The Dynamics of Growing Biolm

University of UtahMathematical Biology

theImagine Possibilities

The Dynamics of Growing BiofilmJ. P. Keener

Department of MathematicsUniversity of Utah

The Dynamics of Growing Biofilm – p.1/30



Biofilms

biofilm fouling of filter fibers

Placque on teethThe Dynamics of Growing Biofilm – p.2/30



Some Interesting Questions

How do gels grow?• P. aeruginisa (on catheters, IV tubes, etc.)• Mucus secretion (bronchial tubes, stomach lining)• Colloidal suspensions, cancer cells• Gel morphology (the shape of sponges)

Why are gels important?• Protective capability• Friction reduction• High viscosity (low washout rate) for drugs• Acid protection




Biofilm Formation in P. Aeruginosa

Wild Type Biofilm Mutant Mutant with autoinducer




Dynamics of Growing Biogels

I: Quorum sensing:• What is it?• How does it work?

II: Heterogeneous structures• How do cells use polymer gel for locomotion?• What are the mechanisms of pattern formation?




I: Quorum Sensing in P. aeruginosa

Quorum sensing: The ability of a bacterial colony to sense itssize and regulate its activity in response.Examples: Vibrio fisheri, P. aeruginosaP. Aeruginosa:• Major cause of hospital infection in the US.• Major cause of death in intubated Cystic Fibrosis patients• In planktonic form, they are non-toxic, but in biofilm they are

highly toxic and well-protected by the polymer gel in whichthey reside. However, they do not become toxic until thecolony is of sufficient size, i.e., quorum sensing.




Stages of Growth

Planktonic




Stages of Growth

Small Dense Colony




Stages of Growth

Biofilm Colony




Biochemistry of Quorum Sensing

lasI

lasR





LasI3−oxo−C12−HSL

lasR

lasI

ALasR





LasR

LasILasR

3−oxo−C12−HSL

lasI

lasR

A

A





LasR

rsaL

LasILasR

3−oxo−C12−HSL

lasI

lasR

A

A

RsaL





LasR

rsaL

LasI

RsaL

LasR

C4−HSL

rhlR

RhlR

RhlR

RhlI

rhlI

3−oxo−C12−HSL

lasI

lasR





LasR

rsaL

LasI

RsaL

LasR

C4−HSL

GacA

Vfr

rhlR

RhlR

RhlR

RhlI

rhlI

3−oxo−C12−HSL

lasI

lasR




Modeling Biochemical Reactions

Bimolecular reaction A+R←→ P

LasR

3−oxo−C12−HSL

A

A

LasR

dP

dt= k+AR− k−P

Production of mRNA P−→ l

LasR A lasI

dl

dt=VmaxP

Kl + P− k−ll

Enzyme production l → LLasIlasI

dL

dt= kll −KLL






LasR

3−oxo−C12−HSL

A

A

LasR

dP

dt= k+AR− k−P


LasR A lasI

dl

dt=VmaxP

Kl + P− k−ll


dL

dt= kll −KLL






LasR

3−oxo−C12−HSL

A

A

LasR

dP

dt= k+AR− k−P


LasR A lasI

dl

dt=VmaxP

Kl + P− k−ll


dL

dt= kll −KLL




Full system of ODE’s

dPdt

= kRARA− kPP

dRdt

= −kRARA+ kPP − kRR+ k1r,

dAdt

= −kRARA+ kPP + k2L− kAA,

dLdt

= k3l − klL,

dSdt

= k4s− kSS,

dsdt

= VsP

KS+P− kss,

drdt

= VrP

Kr+P− krr + r0,

dldt

= VlP

Kl+P1

KS+S− kll + l0

rsaL

LasI

RsaL

A

A

3−oxo−C12−HSL

lasI

lasR

LasR

LasR




Diffusion

E

A

dA

dt= F (A,R, P ) + δ(E −A)

dE

dt= − kEE + δ(A− E)




Diffusion

E

A

dA

dt= F (A,R, P ) + δ(E −A)

dE


rate of change,




Diffusion

E

A

dA

dt= F (A,R, P ) + δ(E −A)

dE


rate of change, production or degradation rate,




Diffusion

E

A

dA

dt= F (A,R, P ) + δ(E −A)

dE


rate of change, production or degradation rate, diffusiveexchange,




Diffusion

E

A

dA

dt= F (A,R, P ) + δ(E −A)

(1− ρ) (dE

dt+KEE) = ρ δ(A− E)

rate of change, production or degradation rate, diffusiveexchange, density dependence.




Model Reduction

Two (possible) ways to proceed:• Numerical simulation (but few of the 22 kinetic parameters

are known),• Qualitative analysis (QSS reduction)

dA

dt= F (A) + δ(E −A), (1− ρ)(

dE

dt+ kEE) = ρδ(A−E)

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

F(A

)




Model Reduction

Two (possible) ways to proceed:• Numerical simulation (but few of the 22 kinetic parameters

are known),• Qualitative analysis (QSS reduction)

dA

dt= F (A) + δ(E −A), (1− ρ)(

dE

dt+ kEE) = ρδ(A−E)

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

A

F(A

)




Two Variable Phase Portrait

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2

3

4

5

Internal Autoinducer

Ext

erna

l aut

oind

ucer

A nullclineE nullcline




A PDE Model

Suppose cells are immobile, so internal variables do not diffuse,but extracellular autoinducer E diffuses

∂A

∂t= F (A,U) + δ(E −A),

∂U

∂t= G(A,U), U ∈ R7,

∂E

∂t= ∇ · (DE∇E)− kEE +

ρ

1− ρδ(A− E)

in Ω with Robin boundary conditions

n ·De∇E + αE = 0

on ∂Ω.



theImagine Possibilities Autoinducer as function of cell

density




A Hydrogel Primer

• What is a hyrogel?A tangled polymer network in solvent.

• Examples of biological hydrogelsMicellar gelsJello (a collagen gel ≈ 97% water)Extracellular matrixBlood clotsMucin - lining the stomach, bronchial tubes, intestinesGlycocalyxSinus secretions




A Hydrogel Primer - II

Functions of a biological hydrogel• Decreased permeability to large molecules• Structural strength (for 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• Can undergo volumetric phase transitions in response to

ionic concentrations, temperature, etc.• Volume is determined by combination of forces (entropic,

electrostatic, hydrophobic, cross-linking, etc).The Dynamics of Growing Biofilm – p.17/30



How gels grow

• Polymerization/deposition

networkmonomers polymers

• Secretion

swollencondensed secretion




Modelling Biofilm Growth

A two phase material with polymer network volume fraction θ∂θ∂t

+∇ · (Vnθ) = gn Network Phase (EPS)

∂θs

∂t+∇ · (Vsθs) = 0 Solute Phase

∂b∂t

+∇ · (Vnb) = gb Bacterial concentration

∂θsu∂t

+∇ · (θs(Vsu−Du∇u) = gu Resource Concentration

where θ + θs = 1,

solute volume fraction, network velocity,

solute velocity, artificial network diffusion.







∂θs


∂b∂t


∂θsu∂t


where θ+θs = 1, solute volume fraction,

network velocity, solute

velocity, artificial network diffusion.







∂θs


∂b∂t


∂θsu∂t


where θ+θs = 1, solute volume fraction, network velocity,

solute

velocity, artificial network diffusion.







∂θs


∂b∂t


∂θsu∂t


where θ+θs = 1, solute volume fraction, network velocity, solute

velocity,

artificial network diffusion.






+∇ · (Vnθ) = gn + ε∇2θ Network Phase (EPS)

∂θs


∂b∂t


∂θsu∂t

+∇ · (θs(Vsu−Du∇u) = gu Resource Concentrationwhere θ + θs = 1, solute volume fraction, network velocity,

solute velocity, artificial network diffusion.




Force Balance

Solute Phase (an inviscid fluid)

hfθθs(Vn − Vs) − θs∇p = 0,

solute-network friction

Network Phase (a viscoelastic material)

Imcompressibility

∇ · (θVn + θsVs) = gn




Force Balance



solute-network friction pressure


Imcompressibility





Force Balance





η∇(θ(∇Vn +∇V Tn )) − hfθθs(Vn − Vs) − ∇ψ(θ) − θ∇p = 0

network viscosity

Imcompressibility





Force Balance






network viscosity solute-network friction

Imcompressibility





Force Balance






network viscosity solute-network viscosity osmosis

Imcompressibility





Force Balance






network viscosity solute-network viscosity osmosis pressure

Imcompressibility





Force Balance






network viscosity solute-network viscosity osmosis pressure

Imcompressibility





Osmotic Pressure

What is the meaning of the term −∇ψ(θ)?ψ

θ

θ

x

ψ′(θ) > 0 gives expansion (swelling)ψ′(θ) < 0 gives contraction (deswelling)

To maintain an edge, ψ(θ) must be of the form ψ(θ) = θ2F (θ)




Osmotic Pressure

What is the meaning of the term −∇ψ(θ)?ψ

θ

θ

x

ψ′(θ) > 0 gives expansion (swelling)ψ′(θ) < 0 gives contraction (deswelling)

To maintain an edge, ψ(θ) must be of the form ψ(θ) = θ2F (θ)




Movement by Swelling

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

xN

etw

ork

Vol

ume

Frac

tion

Movement by SwellingWT Biofilm GrowthMutant Cell Growth




Fingering Instability

"Nutrient Poor" Fingering InstabilityThe Dynamics of Growing Biofilm – p.23/30



Channeling

Modified Network Model: Include elastic strains, σn = γε

η∇(θ(∇Vn+∇V Tn ))+ ∇ · (γθε) −hfθθs(Vn−Vs)−∇ψ(θ)−θ∇p = 0

and displacements D

∂D

∂t+∇ · (VnD) = Vn

where ε is the Cauchy-Green strain tensor.




Channel Formation

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5Below bifurcation point

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5Above bifurcation point

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

20

40

60

80

100

120

140

160

180

200

G

Flux

Flux vs. Pressure Gradient

The "Moses Bifurcation"

Remark: The existence of this channeling "Moses Bifurcation"can be established using singular perturbation arguments.




Summary

• Quorum sensing is via a hysteretic switch involving diffusibleautoinducer

• Fingering and mushrooming may be driven by a substratedeficiency-fingering instability.

• Channeling may be driven by a gel-osmosis "MosesBifurcation".




Acknowledgments

Collaborators• Jack Dockery, Montana State University• Nick Cogan, Tulane University

Notes• Funding provided by a grant from the NSF.• This talk can be viewed at

http://www.math.utah.edu/keener/lectures/biofilmdynamics• No Microsoft products were used or harmed during the

production of this talk.

The End




Structure of the "Moses Bifurcation"

The steady state equation is

εθd

dy

(

dθdy

θ

)

+1

θH(θ, y) = k

subject to∫

1

0

θdy = θ

where

H(y, θ) = G2(y −1

2)2 − θΨ(θ) + θ2

− θ2,Ψ(θ) = κθ2(θ − θref )




Singular Perturbation Analysis

This is a singular perturbation problem.

For ε = 0 (the "outer solution"), we must solve an algebraicequation for θ as a function of y. However, the equationH(θ, y) = kθ has (possibly) multiple solutions.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

θ

h(θ)




Boundary Layer Analysis

The governing equation is a "bistable equation", so transitionlayers can be inserted at certain locations.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

y

θ n

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

y

θ n

It is possible that boundary layer solutions coexist withnon-boundary solutions, as is seen in the bifurcation diagram.

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The Dynamics of Growing Biolm