Conceptual and Experimental Tools to Understand Spatial Effects and Transport Phenomena in Nonlinear Biochemical Networks Illustrated with Patchy Switching

Abstract

Many biochemical systems are spatially heterogeneous and exhibit nonlinear behaviors, such as state switching in response to small changes in the local concentration of diffusible molecules. Systems as varied as blood clotting, intracellular calcium signaling, and tissue inflammation are all heavily influenced by the balance of rates of reaction and mass transport phenomena including flow and diffusion. Transport of signaling molecules is also affected by geometry and chemoselective confinement via matrix binding. In this review, we use a phenomenon referred to as patchy switching to illustrate the interplay of nonlinearities, transport phenomena, and spatial effects. Patchy switching describes a change in the state of a network when the local concentration of a diffusible molecule surpasses a critical threshold. Using patchy switching as an example, we describe conceptual tools from nonlinear dynamics and chemical engineering that make testable predictions and provide a unifying description of the myriad possible experimental observations. We describe experimental microfluidic and biochemical tools emerging to test conceptual predictions by controlling transport phenomena and spatial distribution of diffusible signals, and we highlight the unmet need for in vivo tools.

Keywords: Damköhler number; flow; mass transfer; microfluidics; signaling; state switching.

Figure 1

In patchy switching, local accumulation of diffusible triggers causes transitions between biological states, shown here for OFF and ON (red starburst) states. Triggers accumulate to high concentration (dark blue) only when their removal by transport (green arrows) is slower than their net production (red arrows). The competition between reaction and transport is described by a unitless quantity termed the Damköhler number ($Da$).

Figure 2

Balance of production and removal of triggers controls patchy switching and is described by the Damköhler number ($Da$). In each panel, the plot above depicts the concentration of triggers (C) for each scenario. (a) Three scenarios for the distribution of activators (gray spheres). “No patch” refers to molecules or cells being dispersed. (b) Activators produce triggers (blue) at a fixed rate per activator (red arrows). (c) Diffusion removes triggers (green arrows) with greater effect on smaller patches. (d) For a certain threshold concentration of triggers ($C^{\mathrm{*}}$, gray dotted line), only the large patch accumulates triggers above the threshold and has $Da>1$ (ON, red starburst). The no patch and small patch groups are both below the threshold and have $Da<1$ (OFF).

Figure 3

Switching in response to changes in spatial distribution. (a, i) Blood coagulation (blue) is initiated on micropatterned patches of tissue factor (red) only if they are larger than a critical size, quantified in subpanel ii. Panel a adapted with permission from Reference , © 2006 National Academy of Sciences, USA. (b, i, top) Only localized Ca²⁺ puffs were produced in response to local release of photocaged inositol (1,4,5)-triphosphate (IP₃) in murine pancreatic cells expressing low density of inositol trisphosphate receptors (InsP₃R); (b, i, bottom) global Ca²⁺ (blue) waves were switched on in response to local release of photocaged IP₃ in murine parotid cells expressing high density of InsP₃R (30). (b, ii) A plot showing that parotid cells have a lower threshold IP₃ concentration for maximum Ca²⁺ release than the pancreatic cells. Panel b adapted from Reference with permission. (c, i) Embryoid bodies preferentially differentiated into neural precursor cells (green) with long neurites (red, quantified in subpanel ii) only when they were at least 500 μm in size. Panel c adapted from Reference with permission. (d, i) Plucking hairs on the skin of mice induced amplified hair regeneration only when hairs were plucked above a threshold density, quantified in subpanel ii. Red circles in images show the regions from which 200 hairs were plucked 30 days prior to the photo. Panel d adapted from Reference with permission.

Figure 4

Various mechanisms for inducing patchy switching by changing the Damköhler number. (a) Altering the local concentration of a trigger (blue) by increasing its rate of production by the activators in the patch (gray spheres) can turn a system with threshold kinetics ON (red starburst). Inhibitory binding of a trigger on the extracellular matrix (ECM) also can turn the system OFF, whereas degrading the ECM can restore the ON state. (b) Altering transport (i–iii) by limiting diffusional flux increases the local concentration and can turn the system ON, whereas increasing removal (iv) by increasing the rate of convection or active transport (green arrows) decreases the local concentration and can turn the system OFF. Each of these processes can also occur in reverse.

Figure 5

Controlling patch size and clustering using chemistry, microfluidics, and local drug delivery systems. (a) Two-dimensional patches of human oral squamous carcinoma cells micropatterned using a peel-off perylene stencil. Panel a reproduced from Reference with permission. (b) Three-dimensional (3D) tumor spheroids formed and stained (inset, green) in microfluidic traps. Panel b adapted from Reference with permission. (c) Two-dimensional chemical patterning of 3D microcultures containing rat fibroblasts (green) or human lung adenocarcinoma cells (blue) using DNA templates. Panel c reproduced from Reference with permission. (d) Localized chemical stimulation (red) of a murine lymph node slice cultured in a microfluidic chamber. The slice was immunostained for B cells (green) and counterstained with Hoechst (blue), allowing specific regions to be targeted for stimulation. Panel d reproduced from Reference with permission from The Royal Society of Chemistry. (e) Microchip implanted in a human to locally deliver discrete doses of a hormone fragment, which increased bone formation in women with osteoporosis. Panel e adapted from Reference with permission.

Figure 6

Microfluidic control over confinement and flow near patches. (a) Microfluidic patterning of the extracellular matrix (ECM) using Matrigel with distinct spatial zones visible by fluorescence (top) and phase contrast imaging (bottom). Panel a adapted from Reference with permission. (b) Confinement of single cells or small groups of Pseudomonas aeruginosa in ~0.1 pL droplets activated a fluorescent reporter for the quorum sensing–controlled gene lasB (green fluorescence). Panel b adapted from Reference with permission. (c) Partial confinement of motile Vibrio harveyi (yellow), initially loaded uniformly in a microfluidic maze (white lines) with 100 μm passages, caused accumulation of cells and activation of quorum sensing (blue luminescence) in partially confined regions of the maze. Panel c adapted from Reference with permission. (d) Increased shear of whole blood over patches of collagen and tissue factor (TF) prevented propagation of a fibrin (green) and platelet (red) clot over the middle patch. Panel d adapted from Reference with permission.

Figure 7

Lesions and ectopic follicles in multiple sclerosis (MS) are candidates for patchy switching. (a) Focal demyelinated lesions in the white matter (161) appear consistent with a response to patch size and clustering. The asterisk indicates the center of the lesion identified by the absence of brown staining for proteolipid protein, a component of myelin. Panel a adapted from Reference , available under the Creative Commons Non-Commercial 2.0 UK License (http://creativecommons.org/licenses/by-nc/2.0/uk/). (b) Ectopic B cell follicles located deep inside cortical sulci in secondary progressive MS are consistent with patchy switching in response to confinement of inflammatory signals. The schematic details the geometry of a cerebral sulcus, including the three layers of the meninges (dura, arachnoid, and pia mater). Insets: Immunohistochemistry for B cells (CD20, brown) revealed follicles located inside the sulcus but not at the external surface of the brain. Panel b adapted from Reference with permission. Abbreviations: GM, gray matter; WM, white matter.