Chemotaxis of Escherichia Coli to Controlled Gradients of Attractants(1)

Chemotaxis of Escherichia coli to controlled

gradients of attractants: Experiments and

Mathematical modeling

Submitted in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

in

CHEMICAL ENGINEERING

by

VUPPULA RAJITHA

(Roll No. 04402602)

Supervisor: Prof. Mahesh S. Tirumkudulu

Co-Supervisor: Prof. K. V. Venkatesh

Department of Chemical Engineering

Indian Institute of Technology Bombay

2009

Declaration

I declare that this written submission represents my ideas in my own words

and where others ideas or words have been included, I have adequately cited and

referenced the original sources. I also declare that I have adhered to all principles

of academic honesty and integrity and have not misrepresented or fabricated or

falsified any idea/data/fact/source in my submission. I understand that any vio-

lation of the above will be cause for disciplinary action by the Institute and can

also evoke penal action from the sources which have thus not been properly cited

or from whom proper permission has not been taken when needed.

Vuppula Rajitha

(Roll No. 04402602)

i

Abstract

Chemotaxis is a phenomenon in which a microorganism or multi-cellular organisms

are able to direct their movements in the presence of certain chemicals in their envi-

ronment. This could be either towards favorable chemicals called chemoattractants

or away from unfavorable chemicals called chemorepellents. The understanding

of this phenomena is important in many biological processes including immune

response, embryo-genesis, wound healing, bio-film formation and bio-remediation

of subsurface contaminants. The objective of this thesis is to study the response

of microorganisms in the presence of controlled gradients of chemoattractants us-

ing a micro-capillary. This would involve developing mathematical models that

describe the chemotaxis pathway in a chemotactic cell and to relate the kinetics

to the motion of cell in the presence of chemoattractants. The predictions of the

model will be compared with experiments where the motion of cell in the presence

of chemoattractants will be tracked. The previous studies have reported average

drift velocities for a given gradient and do not measure drift velocities as a func-

tion of time and space. To address this issue, a novel experimental technique was

developed to quantify the motion of E. coli cells to varying concentrations and

gradients of chemoattractant so as to capture the spatial and temporal variation of

the drift velocity. The statistics of single cell motions such as cell velocity, tumbling

frequency, rotational diffusion, drift velocity and translational diffusivity are ob-

tained independently. An existing two state receptor model of Barkai and Leibler

(1997) was used for the intracellular pathway and into this the extra-cellular in-

fluence such as ligand concentration and Brownian motion was incorporated to

predict the response for the experimental conditions. The model predicted the ex-

ii

perimentally observed increase in drift velocity with gradient and the subsequent

adaptation. Our study revealed that the rotational diffusivity induced by the ex-

tracellular environment is crucial in determining the drift velocity of E. coli. The

model predictions matched with experimental observations only when the response

of the intracellular pathway was highly ultra-sensitive to overcome the extracel-

lular randomness. The parametric sensitivity of the pathway indicated that the

dissociation constant for the binding of the ligand and the rate constants of the

methylation/demethylation of the receptor are key to predict the performance of

the chemotactic behavior. The study demonstrates the key role of oxygen in the

chemotaxis response and that the response to a ligand cannot be analyzed in isola-

tion to effects of oxygen. Further, the study noted that the chemotactic response

also depends on the nature of the chemical used as a chemoattractant which in

turn decides the energy status of the cell.

iii

Contents

Abstract ii

List of Figures xii

List of Tables xiii

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Materials and Methods 19

2.1 Experimental Protocols . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Modeling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Mathematical modeling and Experimental validation of chemo-

taxis under controlled gradients of methyl-aspartate in Escherichia

coli 36

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4 Chemotaxis of Escherichia coli to L-serine 56

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

iv

4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Chemotaxis of Escherichia coli in the presence of glucose via phos-

photransferase (PTS) pathway 76

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Conclusions and Future study 88

References 96

v

List of Figures

1.1 Schematic diagram of E. coli with multiple flagella. It is typically

rod-shaped and is about 2 m long and 1 m in diameter. (Eisen-

bach, 1994) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Swimming behavior of E. coli or Salmonella in the absence of

chemotactant. It executes a random walk composed of runs and

tumbles with zero drift. . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Swimming behavior of E. coli or Salmonella in the presence of

chemotactant. It executes a biased random walk composed of runs

and tumbles with significant drift. . . . . . . . . . . . . . . . . . . . 5

1.4 Experimental data using fluorescence correlation spectroscopy: Re-

sponse of individual motors as a function of CheY-P concentration.

(Cluzel et al., 2000) Each data point describes a simultaneous mea-

surement of the motor bias and the CheY-P concentration in an

individual bacterium. CW bias was computed by analyzing video

recordings for at least 1 min. . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Chemotaxis signaling pathways in E. coli (A) in the absence of

chemoattractant, and (B) in the presence of chemoattractant. The

dotted line represents the reactions with reduced rate in response to

ligand binding to the receptor. It can be noted that on adaptation

the pathway returns to steady state as shown in (A). . . . . . . . . 7

2.1 Linear calibration curve: Fluorescence intensity with varying con-

centration of 2-NBDG. . . . . . . . . . . . . . . . . . . . . . . . . . 22

vi

2.2 Micro-capillary experimental setup for establishing glucose gradi-

ents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Measured fluorescence intensity in the capillary for 10 min (solid

line) and 540 min (dashed line) after the start of the experiment.

It can be noted that the gradient was stable for more than 9 hours. 23

2.4 (A) Agar plate containing 0.3 % agar and motility buffer (absence

of glucose) with 107 cells introduced at the center of the plate.The image was taken after 6 hours of incubation. It can be noted

that no ring was formed. (B) Agar plate containing 0.3 % agar,

motility buffer and glucose (5 mM), with 107 cells introducedat the center of the plate. The image was taken after 6 hours of

incubation. Clearly, the cells have migrated forming a distinct ring. 23

2.5 Micro-capillary experimental setup for establishing MeAsp/serine

gradients. x = 0 is located at the end of the pellet with a ligand

concentration, L0. The experimental values of L0 and the estab-

lished gradients (G) are given in Table 2.1 and these conditions

were used in the model simulations. . . . . . . . . . . . . . . . . . . 26

2.6 (A) A magnified view of the micro-capillary set-up (B) Stable linear

gradients; Measured concentration profiles for two gradients: t =

1.5 min (black line) and t = 5 min (blue line) forG = 0.016 M/m,

t = 1.5 min (red line), t = 5 min (green line) and t =30 min (pink

line) for G = 0.16 M/m respectively. . . . . . . . . . . . . . . . . 27

3.1 Mean square displacements (MSD standard error) about themean value (from Gaussian fit) as a function of time. MSD was

obtained using data from all experiments including those with and

without gradients. The solid line represents the linearity obtained

for a diffusive regime. The diffusive regime occurs beyond 3.8 sand is shown by a dotted arrow. . . . . . . . . . . . . . . . . . . . 39

vii

3.2 Measured drift velocity (u0 standard error) as a function of dis-tance in the absence of aspartate using normal buffer (), 100 Maspartate in normal buffer (N), and 10000 M aspartate in normal

buffer (). The solid line represents the average trend for the three

experiments. The empty circle () represents the drift velocity whenthe buffer is saturated with oxygen in the absence of aspartate. . . . 40

3.3 Average run speeds ( standard error) as a function of aspartategradients at: 500 m (+), 1000 m (), and 1500 m (). Thesolid line represents the average value for the entire data set. The

arrow indicates the average run speed of 18 m/s obtained in the

absence of MeAsp. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Drift velocity (uasp standard error) as a function of distance forvarious gradients of aspartate (G) and initial ligand concentration

(L0) (A) G = 0.016 M/m and L0 = 16 M; (B) G = 0.08 M/m

and L0 = 80 M; (C) G = 0.16 M/m and L0 = 160 M; (D)

G = 1.6 M/m and L0 = 1600 M. represents data from ex-periments and solid line represents model prediction. . . . . . . . . 42

3.5 (A) Drift velocity obtained through simulation as a function of as-

partate gradients at four different spatial locations: dotted line in

gray at 100 m, dotted line in black at 500 m, dashed line at

1500 m and solid line at 3000 m, respectively (B) Normalized

CW bias with time for four different gradients: solid line for G =

0.016 M/m and L0 = 16 M, dashed line for G = 0.08 M/m

and L0 = 80 M, dotted line in black for G = 0.16 M/m and L0

= 160 M and dotted line in gray for G = 1.6 M/m and L0 =

1600 M (C) Normalized active receptor concentration with time

for four different gradients: solid line for G = 0.016 M/m and

L0 = 16 M, dashed line for G = 0.08 M/m and L0 = 80 M,

dotted line in black for G = 0.16 M/m and L0 = 160 M and

dotted line in gray for G = 1.6 M/m and L0 = 1600 M. . . . . . 44

viii

3.6 (A) Drift velocity obtained through simulation as a function of lig-

and dissociation constant for a gradient of 0.06 M/m at four

different spatial locations: dotted line in gray at 100 m, dotted

line in black at 500 m, dashed line at 1500 m and solid line

at 3000 m, respectively (B) Drift velocity as a function of ratio

of methylation and demethylation rate constants for a gradient of

0.06 M/m at four different spatial locations: dotted line in gray

at 100 m, dotted line in black at 500 m, dashed line at 1500 m

and solid line at 3000 m, respectively (C) Drift velocity as a func-

tion of gradient with varying Hill coefficients (n) at x = 500 m

(D) Variation of Hill coefficient (n) with rotational diffusivity (Dr)

to obtain a drift velocity of 1 m/s at x = 500 m, for a gradient

of 0.06 M/m. The arrow indicates the experimentally observed

rotational diffusivity, Dr = 0.32 rad2/s, corresponding to n = 50. . . 46

3.7 (A) Measured drift velocity ( standard error) as a function of thespatial gradient of the logarithmic aspartate concentration: G =

0.016 M/m (), G = 0.08 M/m (), G = 0.16 M/m (N)and G = 1.6 M/m () (B) Average value of the drift velocity

( standard error) at each d(lnL)/dx with location. The solid linerepresents a linear curve fit while the dashed line represents the

trend observed by (Kalinin et al., 2009) . . . . . . . . . . . . . . . . 49

ix

4.1 (A) Mean square displacements about the mean value as a function

of time for varying serine concentrations: 0 M (), 250 M (),1000 M (N), 5000 M (), 24000 M () and 240,000 M() (B) The translational diffusivity for various serine concentra-tions. The values of the diffusivities were obtained from rotational

diffusivity and average run speed () and compared with those ob-tained using MSD (). (C) Average run speeds ( standard error)as a function of distance in the absence of serine (), and in thepresence of serine, 5000 M () when the buffer is saturated withair. Average run speeds in the absence of serine (), and in thepresence of serine, 5000 M (N) when the buffer is saturated with

oxygen. (D) Average run speeds ( standard error) as a functionof serine concentration at 500 m. N indicates the average run

speed of 18.65 0.62 m/s obtained in the absence of serine whensaturated with air. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2 (A) Rotational diffusivity as a function of serine concentration at

500 m. N indicates the rotational diffusivity 0.380.05 rad2/sobtained in the absence of serine when saturated with air. (B)

Measured drift velocity ( standard error) as a function of distancein the absence of serine (), and in the presence of serine, 5000 M() when the buffer is saturated with air. Measured drift velocityin the absence of serine (), and in the presence of serine, 5000 M(N) when the buffer is saturated with oxygen (C) Measured drift

velocity ( standard error) as a function of distance for varyingserine concentration when saturated with air: 0 M (), 50 M(), 250 M () and 5000 M (). . . . . . . . . . . . . . . . . 64

x

4.3 The measured drift velocity ( standard error) as a function ofdistance for various gradients of serine (G) and initial ligand con-

centration (L0); (A) G = 0.0016 M/m and L0 = 1.6 M () (B)G = 0.016 M/m and L0 = 16 M () (C) G = 0.16 M/mand L0 = 160 M () (D) G = 1.6 M/m and L0 = 1600

M () (E) G = 16 M/m and L0 = 16000 M ( N) and (F)

G = 160 M/m and L0 = 160000 M (). . . . . . . . . . . . . . 67

4.4 (A) Drift velocity obtained through experiments and simulation as a

function of serine gradients at three different spatial locations: solid

line and at 500 m, dashed line and at 1000 m, dotted lineand at 1500 m respectively. (B) Drift velocity obtained throughexperiments and modified predicted values by addition of an average

drift velocity due to oxygen gradient and serine concentration as a

function of serine gradients at three different spatial locations: solid

line and at 500 m, dashed line and at 1000 m, dottedline and at 1500 m respectively. . . . . . . . . . . . . . . . . . . 68

4.5 (A) Drift velocity obtained through experiments as a function of

serine gradients at 500 m: solid line with represent our experi-mental data while the dotted line with represents that observedby Berg and Turner (1990). (B) Measured drift velocity ( stan-dard error) as a function of the spatial gradient of the logarithmic

serine concentration: G = 0.0016 M/m (), G = 0.016 M/m(), G = 0.16 M/m (), G = 1.6 M/m (), G = 16M/m () and G = 160 M/m (N) (C) Average value of thedrift velocity ( standard error) at each d(lnL)/dx with location.The solid line represents a linear curve fit. . . . . . . . . . . . . . . 70

xi

5.1 (A) The number of E. coli cells as a function of displacement mea-

sured over a time period of 3.3 s for uniform glucose concentra-

tion of 28 M in the capillary: Experimental data:, Gaussianfit:solid line (B) Gaussian fits for zero gradient: solid line, and

0.05 M/m, ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 (A) Mean square displacement (variance from Gaussian fit) for all

experiments (B) Apparent diffusivity as a function of time for var-

ious glucose concentrations and gradients. . . . . . . . . . . . . . . 79

5.3 (A) Measured drift velocity for varying glucose gradients (B) Mea-

sured drift velocity in the absence of glucose. The data excludes the

contribution of non-chemotactic drift (C) Measured drift velocity

for varying glucose concentrations (D) Drift velocity as a function

of effective glucose gradients: for varying glucose concentrationexperiments in the absence of initial gradients (see Figure 5.3 (C)),

for initially established glucose gradients (see Figure 5.3 (A)),and .... for Hill equation obtained by fit. . . . . . . . . . . . . . . . 81

5.4 (A) Measured drift velocity as a function of the spatial gradient of

the logarithmic glucose concentration. The dotted line represents

the linear variation (see equation 5.2) with slope 1000 m2/s. (B)

Average run speeds as a function of effective glucose gradients:for varying glucose concentration experiments in the absence of ini-

tial gradients (Figure 5.3 (C)), for initially established glucosegradients (see Figure 5.3 (A)), for absence of glucose and ....for average value in the presence of glucose. . . . . . . . . . . . . . 83

xii

List of Tables

1.1 Mathematical models overview . . . . . . . . . . . . . . . . . . . . . 14

2.1 Gradients and ligand concentration at x = 0 obtained through ex-

periments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Comparison of chemotactic parameters in E. coli with those re-

ported in literature. (Berg and Brown, 1972; Liu and Papadopou-

los, 1996; Berg and Turner, 1990; Ahmed and Stocker, 2008; Kalinin

et al., 2009) These values were obtained by averaging ( standarddeviation) over all experiments. . . . . . . . . . . . . . . . . . . . . 38

3.2 Parameters used to simulate the model. . . . . . . . . . . . . . . . . 48

xiii

Chapter 1

Introduction

1.1 Background

Chemotaxis is a phenomenon in which a microorganism or multi-cellular organ-

isms are able to direct their movements in the presence of certain chemicals in

their environment. Cells capable of chemotaxis are bacteria, protozoa, amoeba,

cellular slime moulds, sperm and fibroblasts phagocytes. Chemotaxis is used by

organisms to find food by swimming towards the highest concentration of food

molecules, or to flee from poisons. Chemicals which are capable of eliciting such a

response from chemotactic cells are called Chemotactants. Chemotactants may act

as attractants, in which case the chemotactic cell will move towards them, thereby

exhibiting a positive chemotactic response. On the other hand, some chemicals act

as repellents, in which case the chemotactic cell will move away from them, thereby

exhibiting a negative chemotactic response. Examples of chemoattractants include

nutrients such as sugars, amino acids and small peptides, while chemorepellents

include antibiotics and noxious chemicals such as phenol.

Chemotaxis plays an indispensable role in our life from birth to death. During

embryonic process cells migrate to their target destinations to become components

of arms, legs, liver, heart, brain and other organs. In the brain, immature neurons

move from their birthplace to final targets where they make the right connections

and allow functions such as learning and memory. In the process of wound healing,

1

white blood cells migrate towards the site of infection to destroy the foreign organ-

ism. (Condliffe and Hawkins, 2000) In metastasis, cancer cells secrete their own

chemotactic stimulus to direct their migration towards lymphatic vessels using the

lymphatic system as a major route to spread throughout the body. (Carlos, 2001;

Wood et al., 2006; Shields et al., 2007) Cell migration is also important in bio-

film formation and bio-remediation of surface contaminants. (Duffy et al., 1997;

Pandey and Jain, 2002) In all these processes, how the cells sense the environment

and migrate to specific locations is not clear. Bacterial chemotaxis provides a

basic platform for learning how these processes work which in turn will help in

unraveling similar processes in more complex systems.

Escherichia coli is a model organism for the study of bacterial chemotaxis,

because its genetics is comparatively simple, is generally harmless, can be grown

quickly on a wide range of media and much of the biochemical pathway is known.

Thus, it has been used for experimental and theoretical studies of chemotaxis over

the last 40 years. Since the discovery of chemotaxis by Pfeffer and Englemann in

1880, many studies on bacterial chemotaxis have been reported in the literature.

However, the much needed impetus to the study of bacterial chemotaxis began in

the 1960s by Julius Adler who noted that bacteria used specific receptors to recog-

nize the chemicals. Since then, numerous studies have identified these receptors,

unraveled the workings of the receptors and the signaling pathway, recorded the

motion in detail, in the presence of various chemoattractants and, more recently,

tried to integrate this knowledge into detailed mathematical models to predict

the motion of E. coli in the presence of chemicals. Many of these studies will be

reviewed in the following pages of this chapter.

E. coli are extremely tiny creatures. They are rod-shaped, about 2 m long

and 1 m in diameter and are mostly motile. They can achieve motility via the

utilization of specialized structures known as flagella. These flagella allow the

bacteria to exhibit both positive and negative chemotactic responses. A single

bacterial cell may possess multiple flagella as shown in Figure 1.1. (Eisenbach,

1994)

2

Figure 1.1: Schematic diagram of E. coli with multiple flagella. It is typically

rod-shaped and is about 2 m long and 1 m in diameter. (Eisenbach, 1994)

Swimming in the absence of chemotactant: In the absence of chemo-

tactant, E. coli executes a random walk composed of runs and tumbles. Smooth

swimming in a straight line is called a run and an abrupt turning motion is called

a tumble as shown in Figure 1.2. A cell is propelled by a set of several helical

flagellar filaments that arise at random points on its sides and extend several body

lengths out into the external medium. Each filament is driven at its base by a

rotary motor embedded in the cell envelope. The energy for the motor comes from

proton motive force, i.e., by the flow of protons (hydrogen ions) from the outside

to the inside of the cell. During the runs, the filaments coalesce into a bundle

that pushes the cell forward. When viewed from behind the cell, the bundle ro-

tates counterclockwise (CCW), and, to balance the torque, the cell body rotates

clockwise (CW). Tumbles are initiated by CW motor rotation and during this the

reversed filaments come out of the bundle and go through a series of polymorphic

transformations from normal to semi-coiled and then to curly. (Berg, 2004) This

change in course defines the tumble interval. When the motor switches back to

CCW rotation, the filaments regain its normal conformation and rejoin the bun-

3

dle. Turner et al. (2000) noted that tumbling occurs only when 25% or more ofthe flagellar filament on a given cell reverse to clockwise direction.

Figure 1.2: Swimming behavior of E. coli or Salmonella in the absence of chemo-

tactant. It executes a random walk composed of runs and tumbles with zero drift.

Swimming in the presence of chemotactant: An increasing chemoat-

tractant concentration or a decreasing chemorepellent concentration decreases the

probability of CW rotation and, therefore, the probability of tumbles. Thus, the

final outcome is a random walk of the bacterial cell, biased towards the chemoat-

tractant or away from the chemorepellent as shown in Figure 1.3. Segall et al.

(1986) studied the impulsive chemotactant response of the cell and observed that

the cell compares the concentration over the past 1 s with that observed over

the previous 3 s and responds to the difference. This implies that the concen-

tration/gradient sensed by the bacteria is temporal in that the bacteria possess

a memory, which compares the past information with the present information to

make a decision. So the cell decides whether life is getting better or worse. If its

getting better, they continue in the same direction and if its getting worse, they

wont worry about it.

The regulation of this chemotaxis phenomena in the bacteria is achieved by a

network of interacting proteins. (Eisenbach, 1994) The basic mechanism in flagel-

lated bacteria involves a receptor mediated phosphorylation of a cytoplasmic pro-

tein CheY that binds to the flagellar motor and changes the frequency of tumbles.

There are essentially six cytoplasmic proteins viz., CheA, CheB, CheR, CheW,

4

Figure 1.3: Swimming behavior of E. coli or Salmonella in the presence of chemo-

tactant. It executes a biased random walk composed of runs and tumbles with

significant drift.

CheY and CheZ that are needed to process the sensory signals and transmit con-

trol signals to the flagellar motor. E. coli has five methyl-accepting chemotaxis

proteins (MCPs) (referred to as chemotaxis receptors) that mediate responses to

serine (Tsr), aspartate and maltose (Tar), ribose and galactose (Trg), and dipep-

tides (Tap). These MCPs are transmembrane proteins that span from the internal

to the external surface of membrane and are made of 550 amino acids. Of these

receptors, Aer is the most specific one that senses oxygen and energy levels of the

cell. The signaling domain of the MCP modulates the activity of the CheA to

elicit chemotactic response. The CheW protein couples CheA to MCPs to form a

complex. The CheA phosphorylates which in turn phosphorylates two other regu-

lator proteins, CheY and CheB. The phosphorylated CheY activates the flagellar

motor switch protein FliM (M) resulting in increased tumbling frequency. In vivo

experimental studies (Cluzel et al., 2000) have reported CW bias (the fraction of

time spent in CW rotation) as a function of CheY-P concentration (Yp) and is

shown in Figure 1.4. It is known that for a fixed value of the ligand concentration

there exists a fixed concentration of Yp which yields the corresponding CW bias.

This in turn decides the probability of a run or a tumble event. If CheY-P concen-

tration is very low, which means negligible CW value, cell will run and vise versa.

Also, the CheZ promotes the CheY dephosphorylation. The phosphorylated CheB

5

removes the methyl groups from MCPs, where as CheR continuously adds methyl

groups to MCPs. The level of methylation of MCPs decides the tumbling fre-

quency of the organism. (Falke et al., 1997) A schematic diagram of the signaling

pathways in E.coli is shown in Figure 1.5.

Figure 1.4: Experimental data using fluorescence correlation spectroscopy: Re-

sponse of individual motors as a function of CheY-P concentration. (Cluzel et al.,

2000) Each data point describes a simultaneous measurement of the motor bias

and the CheY-P concentration in an individual bacterium. CW bias was computed

by analyzing video recordings for at least 1 min.

When an attractant is added, it binds to the receptor (specific to that attrac-

tant) and the rate of autophosphorylation of CheA decreases. Consequently, the

rate of phosphorylation of CheY and CheB decrease causing a reduction in the ac-

tivity of FliM and decrease in the tumbling frequency. However, the CheR contin-

ues to methylate MCPs, thereby increasing the level of methylation of MCPs. This

progressively reduces the attractant binding affinity to the MCPs and increases

the rate of autophosphorylation of CheA. The phosphorylated CheY and CheB

return to their initial state. Thus, after the introduction of a chemoattractant, the

6

Figure 1.5: Chemotaxis signaling pathways in E. coli (A) in the absence of

chemoattractant, and (B) in the presence of chemoattractant. The dotted line

represents the reactions with reduced rate in response to ligand binding to the

receptor. It can be noted that on adaptation the pathway returns to steady state

as shown in (A).

7

tumbling frequency reduces for a short time, after which the tumbling frequency

increases to the preattractant level. The initial reduction in tumbling frequency

increases the run length and biases the motion of cell towards the attractant. On

the other hand, in the presence of a repellent, the rate of autophosphorylation

of CheA increases resulting in the increase in frequency of tumbles. (Eisenbach,

1994) Quantification of this chemotaxis signaling pathway requires construction

of mathematical models that describe the operation of the whole network.

Experimentally, the chemotactic behavior of E. coli can be quantified using

various parameters such as average run speed, clock-wise bias, drift velocity, cell

diffusivity, rotational diffusivity, etc. Of the four commonly reported parameters,

the first is the run speed that gives the average speed of the cells in between

consequent tumbles. The second is the clock-wise bias which gives the fraction

of time spent by the cells in tumble mode. Of the next two parameters, the

drift velocity measures the mean speed of cells while the translational diffusivity

characterizes its random motion. Note that the random motion is induced by the

collisions of cells with the solvent molecules as well as due to their own tumbling

and is calculated as the mean square distance traversed by the cells upon time.

A related parameter is the rotational diffusivity calculated as the mean square

angular deviation upon time. A wide range of experimental methods have been

developed so far to analyze these chemotactic parameters. (Eisenbach, 1994) They

include capillary (Adler, 1966a), swarming plate or ring forming (Wolfe and Berg,

1989), stopped flow diffusion chamber (Ford et al., 1991), micro-capillary (Liu

and Papadopoulos, 1996), diffusion gradient chamber (Widman et al., 1992) and

microfluidic assays (Berg and Turner, 1990), (Hanbin et al., 2003), (Ahmed and

Stocker, 2008), (Kalinin et al., 2009), all of which are reviewed in the next section.

Experimental methods for analyzing chemotaxis

Capillary assay: This was the first technique developed by Adler (Adler, 1966a) for

measuring bacterial chemotaxis. A capillary tube containing a chemoattractant-

free buffer is placed in a suspension of bacteria. The bacteria enter the cap-

8

illary and accumulate in it. The extent of this accumulation is a measure of

the motility of the bacteria. When the capillary contains a chemoattractant, a

gradient is developed by diffusion. The bacteria follow this gradient and accumu-

late in the capillary. For measuring negative chemotaxis, a capillary containing

chemorepellent-free buffer is immersed in a chemorepellent containing suspension

of bacteria. The bacteria escape from the chemorepellent into the capillary and

accumulate in it. The bacteria in the capillary can be counted by plating them

or by observing them under a microscope. The chemotaxis receptors become sat-

urated when the capillary contains very high concentrations of chemoattractant.

In this case the bacteria were unable to sense the gradient and a sharp drop was

observed in the number of bacteria accumulated in the capillary. This is the most

commonly used method for chemotaxis but the response depends on chemotactant

transport, metabolism, and growth of E. coli.

Swarming plate or Ring forming assay: Bacteria are placed at a certain spot

in a plate containing semisolid agar and a low concentration of a metabolizable

chemoattractant. The bacteria metabolize the chemoattractant and thereby pro-

duce a chemoattractant gradient. The bacteria use up the local supply of chemoat-

tractant and follow the induced gradient. Due to this, a continuous expanding ring

of dense bacteria is formed. This ring marks the boundary between the region that

has been depleted of chemoattractant and the region still containing the chemoat-

tractant. When the plate contains a number of chemoattractants, a number of

expanding rings are formed. Note that this method of quantifying chemotaxis is

restricted to metabolizable chemoattractants only (Wolfe and Berg, 1989). Also,

there is no control over the induced gradient and the final response depends on

both metabolism and growth.

Stopped-flow diffusion chamber assay: In the study by Ford et al. (1991), the

bacterial migration behavior to the gradients was measured using a stopped flow

diffusion chamber. In this assay, two bacterial suspensions differing in stimulant

concentrations were contacted by impinging the two flows. As long as there is

a flow through the chamber, there is no mixing between the two suspensions.

9

However, once the flow is stopped, a transient attractant concentration gradient is

created by diffusion and the variations in bacteria density can then be measured

by light scattering. This method can be used to quantify the bulk properties of

populations such as drift velocity and translational diffusivity.

Micro-capillary assay: The micro-capillary assay used by Liu and Papadopou-

los (1996) consisted of two reservoirs communicating through a long capillary of

50 m inner diameter. The linear concentration profile for the chemoattractant

was achieved by filling one of the reservoirs with motility buffer while the other

one was filled with the chemoattractant solution. It was assumed that after 24

hours, the concentration profile would attain the steady-state linear concentration

profile. Here, the single cell parameters such as run speed, run length, turn angles

etc were measured in the capillary with the help of a microscope.

Diffusion gradient chamber assay: Widman et al. (1992) used a diffusion gradi-

ent chamber assay that consists of a square arena bounded by a reservoir on each

side. Each reservoir was separated from the arena by a semi permeable mem-

brane. From the source and sink reservoirs, mediums containing attractants and

containing no attractant respectively were pumped using a syringe pump. The

system was allowed to establish the gradients partially after which the cells were

inoculated at the center point in the arena of the chamber. Glucose and oxygen

concentrations in the chamber were measured using micro-biosensor and the cell

growth and migration patterns were observed with time. In this study, however,

the concentration of the aspartate which was used as a chemoattractant was not

measured.

Microfluidic assays: Berg and Turner (1990) studied the chemotaxis in glass

capillary arrays. In this assay, two stirred chambers were separated by a micro-

channel plate comprising a fused array of capillary tubes. Here, the cells added to

first chamber migrate to the second chamber via multichannel plate. The density

of these cells were then calculated by measuring the scattering of light beam

incident from a laser diode. The study found that the flux of bacteria increased

on addition of an attractant into the second chamber compared to when there was

10

no attractant. A linear concentration profile of the chemoattractant was assumed

between the chambers. The drift velocity was calculated from the knowledge

of total flux for varying chemoattractant gradients. In this study, the motility

medium consisted of sodium lactate which is a carbon source for E. coli and

whose effect on chemotaxis was neglected. Later, Ahmed and Stocker (2008) used

a micro-channel connected at right angles to a side branch. Initially, a solution

of chemoattractant and fluorescein was injected into the main channel. After the

main channel was completely filled with the solution, the motility buffer containing

cells was injected into the side channel at a constant flow rate so that the cells

migrate to a constant gradient into the main channel established by diffusion. The

trajectories of the bacteria along with the concentration gradient were recorded.

The chemotactic parameter diffusivity was computed from the motion of a single

cell as well as population.

Recently, Kalinin et al. (2009) have used microfluidic assays to quantify the

chemotaxis. In this study, three parallel channels were patterned in an agarose

gel. A fluorescent solution initially flowed through the upper source channel which

was later replaced with the chemoattractant while a blank buffer flowed through

the lower sink channel. This established a linear chemical gradient in the central

channel because of the diffusion that takes place through the agarose gel from the

source channel to the sink channel. The cells were then introduced into the central

channel and their trajectories were recorded. Using the tracked positions, the

cell diffusivity and the chemotactic migration coefficient (CMC), i.e., the average

vertical position of all the cells tracked with respect to the central position of the

channel were calculated. Further, the study suggested that the cells respond to

the spatial gradient of the logarithmic attractant concentration. However, it is

important to note that the motility buffer used in this study contained lactic acid

which is a carbon source and whose influence on chemotaxis was ignored.

11

Mathematical studies on chemotaxis

A number of mathematical models have been developed to describe different as-

pects of chemotaxis at the level of a single cell (Levin et al., 1998; Novere and

Shimizu, 2001; Lipkow et al., 2005) in addition to descriptions of entire bacterial

populations (Keller and Segel, 1970, 1971; Alt, 1980; Rivero et al., 1989; Newman

and Grima, 2004). Most of the population studies used Keller and Segel model

(Keller and Segel, 1970) for analyzing chemotactic population behaviors, which

was basically developed for modeling the movement of slime molds. According to

this model, the flux of cells ~Jb will be given by,

~Jb = ~b+ b~c (1.1)

where b is the cell density, c is the chemical stimulus concentration, is the ran-

dom motility coefficient that measures the translational diffusivity of a population

of bacteria resulting from the random walk behavior, while is the chemotactic

sensitivity coefficient which represents the strength of the attraction of a popula-

tion of bacteria to a given chemical. In physical terms, the bacterial flux comprises

of two parts, namely, diffusion (or random motility) and chemotactic motion char-

acterized by drift velocity.

Lovely and Dahlquist (1975) were the first to relate the individual-cell observa-

tions to the random motility coefficient (), a macroscopic cell transport parameter

describing population-scale motility,

=v2

3(1 cos) (1.2)

where v is the bacterial swimming speed, is the run time and is the turn angle.

Recall that, E. coli chemotaxis relies on temporal sensing mechanism to detect

spatial gradients as they swim through them and this mechanism is governed by

the ability of an individual cell to compare the fraction of bound chemoreceptors

on its cell surface at various points in time. A change in the number of bound

12

chemoreceptors allows a cell to change its CW bias. Experimentally, Berg and

Brown (1972) found that, when cell is traveling in the direction of an increasing

spatial chemical gradient, it extends their run lengths to continue traveling in

that direction. The study suggested that, in the presence of a chemical attractant

the number of bound receptors on a cells surface increases with time resulting in

reduced tumbling frequency. To describe this behavior Rivero et al. (1989) used

the Keller and Segel model. At the population level, the chemotactic migration

of bacteria was described using the drift velocity (vc),

vc =2

3v tanh

(o2v

Kd(Kd + L)2

L

x

)(1.3)

where v is the run speed, o is the chemotactic sensitivity coefficient and L is the

attractant concentration. Kd, the attractant-receptor dissociation constant, is a

measure of the ability of a membrane-bound chemoreceptor on the cell surface

to detect a specific attractant. However, these population studies do not account

for changes in the intracellular pathway to variations in extracellular attractant

concentration.

Chemotaxis intracellular signaling pathway of E. coli provides a unique op-

portunity to identify and develop computational methods required to obtain a

quantitative understanding of the intracellular signaling pathways. Recent models

for chemotaxis incorporate the signaling pathway at molecular level and integrate

it with motor response to predict bacterial motion. Examples include AgentCell

(Emonet et al., 2005), E solo (Bray et al., 2007) and RapidCell (Vladimirov et al.,

2008). AgentCell simulates the complete signaling pathway stochastically in a

single cell and predicts the motion in a three-dimensional environment. Esolo

is a deterministic model that solves ordinary differential equations represent-

ing the signaling reactions in the pathway and predicts bacterial movement in

two-dimensional environment. RapidCell utilizes the Monod-Wyman-Changeux

(MWC) (Mello and Tu, 2005; Keymer et al., 2006) two-state receptor model for

mixed receptor clusters, incorporates the adaptation dynamics and connects the

CheY-P values to the cells tumble/run to predict the motion of E. coli in a

13

Table 1.1: Mathematical models overview

Model Description

Spiro et al. (1997) Used a three methylation state model and simulated the

response to temporal variation of aspartate.

Barkai and Leibler (1997) Model ensured that adaptation is robust and showed

that adaptation time is inversely proportional to

receptor-complex activity.

Levin et al. (1998) Receptor modification reactions catalyzed by CheR and

CheB incorporated in Bacterial ChemoTaxis (BCT)

model. Model explored the consequence of variation

in protein expression CheY-P and the phenomenon of

non-genetic individuality.

Morton-Firth et al. (1999) Stochastic simulation of bacterial chemotaxis

(StochSim): Here, the activity of receptor com-

plexes was determined by free-energy changes due to

both ligand binding and changes in methylation state.

Shimizu et al. (2001) This extended the model of Morton-Firth et al. (1999)

and allowed interactions between neighboring receptor

complexes arranged in a regular lattice according to the

free energy of a receptor complex determined by the

activity states of its immediate neighbors. Model simu-

lated with different lattice sizes and geometries to find

the effect of the coupling energy between neighboring

receptors on the signal to-noise ratio and gain.

Mello and Tu. (2003) Deterministic model used to determine the full set of

conditions under which the system achieves perfect

adaptation.

14

Rao et al. (2004) Used both Barkai and Leibler (1997) and Sourjik and

Berg (2002) models to compare the intracellular path-

ways of E. coli and B. subtilis. Both pathways shown

to be adaptive and robust.

Lipkow et al. (2005) Using the Smoldyn program, model simulated the diffu-

sive movement of individual CheY molecules and their

binding with receptor complexes, CheZ and FliM.

Emonet et al. (2005) AgentCell, an agent-based program that relates stochas-

tic intracellular processes to the behavior of individ-

ual cells and bacterial populations. Cells represented

as agents made up of chemotaxis proteins, motors and

flagella that can move through a three dimensional en-

vironment.

Bray et al. (2007) E solo model which uses ordinary differential equa-

tions of the signaling reactions in the pathway and gives

the graphical display of bacterial movement in two-

dimensional environment.

Vladimirov et al. (2008) RapidCell model uses the Monod-Wyman-Changeux

(MWC) model for mixed receptor clusters, adaptation

dynamics and a model of cell tumbling to give the mo-

tion of E solo in two-dimensional environment.

15

two-dimensional environment. It is only recently that Kalinin et al. (2009) using

MWC model, predicted the bacterial motion to varying chemoatttactant gradi-

ents and compared them with experimental results. The computational single cell

mathematical models used for bacterial chemotaxis are briefly listed in Table 1.1.

In addition to the chemical stimuli, microorganisms also sense other stimuli

such as light, temperature, electric field, etc. In all cases, the name of the response

includes a prefix that describes the stimulus. The suffix taxis means moving to-

wards or away from the stimulus. They include phototaxis movement directed by

light, thermotaxis by temperature changes, electrotaxis by electrical field, mag-

netotaxis by magnetic field, geotaxis by gravity, elasticotaxis by elastic force etc.

(Berg, 2004)

1.2 Motivation

The traditional techniques to characterize chemotaxis were the agarose gel as-

say (Wolfe and Berg, 1989) and the capillary assay (Adler, 1966a; Liu and Pa-

padopoulos, 1996) due to their simplicity. In both these methods, E. coli moves

up a gradient set by the consumption of the attractant and there is no control

over the induced gradients. More recent experiments have been able to establish

controlled gradients using microfluidic techniques (Kalinin et al., 2009) that al-

low the measurements of drift velocities over a wide range of gradients. However,

these techniques fail to capture temporal and spatial variation of the chemotactic

response of E. coli to spatial variations of chemoattractants. In parallel with the

experimental work, a number of extensive mathematical models have been pro-

posed for E. coli chemotaxis, but there exists no study which predicts the bacterial

motion along a gradient and compares them with experiments. Recall that the

motion of E. coli not only depends on the local ligand concentrations but also on

past history of concentration experienced by it. Further, the extent of influence

of oxygen and energy source on the motion is not clear. How the intracellular sig-

naling response reflects the extracellular response characterized by motion is still

16

an active research area. The understanding of the available intracellular signaling

pathway and absence of experimental and theoretical study which quantifies the

chemotaxis in both time and space in controlled gradients motivated us to carry

out the work presented in this thesis.

1.3 Objectives

The main objective of the current study is to quantify the response of microorgan-

isms to varying chemoattractant concentrations/gradients using a micro-capillary.

This would involve developing mathematical models that describe the chemotaxis

pathway in a chemotactic cell and to relate the kinetics to the motion of cell in

the presence of chemoattractants. The predictions of the model will be compared

with experiments where the motion of cell in the presence of chemoattractants will

be tracked.

The overall objectives are summarized below :

Develop a novel technique to quantify the chemotaxis in E.coli in the pres-ence of chemoattractants in controlled environment at the phenotypic level

in a suitable device.

Develop a mathematical model to quantify the motion of E.coli using avail-able signaling pathway and connect it to the motion.

Validate the model results with experimental data to get better insight ofthe chemotaxis mechanism.

1.4 Organization

The work presented in this thesis is organized into six chapters describing the

research methodology and results, followed by conclusions and recommendation

for future work. In chapter 1, an overview of bacterial chemotaxis phenomena is

presented. Chapter 2 describes the various experimental and theoretical methods

17

used in the present work. Here, the methods developed for modeling the motion

of E. coli and experimental techniques used for establishing various controlled

gradients have been described. In chapter 3, we describe the mathematical model

along with experimental validation of chemotaxis under controlled gradients of

methyl-aspartate in E. coli while chapter 4 deals with a similar study in the

presence of a metabolizable attractant, namely, L-serine. Chapter 5 discusses the

phenomena of chemotaxis in the presence of glucose via the phosphotransferase

(PTS) pathway. Finally, the overall conclusions and directions for future work are

presented in chapter 6.

18

Chapter 2

Materials and Methods

2.1 Experimental Protocols

Microorganism

Recall that E.coli has five methyl-accepting chemotaxis proteins (MCPs) that

mediates metabolism-independent chemotactic responses in E.coli. The high-

abundance chemoreceptors of this family (Tsr and Tar) mediate responses to serine

and aspartate, respectively. The low-abundance chemoreceptors (Tap and Trg)

are present at only 10 percent of the concentration of Tsr and while Tap mediates

response to dipeptides, Trg senses galactose and ribose by means of their respective

periplasmic binding proteins. (Li and Hazelbauer, 2004) In this study, we used

Escherichia coli K-12 (MTCC 1302, IMTECH Chandigarh, India) strain and

chemoreceptors characterization was done by Chromous Biotech Pvt Ltd. The

detailed report can be found at the end of this chapter and shows that the strain

used throughout our study has four MCPs (Tsr, Tar, Tap and Aer) excluding Trg

receptor. The culture was revived over monthly intervals.

Rectangular micro-capillaries

Micro-capillaries were obtained from Arte Glass Associates Co., Ltd, Japan. The

dimensions of the capillaries are 5 cm (L) 1000 m (W) 100 m (H).

19

Chemicals

Luria Bertani (LB) broth was obtained from Hi-media company. KH2PO4, K2HPO4,

(NH4)2SO4, MgS04. 7H20, Ehylenediaminetetraacetic acid (EDTA), Polyvinyl

pyrrolidine (PVP), D-glucose (glucose), -methyl-DL-aspartate (MeAsp) and L-

serine (serine) were obtained from Sigma-Aldrich company. 2-(N-(7-nitro-benz-2-

oxa-1,3-diazol-4-yl)amino)-2-deoxy-glucose (2-NBDG) was obtained from Invitro-

gen Corporation. Tryptone (Difco) and Bacto agar (Difco) were obtained from

BD Biosciences company.

Media

The motility buffer (MB) contained (/l of distilled water) K2HPO4, 11.2 g; KH2PO4,

4.8 g; (NH4)2SO4, 2 g; MgS04. 7H20, 0.25 g; PVP, 1 g; and EDTA, 0.029 g. (Adler,

1973) Luria-Bertani (LB) broth contained LB (/l of distilled water) 25 g. Tryptone

medium contained (/l of distilled water) tryptone, 10 g and Nacl, 5 g. Glucose

medium contained (/l of distilled water) glucose, 4 g and the salts (/l distilled wa-

ter): K2HPO4, 11.2 g; KH2PO4, 4.8 g; (NH4)2SO4, 2 g; and MgS04. 7H20, 0.25g.

Different concentrations of chemotaxis medium was prepared by adding different

amounts of chemoattractant to the motility buffer under sterilized conditions.

Growth conditions

The growth conditions of the bacteria are critical to the success of the chemotaxis

experiments. In glucose experiments, the bacteria were grown as per the following

procedure: One loop full of culture from the slant was inoculated into LB media

and allowed to grow for 9 hours in the exponential growth phase. The incubation

was always carried out at 37oC and 240 rpm. After the culture was grown for

9 hours, 1 ml of the culture sample was transferred to the sterilized LB medium

and left to further grow for 6 hours (early exponential phase). Next, 1 ml of the

culture broth from LB medium was transferred to the glucose medium and allowed

to grow for 6 hours (exponential phase). To ensure that the cells are adapted to

20

glucose, 10 ml of sample was transferred to glucose medium and grown again for

4 hours (early exponential phase). The biomass was separated by taking 50 ml of

the culture into sterilized tubes and centrifuged at 4000 rpm for 10 minutes. The

supernatant was decanted and the settled pellet was gently re-suspended in 10 ml

motility buffer. In order to provide energy to the bacteria, 28 M of D-glucose was

added to the motility buffer in all experiments. The above procedure was repeated

three times before introducing the cells into the capillary. High levels of bacterial

motility were observed on viewing the cells under an optical microscope. Finally

for the chemotaxis measurements, cells were introduced gently by touching the

pellet with the mouth of the capillary. Then, concentration of cells in the capillary

were calculated by diluting them in the buffer followed by plating. Approximately

106-107 bacteria/ml were taken into the capillary in all experiments.

We used similar procedure for MeAsp experiments and serine experiments with

some minor protocol modifications. After the culture was grown for 9 hours, 1

ml of the culture broth from LB medium was transferred to tryptone medium

and allowed to grow for 6 hours (exponential phase). To ensure that the cells

are adapted to tryptone medium, 10 ml of sample was transferred to tryptone

medium and grown again for 4 hours (early exponential phase). After this, the

washing procedure is similar except that the motility buffer does not contain any

chemoattractant in the gradient experiments. However, we added chemoattractant

to the motility buffer for observing the motion of E. coli for obtaining uniform

concentrations (zero gradient) of the attractant.

Calibration of intensity for 2-NBDG measurements

The micro-capillaries were first sterilized and were marked with graduations spaced

at 0.5 cm along its length. The concentration gradients were first established using

different concentrations of 2-NBDG (fluorescent glucose) solutions using motility

buffer. In order to calibrate the intensity of 2-NBDG solutions, a 5 cm plug of

2-NBDG solution was drawn into the capillary and the ends were sealed with

wax. Using a microscope with a 4X (NA 0.13) objective lens, the fluorescence

21

intensity of 2-NBDG was measured at the center of the capillary. The fluorescence

intensity was found to vary linearly with 2-NBDG concentration up to 1000 M.

The calibration chart is shown in Figure 2.1.

Figure 2.1: Linear calibration curve: Fluorescence intensity with varying concen-

tration of 2-NBDG.

Establishment of 2-NBDG gradients for glucose experiments

Figure 2.2: Micro-capillary experimental setup for establishing glucose gradients.

Experiments were performed to establish different gradients of 2-NBDG. A 2.5

cm plug containing a fixed concentration of 2-NBDG was drawn into the capillary

followed by a 2.5 cm plug of a lower concentration of 2-NBDG. Approximately 107

22

Figure 2.3: Measured fluorescence intensity in the capillary for 10 min (solid line)

and 540 min (dashed line) after the start of the experiment. It can be noted that

the gradient was stable for more than 9 hours.

Figure 2.4: (A) Agar plate containing 0.3 % agar and motility buffer (absence of

glucose) with 107 cells introduced at the center of the plate. The image wastaken after 6 hours of incubation. It can be noted that no ring was formed. (B)

Agar plate containing 0.3 % agar, motility buffer and glucose (5 mM), with 107

cells introduced at the center of the plate. The image was taken after 6 hours of

incubation. Clearly, the cells have migrated forming a distinct ring.

23

bacteria /ml were taken into the capillary by contacting the pellet (as described

earlier) into the mouth of the capillary. As before, the ends of the capillary

were sealed with wax. A schematic diagram of the micro-capillary experimental

setup is shown in Figure 2.2. Using a microscope with a 4X (NA 0.13) objective

lens, the fluorescence intensity of 2-NBDG was measured along the length of the

capillary as a function of time. The results are shown in Figure 2.3. It is noticed

that the intensity profiles attains a steady state within 10 minutes and negligible

change in the concentration profile was observed up to 540 minutes. These profiles

were robust and easily reproducible and were not affected by the cell movements.

Growth experiments with 2-NBDG demonstrated that there is no change in OD

for 4 hours (results are not shown) indicating that 2-NBDG was not metabolized.

Also, experiments were performed in agar plate assays to check the chemotactic

response of the current strain to glucose. Clear rings were formed within 6 hours

indicating that the E. coli strain lacking Trg shows normal chemotactic behavior

towards glucose (Figure 2.4). Further, for the chemotaxis experiments, normal

D-glucose was used instead of 2-NBDG as the cells did not respond to 2-NBDG.

Establishment of 2-NBDG gradients for MeAsp and Serine

experiments

Initially, a 4.5 cm liquid plug containing a fixed concentration of 2-NBDG was

drawn in the capillary followed by about 0.5 cm plug of a motility buffer without

2-NBDG. Approximately 107 bacteria/ml were taken into the capillary by con-

tacting the pellet (as described earlier) with the mouth of the capillary. Then, the

capillary ends were sealed with wax. A schematic diagram of the micro-capillary

experimental setup is shown in Figure 2.5. Using a microscope with an 4X (nu-

merical aperture = 0.13 ) objective lens, the fluorescence intensity of 2-NBDG was

measured over 0 < x < 1500 m as a function of time shown in Figure 2.6(A)

and the results are shown in Figure 2.6 (B). It is noticed that within the first

1.5 min, the intensity profiles attain the steady state and there is a negligible

change in the concentration profile up to 30 min. These profiles were robust and

24

easily reproducible and were effected neither by the cell movements nor by the

consumption of 2-NBDG by this strain. Further, for the E. coli chemotaxis to

MeAsp/serine, MeAsp/serine was used instead of 2-NBDG. It is assumed that the

gradients obtained with MeAsp/serine is identical to 2-NBDG, since the molecular

weight of both are approximately similar and so the diffusivity in water would also

be approximately the same. The experimental values of L0 and the established

gradients (G) are given in Table 2.1. These conditions were also used in the model

simulation.

Table 2.1: Gradients and ligand concentration at x = 0 obtained through experi-

ments.

S.No Concentration (L0, M) at x = 0 cm Gradient (G, M/m)

1 1.6 0.0016

2 16 0.016

3 160 0.16

4 1600 1.6

5 16000 16

4 160000 160

Quantification of chemotaxis

Image analysis was used to quantify the movements of E.coli. This involves the

analysis of moving objects in image sequences. In this section, specifications of

the optical microscope, image processing and procedure for calculating the cell

movements are discussed.

Specifications of optical microscope: IX71 Inverted Microscope (Olympus, Japan)

was adopted as the optical microscope. Images were taken using Evolution VF

cooled monochrome camera (Media Cybernetics, Japan) in Bright-field (BF) il-

lumination mode with magnification of 40X objective lens (numerical aperture =

0.75).

Image processing procedure: Image-Pro Plus 6.0 image analysis program was

used to locate E.coli in each frame and follow its motion in subsequent frames.

25

Figure 2.5: Micro-capillary experimental setup for establishing MeAsp/serine gra-

dients. x = 0 is located at the end of the pellet with a ligand concentration, L0.

The experimental values of L0 and the established gradients (G) are given in Table

2.1 and these conditions were used in the model simulations.

26

Figure 2.6: (A) A magnified view of the micro-capillary set-up (B) Stable linear

gradients; Measured concentration profiles for two gradients: t = 1.5 min (black

line) and t = 5 min (blue line) for G = 0.016 M/m, t = 1.5 min (red line), t = 5

min (green line) and t =30 min (pink line) for G = 0.16 M/m respectively.

This program uses a single stack TIFF image which contains 500 sequential TIFF

images taken at an interval of 0.11 s. In every experiment two stack images were

recorded at 500, 1000 and 1500 m. Note that the image captured using 40X

magnifications encompassed a physical area measuring 160 120 m2.Procedure for calculating the cell movement: To minimize errors in finding the

cell movements, the following two assumptions were made. First, the cells moving

close to each other or overlapping cell movements were neglected. Secondly, the

cells not in the field of view for the entire time series (minimum 1.1 s) were

neglected.

Using Matlab c, each stack image was thresholded so as to clearly distinguishcells from the image background. Then, with the help of the auto tracking option,

the movement of cells along the gradient direction (x) was tracked in a sequence of

images. The raw data containing spatial location of each cell as a function of time

was exported to an Excel c file. The procedure for calculating the movement ofcell over a period of time was straight forward. Starting from an initial position

in the first frame, the displacements between two consequent frames 0.11 s apart

27

were determined. The tracking data with time was collected from six identical

experiments repeated on different days to capture the variability, if any.

2.2 Modeling Methods

Model for the intracellular pathway

A two state receptor model proposed by Barkai and Leibler (1997) was used to

simulate the intracellular signaling pathway. The model considers the methyl

accepting chemotaxis proteins (MCPs), CheA and CheW, as a single entity (re-

ceptor complex) and assumes that these receptor complexes, whose concentration

is denoted by T , exist in either an active (TA) or an inactive (T I) state. Let

Ti represent the concentration of receptor complexes with i residues methylated

and i(L) denote the probability that the receptor complex Ti is active when the

concentration of chemoattractant is L. The receptor complex can be in one of five

methylation states with i = 0, 1, 2, 3 or 4 methyl groups. The total concentration

of active receptors is given by,

TA =40

i(L)Ti, (2.1)

while the total concentration of inactive receptors is given by,

T I =

40

(1 i(L))Ti. (2.2)

The binding kinetic equation for active receptor complex is given by,

TAF + L [TAL] (2.3)

The total active receptor complex concentration TAT is given by,

TAT TAF + [T

AL] (2.4)

where TAF is free (non-ligand bound) active receptor complex concentration and

[TAL] is the ligand bound active receptor complex concentration respectively. The

28

fraction of free active receptor complex concentration from the above equation is

given by,

TAFTAT

=KL

KL + L(2.5)

where KL is the ligand dissociation constant. Similarly, the fraction of ligand

bound receptor complex concentration is given by,

[TAL]

TAT=

L

KL + L(2.6)

The total probability of the receptor complex being in active state is the sum of

the probabilities of the ligand bound and non-ligand bound receptors being in

active state and is given by,

i(L) =Li L

KL + L+

0iKLKL + L

(2.7)

where the parameters are assigned the following numerical values, L0 = 0, L1 =

0, L2 = 0.1, L3 = 0.5,

L4 = 1,

00 = 0,

01 = 0.1,

02 = 0.5,

03 = 0.75 and

04

= 1. These values are taken from Morton-Firth et al. (1999) and are estimated

from the free energy states of methylation and the ligand occupancy of receptor

complex for MeAsp. The corresponding phosphorylation rate equations with the

corresponding rate constants (Emonent and Cluzel, 2008) are given by,

dApdt

= 23.5(TA)A 100(Ap)Y 10(Ap)B (2.8)

dYpdt

= 100(AP )Y 30(Yp) (2.9)

dBpdt

= 10(Ap)B (Bp) (2.10)

Here, A, AP , Y , Yp, B and Bp represent, respectively, the concentrations of CheA,

phosphorylated CheA, CheY, phosphorylated CheY, CheB and phosphorylated

CheB. Li and Hazelbauer (2004) have measured these chemotaxis protein concen-

trations for wild type and are given by, A + Ap = 5.3 M, B + Bp = 0.28 M,

Y + Yp = 9.7 M and CheR (R) = 0.16 M. The total receptor concentration

29

(Tar+Tsr), T0 + T1 + T2 + T3 + T4 = 17 M and R = 0.16 M reflect the

reported (Sourjik and Berg, 2004) findings that both Tsr and Tar participate in

sensing aspartate.

Barkai and Leibler (1997) model assumes that CheR (R) binds to the inac-

tive receptors (T I) and the phosphorylated CheB (Bp) binds to the active recep-

tors (TA). Assuming that, the methylation and demethylation reactions follows

Michaelis - Menten kinetics, the rate of demethylation and methylation is given

by, respectively,

rB =kbBp

KB + TA(2.11)

rR =krR

KR + T I(2.12)

where, kb = 0.6 s1 and kR = 0.75 s

1 are the rate constants and KB = 0.54 M

and KR = 0.39 M are the Michaelis constants (Emonent and Cluzel, 2008) for

receptor demethylation and methylation, respectively.

The rate of methylation is proportional to the concentration of inactive recep-

tors (1-i(L))Ti, and the rate of demethylation is proportional to the concentra-

tion of active receptors i(L) Ti. For the receptor Ti, the rate of demethylation

is rB i(L) Ti and the rate of methylation is rB (1 i(L)) Ti, the mass balanceequations for the corresponding receptor can be given by,

dT0dt

= rR(1 0(L))T0 + rB1(L)T1 (2.13)

dT1dt

= rR(1 1(L))T1 + rB2(L)T2 + rR(1 0(L))T0 rB1(L)T1 (2.14)

dT2dt

= rR(1 2(L))T2 + rB3(L)T3 + rR(1 1(L))T1 rB2(L)T2 (2.15)

dT3dt

= rR(1 3(L))T3 + rB4(L)T4 + rR(1 2(L))T2 rB3(L)T3 (2.16)

dT4dt

= rR(1 3(L))T3 rB4(L)T4 (2.17)

Using Matlab c, we solved simultaneously the steady state phosphorylationreaction equations and the mass balance equations for the receptors.

30

E. coli motion model

The ligand concentration decides the protein CheY-P concentration which is the

output of the signaling pathway. In vivo experimental studies (Cluzel et al.,

2000) using fluorescence correlation spectroscopy have reported CW and switching

frequency (F ) as a function of CheY-P concentration (Yp). Recall that the CW is

the fraction of time spent in clock-wise rotation of the flagella while the switching

frequency is the number of times the motor switched its direction of rotation per

unit time. Further, Cluzel et al. (2000) used the Hill equation to describe the CW

bias as a function of Yp,

CW =Y np

(Y np ) + (Kn)

(2.18)

where n is the Hill coefficient and K is the half saturation constant. Cluzel et al.

(2000) reported n = 10 and K = 3.1 M for their single cell experiments in-

dicating a highly ultra-sensitive response (Figure 1.4). In our experiments, the

experimentally measured CW bias was used to set the value of K so as to yield

a steady state value of Yp. It was also observed that the switching frequency F

qualitatively behaves as F dCW/d Yp. (Cluzel et al., 2000) So, the inverse ofthe switching frequency gives the time period containing one change of direction

of rotation. The time spent in a single tumble mode is given by,

ttum =CW

F(2.19)

while the time spent in a single run mode, also referred to as the run time is,

trun =1 CW

F(2.20)

Assuming a Poisson process, (Vladimirov et al., 2008) we can now obtain the

probability that E. coli will switch from tumble to run mode and vice versa. If

the E. coli is in run mode, then the probability that it will switch to tumble mode

in time dt is Pruntum = dt/trun while 1 (dt/trun) is the probability that it willcontinue in the run mode after each time step dt. Similarly, Ptumrun = dt/ttum.

31

In the simulations, the E. coli starts from its initial position, x = 0, y = 0

with the ligand concentration varying linearly with x, L(x) = L0 + Gx, where G

is the gradient. At t = 0, the E. coli is made to run for time duration dt along

x, after which Pruntum is determined. A number between 0 and 1 is randomly

generated using a Matlab function that has a uniform probability in that range.

If the number obtained is less than Pruntum, then the E. coli is made to tumble

else it continues to run. In case of tumble, the position of the cell is held constant

and a new direction of the motion is chosen from a gamma distribution of turn

angles that is obtained independently from our experiments. The distribution

yielded a mean turn angle of 71o 1.1 that was close to that observed by Bergand Brown (1972). In case of run, the cell is made to move at a constant run speed

but at an angle chosen from a normal distribution with mean zero angle (about

its previous angle) and a variance2Dr dt, where Dr is the rotational diffusivity

and was determined independently from our experiments. Once the E. coli has

tumbled or run, the above procedure is repeated at the new location for t = t+dt

using the new local ligand concentration. The simulation was obtained for 1000

cells and the mean properties were calculated for comparison with experiments.

This model was used to simulate the motion of E. coli for varying -methyl-DL-

aspartate (MeAsp) and L-serine (serine) concentrations/gradients. It is important

to note that, MeAsp is a non-metabolized chemoattractant whereas serine is a

metabolized (a less source of carbon) chemoattractant. Further, the model results

are validated with the experimental results and are discussed in the next chapters.

The following three pages presents a concise report on the presence of different

chemotaxis receptors in the E. coli strain used in our experiments.

32

Chapter 3

Mathematical modeling and

Experimental validation of

chemotaxis under controlled

gradients of methyl-aspartate in

Escherichia coli

3.1 Introduction

In this chapter, we describe a novel technique to measure the chemoattractant

gradients with respect to space and time in the absence of fluid flow. Unlike the

previous studies, we track the motion of each cell and obtain the local drift velocity

as a function of time and space for a broad range of concentrations and gradients

of -methyl-DL-aspartate (MeAsp, a non-metabolized chemoattractant). We used

an existing two state model (Barkai and Leibler, 1997) for the intracellular path-

way and incorporate the extra-cellular influence such as ligand concentration and

Brownian motion to predict the response for the experimental conditions. The

statistics of single cell motions such as cell velocity, run angle, turn angle, tum-

bling frequency, rotational diffusion, drift velocity and diffusivity are obtained

36

independently. The predicted motion matched with the observation only when

the response of the intracellular pathway was highly ultra-sensitive. The detailed

comparison of the predictions with the observed behavior revealed bounds on the

parameters describing the intracellular pathways. Further, our studies also show

that oxygen plays a key role in the chemotaxis response and the response to a

ligand cannot be analyzed in isolation to oxygen.

Chemotaxis signaling model

A two state receptor model proposed by Barkai and Leibler (1997) was used to

simulate the intracellular signaling pathway. The model considers the methyl ac-

cepting chemotaxis proteins (MCPs), CheA and CheW, as a single entity (receptor

complex) and assumes that these receptor complexes, whose concentration is de-

noted by T , exist in either an active (TA) or an inactive (T I) state. Further, a

receptor is assumed to exist in one of the five methylation state. For a fixed value of

the ligand concentration, the model solution yields the active and inactive receptor

concentrations for each of the methylation states along with the concentrations of

chemotaxis proteins in the phosphorylated and dephosphorylated states. Finally,

the concentration of CheY-P decides the motion of E. coli. The model details of

the intracellular signaling pathway and motion details are given in Materials and

Methods (Chapter 2).

3.2 Results

Gradients of fluorescent glucose (2-(N-(7-nitro-benz-2-oxa-1,3-diazol-4-yl)amino)-

2-deoxy-glucose, 2-NBDG) were established in the capillary and their variation

with time were recorded. It was found that the gradient was established in the

first 1.5 minutes after which the variation was negligible for almost 30 min (see Ma-

terials and Methods). At x = 0, the concentration reaches a constant finite value

within this short time beyond which (x > 0) a stable linear gradient was achieved.

A wide range of gradients could be established by varying the concentration of the

37

ligand in the liquid plug. It can be noted that the gradient experiments with E.

coli were conducted for about 15 min after establishing the gradient. This implies

that the stable gradients could be maintained during the period of the experiment

(see Figure 2.6).

The experiments were conducted with various gradients of MeAsp and with

uniform concentrations in the absence of gradients, and the mean square displace-

ment (MSD) of over 1000 cells from these experiments are presented as a function

of time in Figure 3.1. Interestingly, all the measured points from various exper-

iments collapse to give a unique trend, wherein the MSD increases quadratically

up to t 3.8 s after which the increase is linear. The MSD was further usedto determine the translational diffusivity, ((x )2/2t) at large times (t > 3.8s) yielding a value of 110 26.22 m2/s, irrespective of the external conditions.Here, is the mean displacement in time t obtained using Gaussian distribution.

The measured angular displacement data was fit to a Gaussian distribution and

was used to calculate the rotational diffusivity, given by Dr = 2/2t where is the angular displacement and the angular brackets represent ensemble average.

Table 3.1 presents the various parameters related to E. coli motion obtained from

our experiments and the measured values are close to those reported in previous

studies.

Table 3.1: Comparison of chemotactic parameters in E. coli with those reported in

literature. (Berg and Brown, 1972; Liu and Papadopoulos, 1996; Berg and Turner,

1990; Ahmed and Stocker, 2008; Kalinin et al., 2009) These values were obtained

by averaging ( standard deviation) over all experiments.

Description Present study Reported

Average run speed (v, m/s) 18.6 2.2 14.2 3.4Mean tumble angle (degrees) 71.80 1.1 62 26

Rotational diffusivity (Dr, rad2/s) 0.32 0.07 0.06 - 0.28

Steady state CW bias (CWss) 0.16 0.05 0.1Cell diffusivity (D, m2/s) 110 26.22 125 - 360

38

Figure 3.1: Mean square displacements (MSD standard error) about the meanvalue (from Gaussian fit) as a function of time. MSD was obtained using data

from all experiments including those with and without gradients. The solid line

represents the linearity obtained for a diffusive regime. The diffusive regime occurs

beyond 3.8 s and is shown by a dotted arrow.

Next, the measured drift velocity, in the presence and the absence of MeAsp,

was recorded at three different locations along the capillary and is presented in Fig-

ure 3.2. In the absence of MeAsp, independent experiments were conducted with

only motility buffer that was saturated with air and with pure oxygen. Figure 3.2

shows that in the absence of MeAsp the cells show significant drift velocities (u0)

initially and the velocity drops steadily beyond 1000 m from the entrance. The

high concentration of the cells in the pellet would have rendered the interstitial

fluid devoid of oxygen. Consequently, the cells respond to an oxygen gradient

established when the cells are brought in contact with the motility buffer in the

capillary. However, a significant decrease in drift velocity with increasing distance

was observed beyond 5000 m and was probably due to a lack of endogenous en-

ergy source since the cells become non-motile. Further, the run speeds did not

vary up to 2500 m, suggesting that the endogenous energy source was not limit-

ing for x < 2500 m. Figure 3.2 also includes the drift velocity when the buffer is

saturated with oxygen. The velocities were higher by about 1.5 times everywhere

than that for the normal case although the cells again became non-motile for dis-

tances beyond 5000 m. The higher velocities could be attributed to the higher

39

gradients of oxygen sensed by the cells.

All experiments in the presence of MeAsp were performed at normal levels of

oxygen (saturated with air). Experiments were conducted to first quantify the

chemotactic response in the absence of gradient to uniform MeAsp concentrations

of 100 (N) and 10000 M () and the results are also presented in Figure 3.2. The

measured drift velocities for both the concentrations were similar to that observed

in the absence of MeAsp. This clearly indicates that cells respond solely to oxygen

in the absence of MeAsp gradients. To study the chemotactic behavior of E. coli,

various gradients of MeAsp were established in the capillary. Figure 3.3 presents

the measured average run speed as a function of the gradient of MeAsp at three

different locations in the capillary. The plot also includes measurements in the

absence of MeAsp (indicated by the arrow). It can be noted that the average run

speeds are in the range of 16-21 m/s irrespective of the gradient or the location.

An average value of 18 m/s was used in the model simulations.

Figure 3.2: Measured drift velocity (u0 standard error) as a function of distancein the absence of aspartate using normal buffer (), 100 M aspartate in normalbuffer (N), and 10000 M aspartate in normal buffer (). The solid line represents

the average trend for the three experiments. The empty circle () represents thedrift velocity when the buffer is saturated with oxygen in the absence of aspartate.

The measured drift velocities contain the influence of both oxygen and MeAsp

gradients. In order to separate the influence of oxygen gradient on the total drift

velocity, we subtracted (uo) from measured drift velocity values assuming alge-

40

Figure 3.3: Average run speeds ( standard error) as a function of aspartategradients at: 500 m (+), 1000 m (), and 1500 m (). The solid line representsthe average value for the entire data set. The arrow indicates the average run speed

of 18 m/s obtained in the absence of MeAsp.

braic addition of the responses. (Strauss et al., 1995) Thus, the drift velocity due

to MeAsp alone is given by, uasp = u uo. Note that this procedure not onlyeliminates the effect of oxygen but also the non-chemotactic drift induced by the

no flux condition on the plug side. The measured drift velocities at fixed locations

in the capillary for four different gradients with their corresponding initial con-

centrations are shown in Figure 3.4. For the lowest gradient (G = 0.016 M/m),

the response was minimal indicating that the E. coli had completely adapted

even at x = 500 m. However, on increasing the gradient to G = 0.08 M/m, a

drift velocity of 1.4 m/s was measured at 500 m which monotonically decreased

to 0.8 m/s at 1500 m demonstrating adaptation. On increasing the gradient

further, a similar profile but with higher initial drift velocities was observed. How-

ever, at the highest gradient tested (G = 1.6 M/m), the measured velocities at

500 and 1000 m matched with that for G = 0.08 M/m, but was significantly

lower at 1500 m indicating early adaptation. The model was simulated for the

above mentioned gradients and the predictions are compared with experiments in

Figure 3.4. Note that the simulations do not incorporate the effects of oxygen.

Further, simulations were run in the absence of attractants to determine the non-

chemotactic drift velocity as a function of distance. The non-chemotactic drift

41

velocity decreases gradually from 0.35 m/s at x = 0 to a negligible value at x =

1500 m (results not shown). These values were subtracted from the values ob-

tained for gradients to compare with experimental data, where the effects of both

oxygen and non-chemotactic drift have been eliminated. The spatial variation of

the predicted drift velocities matched reasonably well with the measurements for

all four gradients. Specifically, the model captures the increase in drift velocities

with gradient close to the plug and the subsequent monotonic decrease with dis-

tance. Further, the measured drift velocity indicates that the cells adapt faster at

higher gradients, again in agreement with experiments.

Figure 3.4: Drift velocity (uasp standard error) as a function of distance forvarious gradients of aspartate (G) and initial ligand concentration (L0) (A) G =

0.016 M/m and L0 = 16 M; (B) G = 0.08 M/m and L0 = 80 M; (C) G =

0.16 M/m and L0 = 160 M; (D) G = 1.6 M/m and L0 = 1600 M. represents data from experiments and solid line represents model prediction.

42

The model was used to obtain the drift velocities at different spatial locations

along the capillary as a function of varying gradients (see Figure 3.5 (A)). Below

a gradient of 0.01 M/m, the drift velocity is negligible and does not vary with

distance indicating no response. With increasing gradients, the drift velocity ini-

tially increases at all four locations reaching a maximum after which the velocity

drops. A maximum drift velocity of 1.8 m/s at 100 m was observed close to a

gradient of 0.1 M/m. Note that the peak value of drift velocity also decreases

with spatial distance. A negligible drift velocity is obtained for gradients greater

than 10 M/m but for locations greater than 3000 m due to adaptation.

The model also predicted the normalized CW bias with time for measured

gradients and the corresponding normalized value (with respect to the steady

state value) of the fraction of active receptors, TA/TAss (Figure 3.5 (B) and (C),

respectively). Note that TA decides the concentration of CheY-P and therefore the

CW. Perfect adaptation to a new environment requires that the TA and therefore

the CW return to their steady state values after the initial transients due to change

in the environment. At the lowest gradient tested in our experiments, TA drops

by a small value after which it regains the steady state value suggesting quick

adaptation. Consequently, this small change in TA leads to a step drop of CW

bias drops at x = 0 after which it attains the steady state value within a short time

(200 s). This indicates that the run length is higher than the steady state value

only for a short distance beyond which there is negligible response. At the higher

gradient of 0.08 M/m, the fraction of active receptors decrease further leading

to a slower recovery of the normalized CW bias to its steady state value. The plot

indicates that the cells do not completely adapt even after 2000 s. On increasing

the gradient further, a similar initial response is observed although the normalized

CW bias recovers faster to the steady state value. For very high gradients (G =

1.6 M/m), the bias takes a negligible value up to 500 s indicating very large run

lengths. However, the recovery to the steady state value is achieved within 1500

s, indicating faster adaptation at higher gradients. Note that we were unable to

measure t