Table of Contents
Introduction.......................................................................................................................................
4
Methods............................................................................................................................................
7
Results .............................................................................................................................................
9
Conclusions.....................................................................................................................................
11
Acknowledgements.........................................................................................................................
12
References......................................................................................................................................
13
Appendices.....................................................................................................................................
14
A. Polynomial Curve
Fitting Method....................................................................................
14
B. Integral Method..............................................................................................................
19
C. Simulation Input File........................................................................................................
21
List of Tables
Table 1 Mean squared error in
calculated minimum positions.................................................
10
Table 2 Chi-squared test of calculated
minimum positions...................................................... 10
List of Figures
Figure 1 SPR measurements using
a glass prism in the Kretschmann configuration..................... 4
Figure 2 Schematic of the SPR device.....................................................................................
4
Figure 3 SP reflection coefficient R as
a function of incident angle j..........................................
5
Figure 4 Theoretical graph of the shift
in minimum position over time......................................... 5
Figure 5 Integrals B and A to
determine shift............................................................................
7
Figure 6 Illustration of curve
mixing in an SPR simulation..........................................................
8
Figure 7...... Shifts in minimum
positions over time..........................................................................
9
Introduction
Surface plasmon resonance (SPR) is
a novel method in biochemical research that allows for the real-time study
of the structure and function of biological molecules at the interface between
surfaces with different electrical properties. Measurements taken of this
phenomenon can enable biologists to directly observe the kinetics of interactions
between biomolecules, encompassing a wide variety of applications. These
include use in drug discovery, gene expression studies, and evaluation of
surfaces for biocompatibility [1].
![Text Box: FIGURE 1. SPR measurements using a glass prism in the Kretschmann configuration: (a) focused incident light, (b) CCD detector, (c) biolayer, (d) flow cell [1].](./SPRPaperISEF_files/image001.gif)
![Text Box: FIGURE 2. Schematic of the SPR device. An attached computer controls the pump, valves, temperature controller, and collection of data from the CCD detector [1].](./SPRPaperISEF_files/image002.gif)
Surface
plasmons, or SPs, are electromagnetic waves propagated at the interface between
a metal and a non-metallic medium that have distinct dielectric permeabilities
of opposite sign. In biotechnological applications, they are often excited
using the Kretschmann configuration (Figs. 1, 2); light from a laser or LED
is directed at an angle of incidence j through
a glass prism onto a glass substrate containing a biomolecular layer on top
of thin gold and chromium layers [1]. The light is then reflected from the
biolayer, and its intensity at various angles is recorded by an array of
charge-coupled devices [2]. Over time, as various chemicals are pumped through
a flow cell in the substrate, reactions take place that change the thickness
and the refractive index of the biolayer.
![Text Box: FIGURE 3. SP reflection coefficient R as a function of incident angle j [1].](./SPRPaperISEF_files/image008.gif)
SP
reflectivity at a given time is a function of the incident angle j (Fig.
3); the values of these intensities for a range of angles form a polynomial
curve with a sharp minimum at a specific angle q. The shift of the SPR
minimum position during a reaction is caused by alteration of the refractive
index of the biolayer as its thickness changes. It can be theoretically expressed
by the function
(1)
where qo and qf are
the initial and final minimum positions and K is the rate at which
saturation, or the complete reaction of the biolayer, is achieved (Fig. 4).
Further, the change in thickness is linearly proportional to the shift in
the minimum position for thin (less than 40 nm) films. The rate of reactions
taking place in the biolayer can thus be characterized by calculating the
change in SPR minima as a function of time.
This research developed and investigated
the use of two computational methods for calculating the shift in minimum
position in parallel with collection of SP reflectivity curves in real time.
The first, a modification of a traditional method, fits each curve to a polynomial
function and calculates its minimum from the function’s derivative. The second
directly calculates the shift from the values of points in each curve. These
algorithms were tested with a real-time simulation to determine their accuracy,
precision, and efficiency.
Curve fitting is accomplished by a
weighted version of the least squares method [3]. The reflection coefficient Ri of
the incident angle ji at
a given time can be approximated by a polynomial function such that
. (2)
The least squares method finds values for coefficients a1 through aN that
fit M data points in an SP reflectivity curve to an Nth degree
polynomial so as to minimize the mean squared error, calculated from
. (3)
The coefficients are calculated by constructing
an NxN matrix a and a vector b of length N such
that
(4)
where
(5)
and
. (6)
The normal equations they describe are solved by
Gauss elimination with backward substitution [4]. Weighting of data points
known to have less error, such as those nearer to the minimum of the reflectivity
curve, is done by dividing each item in the summations of akj and bk by
the variance in ji. C++
functions to perform this curve fitting are in Appendix A.
The second method uses numerical integration
to directly determine the shift in minimum position at each time. The area
under an SP reflectivity curve can be divided into two portions, B and A (Fig.
5). After calibrating to choose a division point such that B and A are
equal, the shift in the minimum position from its initial value can be given
by
(7)
where qR is the convergence of incident light, that is,
the range of angles at which light hits the biolayer, and s is an
empirical constant property of that surface [5]. Subroutines for this method
are in Appendix B.
![Text Box: FIGURE 5. Integrals B and A to determine shift. The intensities of B and A are initially balanced (area under solid line). A shift in minimum position (dotted line) results in an imbalance that can be used to calculate it [5].](./SPRPaperISEF_files/image036.gif)
Developing such algorithms for faster and
more accurate data collection and processing can be of great benefit to the
SPR technique, especially in the parallel, multitask environment of multi-channel
imaging systems. Such devices are currently being constructed for use in
high-throughput drug screening, biomarker discovery, assay development and
validation, and protein characterization, function, and interaction studies.
Methods
A simulation of SPR data collection
and processing in the LabWindows/CVI programming environment was employed
for the evaluation of the two algorithms. This application received input
of the initial and final thicknesses and refractive indices of the biolayer
being modeled, calculated initial and final SP reflectivity curves from these
numbers by inserting them into complex matrices, and used them in the generation
of curves over the course of a reaction taking place in real time. As each
curve was generated, the algorithm being tested was used to determine the
shift in minimum position from the initial state at that time [6].
The simulation first ran a one-minute
calibration period in which no reaction took place by generating initial
curves with added noise at each time; the minima of these provided a baseline
from which to calculate the shift that would take place. As the experiment
progressed over time and chemicals were washed over and reacted with the
biolayer, changes in the curve were modeled by mixing the two inputted endpoints;
the reflectivity at time t at an angle j was
(8)
where K was
a user-inputted constant dictating the rate at which the reaction and the
subsequent shift in the reflectivity curve occurred (Fig. 6). As each curve
was generated, its minimum was determined and stored in a data file for later
analysis.
For the two methods of data processing under
consideration, this simulation was run five times, once each with changes
in final thickness of the biolayer of 25, 50, 100, 150, and 200 Å. K for
each trial was given as 5, so that full saturation, signified by a complete
shift, could be achieved in one minute; s for the integral algorithm
was given as 2.11. Refractive indices and other constants used to generate Ro and Rf can
be found in Appendix C.
Results from each simulation were statistically
analyzed for accuracy, precision, and efficiency. The mean squared error
between the actual and calculated minima served as a measure of accuracy.
Precision was derived from taking the chi-squared error in calculated minima
from a smoothed curve fit to them. Efficiency was a function of the number
of curves generated and analyzed during the one-minute simulation of a reaction.
Results
Comparison of simulation
results for the polynomial fit and integral algorithms showed significant
differences in their behavior over the one-minute testing period. Both methods
Figure 7.
Shifts in minimum positions over time: (a) for a change
in final thickness of 25 Å, (b) for a change of 50 Å, (c)
for a change of 100 Å, (d) for a change of 150 Å, (e) for
a change of 200 Å.
|
|
produced graphs of the shift in
minimum position that matched the exponential shape of a theoretical graph
for small shifts of 25 and 50 Å (Figs. 7a, 7b). As the size of the shift
increased, the shape of the graph of minimum positions calculated with integrals
was unaffected. However, the shape of those calculated with the polynomial
fit departed from that of the theoretical graph, with the size of the shift
increasing less rapidly at the beginning of and near saturation in each trial
and more rapidly at mid-saturation (Figs. 7c, 7d, 7e).
Table 1. Mean
squared error in calculated minimum positions.
|
Algorithm
|
Change
in Thickness
|
|
25 Å
|
50 Å
|
100 Å
|
150 Å
|
200 Å
|
|
Polynomial
|
0.000175
|
0.000777
|
0.009444
|
0.117891
|
0.270464
|
|
Integral
|
0.007092
|
0.009589
|
0.001824
|
0.029313
|
0.008228
|
Table 1 shows that
the accuracy of the two methods also varied depending on the change in thickness
being measured. For the smaller shifts of 25 and 50 Å, the polynomial fit
was found to more closely approximate the theoretical values, producing mean
squared errors of a full order of magnitude less than those of the integral
method. Greater shifts, however, resulted in the mean squared error in minimums
found using integrals being less than that of minimums found from polynomial
fits. For the largest shift tested, that of 200 Å, the error in the polynomial
algorithm was of two orders of magnitude greater than that in the integral
algorithm.
Table 2. Chi-squared
test of calculated minimum positions.
|
Algorithm
|
Change
in Thickness
|
|
25 Å
|
50 Å
|
100 Å
|
150 Å
|
200 Å
|
|
Polynomial
|
6.9255 ´ 10-6
|
0.00001
|
0.0008
|
0.00085
|
0.00036
|
|
Integral
|
3.6192 ´ 10-6
|
3.5731 ´ 10-6
|
3.1856 ´ 10-6
|
4.2424 ´ 10-6
|
0.00002
|
Performing chi-squared
tests against smoothed kinetic curves showed the integral algorithm to be
consistently more precise than the polynomial one, as seen in Table 2. For
the smallest thickness of 25 Å, the polynomial algorithm recorded almost
twice the variation in shift in minimum position as the integral. This difference
was more pronounced for larger thicknesses, reaching a maximum of 250 times
greater at a thickness of 100 Å.
The two methods further
differed greatly in running time. The polynomial fit algorithm, on the system
used to run the simulation, was capable of collecting and analyzing an average
of 245 SP reflectivity curves during the one-minute washing period, or approximately
one curve every 0.2449 seconds. The integral algorithm, on the other hand,
collected and analyzed an average of 1923 curves during that same period,
or approximately one curve every 0.0312 seconds; this corresponded to an
increase in speed of 785% over the polynomial fitting.
Conclusions
Based on results with
simulated data, the integral algorithm appears recommendable for use in SPR
experimentation. Its consistently low level of error and high precision in
comparison to the polynomial method give credence to its ability to accurately
process data on actual reactions. Furthermore, its fast processing speed,
an almost eightfold increase over the polynomial method, would enable the
collection and analysis of that many more SP reflectivity curves in a given
period, allowing detection of multiple biochemical reactions in real time.
Testing of this algorithm in actual experimental conditions is now in progress.
The polynomial algorithm
surprisingly displayed high variation in accuracy and precision, especially
as the change in thickness of the simulated biolayer increased (Fig. 7).
Since the algorithm itself is known to accurately find the minimum of a polynomial
curve, discrepancies may lie in the nature of the adsorption process that
is modeled with Eq. 8. Analysis of the reflectivity curves produced by the
simulation program for large changes in thickness reveals that, when the
biolayer is only partially reacted, the curve mixing algorithm used to generate
data results in a predicted curve with two minima. Such a reflectivity curve
can also be detected in real experiments [7].
The integral algorithm
does not display this occurrence because it measures area rather than the
exact minimum; the difference in area that this method relies on remains
the same whether one or two minima are present. The polynomial algorithm,
however, does; at mid-saturation, it shifts from following the movement of
one minimum position to tracking that of the other. The behavior of the simulation
can thus account for differences in shape between the graphs of the theoretical
minima and those determined by the polynomial algorithm, which translate
into decreased measures of accuracy and precision.
Testing under experimental
conditions is therefore necessary to ascertain the true reliability of the
polynomial algorithm. These observations can also be applied in future research
to create an improved simulation of SPR. Such an application could more accurately
predict the behavior of a variety of reactions studied with SPR, allowing
confident modeling of experiments before they are performed.
Acknowledgements
I would like to extend
my gratitude to my mentor, Dr. Vitalii Silin of the National Institute of
Standards and Technology, for providing me with the opportunity to conduct
research with him and with guidance throughout my work. I also wish to thank
Dr. Glenda Torrence, my senior research advisor, for her continuous support.
References
[1] Silin, V., & Plant, A. (1997). Biotechnological
applications of surface plasmon resonance. Trends in Biotechnology,
15, 353-359.
[2] Davies, J., & Faulkner, I. (1996). Surface
plasmon resonance – Theory and considerations. Surface Analytical Techniques
for Probing Biomaterial Processes (Davies, J., ed.). Boca Raton, FL:
CRC Press.
[3] Press, W.H., et al (1988). Numerical recipes in C: The art of scientific
computing. Cambridge: Cambridge University Press.
[4] Sedgewick, R. (1992). Algorithms in C++. Boston: Addison-Wesley.
[5] Tao, N. J., Boussaad, S., Huang, W. L., Arechabaleta, R. A., and D’Agnese,
J. (1999). High resolution surface plasmon resonance spectroscopy. Review
of Scientific Instruments, 70, 12, 4656-4660.
[6] Silin, V.I., Balchytis, G.A., & Yakolev,
V.A. (1993). Study on surface plasmon line broadening due to protein adsorption
on a modified silver surface. Optics Communications, 97, 19-24.
[7] Merle, H. J., Alberti, B., Schwendler, M., & Peterson, I. R. (1992).
Surface plasmon observation of striations in Langmuir-Blodgett films. Journal
of Physics D: Applied Physics, 25, 1556-1558.
Appendices
A Polynomial Curve Fitting Method
/***
FitDataDeg
-
Finds a polynomial of degree porder from the input data (datain[]) using
the Least Squares Method and weighting values
-
Returns the polynomial values in dataout[] and the coefficients in coef[].
-
Returns the minimum and halfwidth.
- Performs calculations
with degrees in the X axis.
***/
int
FitDataDeg(double XData[], double YData[], double dataoutX[], double dataoutY[], double
coef[], double *minimum, double *halfwidth, double *intensity,
int
*numPlotPoints)
{
double
z[DATALEN], yy[DATALEN], weight[DATALEN];
double
mse;
double
*recoef, *widthCoef;
double*
ms=&mse;
int
i,k;
double
f[DATALEN],width=0.0,nwidth=0.0, mserro[DATALEN];
double
SPRmax, SPRmin,anglemin,SPRangleMin;
int
imax, imin, pointNumEnd,ip;
double*
pwidth =&width;
double*
pnwidth =&nwidth;
double
yLimit, widthPosition, calibration;
int
porder;
int
pointNumbeg;
int
err;
err
= GetCtrlVal (gOptionsPanel, Options_porder, &porder);
recoef
= malloc ((porder - 1) * sizeof(double));
widthCoef
= malloc (porder * sizeof(double));
err
= GetCtrlVal (gOptionsPanel, Options_ylimit, &yLimit);
err
= GetCtrlVal (gOptionsPanel, Options_pointnum, &pointNumbeg);
err
= GetCtrlVal (gOptionsPanel, Options_widthpos, &widthPosition);
err
= GetCtrlVal (gOptionsPanel, Options_calibration, &calibration);
MaxMin1D
(YData, DATALEN, &SPRmax, &imax, &SPRmin, &imin);
anglemin=XData[imin];
for
(i=0,ip=0;i<DATALEN;) {
if
(YData[i]>yLimit)
i++;
else
{
dataoutX[ip]=XData[i]-anglemin;
z[ip]=(YData[i]-widthPosition);
if
(YData[i] > (SPRmin + 0.02))
weight[ip]
= 1;
else
weight[ip]
= 1 + (SPRmin + 0.02 - YData[i]) / 0.0001;
i=i+pointNumbeg;
ip++;
}
}
pointNumEnd=ip;
if
(pointNumEnd > porder - 1) {
PolyFit2
(dataoutX, z, weight, pointNumEnd, porder, dataoutY, coef);
for
(i=0;i<pointNumEnd;++i) {
for
(k=0;k<porder;++k) {
if
(k==0)
f[i]=coef[0];
else
{
yy[k]=coef[k]*pow(dataoutX[i],k);
f[i]=f[i]+yy[k];
}
}
}
SPRwidth
(coef, &width, &nwidth, porder);
SPRangleMin=SPRminimum(coef,recoef,width,nwidth,SPRangleMin,
porder);
Reverse
(recoef, porder-1, recoef);
for
(i=0;i<pointNumEnd;++i) {
for
(k=0;k<(porder-1);++k) {
if
(k==0) {
f[i]=recoef[0];
}
else
{
yy[k]=recoef[k]*pow(dataoutX[i],k);
f[i]=f[i]+yy[k];
}
}
}
for(ip=0;ip<pointNumEnd;ip++)
{
dataoutX[ip]=dataoutX[ip]+anglemin;
dataoutY[ip]=(dataoutY[ip]+widthPosition);
z[ip]=z[ip]+widthPosition;
}
for
(ip=0;ip<pointNumEnd;++ip) {
mserro[ip]=z[ip]-dataoutY[ip];
}
*minimum
= (anglemin + SPRangleMin);
*halfwidth
= (width+fabs(nwidth));
*numPlotPoints
= pointNumEnd;
*intensity
= SPRmin;
}
else
{
*minimum
= 0;
*halfwidth
= 0;
*numPlotPoints
= 0;
*intensity
= 0;
}
free
(recoef);
free
(widthCoef);
return
0;
}
/***
PolyFit2
-
Uses normal equations with Gaussian elimination and weighting values to
fit data to a polynomial of order n.
-
Returns the coefficients of the polynomial in coeff[].
***/
void
PolyFit2 (double x[], double y[], double weight[], int n, int order, double
z[], double coeff[])
{
double
*f[DATALEN], *A, *B;
double
t, u;
int
i, j, k;
for
(i = 0; i < order; i++)
f[i]
= malloc(n * sizeof(double));
A
= malloc(order * order * sizeof(double));
B
= malloc(order * sizeof(double));
for
(i = 0; i < order; i++) {
for
(j = 0; j < n; j++) {
f[i][j]
= 1;
for
(k = 0; k < i; k++) {
f[i][j]
*= x[j];
}
}
}
for
(i = 0; i < order; i++) {
for
(j = 0; j < order; j++) {
t
= 0.0;
u
= 0.0;
for
(k = 0; k < n; k++) {
t
+= f[i][k] * f[j][k] * weight[k] * weight[k];
u
+= f[j][k] * y[k] * weight[k] * weight[k];
}
A[i
* order + j] = t;
B[j]
= u;
}
}
LinEqs
(A, B, order, coeff);
for
(i = 0; i < n; i++) {
t
= 0.0;
for
(j = 0; j < order; j++) {
u
= 1.0;
for
(k = 0; k < j; k++) {
u
*= x[i];
}
t
+= coeff[j] * u;
}
z[i]
= t;
}
free(f);
free(A);
free(B);
return;
}
/***
SPRwidth
-
Finds the SPR width. Returns a positive width and negative width. Actual
width is the sum of the two widths (pwidth + fabs(pnwidth))
***/
double
SPRwidth (double coef[], double* pwidth, double* pnwidth, int porder)
{
int
i,k,kk;
double
width, nwidth;
double
* widthCoef = malloc (porder * sizeof(double));
double
* positivewidth = malloc (porder * sizeof(double));
double
* negativewidth = malloc (porder * sizeof(double));
ComplexNum
* widthRoots = malloc (porder * sizeof(ComplexNum));
Reverse
(coef, porder, widthCoef);
CxPolyRoots
(widthCoef, porder, widthRoots);
k=0;
kk=0;
for
(i=0; i<(porder-1); ++i) { if(widthRoots[i].imaginary
!=0.0)
continue;
if
(widthRoots[i].real>0.0) {
positivewidth[k]=widthRoots[i].real;
if
(k==0) {
width=positivewidth[k];
k++
;
continue;
}
if
(width>positivewidth[k]) {
width=positivewidth[k];
k++;
}
continue;
}
if
(widthRoots[i].real<0.0) {
negativewidth[kk]=widthRoots[i].real;
if
(kk==0) {
nwidth=negativewidth[kk];
kk++;
continue;
}
if
(nwidth<negativewidth[kk]) {
nwidth=negativewidth[kk]; kk++;
}
continue;
}
}
*pwidth=width;
*pnwidth=nwidth;
free(widthCoef);
free(positivewidth);
free(negativewidth);
free(widthRoots);
return
0;
}
/***
SPRminimum
-
Chooses the SPR angle minimum
***/
double
SPRminimum (double coef[], double recoef[], double width, double nwidth,
double SPRangleMin, int porder)
{
int
i,k;
double
* tmpCoef = malloc (porder * sizeof(double));
ComplexNum
* polyRoots = malloc (porder * sizeof(ComplexNum));
k=0;
for
(i = 0; i < porder; i++)
tmpCoef[i]
= coef[i];
for
(i=0; i<porder; ++i) {
if(i==0)
continue;
tmpCoef[i]=tmpCoef[i]*i;
k=(porder-1)-i;
recoef[k]=tmpCoef[i];
}
CxPolyRoots
(recoef, porder-1, polyRoots);
for
(i=0; i<(porder-2); i++) {
if
(polyRoots[i].imaginary != 0.0)
continue;
if
(polyRoots[i].real < 0.1*nwidth)
continue;
if
(polyRoots[i].real > 0.1*width)
continue;
if
(fabs(polyRoots[i].real) > 0)
SPRangleMin=polyRoots[i].real;
}
free(tmpCoef);
free(polyRoots);
return
SPRangleMin;
}
/***
B Integral Method
initIntegData
- Must call this
function once before fitting is done. It will initialize variables needed
to check ranges.
***/
int
initIntegData(void)
{
firstfit
= 1;
return
0;
}
/***
IntegData
-
Finds the shift in minimum position between an initial value and the input
(YData[])
-
Returns the minimum and halfwidth.
- Performs calculations
with degrees in the X axis.
***/
int
IntegData(double XData[], double YData[], double *minimum, double *halfwidth,
double *intensity)
{
double
AData[DATALEN], BData[DATALEN];
double
A, B;
double
SPRmax, SPRmin;
double
dt = (gEndDeg - gStartDeg) / DATALEN;
int
curCenter;
int
imax, imin;
int
i;
MaxMin1D
(YData, DATALEN, &SPRmax, &imax, &SPRmin, &imin);
*intensity
= SPRmin;
if
( firstfit == 1 ) {
firstfit
= 0;
*minimum
= initMin = XData[imin];
curCenter
= 3;
A
= 999;
B
= 0;
while
(A > B) {
curCenter++;
for
(i = 0; i < curCenter; i++)
BData[i]
= YData[i];
for
(i = curCenter; i < DATALEN; i++)
AData[i-curCenter]
= YData[i];
NumericIntegration(BData,
curCenter, dt, 3, &B);
NumericIntegration(AData,
DATALEN-curCenter, dt, 3, &A);
}
center
= curCenter;
*halfwidth
= SPRwidth(XData, YData);
}
else
{
for
(i = 0; i < center; i++)
BData[i]=YData[i];
for
(i = center; i < DATALEN; i++)
AData[i-center]=YData[i];
NumericIntegration(BData,
center, dt, 3, &B);
NumericIntegration(AData,
DATALEN-center, dt, 3, &A);
*minimum
= initMin - ((A-B)/(A+B)) * (gEndDeg-gStartDeg) / 2.11;
*halfwidth
= SPRwidth(XData, YData);
}
return
0;
}
/***
SPRwidth
-
Finds the SPR width.
***/
double
SPRwidth (double XData[], double YData[])
{
double
widthPos;
double
lwidth, rwidth;
int
i;
int
err;
err
= GetCtrlVal(gOptionsPanel, Options_widthpos, &widthPos);
i
= 0;
while
(i < DATALEN && YData[i] > widthPos)
i++;
lwidth
= XData[i];
i
+= 10;
while
(i < DATALEN && YData[i] < widthPos)
i++;
rwidth
= XData[i];
return
(rwidth - lwidth);
}
C Simulation Input File
06 8000.0 760. 3 5.2 0
807. 36.0 88. 36.0
173. 6.7 180.9 6.7
473. 26.6 544. 26.6 Constants defining
4000.0 1000. 3 3.5 curve shape
87. 311.0 43.
178. 3.6 7.
544. 1.56 26.
H: .00000010 .00000000 .00000010 Thickness of metal layer in meters
H: .00000540 .00000000 .00000540 Initial thickness of biolayer in meters
H: .00000000 .00000000 .00000000
H: .00000000 .00000000 .00000000
BI: 6600. 1000.0 1000.0
4 1.515 0.00 1. 0.00000001
4 3.6 4.35 1. 0. Refractive indices and dielectric
4 0.16400 3.50 1. 0. constants of chromium, gold, the
4 1.452 0.00 1. 0. biolayer,
and air
4 1.3333 0.000 -16.1 0.775
4 1.3333 0.000 16.00 0.
68.00 0.00977 78.000 Minimum, increment, and maximum incident angle
64.000 76.00
speed and accuracy
of an SPR multi-channel imaging technique. Two algorithms for finding minima
of SPR reflectivity curves and the corresponding change in thickness of
an observed biolayer – a fitting of the curve to a polynomial and a mathematical
function of a differential ratio between two integrals of the curve – were
programmed, used to analyze sets of simulated data, and compared with theoretical
results. The polynomial algorithm was found to be more accurate for changes
of less than 100 Å, while the integral algorithm was more accurate for
larger shifts. The integral method was also up to 250 times more precise
and displayed an almost eightfold increase in speed over the polynomial
method. Differences between the theoretical kinetic curve and that produced
by the polynomial method were attributed to the design of the simulation.
Both are to be further tested under actual experimental conditions.