Effectiveness factor for spatial gradient sensing in living cells

Integrating nanofibers with biochemical gradients to investigate physiologically-relevant fibroblast chemotaxis

Persistent cell migration can occur due to anisotropy in the extracellular matrix (ECM), the gradient of a chemo-effector, or a combination of both. Through a variety of in vitro platforms, the contributions of either stimulus have been extensively studied, while the combined effect of both cues remains poorly described. Here, we report an integrative microfluidic chemotaxis assay device that enables the study of single cell chemotaxis on ECM-mimicking, aligned, and suspended nanofibers. Using this assay, we evaluated the effect of fiber spacing on the morphology and chemotaxis response of embryonic murine NIH/3T3 fibroblasts in the presence of temporally invariant, linear gradients of platelet-derived growth factor-BB (PDGF-BB). We found that the strength of PDGF-mediated chemotaxis response depends on not only the gradient slope but also the cell morphology. Low aspect ratio (3.4 ± 0.2) cells on flat substrata exhibited a chemotaxis response only at a PDGF-BB gradient of 0-10 ng mL^-1. However, high aspect ratio (19.1 ± 0.7) spindle-shaped cells attached to individual fibers exhibited maximal chemotaxis response at a ten-fold shallower gradient of 0-1 ng mL^-1, which was robustly maintained up to 0-10 ng mL^-1. Quadrilateral-shaped cells of intermediate aspect ratio (13.6 ± 0.8) attached to two fibers exhibited a weaker response compared to the spindle-shaped cells, but still stronger compared to cells attached to 2D featureless substrata. Through pharmacological inhibition, we show that the mesenchymal chemotaxis pathway is conserved in cells on fibers. Altogether, our findings show that chemotaxis on ECM-mimicking fibers is modulated by fiber spacing-driven cell shape and can be significantly different from the behavior observed on flat 2D substrata. We envisage that this microfluidic platform will have wide applicability in understanding the combined role of ECM architecture and chemotaxis in physiological and pathological processes.

A novel chemotaxis assay in 3-D collagen gels by time-lapse microscopy

The directional cell response to chemical gradients, referred to as chemotaxis, plays an important role in physiological and pathological processes including development, immune response and tumor cell invasion. Despite such implications, chemotaxis remains a challenging process to study under physiologically-relevant conditions in-vitro, mainly due to difficulties in generating a well characterized and sustained gradient in substrata mimicking the in-vivo environment while allowing dynamic cell imaging. Here, we describe a novel chemotaxis assay in 3D collagen gels, based on a reusable direct-viewing chamber in which a chemoattractant gradient is generated by diffusion through a porous membrane. The diffusion process has been analysed by monitoring the concentration of FITC-labelled dextran through epifluorescence microscopy and by comparing experimental data with theoretical and numerical predictions based on Fick's law. Cell migration towards chemoattractant gradients has been followed by time-lapse microscopy and quantified by cell tracking based on image analysis techniques. The results are expressed in terms of chemotactic index (I) and average cell velocity. The assay has been tested by comparing the migration of human neutrophils in isotropic conditions and in the presence of an Interleukin-8 (IL-8) gradient. In the absence of IL-8 stimulation, 80% of the cells showed a velocity ranging from 0 to 1 µm/min. However, in the presence of an IL-8 gradient, 60% of the cells showed an increase in velocity reaching values between 2 and 7 µm/min. Furthermore, after IL-8 addition, I increased from 0 to 0.25 and 0.25 to 0.5, respectively, for the two donors examined. These data indicate a pronounced directional migration of neutrophils towards the IL-8 gradient in 3D collagen matrix. The chemotaxis assay described here can be adapted to other cell types and may serve as a physiologically relevant method to study the directed locomotion of cells in a 3D environment in response to different chemoattractants.

Formation of morphogen gradients: local accumulation time

Spatial regulation of cell differentiation in embryos can be provided by morphogen gradients, which are defined as the concentration fields of molecules that control gene expression. For example, a cell can use its surface receptors to measure the local concentration of an extracellular ligand and convert this information into a corresponding change in its transcriptional state. We characterize the time needed to establish a steady-state gradient in problems with diffusion and degradation of locally produced chemical signals. A relaxation function is introduced to describe how the morphogen concentration profile approaches its steady state. This function is used to obtain a local accumulation time that provides a time scale that characterizes relaxation to steady state at an arbitrary position within the patterned field. To illustrate the approach we derive local accumulation times for a number of commonly used models of morphogen gradient formation.

Reaction-diffusion constraints in living tissue: effectiveness factors in skeletal muscle design

A mathematical model was developed to analyze the effects of intracellular diffusion of O(2) and high-energy phosphate metabolites on aerobic energy metabolism in skeletal muscle. We tested the hypotheses that in a range of muscle fibers from different species (1) aerobic metabolism was not diffusion limited and (2) that fibers had a combination of rate and fiber size that placed them at the brink of substantial diffusion limitation. A simplified chemical reaction rate law for mitochondrial oxidative phosphorylation was developed utilizing a published detailed model of isolated mitochondrial function. This rate law was then used as a boundary condition in a reaction-diffusion model that was further simplified using the volume averaging method and solved to determine the rates of oxidative phosphorylation as functions of the volume fraction of mitochondria, the size of the muscle cell, and the amount of oxygen delivered by the capillaries. The effectiveness factor, which is the ratio of reaction rate in the system with finite rates of diffusion to those in the absence of any diffusion limitations, defined the regions where intracellular diffusion of metabolites and O(2) may limit aerobic metabolism in both very small, highly oxidative fibers as well as in larger fibers with lower aerobic capacity. Comparison of model analysis with experimental data revealed that none of the fibers was strongly limited by diffusion, as expected. However, while some fibers were near substantial diffusion limitation, most were well within the domain of reaction control of aerobic metabolic rate. This may constitute a safety factor in muscle that provides a level of protection from diffusion constraints under conditions such as hypoxia.

Analysis of reaction-diffusion systems with anomalous subdiffusion

Reaction-diffusion equations are the cornerstone of modeling biochemical systems with spatial gradients, which are relevant to biological processes such as signal transduction. Implicit in the formulation of these equations is the assumption of Fick's law, which states that the local diffusive flux of species i is proportional to its concentration gradient; however, in the context of complex fluids such as cytoplasm and cell membranes, the use of Fick's law is based on empiricism, whereas evidence has been mounting that such media foster anomalous subdiffusion (with mean-squared displacement increasing less than linearly with time) over certain length scales. Particularly when modeling diffusion-controlled reactions and other systems where the spatial domain is considered semi-infinite, assuming Fickian diffusion might not be appropriate. In this article, two simple, conceptually extreme models of anomalous subdiffusion are used in the framework of Green's functions to demonstrate the solution of four reaction-diffusion problems that are well known in the biophysical context of signal transduction: fluorescence recovery after photobleaching, the Smolochowski limit for diffusion-controlled reactions in solution, the spatial range of a diffusing molecule with finite lifetime, and the collision coupling mechanism of diffusion-controlled reactions in two dimensions. In each case, there are only subtle differences between the two subdiffusion models, suggesting how measurements of mean-squared displacement versus time might generally inform models of reactive systems with partial diffusion control.

Membrane-binding/modification model of signaling protein activation and analysis of its control by cell morphology

A mechanism for cell shape control of intracellular signal transduction, whereby the average concentration of activated proteins in the cytosol increases as the height of the cell decreases, has been described recently. An important modification of this analysis is offered, recognizing that signaling proteins are not only activated at the plasma membrane but must first form complexes with signaling molecules that reside there, such as receptors and lipids. With these more realistic boundary conditions, it is shown that the region of parameter space where cell shape amplifies the average cytosolic activity is greatly expanded. Moreover, this model allows for amplification of the activated protein bound at the membrane, which is considered more relevant for certain, spatially driven signaling processes in cell migration.