Multiphase flow models of biogels from crawling cells to bacterial biofilms

A mathematical model of biofilm growth and spread within plant xylem: Case study of Xylella fastidiosa in olive trees

Xylem-limited bacterial pathogens cause some of the most destructive plant diseases. Though imposed measures to control these pathogens are generally ineffective, even among susceptible taxa, some hosts can limit bacterial loads and symptom expression. Mechanisms by which this resistance is achieved are poorly understood. In particular, it is still unknown how differences in vascular structure may influence biofilm growth and spread within a host. To address this, we developed a novel theoretical framework to describe biofilm behaviour within xylem vessels, adopting a polymer-based modelling approach. We then parameterised the model to investigate the relevance of xylem vessel diameters on Xylella fastidiosa resistance among olive cultivars. The functionality of all vessels was severely reduced under infection, with hydraulic flow reductions of 2-3 orders of magnitude. However, results suggest wider vessels act as biofilm incubators; allowing biofilms to develop over a long time while still transporting them through the vasculature. By contrast, thinner vessels become blocked much earlier, limiting biofilm spread. Using experimental data on vessel diameter distributions, we were able to determine that a mechanism of resistance in the olive cultivar Leccino is a relatively low abundance of the widest vessels, limiting X. fastidiosa spread.

Phase Separation Driven by Active Flows

We extend the continuum theories of active nematohydrodynamics to model a two-fluid mixture with separate velocity fields for each fluid component, coupled through a viscous drag. The model is used to study an active nematic fluid mixed with an isotropic fluid. We find microphase separation, and argue that this results from an interplay between active anchoring and active flows driven by concentration gradients. The results may be relevant to cell sorting and the formation of lipid rafts in cell membranes.

Thin-film lubrication model for biofilm expansion under strong adhesion

Understanding microbial biofilm growth is important to public health because biofilms are a leading cause of persistent clinical infections. In this paper, we develop a thin-film model for microbial biofilm growth on a solid substratum to which it adheres strongly. We model biofilms as two-phase viscous fluid mixtures of living cells and extracellular fluid. The model explicitly tracks the movement, depletion, and uptake of nutrients and incorporates cell proliferation via a nutrient-dependent source term. Notably, our thin-film reduction is two dimensional and includes the vertical dependence of cell volume fraction. Numerical solutions show that this vertical dependence is weak for biologically feasible parameters, reinforcing results from previous models in which this dependence was neglected. We exploit this weak dependence by writing and solving a simplified one-dimensional model that is computationally more efficient than the full model. We use both the one- and two-dimensional models to predict how model parameters affect expansion speed and biofilm thickness. This analysis reveals that expansion speed depends on cell proliferation, nutrient availability, cell-cell adhesion on the upper surface, and slip on the biofilm-substratum interface. Our numerical solutions provide a means to qualitatively distinguish between the extensional flow and lubrication regimes, and quantitative predictions that can be tested in future experiments.

Active random forces can drive differential cellular positioning and enhance motor-driven transport

Cells are remarkable machines capable of performing an exquisite range of functions, many of which depend crucially on the activity of molecular motors that generate forces. Recent experiments have shown that intracellular random movements are not solely thermal in nature but also arise from stochasticity in the forces from these molecular motors. Here we consider the effects of these nonthermal random forces. We show that stochastic motor force not only enhances diffusion but also leads to size-dependent transport of objects that depends on the local density of the cytoskeletal filaments on which motors operate. As a consequence, we find that objects that are larger than the mesh size of the cytoskeleton should be attracted to regions of high cytoskeletal density, while objects that are smaller than the mesh size will preferentially avoid these regions. These results suggest a mechanism for size-based organelle positioning and also suggest that motor-driven random forces can additionally enhance motor-driven transport.

Transition from Actin-Driven to Water-Driven Cell Migration Depends on External Hydraulic Resistance

Cells in vivo can reside in diverse physical and biochemical environments. For example, epithelial cells typically live in a two-dimensional (2D) environment, whereas metastatic cancer cells can move through dense three-dimensional matrices. These distinct environments impose different kinds of mechanical forces on cells and thus potentially can influence the mechanism of cell migration. For example, cell movement on 2D flat surfaces is mostly driven by forces from focal adhesion and actin polymerization, whereas in confined geometries, it can be driven by water permeation. In this work, we utilize a two-phase model of the cellular cytoplasm in which the mechanics of the cytosol and the F-actin network are treated on an equal footing. Using conservation laws and simple force balance considerations, we are able to describe the contributions of water flux, actin polymerization and flow, and focal adhesions to cell migration both on 2D surfaces and in confined spaces. The theory shows how cell migration can seamlessly transition from a focal adhesion- and actin-based mechanism on 2D surfaces to a water-based mechanism in confined geometries.

A thin-film extensional flow model for biofilm expansion by sliding motility

In the presence of glycoproteins, bacterial and yeast biofilms are hypothesized to expand by sliding motility. This involves a sheet of cells spreading as a unit, facilitated by cell proliferation and weak adhesion to the substratum. In this paper, we derive an extensional flow model for biofilm expansion by sliding motility to test this hypothesis. We model the biofilm as a two-phase (living cells and an extracellular matrix) viscous fluid mixture, and model nutrient depletion and uptake from the substratum. Applying the thin-film approximation simplifies the model, and reduces it to one-dimensional axisymmetric form. Comparison with Saccharomyces cerevisiae mat formation experiments reveals good agreement between experimental expansion speed and numerical solutions to the model withO ( 1 ) parameters estimated from experiments. This confirms that sliding motility is a possible mechanism for yeast biofilm expansion. Having established the biological relevance of the model, we then demonstrate how the model parameters affect expansion speed, enabling us to predict biofilm expansion for different experimental conditions. Finally, we show that our model can explain the ridge formation observed in some biofilms. This is especially true if surface tension is low, as hypothesized for sliding motility.

A multiphase theory for spreading microbial swarms and films

Bacterial swarming and biofilm formation are collective multicellular phenomena through which diverse microbial species colonize and spread over water-permeable tissue. During both modes of surface translocation, fluid uptake and transport play a key role in shaping the overall morphology and spreading dynamics. Here we develop a generalized two-phase thin-film model that couples bacterial growth, extracellular matrix swelling, fluid flow, and nutrient transport to describe the expansion of both highly motile bacterial swarms, and sessile bacterial biofilms. We show that swarm expansion corresponds to steady-state solutions in a nutrient-rich, capillarity dominated regime. In contrast, biofilm colony growth is described by transient solutions associated with a nutrient-limited, extracellular polymer stress driven limit. We apply our unified framework to explain a range of recent experimental observations of steady and unsteady expansion of microbial swarms and biofilms. Our results demonstrate how the physics of flow and transport in slender geometries serve to constrain biological organization in microbial communities.

Bacteria can grow and thrive in many different environments. Although we usually think of bacteria as single-celled organisms, they are not always solitary; they can also form groups containing large numbers of individuals. These aggregates work together as one super-colony, allowing the bacteria to feed and protect themselves more efficiently than they could as isolated cells.

These colonies move and grow in characteristic patterns as they respond to their environment. They can form swarms, like insects, or biofilms, which are thin, flat structures containing both cells and a film-like substance that the cells secrete. Availability of food and water influences the way colonies spread; however, since movement and growth are accompanied by mechanical forces, physical constraints are also important. These include the ability of the bacteria to change the water balance and their local mechanical environment, and the forces they create as they grow and move.

Previous research has used a variety of experimental and theoretical approaches to explain the dynamics of bacterial swarms and biofilms as separate phenomena. However, while they do differ biologically, they also share many physical characteristics.

Srinivasan et al. wanted to exploit these similarities, and use them to predict the growth and shape of biofilms and bacterial swarms under different conditions. To do this, a unified mathematical model for the growth of both swarms and biofilms was created. The model accounted for various factors, such as the transport of nutrients into the colony, the movement of water between the colony and the surface on which it grew, and mechanical changes in the environment (e.g. swelling/softening). The theoretical results were then compared with results from experimental measurements of different bacterial aggregates grown on a soft, hydrated gel. For both swarms and biofilms, the model correctly predicted how fast the colony expanded overall, as well as the shape and location of actively growing regions.

Biofilms and other bacterial aggregates can cause diseases and increase inflammation in tissues, and also hinder industrial processes by damage to submerged surfaces, such as ships and waterpipes. The results described here may open up new approaches to restrict the spreading of bacterial aggregates by focusing on their physical constraints.

Swimming bacteria power microspin cycles

Dense suspensions of swimming bacteria are living fluids, an archetype of active matter. For example, Bacillus subtilis confined within a disc-shaped region forms a persistent stable vortex that counterrotates at the periphery. Here, we examined Escherichia coli under similar confinement and found that these bacteria, instead, form microspin cycles: a single vortex that periodically reverses direction on time scales of seconds. Using experimental perturbations of the confinement geometry, medium viscosity, bacterial length, density, and chemotaxis pathway, we show that morphological alterations of the bacteria transition a stable vortex into a periodically reversing one. We develop a mathematical model based on single-cell biophysics that quantitatively recreates the dynamics of these vortices and predicts that density gradients power the reversals. Our results define how microbial physics drives the active behavior of dense bacterial suspensions and may allow one to engineer novel micromixers for biomedical and other microfluidic applications.

Computational Investigation of Ripple Dynamics in Biofilms in Flowing Systems

Biofilms are collections of microorganisms that aggregate using a self-produced matrix of extracellular polymeric substance. It has been broadly demonstrated that many microbial infections in the body, including dental plaque, involve biofilms. While studying experimental models of biofilms relevant to mechanical removal of oral biofilms, distinct ripple patterns have been observed. In this work, we describe a multiphase model used to approximate the dynamics of the biofilm removal process. We show that the fully nonlinear model provides a better representation of the experimental data than the linear stability analysis. In particular, we show that the full model more accurately reflects the relationship between the apparent wavelength and the external forcing velocities, especially at mid-to-low velocities at which the linear theory neglects important interactions. Finally, the model provides a framework by which the removal process (presumably governed by highly nonlinear behavior) can be studied.

Multicomponent model of deformation and detachment of a biofilm under fluid flow

A novel biofilm model is described which systemically couples bacteria, extracellular polymeric substances (EPS) and solvent phases in biofilm. This enables the study of contributions of rheology of individual phases to deformation of biofilm in response to fluid flow as well as interactions between different phases. The model, which is based on first and second laws of thermodynamics, is derived using an energetic variational approach and phase-field method. Phase-field coupling is used to model structural changes of a biofilm. A newly developed unconditionally energy-stable numerical splitting scheme is implemented for computing the numerical solution of the model efficiently. Model simulations predict biofilm cohesive failure for the flow velocity between [Formula: see text] and [Formula: see text] m s(-1) which is consistent with experiments. Simulations predict biofilm deformation resulting in the formation of streamers for EPS exhibiting a viscous-dominated mechanical response and the viscosity of EPS being less than [Formula: see text]. Higher EPS viscosity provides biofilm with greater resistance to deformation and to removal by the flow. Moreover, simulations show that higher EPS elasticity yields the formation of streamers with complex geometries that are more prone to detachment. These model predictions are shown to be in qualitative agreement with experimental observations.

An immersed boundary method for two-phase fluids and gels and the swimming of Caenorhabditis elegans through viscoelastic fluids

The swimming of microorganisms typically involves the undulation or rotation of thin, filamentary objects in a fluid or other medium. Swimming in Newtonian fluids has been examined extensively, and only recently have investigations into microorganism swimming through non-Newtonian fluids and gels been explored. The equations that govern these more complex media are often nonlinear and require computational algorithms to study moderate to large amplitude motions of the swimmer. Here, we develop an immersed boundary method for handling fluid-structure interactions in a general two-phase medium, where one phase is a Newtonian fluid and the other phase is viscoelastic (e.g., a polymer melt or network). We use this algorithm to investigate the swimming of an undulating, filamentary swimmer in 2D (i.e., a sheet). A novel aspect of our method is that it allows one to specify how forces produced by the swimmer are distributed between the two phases of the fluid. The algorithm is validated by comparing theoretical predictions for small amplitude swimming in gels and viscoelastic fluids. We show how the swimming velocity depends on material parameters of the fluid and the interaction between the fluid and swimmer. In addition, we simulate the swimming of Caenorhabditis elegans in viscoelastic fluids and find good agreement between the swimming speeds and fluid flows in our simulations and previous experimental measurements. These results suggest that our methodology provides an accurate means for exploring the physics of swimming through non-Newtonian fluids and gels.

A Two-Dimensional Multiphase Model of Biofilm Formation in Microfluidic Chambers

The bacterial pathogen Xylella fastidiosa is the causal agent of many pathological conditions of economically important agricultural crops. There is no known cure for X. fastidiosa diseases, and management of the problem is based solely in controlling the population of insect vectors, which is somewhat effective. The bacterium causes disease by forming biofilms inside the vascular system of the plant, a process that is poorly understood. In microfluidic chambers, used as artificial xylem vessels, this bacterium has been observed to reproducibly cluster into a distinct, regular pattern of aggregates, spatially separated by channels of non-biofilm components. We develop a multiphase model in two dimensions, which recapitulates this spatial patterning, suggesting that bacterial growth and attachment/detachment processes are strongly influential modulators of these patterns. This indicates plausible strategies, such as the addition of metals and chelators, for mitigating the severity of diseases induced by this bacterial pathogen.

Two-Phase Acto-Cytosolic Fluid Flow in a Moving Keratocyte: A 2D Continuum Model

The F-actin network and cytosol in the lamellipodia of crawling cells flow in a centripetal pattern and spout-like form, respectively. We have numerically studied this two-phase flow in the realistic geometry of a moving keratocyte. Cytosol has been treated as a low viscosity Newtonian fluid flowing through the high viscosity porous medium of F-actin network. Other involved phenomena including myosin activity, adhesion friction, and interphase interaction are also discussed to provide an overall view of this problem. Adopting a two-phase coupled model by myosin concentration, we have found new accurate perspectives of acto-cytosolic flow and pressure fields, myosin distribution, as well as the distribution of effective forces across the lamellipodia of a keratocyte with stationary shape. The order of magnitude method is also used to determine the contribution of forces in the internal dynamics of lamellipodia.

Differential growth of wrinkled biofilms

Biofilms are antibiotic-resistant bacterial aggregates that grow on moist surfaces and can trigger hospital-acquired infections. They provide a classical example in biology where the dynamics of cellular communities may be observed and studied. Gene expression regulates cell division and differentiation, which affect the biofilm architecture. Mechanical and chemical processes shape the resulting structure. We gain insight into the interplay between cellular and mechanical processes during biofilm development on air-agar interfaces by means of a hybrid model. Cellular behavior is governed by stochastic rules informed by a cascade of concentration fields for nutrients, waste, and autoinducers. Cellular differentiation and death alter the structure and the mechanical properties of the biofilm, which is deformed according to Föppl-Von Kármán equations informed by cellular processes and the interaction with the substratum. Stiffness gradients due to growth and swelling produce wrinkle branching. We are able to reproduce wrinkled structures often formed by biofilms on air-agar interfaces, as well as spatial distributions of differentiated cells commonly observed with B. subtilis.

An active poroelastic model for mechanochemical patterns in protoplasmic droplets of Physarum polycephalum

Motivated by recent experimental studies, we derive and analyze a two-dimensional model for the contraction patterns observed in protoplasmic droplets of Physarum polycephalum. The model couples a description of an active poroelastic two-phase medium with equations describing the spatiotemporal dynamics of the intracellular free calcium concentration. The poroelastic medium is assumed to consist of an active viscoelastic solid representing the cytoskeleton and a viscous fluid describing the cytosol. The equations for the poroelastic medium are obtained from continuum force balance and include the relevant mechanical fields and an incompressibility condition for the two-phase medium. The reaction-diffusion equations for the calcium dynamics in the protoplasm of Physarum are extended by advective transport due to the flow of the cytosol generated by mechanical stress. Moreover, we assume that the active tension in the solid cytoskeleton is regulated by the calcium concentration in the fluid phase at the same location, which introduces a mechanochemical coupling. A linear stability analysis of the homogeneous state without deformation and cytosolic flows exhibits an oscillatory Turing instability for a large enough mechanochemical coupling strength. Numerical simulations of the model equations reproduce a large variety of wave patterns, including traveling and standing waves, turbulent patterns, rotating spirals and antiphase oscillations in line with experimental observations of contraction patterns in the protoplasmic droplets.

Pattern formation exhibited by biofilm formation within microfluidic chambers

This article investigates the dynamics of an important bacterial pathogen, Xylella fastidiosa, within artificial plant xylem. The bacterium is the causative agent of a variety of diseases that strike fruit-bearing plants including Pierce's disease of grapevine. Biofilm colonization within microfluidic chambers was visualized in a laboratory setting, showing robust, regular spatial patterning. We also develop a mathematical model, based on a multiphase approach that is able to capture the spacing of the pattern and points to the role of the exopolymeric substance as the main source of control of the pattern dynamics. We concentrate on estimating the attachment/detachment processes within the chamber because these are two mechanisms that have the potential to be engineered by applying various chemicals to prevent or treat the disease.

Intracellular mechanochemical waves in an active poroelastic model

Many processes in living cells are controlled by biochemical substances regulating active stresses. The cytoplasm is an active material with both viscoelastic and liquid properties. We incorporate the active stress into a two-phase model of the cytoplasm which accounts for the spatiotemporal dynamics of the cytoskeleton and the cytosol. The cytoskeleton is described as a solid matrix that together with the cytosol as an interstitial fluid constitutes a poroelastic material. We find different forms of mechanochemical waves including traveling, standing, and rotating waves by employing linear stability analysis and numerical simulations in one and two spatial dimensions.

A comparison of computational models for eukaryotic cell shape and motility

Eukaryotic cell motility involves complex interactions of signalling molecules, cytoskeleton, cell membrane, and mechanics interacting in space and time. Collectively, these components are used by the cell to interpret and respond to external stimuli, leading to polarization, protrusion, adhesion formation, and myosin-facilitated retraction. When these processes are choreographed correctly, shape change and motility results. A wealth of experimental data have identified numerous molecular constituents involved in these processes, but the complexity of their interactions and spatial organization make this a challenging problem to understand. This has motivated theoretical and computational approaches with simplified caricatures of cell structure and behaviour, each aiming to gain better understanding of certain kinds of cells and/or repertoire of behaviour. Reaction–diffusion (RD) equations as well as equations of viscoelastic flows have been used to describe the motility machinery. In this review, we describe some of the recent computational models for cell motility, concentrating on simulations of cell shape changes (mainly in two but also three dimensions). The problem is challenging not only due to the difficulty of abstracting and simplifying biological complexity but also because computing RD or fluid flow equations in deforming regions, known as a “free-boundary” problem, is an extremely challenging problem in applied mathematics. Here we describe the distinct approaches, comparing their strengths and weaknesses, and the kinds of biological questions that they have been able to address.

Biofilm growth on rugose surfaces

A stochastic model is used to assess the effect of external parameters on the development of submerged biofilms on smooth and rough surfaces. The model includes basic cellular mechanisms, such as division and spreading, together with an elementary description of the interaction with the surrounding flow and probabilistic rules for extracellular polymeric substance matrix generation, cell decay, and adhesion. Insight into the interplay of competing mechanisms such as the flow or the nutrient concentration change is gained. Erosion and growth processes combined produce biofilm structures moving downstream. A rich variety of patterns are generated: shrinking biofilms, patches, ripplelike structures traveling downstream, fingers, mounds, streamerlike patterns, flat layers, and porous and dendritic structures. The observed regimes depend on the carbon source and the type of bacteria.

Marginal stability and traveling fronts in two-phase mixtures

Mixtures of materials that move relative to each other arise in a variety of applications, especially in biophysical problems where the mixture consists of materials with different material properties. The variety of applications leads to a bewildering array of multiphase models, each with slightly different behaviors and interpretations, depending on the application. Some of the behaviors include phase separation, traveling waves, and linear instabilities. Because of the variability of the predicted behaviors, there has been considerable attention paid to minimal models to determine the fundamental solutions, bifurcations, and instabilities. In this paper we describe a new solution for the simplest two-phase system where both phases are dominated by viscous forces, one-phase response to osmotic forces, and the phases interact through a drag term. The system develops a traveling front separating an unstable, uniform solution from a patterned, phase separated solution. We seek the velocity of the traveling front and show that, for large diffusion, marginal stability gives a simple and accurate prediction for the velocity. For smaller diffusion constants, the front is "pushed," and the linear prediction fails.

Low-Reynolds-number swimming in viscous two-phase fluids

The fluid media surrounding many microorganisms are often mixtures of multiple materials with very different physical properties. The composition and rheology of the mixture may strongly affect the related locomotive behaviors. We study the classical Taylor's swimming sheet problem within a two-fluid model, which consists of two intermixed viscous fluids with different viscosities, with both numerical experiments and analysis. Our results indicate that both the swimming speed and efficiency may be decreased substantially relative to those for a single-phase fluid.

Flow-induced channel formation in the cytoplasm of motile cells

A model is presented to explain the development of flow channels within the cytoplasm of the plasmodium of the giant amoeba Physarum polycephalum. The formation of channels is related to the development of a self-organizing tubular network in large cells. Experiments indicate that the flow of cytoplasm is involved in the development and organization of these networks, and the mathematical model proposed here is motivated by recent experiments involving the observation of development of flow channel in small cells. A model of pressure-driven flow through a polymer network is presented in which the rate of flow increases the rate of depolymerization. Numerical solutions and asymptotic analysis of the model in one spatial dimension show that under very general assumptions this model predicts the formation of channels in response to flow.