Difference between revisions of "Team:Munich/Hardware/Detector"

Revision as of 19:06, 27 October 2017

!-- #919191 Grau1 -->

Fluorescence Detector (Lightbringer)

Our pathogen detection approach relies on Cas13a digesting RNA. A common way of monitoring RNase activities is using commercially available RNaseAlert, consisting of a fluorescent RNA beacon. This is impractical for in-field applications because commercial fluorescence detectors are expensive and inconveniently large. We therefore make our pathogen detection system fit for in-field applications by developing a cheap and handy fluorescence detector. Although many previous iGEM teams constructed fluorescence detectors, we could not find any that had a high enough sensitivity or the ability to measure fluorescence quantitatively. We therefore constructed a detector matching our requirements and compared it to others in a cost vs sensitivity diagram.

Our detector is paper-based and can detect fluorescein concentrations down to 200 nM. The detector is able to automatically measure fluorescence in units of equivalent fluorescein concentrations. It fits in a pipette box and costs less than 15 $. We were able to measure a time trace of Cas13a digesting RNaseAlert with our detector. For comparison we also measured a positive control containing RNase A and a negative control containing only RNaseAlert. The data are displayed in the figure below.

Time lapse measurement of Cas13a digesting RNaseAlert on paper using our detector. The positive control contains RNaseA and RNaseAlert. The negative control contains only RNaseAlert. Data points are connected with lines for the convenience of the eye. Error bars represent the measurement uncertainties of the detector.

The time traces show an enzymatic reaction taking place on filter paper. This proves that our detector is sensitive enough and meets our requirements. However the detector is not limited to our specific application but can be used for the detection of any fluorescence signal in biological or chemical systems. We therefore think that our detector can benefit other iGEM teams and research groups that want to make fluorescence based detection fit for in-field applications.

Overall Design

Light from a blue LED is filtered by a blue filter foil and excites fluorophores on a filter paper. The excitation light is blocked by an orange filter foil while the emission light from the fluorophores passes through the orange filter foil and illuminates a light dependent resistor (LDR). The LDR changes its resistance corresponding to the intensity of the fluorescence light. Finally an Arduino Nano measures the resistance via a voltage divider and calculates the fluorophore concentration. The two figures below show this overall design and the operational detector.

Overall design.

Operational detector.

Components

Micro Controller

We used an Arduino Nano for automatized data collection. This micro controller has analog pins that can measure voltages from 0 to 5 V and gives an integer from 0 to 1023 as output. The micro controller is connected via an USB port with a computer or smart-phone where the data can be processed further.

Light Dependent Resistor (LDR)

For the detection of fluorescence light we used a light depending resistor (LDR). A LDR decreases its resistance R_LDR with increasing light intensity I. The dependence of the resistance R_LDR on the light intensity I is

(1)

where γ is a parameter depending on the type of resistor being used and can even differ for LDRs with the same type designation.

Equation 1 is motivated from the equation

(2)

which is given in the data sheet of the LDR. The denominator is the decadic logarithm of the fraction of two light intensities of 100 Lx and 10 Lx. R₁₀ and R₁₀₀ are the corresponding resistances at these light intensities. The used resistor with the type designation GL5516 NT00183 has a parameter γ of 0.8.

The response of a LDR depends on the wavelength λ of the incoming light. The data sheet provides information on the relative response normalized to the maximal response. The relative response is maximum for a wavelength of 540 nm and is therefore appropriate for detection of green fluorophores.

Circuit for Resistance Measurements

A voltage divider as shown in the figure below is the simplest way to measure resistance.

Voltage divider.

Applying Kirchhoff’s laws we get

(3)

and

(4)

R_LDR and U_LDR are the resistance and voltage drop at the LDR. R_ref and U_ref are the resistance and voltage drop at a reference resistor. U₀ is the supply voltage which we choose to be 5V. This gives

(5)

an equation to calculate R_LDR from U_LDR, which can be measured with the micro controller.

We need to find an equation to choose an optimal resistor R_ref . We want a maximum change of U_LDR for a certain detection range of R_LDR. Therefore equation 5 is solved for U_LDR giving

(6)

The change ∆U_LDR of U_LDR between a maximum value R_max and a minimum value R_min of R_LDR is

(7)

which has a maximum for

(8)

We expect an R_max of approximately 2 MΩ and an R_min of approximately 1 MΩ. By using equation 8 as a guideline we choose R_ref to be 1.5 MΩ.

Light Emitting Diode (LED)

To detect green fluorophores we choose a blue LED with peak emission at 470 nm. For optimum performance we choose the brightest LED we could find. The used LED has a luminous intensity of 12 cd, a maximum current of 20 mA and a forward voltage of 3.2 V.

NPN Transistor

To provide a stable illumination, it is crucial to supply the LED with a constant voltage. We therefore control it via a NPN transistor with type designation BC635. Its base-emitter-on voltage is 2 V.

Control Circuit for the LED

The digital output pin is connected to the base of the transistor via a voltage divider, consisting of the resistor R₁ with a resistance of 1 kΩ and the resistor R₂ with a resistance of 9.1 kΩ. When the output pin is set to 5 V a voltage of 4.5 V is present at the base of the transistor. This is above the base-emitter-on voltage and the LED is turned on. The resistor R₃ with resistance 39 Ω was chosen empirically to limit the LEDs working current and to power it at maximum brightness. The control circuit is illustrated in the image below.

Control circuit for the blue LED.

Light Filter Foils

For filtering the emission and excitation light we used filter foils from LEE filters. The producer provides transmission spectra for every filter foil. For the excitation filter we choose the filter color "TOKYO BLUE" and for the emission filter we choose the filter color "RUST". We used our UV-Vis spectrometer to measure the spectra of different combinations of filter foils. As shown in the figure below, a combination of one orange and two blue filter foils blocks nearly all light up to 700 nm. This combination is therefore ideal for blocking the excitation light of the blue LED from reaching the LDR.

Transmission spectra of the chosen filter foils of our detector. The transmission spectra for the orange filter foil(orange graph) has nearly no overlap with the transmission spectra of the blue filter foil(blue graph). The transmission spectra of two blue and one filter foil(black graph) is therefore nearly 0 up to 700 nm.

Filter Paper

We choose glass fiber filter paper from Whatman Laboratory Products with type designation "934-AH" to detect fluorescence on. In contrast, cellulose or nitrocellulose filter paper is autofluorescent and causes a high background signal.

3D Printed Parts

We intended to put the LED,the fluorescence sample and the LDR in direct proximity, to ensures a maximum use of excitation light and emission light. We therefore chose a sandwich-like design for our sample holder that can be placed into a slot where the detection system snaps in and keeps the sample in position.

In detail, two blue filter foils are stacked and glued with tape in front of two excitation windows of the upper half of the sandwich. One orange filter foil is glued to the lower half of the sandwich. The two detection windows enable us to measure a blank sample and an actual sample with the exact same set-up. We avoid using scotch tape to cover the detection windows because tape is usually autofluorescent and causes a high background signal. A piece of filter paper is placed between the two halves of the sandwich. The upper and lower part of this sandwich are pressed together with magnets to hold the filter paper in position and ensure an user-friendly exchange of filter papers. An explosion drawing of the sample holder is shown in the figure below.

Explosion drawing of the sample holder.

The sandwich can now be inserted into the slot of the detection device. The LED and the LDR are mounted onto beams at opposite sites of the detection device. The sandwich snaps in when the LED and the LDR are at the right position under the excitation and over detection window. Four magnets apply an additional force to the beams and press the LED and the LDR close together. This design ensures that the distance between filter paper, LED and LDR is only limited by the thickness of the filter foils. An explosion drawing of the detection slot is shown in the figure below.

Explosion drawing of detection slot.

STL files of the 3D printed parts can be found here.

List of Materials and Cost Calculation

Used item	Cost in EUR
Geekcreit ATmega328P Arduino compatible Nano	2.51
USB cable	1.50
LDR	0.15
Blue LED	0.10
2 NPN Transistors	0.12
Green LED	0.10
Hole raster board	0.42
8 resistors	0.48
Filter foils	0.10
10 Magnets	4.80
ca. 100 g PLA	3.00
4 M3 screws	0.04
4 M3 nutts	0.04
Total cost	13.36

We kept an eye on using only low cost and easy available items for the construction of our detector.

Calibration

Derivation of a Calibration Function

We want to find an equation that relates fluorophore concentration c and the corresponding resistance R_LDR. The light intensity I at the LDR during a fluorescence measurement is a sum of signal intensity I_s and background intensity I_b:

(9)

We assume that I_b is the intensity for a water sample. Importantly, wet samples give a different background signal than dry ones, suspectedly due to different light scattering on the filter paper. With equation 1 this gives an equation for the resistance R_b of a water sample and for the resistance R_LDR for a fluorescence sample,

(10)

(11)

Then, the normalized resistance can be expressed as

(12)

We assume that the intensity I_s of fluorescence light depends linearly on the concentration c of fluorophores and the light intensity I₀ produced by the LED,

(13)

and that the background intensity I_b depends also linearly on I₀,

(14)

Therefore

(15)

where k is a constant that depends on the transmission spectra of the filter foils, the spectra of the fluorophore, the spectra of the LED and light scattering effects of the filter paper,but not on I₀. k can be assumed to be constant for one specific measurement set-up.

Equation 15 inserted into equation 12 gives the final equation relating c to R_LDR

(16)

Determination of the Calibration Parameter

We want to find a value for k from equation 16.

We therefore prepared 10-fold dilutions from 100 nM to 1 mM. For each measurement we pipetted 30 µl of sample on a fresh filter paper, placed it in the detector and turned on the LED. We measured the resistance R_LDR directly with a multimeter. After each measurement the detector was cleaned gently with ethanol. The first and last measurement of each series was conducted with plain water to determine R_b and to confirm the absence of contaminations. A plot of the normalized resistances is shown in the figure below. We fitted the data with equation 16 to determine a value for k using a value of 0.8 for γ gives

(17)

Normalized resistances R_LDR/R_b vs fluorescein concentration c and corresponding fit function.

To determine that k does not depend on the intensity I₀ we made two measurement series. We changed the resistance R₃ in series to the LED to dim the light Intensity I₀. We used a 39 Ω resistor and a 60 Ω resistor. For the set-up with the 39 Ω resistor we additionally measured a 200 nM sample because this is the expected final working condition of the detector.

Automatized Data Collecting and Processing

We want to verify if a reaction with a fluorescence product is taking place on a filter paper. Therefore we need to measure the resistance R_LDR over a certain time certain period. We choose to measure a data point every 5 min over a time period of 1.5 h to take trace of the reaction kinetics. Before such a measurement series a blank measurement with plain water is performed to determine the resistance R_b.

As a first step the exact value of the supply voltage U₀ needs to be measured. The supply voltage is connected via an additional voltage divider with an analog pin. The voltage divider consists of a 100 Ω resistor and a 910 kΩ resistor. The analog pin measures still a correct value for the supply voltage U₀ because the first resistor is negligible small compared to the second resistor. We did not connect the analog pin directly with the power supply to prevent the micro controller from damage in case of a short circuit or a peak voltage caused by an other component of the overall device. The micro controller measures the supply voltage 50 times with a delay time of 50 ms between measurements. It calculates the average of U₀ and the relative standard error σ_U0 , which is used as measurement uncertainty for further calculations.

To initiate a resistance measurement of the LDR the LED needs to be turned on by setting a digital output pin to 5 V. An additional green LED on the device is turned on as well to indicate that a measurement is taking place. After a waiting time of 30 s the actual measurement starts. This waiting time was determined empirically and is required because of the slow response of the LDR. U_LDR is measured in the same way as U₀. The average of U_LDR and the relative standard error σ_ULDR are calculated. Equation 5 is used to calculate R_LDR from the average of U_LDR. We derived an equation for the propagation of the relative systematic and the relative statistical uncertainty of U₀ and U_LDR. For the relative statistical uncertainty σ_stat of R_LDR we get

(18)

We used a value of 1 digit for the absolute systematic uncertainty for a voltage measurement. The relative systematic uncertainty is 1/U for a measured voltage U. For the relative systematic uncertainty σ_sys of R_LDR we therefore get

(19)

The equation for the total uncertainty σ_RLDR is then

(20)

R_LDR, R_b and their uncertainties are calculated by the micro controller, read by the computer and saved for further analysis in a text file.

Data Analysis and Final Result

With equation 16 and the fitted value for k the measured resistances can be translated into fluorescein concentrations c. Equation 16 solved for c is

(21)

The equation for the relative uncertainty σ_c of the fluorescein concentration c is

(22)

where σ_k is the relative uncertainty from the fit of k. We are now enabled to measure fluorescence in units of equivalent fluorescein concentrations c. We analysed data of a first experiment with these equations. The resulting figure is shown in the beginning of this documentation.

With these quantitative data it is now an easy task to verify if the collateral RNase-activity of Cas13a got activated by trRNA. For the convenience of the end user we choose to extract two informations from the time traces produced with our detector. We check if a reaction has taken place on the filter paper by evaluating if the first 5 data points are monotonous rising. And we check if a threshold of 3 µM equivalent fluorescein concentrations was crossed by looking at the last 3 data points. The two informations are computed in the following way, if the reaction has taken place and the threshold was crossed the detector software will raise the message "Pathogen detected", if the reaction has not taken place but the threshold was crossed the detector software will rise the message "False positive". The other two combinations will lead to the message "no Pathogen detected". The software will also create a graph similar to the graph in the beginning of this documentation to provide a deeper insight for the enduser if needed.

We think that our detector is a more reliable read out system compared to other possibilities because our detector provides deep insight into the reaction dynamics. With an exact knowledge of Cas13a's reaction kinetics and further analytic tools a nearly fail-safe read out can be achieved. A colorimetric readout, a change in color, in contrast is way more primitive and provides no insight in any dynamics.

@@ Line 1: / Line 1: @@
-<!-- #919191 Grau1 -->
+!-- #919191 Grau1 -->
 <!-- #787878 Grau2 -->
 <!-- #51A7f9 Blau1 -->
@@ Line 167: / Line 167: @@
 <div class="equationDiv"><img src="https://static.igem.org/mediawiki/2017/3/33/T--Munich--Hardware_equation1.png"><span>(1)</span></div>
 <p>
-where ? is a parameter depending on the type of resistor being used and can even differ for LDRs with the same type
+where γ is a parameter depending on the type of resistor being used and can even differ for LDRs with the same type
 designation.
 </p>
@@ Line 176: / Line 176: @@
 <p>
 which is given in the data sheet of the LDR. The denominator is the decadic logarithm of the fraction of two light
-intensities of 100 Lx and 10 Lx. <i>R<sub>10</sub></i> and <i>R<sub>100</sub></i> are the corresponding resistances at these light intensities. The used resistor with the type designation GL5516 NT00183 has a parameter ? of 0.8.
+intensities of 100 Lx and 10 Lx. <i>R<sub>10</sub></i> and <i>R<sub>100</sub></i> are the corresponding resistances at these light intensities. The used resistor with the type designation GL5516 NT00183 has a parameter γ of 0.8.
 </p>
 <p>
-The response of a LDR depends on the wavelength ? of the incoming light. The data sheet provides information on
+The response of a LDR depends on the wavelength λ of the incoming light. The data sheet provides information on
 the relative response normalized to the maximal response. The relative response is maximum for a wavelength of 540
 nm and is therefore appropriate for detection of green fluorophores.
@@ Line 220: / Line 220: @@
 <div class="equationDiv"><img class="largeEquation" src="https://static.igem.org/mediawiki/2017/3/3f/T--Munich--Hardware_equation6.png"><span>(6)</span></div>
 <p>
-The change ?<i>U<sub>LDR</sub></i> of <i>U<sub>LDR</sub></i> between a maximum value <i>R<sub>max</sub></i> and a minimum value <i>R<sub>min</sub></i> of <i>R<sub>LDR</sub></i> is
+The change ∆<i>U<sub>LDR</sub></i> of <i>U<sub>LDR</sub></i> between a maximum value <i>R<sub>max</sub></i> and a minimum value <i>R<sub>min</sub></i> of <i>R<sub>LDR</sub></i> is
 </p>
 <div class="equationDiv"><img class="largeEquation" src="https://static.igem.org/mediawiki/2017/9/9b/T--Munich--Hardware_equation7.png"><span>(7)</span></div>
@@ Line 228: / Line 228: @@
 <div class="equationDiv"><img src="https://static.igem.org/mediawiki/2017/c/c0/T--Munich--Hardware_equation8.png"><span>(8)</span></div>
 <p>
-We expect an <i>R<sub>max</sub></i> of approximately 2 MO and an <i>R<sub>min</sub></i> of approximately 1 MO. By using <a href="#equation8">equation 8</a> as a guideline we choose <i>R<sub>ref</sub></i> to be 1.5 MO.
+We expect an <i>R<sub>max</sub></i> of approximately 2 MΩ and an <i>R<sub>min</sub></i> of approximately 1 MΩ. By using <a href="#equation8">equation 8</a> as a guideline we choose <i>R<sub>ref</sub></i> to be 1.5 MΩ.
 </p>
 </td>
@@ Line 255: / Line 255: @@
 <h4>Control Circuit for the LED</h4>
 <p>
-The digital output pin is connected to the base of the transistor via a voltage divider, consisting of the resistor <i>R<sub>1</sub></i> with a resistance of 1 kO and the resistor <i>R<sub>2</sub></i> with a resistance of 9.1 kO. When the output pin is set to 5 V a voltage of 4.5 V is present at the base of the transistor. This is above the base-emitter-on voltage and the LED is turned on.
+The digital output pin is connected to the base of the transistor via a voltage divider, consisting of the resistor <i>R<sub>1</sub></i> with a resistance of 1 kΩ and the resistor <i>R<sub>2</sub></i> with a resistance of 9.1 kΩ. When the output pin is set to 5 V a voltage of 4.5 V is present at the base of the transistor. This is above the base-emitter-on voltage and the LED is turned on.
-The resistor <i>R<sub>3</sub></i> with resistance 39 O was chosen empirically to limit the LEDs working current and to power it at
+The resistor <i>R<sub>3</sub></i> with resistance 39 Ω was chosen empirically to limit the LEDs working current and to power it at
 maximum brightness. The control circuit is illustrated in the image below.
 </p>
@@ Line 468: / Line 468: @@
 We therefore prepared 10-fold dilutions from 100 nM to 1 mM. For each measurement we pipetted 30 µl of sample on
 a fresh filter paper, placed it in the detector and turned on the LED. We measured the resistance <i>R<sub>LDR</sub></i> directly with a multimeter. After each measurement the detector was cleaned gently with ethanol. The first and last measurement
-of each series was conducted with plain water to determine <i>R<sub>b</sub></i> and to confirm the absence of contaminations. A plot of the normalized resistances is shown in the figure below. We fitted the data with <a id="equation16">equation 16</a> to determine a value for <i>k</i> using a value of 0.8 for ? gives
+of each series was conducted with plain water to determine <i>R<sub>b</sub></i> and to confirm the absence of contaminations. A plot of the normalized resistances is shown in the figure below. We fitted the data with <a id="equation16">equation 16</a> to determine a value for <i>k</i> using a value of 0.8 for γ gives
 </p>
 <div class="equationDiv"><img class="largeEquation" src="https://static.igem.org/mediawiki/2017/d/d4/T--Munich--Hardware_equation17.png"><span>(17)</span></div>
@@ Line 479: / Line 479: @@
 </div>
 <p>
-To determine that <i>k</i> does not depend on the intensity <i>I<sub>0</sub></i> we made two measurement series. We changed the resistance <i>R<sub>3</sub></i> in series to the LED to dim the light Intensity <i>I<sub>0</sub></i>. We used a 39 O resistor and a 60 O resistor. For the set-up with the 39 O resistor we additionally measured a 200 nM sample because this is the expected final working condition of the detector.
+To determine that <i>k</i> does not depend on the intensity <i>I<sub>0</sub></i> we made two measurement series. We changed the resistance <i>R<sub>3</sub></i> in series to the LED to dim the light Intensity <i>I<sub>0</sub></i>. We used a 39 Ω resistor and a 60 Ω resistor. For the set-up with the 39 Ω resistor we additionally measured a 200 nM sample because this is the expected final working condition of the detector.
 </p>
 </td>
@@ Line 495: / Line 495: @@
 </p>
 <p>
-As a first step the exact value of the supply voltage <i>U<sub>0</sub></i> needs to be measured. The supply voltage is connected via an additional voltage divider with an analog pin. The voltage divider consists of a 100 O resistor and a 910 kO resistor. The analog pin measures still a correct value for the supply voltage <i>U<sub>0</sub></i> because the first resistor is negligible small
+As a first step the exact value of the supply voltage <i>U<sub>0</sub></i> needs to be measured. The supply voltage is connected via an additional voltage divider with an analog pin. The voltage divider consists of a 100 Ω resistor and a 910 kΩ resistor. The analog pin measures still a correct value for the supply voltage <i>U<sub>0</sub></i> because the first resistor is negligible small compared to the second resistor. We did not connect the analog pin directly with the power supply to prevent the micro controller from damage in case of a short circuit or a peak voltage caused by an other component of the overall device. The micro controller measures the supply voltage 50 times with a delay time of 50 ms between measurements.
-compared to the second resistor. We did not connect the analog pin directly with the power supply to prevent the
+It calculates the average of <i>U<sub>0</sub></i> and the relative standard error <i>σ<sub>U0</sub></i>
-micro controller from damage in case of a short circuit or a peak voltage caused by an other component of the overall
-device. The micro controller measures the supply voltage 50 times with a delay time of 50 ms between measurements.
-It calculates the average of <i>U<sub>0</sub></i> and the relative empirical standard deviation <i>s<sub>U0</sub></i>
 , which is used as measurement
 uncertainty for further calculations.
@@ Line 508: / Line 505: @@
 a waiting time of 30 s the actual measurement starts. This waiting time was determined empirically and is required
 because of the slow response of the LDR. <i>U<sub>LDR</sub></i> is measured in the same way as <i>U<sub>0</sub></i>. The average of <i>U<sub>LDR</sub></i> and the
-relative empirical standard deviation <i>s<sub>ULDR</sub></i> are calculated. Equation 5 is used to calculate <i>R<sub>LDR</sub></i> from the average
+relative standard error <i>σ<sub>ULDR</sub></i> are calculated. Equation 5 is used to calculate <i>R<sub>LDR</sub></i> from the average
 of <i>U<sub>LDR</sub></i>. We derived an equation for the propagation of the relative systematic and the relative statistical uncertainty
-of <i>U<sub>0</sub></i> and <i>U<sub>LDR</sub></i>. For the relative statistical uncertainty <i>s<sub>stat</sub></i> of <i>R<sub>LDR</sub></i> we get
+of <i>U<sub>0</sub></i> and <i>U<sub>LDR</sub></i>. For the relative statistical uncertainty <i>σ<sub>stat</sub></i> of <i>R<sub>LDR</sub></i> we get
 </p>
 <div class="equationDiv"><img class="largeEquation" src="https://static.igem.org/mediawiki/2017/e/e9/T--Munich--Hardware_equation18.png"><span>(18)</span></div>
 <p>
 We used a value of 1 digit for the absolute systematic uncertainty for a voltage measurement. The relative systematic
-uncertainty is 1/U for a measured voltage U. For the relative systematic uncertainty <i>s<sub>sys</sub></i> of <i>R<sub>LDR</sub></i> we therefore get
+uncertainty is 1/U for a measured voltage U. For the relative systematic uncertainty <i>σ<sub>sys</sub></i> of <i>R<sub>LDR</sub></i> we therefore get
 </p>
 <div class="equationDiv"><img class="largeEquation" src="https://static.igem.org/mediawiki/2017/0/09/T--Munich--Hardware_equation19.png"><span>(19)</span></div>
 <p>
-The equation for the total uncertainty <i>s<sub>RLDR</sub></i> is then
+The equation for the total uncertainty <i>σ<sub>RLDR</sub></i> is then
 </p>
 <div class="equationDiv"><img class="largeEquation" src="https://static.igem.org/mediawiki/2017/d/d8/T--Munich--Hardware_equation20.png"><span>(20)</span></div>
@@ Line 537: / Line 534: @@
 <div class="equationDiv"><img class="largeEquation" src="https://static.igem.org/mediawiki/2017/a/a3/T--Munich--Hardware_equation21.png"><span>(21)</span></div>
 <p>
-The equation for the relative uncertainty <i>s<sub>c</sub></i> of the fluorescein concentration <i>c</i> is
+The equation for the relative uncertainty <i>σ<sub>c</sub></i> of the fluorescein concentration <i>c</i> is
 </p>
 <div class="equationDiv"><img class="largeEquation" src="https://static.igem.org/mediawiki/2017/c/c3/T--Munich--Hardware_equation22.png"><span>(22)</span></div>
 <p>
-where <i>s<sub>k</sub></i> is the relative uncertainty from the fit of <i>k</i>. We are now enabled to measure fluorescence in units of equivalent fluorescein concentrations <i>c</i>. We analysed data of a first experiment with these equations. The resulting figure is shown in the beginning of this documentation.
+where <i>σ<sub>k</sub></i> is the relative uncertainty from the fit of <i>k</i>. We are now enabled to measure fluorescence in units of equivalent fluorescein concentrations <i>c</i>. We analysed data of a first experiment with these equations. The resulting figure is shown in the beginning of this documentation.
 </p>
 <p>