Multiphase flow models of biogels from crawling cells to bacterial biofilms

Abstract

This article reviews multiphase descriptions of the fluid mechanics of cytoplasm in crawling cells and growing bacterial biofilms. These two systems involve gels, which are mixtures composed of a polymer network permeated by water. The fluid mechanics of these systems is essential to their biological function and structure. Their mathematical descriptions must account for the mechanics of the polymer, the water, and the interaction between these two phases. This review focuses on multiphase flow models because this framework is natural for including the relative motion between the phases, the exchange of material between phases, and the additional stresses within the network that arise from nonspecific chemical interactions and the action of molecular motors. These models have been successful in accounting for how different forces are generated and transmitted to achieve cell motion and biofilm growth and they have demonstrated how emergent structures develop though the interactions of the two phases. A short description of multiphase flow models of tumor growth is included to highlight the flexibility of the model in describing diverse biological applications.

Figure 1. Different types of cell crawling.

(a) Schematic of a cell crawling over a two-dimensional surface by extending a lamellipodium in the direction of motion through actin polymerization at the front. (b) Some cells are thought to use blebbing to migrate in three-dimensional fibrous environments. When the membrane detaches from the cortex, intracellular pressure causes cytoplasm to stream toward the location of detachment, which inflates the bleb in the direction of motion.

Figure 2. Illustration of fountain flow inside

The arrows indicate the direction of the cytosol motion. The contractile stress within the actin network concentrates the cytoskeleton around the edges of the cell in the ectoplasmic gel layer. The cytoskeleton detaches from the front of the cell, which causes the cytoskeleton to move backward along the edges of the cell. The cytosol is pulled back along the wall by the motion of the cytoskeleton and it flows forward through the channel along the center of the cell.

Figure 3. Snapshots of the solution to the multiphase equations, neglecting the external flow.

The nutrient diffuses from the top (in the blue region) and is consumed by the bacteria within the biofilm region. The colormap shows regions of high growth (red) in the tips of the initial colony. The higher growth leads to higher osmotic pressure, which in turn, moves the biofilm region. Since the tips have access to more nutrient (via diffusion), the perturbation is reinforced leading to a highly heterogeneous structure. The arrows represent the interface velocity.

Figure 4. A schematic of the channeling bifurcation that arises from the interplay of the mechanical and chemical stresses in pressure-driven flow through a gel-filled tube.

(a) Schematic of an osmotic pressure function for which at high volume fractions the chemical stress keeps the mixture of fluid and network stably mixed (solid line), and at low volume fractions this chemical stress drives the mixture to phase separate (dashed line). (b) For modest pressure gradients (i.e., flow rates), the spatially uniform phase moves with a typical velocity that depends only on the vertical position. The faster flow in the middle of the channel causes a stretching of the network in the middle and a compression of network along the edges. (c) At a critical pressure, the network in the middle is stretched to the point when osmotic pressure induces phase separation, which further compresses the network on the edges and opens a flow channel in the middle that is devoid of network. The fluid velocity is much higher in the channel because the fluid is free to move without the additional interphase drag.