Restriction in the membrane diffusion over the dividing septum of Escherichia Coli cells measured by Fluorescence Recovery After Photobleaching

Restriction in the membrane diffusion over the dividing

septum of Escherichia Coli cells

measured by

Fluorescence Recovery After Photobleaching

Restriktion i membrandiffusionen över septum i

Escherichia Coli under celldelning

undersökt med

Fluorescence Recovery After Photobleaching

BILL SÖDERSTRÖM

Master’s Thesis at Experimental Biomolecular Physics

Supervisor: Prof. Jerker Widengren

Examiner: Prof. Jerker Widengren

Abstract

This diploma thesis is a follow up work of a previous paper by Johan Strömqvist et. al. [1]

In their article they presented a novel method on how to determine the Z-ring radius and

the contraction in dividing Escherichia coli. This was done by fluorescent recovery after

photobleaching (FRAP) measurements in the cytosol of E.coli cells transfected to express

the Enhanced Green Fluorescent Protein (EGFP). The E.coli cell was irreversibly bleached

on one side of an already visible invagination in the midcell, and then the fluorescence

recovery was followed in time. Since the fluorescence recovery depends on the cross

sectional area of the septum that information could be used to derive a mathematical

expression to estimate the septal radius of the cell.

Sammanfattning

Detta examensarbete är en fortsättning på en artikel gjord av J. Stömqvist med flera. [1]

I deras artikel presenterades en ny metod att estimera radien på Z-ringen och följa

kontraktionen av densamma på bakterien Escherichia coli (E.coli) under cell delningen. Detta

gjordes med en teknik kallad fluorescence recovery after photobleaching (FRAP).

E.coli-bakterierna var transfekterade på sådant sätt att de uttryckte Enhanced Green

Fluorescencet Protein (EGFP) i cytosolen. Bakterierna blektes sedan irreversibelt på endera

sidan av en redan väl synlig invagination och den efterföljande återhämtningen av

fluorescence monitorerades. Då tiden för fluorescenceåterhämtning beror av öppningens

storlek kunde en matematisk formel härledas för att på så sätt estimera septums radie.

I detta projekt har samma tankesätt gjort det möjligt att utifrån vetskapen om radien

studera diffusionsförmågan hos den membraninbäddade fluoroforen DiD.

Contents

1 INTRODUCTION

1

PART I. BACKGROUND

2

2 BACKGROUND

3

2.1 Biological background 3 2.1.1 Bioluminescence 3

2.1.2 Green Fluorescent Protein (GFP) 3

2.1.3 DiD and the Cell membrane 5

2.1.4 Bacterial Cell Division 6

2.1.5 Escherichia Coli 7

2.2 Background on the physics 8

2.2.1 Introduction to Fluorescence 8

2.2.2 Fluorescence Recovery after Photobleaching (FRAP) 9

2.2.3 Confocal microscopy 10

PART II. MODELS AND EXPERIMENT 12

3.1 MODELS 13

3.1.1 Diffusion coefficient 13

3.1.2 Septal Radius of the dividing cell 15 3.1.3 The membrane models 17

3.1.3.1 The 2-dimensional model 17 3.1.3.2 The 1-dimensional model 17 3.1.3.3 The 3-dimensional models 19 3.1.3.4 Derivation of the flux through the membrane septum 20

3.1.4 Determination of the correction factor

ϕ

21

3.1.4.1

ϕ

( )

r l

,

in the 2 D case 22

)

)

3.1.4.2

ϕ

(

r l

,

in the 1 D case 26

3.1.4.3

ϕ

(

r l

,

in the cytosolic case 26

4 EXPERIMENT 27

4.1 Preparations 27

4.3 Measurements 28

4.3.1 PreFRAP 28

4.3.2 Diffusion coefficient of EGFP from open cells 29 4.3.3 Septal radius determination 29 4.3.4 Diffusion coefficient of DiD 30

4.4 Data analysis and Curve fitting 31

PART III. RESULTS AND CONCLUSIONS 32

1 Introduction 

 

During the last decades the use of fluorescence spectroscopy has increased extremely fast.

This spectroscopic technique takes advantage of the fact that molecules can be tagged with a fluorescent marker and then illuminated with monochromatic laser light in order to stimulate emission in the range from UV to IR. Fluorescence spectroscopy has shown it self to be a useful tool in a wide area of research, such as biomedical research, drug development applications, DNA sequencing, forensic investigations and even in clinical diagnostics.

Fluorescence spectroscopy is relatively harmless to the cell which makes the technique invaluable to monitor molecular interactions and dynamics in live cells.

Though harmless (at the intensities used) to the cell an intense laser light will cause bleaching of the fluorophore that can not be neglected. However, not all bleaching is unwanted. Fluorescence loss in

photobleaching (FLIP) and fluorescence recovery after photobleaching (FRAP) are two methods that use this photobleaching effect for quantitative studies of the mobility inside of living cells in vitro. In FLIP one follows the bleached fluorophore while in FRAP one monitors the recovery of fluorescence in the bleached area. FRAP is a great tool for measurements involving molecular mobilization in the cytoplasm and the membrane of both eukaryotes and prokaryotes. This thesis relies on FRAP measurements.

2 Background 

 

In this chapter the general background of the different topics that are relevant for this thesis are given, starting with a general biological background and then a few lines on the processes involved in bacterial cell division. There is also an introduction to fluorescence spectroscopy, confocal microscopy and fluorescence recovery after photobleaching given in the end of the chapter.

 

2.1 Biological background 

2.1.1 Bioluminescence   

Bioluminescence is an intrinsic chemical process in living animals that give rise to emission of light.

Bioluminescence is most abundant in deep sea life and mostly in the blue-green shifted spectra due to the fact that part of the spectra has the best optical transparency in seawater. But there’s of course a few non-sea living animals that give rise to luminescence as well, most familiar are probably fireflies and “glow worms”.

In the North West pacific outside the coast of America the jellyfish Aequorea Victoria

now so famous for its green fluorescent protein (GFP) gene has its native waters. This almost transparent organism is the origin of two, from a fluorescence point of view, important proteins; Aequorin and Green Flourescent Protein. The discovery of GFP has made such an impact on modern biological research that the scientists that made the discovery and refined the use of GFP in cellular and molecular biology were awarded the 2008 Nobel Prize in chemistry.

The photoprotein Aequorin is composed of the two different subunits; apo-aequorin and coelentreazine. The former is an apoprotein, 189 amino acids long whit an approximate molecular weight of 22 kDa [2]. Coelentreazine is a smaller molecule which upon oxidation (and binding of 3 Ca2+ _{on the apo-protein) is}

undergoing a conformational change into coelenteramide responsible for the emission of blue light typical for Aequorin.

2.1.2 Green Fluorescent Protein (GFP)   

The radiationless energy transfer from the emission of blue light from the protein Aequorin is the source of energy that will make the green fluorescent protein to go into an exited state followed by the subsequent relaxation and emission of radiation of longer wavelength, e.g. green light.

The wild-type green fluorescent protein is an 238 amino acids long polypeptide chain folded into an coaxial α-helix surrounded by an 11-stranded β barrel, forming a nearly perfect cylinder. The chromophore of GFP is consisting of the amino acid triplet Ser65_-Tyr66_-Gly67 _{which is located in the α-helical structure inside of the}

green fluorescent protein was determined in 1996 by M. Ormö et. al. In their study they found that the 11-stranded β barrel structure was 42 Å long with a diameter of 24 Å [3].

Fig. 2.1.Structure of GFP, with the chromophore triplet in the middle of the cylinder. Modified from Wikipedia.fr

The enhanced green fluorescent protein (EGFP) which was used in the experiments has a higher quantum yield and is more photostable than the native GFP. EGFP has its absorption peak at 488 nm. The emission peak is at 508 nm, which is at the lower green part of the visible electromagnetic radiation spectrum, hence the name. This was experimentally established by making a full excitation/emission spectrum of EGFP in the lab using a spectrofluorometer. The EGFP was diluted in PBS buffer to a final concentration of 170 nM.

The unexpected bump around 610-615 nm is probably due to some impurity in the sample.

350 400 450 500 550 600 650 0.0 0.2 0.4 0.6 0.8 1.0 Norm. Fluo re scence ( a .u .) wavelength (nm) Excitation 488 nm 508 nm Emission

Fig. 2.2. Measured excitation/emission spectra of EGFP

2.1.3 DiD and the Cell membrane   

The cell membrane is as highly dynamic structure composed of two lipid bilayers containing phospholipids, glycolipids, cholesterol and proteins [4]. The membrane can be considered as a two dimensional liquid, in which all the proteins and lipids can diffuse more or less freely. The phospholipids have one polar part that is

hydrophilic facing outwards and is composed by a negatively charged phosphate group, “the head”. From this head long fatty acid carbohydrate chains, ”the tails”, are pointing inwards towards the center of the lipid bilayer. These chains are highly hydrophobic. This arrangement makes the membrane a shielding barrier protecting the inside of the cell from the outside (and in some instances vice versa), since it restricts most polar molecules such as proteins, nucleic acids, carbohydrates and ions from diffusion across the membrane but in general allows passively diffusing un-polar molecules to pass [4].

 

Fig. 2.3. Part of a membrane. From Wikipedia.org

The dye 1,1'-dioctadecyl-3,3,3',3'- tetramethylindodicarbocyanine [Molecular Probes Inc. USA.] abbreviated DiD is of highly lipophillic nature which has proven to be an obstacle when it comes to uniform cell labeling in aqueous culture media [5]. This dye is member of a large family of dialkylcarbocyanine probes that are all amphiphilic. Amphiphilic molecules has a charged fluorophore that will locate the probe at the membrane surface and the lipophillic “tail” will be inserted into the membrane functioning as a anchor. The

dialkylcarbocyanines are thus useful to label model membranes and can with advantage be used in tracing assays [17].

Even though there is a bit of a labeling problem the fact that DiD has low cytotoxicity and high resistance to intercellular transfer in addition to an appropriate absorption/emission spectra makes it a well suited probe for my experiments.

500 550 600 650 700 750 800 20 40 60 80 100 Norm. Flu orescence (a .u .) w avelenght (nm ) Em ission Excitation 645nm 669nm  

Fig. 2.4. Measured excitation/emission spectra of DiD

 

2.1.4 Bacterial Cell Division 

Early in the bacterial cell division cycle the formation of a highly dynamic ring complex is accomplished by the prokaryotic FtsZ protein at the septum of the cell (division site) [6].

The analog of FtsZ in eukaryotic organisms is tubulin, a filament structure that is part of the cytoskeleton of the cell [7]. The FtsZ protein assembles into a ring structure, the Z-ring before cell division and recruits other proteins of the Fts family such as FtsW, FtsK and FtsQ [8], and other small, coiled E-coli proteins, ZapA and ZapB [9] in order to proceed with the division. Although little is known about the actual formation and structure of the Z-ring, what is known is that the FtsZ is the first protein in place in prokaryotic cell division and appears to be the driving force throughout the division cycle. The working hypothesis is that each of these Z-ring proteins are recruited in a specific order, thus genetically knocking out one of these should halt the division process and possibly make it stop.

cell before the diffusion in any other part of the cell is affected. Lastly an intermediate state would be possible, that would represent an equal distribution of protein clustering occurring in the membrane and in the cytosolic compartments simultaneously. The tree possible scenarios are shown below. The cell is seen from a cross section at the division site, the outer ring is the membrane, the cylindrical spots are hypothetical proteins and the mid circle is the cross sectional cytosolic opening of the cell.

 

If the fluorescence recovery is slower in the membrane as compared to the cytosol for a specific radius this would be an indication that it is more crowded with proteins and of course vice versa, if the recovery is slower in the cytosol one can draw the conclusion that the proteins primarily are located there instead.

It will as well be interesting to see if, and if so when (at which radial size) the diffusion coefficient will decrease since a decrease in the diffusion coefficient suggests that there is in fact some additional hindrance restricting the diffusional pathway. This would represent the first scenario.

 

2.1.5 Escherichia Coli   

The wild-type Escherichia coli (E.coli) is a rod shaped gram-negative, facultative anaerobic bacteria that is most abundant in the G.I. tract (e.g. the lower intestine ) of warm-blooded species, where it is contributing in

maintaining a normal digestion. Facultative anaerobic means that the bacterium can live in an environment containing O2, making ATP out of the oxygen and if there is no oxygen present the bacterium uses fermentation

2.2 Background on the physics 

2.2.1

 

Introduction to Fluorescence 

 

Fluorophores, or fluorescent probes, are molecules that under influence of a external radiation source are able to be electronically exited into a higher energy state followed by vibrational relaxation back to the lowest

vibrational state of the first excited energy state and eventually to the ground state emitting radiation of a longer wavelength (due to the stokes shift) in the visible (or close to it e.g. IR or UV) part of the electromagnetic spectrum.

A molecule that has the possibility to emit light is termed luminescent. The phenomenon of luminescence can be subdivided into many groups such as bioluminescence (as described above), chemoluminescence (excitation after a chemical reaction), mechanoluminescence (excitation after physical movement of an object) and photoluminescence.

The latter is in turn divided into two groups – fluorescence and phosphorescence.

Both fluorescence and phosphorescence are phenomena where an electron undergoes excitation caused by absorption of one or several photon. This excitation forces an electron to jump from its ground state into a higher energy state. If the excitation energy is high enough this may lead to excitation to a higher singlet state (S2 and

S3 in Fig. 2.5.). Then internal conversion (IC), transitions between two different electronic states with same spin

multiplicity that is non-radiative, will lead to the first singlet state. Due to what is referred to as the Frank-Condon principle the excitation can be regarded as instantaneous (~ 10-15 _{s) meaning that the vibrational levels}

of the electronic states are not adjusted. So, after excitation the electron may end up in a higher vibrational state. In time (~ 10-12 _{s) this will lead to internal vibrational relaxation within the energy state down to the lowest}

vibrational level of the first excited electronic state.

Now when the electron is in the lowest vibrational state of the first excited state it can either drop down directly to the electronic ground state (this is referred to as fluorescence) or cross over to a triplet state (which in turn may lead to phosphorescence) [11].

Fig. 2.5. Jablonski Diagram. Modified from Wikipedia.org

 

2.2.2 Fluorescence Recovery after Photobleaching (FRAP)   

The idea of Fluorescence Recovery After Photobleaching (FRAP) is quite simple and experimentally very straightforward.

The basic concept of FRAP is to use a focused laser beam to irreversibly bleach a small, well defined compartment of an object e.g. a cell and afterwards to monitor the subsequent fluorescence recovery.

Fig. 2.6. Adapted from Wikipadia.org FRAP, fluorescence recovery after photobleaching

The technique has been around for some 40 years and has become a frequently used technique when studying cells in vitro. However it wasn’t until it became possible to genetically transform cells to express GFP that the broad interest for the method increased to make FRAP a standard procedure in research labs around the world. FRAP utilizes the fact the after bleaching of a chromophore the remaining non-bleached molecules outside of the bleached area will re-distribute and also diffuse into the bleached area. Two main things can be examined by this method; the diffusion in general and the immobile fraction of molecules. The initial fluorescence intensity is

and precisely after the photobleaching (at

( )

F i

t

=

0

) the fluorescence drops to the intensity . Due to diffusion bleached and unbleached molecules will equilibrate between the bleached area and the surrounding

non-bleached areas and thus fluorescence recovery in the bleached area will occur. The total fraction of recovery is dependent on how well the fluorescent molecules can move freely or if they are immobilized due to some biophysical property, for example binding to cytoskeleton. In theory the total fluorescence recovery (at

t

= ∞

)

will be almost equal to the initial fluorescence (however if no new fluorophore is expressed during the measurement, then of course will be slightly lower than

( )

F

∞

( )

F

∞

F i

_{( )}

) even if all the molecules in the bleached area and the surrounding areas are fully mobile [13].

 

Fig 2.7. Adapted from Wikipedia.org  A, Initial intensityF i

( )

. B, Post bleach intensity F 0

( )

  C, Fluorescence recovery. D, Intensity at 

t

= ∞

F

( )

∞

 

In most cases the area of the cell that is bleached is a small focal spot, in those cases the fluorescence intensity at a late time will be approximately the same as the initial intensity but in the case of bleaching one half of the cell the recovery will return to no more than a maximum of 50 percent of the initial intensity (since no additional fluorophores are added). Consequently, in my analysis the normalized relative fluorescence recovery after photobleaching was studied.

 

2.2.3 Confocal microscopy   

In traditional fluorescence microscopy the light that hits the detector originates from the entire sample; this will create a blurry out-of-focus picture of the specimen. In the old days this was a problem, but blurriness can nowadays be solved by using the Confocal Microscopy technique. Confocal microscopy is used worldwide in labs studying fluorescence.

The principle for confocal microscopy is as follows:

emitted from the fluorophore and a pinhole to let light from that particular focal plane to pass it but to restrict all or most of the light from any other plane in the specimen, as shown in fig.2.8.

Fig.2.8. Principle of Confocal Microscopy. Modified from Wikipedia.org

The beam splitter is a dichroic mirror which is transparent for radiation above a user-defined wavelength. Compared to traditional fluorescence microscopes, the confocal microscope has great advantages, including:

• Light emitted outside of the focal plan will not be detected.

• Scanning can be performed in the x/y- plane as well as in the z-directions allowing the sample to be viewed in all three dimensions.

PART II. 

3.1 Models  

 

This chapter contains the mathematical methods and the models that were used to describe the processes that were studied, i.e. the diffusion, reflected by the fluorescence recovery, in turn influenced by the septal radii of the cells. In total there are four different models with accompanying mathematics described here, one model to determine the diffusion coefficient, one model to estimate the septal radius of the cell and two different models that simulate the membrane diffusion.

3.1.1 Diffusion coefficient 

 

D

The methodology is here shown for the determination of the diffusion coefficient of DiD but can by applied to determine the diffusion coefficient of EGFP as well since it is scaled down to the one dimensional case. For a thorough derivation of the expression of the diffusion coefficient consult the appendix.

In order to calculate the diffusion coefficient an analytical expression for the concentration in the cell after photobleaching was derived starting with Ficks second law of diffusion in one dimension:

2

c

c

2

D

t

x

∂

₌

∂

∂

∂

      

[3.1]

With boundary conditions:

(0, )

( , )

c

t

c L t

∂

∂

0

x

=

x

=

∂

∂

[3.2]

due to the fact the that the cell is a “closed compartment”. Where C is the concentration of DiD at position x at time t, D is the diffusion coefficient, x is the x-direction, t is the time and L is the length of the cell. The initial condition (at ) was set to a step function following from that the photobleaching pulse at is much shorter (in time) than the subsequent fluorescence recovery:

0

t

=

t

=

0

1 2

( ,0)

;

2

C x

c

L

x

= ⎨

c

;0

x

L

2

L

< <

⎧

< <

⎩

[3.3]

2

c

represents the concentration in the unbleached side of the cell and

c

is the concentration in the bleached side.

Fig. 3.1. Schematic figure of the situation directly after the bleaching pulse, Left side irreversible bleached

The partial differential equation of equation 3.1 has homogenous Neumann conditions so the general solution is in the form:

( , )

_k

( ) cos(

)

k o

k

u x t

u t

x

L

π

∞ =

=

∑

[3.4]

Including the boundary and initial conditions the general analytical solution to the diffusion equation of eq. 3.1 is given by:

(

_{) (}

₎

( )2 1 2 2 2 1

sin

2

( , )

2

cos

2

k L D t k

k

c

c

k

C x t

c

c

e

x

k

L

π

π

π

π

∞ − ⋅ ⋅ =

⎛

⎞

⎜

⎟

+

_⎝

_⎠

_⎛

=

+

−

⋅

⋅

_⎜

⎝

⎠

∑

⎞

⎟

[3.5]

Further assuming that the fluorescence is proportional to the concentration the relative fluorescence recovery in the bleached half of the cell can be described as:

( )

( )

( )

( )

( )

( )

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 3 5 7 1 2 1 1 2 2 1 9 11 13 15

0

0

1

2

1

1

1

2

2

0

0

3

5

7

1

1

1

1

9

11

13

15

L D t L D t L D t L D t L D t L D t L D t L D t

F t

F

F

e

e

e

e

F t

F t

F

F

e

e

e

e

π π π π π π π π

π

− ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅

−

_⎡

= −

⋅

⋅

_⎢

+

−

−

+

+

+

⎣

⎤

+

+

−

−

⎥⎦

2

[3.6]

Where only the eight first terms of the infinite sum in [3.5] is included since the rest of the terms are neglictible in comparison.

In the same way the relative fluorescence decay in the other half of the cell will follow:

These equations were fitted simultaneously to the experimentally measured recovery curves from cells that lacked a visible invagination, thus regarded as open and not in a division state. The only free parameter in the equation is the diffusion coefficient

D

. For numerical values see the results section.

3.1.2 Septal Radius of the dividing cell 

 

To determine the radius

r

, expressed as

A

=

π

r

2

, of the division site in the cell Fick´s first law was used. This

law states that the flow per time equals the diffusion coefficient times the cross sectional area of the opening times the concentration gradient:

c

D A

t

x

c

∂

_{= ⋅ ⋅}

∂

∂

∂

[3.8]

The concentration coefficient was approximated with:

( ) (

_,

c

2

c

1

)

c

r l

t

ϕ

l

−

∂

₌

_⋅

∂

[3.9]

The correction factor

ϕ

(

r l

,

)

is the ratio between the real concentration gradient

c

t

∂

∂

and an approximation

(

c

2

c

1

)

l

−

representing a straight line (linear difference in concentrations) between the two cell compartments some length

l

apart.

If there are N molecules in the cell in total and of those are bleached, then the number of unbleached

molecules is .

The concentrations are

2

N

1 2

N

= −

N

N

1 1

c

=

1

N

V

and 2 2 2

N

c

V

=

in the unbleached and bleached compartment, respectively and where

V

_i represents the volume in compartment i. Rewriting the concentration gradient as:

Ficks law can be modified to say that the number of molecules that enters the bleached area per time is:

( )

1 2

_{( )}

( )

1 2 1 1 2 1 2 1 2 1 2

,

N

V

V

,

N

,

V

V

D A

r l

N

D A

r l

D A

r l

N

V

V V

V V

dN

V

dt

l

l

l

ϕ

⎛

⎛

+

⎞

⎞

ϕ

ϕ

⎛

+

⎞

⋅ ⋅

⋅

⎜

−

⎜

_⋅

⎟

⎟

⋅ ⋅

⋅

⋅ ⋅

⋅

⎜

_⋅

⎟

⎝

⎠

⎝

⎠

⎝

=

=

−

⎠

[3.11]

This is a first order linear differential equation, for which the general solution is:

( ) ( )

( )

( )

f x dx f x dx

( )

y

′ +

f x y

=

g x

⇒ =

y

e

−

∫

⎛

⎜

e

∫

g x dx C

+

⎝

∫

⎠

⎞

⎟

[3.12]

Thus the number of molecules entering compartment 1 per time can be expressed as:

( )

( ) 11 22 , 1 1 1 2 D A r l V V t l V V

V

N t

N

C e

V

V

ϕ ⋅ ⋅ ⎛ + ⎞ − ⎜ _⋅ ⎟ ⎝

⎛

⎞

= ⋅

_⎜

_⎟

+ ⋅

+

⎝

⎠

⎠

_[3.13]

Where C is an arbitrary constant and

l

is the width of the septum, which was set to 50 nm from earlier studies with electron microscopy [15].

Since the fluorescence

F t

( )

is proportional to the concentration of fluorescent molecules one can write:

( )

( )

(

)

( ) 1 2 1 2 * , 1 1 1 2 1 2 1 2 1 2 2 1 D A r l V V t l V V

F t

N V

N V V

V

e

f t

N V

N

N

V

N

V

V

ϕ

β

⋅ ⋅ + − ⋅ ⋅ ⋅

⋅

=

=

=

+ ⋅

+

−

+

[3.14]

where β is an arbitrary constant and

( )

( )

1

( )

1 2

V

2

f t

F t

F t

V

=

⋅

+

is a correction function that corrects for drift and bleaching during the measurements [1].

The diffusion coefficient

D

*

in equation [3.14] is:

2 3

*

1

a

1 2.1044

a

2.089

a

0.948

a

D

D

r

r

r

r

⎛

⎞

⎛

⎞

⎛ ⎞

⎛ ⎞

⎛ ⎞

= −

_⎜

_⎟

⋅ −

⎜

_⎜

_{⎜ ⎟}

+

_{⎜ ⎟}

−

_{⎜ ⎟}

⎟

_⎟

⋅

⎝

⎠

_⎝

⎝ ⎠

⎝ ⎠

⎝ ⎠

_⎠

5

m

[3.15]

Where is the average hydrodynamic radius of the EGFP molecule [3]. The first factor is due to steric and the second factor to hydrodynamic hindrance, the so called centerline approximation valid when

2.4

a

=

n

0.4

r

>

6

a

r

<

m

)

or

n

[20 ] [21].

The function

ϕ

(

r l

,

was determined independently by FEM simulations described below.

3.1.3 The membrane models 

Four different models were tried in this work, one 2-dimensional, one 1-dimensional and two 3-dimensional.

3.1.3.1 The 2­dimensional model   

The 2-dimensional model consisted of two rectangles with a connection in between representing the septal region. The two rectangular areas represented the total surface area of the two cell compartments and the small connection the total septal surface area. The height of the right figure below (figure 3.2) is 2 R

π

₀ and the length is L, this means that the total area of that square is equal to the surface area of one of the shell cylinders of the left figure.

Based on the values used from [1] the length of the interconnecting rectangle between the two cell compartments were set to be

50

.

The total area covered by the two cylindrical disks facing towards the septum (in the left figure) is equal to the area spanned by the two trapezoids in the right figure. In this way the total amount of molecules that diffuses from the left side to the right side after bleaching is equal in the two figures.

The distance the molecules have to diffuse, from the edge to the septum in the model is

nm

(

R

r

)

π

⋅

−

, while in reality this distance is closer to

(

R

−

r

)

, where

R

is the constant radius of the compartments and is the radius of the septum. This is a quite crude approximation; however as a first approximation this turned out to be a close enough approximation to reality, as the results will show.

r

Fig. 3.2. 2-Dimensional model

3.1.3.2 The 1­dimensional model 

For the 1-dimensional model the geometry consisted of a line with a variable diffusion coefficient due to the difference in the cross sectional area in the different parts of the cell. This diffusion coefficient varied from

12 2

s

−

0

3.4

D

=

e

μ

m

in the cell compartments (of constant length 2500 nm) with constant radius

R

₀ down to

min 0

0

r

D

D

R

m

septal region. was set in the in the middle of (see figure 3.3) to gain a symmetrical model in respect to x. The length of the septum in this model was based on estimations of the thickness of the membranes which was approximated to , from inspection of electron micrographs of E.coli cells [15] and reference values from the CyberCell Database [18]. The difference in the septum lengths in the two models is based on that different ideas on how to specify the length of septum was used. In the 2-dimensional model the length is spanned by the membrane it self (maximal, including the thicknesses of the membranes, separation between the two daughter cells) while in the 1-dimensional model the length is set to be the minimal free length between the two membranes in the septum cleft. This reasoning probably makes the 1-dimensional model more biologically true since it was assumed that the dye was incorporated in the outer membrane based on [18].

0

x

=

D

_min

20

l

=

n

0

D

D

_mi

The line connecting and _n is a straight line described by

D x

( )

=

D

_min

+

k x

(

−

x

₀

0

)

. The slope

k

depends on the septal radius and obviously on which side of

x

=

you are. Furthermore, the length of the interconnecting line is

R

0

−

r

, meaning that for smaller a septal radius

r

the line will be longer.

The slope is:

k

0 mi 1 0

D

D

k

n

x

_±

x

_±

−

=

−

where

D

₀ and

D

_min is as described above.

x

₁₊

=

(

R

₀

− +

r

)

10

nm and

x

₀₊

=

10

nm on the positive x-axis whereas

x

₁₋

= −

(

(

R

₀

− +

r

)

10

)

nm and

x

₀₋

= −

10

nm on the negative x-axis.

The actual slope (shape) of the invagination (figure 3.4.) was of less importance based on the results from the 3-dimensional models. In those models it was shown that the geometry of the invagination not is the limiting factor in the diffusion process.

The span of simulated radii was from 6 nm to 400 nm, thus the length of the line between and varied between 494 nm and 100 nm.

0

D

D

_min

The total length over which diffusion of DiD takes place is:

[ ]

0

2 2500

2 (

)

tot

L

= ⋅

+

20

+ ⋅

R

− r

nm

Since the radial dependence of the diffusion coefficient could be factored out the model could be realized as a line in one dimension, with decreasing diffusion coefficients towards the center of the line at where the total membrane surface area is the smallest.

For the interested reader a step-by-step derivation of the model is found in appendix 2.

0

In short the diffusion coefficient can be written as:

( )

(

)

(

)

(

)

(

)

0 0 min 0 0 min min 0 0 0 0

10

10

10

10

10

[

]

10

10

10

D

x

R

D

k x

x

R

x

D x

D

x

nm

D

k x

x

x

R

D

x

R

− +

< −

+

⎧

⎪

+

−

−

+

< < −

⎪

⎪

=

_⎨

−

< <

⎪

₊

₋

_{< <}

₊

⎪

⎪

>

+

⎩

A figure to describe the model is inserted below. The upper part is the model and the lower part shows how D varies along the dividing cell, D(x).

Fig. 3.3. 1-Dimensional “membrane” model, r* is the variable radius

 

3.1.3.3 The 3­dimensional models   

As an additional study two different 3 dimensional models were implemented; one with its invagination shaped like a V ( left hand side of figures 3.4. and 3.5.), and the other with the invagination closer and tighter shaped, more like a U (right hand side of figures 3.4. and 3.5.). The thickness of the membrane was 10 nm based on reference values from The CyberCell Database [18].

The two models were tested by a FEM program, calculating the resulting fluorescence recovery curves based on equations 3.14 and 3.21.

It was found that the differences from the 1-dimensional model were very small in both cases. In addition the difference in the simulated recovery curves between the two 3-dimensional models was hardly noticeable indicating that the geometry in the septal part was of less importance. Therefore, the decision was made to focus on the 1-dimensional model instead. This decision was based on that there was no additional information gained from the 3- dimensional models.  

 

Fig.3.5. 3-dimensional models

 

3.1.3.4 Derivation of the flux through the membrane septum  

To derive an analytical expression for the recovery of the fluorescence in the membrane a modified Fick´s first law was used again:

c

D

t

x

c

∂

∂

= ⋅∅ ⋅

∂

∂

[3.16]

where

∅

is the circumference

2

π

⋅

r

of the septal region.

Equation 3.16 resembles equation 3.8, but in this case the molecules are diffusing through a “line” instead of through an area as in the cytosolic case.

The concentration coefficient was again approximated with:

( )

_,

c

2

c

1

c

r l

t

ϕ

l

−

∂

=

⋅

∂

[3.17]

If there are N molecules in the cell in total and of those are bleached then the number of unbleached

molecules is .

The concentration this time is

2

N

1 2

N

= −

N

N

1 1 1

N

c

A

=

and 2 2

=

2

N

c

A

in the unbleached and bleached compartment

respectively where is representing the area of the membrane of compartment i, rewriting the concentration gradient as:

i

( )

( )

( )

2 1 1 1 1 2 2 2 1 2 1

,

,

,

A

A

N

N

N

N

N

A

A A

c

c

A

A

dc

r l

r l

r l

dx

ϕ

l

ϕ

l

ϕ

l

⎛

+

⎞

−

₋

₋

⎜

_⋅

⎟

−

_⎝

=

⋅

=

⋅

=

⋅

1

⎠ [3.18]

Ficks law says that the number of molecules that enters the bleached area per time is:

( )

1 2

_{( )}

( )

1 2 1 1 2 1 2 1 2 1 2

2

,

,

,

DiD DiD

A

A

N

N

A

A

D

r

r l

N

D A

r l

D A

r l

N

A

A A

A A

dN

A

dt

l

l

l

π ϕ

⎛

⎛

+

⎞

⎞

ϕ

ϕ

⎛

+

⎞

⋅

⋅

⋅

⎜

−

⎜

_⋅

⎟

⎟

⋅ ⋅

⋅

⋅ ⋅

⋅

⎜

_⋅

⎟

⎝

⎠

⎝

⎠

⎝

=

=

−

⎠

[3.19]

Thus can the number of molecules entering compartment 1 per time be expressed as:

( )

( ) 11 22 , 1 1 1 2 D A r l A A t l A A

A

N t

N

C e

A

A

ϕ ⋅ ⋅ ⎛ + ⎞ − ⎜ _⋅ ⎟ ⎝

⎛

⎞

= ⋅

_⎜

_⎟

+ ⋅

+

⎝

⎠

⎠

_[3.20]

Where C is an arbitrary constant and

l

is the width of the septum.

Since the fluorescence

F t

( )

is proportional to the concentration of fluorescent molecules this can be written as:

( )

( )

(

)

( ) 1 2 1 2 , 1 1 1 2 1 2 1 2 1 2 2 1 D A r l A A t l A A

F t

N A

N

A A

A

e

f t

N A

N

N

A

N

A

A

ϕ

β

⋅ ⋅ + − ⋅ ⋅ ⋅

⋅

=

=

=

+ ⋅

+

−

+

[3.21]

where β is a arbitrary constant and

( )

₁

( )

1

( )

2

A

2

f t

F t

F t

A

=

⋅

+

is a correction function that corrects for drift and bleaching during the measurements [1].

3.1.4 Determination of the correction factor ϕ  

 

The Finite Element Method (FEM) is a method to approximate numerical solutions to partial differential equations (PDE) [19]. In COMSOL multiphysics there is an interface “Chemical Species Transport” that houses all essential tools one needs to solve diffusion problems in 1, 2 or 3 dimensions.

The idea of how to derive this correctional function

ϕ

( )

r l

,

was gained from knowledge on how it was derived in ref 1. The basic idea is to initially approximate the concentration gradient

c

x

∂

with

c

2

c

1

l

−

where c2and c1 is the average concentrations in the respective compartment.

By introducing the correctional function

( )

r l

,

c

c

2

c

1

x

l

ϕ

=

∂

−

∂

the concentration gradient can be written as:

( )

_,

c

2

c

1

c

r l

x

ϕ

l

−

∂

₌

_⋅

∂

.

ϕ

( )

r l

,

is a function which is correcting for the error in approximating the concentration gradient with the overall concentration difference in the two cell compartments.

   

3.1.4.1 

ϕ

(

r l

,

)

 in the 2 D case 

 

For the membrane model the expression for the correction function was derived as follows;

First, “Cells” are implemented in COMSOL multiphysics in 2D, trying to resemble the membrane diffusion as well as possible. The “Chemical Species Transport” package in COMSOL uses the predefined diffusion equation [eq. 3.22] with user defined boundary- [eq. 3.23] and initial conditions.

The geometry was as described in section 3.1.3.1, two rectangles (with side

2

π

⋅

R

and length L) with additional areas formed as trapezoids accounting for the faces pointing towards the septum interconnected by a smaller rectangle of variable radius. The cells were given different septum radii varying from 3.2 nm to 400 nm. Then the two-dimensional diffusion equation was applied on this problem:

( , , )

( , , )

D

( , , ) ( , , )

c x y t

D c x y t

k

x y t c x y t

t

∂

_{= Δ}

₋

_⋅

∂

[3.22]

With the boundary condition:

ˆ

( , , )

0

c x y t n

∇

⋅ = [3.23]

Which is valid since no diffusion out of the cell surface takes place, is the surface normal.

ˆn

The initial concentration was set as since it was presumed that the membrane fluorophore DiD was evenly distributed over the whole cell membrane and since it was the normalized fluorescence recovery was simulated.

0

1

c

=

In COMSOL the following parameters were used:

-12 2

3.4 10

DiD

D

=

⋅

μ

m

s

Diffusion coefficient of DiD

10 9

1

1.5 10

k

=

₋

⋅

s

Fluorescence lifetime of DiD

6

2 10

D

−

Φ

=

⋅

Photobleaching quantum yield of DiD

-3

99 10

t

B

=

⋅

s

Duration of the bleach pulse

S

max

= 7% Average steady state population of the first

exited state of DiD in the bleach pulse during

B

_t

max

S

is dependent on the laser intensity and the shape of the laser beam, and was calibrated from the experimental data.

The simulated laser beam was a step function that was zero for times larger than

B

_t and for all x >0.

By giving a value of 7% it was made sure that the whole part of the cell that was supposed to be bleached indeed was bleached during

max

S

t

B

. This was checked empirically and in line with [1].

The rate of degradation

k

_Dwas only non-zero during the bleach time

B

_t and on the negative x-axis.

S

₁

=

S

_max

⋅ < ⋅ <

(

t B

t

) (

x

0

)

Steady state population of the exited state of

DiD at times smaller than

B

_t and for x smaller then zero.

10 1

D D

k

=

k

⋅ Φ ⋅

S

Bleaching rate

Fig.3.6. Example “cell” with radius 16 nm

Each of these different geometries (radii) generated a simulated recovery curve that was later analyzed in the software package Origin 8.

0 10 20 30 40 50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 N o rm . Fl u o re s c e n c e R e c o v e ry (a .u .) Time (s)

Fig. 3.7. Simulated recovery/decay curves

The equation that was fitted into the curve was, as derived in section 3.1.3.2. :

(

)

11 22 2 2 1 2

, ,

,

DiD DiD D r A A t l A A DiD

A

F r t D

A

e

A

A

π ϕ

β

⋅ ⋅ + − ⋅ ⋅ ⋅

=

+ ⋅

+

[3.24]

Where

2

π

⋅

r

is the circumference of the septum, and is the area of each of the two cell compartments on each side of the septum,

t

is the time,

1

A

A

₂

DiD

Fitting the curves generated by the FEM model above to equation 3.24 resulted in a table showing for each

r

a unique

ϕ

:

r [nm]

φ

3.2 0.427

6.37 0.336

16 0.166

32 0.110

63.7 0.0734

99 0.0560

156 0.0450

250 0.036

318 0.033

400 0.029

   

The expression that was obtained from the dependence of

ϕ

on was a 2 exponential decay curve:

r

(

)

₂ 109 109 8.39316 83.65898 2

,

50

2.99 10

0.45583

0.09354

r r DiD

r l

nm

e

e

ϕ

₌

₌

_⋅

−

₊

− ⋅

₊

− ⋅

[3.25]

The expression above [3.25], together with the values of

ϕ

versus

r

in the table is shown in figure 3.8.:

0 100 200 300 400 0.0 0.1 0.2 0.3 0.4 0.5 ϕDi D Radie (nm)  

)

3.1.4.2 

ϕ

(

r l

,

 in the 1 D case 

 

As for the 2D case above the correction function for the 1-dimensional case was determined.

In this model the representation of the “cells” was a 1-dimensional line of variable length, due to variable radii. The diffusion coefficient varied along the line depending on where on the cell membrane it was represented, as described in 3.1.3.2. In the FEM simulations, half of the line was bleached and the subsequent recovery was calculated whereby a recovery curve could be extracted from the data.

The derived equation for the fluorescence recovery

( )

1 1 2 1 2 2 2 1 2 DiD DiD D r A A t l A A

A

F t

e

A

A

π ϕ

β

⋅ ⋅ + − ⋅ ⋅ ⋅

=

+ ⋅

+

was then fitted

to the curves to get . and are the surface areas that the model line spans if revolved 360 degrees around its base axis.

( )

r l

,

ϕ

A

₁

A

₂

The recovery curves calculated by the FEM program for different were fitted to equation 3.24 and yielded the following

r

DiD

ϕ

versus dependence:

r

(

)

₂ 109 109 6.67145 56.8825 1

,

20

1.415 10

0.31398

0.09307

r r DiD

r l

nm

e

e

ϕ

₌

₌

_⋅

−

₊

− ⋅

₊

− ⋅

[3.26]

  3.1.4.3 

ϕ

(

r l

,

)

 in the cytosolic case   

The correction factor

ϕ

_GFPfor the cytosolic simulations was derived in the same way as for the other cases but with a geometry consisting of two solid cylinders both with radius

r

=

500

nm

interconnected with a third smaller cylinder with a variable radius .The resulting correction function was determined to:

r

4 Experiment  

4.1 Preparations 

 

In order to make the measurements the first thing that had to be done was to genetically transform E.coli cells to express the gene for Enhanced Green Fluorescent Protein (EGFP). This was done at the department of

Biochemistry and Biophysics (DBB), at the Arrhenius Laboratories at Stockholm University.

A 100 μl solution of E.coli cells was mixed with 1 μl of EGFP plasmids in solution (containing as well the gene for AmpR _{) and then putted on ice for 30 minutes. Thereafter, the cells were heat shocked in a water bath (42 ºC)}

for 1 minute and then back on ice for another 2 minutes.

After diluting the sample with 500 μl LB media it was incubated at 37 ºC for half an hour before it was set on an overnight culture. The LB media was added to provide the cells with sufficient nutrition in order to make them grow and multiply in the overnight culture.

The following day a 50 μl sample of cells was harvested from the overnight culture, with 1ml LB medium and 1 μl of ampicillin [100 µg ml-1_{]. The mixture was set on continuous shaking for 3 hours at 37 ºC. The adding of}

ampicillin was to eliminate all the cells that not had taken up the plasmid in to their genome and thus not expressing the ampicillin resistance gene nor the GFP gene.

After incubation, the sample was centrifuged at 850 g for 5 minutes and washed in a PBS buffer. Lastly, 10 μl of the highly lipophillic carbocyanine fluorophore DiD was added in order to label the cell membrane, incubated for another 20 minutes and washed three times in the PBS buffer. The cells were then placed back in LB medium.

 

 

4.2 Microscopy 

 

The fluorescence microscopy measurements were performed at the Centre for Molecular Medicine (CMM) at Karolinska Institute on a Carl Zeiss ConfoCor 3 confocal laser scanning microscope. The objective used was a C-Apochromat 40x with a numerical aperture of 1.2.

The VIS-laser module held one argon ion laser (458 nm, 477nm, 488nm and 514 nm), one HeNe 543 nm laser and one HeNe 633 nm laser for continuous-wave excitation.

In order to excite EGFP and DID two different lasers was used, the 488 nm line of the argon ion laser to excite the cytosolic EGFP and for the excitation of the DiD-tagged outer cell membrane the 633 nm HeNe laser was used.

 

4.3 Measurements 

 

Cells (~ 2 μl) were mixed together with 1 μl agarose (1%) on a cover glass and thereafter placed in the

microscope. Cells that were emitting both green and red light were quite rare (all of the cells were green but just a few were red). However, there were some that did show those characteristics. There is some space for

optimization in the tagging procedure of the membrane dye, but as time did not allow that for this work and the fact that some cells were tagged with the dye the optimization had to wait and will be a thing to solve in the future. Consequently, quite some time was spent on finding candidate cells for further examination.

When chosen, the cell was further examined using the ConfoCor 3 software on the computer. In total three kinds of measurements were performed, one to determine the diffusion coefficient of EGFP and DiD of open cells, one to determine the septal radius of cell and one to determine the diffusion coefficient on cells under division.

   

4.3.1 PreFRAP   

For all kinds of measurements performed the first task was to make sure the cell was alive and functionally normal. To enable that a “good” cell had been chosen a “preFRAP” measurement was performed consisting in a partial bleaching of one of the two sub compartments with a laser power of 1.4 mW. The length of the sub compartment was set to 0.6 µm and if fluorescence recovery was achieved within ~0.8 s, the cell was considered a candidate for further examination.

 

4.3.2 Diffusion coefficient of EGFP from open cells   

For the determination of the diffusion coefficient the cells that were chosen had no visible septum in order to represent free diffusion. The next step was to irreversibly photobleach (laser excitation power was 1.4 mW here as well) on half of the cell and then to follow the fluorescence recovery. In those measurements a bleaching time of ~ 45 ms was used. After the bleach pulse the cell was scanned every 50 ms with a lower laser intensity, 42 µW, to determine the diffusion coefficient.

Fig 4.2. FRAP measurements in order to determine the diffusion coefficient of EGFP

 

4.3.3 Septal radius determination     

To determine the septal radius of the dividing cell I chose cells that had a well visible invagination and which were healthy by the requirements stated above.

In order to meet with the limitations of the model the cells that were chosen had to have as small radius as possible, preferably less than 200 nm.

This selection was not an evident and trivial matter. The picture below shows 4 different cells with radii varying from 14 to 282 nm and it is not obvious just by visual inspection to tell which one is which. At last the

conclusion was to only pick cells that had an invagination large enough to not be able to by visible inspection determine whether they were still open or had already been sealed off into two new daughter cells. In other words cells were selected where the determination of the division stage was limited by the spatial resolution.

Once a cell was chosen it was further analyzed by equation 3.14. The information that equation 3.14 gives us is the radius of the cell. However since there was some uncertainty in the diffusion coefficient of EGFP this will propagate into an uncertainty in this radius. Moreover, the correction factor

D

GFP

ϕ

determined for the

simulations was most accurate (closer to one) when the radius was small and the error that comes from that approximation will be smaller for small radii. Hence the uncertainty in the radius determination will be smaller for small radii.

4.3.4 Diffusion coefficient of DiD

To determine the diffusion coefficient of DiD the same cell that has been used to determine the radius was examined, but instead of following the intensity in the green channel from EGFP the intensity (recovery after photo bleaching) in the red channel from DiD was followed.

Fig. 4.4 DiD fluorescence.

To the cytosolic radius of EGFP was added some extra 20 nm to account for the thickness of the inner membrane, periplasma and the outer membrane. This was done since it was assumed that the dye was mainly located in the outer membrane. The idea was that if there is only free diffusion in the membrane then the diffusion coefficient would not decrease when the radius is decreasing, but if the diffusion coefficient in fact would decrease then there is a strong indication on that there is some restricting factor that comes in to play. This restriction in diffusion is then likely due to some protein complexes that are recruited during the division

4.4 Data analysis and Curve fitting 

To extract the measured data the software ImageJ (National Institute of Mental Health (NIMH), Bethesda, USA) was used. When the data was extracted the curve fitting in order to determine the diffusion coefficient and septum radius from the EGFP data and the diffusion coefficient from the DiD data was done in Origin 8 (OriginLab Corporation, Northampton, USA) software.

The order of steps that were preformed can be summarized to;

1. Determine the diffusion coefficient of EGFP and DiD in open cells respectively. The equation;

( )

( )

( )

( )

( )

( )

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 3 5 7 1 2 1 1 2 2 1 9 11 13 15

0

0

1

2

1

1

1

2

2

0

0

3

5

7

1

1

1

1

9

11

13

15

L D t L D t L D t L D t L D t L D t L D t L D t

F t

F

F

e

e

e

e

F t

F t

F

F

e

e

e

e

π π π π π π π π

π

− ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅ − ⋅ ⋅

−

_⎡

= −

⋅

⋅

_⎢

+

−

−

+

+

+

⎣

⎤

+

+

−

−

_⎥⎦

2

[4.1]

was fitted to the recovery curves for a fixed length of a cell leaving the diffusion coefficient

as the

only free parameter.

D

2. With the found value of the diffusion coefficient for EGFP, the analytical expression for the relative fluorescence recovery;

( )

( )

( ) 1 2 1 2 2 2 1 EGFP EGFP V V _t l V V

F t

V

e

f t

V

V

β

, D ⋅ ⋅Aϕ r l + − ⋅ ⋅ ⋅

=

+ ⋅

+

[4.2]

was fitted to the experimental recovery curves of the EGFP in order the extract the radii of the corresponding cells.

3. Then, assuming that the radius of the membrane is approximately the cytosolic radius plus 20 nm for the membrane thickness, the next step was to fit the derived analytical expression for the fluorescence recovery in the membrane for a fixed radius, in order to extract the diffusion coefficient

D

_DiD;

( )

1 11 22 2 2 1 2 DiD DiD D r A A t l A A

A

F t

e

A

A

π ϕ

β

⋅ ⋅ + − ⋅ ⋅ ⋅

=

+ ⋅

+

[4.3]

 

 

PART III. 

5 Results 

 

The total number of cells that were examined was just above 200. Half of those (103) were neglected from further study due to that they were already closed and did not show any recovery. Of the reaming cells another 60 % were excluded due to failure to meet with the “pre-FRAP” condition described in section 4.3.1.

The final number of cells that were examined was 40. The diffusion coefficient D of the enhanced green fluorescent protein was found to be

D

=

4.3 0.6

±

μ

m

2

s

, which is quite well in agreement with previous experimental data [16] [1]. The diffusion coefficient of the membrane bound probe DiD was

2

3.4 1.8

D

=

±

μ

m

s

. The total number of cells that where examined to extract the diffusion coefficients where 19 and 13 for EGFP and DID, respectively. Not all cells had a high enough DiD concentration to yield a sufficient signal (in the time resolution needed) to extract the diffusion coefficient. That is the reason why there is more data on EGFP than on DiD.

GFP DiD -1 0 1 2 3 4 5 6 7 8 9 D = 3.4±1.8μm2 /s Di ff usi on C oef fi ci ent [ μ m 2/s ] D=4.3 ± 0.6μm2 /s  

Fig 5.1. Diffusion Coefficients  

The model used for determination of the Z-ring radius is most valid for small radii as can be seen from fig. 5.2. For radii over ~200 nm the uncertainty becomes quite large. This is due partially to the uncertainty in the diffusion coefficient of EGFP and partially to the correction factor derived from the computer simulations.

0 10 20 30 40 0 100 200 300 400 500 R adi u s (n m) Cell number  

In figure 5.3, typical recovery/decay curve of EGFP (left) for determination of the septal radius and corresponding recovery/decay curves of DiD (right) are shown.

0 20 40 60 80 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 N o rm . re lat iv e f luo re sc en ce r e c o v e ry /dec ay ( a .u .) Time (s) r = 10.7 (+0.5/-0.7) nm 0 20 40 60 80 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 N o rm . re la ti ve fl u o re sce n c e re co ve ry/ d e c a y ( a .u .) Time (s) 0 5 10 15 20 25 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 N o rm . re lat iv e f luo re sc en ce r e c o v e ry /dec ay ( a .u .) Time (s) r = 36.6(+3.2/-2.8) nm 0 5 10 15 20 25 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 N o rm . rel a ti ve fl uo re sce nce re cov e ry /d e c a y (a.u .) Time (s) 0 10 20 30 40 50 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 No rm . re la ti ve f lu o re s c e n c e re co ve ry/d e c a y (a .u .) Time (s) r = 73.5 (+10/-7) nm 0 10 20 30 40 50 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 No rm . re la ti ve f lu o re s c e n c e re co ve ry/d e c a y (a .u .) Time (s)  

Fig. 5.3. Three sets of experimentally recorded recovery/decay curves.

In fig. 5.4 and 5.5 the relation between the cell radius and the diffusion coefficient of DiD is shown. The results are from two different models but show a similar trend in their slopes. Error bars indicate the

uncertainty in radius (in x) and diffusion coefficient (in y).

0 100 200 300 400 500 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Dif fus ion coe ff icie n t of DiD ( μ m 2 /s ) Raduis of septum (nm)  

Fig 5.4. Radius vs. Diffusion coefficient From the 2-D model    0 50 100 150 200 250 300 350 400 450 500 550 0 1 2 3 4 5 6 Dif fusion C oef fi cient DiD [ μ m 2 /s ]

Radius, outer membrane [nm]

 

Fig. 5. 5 Diffusion coefficients as a function a the radius From the 1-D model