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This is a review of the article Role of the Biofilm Matrix in Structural Development

Executive Summary

This article creates a mathematical model of how the EPS matrix within the biofilm redistributes itself due to changes in osmotic pressure as well as the standard growth and diffusion. The biofilm is modeled as the EPS network and the solvent. The governing equations of the biofilm are derived from fundamental principles and include the momentum balances on the network and the solvent, and mass balances on the bacteria concentration, polymer network, and substrate. The equations are solved numerically after a number of simplifications, and are also analyzed linearly.

The model ends up demonstrating numerically that the swelling mechanism of the biofilm sometimes induces a mushrooming effect. The effect is reduced when there is a high availability of substrate. The results agree with experimental data performed by other groups.

Abstract

Although the initiation, development and control of biofilms has been an area of experimental investigation for more than three decades, the role of extra-cellular polymeric substance (EPS) has not been well studied. We present a mathematical description of the EPS matrix to study the development of heterogeneous biofilm morphology. In developing the model, we assume that the biofilm is a biological gel composed of EPS and water. The bacteria are enmeshed in the network and are the producers of the polymer. In response to external conditions, gels absorb or expel solvent causing swelling or contraction due to osmotic pressure gradients. The physical morphology of the biofilm depends on the temperature, solvent composition, pH and ionic concentrations through osmotic pressure. This gives a physically based mechanism for the redistribution of biomass within the biofilm. Analysis of a reduced model indicates that biomass redistribution, through the mechanism of swelling, may induce the formation of isolated towers or mushroom clusters by spatial variation in EPS production which leads to gradients in osmotic pressure.

Keywords:biofilm; EPS; gel; model; viscoelasticity; osmotic pressure.

Background Information

Biofilms are communities of microorganisms which form on living and non-living substances. The microorganisms attach to the surface and to each other and produce a matrix of extracellular polymeric substance (EPS). Biofilms are formed when free-floating organisms attach to a surface. Next, the cells form a matrix to hold the biofilm together and to recruit more organisms. The steps in the picture below are as follows:

  1. Initial attachment
  2. Irreversible attachment
  3. Maturation 1
  4. Maturation 2
  5. Dispersal

Biofilm.jpg

Because of their unique physical properties, biofilms are resistant to anti-microbial agents. The formation of biofilms can lead to contamination or corrosion of equipment. The situations in which biofilms can form are diverse - from contamination of catheters to adhesion to rocks in streams to corrosion of pipes in the oil industry.

Mathematical Model

Initial Assumptions

The model is set up to include a polymer network region (EPS) and a fluid solvent region. The bacteria and substrate are also involved, but it is assumed that the relative volume of bacteria and of substrate is negligible, so the volume fraction of the polymer network, \theta _{{n}} and the volume fraction of the solvent, \theta _{{s}}, sum to one. The polymer network is assumed to be a constant density, viscoelastic material. The solvent is assumed to be a Newtonian fluid with a significantly lower viscosity than the EPS network.


There are four forces that act on the network:

  1. Surface forces
  2. Frictional drag caused by the interactions between the network and the fluid
  3. Osmotic pressure
  4. Hydrostatic pressure

Surface Forces

The surface forces are divided into viscous stress, \sigma _{{v}}, and elastic stress, \sigma _{{e}}. Those sum to the network stress tensor, \sigma _{{n}}. The surface forces can be described mathematically by \nabla \bullet (\theta _{{n}}\sigma _{{n}})

Friction

Frictional drag is modeled by h_{{f}}\theta _{{n}}\theta _{{s}}(U_{{n}}-U_{{s}}).

U_{{n}} is the network velocity. U_{{s}} is the solvent velocity. h_{{f}} is the coefficient of friction.

Osmotic Pressure

Osmotic pressure gradients cause a force on the polymer. According to Flory-Huggins theory, it can be described using the following formula:

\psi =-{\frac  {k_{{B}}T}{v_{{1}}}}(ln(1-\theta _{{n}})+(1-1/m)\theta _{{n}}+\chi _{{1}}\theta _{{n}}^{{2}}).

In the formula, m is the ratio of solvent volume to polymer volume, \chi _{{1}} is the Flory interaction parameter which indicates the strength of attraction between the chains within the polymer, k_{{B}} is Boltzmann's constant, v_{{1}} is the volume occupied by one monomer of the EPS, and T is the temperature.

The ratio between solvent volume and polymer volume is large, so the 1/m term can be taken out. Next, the logarithmic term can be expanded using the Taylor series about \theta _{{n}}=0. The equation simplifies to:

\psi =-{\frac  {k_{{B}}T}{3v_{{1}}}}\theta _{{n}}^{{2}}(\theta _{{n}}-3(\chi _{{1}}-1/2)).

Since parameters \chi _{{1}} and v_{{1}} are not easily determined through experiment, the model has to be simplified further. \theta _{{ref}} is defined as the reference volume fraction, so that the swelling pressure is 0 when \theta _{{n}}=0 or \theta _{{n}}=\theta _{{ref}}. This method ignores the water contained in bacteria. The final equation comes out to be:

\psi =\xi _{{OS}}\theta _{{n}}^{{2}}(\theta _{{n}}-\theta _{{ref}})

Where \xi _{{OS}} is a parameter that combines the effects of ionic environment, polymer structure, and solvent concentration.

Hydrostatic Pressure

P is the total amount of hydrostatic pressure acting on the volume. The amount of force on the network due to the hydrostatic pressure is \theta _{{n}}\nabla P.

Governing Equations for a Growing Biogel

The governing equations are created from momentum balances on the polymer network and the solvent, and mass balances on the polymer network, bacteria concentration, and the amount of substrate.

Momentum Balances

The polymer network momentum balance combines the four forces on the network discussed above. It ignores inertial effects.

\eta _{{n}}\nabla \bullet (\theta _{{n}}(\sigma _{{v}}+\sigma _{{e}}))-h_{{f}}\theta _{{n}}\theta _{{s}}(U_{{n}}-U_{{s}})-\nabla \psi -\theta _{{n}}\nabla P=0

The solvent momentum balanced is derived similarly.

\eta _{{s}}\nabla \bullet ({\frac  {\theta _{{s}}}{2}}(\nabla U_{{s}}+\nabla U_{{s}}^{{T}})+h_{{f}}\theta _{{n}}\theta _{{s}}(U_{{n}}-U_{{s}})-\theta _{{s}}\nabla P=0

Notice that the solvent balance leaves out the elastic stress term because the solvent is a Newtonian fluid.

Mass Balances

The mass balance on the network includes the movement of the network, by including the velocity term U_{{n}}. The term that accounts for creation of the network by bacteria is g_{{n}}.

{\frac  {\partial \theta _{{n}}}{\partial t}}+\nabla \bullet (\theta _{{n}}U_{{n}})=g_{{n}}

The growth rate of the network can be modeled by Monod kinetics, after taking into account that c, concentration of substrate, is very small. g_{{n}}={\frac  {\epsilon \mu \theta _{{n}}c}{K_{{c}}}}. In the equation, K_{{c}} is the half-saturation constant, \mu is the maximum production rate, and \epsilon is the scalar that accounts for the difference in time scale between network production and network motion.


The mass balance on bacterial concentration (B) is also necessary. The bacteria are entangled within the network so the movement of the network has to be accounted for. They also reproduce with a rate g_{{b}}.

{\frac  {\partial B}{\partial t}}+\nabla \bullet (BU_{{n}})=g_{{b}}

It is assumed that network density is proportional to bacterial concentration, which allows the bacterial concentration equation to be ignored.

The concentration of substrate, c, is changed because it is consumed by the bacteria. The substrate is dissolved in the solvent, so the flux is due to solvent motion as well as diffusion.

{\frac  {\partial (\theta _{{s}}c)}{\partial t}}+\nabla \bullet (cU_{{s}}\theta _{{s}}-D\theta _{{s}}\nabla c)=-g_{{c}}

Diffusion of substrate occurs much faster than the network movement, so the substrate can be assumed to be in a quasi-steady state. The consumption of substrate is related to network growth. Therefore, the substrate equation becomes:

D\nabla \bullet ((1-\theta _{{n}})\nabla c)=A\theta _{{n}}c


Analysis and Results

The interface between the EPS region and the fluid region is analyzed. If the interface is flat, the bacterial concentration depends only on the depth and the problem is one-dimensional. However, the interface is not flat and uniform, so the growth is not uniform. This is a result of substrate diffusion into the peaks happening faster than in the middle - the bacteria in the peaks have easier access to substrate. This causes higher growth rates in the peaks, increasing osmotic pressure and increasing the velocity of the network at the peaks.

The analysis of the network can be done using two separate methods - linear analysis of the interface as well as solving the equations to determine non-linear behavior.

Simplifying Assumptions

Due to low solvent flow, it can be assumed that the friction coefficient, h_{{f}} is negligible. The fluid velocity can be assumed to be zero. Because the network takes a much longer time to grow than to relax, the stress is dominated by the viscous component and the elastic component can be ignored. Those factors can be used to simplify the momentum balance on the network.

\eta _{{n}}\nabla \bullet (\theta _{{n}}\sigma _{{v}})-\nabla \psi =0

On the interface between the network and the solvent, the normal component of interface velocity must match the normal component of network velocity.

{\frac  {\partial \Gamma }{\partial t}}\bullet n=U\bullet n

(\eta _{{n}}{\frac  {\theta _{{n}}}{2}}(\nabla U_{{n}}+\nabla U_{{n}}^{{T}})-\psi I+\kappa \nabla \bullet nI)\bullet n=0

In the equation, the term \nabla \bullet nI corresponds to the stress due to surface tension while \kappa is the surface tension constant.

Results

The linear analysis relates the growth rate \lambda to perturbation frequency \alpha for different values of proportionality constants \kappa . The examples of the results are shown below. The surface tension acts as a stabilizer. When it is equal to zero, all of the growth rates are above zero, which means that the system is unstable. As \kappa increases, the area where \lambda is above zero gets smaller for different sizes of perturbations.

Linear.jpg

The nonlinear analysis produces a plot of \theta _{{n}} profile and contours of growth rate. When the growth rate is high, the volume fraction of network is increased, which increases osmotic pressure locally.

Stuff.jpg

After a period of 3-6 days, the perturbation increases in size even more.

Perturbation.jpg

When the initial interface has a higher peak, there is a mushrooming effect that is visible.

Mushrooming.jpg

Mushrooming effects also occur when the perturbation has a higher initial frequency.

Mushrooms.jpg

One of the theories that was tested using the model that was created is that a rougher biofilm is created when there is less substrate due to resource competition. The theory was tested by simulating a linearly growing biofilm cluster using first order kinetics. Then, when the kinetics are switched to saturated growth, zero-order kinetics, the biofilm is smoothed out.

Comparison.jpg

Discussion of Recent Paper

Isaac Klapper and Jack Dockery give an overview of microbial communities in their paper, Mathematical Description of Microbial Biofilms located in the SIAM REVIEW Vol. 52, No. 2, pp. 221–265. The paper can be found here. Klapper and Dockery begin with an introduction noting the magnitude, diversity, and the prevalence of microbial life on Earth. The article gives a broad overview of the stages of a biofilm and begins to discuss different mathematical models for biofilm development, biofilm growth, quorum sensing, biomechanics and antimicrobial tolerance. Quorum sensing , cell to cell singling in a biofilm, allows microbes to express certain genes only after the biofilm community reaches a certain concentration. The quorum sensing is modeled by assuming the there is sufficient substrate to produce signaling molecules and that these molecules are produced following Michaelis–Menten kinetics and travel by diffusion. Looking at the steady states of promoting and retarding signals a bifurcation takes place once the biofilm population reaches a critical value. There is a small unstable region between the two steady states.



References

N.G. Cogan and J. P. Keener. "The role of the biofilm matrix in structural development." Mathematical Medicine and Biology 21(2):147-166, 2004.

Osada, Y., Kajiwara, K. Gels Handbook Vol. 1, Chapters 1-4. New York: Academic.

www.wikipedia.org, Biofilm