Mathematical models for biofilms on the surface of monuments Magali - - PowerPoint PPT Presentation

Mathematical models for biofilms on the surface of monuments

Magali Ribot

Laboratoire J.-A. Dieudonn´ e Universit´ e de Nice, UMR 6621 CNRS in collaboration with

• F. Clarelli (Univ. dell’Aquila & I.A.C. – C.N.R. Roma),
• C. Di Russo (Univ. Roma 3 & I.A.C. – C.N.R. Roma),
• R. Natalini (I.A.C. – C.N.R. Roma)

Thursday, September 4 th, 2008

Outline

• 1. What is a biofilm ?
• 2. Previous mathematical models for biofilms
• 3. A new multidimensional model for biofilms
• 4. Numerical scheme
• 5. Numerical results
• 6. Perspectives

Definition of biofilm

◮ A biofilm is a complex mix of microorganisms (bacteria,

cyanobacteria, algae, protozoa and fungi) that are associated

• r embedded within a polymer matrix and which are

colonizing a certain surface.

◮ At one time, this term was confined to surfaces in constant

contact with water (solid/liquid interface), but now has been extended to any interface (air/solid, liquid/liquid or air/liquid), where growth of microorganisms occurs.

Staph Infection (Staphylococcus aureus biofilm) of the surface of a catheter.

Biofilm formation

• 1. Attachment : Bacteria approach the surface and get attached.
• 2. Colonization : Bacteria lose flagella and produce EPS.
• 3. Growth : Bacteria build the 3D biofilms and then specialize

(fixed cells, motile cells, monolayer cells).

Biofilm properties

◮ Combination of moisture, nutrients and surface. ◮ Surface attachment.

Biofilm properties

◮ Combination of moisture, nutrients and surface. ◮ Surface attachment.

⊲ Complex community interactions (cooperation between many species of bacteria) and mechanism. ⊲ Formation of an Extracellular matrix of Polymeric Substances (EPS).

Biofilm properties

◮ Combination of moisture, nutrients and surface. ◮ Surface attachment.

⊲ Complex community interactions (cooperation between many species of bacteria) and mechanism. ⊲ Formation of an Extracellular matrix of Polymeric Substances (EPS). ◮ EPS : Resistance to antibiotics and immune system, disinfectants and cleaning fluids. ◮ Possible bioremediation of waste sites, water-cleaning systems, formation of biobarriers to protect soil from contamination.

Multispecies biofilm growing on a biofilm carrier

◮ Finger type structures.

Stone degradation

Two types of degradation :

◮ Epilithic zones : De-cohesion and loss of substrate material

from the surface of the monuments (granular disaggregation, flaking and pitting).

◮ Endolithic zones : Degradation of internal structure

(penetration, geomechanical effects).

Cyanobacteria [Tiano]

◮ Develop on a water film on the stone’s surface. ◮ Die during cold and dry seasons & deposit of dead cells, which

lead to rapid new growth at warm time.

◮ Need warmth and light to develop. ◮ Nutrients : CO2, N2 and salt minerals traces. ◮ Anaesthetic coloured patinas.

Outline

• 1. What is a biofilm ?
• 2. Previous mathematical models for biofilms
• 3. A new multidimensional model for biofilms
• 4. Numerical scheme
• 5. Numerical results
• 6. Perspectives

Delft team’s models [Picioreanu, Noguera et al.]

Characteristics :

◮ Multi-dimensional, multispecies and multi-substrates ODEs

model.

◮ Discrete models : Individual-based approach (hard spheres) or

Cellular Automata.

◮ Simulation of detachment.

Problems :

• Large system of equations .
• Very sensitive to initial conditions.
• Difficult to take into account spatial behaviour of particles.

Delft team’s models(2)

A multiD & multispecies model [Alpkvist & Klapper, 2007]

◮ Extension of the one-dimensional model of Wanner & Gujer

[Wanner & Gujer, 1986].

◮ Multi-dimensional, multispecies and multi-substrates PDEs

model.

◮ Biofilms are divided into two phases : biomass and liquid. ◮ Equations :

• Elliptic equations for substrates and transport equations for

biomass.

• Transport of all biomass species with the same velocity u that

follows Darcy’s law.

◮ Numerical simulations : front tracking and level set

methods.

Alpkvist & Klapper’s model (2)

Problems :

◮ Difficult to obtain sharp interfaces and finger like structures.

⇒ Implementation of level set methods.

A 1D model with quorum sensing [Anguige, King & Ward, 2006]

◮ Medical setting (Pseidomonas aeruginosa biofilm). ◮ One-dimensional, multispecies PDEs model. ◮ 4 phases : live cells(C), dead cells(D) , EPS(E) and liquid(W). ◮ Influence of nutrients (oxygen) and of Quorum Sensing (QS). ◮ Tests of different treatments (antibiotic and antiQS drugs). ◮ Equations :

• Hyperbolic equations for B, D, E and W .
• 1 common velocity for B, D and E and 1 velocity for W .
• Advection-diffusion equations for nutrients, antibiotics,

antiQS...

• BUT closure of system by condition : W = W0 + αE.

◮ Travelling waves analysis.

A 2D fluid multicomponents model[Zhang, Cogan, Wang, 2008]

◮ 2 phases : polymer network and solvent(=substrate with

nutrients ?).

◮ Nutrients are coming from the substrate. ◮ Equations :

• Momentum equation for the average velocity v.
• Transport equation for nutrient concentration
• Transport equation for polymer networks with a modified

velocity (modified Cahn-Hilliard equation).

◮ Numerical simulations of pitching-off (or detachment) and

shedding.

Zhang, Cogan & Wang’s model (2)

Outline

• 1. What is a biofilm ?
• 2. Previous mathematical models for biofilms
• 3. A new multidimensional model for biofilms
• 4. Numerical scheme
• 5. Numerical results
• 6. Perspectives

Mass balance equations

Unknowns :

• B,D,E, L = volume ratios of cyanoBacteria, Dead

cyanobacteria, EPS and Liquid.

• vS=velocity of cells and EPS ; vL =velocity of liquid.
• ΓB, ΓD, ΓE, ΓL= mass exchange rates of B, D, E, L.

◮ Equations of mass balance are :

∂tY + ∇ · (Y vS) = ΓY , (Y = B, D, E), (1) ∂tL + ∇ · (LvL) = ΓL. (2)

◮ Volume constraint :

B + D + E + L = 1. (3)

Mass exchange rates

• I=lux intensity ; θ=temperature ; N=nutrients.

◮ Conservation of mass :

ΓB + ΓD + ΓE + ΓL = 0. (4)

◮ Mass exchange rate for B (cyanobacteria) :

ΓB = kB(I, θ, N) BL

• birth term for bacteria

− kD(I, θ, N) B

• death term for bacteria

.

◮ Mass exchange rate for D (dead bacteria) :

ΓD = αkD(I, θ, N) B

• from death of bacteria

− kN(θ) D

natural decay

.

◮ Mass exchange rate for E (EPS) :

ΓE = kE(θ) BL

• EPS production

− εE

• natural decay

.

Force balance conservation [Preziosi et al, ...]

• P=hydrostatic pressure.
• Σ=stress function (Σ = γ(1 − L)).
• Assumption : interaction forces for the liquid as Darcy’s

law : −M(vL − vS).

◮ Force balance for B, D and E :

∂t((1 − L)vS) + ∇ · ((1 − L)vS ⊗ vS) = −(1 − L)∇P + ∇Σ + M(vL − vS) − ΓLvL, (5)

◮ Force balance for L :

∂t(LvL) + ∇ · (LvL ⊗ vL) = −L∇P − M(vL − vS) + ΓLvL. (6)

Closure of the system & Boundary conditions

Closure of the system :

◮ Sum of the 4 mass balance equations (1), (2) & Volume

constraint (3) & Conservation of mass (4) give : ∇ · ((1 − L)vS + LvL) = 0. (7)

◮ Divergence of the sum of the 2 force balance equations (5),

(6) & Equation (7) give : −∆P = ∇ · (∇ · ((1 − L)vS ⊗ vS + LvL ⊗ vL)) − ∆Σ. Boundary conditions :

• No-flux BC for the velocities : vS · n|∂Ω = vL · n|∂Ω = 0.
• Neumann BC for the volume ratios :

∇B · n|∂Ω = ∇E · n|∂Ω = ∇D · n|∂Ω = 0.

Final system

◮ Equations for the volume ratios :

∂tB + ∇ · (BvS) = B (LkB(I, θ, N) − kD(I, θ, N)) , (S1) ∂tD + ∇ · (DvS) = αBkD(I, θ, N) − DkN(θ), (S2) ∂tE + ∇ · (EvS) = BLkE(θ) − εE, (S3) L = 1 − (B + D + E) (S4)

◮ Equations for the velocities :

∂t((1 − L)vS) + ∇ · ((1 − L)vS ⊗ vS) + (1 − L)∇P = ∇Σ + (M − ΓL)vL − MvS, (S5) ∂t(LvL) + ∇ · (LvL ⊗ vL) + L∇P = −(M − ΓL)vL + MvS, (S6)

◮ Equation for the pressure :

−∆P = ∇ · (∇ · ((1 − L)vS ⊗ vS + LvL ⊗ vL)) − ∆Σ. (S7)

Remarks about the model

◮ Simplifications in the 1D case :

• Expression of ∂xP on behalf of vS and vL :

−∂xP = ∂x((1 − L)vS

2 + LvL 2) − ∂xΣ.

• Relation between vL and vS :

(1 − L)vS + LvL = 0.

◮ {(B, D, E) ∈ [0, 1]3; 0 ≤ B + D + E ≤ 1} is an invariant

domain under the following conditions : kD, kE ≥ 0, [(1 − α)B + D]kD0 + εE + (vS − vL)∇L ≥ 0, if L = 0.

Extension of the model with the influence of light, temperature and nutrients

kB(I, θ, N) = kB0 ∗ p(I) ∗ N ∗ e− (θ−θc )2

d2

, kD(I, θ, N) = kD0

• 1 − 0.4 ∗ p(I) ∗ N ∗ e− (θ−θc )2

d2

• ,

kE(θ) = kE0 ∗ e− (θ−θc )2

d2

, kN(θ) = kN0 ∗ e− (θ−θc )2

d2

. where

◮ p(I) = (b + 2√ac) ×

I aI 2 + bI + c ,

◮ I(x, z, t) = I0(t) exp

• −

Lz

z

g(1 − L(x, s, t))ds

• ,

◮ N satisfies a reaction-diffusion equation.

Outline

• 1. What is a biofilm ?
• 2. Previous mathematical models for biofilms
• 3. A new multidimensional model for biofilms
• 4. Numerical scheme
• 5. Numerical results
• 6. Perspectives

Numerical scheme

Notations : δt= time step ; tn = nδt= iteration time ; Y n=approximation of function Y at time tn.

• 1. Resolution of equations (S1), (S2), (S3) as (Y = B, D, E) :

Y n+1 = Y n − δt (∇ · (Y vS))n + δt Γn

Y .

• 2. Computation of Ln+1 = 1 − (Bn+1 + Dn+1 + E n+1).
• 3. Resolution of equations (S5), (S6) (Φ = (M − ΓL)vL − MvS) :

(1 − Ln+1)vSn+1 = (1 − Ln)vSn − δt (∇ · ((1 − L)vS ⊗ vS))n − δt ((1 − L)∇P)n + δt (∇Σ)n+1 + δtΦn. Ln+1vLn+1 = LnvLn − δt (∇ · (LvL ⊗ vL) + L∇P)n − δtΦn.

• 4. Resolution of equation (S7) :

−∆Pn+1 = ∇·(∇ · ((1 − L)vS ⊗ vS + LvL ⊗ vL))n+1−∆Σn+1.

An implicit scheme

Problem :

• Computed : (1 − Ln+1)vSn+1 and Ln+1vLn+1.
• Needed : vSn+1 and vLn+1.

How we compute vSn+1 if Ln+1 = 1 (resp. vLn+1 if Ln+1 = 0) ? Trick : use an implicit scheme, i. e. :

• 1. System of equations for B,D and E :

Y n+1 = Y n − δt (∇ · (Y vS))n + δt Γn+1

Y

.

• 3. System of equations for vS and vL (Φ = (M − ΓL)vL − MvS) :

(1 − Ln+1)vSn+1 = (1 − Ln)vSn − δt (∇ · ((1 − L)vS ⊗ vS))n − δt ((1 − L)∇P)n + δt (∇Σ)n+1 + δtΦn+1. Ln+1vLn+1 = LnvLn − δt (∇ · (LvL ⊗ vL) + L∇P)n − δtΦn+1.

Outline

• 1. What is a biofilm ?
• 2. Previous mathematical models for biofilms
• 3. A new multidimensional model for biofilms
• 4. Numerical scheme
• 5. Numerical results
• 6. Perspectives

One-dimensional case

One-dimensional case (2)

Two-dimensional case

Long-time behaviour. Evolution of B, E, L.

Two-dimensional case (2)

Short-time behaviour. Evolution of B.

Two-dimensional case (3)

Short-time behaviour. Evolution of B.

Two-dimensional case (4)

Short-time behaviour. Evolution of E.

Two-dimensional case (5)

Short-time behaviour. Evolution of B.

Outline

• 1. What is a biofilm ?
• 2. Previous mathematical models for biofilms
• 3. A new multidimensional model for biofilms
• 4. Numerical scheme
• 5. Numerical results
• 6. Perspectives

Perspectives

◮ Mathematical analysis of the model and of the numerical

scheme.

◮ Different conditions of temperature, humidity, light, nutrients

(CO2 and O2) etc...

◮ Confrontation with experiments of ICVBC (Istituto per la

Conservazione e la Valorizzazione dei Beni Culturali) of Firenze.

◮ Study of the influence of signalization. ◮ Coupling with chemical deterioration of monuments.