1. T HE ROLE OF MOTILITY AND NUTRIENTS IN BACTERIAL COLONY FORMATION AND COMPETITION 1 Silogini Thanarajah Guest Lecture

2. O UTLINES Introduction General model Mathematical theorems Numerical simulation Conclusion Extended model 2 Single competition Agar Liquid

3. I NTRODUCTION Bacterial competition and colony formation are an important part in medicine and plant roots colonization. 3

4. 4 Bacterial colonies. Different colony shapes. Many existing theoretical studies assume that bacteria have to move in the direction of nutrients. “Random walk” movement observed for bacteria in the absence of chemotaxis. Undirected motility must have evolved before chemotaxis.

5. 5 Directed movement (chemotaxis) and undirected movement. The undirected motility was thought to be not important. Undirected motility can be more important in resource- homogeneous environments. When the chemicals are not chemotactic stimulus. Wei et al.

6. PDE model in the explicit consideration of nutrient and different bacterial strains characterized by motility. Two types of bacterial strains: motile and immotile Agar vs liquid

7. Motile: Moving or having power to move spontaneously. Immotile: Almost not moving or lacking the ability to move. Agar media: A dried hydrophilic, colloidal substance extracted from various species of red algae; used in solid culture media for bacteria and other microorganisms. Liquid media:Chemically defined basal liquid media are used to provide nutrients for cell growth in research. 7

8. R ANDOM VS B IASED RANDOM WALK 8

9. 9 Reaction - diffusion system

10. O THER BACTERIA MODELS Lauffenburger modelWei model Mimura model Tyson model 10

11. E FFECT OF CHEMOTAXIS IN COMPETITION - CONFINED NONMIXED 11  Focusing on the influence of the random motility (μ) and the chemotaxis(Χ)  There is a minimum value of Χ necessary for a chemotactic population to have a competitive advantage over an immotile population in a confined nonmixed system.  Chemotaxis does not automatically provide a competitive advantage Conclusion: (1)Both die out; species 1 exist alone, species 2 exist alone; both coexist (2) well-mixed: slower growing can coexist and even exist alone if it possesses sufficiently superior motility and chemotaxis.

12. E FFECT OF RANDOM MOTILITY IN COMPETITION - CONFINED NONMIXED 12 Assumption:  differing growth kinetic and motility properties  Diffusible growth-rate-limiting chemical nutrient entering from the boundary Conclusion: (1)Species 1 survives, 2 dies out ; species 2 survives, 1 dies out; both coexist (2) Smaller maximum growth rate may grow to a larger population than he other

13. A GAR METHOD VS L IQUID METHOD (B RUCE L EVIN ’ S GROUP EXPERIMENT ) Observation from experiments results: In agar, the motile strain has higher total density. In liquid, both have the same total density. 13 Non-motilemotile agar The population dynamics of bacteria in physically structured habitats and the adaptive virture of random motility, Wei et al., PNAS day

14. G ENERAL MODEL - R EACTION - DIFFUSION MODEL 14 Where h i (N)=α i N or α i N/(k i +N) satisfy h i (0)=0,h i ’(t)>0 and h i ’’(t)≤0 B i – Density of bacterial strains; N- Density of nutrient D i –Diffusion coefficients; δ i – Mortality rates Ϫ i – The yield coefficients; α i – resource uptake rate K i - Half-saturation constants (nutrient uptake efficiencies) with initial and zero flux boundary conditions. B 1 -motile strain B 2 -immotile strain

15. S INGLE BACTERIAL SPECIES 15 Bacteria-Substrate model without nutrient diffusion

16. 16 Spatially uniform steady states (1-D,2-D) Spatially uniform steady states: independent of time and space

17. M ATHEMATICAL THEOREMS 17

18. C OMPETITION CASE - A GAR 18 (5) These bacterial strains are genetically identical except for their motility: α 1 =α 2, δ 1 =δ 2, ϫ 1 = ϫ 2, k 1 =k 2 with D 1 >>D 2

19. C OMPETITION CASE - L IQUID 19 These bacterial strains are genetically identical except for their motility: D 3 – diffusion constant for nutrient. α 1 =α 2, δ 1 =δ 2, ϫ 1 = ϫ 2, k 1 =k 2 with D 3 >>D 1 >>D 2 (6)

20. 20 Theorem III.4. The equilibrium line (0,0,ζ), where ζ is an arbitrary, is globally attracting. Theorem III.5. The necessary condition for existence of traveling wave solutions for (5) is

21. 21 Numerical Simulations for 1-dimensional space-Agar (Motile vs immotile)

22. M OTILE STRAIN AND IMMOTILE STRAIN TOTAL POPULATION OVER THE SPACE 22 о-motile ■-immotile

23. 23 Numerical Simulations for 1-dimensional space-Liquid (Motile vs immotile)

24. 24 day Motile strain and immotile strain total population over the space о-motile ■-immotile

25. A GAR CASE VS LIQUID CASE In agar, the density of the motile strain is high on the boundary of the petri dish while the density of the immotile strain is high in the middle of the petri dish. In liquid, bacterial motility is not that important because liquid nutrient moves almost infinitely fast compared to bacterial movement. 25

26. 26 Numerical simulations for 2-dimensional space-Agar

27. 27

28. C ONCLUSION Bacteria always go extinct due to lack of nutrient after a long time while some nutrient will always be remaining. From existence of traveling wave solutions: As the motility of motile bacteria increases, traveling waves propagate faster, thus it takes less time for motile bacteria to occupy the non-center region of petri dish. In agar media the motile strain is dominant in total density, while in liquid media bacterial motility is not that important. 28

29. In 2-D agar case motile strain is dominant in total density. Simulation results are consistent with Bruce Levin’s group experimental results. Simulation and experimental results illustrate the advantage of undirected motility in agar media and in absence of chemotaxis. Undirected motility gives bacteria with a selective advantage at which they compete in nutrient-limited enivironment. 29

30. 30 Competition of fast and slow movers for renewable and diffusive resource

31. I NTRODUCTION We extend the model to incorporate renewal diffusive resource to discuss the competition of slow and fast movers. The environment is assumed to be continuous but not homogeneous. species are genetically identical except their moving speeds. The resource uptake functions are assumed to be linear or nonlinear. 31

32. T HE M ODEL 32

33. O THER RD M ODELS 33 Gray-Scott Model Tsoularis Model Inkyung Inn Model J Dockery et. Al model

34. 34 First we consider J.Dockery et.al model:

35. 35 Conclusion: Biologically, evolution always favors the slowest diffuser.

36. C ONVERT OUR MODEL TO THEIR MODEL FORMAT 36 Same as Dockery model

37. S IMULATION RESULTS : LINEAR, NONLINEAR 37 Red-fast Black-slow h(N)=αN h(N)=αN/k+N

38. O BSERVATIONS Linear case: The fast mover goes extinct but the slow mover survives at a positive constant level, or both species go extinct. Non-linear case: (New outcomes) The fast mover goes extinct but the slow mover survives at oscillations, or both species survive at oscillations. 38

39. 39 B IFURCATION D IAGRAM Nutrient uptake rate of fast mover (α 1 ) α2α2

40. A CKNOWLEDGEMENTS Dr.Hao Wang, Department of Mathemetics & Statistical Sciences, University of Alberta Dr.Bruce Levin, Emory University 40

41. T HANK YOU ! 41