Dual Gradient Chemotaxis Chamber System

Dual Gradient Chemotaxis Chamber System

Thesis by Rodolfo Amezcua

In partial fulfillment of the Requirements for the University Honors Program Certificate

California State University, Long Beach

December 2016

Acknowledgements

I would like to thank Dr. David Stout for his exceptional guidance throughout the devel-

opment of this thesis, and my undergraduate career. Without his mentorship, I would not

have been exposed to the wonders of research. Furthermore, I would like to recognize all the

programs that have provided funding for this research: HSI-STEM, the Office of Research

and Supported Programs, the Louis Stokes Alliance for Minority Participation program, and

the University/Engineering Honors Program.

Abstract

Neutrophils are a vital part of the innate immune response due to their capability of migrat-

ing towards and eradicating pathogenic bacteria. However under septic conditions, it has

been observed that neutrophil expression of chemotactic receptors is down-regulated thereby

impairing usual migratory behavior. Beyond this knowledge, there is a lack of understand-

ing as to how septic conditions impact neutrophil migration. To address this problem we

previously proposed a planar gradient diffusion system coupled with digital volume corre-

lation capable of inducing neutrophil chemotaxis and quantifying the forces generated by

the cell during chemotaxis. Our current work uses this system but now introduces a dual

chemoattractant gradient. A dual gradient will be insightful as it has been observed that

neutrophils exhibit intracellular signaling hierarchy in competing gradients. Using confo-

cal time-lapse microscopy and custom image processing techniques we have experimentally

found the diffusion coefficient and characterized the gradient of rhodamine diffusing through

a 3D collagen gel. We have also created a mathematical model of the dual diffusion process

and observed that a long-lasting gradient, a necessary condition for chemotaxis, is formed

once steady-state is reached.

Contents

1 Introduction 1

1.1 Role of cell migration in sepsis . . . . 1

1.2 Chemotaxis assays . . . . 2

1.3 Competing chemoattractant gradients . . . . 4

2 Development of a Matlab Script for image analysis 6 2.1 Previously collected data . . . . 6

2.2 Image processing . . . . 6

2.3 Extraction of intensity values . . . . 7

2.4 Determining the diffusion coefficient . . . . 8

2.5 Simulation of a dual gradient . . . . 9

3 Application of the diffusion equation for gradient characterization 10 3.1 The governing equation . . . . 10

3.2 Solution of the diffusion equation . . . . 11

4 Results 14 4.1 Diffusion Coefficients . . . . 14

4.2 Intensity Values . . . . 15

4.3 Linear Gradient . . . . 15

4.4 Dual Gradient . . . . 17

5 Conclusion 19 5.1 Summary . . . . 19

5.2 Future work . . . . 20

Chapter 1

Introduction

1.1 Role of cell migration in sepsis

Sepsis is defined as a systemic inflammatory response to infection, and is the cause of more than 250,000 annual deaths in America [1]. An even more alarming statistic is the 50%

mortality rate under severe conditions [2]. The reason for such figures is due to the difficulty in diagnosing sepsis within an adequate time period because any infection has the potential to cause sepsis, and the symptoms that characterize it are also common with several other medical conditions. Once clinicians verify the patient’s septic state, they must conduct a trial-and-error process with a variety of anti-bacterial drugs to determine the appropriate medication for that patient. Meanwhile, because there is no immediate cure the body’s immune system recruits excessive white-blood cells in an attempt to combat the infection.

However, literature has shown that neutrophil behavior is altered during septic conditions,

and ultimately contribute to organ failure. Specifically, neutrophil chemotaxis is impaired

thus hindering their proper migratory behavior. Instead of migrating towards the pathogenic

bacteria, they damage neighboring tissue resulting in organ failure, a common feature of

sepsis.

White blood cells play a key role in the initial response to infection, particularly the neutrophil. Neutrophils are a vital part of the innate immune response due to their capability of migrating towards and eradicating pathogenic bacteria. To do so, it releases extracellular traps (NETs) [3]—the release of neutrophil DNA into a net consisting of toxic molecules [4]—

or by phagocytosis—engulfing the bacteria, reducing the available oxygen, and then using the toxins produced to kill the bacteria [5]. The procedure in which neutrophils undergo to reach the site of infection is commonly characterized into four phases: mobilization, tethering and rolling, transmigration, and mechanotactic and chemotactic guidance [4, 6]. However, under septic conditions, it has been observed that neutrophil expression of chemotactic receptors is down-regulated thereby impairing usual chemotactic behavior [7]. This impaired behavior is illustrated in Figure 1.1. Beyond this knowledge, there is a lack of understanding as to how septic conditions impact neutrophil migration after transmigration.

1.2 Chemotaxis assays

To advance fundamental understanding of neutrophil migration, particularly chemotaxis, re- searchers have resorted to creating systems that replicate in vivo chemical gradients. Chemo- taxis studies began five decades ago, and have progressively increased in reliability and re- producibility. In recent years, literature reports microfluidic systems capable of producing various gradient profiles such as linear, root, and power [8]. While microfluidics offer a facile method of creating these complex gradients, it comes at various costs: the fluid flow within the system exposes cells to shear forces thus creating data noise, and also washes away se- cretion factors that may reveal pertinent information; cell migration is confined to either 1D or 2D in a polyacrylamide gel variant while the physiologically relevant scenario is in 3D and a biomaterial such as collagen or fibrin.

To improve upon these drawbacks, free diffusion systems have been designed. These

Figure 1.1: Illustration of how neutrophils migrate to eradicate pathogenic bacteria. A shows

migration within a healthy patient. 1) Mobilization 2) tethering 3) rolling 4) transmigration

through the endothelial layer 5) chemotactic guidance by chemokine production. B shows

altered neutrophil chemotactic behavior under septic conditions. 5a) and 5b) illustrate how

not all neutrophils successfuly migrate towards the pathogenic bacteria, but instead damage

organs, which can then lead to organ failure.

Figure 1.2: (1) The system consists of a bottom housing, a hydrophilic 22mm cover slip, and PDMS mold. (2) Dimensions of PDMS mold. (3) Mold with collagen gel with hydrophobic cover slips on each side creating a 3 well system. Before imaging the diffusion process, the hydrophobic cover slips are removed. Next, the first well is filled with 100 µl of Leibovitz’s L- 15 with 2 mg/ml glucose. Afterwards, the third well is filled with 100 µl of 5 µM rhodamine diluted with deionized water. Rhodamine was selected since it allows for fluorescence imag- ing and has similar molecular weight as Formyl-Methionyl-Leucyl-Phenylalanine (fMPL), a chemoattractant released by foreign bacteria.

designs typically use a three-well approach: reservoir, biomaterial, and sink. By keeping the reservoir and sink at constant chemical concentrations, a linear gradient across the biomaterial will be generated under steady-state conditions [9]. This design is simple, allows for 3D migration, and use of biomaterials as shown in Figure 1.2.

1.3 Competing chemoattractant gradients

Our current work uses the free diffusion system previously describe, but now introduces a

competing dual chemoattractant gradient. A dual gradient will be insightful as it has been

observed that neutrophils exhibit intracellular signaling hierarchy in competing gradients

[10]. Using confocal time-lapse microscopy and custom image processing techniques we have

experimentally found the diffusion coefficient and characterized the gradient of rhodamine

diffusing through a 3D collagen gel. We have also created a mathematical model of the

dual diffusion process and confirmed that a long-lasting gradient, a necessary condition for

chemotaxis, is formed once steady-state is reached. These results will serve for future in

vitro experiments for inducing and quantifying neutrophil chemotaxis. This data can then

be used to study how signaling pathways might differ in healthy and septic neutrophils.

Chapter 2

Development of a Matlab Script for image analysis

2.1 Previously collected data

Time-lapse confocal microscopy images at three second intervals of a tracer dye named rho- damine diffusing through a 3D collagen gel were previously acquired. A field of view of 512 x 512 pixels, and the chemotaxis system shown Figure 1.2 were used. Rhodamine was selected since it allows for fluorescence imaging and has similar molecular weight as Formyl- Methionyl-Leucyl-Phenylalanine (fMPL), a chemoattractant released by foreign bacteria that induces neutrophil chemotaxis. Image stacks for several collagen concentrations were cap- tured.

2.2 Image processing

To prepare the image stacks, image analysis must be done to remove noise from the data.

To do so the data was loaded onto Matlab and converted into gray scale images. An average

(a) (b)

Figure 2.1: Comparison of (a) an unfiltered grayscale image and (b) a processed image after applying an average filter and normalizing by maximum intensity. Collagen concentration density is 2.2 mg/ml.

filter was then applied to each image within the stack and normalized by maximum intensity as shown in Figure 2.1. Because it is difficult to begin confocal imaging at the exact time that the diffusion process starts, the Matlab program loads the first image within the stack, and asks the user to choose six points that defines the boundary of the tracer dye. Using the locations of the user-defined points, a cubic spline is fit to these points as shown in Figure 2.2. To confirm an accurate selection, the Matlab program displays the area of under the spline, and asks the user whether or not to continue.

2.3 Extraction of intensity values

In order determine the diffusivity of the tracer dye within the collagen gel, the concentration

at specific points must be observed over time. The Matlab program asks the user to select the

range along the diffusion direction to create ten equally spaced points as shown in Figure

(a) (b)

Figure 2.2: (a) The first picture of the image stack of 1.0 mg/ml collagen concentration. The edge of the chemoattractant and ten point locations for analysis are shown. (b) The last picture of the image stack showing how the chemoattractant has diffused through the gel.

2.2 (a). At each of these points, the concentration over time is extracted and smoothed according to a robust version of the loess method, a local regression method using weighted linear least squared and a second degree polynomial model, that assigns lower weight to outliers in the regression. The method assigns zero weight to data outside six mean absolute deviations.

2.4 Determining the diffusion coefficient

To determine the diffusion coefficient of rhodamine in the 3D collagen gel, physical param- eters must be specified: initial concentration, the time interval between each image, total time elapsed, pixel to micrometer conversion factor, length of the entire sample, and inten- sity values at ten specified points over time. The first step is to provide an initial guess for the value of the diffusivity of rhodamine; this can be accomplished via a literature search.

This guess is provided to a Matlab optimization function that finds the minimum of an

unconstrained function using a derivative-free method. In this case, it finds the diffusion

coefficient that minimizes residual sum of squares of the error between concentration values extracted from the image stacks and theoretical concentration values using the approximated diffusion coefficient. The output is a diffusion coefficient for each of the ten selected points.

The median of this set was chosen as the final value for rhodamine given a specific collagen concentration density.

2.5 Simulation of a dual gradient

The experimental data only captures a single chemoattractant diffusing through a collagen

gel. However, a dual gradient will be insightful as it has been observed that neutrophils

exhibit intracellular signaling hierarchy in competing gradients. Using the experimentally

found diffusion coefficient of rhodamine and multiplying this value by a factor of four to mimic

a second chemoattractant, a simulation characterizing the concentration of a dual gradient

was executed. To find the theoretical concentration along the sample, the solution of the one-

dimensional diffusion equation was used. Treating the system as a one dimensional problem

is valid because confocal images at the top-plane, mid-plane, and bottom-plane along the

vertical thickness of the gel sample returned intensity values of negligible difference.

Chapter 3

Application of the diffusion equation for gradient characterization

3.1 The governing equation

The system can be modeled by the diffusion equation with prescribed boundary (BC) and initial conditions (IC); the problem is presented in Equation 3.1. Since the BCs are inhomo- geneous, modification—subtraction of any known function that satisfies the BCs—is needed before solving by separation of variables. The coefficients are then determined by Fourier series using the IC; the solution is provided in Equation 3.2.

u _t = ku _xx (0 < x < L, 0 < t < ∞) u(0, t) = A, u(L, t) = B

u(x, 0) = φ(x)

(3.1)

u(x, t) = [A + x

L (B − A)] − 2 π

∞

X

n=1

1

n [A − B cos(nπ)] sin( nπx

L )e ⁻ⁿ

^π

^kt/L

(3.2)

3.2 Solution of the diffusion equation

There are multiple ways to solve the problem illustrated in Equation 3.1. The method described here will use separation of variables. This method consists of building the general solution as a linear combination of ones that are easy to find. Before doing this, we must find a function w(x, t) such that

w(0, t) = A, w(L, t) = B

so that the difference v = u − w solves



 

 

 



 

 

 



v _t = kv _xx − w _t + kw _xx (0 < x < L, 0 < t < ∞) v(0, t) = 0 v(L, t) = 0

v(x, 0) = φ(x) − w(x, 0)

where u is the solution to the problem posed in Equation 3.1. This way, the problem becomes a partial differential with homogeneous boundary conditions that can be solved by separation of variables. To solve for w(x) we first note that

−w _t + kw _xx = 0, w(0, t) = A, u(L, t) = B

and that the boundary conditions are not dependent on time. Thus we can simply solve the ordinary differential equation

w ⁰⁰ (x) = 0, w(0) = A, w(L) = B

and arrive at

w(x) = A + x

L (B − A) (3.3)

Now we solve or v(x, t) by means of separation of variables. One solution of the differential equation with homogeneous boundary conditions can be written as

v(x, t) = X(x)T (t)

by plugging it into Equation 3.1, we get

X(x)T ⁰ (t) = kX ⁰⁰ (x)T (t) if we separate the variables we arrive at

T ⁰

kT = X ⁰⁰

X = −λ

where λ is a constant because it does not depend on x, as shown by the first expression, and it does not depend on t, as shown in the second expression. Thus it must be a constant. The reason for the negative is simply for convenience in later calculations. We can now form two ordinary differential equations.

X ⁰⁰ + λX = 0 and T ⁰ + kλT = 0 It can readily be seen that the solution of these two ODEs are

X(x) = C ₁ cos √

λx + C ₂ sin √ λx

T (t) = C ₃ e ^−λkt

where C ₁ , C ₂ , and C ₃ are constants. By imposing the boundary conditions X(0) = 0 = X(l)

we see that

X(0) = 0 = C ₁ , X(l) = 0 = C ₂ sin √ λl

To avoid the trivial solution where C ₂ = 0, the argument of sin must be nπ, thus λ =  nπ l

 2

and

X _n (x) = sin nπx

l , (n = 1, 2, 3, ...) Thus the general solution is

v(x, t) =

∞

X

n=1

A n e ^−(nπ/l)

^kt sin nπx

l (3.4)

provided that the initial condition satisfies

φ(x) =

∞

X

n=1

A _n sin nπx

l (3.5)

To solve for A _n , the Fourier sine series will be used.

A _n = 2 l

Z l 0

φ(x) sin nπx l dx

A n = − 2 l

Z l 0

h A + x

L (B − A) i

sin nπx l dx

Solving for A n and noting that u = v + w, we arrive at the solution shown in Equation 3.2.

Chapter 4

Results

4.1 Diffusion Coefficients

Using the Matlab script previously described, the diffusion coefficient of rhodamine in various collagen gel densities was found and are presented in Table 4.1. At a glance, these results are logically sound because as the gel concentration increases, the diffusion coefficient decreases.

Only the 1.5 mg/ml concentration showed otherwise, which could potentially be due to the difficulty in acquiring time-lapse images while simultaneously beginning the diffusion process.

The median coefficient from the ten user-defined points shown in Figure 2.2 was used as the final value. The standard deviation was found by running the script multiple times for each collagen density.

Collagen Gel Concentration Diffusion Coefficient( m ² /s) 2.5 (mg/mL) 2.1433E-11 ± 4.3041E-12 2.2 (mg/mL) 7.6398E-11 ± 2.1227E-11 2.0 (mg/mL) 5.3875E-10 ± 1.3360E-10 1.5 (mg/mL) 7.0903E-11 ± 6.0252E-12 1.0 (mg/mL) 7.9766E-10 ± 3.5204E-11

Table 4.1: Diffusion coefficients of rhodamine in various collagen gel densities.

4.2 Intensity Values

To determine if our diffusion system truly follows the mathematical model previously de- scribed, the normalized intensity values at a given user-defined point were compared with its theoretical value. As seen in Figure 4.1(a) the true intensity values (before smoothing) follow the same curve as the theoretical values, but reach a steady-state value of about 0.3 less than that of the theoretical value. In Figure 4.1(b), the smoothed values are compared to the theoretical values, and show an error of 0.2 at 2 hours, which is sustained for an additional hour.

A similar saturation curve behavior for the intensity values of the remaining user-defined points is also observed, which can be seen in Figure 4.2. The experimental concentration profile agrees with the theoretical concentration profile, and holds a similar margin of error as previously discussed in Figure 4.1. It is also interesting to note that, theoretically, the user-defined point closest to the source of diffusion has a faster rise time than that of the remainder points. Although this is not the case with the experimental graphs, it may actually serve to our benefit because this way we can denote a single settling time for all points.

4.3 Linear Gradient

The goal of our diffusion system is to create a linear concentration profile across the collagen gel once steady-state is reached. The Matlab script confirms this accomplishment as shown in Figure 4.3. It is important to note that each blue circle represents one user-defined point. Both the experimental and theoretical linear gradients along with their corresponding coefficient of determination are shown. As expected the theoretical R ² value is nearly one, but the experimental R ² value confirms our system is capable of creating a linear gradient.

The normalized concentration values at each of the ten points were determined by taking

(a) (b)

Figure 4.1: (a) Scatter plot showing the true vs theoretical intensity values of the 1.0 mg/ml image stack through the third selected point. (b) Scatter plot comparing the smoothed vs theoretical intensity values also for 1.0 mg/ml.

(a) (b)

Figure 4.2: Graphs showing the diffusion profile for 1.0 mg/ml collagen concentration at

each selected point over 2.8 hours. Each line represents a different point.

(a) (b)

Figure 4.3: Graphs showing the linear concentration profile across the 1.0 mg/ml sample after 2.8 hours. Each blue circle represents one of the ten selected points.

the mean intensity values at each point for the last ten frames on the image stack.

4.4 Dual Gradient

Exposing seeded cells in the collagen gel to a dual gradient will more closely simulate in vivo conditions. However, in order to determine which chemoattractant is affecting a given cell at a certain time point, the concentration of each chemoattractant must be known.

Due to the simple design of our chemotaxis system, the process for characterizing a second

chemoattractant is the same as the process illustrated for rhodamine. A key feature of

the Matlab script is its capability of determining the time point and location that both

chemoattractants meet (Figure 4.4). This is of special interest because cell behavior can

then be more closely monitored from this time point onward.

(a)

(b)

Figure 4.4: The diffusion coefficient was taken to be 4 times that of rhodamine for the second

chemical in this figure. (a) 3D rendition of dual diffusion at the time when both chemoattrac-

tants meet. A 2.2 mg/ml collagen concentration was used to illustrate how the Matlab code

can accommodate non-linear initial edges as compared to Figure. Color bar shows normal-

ized concentration. (b) Graph showing the concentration of both chemoattractants across

the gel. The time point in which the normalized value of 0.1 from both chemoattractants

meet was selected.

Chapter 5

Conclusion

5.1 Summary

Sepsis has a large impact throughout the world, but especially in developing countries where intensive care units to treat patients are scarcely available. In addition, clinicians are con- stantly facing a limited time window to diagnose and treat sepsis. To address the world’s need for adequate medical technology against this condition, the proposed chemotaxis sys- tem is an innovative approach to advancing fundamental knowledge of neutrophil migration.

Once established, this knowledge will be directly applicable for the development of enhanced drug treatment for sepsis. Additionally, the chemotaxis system can also accommodate a variety of cell types, therefore making it accessible to a multitude of biomedical research groups. Cancer cells, neurons, macrophages, and fibroblasts could all be investigated using the proposed system thus this system will have far reaching medical impacts.

The work presented here is a mathematical model that represents the diffusion process of

a chemoattractant, and its implementation in a Matlab script. The output of this script is the

diffusion coefficient of the chemoattractant under analysis. It is also capable of characterizing

a competing dual chemoattractant gradient.

5.2 Future work

There are many potential avenues for future research. The next step would be to choose a tracer dye representing another vital chemoattractant in neutrophil migration, such as interleukin-8 (IL-8). IL-8 is a chemoattractant released by endothelial cells and macrophages to recruit neutrophils. Although studies exposing neutrophils to fMLP and IL-8 already exist, none report quantitative measures of its migration. A common approach to determining these quantitative measures is by traction force microscopy. However, this is only applicable when the material properties of the gel are known. Collagen does not exhibit well behaved material properties, thus a different approach would be required.

One such approach embeds fluorescent microspheres within the collagen gel. As the cell

migrates, the microspheres also move, thus creating a displacement field. Using a digital

volume correlation algorithm, this displacement field can be determined [11]. Researchers

have then used this displacement field to characterize the kinematics behind cellular migra-

tion, therefore circumventing the need to use traction force microscopy to describe migratory

behavior [12]. Applying this approach to determine the contractility, volume change, and ro-

tation of a cell exposed to a dual chemoattractant gradient could reveal pertinent information

on how to combat sepsis.

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