Analysis of exponential data using a noniterative technique: application to surface plasmon experiments.

Abstract

The analysis of experimental data of exponential type plays a central role in many biophysical applications. We introduce a novel noniterative algorithm to analyze the association phase and dissociation phase of surface plasmon resonance experiments. It is shown that this algorithm can determine kinetic constants with a high level of accuracy in the presence of significant levels of noise. This algorithm should provide a valuable alternative to existing data analysis techniques.

Full text

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Analysis of exponential data using a noniterative
technique: application to surface plasmon experiments
Raimund J. Ober,a,bJeffrey Caves,cand E. Sally Wardb,c,*
aCenter for Systems, Communications and Signal Processing, Eric Jonsson School of Electrical Engineering and Computer Science,
University of Texas at Dallas, Richardson, TX 75083-0688, USA
bCancer Immunobiology Center NB9.106, University of Texas Southwestern Medical Center, 6000 Harry Hines Boulevard,
Dallas, TX 75390-8576, USA
cCenter for Immunology, University of Texas Southwestern Medical Center, 6000 Harry Hines Boulevard, Dallas, TX 75390-8576, USA
Received 8 July 2002
Abstract
The analysis of experimental data of exponential type plays a central role in many biophysical applications. We introduce a novel
noniterative algorithm to analyze the association phase and dissociation phase of surface plasmon resonance experiments. It is
shown that this algorithm can determine kinetic constants with a high level of accuracy in the presence of significant levels of noise.
This algorithm should provide a valuable alternative to existing data analysis techniques.
? 2003 Elsevier Science (USA). All rights reserved.
Keywords: Surface plasmon resonance; Subspace algorithm; Parameter estimation
The analysis of empirical data is often a key aspect of
an experiment. The advent of high-powered and rela-
tively inexpensive computers has brought with it re-
newed interest in the use of more complex algorithmic
tools that had earlier not been possible to implement.
One characteristic of the data of many biophysical ex-
periments is that the data are in effect of exponential
type, i.e., that the data are the sum of data points that
can be represented as an exponential function with ap-
propriately chosen coefficient and exponent. Surface
plasmon resonance experimental data, in the absence of
mass transport, is the example that will be investigated
here. Liquid-phase NMR (see, e.g., [1–3]), fluorescence
life time (see, e.g., [4]), and sedimentation equilibrium
(see, e.g., [5]) experiments also lead to this type of data.
Early approaches to the analysis of BIAcore dissoci-
ation data for simple 1:1 interactions relied on taking
the logarithm of the data and on determining the dis-
sociation constant through a simple linear regression
analysis [6–8]. This approach is not very satisfactory for
a number of reasons. For example, taking the logarithm
destroys many basic statistical properties of the noise,
which can lead to nonoptimal estimates. This approach
can also be applied only to data resulting from 1:1 in-
teractions with nonnegative data points. Therefore, use
of a nonlinear least squares optimization routine to fit a
suitable model to the data is now often advocated [6,9].
In fact, we [10] have shown that this approach leads to
very good estimates in the sense that the variance of the
parameter estimates is consistent with the Cramer Rao
lower bound. One major disadvantage is that the opti-
mization routine is iterative and therefore suffers from
the standard convergence problems of gradient-based
iterative schemes. First, initial conditions have to be
given which requires another algorithm. Second, if the
initial conditions are not close to the actual global
minimum of the optimization problem the algorithm
might converge to a local minimum which corresponds
to an incorrect parameter estimate.
An ever greater demand for experimental throughput
also puts higher demands on the capabilities of the
data analysis software. With existing tools it is not too
difficult to analyze individual sensorgrams since poten-
tial shortcomings in the data analysis algorithms can
be overcome by user interaction with the software.
*Corresponding author. Fax: 1-214-648-1259.
E-mail address: sally.ward@UTSouthwestern.edu (E. Sally Ward).
Analytical Biochemistry 312 (2003) 57–65
www.elsevier.com/locate/yabio
ANALYTICAL
BIOCHEMISTRY
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High-throughput experimentation does, however, re-
quire algorithmic tools that depend less on user input.
A recent advance in the development of data analysis
algorithms has received a considerable amount of at-
tention in the signal processing field. The so-called
subspace algorithms (see, e.g., [11–15]) are noniterative
algorithms that exhibit high-quality estimates. In this
paper we investigate the subspace algorithm that ad-
dresses the data analysis problem encountered in the
analysis of surface plasmon resonance data as acquired
for example on a BIAcore instrument.
It is the expectation that this algorithm can become a
routine technique to analyze surface plasmon resonance
data.
Materials and methods
The experimental setup and data have been described
earlier[10].Themainaspectsarebrieflysummarizedhere.
Reagents
The mouse antibody (IgG1) was purified from culture
supernatants of the 9E10 hybridoma [16] using protein
A–Sepharose and standard methods. The myc peptide
(EQKLISEEDLN) was synthesized in the peptide syn-
thesis facility of the Howard Hughes Medical Institute
and Department of Biochemistry, University of Texas
Southwestern Medical Center.
Immobilization
The 9E10 antibody was coupled at various densities to
CM5 chips using amine coupling. Relatively high cou-
pling densities (ca. 5000–12,000 resonance units (RU))
were found to be necessary to obtain measurable signals
with the myc peptide. However, to avoid unacceptable
baseline drift these high coupling densities necessitated
the use of extensive equilibration times prior to running
experiments. For use as a control (reference) surface,
flow cell 1 of each sensor chip was treated with the
coupling chemistry using buffer without added ligand.
SPR experiments
Experiments were carried out using a BIAcore 2000
instrument (see [6]). Experiments were run using pro-
gramed methods and the BIAcore control software. Myc
peptide was used at a concentration of 38lM in phos-
phate-buffered saline, pH 7.2, containing 0.01% Tween
20 (PBST). All experiments were run at 25?C and in-
jections of analyte were carried out using 240ll and the
kinject command. Dissociation phases were sufficiently
long to allow complete dissociation of analyte from the
flow cells, therefore avoiding the need to use regenera-
tion injections. A flow rate of 20ll/min was used in all
experiments. For the present data 11 repetitions of the
experiment were analyzed.
Data analysis
Data models
The standard equation that describes the result of a
measurement in a flow cell for a 1:1 interaction is given
by (see, e.g., [6])
dR
dtðtÞ ¼ konCðtÞðRmax? RðtÞÞ ? koffRðtÞ;
where RðtÞ is the measured signal in resonance units
(RU), kon is the association rate constant, koff is the
dissociation rate constant, Rmaxis the maximum analyte
binding capacity in (RU), and CðtÞ is the concentration
of the analyte that flows over the chip.
During the association phase the analyte concentra-
tion is assumed to be constant, i.e., CðtÞ ¼: C0for
t06t6t1. During the subsequent dissociation phase the
injected analyte concentration is then set to zero; i.e.,
CðtÞ ¼ 0 for tPt1. Hence for the association phase the
signal is given by
RaðtÞ ¼ Reqð1 ? e?tkobsÞ
where kobs:¼ konC0þ koff and Req:¼ ðkonC0RmaxÞ=kobs;
for the dissociation phase it is given by
for 06t6t1;
RdðtÞ ¼ Raðt1Þe?ðt?t1Þkoff
It is important to note for the subsequent development
that both the association and the dissociation phase data
are of exponential type; i.e., they are linear combina-
tions of exponential functions. Here we also interpret
the constant function as an exponential function with
zero exponent. Specifically, if we set
Ad:¼ koff;
then the dissociation data is easily represented as
RdðtÞ ¼ cdeðt?t1ÞAdbd
Similarly, if we set
0
@
for tPt1:
bd:¼ Raðt1Þ;
and
cd:¼ 1
for tPt1:
Aa:¼
0
0
0
?kobs
1
A;
ba:¼
Req
?Req
0
@
1
A;
ca:¼ ð11Þ
then the association phase data can be represented as
RaðtÞ ¼ caetAaba
where we used here the standard definition of the matrix
exponential (see, e.g., [17]).
The measured data are of course not of the contin-
uous time type described here but are sampled data with
constant sampling intervals and are corrupted by noise,
i.e., the data are given by
for tP0;
yðkTÞ ¼ cekTAcb þ wðkÞ ¼ cAkb þ wðkÞ
for k ¼ 0;1;2;...;
ð1Þ
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for some vectors c, b, and matrix Ac, with A ¼ eTAc,
sampling interval T, and noise sequence wðkÞ for k ¼
0;1;2;....
Subspace algorithm for exponential data
The specific subspace algorithm (see, e.g., [12,13])
that we will review here determines the A, b, and c in Eq.
(1), given a data sequence yð0Þ;yð1Þ;...;yðNÞ.
First, the original data is organized into a Hankel
matrix
Hdata¼
yð0Þ
yð1Þ
...
...
...
yð1Þ
yð2Þ
..
...
..
..
..
......
.
.
.
..
.
.
yðN ? 2Þ
yðN ? 1Þ
yðN ? 1Þ
yðNÞ
2
666666664
3
777777775
:
Second, a singular value decomposition Hdata¼ USV
of the Hankel matrix is performed, where U and V are
unitary matrices, i.e., U?U ¼ I and VV?¼ I, and S is a
diagonal matrix with positive entries, the singular val-
ues, that are ordered with respect to magnitude, i.e., the
largest singular value is the first diagonal entry, etc.
The subsequent steps depend on the number of com-
ponents that make up the signal that is to be estimated.
Let k be the number of exponential components that
make up the data. In the situation discussed above, a 1:1
interaction, k ¼ 1 for the dissociation data and k ¼ 2 for
the association data. We define reduced matrices of U, V,
and S as follows. Ureducedis given by the first k columns of
U, Vreducedis given by the first k rows of V, and Sreducedis
given by the first k rows and columns of S.
The final estimates^A A,^b b, and ^ c c of A, b, and c are now
obtained as follows. The estimate ^ c c is given by the first
row of UreducedS1=2
column of S1=2
reduced, the estimate^b b is given by the first
reducedVreduced, and^A A is given by
^A A ¼ U"
where U"
first row of Ureduced, and U#
by removing the last row of Ureducedand ??Rstands for a
right inverse of ?, i.e.^A A can be obtained by solving the
associated system of linear equations.
As a last step the desired parameters have to be ex-
tractedfromtheestimated^A A,^b b,and^ c c.TothisendletWbe
adiagonalizingtransformationof^A A;i.e.,AD¼ W^A AW?1is
a diagonal matrix. Also define bD¼ W^b b and cD¼ ^ c cW?1.
Of course, for the dissociation data discussed above this
stepisnotnecessary since^A Aisascalar. In this casewecan
immediately set AD:¼^A A, bD:¼^b b, and cD:¼ ^ c c.
For the dissociation data the dissociation constant
koffis estimated by
^k koff¼ ?logðADÞ
T
reducedS1=2
reducedðU#
reducedS1=2
reducedÞ?R;
reducedis obtained from Ureducedby removing the
reducedis obtained from Ureduced
;
where T is the sampling interval. The analogous step for
the association data is slightly more delicate. For con-
sistency of notation, we assume that the first diagonal
entry of the 2 ? 2 matrix ADis closest to 1. If this is not
the case, the diagonal entries of AD, the entries of bD,
and the entries of cDneed to be transposed. Let now a1
be the first and a2be the second diagonal entries of AD.
An estimate of kobs¼ konC0þ koffis given by
^k kobs¼ ?logða2Þ
T
and an estimate,^R Req, of the equilibrium value Reqis gi-
ven by
;
^R Req¼ bDð1ÞcDð1Þ:
The analysis of data for other interaction models such
as the parallel association and dissociation of two li-
gands is carried out analogously.
Data analysis based on logarithm of data
To analyze the data in the ‘‘classical’’ approach [7]
the logarithm of all the dissociation data points is taken.
For the association data the equilibrium value Req is
estimated by averaging a number of data points for
which the equilibrium value of the association phase has
been attained. This equilibrium value is then added to
each of the association data points after these data have
been multiplied by )1. The logarithm of the resulting
data points is taken.
The observed association constant kobsand dissocia-
tion constant koffare then easily estimated by solving a
linear regression problem.
Parameter estimation by least squares minimization
The initial conditions for the gradient-based least
squares algorithm were given by a subspace estimate.
The gradient-based search algorithm to minimize the
least squares modeling error was of the Marquardt–
Levenberg type [18].
Simulation of BIAcore data
Data were simulated using the programming language
Matlab [19]. Zero mean independent Gaussian noise was
added to each data point. The standard deviation was
chosen to be 0.25 RU in line with the measured noise
level in our BIAcore 2000 instrument (see [10]), which is
slightly lower than the noise standard deviation of 0.3
RU as reported in the instrument manual [6].
Data processing and software environment
All data were analyzed and processed using custom-
written software in the programming language Matlab.
The sensorgrams were all zero-adjusted andreference cell
data were subtracted. For the analysis of the association
R.J. Ober et al. / Analytical Biochemistry 312 (2003) 57–65
59

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and dissociation phases suitable data segments were cut
out from the sensorgrams. The software can be obtained
by contacting the first author (ober@utdallas.edu).
Results
To evaluate the performance of the proposed sub-
space algorithm a data set with relatively low signal
levels was chosen. Such data sets are typically more
difficult to analyze since the signal-to-noise ratio is lower
and potential artifacts are more pronounced than those
in data sets with larger signal levels. Fig. 1 shows an
overlay of the 11 sensorgrams for the myc peptide–9E10
antibody interaction [10,16].
All of the sensorgrams in this data set were analyzed
using the methods presented above. Fig. 2 shows a
representative association data segment analyzed using
Fig. 1. An overlay of 11 sensorgrams of the myc peptide–9E10 antibody interaction [10] is shown. The measured sensorgrams are zero-adjusted and
aligned along the time axis, and a background reference sensorgram is subtracted. Myc peptide was injected at a concentration of 38lM and a flow
rate of 20ll/min.
A
B
Fig. 2. (A) The association phase of one of the sensorgrams of the data presented in Fig. 1. The data fit based on the subspace method is shown in
heavy solid line. (B) The residuals of the fit in A.
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the subspace method. The plot of the residuals shows
that the fit is good overall. A measure of the quality of
the fits is that a subsequent adjustment of the parame-
ters using a gradient-based optimization routine did not
significantly reduce the mean square errors. This is
shown in Figs. 3 and 4, where the values of the estimates
are shown for the rate constants for the individual sen-
sorgrams and the corresponding mean square errors.
The results are summarized in Table 1. They show that
use of the subspace-based method generally produces
A
B
Fig. 3. (A) Estimates of kobs¼ konC0þ kofffor the association phases of the sensorgrams in Fig. 1. The parameters from three estimation methods are
shown: linear regression of the log of the data (?), subspace method (s), and optimization method using the results of the subspace method as initial
conditions (M). (B) The mean square errors for the fits that yielded the estimates in A. The symbols correspond to those in A.
A
B
Fig. 4. A Estimates of the dissociation constant kofffor the dissociation phases of the sensorgrams in Fig. 1. The parameters from three estimation
methods are shown: linear regression of the log of the data (?), subspace method (s), and optimization method (M). (B) The mean square errors for
the fits that yielded the estimates in A. The symbols correspond to those in A.
R.J. Ober et al. / Analytical Biochemistry 312 (2003) 57–65
61

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good results that are often better than those obtained by
use of the linear regression-based method. The optimi-
zation-based results consistently show the smallest mean
square error. This is not surprising because the sub-
space-based estimates were used as initial conditions for
the optimization algorithm, and the algorithm mini-
mizes mean square error.
The linear regression-based method is applicable
only for 1:1 interactions, whereas the subspace method
can be used to analyze exponential data that arise
from more complicated interactions. To demonstrate
this, noisy data were simulated for a parallel associ-
ation and dissociation interaction for two ligands (see
Fig. 5 for a representative simulated sensorgram).
Using the subspace-based method a data set of 20
sensorgrams was analyzed. The two association con-
stants and dissociation constants, the corresponding
coefficients, and the equilibrium value Req were esti-
Table 1
Myc–9E10 antibody interaction
Estimation methoda
Mean kobsðs?1Þ
0.034334
0.04177
0.037254
Var kobsðs?2Þ
4.1767e)005
1.4059e)005
6.0447e)006
Mean Req(RU) Var Req(RU2)Average mean square error (RU2)
Linear regression
Subspace
Optimized
14.67
14.2638
14.642
0.040272
0.099033
0.032477
0.096408
0.17685
0.06018
Estimation methodb
Mean kdðs?1Þ
0.0047508
0.0049176
0.0047818
Var kdðs?2Þ
3.0753e)008
3.6375e)008
2.749e)008
Mean R0(RU)Var R0(RU2)Average mean square error (RU2)
Linear regression
Subspace
Optimized
13.7255
13.801
13.801
0.18459
0.14189
0.13984
0.063886
0.054625
0.053033
aResults of estimates for the analysis of the association phase of the myc peptide–9E10 antibody interaction (Fig. 1). For each of the 11 as-
sociation segments in this data set, the observed association constant kobsand the equilibrium value Reqare estimated with the linear regression
method, the subspace method, and the optimized nonlinear least squares method using the estimates of the subspace model as initial conditions. For
each estimation method, the mean and variance of the 11 kobsand Reqestimates are tabulated. In addition, the average mean square error of the fits is
shown for each estimation method. The length of the association segment is varied to use the most suitable segment for each type of estimation
method. In particular, for the linear regression approach only very short segments could be used. Otherwise, negative values in the difference between
the data and the estimated equilibrium values would cause the logarithm operation to fail.
bResults of estimates for the analysis of the dissociation phase of the myc peptide–9E10 antibody interaction (Fig. 1). For each of the 11 dis-
sociation segments in this data set, the dissociation constant kdand the initial value R0are estimated with the linear regression method, the subspace
method, and the optimized nonlinear least squares method using the estimates of the subspace model as initial conditions. For each estimation
method, the mean and variance of the 11 kdand R0estimates are tabulated. Again, the average mean square error of the fits is shown for each
estimation method. The length of the dissociation segment is 70 data points in all cases.
Table 2
Parallel interaction with two ligands
Parameter value
SubspaceOptimized
Mean VarianceMean Variance
Dissociation phase
koff1ðs?1Þ
koff2ðs?1Þ
R01(RU)
R02(RU)
0.1
0.02
10
5
0.10097
0.020217
9.9401
5.0723
3.6117e)005
1.3281e)006
0.15804
0.189
0.099882
0.020008
10.0008
4.9971
3.137e)005
1.1633e)006
0.13583
0.15925
Association phase
kobs1ðs?1Þ
kobs2ðs?1Þ
Req1(RU)
Req2(RU)
Req1þ Req2(RU)
Req(RU)
0.02
0.5
10
5
15
15
0.020641
0.50848
9.9182
4.7914
14.7096
14.8721
3.701e)006
0.0092195
0.39673
0.11445
0.65333
0.52575
0.019696
0.5215
10.1084
4.9759
15.0843
1.7156e)006
0.0031465
0.12752
0.032272
0.079072
——
Results of the analysis of the simulated parallel association and dissociation interaction (Fig. 5) are shown. The parameter value column gives the
values used to simulate the association and dissociation data. Independent Gaussian noise with zero mean and standard deviation of 0.25 RU has
been added to the data points. The subspace method and optimized nonlinear least squares method (using the estimates of the subspace method as
initial conditions) have been used to estimate the parameters. The simulation has been repeated 20 times with randomly generated noise. For each
estimation technique, the means and variances of the various parameters estimated for the 20 sensorgrams are tabulated. Note that in this model
there are two ways to estimate the equilibrium value Reqof the association phase of the sensorgram. First, the coefficients Req1and Req2can be added.
Second, Reqcan be estimated directly as was done in the analysis of the 1:1 interaction (see Materials and methods). The results of both approaches
are given. They agree well, but not surprisingly the variance of the direct estimate of Reqis slightly lower than that obtained as the sum of Req1and
Req2. The average mean square error for the subspace models is 0.29313 RU2for association and 0.06369 RU2for dissociation. For the optimized
models, the corresponding error measures are 0.07027 RU2for association and 0.06363 RU2for dissociation.
62
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mated. Fig. 6 shows a representative fit of the asso-
ciationphase ofa sensorgram.
summarized in Table 2. The estimates using the sub-
Theresultsare
space-based method generally show only a small dif-
ference compared with the actual values of the
parameters. These estimates were further improved by
Fig. 5. A simulated sensorgram for a parallel association and dissociation interaction of two ligands, i.e., the sensorgram is given by
RðtÞ ¼
0
Req1ð1 ? e?ðt?t1Þkobs1Þ þ Req2ð1 ? e?ðt?t1Þkobs2Þ
R01e?ðt?t2Þkd1þ R02e?ðt?t2Þkd2
with t1¼ 200s, t2¼ 600s, kobs1¼ 0:02s?1, kobs2¼ 0:5s?1, Req1¼ 10 RU, Req2¼ 5 RU, koff1¼ 0:1s?1, koff2¼ 0:02s?1, R01¼ 10 RU, R02¼ 5 RU.
Thus, the association phase is simulated for 400s and subsequent dissociation is simulated for 500s. Gaussian noise with a standard deviation of 0.25
RU is added to the data points.
t < t1
t16t < t2
tPt2
8
<
:
A
B
Fig. 6. (A) The association phase of the simulated sensorgram described in Fig. 5 with data fit based on the subspace method (heavy solid line). (B)
The residuals of the fit in (A).
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using them as initial conditions for a gradient-based
optimization algorithm.
Discussion
Exponential data arises in many biophysical appli-
cations. Here our particular interest is data arising from
surface plasmon resonance experiments. However, the
approach and results are easily applied to many other
areas of biophysical data analysis.
The analysis of such exponential data is not without
problems. If no measurement noise were present a
number of more or less elementary schemes would easily
lead to a solution. It is, however, the presence of noise
that complicates the analysis. In most cases the param-
eters of interest are decay rates. This means that the
parameters to be estimated are nonlinear in the data.
Currently two data analysis methods are primarily used.
In one method, the logarithm of the data points is taken
[7]. This translates the data analysis problem into a
linear regression problem if the data are made up of only
one exponential component. In the other standard
method, gradient-based optimization algorithms are
used to fit a model to the data [6,9]. Both algorithms
have their advantages and disadvantages. The first
method is restricted to the analysis of 1:1 interactions. In
addition, the logarithm introduces nonlinear distortions
in the noise and changes the noise characteristics in a
nonuniform fashion. This makes a statistical analysis of
the estimation procedure difficult and can lead to biased
estimates. Furthermore, when low signal levels are en-
countered, measurement of noise might lead to negative
data points for which the logarithm cannot be taken.
The gradient-based optimization methods on the other
hand often have statistically optimal or near optimal
properties (see, e.g., [10]). Their disadvantages lie in the
fact that convergence is not guaranteed. Depending on
the initial conditions of the search the algorithm could
converge to a local minimum rather than a global one.
This highlights another potential problem with iterative
algorithms, i.e., that good initial conditions are needed.
An additional algorithm is needed to determine these
initial conditions.
In the signal-processing literature so-called subspace
algorithms have received a significant amount of atten-
tion (see, e.g., [11–15]). Here we have analyzed simulated
and experimental surface plasmon resonance data ac-
quired on a BIAcore 2000 instrument using a particular
subspace algorithm for exponential type data. Subspace
algorithms have the advantage that they are noniterative
and therefore do not require initial conditions and do
not have the convergence problem of iterative algo-
rithms. In contrast to the linear regression-based algo-
rithm that relies on taking the logarithm of the data
points, subspace algorithms are not restricted to data
that arise from 1:1 interactions and can easily deal with
negative data points. We have adapted the existing
subspace algorithm to analyze the present data and
evaluated its performance using both experimental and
simulated data.
We have examined the use of this algorithm to pro-
vide estimates for association and dissociation data. We
found that the estimates are very accurate. In fact, if the
estimates are used as starting points for a subsequent
gradient-based optimization routine the quality of the
parameter estimates does not improve significantly
through the further optimization step.
An examination of the performance of the subspace
algorithm applied to simulated data for a parallel as-
sociation and dissociation interaction of two ligands
shows that the algorithm also performs well in this more
complicated situation. Such an interaction cannot be
adequately analyzed with the standard method which
relies on turning the problem into a linear regression
problem by taking logarithms of the data.
The use of this subspace algorithm is, however, also
not without potential pitfalls. The algorithm can easily
handle very high noise levels, but care should be taken
with data sets that have outliers or other artifacts. These
should be removed; the resulting gaps can be filled with
values interpolated from neighboring data points.
A central step in the subspace method is a singular
value decomposition. This is typically a computationally
expensive operation. However, with current computers
this is not a problem for the typical lengths of data sets
that are being analyzed. In fact, in our experience the
time taken to analyze a data set using the subspace
method was significantly shorter than the time taken by
the gradient-based method.
This algorithm, like most others, will not produce
reliable estimates if the data does not comply with the
model. For example, in the estimation of dissociation
data for a 1:1 interaction a misleading estimate could be
expected when the data have a nonzero offset that is not
accounted for in the model. To account for such an
offset is in principle straightforward and would be a
minor modification of the procedure used to estimate
the equilibrium value Reqin the analysis of the associ-
ation data.
The newly proposed method is a complement to ex-
isting methods and provides an important tool for the
analysis of surface plasmon resonance experiments of
exponential type. The availability of noniterative algo-
rithms is of particular importance for automated data
analysis problems.
Acknowledgment
This research was supported in part by a grant from
the National Institutes of Health (NIH GM58538).
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