D plot in Cameron’spaper) and show interesting variation in the asymmetry depending on the specific initial condition used (addmore examples of this?).

3.4.4 Post Division Simulation

At this point our model can simulate the changes in RomR concentration profile along the bacteria cellin the period between two division times, uses a growing domain doubling in size during this time, andreproduces division effects at its center such as reduction of diffusibility and production of new receptors.However, one missing aspect is what happens once the two daughter cells are completely divided. At the

18

end of division the two cells have now completely developed their new cell poles and are separated by thefully formed membranes. Moreover, in experiment has been shown that after division the two daughter cellstend to separate going in different directions. For all this reason we can than assume that the two new cellswill have no exchange of protein. However, note that in certain condition cells can exchange protein throughcontact via their membranes. We now want to be able to simulate the concentration profile for RomR inthe two daughter cells originating from the cell that was simulated up to the end of its division cycle. Todo this we will separate the two halves of the initial cell solution at the final time, with left part [0, 0.5]and right part [0.5, 1] of the normalized cell length. Then we will obtain a Spline approximation, as in thecase of the IC obtained from previous data, and will rescale the domain of the ensuing functions accordingto the following composition Fl = flogl and Fr = frogr, where fl, fr are left and right spline-reconstructedhalf profiles, Fl, Fr will be the corresponding domain rescaled functions, and gl, gr are the following domaintransformations:

gl(x) =x

2;x ∈ [0, 1] left side (11)

gr(x) =x

2+

1

2;x ∈ [0, 1] right side (12)

Therefore the two functions Fl and Fr are now two profile functions defined on the normalized cell lengthdomain [0, 1] that gives us the values of the profiles corresponding to left and right side respectively. Usingthis approach we were able to successfully simulate the concentration of RomR not only for one cell betweentwo divisions, but also do the same for its two daughter cells (see Fig. 31). However, as we will illustrate inthe result section, not all daughter cells will preserve oscillation through this process (see Fig. 32, and resultsection for more details).

4 Model Results

4.1 1D Model Results

At the present stage the 1D model can qualitatively represent the periodic reversal in brightness between thetwo poles, similarly to what was seen in experiments, but the reversals times are purely deterministic anddepends on diffusion coefficients and length of the cell. When this are fixed the period is fixed. To achievebetter agreement with the simulation we should introduce some stochastic component in our equation.This will allow us to get a more reasonable distribution of reversal periods similar to what is observedin experiment. We will need to discuss possibilities for this component, but one option could be to havenoise in our diffusion coefficients to indicate a non completely homogeneous media. An other result of thesimulations was the fact that different random initial conditions all produce oscillations, but they differ inthe starting time of the oscillations. This could reflect the fact that two daughter cells after division arenot perfectly identical, and having essentially randomized quantities and concentration profile of proteins,start their reversal mechanism at different moments. The last result was the appearance of asymmetry inleft and right concentration at the division site, as seen in Cameron’s Paper. This phenomenon is onlyqualitative at this point, but seem to agree with what we saw in the experiments. It is most likely due tothe randomized start of reversal combined with the decrease in diffusion at the center due to the formationof the new boundary.

4.2 2D Model Results

To obtain relevant results we are considering a number of particles of 10000 for each simulation and we arealso averaging our measures over 10000 simulations. Some of the preliminaries results can be observed inthe following pictures.

These plots seam to indicate that the mobility of particles close to the center is indeed more and moreimpaired the more we progress through the division, this is both stopping the particles to move from oneside to the other (valley at the center) and is also slowing them down close to the center (peaks). To show

19

Figure 16: Simulation of RomR concentration before and during division. After half of the simulationdivision is initiated through change in density and addition of receptors near the center.

Figure 17: Concentration Change during division.Series of frames equally spaced during division showing the formation of two peaks close to the center where a valley is formed.

Figure 18: Change in average number of particle per position.Series of frames equally spaced during division showing the formation of two peaks close to the center where a valley is formed

similarly to previous picture, but here the valley is significantly lower than the constant base level away from the center.

20

Figure 19: Space derivative of concentration.Series of frames equally spaced during division showing the change in space derivative of the concentration due to the change

in the domain during division.

Figure 20: 3D representation of change in concentration.

21

Figure 21: 3D representation of change in average number of particles.

Figure 22: 3D representation of change in space derivative of the concentration.

22

this result we can calculate the effective diffusion coefficient reversing the relation between Variance of thestep for a randomly moving particle. In fact, from the description of the model we have that

V ar(xt+1 − xt) = 2D∆t⇒ D =V ar(xt+1 − xt)

2∆t(13)

therefore if we change xt+1 (new coordinate in absence of boundary) with x̄t+1 (new coordinate after bound-ary condition is applied), we obtain

D̄ =V ar(x̄t+1 − xt)

2∆t(14)

that gives us the effective diffusion coefficient. We can see a plot of this in the following figure and we can

4.5 5 5.50.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

0.026

Figure 23: Effective diffusion coefficient during division

compare this with the hypothesis mad for the 1D model. [I have the results also with the added receptorsas described above, but seeing the plot I am obtaining I think the receptors model need to be revised (asexpected).

23

4.2.1 Idea to avoid receptors in 1D model

In the 1D model we decided to include receptors due to the impression from experimental data that theRomR particles move ”less” at the poles. However, we lack any concrete evidence for their existence. Basedon the results obtain with the 2D model we can now see that at the division site an higher concentrationnaturally forms only due to the change in the domain shape with no need to add receptors. This lead us tothink that possibly a similar effect may also be achieved at the two original poles due to the different geometry(local curvature) of the domain in those regions (as can be seen in Fig.12). In this figure a symmetric polarcondition was used instead of a uniform concentration, and the two poles, retain at least partially, thosepeaks along the simulation. Therefore we are advancing the hypothesis that the domain shape may beenough for the peaks to form at the two poles under certain initial conditions. [This need more investigationstill] Furthermore we are also advancing the hypothesis that the skewness in the RomR concentration seenin the experiments may be a consequence of the movement instead of the other way around (as stated inthe literature) and that if a directional advective term is added to our model that asymmetry can easily beachieved.

4.3 Additional 1D Model Components Results

4.3.1 1D model Growth Results

As described above we performed simulation with the 1D model in a growing domain. Additionally in thisimplementation we are keeping the actual size of the poles constant through the simulation (1 micron=0.5micron of pole+0.5 micron of near pole transition). This means that the relative size with respect to thetotal length also changes linearly like the new spatial variable (see pictures). Also, note that at the momentwe have not yet introduced an increase in receptors at the center (formation of new poles) with this newimplementation (but this can be done with minimal effort, and will be done soon). A typical example of thesimulation obtained using the model is shown in the following picture. As expected the oscillation period

Figure 24: RomR concentration simulated for a period of 4 hours (time between consecutive division events).The x-axis correspond to the relative position with respect to the total domain size (new spatial variable),the y-axis correspond to time (in second), and the z-axis (color scale) corresponds to the concentration ofRomR (blue low, yellow high). The typical simulation (starting from random initial condition) shows atransitional period of seemingly symmetric behavior followed by the emergence of oscillatory behavior.

increases in time. However there is an apparent asymmetry in the times for half oscillations from left toright and right to left (see table). Moreover, toward the end of the simulation the period increase is notmonotonic. Most probably this is due to an effect observed in the model for MinCDE system used as abasis of our model (and similar effects in pattern formations models). These models have wave like solutionsoriginating from small perturbation whose wavelength is allowed to survive due to its relation with the sizeof the domain. Usually the size of the domain is fixed and therefore only fixed wavelength survive. If none

24

Figure 25: Time sequence of left and right peak and time between them in seconds (from previous simulation).

survive (domain too small) we obtain a constant solution (symmetric between the poles, in our case) ifthe domain increases we find a regime in which a stable oscillation appears, if the domain keeps growingadditional oscillation will start to appear and will interfere with each other (due to the non-linear natureof our system). In the next plot we can observe a typical plot of the way concentration changes at the twosides of the cell. As we can easily see, the main peaks are easy to identify. However, additional smaller

Figure 26: Concentration at the two sides of the cell. The x-axis is time and the y-axis is RomR concentration.Red line concentration at x=1 (right pole), Blue line concentration at x=0 (left pole)

peaks start to emerge soon and become more and more prominent toward the end of the simulation. We willperform some calculation to determine when these effects occur. At present we are getting some statisticon different aspects of the observed oscillations. The first significant one was to determine if there is a welldefined probability distribution for the starting time of oscillation (i.e. the time at which the first relevantpeak on either side appears). For this the following histogram plot shows that a log-normal distribution islikely. We have implemented a routine to find relevant peak times, and we are currently looking at a wayto analyze the results obtain through simulation statistically and from a qualitative and quantitative pointof view. It is relatively easy to produce a big number of simulation with different random initial conditionand analyze different statistics. However, random initial conditions are not biologically sound. Currentlywe are working to produce initial condition closer to the experiment while maintaining a reasonable randomcomponent. From these newer simulation a few interesting aspects emerge. First, it appears that based onsome randomicity the cell is likely to have one preferential direction, expressed by a faster reversal time fromleft to right pole or vice versa. Second, the period between peak switches increases when we get closer todivision event, corresponding to the concentration at the poles spending more time at intermediate levels,and hence reducing the difference between poles and possibly the motility of the cells. Lastly reversals seem

25

Figure 27: Histogram plot of starting time for oscillation. The x-axis is the time in seconds, the y-axis isthe cumulative bin count. It is possible to distinguish a log-normal tendency.

to become more inconsistent close to division time due to the emergence of new wave length of the solution.These last two aspects in particular may correspond to the lack of mobility observed in the last 20 minutesbefore the separation into two daughter cells. Also, extended periods for the oscillation may increase theprobability of an equal splitting of chemicals between the two daughter cells, hence explaining their mobilityalmost immediately after division.

4.3.2 Bigger Central Zone

When we expanded the area of effect of division, around the center of the cell, the asymmetry in concentrationacross the midpoint of the domain became more evident, and the central profile of RomR became smootherand more similar to the one at the old poles. These result more closely resemble the experimental evidencethat the new poles are not symmetric (see Cameron’s paper) and the fact that the concentration of the newand old poles during division appear to have a very similar profile.

Figure 28: Simulation of RomR concentration before and during division. In the last 20 minutes of simulatedtime division is initiated through change in density and addition of receptors near the center.

26

4.3.3 New IC Results

The new IC, while more biologically accurate, did not significantly change the qualitative behavior of thesimulated solution. However, we determined through testing of other ICs that not all of them will necessarilyproduce oscillations even if the proportion of chemicals is preserved.

Figure 29: Simulation of RomR concentration with skewed IC. After a brief initial period the RomR con-centration set to a symmetric profile and remains relatively stable, showing no oscillations for the rest of thesimulation.

Figure 30: Simulation of RomR concentration with IC taken from a random pre-division time in a simulationlike the one shown in Fig. 28. Almost immediately the RomR concentration starts to oscillate periodicallybetween the two poles, as previously obtained with random IC.

4.3.4 Post Division Simulation

Being able to simulate the concentration of RomR in the two daughter cells, after the first division event,allowed us to prove that our model is able to preserve oscillations even after division. However, is easy tosee how in Fig. 31 the two daughter cells do not have the same oscillation frequencies and how these alsodiffer from the one of the mother cell. Furthermore, depending on initial conditions the oscillation sometimeare not preserved in one or both daughter cells (see Fig. 32). These effects are most likely due to theway that RomR and the other chemicals interacting with it (MinCDE-like system) get separated in the twohalf of the initial cell during the division process. In fact, we have already discussed how in the attractantrepellent system used to represent the chemicals an important factor that determines oscillation emergenceand frequencies are the proportions of each chemical. Therefore, it is not hard to imagine that the cut offhappening when cells separate could leave one or both cells with the wrong proportions of chemicals foroscillations to occur, or different proportions that will produce different frequencies. One way of avoidingthe problem of daughter cells not oscillating could be to add a production component to the model. In fact,

27

in our model there is no active production of chemicals. Is easy to understand that this cannot be what celldo in experiments, or each generation of cells will have a more and more depleted concentration of chemicals.Moreover, because of the compartmentalization happening at division the daughter cells will most of thetime have not symmetric concentration, and is reasonable to expect that the cell will try to re-equilibrateits chemicals level to a reasonable middle level after division. The effect of a production component as well

Figure 31: Simulation of RomR concentration with skewed IC. After a brief initial period the RomR con-centration set to a symmetric profile and remains relatively stable, showing no oscillations for the rest of thesimulation.

Figure 32: Simulation of RomR concentration with skewed IC. After a brief initial period the RomR con-centration set to a symmetric profile and remains relatively stable, showing no oscillations for the rest of thesimulation.

as better study of the profiles that generate oscillating and non oscillating daughter cells will be done soon.

5 Conclusions

28