Intracellular Concentration Gradients That Mirror External Gradients in Microfluidic Flows: A Computational Analysis

Abstract

The generation of stable intracellular concentration gradients is a useful method for local control of cell function, selective manipulation of cellular structures and testing hypotheses related to dynamical intracellular processes. Cell culture in a microfluidic device allows the presentation of a stable gradient of small molecules across a single cell. This method has been used to selectively label mitochondria in portions of the cell, trypsinize specific cellular domains, and trigger receptor-mediated endocytosis in specific portions of the cell. Given the small length scales of a typical cell (~30 μm) and short cytoplasmic diffusive time scales of small molecules, it is surprising that cells can be labeled locally with this method. Here we developed models to explore the parametric space over which stable intracellular concentration gradients can be maintained in a microfluidic device. We show that gradients can develop and be maintained indefinitely for high rates of mass transfer across the membrane compared with diffusion, that is, for Sherwood number greater than 1. We show how these gradients can result in gradients in ligand–receptor binding and enzyme substrate binding. This analysis can help interpret and design microfluidic experiments for cytoplasmic partitioning.

Electronic supplementary material

The online version of this article (doi:10.1007/s12195-016-0474-3) contains supplementary material, which is available to authorized users.

Introduction

The generation of spatially varying concentration of small molecules in the cytoplasmic space of living cells can allow the selective manipulation of cellular structures, control over cell function and the testing of hypotheses related to dynamical intracellular processes.20 One approach pioneered by the Whitesides and Ingber groups is to culture single cells in microfluidic devices. By including small molecules in only one of two laminar streams that flow over the cell, it is possible to impose a step gradient in concentration of these molecules over a single cell. Whitesides and coworkers demonstrated the labeling of mitochondrial structures over specific spatial domains in the cell over long periods of time (minutes).20 They similarly used this method to detach portions of the cell and to disrupt cytoskeletal structures over pre-chosen domains.21

Since then, microfluidics-based biological labeling has found use in a number of studies. Local damage of neuritis with a detergent was in a microfluidic device to study subsequent repair.9 Another study treated cells locally with epidermal growth factor (EGF), and studied the spatial and temporal propagation of EGF signaling.17 Multiple microfluidic streams have been used to apply a continuous gradient of growth factors (GF) to populations of neural stem cells in order to investigate the correlation of differentiation or proliferation with growth factor concentrations.10 Confluent epithelial cells have been exposed to chemical gradients in microfluidic devices in order to selectively induce gene expression in a band of cells.5 Temperature gradients have also been created in microfluidic devices to study patterning in drosophila2 and multiples streams have been used for generating electricity from organic matter.11

As these studies show, there is a broad range of applications of microfluidic devices in treating cells with gradients of soluble factors. In this paper, we are interested in the development and maintenance of sub-cellular gradients. Previous studies Refs. 20, 21 that sought to create intracellular gradients do not report measurement of the cytoplasmic gradient of the freely diffusing small molecules. This is likely due to the low cytoplasmic fluorescence compared with the localized fluorescence from sub-cellular organelles that bind (and therefore concentrate) small molecules. While it is reasonable to deduce from the spatially distributed labeling, or spatially selective action of small molecules (such as local cytoskeletal disruption) that the small molecule must partition spatially in the cytoplasmic space, it would appear challenging to maintain cytoplasmic gradients for significant lengths of time. Unlike the external gradients which are maintained because of laminar flow in the microfluidic device, there is no flow in the cell. Moreover, the length scales of cells are small (~30 μm), and diffusion coefficients in the cytoplasm of small molecules are large (~100 μm²/s).12

Here we formulated models that account for mass transfer across the cell membrane, cytoplasmic diffusion through the cytoplasm, and reaction. We show that the intracellular gradient depends on the relative rates of mass transfer to diffusion. We predict the development of gradients in the cytoplasm of ligand-cytoplasm receptor binding and enzyme substrate binding. This model can provide guidance on the interpretation and design of microfluidic experiments for sub-cellular partitioning.

Model

Ligand Concentration When No Reaction

Because the length of the cell is much greater than the height, we modeled the concentration of the ligand in the cytoplasm with a one dimensional reaction diffusion equation (Eq. (1))

$$\begin{array}{cc}\hfill \frac{\mathit{\partial }{C}_L}{\mathit{\partial }t}& =D_L\frac{{\mathit{\partial }}^2{C}_L}{\mathit{\partial }{x}^2}+k_m\left[C_{L0}\mathrm{H}\left(l^{\ast }-x\right)-C_L\right]C_L\left(x,0\right)=0\hfill \\ \hfill \mathrm{H}\left(l^{\ast }-x\right)& =1,l^{\ast }\ge x\hfill \\ \hfill & =0,l^{\ast }<x\hfill \end{array}$$ 1

Equation (1) describes the spatial and temporal variation in the cytoplasmic concentration of the ligand. Here $C_L$ is the concentration of the ligand in the cytoplasm, $D_L$ is the cytoplasmic diffusion coefficient of the ligand, $k_m$ is the modified membrane mass transfer coefficient, $l^{\ast }$ is the length of the treated part of the cell) over which there is flux of the ligand into the cell and $C_{L0}$ is the concentration of the ligand in the external stream.

The length of the cell ($l$) was chosen as the length scale, concentration of the ligand in the external stream ($C_{L0}$) as the concentration scale, and the characteristic cytoplasmic diffusion time of the ligand ($t_L=\frac{l^2}{D_L}$) as the time scale for non-dimensionalizing Eq. (1). The non-dimensionalized equation is

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{\mathit{\theta }}_L}{\mathit{\partial }\mathit{\tau }}=\frac{{\mathit{\partial }}^2{\mathit{\theta }}_L}{\mathit{\partial }{\mathit{\xi }}^2}+Sh\left[\mathrm{H}\left(\mathit{\omega }-\mathit{\xi }\right)-{\mathit{\theta }}_L\right]\\ \hfill {\mathit{\theta }}_L\left(\mathit{\xi },0\right)=0\\ \hfill \end{array}$$ 2

here ${\mathit{\theta }}_L=\frac{C_L}{C_{L0}}$, $\mathit{\tau }=\frac{t}{t_L}$, $\mathit{\xi }=\frac{x}{l}$, $\mathit{\omega }=\frac{l^{\ast }}{l}$ and $Sh=\frac{k_m{l}^2}{D_L}$. $Sh$ is the Sherwood number (based on the modified mass transfer coefficient) and it describes the relative rate of mass transfer across the cell membrane with respect to diffusion in the cytoplasm.

Receptor Ligand Binding

We extended the model developed in the previous section to reversible receptor–ligand binding in the cytoplasm. This model could apply (for example) to cytoplasmic hormone receptors, or membrane bound receptors like IP3, which is located on the ER membranes.

$$\begin{array}{cc}\hfill \frac{\mathit{\partial }{C}_L}{\mathit{\partial }t}& =D_L\frac{{\mathit{\partial }}^2{C}_L}{\mathit{\partial }{x}^2}\hfill \\ \hfill & +k_m\left[C_{L0}\mathrm{H}\left(l^{\ast }-x\right)-C_L\right]\hfill \\ \hfill & +k_{off}{C}_P-k_{on}{C}_R{C}_L\hfill \\ \hfill C_L\left(x,0\right)& =0\hfill \end{array}$$ 3

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{C}_R}{\mathit{\partial }t}=D_R\frac{{\mathit{\partial }}^2{C}_R}{\mathit{\partial }{x}^2}+k_{off}{C}_P-k_{on}{C}_R{C}_L\\ \hfill C_R\left(x,0\right)=C_{R0}\\ \hfill \end{array}$$ 4

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{C}_P}{\mathit{\partial }t}=D_P\frac{{\mathit{\partial }}^2{C}_P}{\mathit{\partial }{x}^2}-k_{off}{C}_P+k_{on}{C}_R{C}_L\\ \hfill C_P\left(x,0\right)=0\\ \hfill \end{array}$$ 5

Equations (3)–(5) describe the cytoplasmic concentration profiles of the ligand ($C_L$), receptor ($C_R$), and bound receptor ($C_P$) respectively. $D_L$,$D_R$, and $D_P$, and are their respective diffusion coefficients, $k_{off}$ is the dissociation rate constant, and $k_{on}$ is the binding rate constant. $K_d=\frac{k_{off}}{k_{on}}$ is the equilibrium dissociation constant for the reaction.

Choosing $l$ as the length scale, cytoplasmic diffusion time of the receptor $t_R=\frac{l^2}{D_R}$ as the time scale, $C_{L0}$ and $C_{R0}$ as the concentration scales for ligand and receptor respectively, and assuming $D_P\approx D_R$ (for example, Ref. 7), Eqs. (3)–(5) were non-dimensionalized to:

$$\begin{array}{cc}\hfill \frac{\mathit{\partial }{\mathit{\theta }}_L}{\mathit{\partial }\mathit{\tau }}& =a_L\frac{{\mathit{\partial }}^2{\mathit{\theta }}_L}{\mathit{\partial }{\mathit{\xi }}^2}+a_{L}Sh\left[\mathrm{H}\left(\mathit{\omega }-\mathit{\xi }\right)-{\mathit{\theta }}_L\right]\hfill \\ \hfill & +D{a}_{off}\mathit{\gamma }{\mathit{\theta }}_P-D{a}_{on}\mathit{\gamma }{\mathit{\theta }}_L{\mathit{\theta }}_R\hfill \\ \hfill {\mathit{\theta }}_L\left(\mathit{\xi },0\right)& =0\hfill \\ \hfill \end{array}$$ 6

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{\mathit{\theta }}_R}{\mathit{\partial }\mathit{\tau }}=\frac{{\mathit{\partial }}^2{\mathit{\theta }}_R}{\mathit{\partial }{\mathit{\xi }}^2}+D{a}_{off}{\mathit{\theta }}_P-D{a}_{on}{\mathit{\theta }}_L{\mathit{\theta }}_R\\ \hfill {\mathit{\theta }}_R\left(\mathit{\xi },0\right)=1\\ \hfill \end{array}$$ 7

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{\mathit{\theta }}_P}{\mathit{\partial }\mathit{\tau }}=\frac{{\mathit{\partial }}^2{\mathit{\theta }}_P}{\mathit{\partial }{x}^2}-D{a}_{off}{\mathit{\theta }}_P+D{a}_{on}{\mathit{\theta }}_L{\mathit{\theta }}_R\\ \hfill {\mathit{\theta }}_P\left(\mathit{\xi },0\right)=0\\ \hfill \end{array}$$ 8

here ${\mathit{\theta }}_R=\frac{C_R}{C_{R0}}$, ${\mathit{\theta }}_P=\frac{C_P}{C_{R0}}$, $\mathit{\tau }=\frac{t}{t_R}$, $\mathit{\gamma }=\frac{C_{R0}}{C_{L0}}$, $a_L=\frac{D_L}{D_R}$, $D{a}_{on}=\frac{k_{on}{C}_{L0}{l}^2}{D_R}$ and $D{a}_{off}=\frac{k_{off}{l}^2}{D_R}$. The dimensionless equilibrium dissociation constant is $K_d^{{}^{\prime }}=\frac{K_d}{L_0}$. $D{a}_{off}$ and $D{a}_{on}$ are the Damkohler number for dissociation and binding. They describe the relative rate of dissociation and binding (respectively) with respect to the cytoplasmic diffusion rate of the receptor.

Membrane-Receptor Ligand Binding

A membrane embedded receptor is assumed to bind reversibly with the extracellular ligand to form a receptor ligand complex. The ligand is assumed to be membrane impermeable. This is different from before because there is no mass transfer across the cell membrane. The equations describing the spatially and temporally varying concentration of the free receptor (Eq. (9)) and the bound receptor (Eq. (10)) are below:

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{C}_R}{\mathit{\partial }t}=D_R\frac{{\mathit{\partial }}^2{C}_R}{\mathit{\partial }{x}^2}-k_{on}\mathrm{H}\left(l^{\ast }-x\right)C_R{C}_{L0}+k_{off}{C}_P\\ \hfill C_R\left(x,0\right)=C_{R0}\\ \hfill \end{array}$$ 9

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{C}_P}{\mathit{\partial }t}=D_P\frac{{\mathit{\partial }}^2{C}_P}{\mathit{\partial }{x}^2}+k_{on}\mathrm{H}\left(l^{\ast }-x\right)C_R{C}_{L0}-k_{off}{C}_P\\ \hfill C_P\left(x,0\right)=0\\ \hfill \end{array}$$ 10

In Eqs. (9) and (10), $C_R$ is the membrane concentration of the receptor and $C_P$ is the membrane concentration of the bound receptor. $D_R$ and $D_P$ are the membrane diffusion coefficients of the free receptor and the bound receptor respectively, $k_{off}$ is the dissociation rate constant, $k_{on}$ is the binding rate constant. $K_d=\frac{k_{off}}{k_{on}}$ is the equilibrium dissociation constant for the reaction.

Using the same scaling scheme defined in the receptor ligand binding section (the difference is that the parameters refer to properties of membrane-embedded receptors) and assuming that $D_P\approx D_R.$ Eqs. (9) and (10) were non dimensionalized to:

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{\mathit{\theta }}_R}{\mathit{\partial }\mathit{\tau }}=\frac{{\mathit{\partial }}^2{\mathit{\theta }}_R}{\mathit{\partial }{\mathit{\xi }}^2}-D{a}_{on}\mathrm{H}\left(\mathit{\omega }-\mathit{\xi }\right){\mathit{\theta }}_R+D{a}_{off}{\mathit{\theta }}_P\\ \hfill {\mathit{\theta }}_R\left(\mathit{\xi },0\right)=1\\ \hfill \end{array}$$ 11

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{\mathit{\theta }}_P}{\mathit{\partial }\mathit{\tau }}=\frac{{\mathit{\partial }}^2{\mathit{\theta }}_P}{\mathit{\partial }{\mathit{\xi }}^2}+D{a}_{on}\mathrm{H}\left(\mathit{\omega }-\mathit{\xi }\right){\mathit{\theta }}_R-D{a}_{off}{\mathit{\theta }}_P\\ \hfill {\mathit{\theta }}_P\left(\mathit{\xi },0\right)=0\\ \hfill \end{array}$$ 12

The mathematical form of the parameter groups in Eqs. (11) and (12), is the same as the previous section. $D{a}_{on}$ and $D{a}_{off}$ under this situation describe the relative rate of binding and dissociation (respectively) to the membrane diffusion rate of the receptor.

Enzyme Substrate Binding

Substrate diffuses from the external stream across the membrane and is catalyzed by a cytoplasmic enzyme to a product. The product is assumed to be able to diffuse across the cell membrane. Equations (13) and (14) describe the cytoplasmic concentration profile of the substrate and the product respectively.

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{C}_S}{\mathit{\partial }t}=D_S\frac{{\mathit{\partial }}^2{C}_S}{\mathit{\partial }{x}^2}-\frac{R_{max}{C}_S}{K_M+C_S}+k_m\left[\mathrm{H}\left(l^{\ast }-x\right)C_{S0}-C_S\right]\\ \hfill C_S\left(x,0\right)=0\\ \hfill \end{array}$$ 13

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{C}_P}{\mathit{\partial }t}=D_P\frac{{\mathit{\partial }}^2{C}_P}{\mathit{\partial }{x}^2}+\frac{R_{max}{C}_S}{K_M+C_S}-k_{mp}{C}_P\\ \hfill C_P\left(x,0\right)=0\\ \hfill \end{array}$$ 14

here $C_S$ and $C_P$ are the cytoplasmic substrate and the cytoplasmic product concentration respectively, $D_S$ and $D_P$ are the cytoplasmic diffusion coefficients of the substrate and the product respectively, $k_{ms}$ and $k_{mp}$ are the modified cell-membrane mass transfer coefficients of the substrate and product respectively. The term $k_{mp}P$ refers to the efflux of product from the cell. $K_M$ is the Michaelis–Menten constant, and $R_{max}$ is the maximum rate. Choosing $C_{S0}$ as the concentration scale, $l$ as the length scale, $t_S=\frac{l^2}{D_S}$ as the time scale and assuming $D_S\approx D_P$ and $k_{mp}\approx k_m$, Eqs. (13) and (14) were non-dimensionalized to

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{\mathit{\theta }}_S}{\mathit{\partial }\mathit{\tau }}=\frac{{\mathit{\partial }}^2{\mathit{\theta }}_S}{\mathit{\partial }{\mathit{\xi }}^2}-\frac{D{a}_M{\mathit{\theta }}_S}{{\tilde{K}}_M+{\mathit{\theta }}_S}+Sh\left[\mathrm{H}\left(\mathit{\omega }-\mathit{\xi }\right)-{\mathit{\theta }}_S\right]\\ \hfill {\mathit{\theta }}_S\left(\mathit{\xi },0\right)=0\\ \hfill \end{array}$$ 15

$$\begin{array}{c}\hfill \frac{\mathit{\partial }{\mathit{\theta }}_P}{\mathit{\partial }\mathit{\tau }}=\frac{{\mathit{\partial }}^2{\mathit{\theta }}_P}{\mathit{\partial }{\mathit{\xi }}^2}+\frac{D{a}_M{\mathit{\theta }}_S}{{\tilde{K}}_M+{\mathit{\theta }}_S}-Sh{\mathit{\theta }}_P\\ \hfill {\mathit{\theta }}_P\left(\mathit{\xi },0\right)=0\\ \hfill \end{array}$$ 16

In Eqs. (15) and (16) ${\mathit{\theta }}_S=\frac{C_S}{C_{S0}}$, ${\mathit{\theta }}_P=\frac{C_P}{C_{S0}}$, $D{a}_M=\frac{R_{max}{l}^2}{C_{S0}{D}_S}$, ${\tilde{K}}_M=\frac{K_M}{C_{S0}}$ and $Sh=\frac{k_m{l}^2}{D_S}.$

Finite Element Calculation

COMSOL 5.1 was used to solve the equations. Transport of dilute species (tds) module was used to generate spatially and temporally dependent concentration profiles for different species. Steady state was verified by ensuring that the spatial concentration profiles overlapped completely for longer time instants.

Results

External Gradients of Membrane-Permeable Ligands can Result in Intra-cellular Gradients

An elongated cell is assumed to be adherent to the bottom of a microfluidic device and oriented perpendicular to the direction of flow (Fig. 1a). Of the two streams flowing over the cell, only one contains the ligand. Negligible lateral mixing in the channel is assumed,19 such that the external concentration profile across the cell can be assumed to be a step function. The ligand is modeled to enter the cell by transport through the cell membrane with a modified mass transfer coefficient k _m $(k_m=k/h$, here $k$ is the mass transfer coefficient across the membrane, and $h$ is the height of the cell) and then to diffuse within the cytoplasm with diffusion coefficient $D_L$. Here we assume that the cell is much thinner vertically than its length, which allows us to ignore gradients in the vertical direction.21

Figure 1.

Cytoplasmic concentration of ligand under an external gradient. (a) Schematic of the experimental system. Ligand is present in one of the streams. Diffusional mixing between two streams is assumed to be negligible. (b) Cytoplasmic concentration of the ligand for different values of the Sherwood number ($\mathit{\omega }=0.3$; this is the fractional treated length of the cell). A steeper gradient is observed for higher values of the Sherwood number. (c) Variation in the gradient for changing values of the treated cell length ($\mathit{\omega }$ is the fractional treated length of the cell, circles indicate the location of the interface). The difference in intracellular concentration across the length of the cell is maximum when $\mathit{\omega }=0.5$. $Sh=100$ for the calculations in (c).

To understand the behavior of the ligand in the cytoplasm in the absence of any ‘reaction’, the one dimensional transport equation was solved at steady state under an external step gradient. 30% of the cell length was assumed to be treated with ligand-containing flow.21 Shown in Fig. 1b is the predicted concentration profile for different values of $Sh$ (the Sherwood number). $Sh$ quantifies the relative time scale of transport across the membrane to the diffusion time scale across the length of the cell ($Sh=\frac{k_m{l}^2}{D_L}$, $l$ is the length of the cell, see modeling section). For $Sh<1$, transport across the membrane is slow compared to diffusion inside the cell, and the concentration profile in the cytoplasm is nearly uniform. For $Sh>>1$, membrane transport becomes faster than intracellular diffusion, and the concentration profile approaches the step-function of the extracellular concentration profile. The rapid decay in the intracellular concentration at high $Sh$ values is because of the high efflux rate across the untreated cell membrane relative to the rate of diffusion along the cell length. Significantly, such gradients can be maintained indefinitely; also it is possible to achieve $Sh>1$ for biological ligands (Tables 1 and 2). Figure 1c shows the effect of the fractional treated length of the cell ($\mathit{\omega }=\frac{l^{\ast }}{l}$, $l^{\ast }$ is the length of the treated part of the cell) on the concentration profile. The difference in intracellular concentration across the length of the cell is maximized when the interface is positioned at the mid-point of the cell.

Table 1.

Experimental values of parameters.

Name	Symbol used	Value or range	source
Length of the cell	$l$	~30 μm	Measured from images taken from Ref. 21
Treated length	$l^{\ast }$	~10 μm	Measured from images taken from Ref. 21
Diffusion coefficient of the ligand/substrate	$D_L$ or $D_S$	~100 μm2 s−1	For example, Ref. 12
Membrane mass transfer coefficient of the ligand/substrate	$k_m$	~10 s−1	Estimated from Ref. 14
Diffusion coefficient of the receptor (cytoplasmic)	$D_R$	~10 μm2 s−1	For example, Ref. 18
Diffusion coefficient of the receptor (membrane)	$D_R$	~1 μm2 s−1	For example, Ref. 8
Binding rate constant	$k_{on}$	~104–107 M−1s−1	For example, Refs. 6, 1
Dissociation rate constant	$k_{off}$	~1–10−3 s−1	For example, Refs. 6, 16
Initial receptor concentration	$C_{R0}$	~1 μM	Estimated from Ref. 24
Ligand/substrate concentration in the external stream	$C_{L0}$, $C_{S0}$	~1 μM	Ref. 21
Max enzyme rate	$R_{max}$	0.01–10 μM s−1	See for example, Ref. 23
Michaelis–Menten parameter	$K_M$	~1–100 μM	For example, Ref. 23

Table 2.

Values of dimensionless parameter groups.

Name	Symbol used	Expression	Range
Ratio of diffusion coefficients	$a_L$	$a_L=\frac{D_L}{D_R}$	10
Fractional treated length of the cell	$\mathit{\omega }$	$\mathit{\omega }=\frac{l^{\ast }}{l}$	0.3
Sherwood number	$Sh$	$Sh=\frac{k_m{l}^2}{D_L}$	100
Damkohler number for binding	$D{a}_{on}$	$D{a}_{on}=\frac{k_{on}{C}_{L0}{l}^2}{D_R}$	1–1000
Damkohler number for dissociation	$D{a}_{off}$	$D{a}_{on}=\frac{k_{off}{l}^2}{D_R}$	0.1–100
Damkohler number for binding (membrane)	$D{a}_{on}$	$D{a}_{on}=\frac{k_{on}{C}_{L0}{l}^2}{D_R}$	10–10,000
Damkohler number for dissociation (membrane)	$D{a}_{off}$	$D{a}_{on}=\frac{k_{off}{l}^2}{D_R}$	1–1000
Dimensionless equilibrium dissociation constant	$K_d^{{}^{\prime }}$	$K_d^{{}^{\prime }}=\frac{D{a}_{off}}{D{a}_{on}}$	0.01–10
Concentration ratio	$\mathit{\gamma }$	$\mathit{\gamma }=\frac{C_{R0}}{C_{L0}}$	1
Damkohler number for the enzyme reaction	$D{a}_M$	$D{a}_M=\frac{R_{max}{l}^2}{C_{L0}{D}_S}$	1–100
Non dimensionalized Michaelis–Menten constant	${\tilde{K}}_M$	${\tilde{K}}_M=\frac{K_M}{C_{S0}}$	1–100

The parameters in Table 1 were used to estimate the values of dimensionless parameter groups in Table 2

Receptor Ligand Binding

We extended the model developed in the previous section to also account for cytoplasmic receptor–ligand binding. The ligand was assumed to diffuse across the membrane and bind reversibly to a receptor present in the cytoplasm. $Sh$ was assumed to be greater than 1 to ensure a spatial gradient in the ligand concentration (see analysis above). The rate of binding relative to the rate of diffusion is expected to affect the gradients in bound receptor concentration. We therefore calculated bound receptor concentration profiles for different values of the Damkohler number $D{a}_{on}$ ($D{a}_{on}=\frac{k_{on}{C}_{L0}{l}^2}{D_R}$, reflects the rate of binding relative to the rate of diffusion). We chose parameter values such that the receptor is not in large excess to the ligand (because then the ligand cannot impact receptor concentration) and the values were biologically reasonable (the range of parameters used here is consistent with experimentally measured parameters, see Tables 1 and 2). As seen in Fig. 2a, larger concentration differences in bound receptor concentration across the length of the cell are predicted for larger values of $D{a}_{on}$ at steady state which reflect the larger spatially dependent net reaction rate (rate of binding − rate of dissociation, see Eq. (4)). Here, $D{a}_{off}$ which is the Damkohler number for dissociation ($D{a}_{off}=\frac{k_{off}{l}^2}{D_R}$) is fixed at 100 ($D{a}_{off}$ is also an important parameter that determines the gradient, which we explore in Fig. 2b). For $D{a}_{on}=1000$, that is for high $D{a}_{off}$ and high $D{a}_{on}$ the reaction is close to local equilibrium (Figure S2A, green curve) which determines the spatial gradient.

Figure 2.

Effect of external ligand gradient on cytoplasmic receptor ligand binding. (a) Left: Concentration profile of the bound receptor at steady state for different values of $D{a}_{on}$ ($D{a}_{off}=100$). Differences in the concentrations are smaller for smaller values of $D{a}_{on}$. Right: Corresponding net reaction rate $D{a}_{on}{\mathit{\theta }}_R{\mathit{\theta }}_L-D{a}_{off}{\mathit{\theta }}_P$ (here ${\mathit{\theta }}_L$, ${\mathit{\theta }}_R$ and ${\mathit{\theta }}_P$ are non dimensionalized concentration of the ligand, free receptor and bound receptor respectively); The net reaction rate is higher for high values of $D{a}_{on}$. (b) Left: Concentration profile of the bound receptor at steady state for different values of $D{a}_{off}$($D{a}_{on}=1000$). No gradients are observed when $D{a}_{off}<1$. Right: Corresponding net reaction rate.

Figure 2b shows the effect of larger values of the Damkohler number based on the dissociation rate constant for fixed $D{a}_{on}$. Despite large $D{a}_{on}$ (binding much faster than diffusion), no gradients are observed at steady state for $D{a}_{off}<1$ (dissociation much slower than diffusion). Because the bound receptor preferentially diffuses rather than dissociates, most of the receptor is in a bound state regardless of the spatial position (the reaction is not at equilibrium as indicated by the plot of net relative rates, Figure S2B, $D{a}_{off}<1$). Thus, for significant steady state gradients, both $D{a}_{on}$ and $D{a}_{off}$ have to be greater than 1. An example of this is mitochondrial labeling with red and green Mitotracker dye (Fig. 1C–D in Ref. 20). According to the analysis above $D{a}_{on}>1$ (slow diffusivity/motion of mitochondria3) and $D{a}_{off}\approx 0$ (nearly irreversible binding of Mitotracker to mitochondria15). As $D{a}_{off}$ is not greater than 1, the initial local labeling (Fig. 1C in Ref. 20) owing to fast binding compared to cytoplasmic diffusion is not observed to persist due to subsequent mixing over hours (Fig. 1D in Ref. 20). Also, $K_d^{{}^{\prime }}$ ($K_d^{{}^{\prime }}=\frac{D{a}_{off}}{D{a}_{on}}$) should be smaller than 1, as otherwise, there is little binding and very low levels of bound receptor in the cytoplasm (Figure S3, $K_d^{{}^{\prime }}=10$).

We performed the same analysis for membrane receptor ligand binding (see model section for the model equations). This scenario is different from the above case as the ligand is membrane-impermeable. We assume a step gradient in extracellular ligand as before. This implies that the membrane receptor can be bound only in the treated part of the cell membrane whereas the dissociation from the (diffusing) bound receptor can occur throughout the cell length. Figure 3 shows the concentration of the bound receptor and net reaction rates for different values of the $D{a}_{on}$ and $D{a}_{off}$. As expected, both $D{a}_{on}$ and $D{a}_{off}$ have to be larger than one for spatial gradients to exist.

Figure 3.

Effect of external ligand gradient on membrane receptor ligand binding. (a) Left: Concentration profile of the bound receptor at steady state for different values of $D{a}_{on}$ $D{a}_{off}=100$). Right: Corresponding net reaction rate, bound receptor is formed only in the treated part of the cell. (b) Left: Concentration profile of the bound receptor at steady state for different values of $D{a}_{off}$($D{a}_{on}=1000$). Right: Corresponding net reaction rate.

Enzyme Substrate Reaction

We next explored how extracellular substrate gradients could result in spatially distributed enzyme catalyzed reactivity and spatial gradients in the product of reaction at steady state. We considered a model in which substrate is present in one stream, crosses the membrane into the cytoplasm, and is converted into product by an enzyme (see modeling section for details). The product is assumed to cross the membrane and leave. An example of such a type of reaction is glycogenolysis performed by liver cells.13,22

Two parameter groups determine the steady state substrate concentration in the cytoplasm—the Damkohler number $D{a}_M$ based on the maximum reaction rate ($D{a}_M=\frac{R_{max}{l}^2}{C_{S0}{D}_S}$, here $C_{S0}$ is the concentration of the substrate in the external stream, $R_{max}$ is the maximum rate, and $D_S$ is the cytoplasmic diffusion coefficient of the substrate, and $l$ is the length of the cell, see model) and the Sherwood number $Sh$ for transport of substrate across the membrane. Figure 4a shows the concentration profile of the substrate for different values of the Damkohler number and Sherwood number. For $Sh=10$, the substrate concentration is spatially dependent because of fast mass transfer across the membrane compared to cytoplasmic diffusion. The parameter group $Sh/D{a}_M$ determines the nature of the spatial gradient here. For $\frac{Sh}{D{a}_M}>1$, where mass transfer dominates reaction, the substrate concentration is insensitive to reaction and little product is formed (Figs. 4a and 4b). For low $Sh/D{a}_M$, the product concentration and hence the spatial gradient is larger.

Figure 4.

Effect of an external substrate gradient on cytoplasmic enzyme substrate reaction. (a) Cytoplasmic concentration of the substrate for different values of $\frac{Sh}{D{a}_M}$; $Sh=10$. The reaction affects substrate concentration only when $\frac{Sh}{D{a}_M}\le 1$ (compare with Fig. 1b). (b) Cytoplasmic concentrations of the product corresponding to the curves in (a).

Discussion

The creation of intracellular gradients by applying an extracellular step gradient in ligand concentration using a microfluidic device has been demonstrated in the past.20,21 This method has been used to label intracellular organelles, detach selected portions of the cell, and cause spatially localized receptor internalization. Existence of cytoplasmic ligand gradients has been primarily inferred from selective binding of fluorescent ligand molecules to membrane-embedded receptors (e.g. in mitochondria,20). However, due to low fluorescence of diffuse ligand in the cytoplasm compared to bound organelles, cytoplasmic concentration gradients of free ligand need to be inferred from observations of spatially local phenomena, such as binding or cell detachment.

To address this, Takayama et al. analyzed a mathematical model in which concentration of the small molecule was fixed at an intracellular plane of a one-dimensional cell, and solved a spatially-dependent partial differential equation for cytoplasmic concentration.21 This model accounted for mass transfer across the membrane and cytoplasmic diffusion, but it made the unphysical assumption that there will be a constant concentration in the treated portion of the cell. We did not make this assumption but rather modeled the entire cell (i.e. the portion in the treated stream as well). A novel aspect of this work is that we identified the Sherwood number as the pertinent parameter which governs the existence of intracellular ligand gradients (refer to Tables 1 and 2 for the value of the parameters and the definition of the non-dimensional groups, and to Tables 3 and 4 for the symbols used).

Table 3.

Symbols used.

Variables	Symbol used
Concentration of the ligand	$C_L$
Concentration of the receptor	$C_R$
Concentration of the bound receptor/enzyme-substrate reactivity product	$C_P$
Concentration of the substrate	$C_S$
Time	$t$
Space	$x$

Table 4.

Symbols used for dimensionless variables.

Dimensionless variables	Symbol used
Concentration of the ligand	${\mathit{\theta }}_L$
Concentration of the receptor	${\mathit{\theta }}_R$
Concentration of the bound receptor/enzyme-substrate reactivity product	${\mathit{\theta }}_P$
Concentration of the substrate	${\mathit{\theta }}_S$
Time	$\mathit{\tau }$
Space	$\mathit{\xi }$

The model by Takayama et al. did not account for reaction in the cytoplasm. Yet, the purpose of creating ligand gradients is to cause local binding in the cytoplasm. Such reactions can (for example) be used to disrupt cytoskeletal structures locally (for example, Refs. 4,21) In this paper, we computationally investigated the development of gradients in receptor concentrations in response to extracellular step gradients in a microfluidic device. Our results demonstrate that steady-state receptor binding gradients can be established with the microfluidic technique, provided $Sh>1$, and $Da>1$. We showed that the Damkohler number (both based on binding rate and dissociation rate), is a second parameter that affects gradients in receptor concentration. We also show that it is possible to create steady-state gradients in enzyme-catalyzed reactivity and product concentration. The values of $Sh$, $D{a}_{on}$, $D{a}_{off}$ or $D{a}_M$ under which gradients develop are reasonable for biological applications (see Tables 1 and 2) based on measured parameters (Table 1). We hope that this work will provide new impetus to the use of microfluidic devices for sub-cellular spatial patterning and the spatial probing of sub-cellular processes.

Electronic supplementary material

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