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8/3/2019 Biofilm Dynamics
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The Dynamics of Growing Biofilm
J. P. Keener (Utah),
with lots of help from
Nick Cogan (Tulane), Jack Dockery (Montana SU)
and the NSF
U
U
The Power ofMathematics
The Excitement ofBiology
8/3/2019 Biofilm Dynamics
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Biofilms
Biofilm fouling of fiber filter
Placque on teeth
U
U
The Power ofMathematics
The Excitement ofBiology
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Some of the Interesting and Important Questions
P. Aeruginosa (on catheters, IV tubes, etc.)
How do gels grow?
Mucus secretion (bronchial tubes, stomach lining)
Colloidal suspensions of cells (cancer cell growth)
Gel Morphology (The cellular shape of sponges)
Why are gels important?
Protective capability
High viscosity (low washout rate) -for drugs, acid protection
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Biofilm formation inPseudomonas Aeruginosa
Wild type Biofilm mutant Mutant with autoinducer
Heterogeneous structures
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The Dynamics of Growing Biofilm
Quorum Sensing:
What is it?
How does it work?
Heterogeneous structures:
How do these cells use polymer gel (EPS) for locomotion?
What are the mechanisms of pattern (structure) formation?
Why is polymer gel so effective as a protective environment?
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Quorum Sensing inP. Aeruginosa
P. Aeruginosa
Major cause of hospital infection in the US.
Major cause of deaths in intubated CF patients, and IV fed patients.
Quorum Sensing: The ability of a bacterial colony to sense itssize and regulate its activity in response.
Examples: Vibrio fisheri, myxococcus, P. Aeruginasa
P. Aeruginosa in planktonic (non-colonized) form are non-toxic, but
as a biofilm, they are highly toxic and well protected by the polymer
gel in which they reside. However, they do not become toxic or beginto form polymer gel until the colony is of sufficient size to overwhelm
the immune system. Before this, they cannot be detected by
the immune system.
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Quorum sensing inP. Aeruginosa
Planktonic
Loosely Bound EPS secreting
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Wall Sensing inP. Aeruginosa
Wall Sensing: The ability of bacteria to differentiate in response toContact with a wall (the substratum).
Planktonic Loosely Bound EPS secreting
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The Dynamics of Growing Biofilm
Quorum Sensing:
What is it?
How does it work?
Heterogeneous structures:
How do these cells use polymer gel for locomotion?
What are the mechanisms of pattern (structure) formation?
Why is polymer gel so effective as a protective environment?
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The Biochemistry of Quorum Sensing
ALasR
lasR
lasI
LasR LasIA
3-oxo-C12-HSL
rsaL LasR
B
C4-HSL
RhrR B
RhlI
rhlI
GacA
Vfr
rhlR
RhlR
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An ODE Model
Variable Concentration
R LasR
A 3-oxo-C12-HSL
P Dimer complex
L LasI
S RsaL
r lasR mRNA
l lasI mRNA
s rsaL mRNA
lasI
LasI
ALasR
lasRLasR A
3-oxo-C12-HSL
rsaL LasR
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Modeling Biochemical Reactions with ODEs
dP
dt k
AR
Rate of production
kP
Rate of degradation
dL
dt kll
Rate of production
KLL
Rate of degradation
dl
dt
VmaxP
Kl Pk
ll
l LProduction of enzyme from mRNA
lasI
LasI
A R PBimolecularreaction
ALasR
LasR A3-oxo-C12-HSL
P lProduction of mRNA
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dP
dt kRARA kPP,
dR
dt kRARA kPP kRR k1r,
dA
dt kRARA kPP k2L kAA,
dL
dt k3l klL,
dS
dt k4s kSS,
ds
dt
VsP
KsP
kss,
dr
dt Vr
P
Kr P krr r0,
dl
dtVl
P
Kl P
1
KS S kll l0.
The full system of differential equations for the biochemistry:
lasI
LasI
ALasR
lasRLasR A
3-oxo-C12-HSL
rsaL LasR
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Modeling diffusion across cell membrane
Does this model exhibit quorum sensing?
Two ways to proceed:
1) Numerical simulation (but few of the 22 parameters are known)
2) Qualitative analysis
dA
dt kRARA kPP k2L kAA
dE
dt kEE
A
E
where is the cell volume fraction.
( )
(1)( )
1 (E A)
(A E)
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Quasi-steady state analysis: Assume that all fast variables are in quasi-steady state
dR
dtVR
P
KR P kRR R0,
dA
dtVA
P
KA P A0 d()A,
PkRARA
kP, d() kA
(
kE(1)
kE(1)).
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An even simpler model
Set R=A (which is not correct), to find
dA
dtVA
A2
KA A2 A0 d()A, d() kA
(
kE(1)
kE(1)).
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A Simple Model for Wall Sensing
V
t G(V,A), V
7
A
t F(A,V) d(E(x L) A),
E
t DE
2E
x2 kEE d(x L)(A E)( 0)
Cells are immobile, so internal variables A and V do not diffuse,
but extracellular autoinducer E diffuses (rapidly) so
L
E
x
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A Simple Model for Wall Sensing - contd
V
t G(V,A), V
7
A
tF(A,V)
d
p
1 pA,
pD
d(1 tanh(
DL))
We can solve for E(x=L) assuming a quasi-equilibrium, to find
Since p decreases as L 0, this has a chance of up-regulating for small L.
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A Proper (PDE) Model
V
t G(V,A), V
7
A
tF(A,V)
(EA),
E
t
2E kEE
1(A E),
Cells are immobile, so internal variables A and V do not diffuse,
but extracellular autoinducer E diffuses so
on the domain , subject to the (Robin) boundary condition
nDEE0, on
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Autoinducer with fixed colony size, variable density
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Autoinducer concentration; fixed density, variable
size colony
QuickTime and aAnimation decompressor
are needed to see this picture.
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Heterogeneous Structures
How do these cells use polymer gel for locomotion?
What are the mechanisms of pattern (structure) formation?
Why is polymer gel so effective as a protective environment?
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A Hydrogel Primer
What is a hydrogel?A tangled polymer network in solvent
Examples of biological hydrogels:
Micellar gels
Jello (a collagen gel ~ 97% water)
Extracellular matrixBlood clot
Mucin - lining the stomach, bronchial tubes, intestines
Gycocalyx - lining epithelial cells of blood vessels
Sinus secretions
Other examples of gel-like structures
Cell colonies - colloidal suspensions
Gel morphology - sponges, jellyfish
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8/3/2019 Biofilm Dynamics
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A Hydrogel Primer - II
Function of a biological hydrogelDecreased permeability to large molecules
Structural strength (for epithelial cell walls)
Capture and clearance of foreign substances
Decreased resistance to sliding/gliding
High internal viscosity (low washout)
Important features of gels
Usually comprised of highly polyionic polymers
Often exhibit large volumetric changes
eg. Highly compressed in secretory vessicle and expand rapidly
and dramatically on release
Can undergo volumetric phase transitions in response to ionicconcentrations (Ca++, H+), temperature, ..
Volume is determined by combination of attractive and
repulsive forces:
-- repulsive electrostatic, hydrophobic
-- attractive, hydrogen binding, cross-linking
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How gels grow
Polymerization/deposition
(blood clots)
Secretion
High Ca++
Ca++ Na+
Highly condensedDecondensed (swelled)
Interesting facts:
- Diffusion coefficient can vary substantially as a function of Ca++
- Expansion does not occur in distilled water.
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Modeling Biofilm Growth
A two phase material with polymer volume fraction
t (Vn) gn Network Phase
(EPS)
(1)
t (Vs(1)) 0 Solute Phase
b
t (Vnb) gb Bacterial Concentration
(1)u
t ((1)(VsuDu) gu Resource
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Force Balance
Solute phase hf(1)(Vn Vs)(solute-network friction)
(
n(V
n V
n
T))
(viscosity)
(pressure)
(1)p 0
Network phase
(pressure)
p0
(solute-network friction)
hf
(1)(Vn
Vs
)
(osmosis)
()
Incompressibility
(VnVs(1)) gnt (Vn) gn (1
)
t (Vs(1)) 0
Remark: Models of this form for the network phase have been used by lots of people
(Dembo and He, Lubkin and Jackson, Wolgemuth, Oster, Mogilner, )
Remark 2: A better model might keep track of Ca++ and Na+ concentrations as well
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Osmotic Pressure
What is the meaning of the term
In some formulations
(osmosis)
() ?
()F'() where F()
F() ()
is the Free Energy
'()> 0 gives expansion (swelling)
< 0 gives contraction (deswelling){() kT(k1(1)ln(1)(1)) From Flory-Huggins theory
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Phase Separation
Double-welled potentials give phase transitions and phase separation, similar to Cahn-Hilliard
()
(x)
x
To maintain an edge, () must be of the form ()2G()
()
0
0
0
(x)
x
(This is a theorem)
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Movement by Swelling
QuickTime and a
Animation d ecompressorare needed to see this picture.
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Propulsion by Swelling
QuickTime and a
Animation decompressorare needed to see this picture.
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No gel; no swell
QuickTime and a
Animation decompressorare needed to see this picture.
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Heterogeneous Structures
How do these cells use polymer gel for locomotion?
What are the mechanisms of pattern (structure) formation?
Why is polymer gel so effective as a protective environment?
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Limited Resource Instability
Remark: This instability does not occur in a resource rich environment.
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Channeling
A Pouiseille flow
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Channeling
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Channeling
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Conclusions
Quorum Sensing:
What is it? A biochemical switch
How does it work? Extracellular autoinducer fails to diffuse away.
Heterogeneous structures:
How do these cells use polymer gel for locomotion? Expansion byswelling
What are the mechanisms of pattern (structure) formation? Limited resource fingering Friction driven channeling
Others?(Is sloughing related to a Kelvin-Helmholz instability?)
Why is EPS so effective as a protective environment? Restricted pore size, chemical reactivity???
8/3/2019 Biofilm Dynamics
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More Questions (than answers)
Why does P.Aeruginosa attack only immune-compromised individuals?
(burn victims, post-surgery)
Why does P.Aeruginosa find a ready host in CF patients? (Decreased diffusion???)
What is it about the CFTR mutation in CF patients that makes their mucin so thick?
Is there cross talk between P. Aeruginaosa and other pathogens and if so, how?