Modeling competition between yeast strains

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Modeling competition between yeast strains Maarten de Gee, Hilda van Mourik, Arjan de Visser, and Jaap Molenaar Citation: AIP Conference Proceedings 1723, 020001 (2016); doi: 10.1063/1.4945057 View online: http://dx.doi.org/10.1063/1.4945057 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1723?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A model with competition between the cell lines in leukemia under treatment AIP Conf. Proc. 1637, 1325 (2014); 10.1063/1.4907298 Bifurcation analysis of a model of the budding yeast cell cycle Chaos 14, 653 (2004); 10.1063/1.1780011 Competition between strain-induced and temperature-controlled nucleation of InAs/GaAs quantum dots J. Appl. Phys. 95, 2998 (2004); 10.1063/1.1645637 Ultrathin epitaxial ferroelectric films grown on compressive substrates: Competition between the surface and strain effects J. Appl. Phys. 91, 2247 (2002); 10.1063/1.1427406 Mathematical model of the cell division cycle of fission yeast Chaos 11, 277 (2001); 10.1063/1.1345725

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Modeling Competition between Yeast Strains Maarten de Gee1, a), Hilda van Mourik2, Arjan de Visser3 and Jaap Molenaar1 1 Biometris, Wageningen UR, Droevendaalse steeg 1, 6708PB Wageningen, the Netherlands PRI Bioscience, Wageningen UR, Droevendaalse steeg 1, 6708PB Wageningen, the Netherlands. 3 Laboratory of Genetics, Wageningen UR, Droevendaalse steeg 1, 6708PB Wageningen, the Netherlands. 2

a)

Corresponding author: [email protected]

Abstract. We investigate toxin interference competition between S. cerevisiae colonies grown on a solid medium. In vivo experiments show that the outcome of this competition depends strongly on nutrient availability and cell densities. Here we present a new model for S. cerevisiae colonies, calculating the local height and composition of the colonies. The model simulates yeast colonies that show a good fit to experimental data. Simulations of colonies that start out with a homogeneous mixture of toxin producing and toxin sensitive cells can display remarkable pattern formation, depending on the initial ratio of the strains. Simulations in which the toxin producing and toxin sensitive species start at nearby positions clearly show that toxin production is advantageous.

INTRODUCTION The two most important types of competition between species are exploitative and interference competition. Exploitative competition is an indirect negative interaction caused by the use of a common resource. Interference competition is a direct negative interaction involving active suppression of a rival population. Here we deal with both types of competition between two strains of S. cerevisiae: the “normal” or sensitive strain, and a toxic strain. The killer system in a toxic S. cerevisiae strain is controlled by two RNA viruses: one encoding the killer toxin (killer virus), the other responsible for stable replication and maintenance of the toxin producing virus (helper virus) (Schmitt and Breinig, 2006). Once excreted into the growth medium, the toxins attack sensitive yeast cells. In low concentrations the toxin induces apoptotic cell death, whereas high toxin concentrations cause necrotic cell death. Toxin production not only incurs costs to the recipient but also to the producer (Gardner et al., 2004; West et al., 2006). It reduces the growth rate of its producer, making it an inferior exploitative competitor in comparison to nonproducing individuals. In order to become competitive against sensitive non-producers, producers must compensate these costs by benefits resulting from killing (Pintar and Starmer, 2003): there is a trade-off between interference competition and growth. Apart from the characteristics of the producer and sensitive strains, the outcome of the competition is strongly affected by environmental conditions, such as the availability of nutrients, and the motility of yeast, nutrients and toxin. In general, liquid environments are associated with high dispersal rates. In this work we focus on interference competition in environments with low dispersal and limited nutrients, like rotting fruits and cacti. Taking agar plates as a model environment we developed a spatially explicit mathematical model for colony growth and for the competition between producer and sensitive yeasts. As nutrient availability affects the outcome of the competition, its dynamics were modelled explicitly, including the nutrients freed from the killed yeast cells. In this way, the competing yeasts directly influence their habitat’s quality, as in nature. The newly developed model is used to study the shapes of competing colonies, especially as a function of varying initial numbers of microbes.

MODEL DESCRIPTION We developed a mathematical model for the competition between toxin producing and sensitive yeasts. Its state variables are the densities of the toxin producer strain (P) and the toxin susceptible strain (S), and the concentrations of nutrients (N) and toxin (T), depending on the two-dimensional position x on the agar plate, and the time t. By common assumptions on population dynamics and a simple hypothesis on the spatial dispersal of the yeast cells, colony formation becomes an emergent property of our model. Nearly all parameters could be derived from existing literature, only one was fitted to our experimental data.

Symposium on Biomathematics (Symomath 2015) AIP Conf. Proc. 1723, 020001-1–020001-5; doi: 10.1063/1.4945057 © 2016 AIP Publishing LLC 978-0-7354-1370-2/$30.00

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Single Strain Colonies The local concentration of yeast cells in a colony changes in time because of the balance between newly formed cells and dying cells, and the dispersive motion of cells that are being pushed away by newly formed cells. In a yeast colony on a solid medium, cells derive their nutrients from that medium. These nutrients move upward by diffusion, but on the way up they are consumed. Therefore the nutrients reach up only to a certain level in the colony, which is called the penetration depth. This behavior is modeled by logistic growth, where the growth rate per capita decreases linearly with the population density. Nutrient uptake increases with the nutrient concentration up to a saturation point (Holling type II), and natural mortality is proportional to the amount of yeast cells. As S. cerevisiae cells do not have an active mode of movement, expansion of the colony is determined by generation of new cells. This process is modelled as growth driven diffusion: · § J N ( x, t ) § y ( x, t ) · w ¨ ¨1  ¸  P ¸ y ( x, t )  ’ ˜ D y ’ y ( x, t ) , y ( x, t ) ¸ ¸ ¨ D K  N ( x, t ) ¨ wt Y max ¹ © ¹ ©





· § J N ( x, t ) § y ( x, t ) · ¸  P ¸ y ( x, t ) . ¨1  R¨ (1) ¸ ¸ ¨ D K  N ( x, t ) ¨ Y max ¹ © ¹ © Here y(x, t) is the superficial density of yeast at position x and time t, Ymax is the maximum density of yeast, and P is the natural mortality rate. N(x, t) is the nutrient concentration at position x and time t, J is the maximum nutrient uptake rate, D is the amount of nutrients per cell division, K is the nutrient uptake saturation constant. The nutrient concentration is affected by the uptake of nutrients by viable cells, the return of nutrients from cells that die by apoptosis, and by diffusive transport in the solid medium: § · y ( x, t ) w N ( x, t ) § y ( x, t ) · ¨1  ¸  EP¸ (2) N ( x, t ) ¨  J  D N 'N ( x, t ) , ¨ ¸ ¨ ¸ h K  N ( x, t ) © Ymax ¹ wt © ¹ where h is the thickness of the agar layer, DN is the diffusion coefficient of glucose in the solid medium, and E is the amount of nutrients released per apoptotic cell. In general, E < D. Dy

Exploitative and Interference Competition Grown on a solid medium, two strains of S. cerevisiae may form a mixed colony in which they engage in exploitative competition for nutrients. As the strains have comparable characteristics, their growth can be modelled with the same set of equations, differing only in the amount of nutrients needed per cell division (D). However, the carrying capacity of a mixed colony is not limiting to each strain separately, but to their combination. Also the diffusion rate depends on the net growth rate of both species. When a sensitive strain and a toxin producing strain coexist, interference competition comes into play. The toxin concentration, denoted by T, is affected by toxin production, degradation and uptake, and diffusion. The production rate is modelled as proportional to the gross growth rate of the producer strain (Ramon-Portugal et al., 1997a). The rate of toxin uptake by sensitive cells is modelled as mass action, and the degradation is proportional to the toxin concentration. Finally, the toxin spreads through the agar layer by molecular diffusion. This leads to w 1 (3) V ˜ (gross growth rate of P( x, t ))  G S ( x, t ) T ( x, t )  OT ( x, t )  DT 'T ( x, t ) . T ( x, t ) wt h Here T(x, t) is the toxin concentration at position x and time t, and S(x, t) and P(x, t) are the densities of the sensitive and the toxin producing strain respectively. V is the toxin production factor, O is the toxin degradation rate, G is the toxin uptake rate, and DT is the diffusion coefficient of the toxin. Assuming that the rate of cell killing by the toxin is proportional to the rate of consumption of the toxic protein, the model for the yeast strains becomes

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w S ( x, t ) wt

§ J · N ( x, t ) § S ( x, t )  P ( x, t ) · ¨ ¨1  ¸  P  HT ( x, t ) ¸ S ( x, t ) ¸ ¨ D K  N ( x, t ) ¨ ¸ Ymax © ¹ © S ¹ S ( x, t )  ’ ˜ D y ’( S ( x, t )  P ( x, t )) , S ( x , t )  P ( x, t )



(4)



· § J N ( x, t ) § S ( x , t )  P ( x, t ) · ¨ ¨1  ¸  P ¸ P ( x, t ) ¸ ¸ ¨ D K  N ( x, t ) ¨ Ymax © ¹ ¹ © P P ( x, t )  ’ ˜ D y ’( S ( x, t )  P ( x, t )) , S ( x, t )  P ( x , t ) where H is the lethality coefficient of the toxin. w P ( x, t ) wt



(5)



RESULTS Single Strain Colonies First we simulated the development of a single strain colony and compared its pattern with observed shapes. Figure 1 shows typical side views of Saccharomyces cerevisiae colonies grow on a 2% YPD agar medium. The pictures are taken 2, 3, 4 and 6 days after inoculation. Initially, S. cerevisiae colonies grow as parabolic caps on the solid-medium. As time increases, the characteristic nipple shape appears. In our in vivo experiments, the yeast colonies show contact angles between 35° and 50°, depending on the time after inoculation and the colony size. We used this to calibrate the reorganization coefficient R in our model. With R = 6.0·10–12, the angle with the solid C that, it slowly declines D A B of 47° after 3 days. After medium reaches its maximum to 34° at day 8.

Figure 1: Side view of colonies grown on solid agar, from left to right 2, 3, 4 and 6 days after inoculation. While the actual colony height increases, the scale of the image decreases accordingly. The downward shape is a reflection in the agar plate. Photos by Hilda van Mourik. Single colony: yeast

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Figure 2: A simulated single strain colony for 1 up to 8 days. Note that the axes are scaled differently.

Cross-sections of a single strain colony simulated with our model are shown in Figure 2, for eight consecutive days. The model correctly predicts the characteristic nipple shape.

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Interference Competition in a Mixed Colony Next we simulated a single colony inoculated with a homogeneous mixture of sensitive and toxin producing cells. Figure 3 shows a simulated colony that starts with 1000 cells, with 1 producer for each 15 sensitive cells. After 8 days, we see a remarkable pattern: a central island that is separated by a moat from an outer wall. For the first six days, both strains form a mixed colony. However, on day 7 a gap is forming, and on day 8 the strains are almost segregated: the outer ring consists of producer cells, while the central island consists for a large part of sensitive cells, and for a small part of producer cells.

Figure 3. A 3D view of a single colony after 8 days, staring with mixed strains in a ratio 1:15.

The occurrence of a pattern as in Figure 3 depends critically on the parameters. Simulations starting with the same number of sensitive cells, and a varying number of producer cells yield different outcomes. Figure 4 shows how producers and sensitives develop in time, when the starting population of producers is 1:10, 1:15, or 1:100. In the simulation with an initial ratio p:s = 1:10, the sensitive strain’s advantage is shortly lived. After a short while, the toxin concentration becomes high enough to kill sensitive cells. The amount of killer proteins quickly increases , leading to complete extinction of the sensitive cells in the colony in the third day. In the simulation with an initial ratio p:s = 1:15 the producers still win the competition, but it takes longer than the 8 days of this simulation. Furthermore, an island in the center is left where sensitives and producers coexist. The nutrients are depleted here, so there is no production of new cells, but only natural death, equally in both strains. In the simulation with an initial ratio p:s = 1:100, the producers cannot win the competition anymore. At any time, the number of sensitive cells increases faster than the number of producer cells. While the producer strain hangs on tenaciously, it is marginalized ever more. Mixed colony: yeast

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Figure 4. Simulation of a mixed colony of S. cerevisiae with a sensitive and a toxin producing strain. The initial ratios between producer and sensitives are 1:10, 1:15 and 1:100 respectively.

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Interference Competition between Separated Colonies To predict the outcome of competition between two separate colonies, simulations were run starting with two inoculums placed 0.2 cm apart. One of these colonies consisted of a toxin producing strain, the other of a sensitive strain. In all cases, the sensitive’s colony was inoculated with about 2700 cells. In several runs, the opposing toxic colony started with a varying number. A typical outcome is shown in Figure 5, for an initial ratio p:s = 1:10. In this simulation, the colonies develop about equally fast for the first three days, the sensitive strain having a slight advantage; the gap between the colonies fills up quickly, and to the eye the colonies are merged. After 3 days, the toxic strain starts catching up slowly, and a new gap is formed as a result of the toxic interference; at day 8, the colony sizes are about equal. The right part of Figure 5 displays a spatial view of the colonies, with the sensitive strain to the right, the toxic one to the left, after 8 days. We observe two separated colonies, about equal in size, divided by a distinct cleavage. This cleavage gradually moves in the direction of the sensitive strain colony. Sensitive cells are being killed by the toxin, freeing nutrients in an apoptotic death in an elsewise nutrient depleted environment. Interference: yeast

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Figure 5: To the left: time course of sensitive cells (solid) and toxin prodicers (dashed). To the right: 3D image of competing colonies with initial ratio p:s = 1:10. Note that the vertical axis is not to scale.

REFERENCES Gardner, A., West, S. A., Buckling, A., 2004. Bacteriocins, spite and virulence. Proc Biol Sci 271, 1529-1535, doi:10.1098/rspb.2004.2756. Nguyen, B., Upadhyaya, A., van Oudenaarden, A., Brenner, M. P., 2004. Elastic Instability in Growing Yeast Colonies. Biophysical Journal 86, 2740-2747. Pintar, J., Starmer, W. T., 2003. The costs and benefits of killer toxin production by the yeast Pichia kluyveri. Antonie Van Leeuwenhoek 83, 89-97, doi:10.1023/A:0000000089097. Ramon-Portugal, F., Delia-Dupuy, M. L., Pingaud, H., Riba, J. P., 1997a. Kinetic study and mathematical modelling of the growth of S. cerevisiae 522D in presence of K2 killer protein. Journal of Chemical Technology and Biotechnology 68, 195-201. Ramon-Portugal, F., Delia-Dupuy, M. L., Pingaud, H., Carrillo-Ieroux, G. A., Riba, J. P., 1997b. Kinetic study and mathematical modelling of killer and sensitive S. Cerevisiae strains growing in mixed culture. Bioprocess Engineering 17, 375-381. Schmitt, M. J., Breinig, F., 2006. Yeast viral killer toxins: lethality and self-protection. Nat Rev Microbiol 4, 212221, doi:10.1038/nrmicro1347. West, S. A., Griffin, A. S., Gardner, A., Diggle, S. P., 2006. Social evolution theory for microorganisms. Nature Reviews Microbiology 4, 597-607.

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