On the growth of non-motile bacteria colonies: an agent-based model for pattern formation
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On the growth of non-motile bacteria colonies: anagent-based model for pattern formation
Lautaro Vassallo , David Hansmann , and Lidia A. Braunstein , Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, and Instituto deInvestigaciones Físicas de Mar del Plata (IFIMAR-CONICET), Deán Funes 3350, 7600 Mar del Plata, Argentina Physics Department and Center for Polymer Studies, Boston University, Boston, Massachusetts 02215, USAthe date of receipt and acceptance should be inserted later
Abstract.
In the growth of bacterial colonies, a great variety of complex patterns are observed in exper-iments, depending on external conditions and the bacterial species. Typically, existing models employsystems of reaction-diffusion equations or consist of growth processes based on rules, and are limited to adiscrete lattice. In contrast, the two-dimensional model proposed here is an off-lattice simulation, wherebacteria are modelled as rigid circles and nutrients are point-like, Brownian particles. Varying the nutri-ent diffusion and concentration, we simulate a wide range of morphologies compatible with experimentalobservations, from round and compact to extremely branched patterns. A scaling relationship is foundbetween the number of cells in the interface and the total number of cells, with two characteristic regimes.These regimes correspond to the compact and branched patterns, which are exhibited for sufficiently smalland large colonies, respectively. In addition, we characterise the screening effect observed in the structuresby analysing the multifractal properties of the growth probability.
PACS.
XX.XX.XX No PACS code given
The concept of active matter is relatively new within softmatter physics; the fundamental units of this type of mat-ter, called active agents, have the particularity of absorb-ing energy from their environment and dissipating it in or-der to move, grow or replicate, among other activities [1].Most of the examples of active matter are biological, suchas bacteria. Although they can be seen as the simplest liv-ing organisms, they present interesting behaviours, bothindividually and collectively.Bacteria exhibit many different types of movement, de-pending on the species and the environment, which de-termines the macroscopic appearance of the colony. Ac-cording to Henrichsen [2], six types of motility can beidentified: swimming, swarming, twitching, darting, glid-ing (newer studies subdivide this category [3]) and slid-ing. We focus on the last one, which is a mechanism pro-duced by the expansive forces of the colony, in combina-tion with special properties of the cell membranes charac-terised by low friction with the substrate on which theygrow; the bacteria do not move by their own motors, butpush each other by duplicating themselves and competingfor the same spaces. Despite its simplicity, its importancehas been pointed out in more complex bacterial processes,such as in the formation, dispersion and restructuring ofbiofilms [4]. Although we will not strictly enter the biofilmsfield in this work, it is important to mention they are the focus of numerous studies, due to the complexity ofthe processes involved, their resistance to hostile environ-ments, and the challenge they present to medicine [4–8].At the end of the 80s, Matsuyama [9], Fujikawa andMatsushita [10], showed that the patterns of bacterialcolonies obtained in the laboratory could be fractal ob-jects. The properties of their patterns depend on twomain factors: the concentration of nutrients, which in-fluences the growth rate of the colony, and the con-centration of agar, which determines the hardness ofthe substrate, and therefore, the mobility of the bac-teria. In the absence of special forms of motility, thepatterns were classified in a two-dimensional phase di-agram in which five characteristic patterns were identi-fied: diffusion-limited aggregation-like (DLA-like), Eden-like, dense branching morphology (DBM), concentric ringand homogeneous disk-like. The experiments were per-formed mainly with the species Bacillus subtilis [10–13].Without self-propulsion, only DLA and Eden-like patternsare expected.At the theoretical level, continuous models are themost traditional and extended way of studying the pat-terns exhibited by bacteria colonies. In them, both bac-teria and nutrients (or any other variable of interest) arerepresented by density functions per unit area, and thespatio-temporal evolution of the system is described bysystems of reaction-diffusion equations [11, 14–23]. Thesemodels are successful in describing a wide range of pat- a r X i v : . [ phy s i c s . b i o - ph ] N ov s part of the Springer Nature SharedIt initiative, a full-text view-only version of the paper can be accessed clicking here terns, although they are valid only at a mesoscopic scale.To represent growth at the microscopic level, microor-ganisms must be represented by discrete mobile entities(agents) [24–27]. In this scheme, on-lattice [26] and off-lattice [24, 25, 27, 28] approaches can be chosen.The motivation of this work is to propose a micro-scopic model that can explain the experimental observa-tions, based on the fact that sliding is dominated by themechanical interaction between the bacterial cells. As hasbeen said, continuous models work well only on a meso-scopic scale, whereas in agent-based models, if space isdiscretised as Euclidean networks, mechanical laws can-not be used. In spite of being computationally expensive,in this work, we choose an off-lattice model, in order torepresent our agents as rigid bodies governed by laws ofmechanics. Thus, we can analyse the growth of bacterialcolonies on a microscopic level. The off-lattice approachalso avoids anisotropies in the patterns exhibited by thecolonies induced by the discretization of the space.A typical way to characterise the complex structuresthat arise in surface growth is by means of the Hausdorffdimension, often referred to as the fractal dimension. How-ever, the fractal dimension is not a unique descriptor, as itwas shown that two structures may have the same fractaldimension but are fundamentally different [29]. In order todescribe structures more deeply and unequivocally, the de-termination of the multifractal properties of an associatedmeasure (e.g. growth probability) offers a suitable supple-ment to the sole measurement of their fractal dimension.This is an entropy-based approach [30], classified this wayto differentiate it from other analyses that rely only onmetric concepts. Here the scaling properties are analysedfor variations in different parts of the pattern, which areoverlooked by a simple measurement of the fractal dimen-sion. As it is arduous to treat growth models with nonlocalrules analytically [31], it may be interesting to characterisean associated measure such as the growth probability togain some insight about the process.A way to describe the multifractal behaviour isthrough the generalised dimensions D q (also known asRényi dimensions). If one covers the support of the mea-sure (set of all points where the measure is positive) witha set of boxes of size l and defines a probability P i ( l ) (integrated measure) in the i th box, the generalised di-mensions D q correspond to the scaling exponents for the q th moments of P i , defined by (cid:80) i P qi ( l ) ∼ l ( q − D q . In thiscontext, q is typically referred to as the order q of the gen-eralised dimension D q . Solving for D q and taking the limitof l → , the conventional expression for the generaliseddimensions is given by D q = 1( q − lim l → ln (cid:80) i P qi ( l )ln l . For the case q = 1 , the L’Hôpital’s rule must be used;thus, D = lim l → (cid:80) i P i ( l ) log P i ( l )ln l . The generalised dimensions are exponents that charac-terise the non-uniformity of the measure; the positive or-ders q accentuate the regions with higher probabilitieswhile the negative q ’s the opposite.In early works, D q was only defined for q ≥ [32], sothe previous definition had the particularity that the firstvalue of the set, i.e. D q =0 , corresponded to the Hausdorffdimension of the support (because all the boxes have thesame weight). There are also specific names for other cer-tain values. For example, D q =1 is known as the informa-tion dimension, which is interesting in the case of diffusion-limited aggregations, since it can be physically interpretedas the fractal dimension of the active region, i.e., the un-screened region [29]; D q =2 is known as the correlationdimension; D q → ± ∞ are known as the Chebyshev dimen-sions, which are calculated with the maximum and min-imum probabilities, respectively; equivalences with otherdimensions definitions can be made, even for fractional q values [30].This is not the only multifractal analysis possible. Inother works, the singularity spectrum is computed [33,34],which is closely related to the Rényi dimensions by a Leg-endre transform. Temporal fractals can also be studied,where the local scaling properties are now related to timebehaviour [35]. We will use the generalised dimensions tocharacterise quantitatively the patterns produced by ourmodel. In this paper, we model the growth of non-motile bacterialcolonies under different environmental conditions, specif-ically, nutrient concentration and nutrient diffusion. Thegrowth rules are inspired by biology, as we capture the es-sential characteristics of bacteria without losing simplicity.We consider a two-dimensional and off-lattice space, whichallows us to consider mechanical interactions between theagents, as we will explain below.There are two kinds of particles in the model, nutri-ent particles and bacterial cells. Both of them have phys-ical properties such as size, mass, position, velocity andmight have applied forces. Nutrient particles are idealisedas Brownian particles, so its initial velocities follow theMaxwell-Boltzmann distribution and evolve according toa Langevin equation of the form m ˙ v ( t ) = − κTD v ( t ) + f ( t ) , where κ is the Boltzmann’s constant, T is the tempera-ture and D is the diffusion coefficient. The function f ( t ) is a stochastic force whose components follow a Gaussianprobability distribution with mean zero and standard de-viation σ = κT (cid:112) /D . The Langevin equation is numer-ically integrated using a small time step ∆t , followingthe explanations in The Fokker-Planck Equation by H.Risken [36]. Nutrient particles are considered point-like,non-interacting with each other and with a small mass.The bacterial cells are modelled as rigid circles with radius r b , so they can interact with each other through normalforces. The numerical values used in the simulations canbe found in Table 1.s part of the Springer Nature SharedIt initiative, a full-text view-only version of the paper can be accessed clicking here Lautaro Vassallo et al: On the growth of non-motile bacteria colonies 3
Table 1: Numerical values used for the simulation.
Variable Value [arbitrary units] kT ∆t . r b m (nutrient mass) . In addition to the physical properties, bacterial cellshave two biological characteristics: they can be fed andreproduce. The first one is an interaction with the nutri-ents, which are absorbed by the cell when they are in con-tact. Reproduction is the process by which the bacteriumduplicates: an identical copy of the cell is generated inthe same position as the original, so they overlap. Then,they are disaggregated by opposing velocities in a ran-dom direction. As a consequence, these cells may collideelastically with neighbours, according to the mechanics ofrigid bodies, as shown in Fig. 1. When the cells stop over-lapping, they stop moving and become static because themedium is considered to be very viscous so that the mo-mentum gained by the collisions is immediately dissipated.All the calculations to resolve collisions and overlaps arebased on an iterative constraint solver introduced by ErinCatto [37].Fig. 1: Feeding and reproduction. Bacteria cells (circles)can absorb the particles diffusing in the medium (dots)and duplicate, causing collisions with neighbours, whichmove in the direction pointed by the arrows. The doubleblue arrow (colour online) points the direction in whichthe newborn bacteria disaggregate.In summary, reproduction causes both the movementof newborn cells and their neighbours, representing slidingmotility.The simulation begins with a single cell in the origin ofcoordinates in a 2D substrate and a given quantity of nu-trient particles diffusing in space, according to the concen-tration and diffusion coefficient specified. When a nutrientparticle touches the cell, it is absorbed and the bacteriaduplicates. Now the colony is formed by two cells, whichcan absorb nutrients and reproduce. The process contin-ues in this way and the colony grows progressively. Allthe bacteria have a time delay (20 integration time steps),during which they can’t duplicate; this rule ensures that Fig. 2: A ring surrounding the colony (in red) acts as a nu-trient reservoir. The concentration of nutrient is constantinside it, but decreases in the proximity of the colony, be-cause of the feeding. The distance between the ring andthe colony is r b at least.no duplication occurs while newborn cells are still over-lapping and it is consistent with biological observations,e.g., Bacillus subtilis species has a delay of ∼ minutesbetween duplications [17].In order to keep the nutrient concentration constant,there is a ring that acts as a nutrient reservoir located at agiven distance from the most external position of the bac-teria. This distance increases progressively as the colonygrows, so the separation is r b at least. The nutrient con-centration within the ring ( r > r b ) is kept constant ata specified value. But, closer to the colony ( r < r b ),the concentration drops due to nutrient absorption by thebacteria. Periodic boundaries conditions are considered forthe outer side of the ring (Fig. 2).The growth stops when the colony reaches a radiusof r b , when characteristic patterns are fully developed.This implies that we have up to half a million cells formingthe colony, depending on the parameters. In order to see the variety of morphologies that the modelcan produce, we choose several different values of nutrientconcentration and nutrient diffusion, and register the posi-tion of the bacteria along the perimeter of the colony overtime and average over a hundred realizations for each setof parameters. A morphology diagram is shown in Fig. 3a,where it can be seen that it is possible to generate roundand compact colonies, as well as ramified, going througha variety of intermediate patterns. Similar morphologicalcrossover can be seen in Figs. 3b–e, which corresponds tothe experiments carried out in [11,12]. The fractal dimen-sion of intermediate patterns in experiments was reportedfor the case of the most ramified one. In Table 2, we sum-marise the results found in the bibliography [10, 12] andours (for the most ramified cases), which are in good agree-ment.s part of the Springer Nature SharedIt initiative, a full-text view-only version of the paper can be accessed clicking here
Fig. 3: Examples of different morphologies. In (a) there is a sample of the model predictions. On the X -axis, the diffusioncoefficient D is varied (in arbitrary units), while on the Y -axis the nutrient concentration is varied (measured as thenumber of particles per unit area). Each curve, in a different shade of grey, corresponds to a different time. All othersub-figures correspond to experiments. The parameter C a corresponds to the agar concentration, which determinesthe hardness of the substrate, and C n corresponds to the nutrient concentration. (b) C a = 10 g/l ; C n = 20 g/l . (c) C a = 9 g/l ; C n = 4 . g/l . (d) C a = 8 g/l ; C n = 3 g/l . (e) C a = 9 g/l ; C n = 1 g/l . Figure (b) is from [11], reprinted withpermission from Elsevier. Figures (c), (d) and (e) are from [12], ©(1992) The Physical Society of Japan, reproducedwith permission.Table 2: Fractal dimension D f . Values reported in exper-imental works, our model and DLA (the error is the stan-dard deviation). Only the results for the most branchedcases are included with C = 1 / , D = 512 and C =1 / , D = 512 . Greater values than for the case of DLAare expected, where only one particle diffuses at a time,unlike our case where we have many ( C > ). D f Model [ C = 1 / D = 512 ] . ± . Model [ C = 1 / D = 512 ] . ± . Experiment [10] . ± . Experiment [12] . ± . DLA [29, 31] . ± . Despite using different values for the parameters, it is ob-served that the curves of the number of bacteria at the interface S versus the total number N show two power-lawregimes. The first regime corresponds to initial compactstructures S ∼ N / , while the second regime correspondsto ramified structures with S ∼ N , as shown in Fig. 4a.These two behaviours are characteristic of the Eden andDLA models, respectively.To characterise the crossover between regimes, wecompute the total number of bacteria N ∗ at which thecrossover takes place. To achieve this, we simply fit power-law functions in the tails of the S vs N curves and computethe intersection. More sofisticated methods for an auto-matic determination of crossovers can be found in [38].After dividing N by N ∗ in each of the data sets, the Y -axis is divided by some value S ∗ looking for a satisfactorycollapse of the curves. We found that the best collapseoccurs when N ∗ = S ∗ , as shown in Fig. 4b.It can be seen that N ∗ depends on the diffusion D and the concentration C , having an increasing relationshipwith both (Fig. 5).s part of the Springer Nature SharedIt initiative, a full-text view-only version of the paper can be accessed clicking here Lautaro Vassallo et al: On the growth of non-motile bacteria colonies 5(a)(b)
Fig. 4: Power-law regimes. (a) Results, averaging over allrealizations, of the number of bacteria in the interface S versus the total number of cells N . (b) Collapsed curvesof S versus N . Both plots are double-logarithmic. (a)(b) Fig. 5: Relationship between N ∗ and the parameters. (a) N ∗ is plotted against the nutrient concentration C , leavingthe diffusion coefficient D constant; (b) the same but for D . Taking these observations into account, an attempt ismade to establish a scaling law. We know that the be-haviour of S is: S ∼ (cid:40) N / , N (cid:28) N ∗ N, N (cid:29) N ∗ , (1)where N ∗ = N ∗ ( C, D ) . The curves collapse dividing N and S by N ∗ and S ∗ , respectively, so: S/S ∗ ∼ (cid:40) ( N/N ∗ ) / , N/N ∗ (cid:28) N/N ∗ , N/N ∗ (cid:29) . (2)Then, having validated the relationship N ∗ = S ∗ andproposing the scaling function f ( x ) ∼ (cid:40) const, x (cid:28) x, x (cid:29) , (3)the relation between N and S can be written as: S = N / f [( NN ∗ ) / ] . (4)This result suggests that if we allow N to grow suffi-ciently, branches will always be generated, after a criticalnumber of cells is reached, dependent on the parameters C and D . The growth probability of each region of the colony cangive information about why a certain pattern displays. Ev-ery cell duplicates when a nutrient particle is captured, sothe growth probability is associated with the probabilitythat a diffusing particle reaches the site where the cell is.We use two methods to estimate this probability, focus-ing on the final stage of the colony. The first one consistson counting how many nutrient particles are absorbed byeach cell without letting it duplicate, i.e., the colony is"frozen" and the growth probability of each cell is com-puted dividing this counting by the total of particles in-corporated by the whole colony (we use approximately particles). We will refer to this method as C.M. The disad-vantage with this method is that it does not estimate lowprobabilities well, because several million particles maybe captured by the colony in total, but the internal re-gions may hardly incorporate any. Due to this, we alsosolve the Laplace equation ∇ φ = 0 , where φ representsthe nutrient concentration, by the relaxation method [39],where φ = 1 at infinity and φ = 0 along the perimeter ofthe colony, as it can be seen that the growth probabilityis proportional to the gradient of the potential ∇ φ [29].We use an iteration error of − , after checking that themultifractal curves do not vary appreciably. In order touse this method, referred to as L.M. henceforth, properly,space has to be discretised, so some differences with theC.M. are expected.In Fig. 6, it is shown how uneven is the number ofnutrient particles consumed between the outer and inners part of the Springer Nature SharedIt initiative, a full-text view-only version of the paper can be accessed clicking here > (b) Absorption > (c) Absorption > Fig. 6: Example of the C.M. results for a branched colony.After particles are captured, the green colour marksthe cells who absorbed more than (a) 1 particle, (b) 20particles and (c) 80 particles.regions of a ramified colony. This phenomenon is usuallyreferred to as ‘shadowing’ or ‘screening’ effect and is moreor less noticeable depending on the parameters. As thestructures that emerge from the simulations are fractal,the proper way to study this effect is by the multifractalformalism, explained in the first section.In Fig. 7, the generalised dimension D q> curve is plot-ted for different morphologies. It can be seen that theprobability associated with a ramified colony presents astrong multifractality since D q varies significantly with q .It is also included in Fig. 7 the curve for a diffusion-limitedaggregate [40] for comparison. The standard deviations arepresented in Table 3. Unfortunately, this analysis cannotbe carried out in very compact colonies since the fractalregime is very short to be reliable or it is not observable.Nor can it be done in the early stages of the growth processfor the same reason.In Fig. 8, the generalised dimension is plotted again,but now including the q < interval. Only the resultsobtained by the L.M. can be used in this interval. Thevalue of D q (cid:28)− is very interesting, because it quantifiesthe shadowing effect, making evident the differences be-tween different morphologies. It is worth noting that D q =0 should not be equal to the fractal dimension D f presentedin the previous subsection because D q =0 is the fractal di-mension of the support of the measure, i.e., the perimeterof the colony, while D f is the fractal dimension of thearea. All of these characteristic points are summarised inTable 3, which also includes the values corresponding todiffusion-limited aggregation as a comparison [29,40]. Theasymptotic values are estimated with q = 25 and q = − .Note that the curves of the generalised dimension arealways above the case of DLA. In the region of q > ,taking into account the calculated standard deviations, (a)(b) Fig. 7: Generalised dimension curve for q > . Differenticons correspond to different values of nutrient concentra-tion C and diffusion coefficient D . Red dashed line corre-sponds to DLA (colour online). Results using the (a) C.M.and (b) L.M.Fig. 8: Generalised dimension curve including negativevalues of q . Only the L.M. results are plotted. Differenticons correspond to different values of nutrient concentra-tion C and diffusion coefficient D . Note that the curves donot collapse for q > , the difference between them cannotbe seen on this scale (see Fig. 7).the differences are not as noticeable, but they are in theregion of q < . In this region, where the measurementbest distinguishes each case, they depart notoriously fromthe case of DLA. Always considering branched cases, itis observed that higher values are associated with higher C and D values, which can be understood if we associatethis measure with the screening phenomenon. The largerare C and D , the thicker and narrower the branches andthe fjords become, respectively, so the screening increasesto the interior areas. Nevertheless, note that there is as part of the Springer Nature SharedIt initiative, a full-text view-only version of the paper can be accessed clicking here Lautaro Vassallo et al: On the growth of non-motile bacteria colonies 7
Table 3: Some characteristic values of the generalised dimension D q (the error is the standard deviation). DLA resultsfound in [29, 40] are included for comparison. C = 1 / D = 512 C = 1 / D = 512 C = 1 / D = 1024 C.M. L.M. C.M. L.M. C.M. L.M. D q =0 − . ± . − . ± . − . ± . D q =1 . ± .
01 1 . ± .
05 1 . ± .
01 1 . ± .
09 1 . ± .
01 1 . ± . D q =2 . ± .
02 1 . ± .
09 1 . ± .
02 1 . ± .
12 1 . ± .
01 1 . ± . D q (cid:29) . ± .
04 0 . ± .
09 0 . ± .
04 0 . ± .
15 0 . ± .
04 0 . ± . D q (cid:28)− − . ± . − . ± . − . ± . C = 1 / D = 512 C = 1 / D = 1024 DLAC.M. L.M. C.M. L.M. [29] [40] D q =0 − . ± . − . ± .
01 1 . ± . D q =1 . ± .
01 1 . ± .
07 1 . ± .
01 1 . ± .
06 1 . ± .
01 1 D q =2 . ± .
02 0 . ± .
11 1 . ± .
01 1 . ± . − . D q (cid:29) . ± .
03 0 . ± .
14 0 . ± .
03 0 . ± .
13 0 . ± .
03 0 . D q (cid:28)− − . ± . − . ± . (cid:39) − limit on how large the parameters can be. If C and D aretoo large, so the branches disappear, the screening effect isbarely noticeable. The growth probability becomes almostuniform, and the multifractality should be lost. The goal of this work is the construction of a model basedon basic theories of physics, capable of generating a va-riety of complex patterns observed in bacteria colonies.Under the hypothesis that there is a single collective move-ment mechanism behind the different morphologies (slid-ing), the different results are achieved by varying parame-ters of the environment outside the colony, without chang-ing the behaviour of the agents. Under these precepts, wemanage to generate patterns from the most round andcompact to extremely ramified, going through differentintermediate morphologies. The different approaches usedto characterise the structures also allow comparison withthe two most studied models that predict patterns of bac-terial colonies, the Eden model and DLA.The fractal dimension analysis of simulations withbranched colonies shows a fractal dimension compatiblewith experiments [10]. Although there are distinct dif-ferences between the present and the DLA model, theirfractal dimensions are in good agreement, which suggeststhat the most ramified cases considered are close to thediffusion limit.On the other hand, the characterization using scal-ing laws show that there are two characteristic growthregimes, one compact and one branched. According to therelation found, the crossover between regimes occurs at acritical number of cells, that depends on the parametersfor nutrition concentration C and nutrition diffusion D .Thus, branches will always be generated for finite valuesof these parameters.The multifractal measurement shows strong multifrac-tality for the ramified cases. In case the order q of the generalised dimension D q is q > , the curves of D q ofthe simulations of different values of C and D are closeto each other, but in the region of q < , their differencesbecome notoriously. All these curves are, however, alwaysabove the DLA curve and, as expected, approach to theDLA curve for lower values for C and D . Higher valuesare associated with higher C and D values because fjordsinto the interior are narrow.Unfortunately, a more in-depth comparison betweensimulation and experimental data cannot be carried outdue to the lack of quantitative experimental data. To date,there are almost exclusively qualitative characterizationsof the morphologies of bacterial colonies, which only insome rare cases provide additional information on thefractal dimension (which is not a complete indicator). Al-though the methods that we use in this work to calculatethe generalised dimensions cannot be used in experiments,there are other methods that might be used, such as theone described by Ohta and Honjo [41], based on associ-ating probabilities according to the variation of the areaoccupied by the colony in a certain section. Only such adeeper experimental analysis would offer a complete char-acterization of the processes involved in the structure for-mation of bacterial colonies and would allow contrastingthe proposed model with experiments. Conflict of interest
Authors declare that there are no conflicts of interest.
Acknowledgements
L.A.B. and L.V. thanks UNMdP and CONICET (PIP00443/2014) for financial support. D.H. thanks UNMdPand CONICET (PIP 0100629/2013) for financial support.s part of the Springer Nature SharedIt initiative, a full-text view-only version of the paper can be accessed clicking here
Author contribution statement
L.V. and D.H. designed the model. L.V. implemented thecomputational framework. All authors contributed to theanalysis of the data and in the writing of the manuscript.
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