Classification of PDEs First order vs. Second order (depending on the highest order derivative) Linear vs. Nonlinear vs Quasilinear (depending on the dependent variable ‘u’) Second order: Parabolic (heat equation or reaction-diffusion equation), Hyperbolic (wave equation), Elliptic (Laplace equation or steady-state equation of a hyperbolic or parabolic equation)

Analytical methods Key methods (idea: PDE  ODE): Method of characteristics (usually for first-order PDE) Separation of variables (for finite domain) Fourier transform (for infinite domain) Additional methods: Change of variables (to obtain a simpler PDE) Fundamental solution (taking the convolution with the B.C.’s to obtain the solution Superposition principle (also used in key methods) Methods for nonlinear PDEs

Numerical methods Finite element methods Finite volume methods Finite difference methods

Case study: nutrient-based bacterial competition and colony formation Herbert Levine, UCSD, presentation

Phase diagram for Bacillus Herbert Levine, UCSD, presentation

Why do theory? Identification of pattern formation –The first step in analyzing a biological pattern is to place it within a specific schema –Formation of bacterial colonies is within the diffusive nutrient-limited growth class Detailed understanding of the generic behavior of this type of process –Predictions which can be verified independent of knowing the precise underlying mechanism –Identifies the roles of key features in the process of colony formation and bacterial competition Herbert Levine, UCSD, presentation

Dr.Hao Wang & Silogini Thanarajah The role of motility and nutrients in a bacterial colony formation and competition

Outlines Definitions Introduction Model of bacterial competition in a petri dish Mathematical Analysis Theorems Two bacterial competition in a petri dish model. Simulation in 1-D, 2-D space Conclusion

Definitions Motile: Moving or having power to move spontaneously. Immotile: Not moving or lacking the ability to move. Biofilm: A very thin layer of microscopic organisms that covers the surface of an object. Over 90% of all bacteria live in biofilms. Agar: a dried hydrophilic, colloidal substance extracted from various species of red algae; used in solid culture media for bacteria and other microorganisms.

Shapes of Bacteria

Introduction In most natural environments, bacteria fight with neighbors for space and nutrients. Most are harmless, some are beneficial and a few become a threat to our health when they grow and reproduce. Many but not all bacteria exhibit motility, i.e. self-propelled motion, under appropriate circumstances. Motility is an important part in the colonization of plant roots by bacteria. Also, colony formation could help clarify factors influencing biofilm formation and illuminate how groups control the fitness of bacteria. Naturereviewsamicrobiology

Examples of Biofilms Biofilms are also present on the teeth of most animals as dental plaque, where they may become responsible for tooth decay and gum disease.teeth dental plaque tooth decaygum disease Biofilms are found on the surface of and inside plants. Biofilms can grow in showers very easily since they provide a moist and warm environment for the biofilm to thrive. Biofilms have been found in the body such as urinary tract infections, middle-ear infections, formation of dental plaque, coating contact lenses. urinary tract infectionsmiddle-ear infectionsdental plaquecontact lenses En.wikipedia.org/wiki/biofilm

Uses of Biofilms Often used to purify water in water treatment plants. Used to break down toxic chemicals. Bacterial biofilms impair cutaneous wound healing and reduce topical antibacterial efficiency in healing or treating infected skin wounds (Journal of Applied Microbiology, Jan 4 th 2010).

Bacteria display kinds of colony patterns according to the substrate softness and nutrients concentration. Previous studies showed four different colony shapes and recognized a morphological diagram by dividing into four regions like diffusion-limited aggregation-like, eden-like, concentric-ring and fluid spreading. Pnas.org

Purpose of this paper is to use bacteria as model organism to study competition and determine which strain will “win” in competition with other strain when the two are mixed in a petri dish. We plug these biological characteristics into simulation programs and observe the outcomes.

Agar method vs Liquid method (Bruce Levin’s group experiment) Observation from experiments results: For agar case, motile strain dominates the community. For liquid case, immotile strain dominates the community. Ratio of the 2 strains immotile/motile T T Ratio of the 2 strains immotile/motile T T

Bacterial competition in a petri dish model B 1 -motile strain B 2 -immotile strain

Bacteria-substrate model without nutrient diffusion

Theorems

Competition of two bacterial strains in a petri dish model

Motile strain

Immotile strain

Resource

Motile strain and immotile strain total population over the space

Simulations for 2-D space We placed motile strain in the middle and the Immotile strain little far from the middle of the agar plate and observed the pattern formation after 1hr, 5hrs, 8hrs and 15hrs.

Observarion at t=1: Motile and immotile strains are start to grow on the same position, we placed. Some of the nutrients consume by bacterial strains on the same position.

Observation at t=5: Motile strain move and grows around the middle of the petri dish and immotile strain grows on the same position, like narrow. Nutrients consume around the middle of the petri dish.

Observation at t=8: Motile strain move fast and grows to over lab immotile strain and immotile strain face for the competition with motile strain for nutrients. More and more nutrients used by bacterial strains surrounding the middle of the petri dish.

Observation at t=15: Motile strain grows everywhere even over immotile strain and immotile strain don’t have enough nutrients to eat and survive. Almost all nutrients are used but some are still there.

Total populations of both bacterial strains Total population of both strains up to 5 hours look same but after that motile strain dominate.

Depends on the initial conditions we will get different pattern formation.

Conclusion Bacteria always go extinct due to lack of nutrient after a long time while some nutrient will always be remaining. If we incorporate a nutrient input as chemostat-type models, then the bacterial community can be sustained (“closed”->”open”). From computer stimulations (1-D case): If we put motile and immotile bacterial strains on the middle of the petri dish: initially motile strain move fast and grow everywhere but the immotile strain grow fast on the middle and finally both will die out. In this case motile strain is dominate. It is consistent to Bruce Levin’s group agar case. For liquid case we have to choose different nutrients equation (Liquid is move everywhere).

From 2-D case: If we put motile strain on the middle and the immotile strain little far from the middle of the petri dish: initially both strains grow on the same position as we placed; later, they overlap in some place, then they compete for nutrients such that a some strange patterns occur; after a long time, motile strain passes over immotile one and thus moves fast and grows everywhere and dominate the bacterial community; Finally (not shown in 2-D simulations), all bacteria go extinct due to “closed” system (no nutrient input).

Question 1.How do bacteria move in the absence of flagella propellers? 2.what are 3 factors that limit bacteria growth? 3.Why are people so worried about bacteria? 4.How does bacteria cause food poisoning? 5.Does bacteria grow on jam? 6.How does bacteria help?