Download Presentation1

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,
Phase diagram for Bacillus
Herbert Levine, UCSD,
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,
The role of motility and nutrients in a bacterial
colony formation and competition
Dr.Hao Wang & Silogini Thanarajah
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.
• 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.
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 4th 2010).
www.nature.com,scienceblogs.com,pasteur.fr
• 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)
Ratio of the 2
strains
immotile/motile
Ratio of the 2
strains
immotile/motile
T0
0.9103
T0
0.9057
T24
0.1714
T24
3.3218
Observation from experiments results:
• For agar case, motile strain dominates the community.
• For liquid case, immotile strain dominates the community.
Bacterial competition in a petri dish model
B1-motile strain
B2-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?