A model of chloroplast growth regulation in mesophyll cells
Kelly M. Paton, Lisa Anderson, Pauline Flottat, Eric N. Cytrynbaum
BBulletin of Mathematical Biology manuscript No. (will be inserted by the editor)
A model of chloroplast growth regulation in mesophyll cells
Kelly M Paton · Lisa Anderson · Pauline Flottat · Eric N Cytrynbaum
Received: date / Accepted: date
Abstract
Chloroplasts regulate their growth to optimize photosynthesis. Quantitative datashows that the ratio of total chloroplast area to mesophyll cell area is constant across dif-ferent cells within a single species, and also across species. Wild-type chloroplasts exhibitlittle scatter around this trend; highly irregularly-shaped mutant chloroplasts exhibit more scatter. Here we propose a model motivated by a bacterial quorum-sensing model consist-ing of a switch-like signalling network that turns off chloroplast growth. We calculated thedependence of the location of the relevant saddle-node bifurcation on the geometry of thechloroplasts. Our model exhibits a linear trend, with linearly growing scatter dependent onchloroplast shape, consistent with the data. When modelled chloroplasts are of a shape thatgrows with a constant area to volume ratio (disks, cylinders) we find a linear trend with min-imal scatter. Chloroplasts with area and volume that do not grow proportionally (spheres)exhibit a linear trend with additional scatter.
Keywords chloroplasts · ordinary differential equations · bifurcation · model · growthregulation · switch Chloroplasts are the organelles within plant cells which convert sunlight to usable energyvia photosynthesis. Their exposure to sunlight is integral to optimal photosynthetic output,so their coverage of a cell’s surface area is important. There is evidence that chloroplastgrowth and replication is regulated to control the percentage of a mesophyll cell’s surfacearea that is occupied by chloroplasts. This is quantified by comparing the plan areas of thechloroplasts and the cell, defined as the areas taken up by the chloroplasts and the cell whenviewed from above (Pyke and Leech, 1991). Several studies support this claim.Ellis and Leech (1985) found that leaves of
Triticum aestivum L. and
T. monocoecum L. demonstrated a trade-off between chloroplast size and number that resulted in a similar total
E. CytrynbaumInstitute of Applied Mathematics, University of British ColumbiaTel.: 1-604-822-3784Fax: 1-604-822-6074E-mail: [email protected] a r X i v : . [ q - b i o . S C ] S e p Kelly M Paton et al. T o t a l c h l o r op l a s t p l a n a r ea p e r m e s ophy ll ce ll ( mm ) T o t a l c h l o r op l a s t p l a n a r ea p e r m e s ophy ll ce ll ( mm ) Mesophyll cell plan area ( μ m ) Mesophyll cell plan area ( μ m ) wild-type arc1arc2 arc3 (a) (b)(c) (d) Fig. 1
Figure reprinted (with permission) from Pyke and Leech (1992), figure 3, showing the linear relation-ship between total chloroplast plan area per cell and cell plan area for each of the wild type and three mutantsof
Arabidopsis thaliana . The slope of all four lines is similar. There are two layers of chloroplasts within acell, one at the top and one at the bottom, so “full” cells would exhibit a slope of 2. Scatter varies, with thewild-type and arc1 mutants having the least scatter ( R = .
92 and R = .
93, respectively, in (a) and (b))and arc2 and arc3 mutants showing more scatter ( R = .
81 and R = .
61, respectively, in (c) and (d)). chloroplast plan area per unit cell plan area. Pyke and Leech (1992) then examined meso-phyll cells from a wild type and three chloroplast replication and growth mutants (termed“arc” mutants) of
Arabidopsis thaliana and consistent with the observations of Ellis andLeech, they found an inverse relationship between the mean individual chloroplast area andthe total number of chloroplasts (see Figure 1). For each of the wild-type and the three mu-tants, the relationship between total chloroplast plan area and cell plan area was linear, nomatter the individual size or shape of the chloroplasts. Moreover, the slope of the data plottedin the “plan area plane” (total chloroplast plan area versus cell plan area) was quantitativelysimilar for all four data sets. Two of the mutants showed considerably more scatter than thewild type and the other mutant, and the mutant with the most scatter was observed to haveirregularly shaped chloroplasts.Osteryoung et al (1998) unearthed more evidence of this phenomenon from a collectionof
Arabidopsis cells of both the wild type and two transgenic types. Their figure plotting the model of chloroplast growth regulation in mesophyll cells 3
Fig. 2
Figure reprinted (with permission) from Osteryoung et al (1998), figure 5, showing the chloroplastplan areas versus cell size for wild type (WT, open circles) and two different transgenic types (1-1, filledboxes, and 2-1, filled triangles) of
Arabidopsis . All three relationships are linear with similar slopes; seecalculated slopes and R values. total chloroplast plan area versus the cell plan area for each cell is duplicated here in Figure2. Similar to the data in Pyke and Leech (1992), each of the three types displayed linearrelationships of similar slope, with scatter that scaled with cell size. The transgenic types– with larger and more irregularly shaped chloroplasts than the wild type – presented withmore scatter; see the R values in Figure 2.Pyke (1999) further showed that this regulated chloroplast density persists not onlyacross different cells, but across different species. He found a linear relationship betweentotal chloroplast area and cell plan area when considering an average data point from eachof eight different species and several Arabidopsis mutants. He noted that the signalling meth-ods used by chloroplasts to regulate their growth and division are not known, but concludedthat “chloroplast density in relation to the size of the cell seems to be involved”.The most notable feature in the data sets shown in Figures 1 and 2 is the linear rela-tionship between total chloroplast plan area and cell plan area; this is backed up by the datafrom Pyke (1999). The second notable feature is that the scatter about the relationship growssomewhat linearly with cell plan area, more like a percentage of the plan area ratio ratherthan an absolute measure. This data can be resummarized in the context of the ratio of planareas, which we’ll call α , defined as the total chloroplast plan area divided by the cell planarea. These two observations – the linearity of the relationship and the linearly increasingscatter – can be reduced down to the observation that the range of ratios of plan areas is Kelly M Paton et al. constant for all cell plan areas. The third important observation is that irregularly shapedchloroplasts have more scatter, which means that chloroplast geometry matters.Here we propose a mechanism to explain this observed chloroplast growth regulationphenomenon. The ability to sense the collective density of one’s own population is foundin certain bacteria and is referred to as “quorum sensing” (Dockery and Keener, 2001).Much work has been focused on understanding quorum sensing; Dockery and Keener (2001)modelled the biochemical quorum-sensing switch in
Pseudomonas Aeruginosa .At the core of our model is the Dockery-Keener biochemical switch. In bacteria, flip-ping this switch triggers a shift in gene regulation allowing a sufficiently large population tocollectively change behaviour. These triggered behaviours include producing exopolysac-charide to form a biofilm, secreting toxins and turning on bioluminescence. In chloroplasts,we propose that a similar switch turns off chloroplast growth.Although chloroplasts may have a quorum-sensing biochemical pathway homologousto that of bacteria, we prefer to think of this quorum-sensing model simply as a biochem-ical system with switch-like behaviour. This switch-like behaviour is the important featurefor our results. Even if the chloroplast growth regulation pathway operates differently, anyswitch-like behaviour involving secretion of the signal ought to behave similarly to what wedescribe here. We find that the model, when applied to a system of chloroplasts within a cell,can capture both the linear trend and the linearly growing scatter in the data. In the model,chloroplast shape influences the extent of the scatter. ρ , and the chloroplasts’effective membrane permeability, δ – which is dependent on surface area – to use in thequorum-sensing model.Consider a cell of fixed size that contains a population of chloroplasts. A leaf mesophyllcell of A. thaliana is largely occupied by one or more vacuoles, leaving only a thin layer ofthe cytosol available for the chloroplasts at the surface (Pyke, 2009, 1999); see Figure 3. Wemodel this available cytosolic space as two thin slices that can each hold a single layer ofchloroplasts, with a top surface area of S and a thickness of τ . The volume of a cell is thus V = S τ , and its plan area (area viewed from above) is S . We look at a range of cell sizes byconsidering a range of S values; we keep the thickness of each cytosolic space constant at τ .Chloroplast shape is more complicated. Chloroplasts typically resemble flattened foot-balls, although this varies widely (Pyke, 2009). The variation in their morphology is bothcharacteristic of different plant species or cell types, and dynamic in response to the environ-ment. For example, the arc6 mutant of A. thaliana has highly irregularly shaped chloroplastsin comparison to the wild-type chloroplasts (Pyke and Leech, 1994). Dynamically, chloro-plasts can deform in response to being squeezed against neighbouring plastids or vacuolesor the cell wall, which can happen over long time scales as the cell grows, or in shortertime scales as chloroplasts move in response to changes in light (Pyke, 2009; Briggs andChristie, 2002). Chloroplast morphology also changes during division when the dividing model of chloroplast growth regulation in mesophyll cells 5 vacuolechloroplasts (a)(b)
Fig. 3
This cartoon summarizes the features of
A. thaliana leaf cell images (see examples in Pyke et al(1994) and Hall and Langdale (1996)) from both (a) an above view and (b) a transverse section view. Thelarge vacuole occupying the middle of the cell pushes the chloroplasts into a thin layer against the cell wall,usually concentrated on the top and bottom for better access to light sources. Based on this cartoon, we modelthe relevant cytosolic space as two thin layers of thickness τ and top surface area S that can each hold a singlelayer of chloroplasts, one at the top and one at the bottom. Figure created in OmniGraffle. (a) (b) (c) l l l τ Fig. 4
Chloroplasts are modelled either as all cylinders as in (a); all disks as in (b); or all spheres as in (c).Orientation is as shown, within a layer of fixed thickness τ . Each shape grows in the direction labelled (cid:96) ;all other dimensions are fixed. Our model contains two such layers, of top surface area S . Figure created inAdobe Illustrator. chloroplast goes through a “dumbell-shaped” stage (Pyke, 2009). Since there is such vari-ability in chloroplast shape, we represent chloroplasts as either all cylinders, all thin disks,or all spheres, and we fix all the dimensions except for one which we refer to as the growthdirection, (cid:96) . Cylinders are of a fixed radius τ / (cid:96) ; disks are a fixed thick-ness τ with variable radius (cid:96) ; spheres have a variable radius (cid:96) < τ /
2. See Figure 4.For any of these three basic chloroplast shapes, we can define the surface area of eachchloroplast as S c , and the volume of each chloroplast as V c . In a cell of volume V = S τ , thechloroplast density is then ρ = nV c / S τ and the effective permeability is δ = nS c ˜ δ , where n is the number of chloroplasts and ˜ δ is the per surface area permeability of the chloroplastmembrane. Kelly M Paton et al. ρ , δ ).Dockery and Keener (2001) based their quorum sensing model on the eight componentgene-regulatory system described in Van Delden and Iglewski (1998). They simplified itdown to two ordinary differential equations using time scale assumptions. We adopt the twovariable model in this paper but consider chloroplasts within a cell in place of cells within amatrix, and chloroplast density instead of cell density. The first equation tracks the concen-tration of an autoinducer ( A ), a signalling molecule which is secreted and can be sensed byall chloroplasts. The second equation tracks the concentration of a protein ( R ) that dimerizeswith the autoinducer to form an activator of autoinducer production, thereby generating apositive feedback loop. When the autoinducer reaches a critical level, we assume (but do notexplicitly model) that a downstream pathway is triggered and chloroplast growth is shut off.The equations for A and R are dRdt = V R RAK R + RA + R − k R R , (1) dA dt = V A RA K A + RA + A − dA , (2) where d = k A + δρ k E ( − ρ ) k E ( − ρ ) + δ . (3)Here k E and k A are degradation rates of the signalling molecule outside and inside thechloroplast, respectively; k R is the degradation rate of the protein R inside the chloroplast; V R , V A control the maximum rate of Michaelis-Menten-type production, and K A , K R controlthe shape of the production curve. A and R are background production rates. The parameter d depends on ρ , the relative volume density of chloroplasts (with maximum 1), and δ , theeffective permeability of the chloroplast membrane. This complicated expression arises fromthe reduction of the system from eight variables down to two. It consists of two terms. Thefirst term is the degradation rate constant for autoinducer inside the chloroplast. The secondterm gives the secretion rate when the cytosolic concentration of autoinducer is assumed tobe in quasi-steady state. We use d as the parameter for our bifurcation analysis because it isthe aggregate parameter through which the system depends on chloroplast geometry ( δ ) andpopulation density ( ρ ). Using the geometry defined in Section 2.1, the bifurcation parameter d can be written as a function of (cid:96) and n for each fixed cell surface S (denoted by a subscript S ): d S ( (cid:96), n ) = k A + S τ S c ( (cid:96) ) ˜ δ V c ( (cid:96) ) k E (cid:16) − nV c ( (cid:96) ) S τ (cid:17) k E (cid:16) − nV c ( (cid:96) ) S τ (cid:17) + nS c ( (cid:96) ) ˜ δ , (4)where S c and V c will vary depending on the shape and size of the model chloroplasts. When d is written in this form it becomes clear that chloroplast geometry and growth plays animportant role in the dynamics of this system.The steady states are found by solving a cubic in RA . Figure 5 shows a bifurcation plotof the steady state value of A versus d . Each of the two “knees” indicates a saddle-nodebifurcation. As the chloroplasts grow, d decreases, so the biologically relevant bifurcation is model of chloroplast growth regulation in mesophyll cells 7 bifurcation parameter, d s i gn a lli ng m o l ec u l e , A d = d*A = A* Fig. 5
The solid curve is a bifurcation plot showing the concentration of signalling molecule A versus the bi-furcation parameter d . A cell with low chloroplast density will begin on the right side of the lower branch andmove left ( d will decrease) as the chloroplasts grow. When d reaches the bifurcation point at d ∗ , the systemwill fall off the knee and the concentration of signalling molecule A will rise above the critical value A ∗ (indi-cated by a horizontal dashed line). This transition is the end-of-growth bifurcation at which the chloroplastswill stop growing. The bifurcation diagram was calculated using AUTO and plotted using MATLAB. the one at the lower d value; we label this d -value as d ∗ . A cell with low chloroplast density ρ will begin on the lower branch at the right-hand side (the low steady state) and then trackleft as the chloroplasts grow, falling off the knee onto the upper branch (the high steadystate). This bifurcation – depicted in the phase plane in Figure 6 – is a biochemical switch.We assume that the chloroplasts’ growth is binary (on or off) dependent on the magni-tude of A . While A is below some critical value A ∗ , they remain in the growth regime. When A rises above A ∗ , the chloroplasts stop growing. We refer to this switch as the end-of-growthbifurcation.We are interested in determining the relationship between size, number, and shape ofchloroplasts at the critical d ∗ value, and how well that relationship predicts the data pre-sented by Pyke and Leech (1992) and Osteryoung et al (1998). In particular we aim todetermine how the linear trend and linearly growing scatter of the data, and the variance inthe amount of scatter, depend on chloroplast geometry. S and denote dependence on cell size by a subscript S where relevant.Since the chloroplasts vary in only one dimension the bifurcation parameter d S is a functionof only two variables: the growth dimension, (cid:96) , and the number of chloroplasts n . The formvaries depending on which geometry we consider. Our goal is to find which combinations Kelly M Paton et al. d < d*d = d*d > d*protein, R s i gn a lli ng m o l ec u l e , A d < d*d = d*d > d* Fig. 6
The nullclines in the phase plane of the signalling molecule A vs the protein R , plotted for three valuesof the bifurcation parameter d , show the passage through the first saddle-node bifurcation (the end-of-growthbifurcation). As chloroplast density increases in a cell of fixed size, d will begin above d ∗ and decreasethrough the bifurcation point at d ∗ . The curve A (cid:48) = R (cid:48) = d . Figure created in MATLAB. of size and chloroplast number correspond to the bifurcation point at d S = d ∗ , and thencalculate the corresponding plan areas of those size–number combinations. Using those datapoints we can then draw the plan area plane (total chloroplast plan area versus cell plan area S ) for each case.Finding the bifurcation point requires solving the equation d S ( (cid:96), n ) = d ∗ . We nondimen-sionalize this equation and reformulate it as a search for the zeros of the function g S ( (cid:96), n ) defined by g S ( (cid:96), n ) = ( D ∗ − K ) − S c ( (cid:96) ) τ V c ( (cid:96) ) (cid:16) − nV c ( (cid:96) ) S τ (cid:17) β S (cid:16) − nV c ( (cid:96) ) S τ (cid:17) + nS c ( (cid:96) ) S , (5)where D ∗ is the nondimensional bifurcation point defined as D ∗ = d ∗ / k E , β S = k E / S ˜ δ , and K = k A / k E . Each ( (cid:96), n ) pair corresponding to the bifurcation point describes a size (cid:96) andnumber n of chloroplasts at which growth shuts off. The allowable pairs are restricted bythe physical requirement that the chloroplasts must fit within the cell: nV c ( (cid:96) ) S τ <
1. Note that n is discrete, so the bifurcation “curve” in the (cid:96) − n plane consists of a collection of discretepoints.Once we have a collection of these ( (cid:96), n ) pairs that correspond to the bifurcation curve,we can calculate the ratio of the total chloroplast plan area to the cell plan area – the slopeof the line through the origin and data point in the plan area plane – for each pair. We labelthis ratio generally as α S ( (cid:96), n ) ; an explicit definition depends on the chloroplast geometrythat we’re working with. We then plot the points ( S , α S · S ) in the plan area plane.We repeat these calculations over a range of fixed S values to generate the full data set.The scatter and linearity of the data can also be examined analytically for each geometriccase by looking at the expressions for α S ; we include this below on a case by case basis. The model of chloroplast growth regulation in mesophyll cells 9 range of α S values dictates the vertical scatter of the data at that fixed S -value on the x-axis.It is possible that different ( (cid:96), n ) pairs return the same α S value so the vertical scatter doesnot necessarily correlate to the number of bifurcation points. The variance in the range of α S values across the different cell sizes S dictates the shape or trend of the data as a whole. Ifthe range of values of α S is invariant in S (i.e., α S = α ), then the predicted chloroplast planarea to cell plan area relationship is linear with linearly growing scatter.3.2 Cylinder-shaped chloroplastsA right cylinder of variable length (cid:96) and fixed diameter τ has a surface area of S c = πτ / + πτ (cid:96) and volume V c = πτ (cid:96)/
4. To keep the chloroplast dimensions in a physically reasonableregime we consider (cid:96) > τ /
2. When the chloroplasts take this shape in a cell of fixed surfacearea S and thickness τ , the function in (5) becomes g S ( (cid:96), n ) = ( D ∗ − K ) − (cid:16) τ (cid:96) + (cid:17) (cid:0) − n πτ (cid:96) S (cid:1) β S (cid:0) − πτ (cid:96) n S (cid:1) + πτ (cid:96) nS ( τ (cid:96) + ) . (6)To find the roots – the ( (cid:96), n ) pairs that correspond to the end-of-growth bifurcation for this S – we set g S ( (cid:96), n ) =
0. This can be solved numerically using Newton’s method (for the generalcase), or in this case analytically as a quadratic equation in (cid:96) (taking only the positive root).Then the plan area ratio for cylinders, defined as α S cylinder = τ (cid:96) nS , (7)is calculated for each ( (cid:96), n ) pair at the end-of-growth bifurcation. Each unique α S cylinder valuecorresponds to a point ( S , α · S ) in the plan area plane. The range of α · S values provides thevertical distribution of the data at that cell size S .We repeat this process for a range of S values, generating a set of data points at each S . Asample plot showing the scatter of plan areas for the full distribution of predicted chloroplastsize-number pairs for cylinder-shaped chloroplasts at the end-of-growth bifurcation is shownin Figure 7. The apparent slope of the model data ( S , α · S ) depends on as yet unconstrainedparameter values. In the figures, we have chosen k A so as to get a slope close to what isobserved in mesophyll cells. All parameter choices are listed in Appendix A.To gain further insight into the scatter and linear trend of this distribution, we solve (7)for n and replace n in g S = α S cylinder in termsof (cid:96) : α S cylinder = π ( + τ (cid:96) ) − β S ( D ∗ − K ) ( + τ (cid:96) )( D ∗ − K + ) − β S ( D ∗ − K ) (8)This is subject to the restrictions D ∗ − K > β S >
0, and we require α >
0. To examinethe scatter of points at each S value, consider varying (cid:96) . Under the domain constraint (cid:96) > τ / (cid:96) and thus only a small range of α valuesfor each S . This translates to the small amount of scatter that we see in Figure 7. Next, toexamine the overall trend of the data as cell size S varies, we note that the only S -dependenceis in β S = k E / S ˜ δ . Since S is large with respect to k E / ˜ δ , β S is small for any S . This meansthat there is little variance in α S cylinder with respect to S , which explains the linear trend seenin Figure 7. To see both of these observations more clearly, we use the fact that β S is smallto express α S cylinder to leading order in β S as α S cylinder ∼ π ( D ∗ − K + ) . (9) c h l o r op l as t p l a n a r ea cylinder Fig. 7
A sample of calculated cylindrical chloroplast total plan areas that correspond to the end-of-growthbifurcation, over a range of cell plan areas. Note the linear trend, with little scatter. Areas are in µ m . Figurecreated in MATLAB. [Parameters: ˜ δ = k E = . k A = . This leading order α S cylinder is constant in both (cid:96) and S , which predicts a linear trend withlinearly growing but minimal scatter. These analytical observations match what we observein our calculated data in Figure 7.3.3 Disk-shaped and spherical chloroplastsWe perform the above analysis for disk-shaped chloroplasts, with a fixed thickness of τ anda variable radius of (cid:96) , and spherical chloroplasts with a variable radius (cid:96) restricted to be (cid:96) < τ /
2. The disk shape has a surface area of S c = π (cid:96) ( (cid:96) + τ ) and a volume V c = π (cid:96) τ ; thesphere has surface area S c = π (cid:96) and volume V c = π (cid:96) . Both disks and spheres look thesame from above, resulting in identical plan area ratio definitions: α S disk = π (cid:96) nS , α S sphere = π (cid:96) nS . (10)Again, we explicitly calculate the ( (cid:96), n ) pairs that correspond to the bifurcation point byfinding the roots of (5) with the appropriate S c ( (cid:96) ) and V c ( (cid:96) ) expressions, and then calculatethe α S values for each ( (cid:96), n ) pair. We repeat this for each cell size S , plotting all ( S , α S · S ) points in the plan area plane. See this calculated data for disk-shaped chloroplasts in Figure8 and spherical chloroplasts in Figure 9.To explain the observed scatter and the linear trend in these figures we look at the ex-pressions for α S . As with the cylinders, we use the definition of α S in each case (10) andeach geometry’s version of g S = α S as a function of only (cid:96) , analogous to(8). model of chloroplast growth regulation in mesophyll cells 11 The calculated expression for α as a function of (cid:96) for the disk is α S disk = ( + τ (cid:96) ) − β S ( D ∗ − K ) ( + τ (cid:96) )( D ∗ − K + ) − β S ( D ∗ − K ) . (11)The disk expression has slightly more variance with (cid:96) than the cylinder due to the factor of ( + τ (cid:96) ) instead of the cylinder’s ( + τ (cid:96) ) ; this shows up as slightly more scatter in the data.The scatter is still minimal, though, which is evidenced by the leading order expression α S disk ∼ ( D ∗ − K + ) , (12)which is independent of (cid:96) . As in the cylinder case, the scatter in the disk case is also restrictedto the higher order terms. The sphere expression, on the other hand, is α S sphere = τ (cid:96) τ (cid:96) − β S ( D ∗ − K ) τ (cid:96) ( D ∗ − K + ) − β S ( D ∗ − K ) (13)which is similar in form to the other two but multiplied by τ (cid:96) . That factor appears due to theway the δ and ρ factors simplify. It translates to a larger range of α S values for each cell size S due to the stronger dependence on (cid:96) , which means that the spherical chloroplasts exhibitconsiderably more scatter than the cylindrical or disk-shaped chloroplasts.The leading-order expression for the sphere α S is α S sphere ∼ τ (cid:96) ( D ∗ − K + ) , (14)from which we clearly see that there is (cid:96) -dependent scatter even at leading order. However, α S sphere is still independent of S at leading order, which agrees with the linear trend of thedata. Here we’ve proposed a model to understand the linear trend and scatter seen in the data fromPyke and Leech (1992) and Osteryoung et al (1998). Our model explains both the linearityand scatter as being dependent on the geometry of growth. For cylinders or disks that growin one dimension, our model predicts a linear trend with minimal scatter. For spheres, we seemore scatter. This is consistent with mutant phenotypes that are characterized by irregularshapes and greater scatter when compared to the more cylindrical wild-type chloroplasts.4.1 Geometry effectsIf the geometry of a chloroplast has a surface area, plan area, and volume that scale togetheras it grows, at the end-of-growth bifurcation we get a narrow range of plan area ratios α foreach cell size for any reasonable range of parameter values. For example, the cylindrical-and disk-shaped chloroplasts, whose surface area, plan area, and volume all scale with (cid:96) and (cid:96) , respectively, show minimal scatter in the model. If the areas and volume do not scaletogether, then we get a wide range of plan area ratios for each cell size, which shows up asscatter. This appears in the case of spherical chloroplasts, which have a volume that scaleswith (cid:96) but surface and plan areas that scale with (cid:96) . In all cases, whether the range of α values is small or large, the actual α values have little dependence on cell size S . As a resultwe observe a linear relationship between total chloroplast plan area and cell plan area, withlinearly growing scatter. c h l o r op l as t p l a n a r ea disk Fig. 8
A sample of calculated disk-shaped chloroplast total plan areas that correspond to the end-of-growthbifurcation, over a range of cell plan areas. Note the linear trend, with some scatter. Areas are in µ m . Figurecreated in MATLAB. [Parameters: ˜ δ = k E = . k A = . c h l o r op l as t p l a n a r ea sphere Fig. 9
A sample of calculated spherical chloroplast total plan areas that correspond to the end-of-growthbifurcation, over a range of cell plan areas. Note the linear trend, with significant scatter. Areas are in µ m .Figure created in MATLAB. [Parameters: ˜ δ = k E = . k A = . R values for the data presented by Pyke and Leech (1992) indicate that although allthe data sets have a linear trend, the amount of scatter varies amongst the wild-type andthe mutants (see Figure 1). Our model can explain the variance in scatter by consideringgeometry. The data sets with little scatter – the wild type, with R = .
92, and the arc1 mutant with R = .
93 – can be explained by chloroplasts that have areas and volume thatscale together as they grow, such as the cylinders or disks examined here. The data sets withadditional scatter – the arc2 mutant, with R = .
81, and the arc3 mutant with R = .
61 –can be explained by chloroplasts that have areas and volumes that scale disproportionally,such as spheres. All three shapes considered here produce linear trends as seen in the data.Revisiting the original data and measuring either the number of chloroplasts in each cellor the size of the chloroplasts, for each data point, would allow us to predict where thosepoints should fall in the plan area plane based on our model. A comparison of the residualsfor a least squares fit could tell us whether our model accounts better for the scatter in thedata than noise would.The features of the data from Osteryoung et al (1998) are comparable to those from Pykeand Leech (1992), and can be similarly explained by our model if the transgenic types withhigher scatter have chloroplast shapes with volume and area that scale disproportionally.
A Parameter choices
The ODE system constants in equations (1) and (2) were taken directly from Dockery and Keener (2001): V R = V A = K R = K A = R = . A = . k R = .
7. With these constants, the relevant bifurcationvalue of d is at d ∗ = . τ and surface area S , rates k A and k E , and permeability ˜ δ , that determinethe value of the bifurcation parameter d are chosen as follows.Cell size S is variable but of order 10 µ m (see Figure 1 as duplicated from Pyke and Leech (1992))and cell thickness τ is constant and of order 1 µ m (Pyke, 1999). We take τ = µ m .The expression for d requires d ∗ − k A >
0; this provides an acceptable range for choosing k A < . k A and k E represent the degradation of the autoinducer (inside and outside the chloroplasts, respec-tively), we assume they are of the same order of magnitude. In addition, we need to choose ˜ δ such that4 Kelly M Paton et al. S ˜ δ >> k E ; this is the regime in which the autoinducer can collect to a sufficient density outside the chloro-plasts to trigger the switch in d . Final specification of k A , k E and ˜ δ is subject to a rough fit of the observedslope of the data in Pyke and Leech (1992) to the leading order expression for α for each geometry. Wechoose to fix k E = . δ = k A to adjust the slope for each geometry. Thereis some flexibility in choosing these parameters within an order of magnitude or more. Acknowledgements
K Paton thanks NSERC and UBC for their financial support. E Cytrynbaum was sup-ported by an NSERC Discovery Grant 298313-09.