Resumen de Modeling and simulation of bacterial biofilm growth
Sergei Iakunin
What are biofilms? ================== In now days almost everyone has heard about bacteria. Speaking about these microorganisms people usually consider them in relation to health both as dangerous pathogens spreading disease or as a help for digestion process. In any case bacteria in common opinion are supposed to be a secondary attribute of our living activity whereas in reality they play a fundamental role for all life on our planet. These microorganisms perform the nitrogen cycle necessary for all organisms so everywhere on Earth where we can find life, from volcanic geysers of Yellowstone National Park to subglaciar antarctic lakes, we also find bacteria. One can doubt that how small and fragile microorganisms can survive even in quite aggressive environment and besides create a basement for more complex life forms? The answer is that only less than 10% of bacteria live as isolated cells whereas the rest of them exist in communities known as biofilms, [2], where bacteria separate their function in order to better resist to hazardous environment. Biofilms usually emerge on interfaces between surfaces and fluids or between two fluids. The formation starts when a few planktonic bacteria adhere to the surfaces. After the attachment, their phenotype and their behavior change: bacteria start to produce extracellular matrix (ECM) -- a mixture of proteins, extracellular DNA, exopolysacharids, and water. The ECM is the most important component of a biofilm, which has many fascinating properties and makes bacterial colonies, in some sense, predecessors of multicellular organisms. Firstly, the ECM holds the cells close to each other, facilitating cell-to-cell communication and horizontal gen transfer that help the bacterial colony responding quickly to external irritants. Besides, the matrix keeps enzymes close to bacteria and plays a role of external digestive system, [9]. Secondly, the ECM prevents penetration of antibiotics inside the biofilm, which makes these structures a thousand times more resistent to different types of drugs than solitary bacteria, [2]. Finally, the matrix makes the bacterial colony adopt a complex 3D shape with wrinkles, channels, and pores. Theses peculiarities can be used by bacteria as a transport network for nutrients and also in order to increase the surface of the colony and, therefore, oxygen absorption. Thus, thanks to ECM, biofilms from just a set of cells turn into "cities of microbes" with a complex infrastructure where bacteria adapt to external environment and distribute functions in order to survive and spread. Biofilms have a huge impact in all human activity. On one hand, they may be quite dangerous. Bacterial colonies are responsible for 80% of chronic infections and can attach to heart valves and protheses causing hospital-acquired infections, [14]. Due to the high tolerance of biofilm to antibiotics, there is not yet an efficient therapy to cure these infections. On the other hand, bacteria can also be beneficial in many applications. Particularly biofilms can be used for plant protection, [11], drug and fuel production, [3], and for fermentation processes, [10].
In order to better understand formation and development of biofilms, get rid of them when they are harmful, and enhance their growth when they are beneficial, we have studied them using mathematical simulations and analysis. Since there are many mechanical, chemical, and biological processes occurring in bacterial colonies, there are many different models focusing on specific peculiarities of biofilm development. In this thesis, we study the formation of rugose patterns in Bacillus subtilis colonies growing on agar substratum. Parts of the presented results have been published in Ref. [13].
State of art ============ The first models of biofilm development were proposed in the 1980s. Wanner and Gujer in 1986 presented a simple 1D model for growth of bacterial biomass, [19], which nevertheless can be used to estimate efficiency of bioreactors, [16]. The model consists of 4 main components: (i) bacterial growth due to utilization of nutrients; (ii) expansion of biomass; (iii) diffusion of nutrients; (iv) attachment and detachment of cells.
Erbel et al. [7] generalized this model to 2D by adding a description of the external flow by means of the Navier-Stokes equations. Another generalization was proposed by Dockery and Klapper [6], who replaced Darcy's law instead of the diffusion equation previously used to describe biomass spreading. The main challenge in application of these models consists in that the boundary between biofilm and surrounding fluid is not known and has to be determined as part of the solution. One possible way to get rid of this trouble is to use a discrete approach instead of the apparatus of differential equations. It was done by Piciorenau et al. who proposed a cellular automaton model describing growth of biomass in bacterial biofilm, [17]. Later, Lapsidou and Rittman generalized this model by adding more components that catch the heterogeneity of biofilms, [15].
The previous models describe very well biomass production and spreading, but the mechanical behavior of the biofilm is usually neglected or simplified. However, experiments show that mechanical effects have a great influence to morphogenesis of bacterial colonies, their infrastructure, ability to resist external forces and attach to the surface. Simulation of these mechanical effects is quite challenging, which is why in the present thesis we focus on wrinkling patterns that emerge in species of bacteria growing on agar substrata (e.g. Bacillus subtilis). In these biofilms, complex networks of wrinkles emerge as a result of mechanical instability. Firstly, the framework of solid mechanics has been applied to simulation of biofilms by Trejo et al. They consider a bacterial colony as a thin plate bonded to liquid substratum and deformed under effect of compressing force, [18]. Even though Trejo et al. consider only the 1D case, their approach can be generalized to the geometrically nonlinear deformation of thin plates described by Foppl-von Karman equations (FvKEs). These equations have rugose patterns and wrinkles as solutions, which are of interest in applications such as crumpling of paper, [1], morphogenesis of plant leafs, [5], or deformation of bilayer structures in microelectronic devices, [12]. A growing biofilm can be modeled by FvKEs that include growth in the strain tensor [5,8,20].
Proposed model and results ========================== We consider a bacterial biofilm growing on an agar substratum as a thin elastic heterogeneous film bonded to a viscoelastic layer, but we do not use cellular automata to model bacterial growth as in Ref. [8]. The source of deformation is the internal growth defined as an arbitrary tensor that is independent of the vertical coordinate but it is heterogeneous along the central surface of the biofilm. This tensor may turn the initial domain occupied by the biofilm into a domain containing gaps or overlaps that do not correspond to any displacement field. In this case, the growth is called incompatible, and elastic strain closes the gaps and discontinuities provoking wrinkling with respect to the flat state. Otherwise, the growth is compatible and no wrinkles are observed. We follow Ref. [5] to define geometrically nonlinear stress and strain of the biofilm and apply the principle of virtual work to obtain modified FvKEs. Next we combine these equations with the ones for a viscoelastic substratum using ideas of Ref. [12], but we propose a simpler derivation based on a proper scaling. The final system contains three time-dependent PDEs -- one for each component of displacement. Since the film is very thin, the geometrical nonlinearity influences only the vertical deflection. Then only one of equations is nonlinear whereas the two remaining ones are linear. Analysis of the resulting system demonstrates that mechanical relaxation in the in-plane direction happens much faster than in the vertical one, [13]. This allows us to simplify the equations of motion removing time derivatives of in-plane displacements. The equations can be simplified even more if the substratum is soft or purely viscous. In this case we can use the Airy potential to solve the two linear equations for the in-plane displacements and reduce number of unknowns to two (vertical deflection and Airy potential) instead of the three displacements, [13].
It is not possible to solve the modified FvKEs analytically. However, in a simple case of homogeneous radial or azimuthal growth of a round biofilm, we can perform an asymptotic analysis. In this case, if thickness of the biofilm tends to zero the system becomes the Monge-Ampere equation (MAE), which describes the pure geometrical transformation of an infinitely thin surface. For homogeneous radial growth, the solution is a cone whereas for azimuthal growth we have a series of solutions, the first one being a saddle. These shapes provides a good approximation of FvKEs solution. However, there are two problems: these shapes are not smooth at the origin, and they do not satisfy the boundary conditions. In order to resolve these troubles, we have inserted corner and boundary layers, [13]. For the conical solution, that is for the case of radial symmetry, it is possible to simplify the FvKEs near the boundary and obtain an approximate solution there that matches the outer solution. Near the origin, all terms in the FvKEs are of the same order, hence we have approximated the solution in the corner layer by patching. We also perform a numerical simulation in the case of radial symmetry comparing the obtained approximations with the numerical solution. It is important to note that the thickness of boundary and corner layer depends on the thickness of the biofilm: the thinner the film is, the thinner these layers are, [13]. For the general case we reformulate the FvKEs in weak form. To this end, we use that the modified FvKEs are nonlinear only respect to the vertical deflection. This allows us to express in-plane displacements (or Airy potential in the case of pure viscous substratum) in terms of the vertical deflection, thereby leaving only one unknown. The final simplified equations can be obtain from a variational problem of minimization of a target functional with respect to a vertical deflection that belongs to a Sobolev space. We can distinguish three parts in this functional: the pure geometrical deformation of an infinitely thin surface; a smoother of vertical displacement represented by the bending energy; and the influence of the substratum that tries to cancel large displacements. The problem of minimization of only the first term is ill-posed and has to be regularized by adding constrains or regularization. Luckily, we have the bending energy, which does not allow sharp bends and hence regularizes the ill-posed geometrical problem. Thus we have a set of well defined, separated solutions from which we should pick one according to the stiffness of the substratum. For pure viscous agar, the best solution is that with the smallest number of wrinkles, whereas for stiffer agar wrinkles emerge because the deflection amplitude should decrease and the frequency increase. Apart from this analysis, we can prove existence of solution of modified FvKEs in a case of small growth following Ref. [4]. The proof is based on reformulation of the system in operator form and application of fixed point theorem.
We also use the weak formulation to develop a numerical method based on finite elements. We use Zienkiewicz triangular elements, [21], because they are widely used for simulation of bending of thin plates and can discretize domains of any shape. The developed solver yields results which coincide with theoretical prediction for the simple cases of homogeneous radial or azimuthal growth of circular biofilm. However, numerical simulation allows us to resolve cases of complex heterogeneous growth. In particular, we find that the combination of radial and azimuthal growth yields a combination of conical and corona shapes as a solution. Furthermore, we perform a numerical study of two interesting bifurcation cases in this thesis. The first one corresponds to heterogeneous growth of biofilm on a pure viscous substratum. If we start to increase the growth factor, the film buckles into a simple shape with radial symmetry after a certain critical value. Additional increment leads to more complex asymmetric shapes. In the second bifurcation case, we study the influence of the stiffness of the agar layer on the emergence of rugose patterns. If the substratum is soft, homogeneous growth is compatible and does not provoke any wrinkles. However, if we increase the stiffness of the agar to a certain critical value, then a complex network of wrinkles emerges, which is similar to one that is seen in experiments, [11].
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