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# Statistical modeling of large microarray data sets to identify stimulus-response profiles

Communicated by Leland H. Hartwell, Fred Hutchinson Cancer Research Center, Seattle, WA (received for review August 3, 2000)

## Abstract

A statistical modeling approach is proposed for use in searching large microarray data sets for genes that have a transcriptional response to a stimulus. The approach is unrestricted with respect to the timing, magnitude or duration of the response, or the overall abundance of the transcript. The statistical model makes an accommodation for systematic heterogeneity in expression levels. Corresponding data analyses provide gene-specific information, and the approach provides a means for evaluating the statistical significance of such information. To illustrate this strategy we have derived a model to depict the profile expected for a periodically transcribed gene and used it to look for budding yeast transcripts that adhere to this profile. Using objective criteria, this method identifies 81% of the known periodic transcripts and 1,088 genes, which show significant periodicity in at least one of the three data sets analyzed. However, only one-quarter of these genes show significant oscillations in at least two data sets and can be classified as periodic with high confidence. The method provides estimates of the mean activation and deactivation times, induced and basal expression levels, and statistical measures of the precision of these estimates for each periodic transcript.

Advances in microarray technologies (1–5) have enabled investigators to explore the dynamics of transcription on a genomewide scale. The current challenge is to extract useful and reliable information out of these large data sets. A common, first approach is cluster analysis. The primary objective of cluster analysis is to group genes that have comparable patterns of variation. This method is valuable for reducing the complexity of large data sets and for identifying predominant patterns within the data (6). However, additional methods are needed to extract information about individual genes from these large data sets.

Toward these goals we consider a statistical method to identify genes whose transcript profiles respond to a stimulus. In general terms, this approach involves modeling the association of a generic response with a specific experimental variable, for example, timing, cell type, temperature, or drug dosage, using a set of interpretable parameters. One objective is to estimate pertinent parameters for individual transcripts, with the goal of testing specific hypotheses concerning transcript response to the stimulus. If the statistical model provides an adequate representation of the expression data for a specific gene, then the corresponding model parameter estimates can provide certain response characteristics for that gene. For example, model parameters may describe the magnitude, duration, or timing of the response. This modeling strategy can be used for two group comparisons, where the objective may be to identify genes that are differentially expressed between normal and abnormal tissues, or in drug discovery studies, where the objective may be to identify transcripts affected by drug dosage.

To demonstrate the utility of this approach, we formulated a model for
identifying periodically transcribed genes of the budding yeast
*Saccharomyces cerevisiae.* In this case, the stimulus is
synchronous resumption of the cell cycle by releasing the cells from a
fixed arrest point. The response is a pulse of transcription, and the
key experimental variable is cell cycle timing (7–9).

Four synchronized cell cycle data sets have been generated and made available for general exploration (8, 9). These large data sets have been analyzed by visual inspection (8), Fourier transform and hierarchical clustering (9), k-means (10) and QT clustering (11), self-organizing maps (12), and singular value decomposition (13, 14). Fourier transform analysis of three data sets, where the threshold for periodicity was based on the behavior of known periodic genes, led to a report that there are 800 periodically transcribed genes (9). Later, k-means clustering was applied to one data set, and five periodic clusters with 524 members were identified (10). However, only 330 genes were identified by both approaches. For comparison, we have used statistical modeling to look for regularly oscillating profiles within these large data sets. This approach complements clustering methods in that rather than seeking to group together genes having similar expression patterns it aims to directly identify transcripts affected by a given stimulus and to provide specific information regarding individual response patterns. As amplified below, the method also allows for heterogeneity in response patterns among samples, with expected robustness of inferences on response parameters to certain types of experimental variations.

## Methods

### A Modeling Framework.

Let *Y*_{jk} denote the expression level for the
*j*th gene in the *k*th sample in a stimulus
experiment. The number, *J*, of genes studied often will be of
high dimension, typically in the thousands, while the number of
samples, *K*, may be comparatively few. A standard statistical
approach would relate the mean of the vector response,
*Y*′_{k} = (*Y*_{1k}, … ,
*Y*_{Jk}) for the *k*th sample to a corresponding
vector of *p* covariates *x*_{k} =
(*x*_{1k}, … , *x*_{pk}) that codes the
stimulus categories and possible other characteristics of the
*k*th sample using a regression function, say
Δ(*x*_{k}, θ)′ = {Δ_{1k}(*x*_{k},
θ), … , Δ_{Jk}(*x*_{k}, θ)}, where
θ′ = (θ_{1}, … , θ_{J}) may
include gene-specific and other parameters and is to be estimated.
Under such a regression model the elements of the vector of differences
*Y*_{k} − Δ_{k}(*x*_{k}, θ)
have mean zero, but can be expected to be correlated due, for example,
to variations in mRNA extraction, amplification, and assessment among
samples. Such variations can be acknowledged by introducing additional
parameters, which we refer to as heterogeneity parameters into the
model for the mean of *Y*_{k}. In fact, for sample
*k*, one can introduce both an additive heterogeneity
parameter δ_{k} and a multiplicative
heterogeneity parameter λ_{k} giving a model,
δ_{k} +
λ_{k}Δ_{jk}(*x*_{k}, θ) for the
expectation of *Y*_{jk}. The average of the
δ_{k}s and λ_{k}s are
restricted to be zero and one, respectively, to avoid possible
identifiability problems relative to the regression parameters θ of
primary interest. The high dimension of *Y*_{k} will
allow those heterogeneity parameters to be precisely estimated in many
applications. The inclusion of these parameters may make plausible an
assumption that *Y*_{k} given
*x*_{k} are nearly independent, especially for
*in vitro* experiments. Under such an assumption, one can
simplify the modeling and numerical procedure for the estimation of
θ.

Following the approach described in the seminal statistical paper by
Liang and Zeger (15), estimation of the mean parameter vector
η′ = {δ_{1}, … , δ_{K},
λ_{1}, … , λ_{K}, θ} can proceed
by specifying a “working” covariance matrix for
*Y*_{k}, which under the above independence
assumption will be approximated by a diagonal matrix, written as
*V*_{k} = diag(*v*,
… , *v*), so that the expression level for
each of the *J* genes is allowed to have a distinct variance.

Estimates of the vector of mean parameters η can now be estimated as
η̂′ = {δ̂_{1}, … ,
δ̂_{K}, λ̂_{1}, … ,
λ̂_{K}, θ̂}, a solution to the estimating
equation,
1
where *D*_{k} is the matrix of partial
derivatives of the mean of *Y*_{k} with respect to
the parameter η, V̂_{k} denotes
*V*_{k} with each *v*
replaced by a consistent estimate v̂,
and 1 denotes a column vector of ones of length *J*. Under the
above modeling assumptions, η̂ will be approximately jointly
normally distributed provided both *J* and *K* are
large, and the variance of η̂ can be consistently estimated (as
*J* and *K* become large) by a standard sandwich
formula (15, 16).

The mean parameter estimation procedure just outlined is expected to be
useful in various types of microarray data sets. It will allow the
estimation of meaningful gene-specific parameters to characterize
expression levels in response to a stimulus, and, in that sense, is
complementary to cluster analysis that seeks to group genes having
similar expression patterns, with less emphasis on pattern
characteristics. For example, in the context of comparing the
expression patterns between diseased and nondiseased tissues, one may
define a binary indicator *x*_{k} that takes value
zero for nondiseased tissue samples and one for diseased tissue
samples, and specify a regression function,
Δ_{jk}(*x*_{k}, θ) =
θ_{j0} + θ_{j1}*x*_{k}, under
which the *j*th gene would be differentially expressed between
normal and abnormal tissues, whenever θ_{j1} ≠
0. The regression variable *x*_{k} also could be
expanded to allow the regression function to depend on other measured
characteristics of the *k*th sample (or the *k*th
study subject). Similarly, in a study of variations of expression over
time one would define *x*_{k} = *t*_{k},
the timing of the *k*th sample to be gathered, and one can
choose a linear or other functional form to model the regression
function Δ_{jk}(*x*_{k}, θ).

In the remainder of this paper we focus on the specific setting of transcript response patterns when cells are released from cell cycle arrest. Our particular interest will be the identification of genes having periodic expression level changes over multiple cell cycles. A key data analysis challenge relates to the possibility of a single nonoscillating pulse of transcription for some genes, after the resumption of cell cycling.

### Modeling Periodic Transcription.

Consider a synchronized experiment to identify mRNAs that are
transcribed once per cell cycle. Suppose that when activated, the
*j*th mRNA reaches an elevated value
(α_{j} + β_{j}) and when
deactivated it falls to a basal expression level
(α_{j}) (Fig. 1).
Naturally, β_{j} is interpreted as the difference
between averaged peak and trough expression levels. Considering
multiple copies of the *j*th mRNA, transcribed and dissipated
at consecutive times in multiple cells with imperfect synchronization,
the mean expression level of the *j*th transcript at the time
*t*_{k} may be modeled as:
in which *j* = 1, 2, … , *J* and *k* =
1, 2, … , *K* for all *J* transcripts at all
*K* time points, where ζ_{j},
ξ_{j} are the activation and deactivation times for the
*j*th gene, respectively, *t*^{*}_{k} =
*t*_{k} + τ, where τ denotes the difference of
actual cell cycle timing and observed timing and is typically known as
phase, Θ is the cell cycle span, and the summation is over multiple
cell cycles, *c* = 0, 1, 2, … . The standard
deviation, σ_{k}, depicts the variation of
“true” cell-specific timings around *t*_{k},
which we assume to follow a normal distribution with mean
*t*_{k}, resulting in the cumulative normal
distribution function φ(.) in the mean model. Also,
δ_{k}, λ_{k} are the additive and
multiplicative heterogeneity parameters for the *k*th sample
as described above, and here *x*_{k} =
*t*_{k}. The above single-pulse model (SPM) specifies a
model for the mean expression of each gene as the cell cycle proceeds.
Gene-specific activation and deactivation times as well as the
background and elevated expression levels are estimated for each gene.
SPM also allows for variation between samples, for the fact that the
synchrony is imperfect and, as described below, for the synchrony to
deteriorate over time (Fig. 1). Further detail on the development of
the SPM is given in the *Appendix*, which is published as
supplemental material on the PNAS web site, www.pnas.org. The resulting
mean expression model has been shown visually to reproduce the profiles
observed for periodic transcripts measured by conventional means (17).

The SPM described above can be applied by using the mean model
estimation procedure outlined above. To simplify numerical aspects we
have used a multistage procedure: (*i*) heterogeneity
parameters, (δ_{k}, λ_{k}), *k* = 1,
… , *K*, are estimated by using all genes when the pulse
heights are set to zero, (*ii*) the cell cycle span, Θ, is
estimated by using a group of known cell cycle genes under a pulse
model, (*iii*) the synchronization variability,
σ_{k}, *k* = 1, … , *K*, is estimated by
using the same group of known genes, and (*iv*) gene-specific
parameters (α_{j}, β_{j}, ζ_{j},
ξ_{j}), *j* = 1, … , *J*, are estimated while
other estimated parameters are treated as fixed at their estimated
values. Although a simultaneous estimation approach using the
estimating Eq. 1 would be preferable, the impact on
estimation of the gene-specific parameters of their variance estimates
is likely to be minimal as gene-specific parameters are weakly
correlated with other parameters. Fixing the cell cycle span and
sample-specific parameters allows a separate simple calculation of the
gene-specific parameter estimates, and of their variance estimates, for
each of the *J* genes. Further detail on these calculations is
given in the *Appendix.*

To test the fit of the SPM we introduced additional polynomial
functions of time in the mean model and tested the hypothesis that the
polynomial coefficients were identically zero. Specifically, the SPM is
augmented and written as
allowing a departure from the SPM. A score-type test statistic for
(γ_{j1}, γ_{j2}, γ_{j3}) =
(0, 0, 0) is then constructed using the asymptotic normal theory
described above. This score statistic,
χ, will have an approximate chi-square
distribution with three degrees of freedom under the SPM model, for
sufficiently large *J* and *K*. We choose 11.3, the
1% upper percentage of this chi-square distribution to identify genes
with patterns that depart significantly from the SPM. For the
*cdc28* data set, for example, only 262 genes give test
statistics that exceeded the critical value. Note that other deviations
than those polynomial terms could be specified but are not pursued
here.

For those genes for which the expression pattern does not depart
significantly from SPM, we estimate activation time
(ζ_{j}), deactivation time
(ξ_{j}), basal expression level
(α_{j}), and elevation in expression level during
the interval (β_{j}), along with their estimated
standard deviations. Under the SPM, expression levels are cell cycle
regulated if and only if β_{j} ≠ 0. We
choose a critical value of 5 for the absolute value of each
*Z*_{j}, the ratio of the estimate of
β_{j} to its estimated standard deviation, to
reject the null hypothesis. This value, far in the tail of normal
distribution, is expected to preserve a genomewide significance level
of about 0.3% (two-sided) even with as many as 6,000 genes under
study. Some of genes that showed evidence of departure from SPM also
may have expression patterns that vary with the cell cycle. One could
test β_{j} = 0 also for those genes in the
context of the augmented mean model,
μ̃_{j}(*t*_{k}), described above,
though the interpretation of such a test would be conditional on the
adequacy of the augmented model.

### Nature of the Data.

Three data sets were used in this analysis. The *cdc28* data
set was generated by Cho *et al.* (8), and synchrony was
established by using a temperature-sensitive *cdc28* mutation
to reversibly arrest cells in G_{1}. Oligonucleotide arrays
(Affymetrix, Santa Clara, CA) were hybridized to fluorescently labeled
cDNAs made from each sample, and the absolute fluorescence intensity
values are assumed to be proportional to the amounts of each transcript
in each target sample (7). These data were downloaded from
http://genomics.stanford.edu. The two other sets of data (alpha
factor and *cdc15*) were generated by Spellman *et
al.* (9) using an alpha factor-mediated G_{1} arrest and a
temperature-sensitive *cdc15* mutation to induce a reversible
M-phase arrest, respectively. In this case, fluorescently labeled cDNAs
were made from RNA from each time point and a second fluorescent dye
was used to label cDNA made from an asynchronous control culture.
Control and test cDNAs were mixed and hybridized to arrays of
PCR-amplified yeast ORFs. Fluorescence intensity values of both dyes
were measured, and logarithmic ratios of test versus control values
were generated. Obtained ratios are assumed to approximate the
corresponding true ratios of test versus control mRNA levels (9). These
data and the *cdc28* data, rescaled to mimic the ratio data,
were accessed from the public domain site
http://cellcycle-www.stanford.edu. Our results are based on the
analysis of these data sets and as such will be influenced by all
sources of variation involved in the preparation and processing of
these arrayed samples.

## Results and Discussion

The primary assumptions of SPM are that cell cycle-regulated
transcripts will peak only once per cycle and that these pulses occur
at invariant times in consecutive cycles. SPM includes terms that
enable additive and multiplicative heterogeneity across samples to be
accommodated. Fig. 2 shows additive
heterogeneity estimates for each data set. Additive heterogeneity is
minimal when logarithmic ratios are used. When the absolute intensity
is considered for the *cdc28* data set, the additive
heterogeneity is most evident at the 90-min time point. This confirms
the concern over this particular time point (8) and provides a means of
correcting for its heterogeneity.

We estimated cell cycle span for each data set by using a set of 104
known cell cycle-regulated genes and profiling over a range of possible
cell cycle spans (see *Appendix*). As expected, the cycle span
differs for each synchrony method. Cycle spans for the alpha factor and
*cdc15* data sets show bimodal distributions (Fig. 2). These
may be due to recovery artifacts that differentially affect the first
cycle and alter the timing of a subset of the transcripts. We have used
the estimated cell cycle span that minimizes a certain weighted sum of
squares, giving a value of 58 min for the alpha factor synchrony, 115
for the *cdc15* cells, and 85 for the *cdc28*
culture. Fig. 2 also shows the estimated standard deviations associated
with loss of synchrony over time. Once these values have been obtained,
the χ values are calculated for the
*j*th gene for *j* = 1, … , *J*, and
gene-specific parameters are estimated for all genes having
transcription patterns consistent with the SPM (i.e.,
χ values less than 11.3). Gene-specific
parameters include the mean activation and deactivation times and the
basal and elevated levels.

Fig. 3 shows the microarray data (solid
lines) for five periodic genes and the fitted SPM to these profiles
(dotted lines). Clearly, the model closely approximates the profile of
the data and provides mean activation and deactivation times (in
brackets) that are consistent with the patterns observed. The
*Z* values for these oscillations vary from about 18 for
*RFA1* in the *cdc15* data set to about 3.5 for
*MCM3* in the alpha factor data set. The fact that the
periodic behavior of *MCM3* is still evident gives us
confidence that we have set a sufficiently conservative threshold for
each *Z*_{j}. The top three transcripts have been
classified as G_{1}-specific, MCB-regulated genes (9).
However, the *PDS1* pulse is delayed compared to the other
two. *RFA1* and *CLB6* are activated at about the
same time, but the pulse of *CLB6* mRNA is shorter lived.
These differences are reflected in the activation and deactivation
times calculated for each gene by SPM and can be used to identify
coordinately regulated transcripts.

### Identification of Cell Cycle-Regulated Transcripts Using SPM on the
*cdc28* Data Set.

A total of 607 genes met SPM thresholds for periodicity (i.e.,
|*Z*_{j}| value of 5 or greater) using absolute
fluorescence intensity measurements directly from the *cdc28*
data (8). We obtained about the same number of genes by using either
the logarithm of the intensity or the logarithmic ratios of intensities
as generated by Spellman *et al.* (1, 2, 9). However, only
about 500 genes were identified in all three analyses. Thus, any single
data transformation may miss about 20% of the potential positives, due
to *Z* values that are close to our threshold. In all
subsequent analysis, we have used logarithmic ratios of the
*cdc28* data to be consistent with the alpha factor and
*cdc15* data.

Lists of cell cycle-regulated genes in the *cdc28* data set
have been compiled by visual inspection (8) and k-means clustering
(10). SPM analysis confirms a majority of these assignments and
identifies many more candidate oscillating transcripts. The application
of the k-means approach provided by Tavezoie *et al.* (10)
used an initial filtering strategy to select the 3,000 yeast genes,
which showed the highest coefficient of variation over the time course.
Then, the iterative k-means procedure was used to partition all 3,000
profiles into 30 clusters. The requirement that all 3,000 profiles fit
into one of 30 clusters necessitated the assembly of large clusters
with loosely correlated patterns of expression. Five of these clusters
had mean temporal profiles that were clearly periodic over two cell
cycles. However, only about half of the profiles of the 524 cluster
members exceeded the thresholds for periodicity in SPM.

To see whether SPM could identify a tight cluster of periodic
genes, we computed the χ^{2} and *Z* values for a
cluster of G_{1}-specific transcripts which was assembled at
three different thresholds by Heyer *et al.* (11), using the
QT_Clust algorithm. In this case, we find that all the tightest
cluster members either exceed or come very close to the threshold for
periodicity set in SPM (Fig. 4
*Top*). Inspection of the borderline cases indicates that they
are likely to be periodic and thus our *Z* value threshold is
conservative. When the cluster threshold is set lower, membership
doubles and again nearly all profiles are at the SPM threshold or well
above it (Fig. 4 *Middle*). However, as noted by Heyer
*et al.* (11) further relaxation of the cluster threshold to
include 272 profiles leads to the inclusion of many poorly matching
patterns that also have low *Z* values by SPM (Fig. 4
*Bottom*). This finding indicates the efficiency of both
approaches in identifying the most periodic transcripts. It also
illustrates the value of having two completely different methods of
analyzing the data to establish meaningful thresholds and characterize
the less robust response patterns.

Another feature of SPM is its estimation of gene-specific parameters. Fig. 4 also shows how the distributions of activation and deactivation times broaden as cluster membership increases. This broadening indicates that in addition to containing nonperiodic profiles, this group contains genes with different kinetics of expression. Thus, SPM enables these clusters of similar expression patterns to be further subdivided, depending on the question of interest.

### Using All the Data Sets to Estimate the Number of Periodic Genes.

One limitation of these cell cycle data sets is the small number of
samples and the lack of multiple measurements at any time point, which
makes the identification of false positives and false negatives
problematic. To mitigate this problem, we have used SPM to identify
periodic transcripts from the *cdc28*, *cdc15*, and
alpha factor data sets separately and then compared the results. SPM
identifies about twice as many periodic genes in the *cdc28*
data set as in either of the other two synchronies (Fig.
5), and overall there are 1,088 genes
that show significant oscillations in at least one data set. Included
among the 1,088 candidate periodic genes identified by SPM are 81% of
the 104 known periodic genes. A total of 254 genes oscillate
significantly in at least two databases, which represents 4% of all
genes, but includes 46% of the known periodic genes. Thus, SPM
identifies the known periodic transcripts well above the level expected
by chance. Only one-quarter of the known periodic genes are among the
71 genes that score as periodic in all three data sets. A total of 834
genes appear periodic in only one data set and as such further data
collection will be required before this large group of genes can be
unambiguously classified. Complete lists of the periodic transcripts
identified by SPM are available at our web site
(www.fhcrc.org/labs/breeden/SPM).

Spellman *et al.* (9) used Fourier analysis of the combined
data from the same three data sets to identify periodic transcripts.
Using the known periodic genes as a guide for setting their threshold,
they estimated that 799 genes are periodic. Only 65% of these genes
also are picked up by SPM as being periodic in at least one data set.
This difference may be explained, in part, by our conservative
threshold for *Z* because reducing the threshold value for
*Z* to 4.0 enables 79% of these genes to be classified as
periodic in at least one data set.

Nearly all of the genes that exceed the threshold for periodicity by
SPM in at least two data sets also are recognized by the method of
Spellman *et al.* (9). Here again, as with clustering, the
most robust periodic patterns are identified by both methods. However,
there are 571 genes that appear to be periodic by SPM criteria in at
least one data set but are not classified as such by Spellman *et
al.* (9). As noted above, these cannot be unambiguously classified
as periodic without further corroborating data. They are either false
negatives in two data sets or false positives in one. Experimental
variation is much more likely to result in a nonperiodic pattern than
it is to produce a smoothly oscillating profile. With SPM, the peaks
also must occur at the same time in consecutive cell cycles and peaks
and troughs are not recognized if they are represented by a single
point in the profile (see *Appendix*). These restrictions
should reduce the impact of noise and result in a lower false positive
error rate. However, we cannot eliminate the impact of noise in the
data and, with so few data points to base these assignments upon, many
remain ambiguous. The 254 genes that score as periodic in two data sets
can be considered periodic with reasonably high confidence, but they
include only about half of the known periodic genes and as such clearly
underestimate the number. Unless more data are generated,
classification of the other transcripts will remain ambiguous. In other
words, in spite of the accumulation of nearly one half million data
points, we can identify only about half of the periodic transcripts of
budding yeast with high confidence. These ambiguities, combined with
the fact that statistical methods are most reliable when there is a
large number of independent samples, leads us to conclude that another
data set, traversing two full cell cycles and having closer time points
will be required to more completely identify and order the periodic
transcripts of this important model organism.

If even half of these 1,088 genes are actually periodic (see Fig. 5), they would comprise about 10% of all budding yeast genes, which could be viewed as an enormous regulatory burden to the cells, especially if there are many different ways in which this regulation is accomplished. On the other hand, if there are only 20 different circuits that achieve this regulation and gene products have evolved into these limited expression patterns based on the cell's need for them, one could view it as a highly parsimonious strategy for limiting the biosynthetic load on the cell.

## Conclusion

In this report we use a statistical model (SPM) to identify and characterize single pulses of transcription that occur at invariant times in consecutive cell cycles. SPM is a specific application of statistical modeling, but the basic strategy can be applied to any large data set to identify genes undergoing a transcriptional response to a stimulus. Due to its relative simplicity, statistical modeling can be used to interrogate large data sets without using additional filters to reduce the number of genes to be analyzed. It also includes heterogeneity parameters that will tend to reduce the impact of noise in the data sets. SPM identifies regularly oscillating transcripts without regard to the abundance of the transcript or the height or timing of the peak and provides estimates of the mean time of activation and deactivation. These values are only estimates, but they are unbiased under the assumed SPM and can be considered defining characteristics of individual genes. SPM also provides statistical measures of the precision of the parameter estimates so that optimal groupings can be made and subjected to further analysis. These features of statistical modeling complement and augment the other methods used to analyze microarray data.

## Acknowledgments

We thank Drs. M. Campbell, J. Cooper, and J. Sidorova for helpful discussions about the data. We also thank the investigators who contributed their yeast expression data to the public domain, and L. Heyer, S. Kruglyak and G. Church for providing cluster information for comparison. This research is supported by grants from the National Institutes of Health (GM41073 to L.B. and PO1 CA53996 to L.P.Z. and R.P.).

## Footnotes

↵† To whom reprint requests should be addressed. E-mail: lbreeden{at}fhcrc.org or lzhao{at}fhcrc.org.

## Abbreviation

- SPM,
- single-pulse model

- Received August 3, 2000.
- Accepted January 4, 2001.

- Copyright © 2001, The National Academy of Sciences

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