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PLoS Comput Biol. 2009 January; 5(1): e1000257. Published online 2009 January 2. doi: 10.1371/journal.pcbi.1000257. | PMCID: PMC2599889 |
Copyright Airoldi et al. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original author and
source are credited. Predicting Cellular Growth from Gene Expression Signatures Edoardo M. Airoldi,#1,2 Curtis Huttenhower,#1,2 David Gresham,1,3 Charles Lu,1,3 Amy A. Caudy,1 Maitreya J. Dunham,4 James R. Broach,3 David Botstein,1,3* and Olga G. Troyanskaya1,2* 1Lewis-Sigler Institute for Integrative Genomics, Carl Icahn Laboratory,
Princeton University, Princeton, New Jersey, United States of
America 2Department of Computer Science, Princeton University, Princeton, New
Jersey, United States of America 3Department of Molecular Biology, Princeton University, Princeton, New
Jersey, United States of America 4Department of Genome Sciences, University of Washington, Seattle,
Washington, United States of America Andrey Rzhetsky, Editor University of Chicago, United States of America Received September 2, 2008; Accepted November 18, 2008. Maintaining balanced growth in a changing environment is a fundamental
systems-level challenge for cellular physiology, particularly in microorganisms.
While the complete set of regulatory and functional pathways supporting growth
and cellular proliferation are not yet known, portions of them are well
understood. In particular, cellular proliferation is governed by mechanisms that
are highly conserved from unicellular to multicellular organisms, and the
disruption of these processes in metazoans is a major factor in the development
of cancer. In this paper, we develop statistical methodology to identify
quantitative aspects of the regulatory mechanisms underlying cellular
proliferation in Saccharomyces cerevisiae. We find that the
expression levels of a small set of genes can be exploited to predict the
instantaneous growth rate of any cellular culture with high accuracy. The
predictions obtained in this fashion are robust to changing biological
conditions, experimental methods, and technological platforms. The proposed
model is also effective in predicting growth rates for the related yeast
Saccharomyces bayanus and the highly diverged yeast
Schizosaccharomyces pombe, suggesting that the underlying
regulatory signature is conserved across a wide range of unicellular evolution.
We investigate the biological significance of the gene expression signature that
the predictions are based upon from multiple perspectives: by perturbing the
regulatory network through the Ras/PKA pathway, observing strong upregulation of
growth rate even in the absence of appropriate nutrients, and discovering
putative transcription factor binding sites, observing enrichment in
growth-correlated genes. More broadly, the proposed methodology enables
biological insights about growth at an instantaneous time scale, inaccessible by
direct experimental methods. Data and tools enabling others to apply our methods
are available at http://function.princeton.edu/growthrate. A major challenge for living organisms is the regulation of cellular growth in a
fluctuating environment. Sudden changes in nutrient availability or the presence
of stress factors typically require rapid adjustments of cellular growth. The
misregulation of growth control in higher organisms is a major factor in the
development of cancer. A statistical characterization of cellular growth based
on gene expression levels provides a quantitative perspective to understand the
regulatory network that controls growth. We develop a model of cellular growth
in the yeast Saccharomyces cerevisiae, grounded in the
expression levels of a small set of genes. The model is able to predict the
growth rate of new cellular cultures from expression data and is robust to
changing biological conditions, experimental methods, and technological
platforms. The predictions are informative about changes in growth at very short
time scales, which direct experimental methods cannot generally access. The
model also predicts growth rates in Saccharomyces bayanus and
in Schizosaccharomyces pombe, a yeast diverged by approximately
a billion years of evolution. Our findings suggest that the model describes
fundamental characteristics of the unicellular eukaryotic growth regulatory
program. A case study explores the role of nutrient sensing in the yeast growth
regulatory system. Proper regulation of growth rate is a key systems-level challenge for all cells,
particularly microorganisms facing a fast-changing and often hostile environment.
Cell growth, defined as an increase in cellular biomass due to biosynthetic
processes, is one of the primary functions that must be coordinated with the
environment in order for cells to maintain viability and reproduce. The
determination of how cells integrate information from the external environment with
information from their internal state to mount an appropriate
response—growing in the presence of nutrients, arresting growth when
stressed, and resuming afterwards— is of central importance to our
understanding of basic biology. From a genomic perspective, growth also raises the
issue of disentangling correlated systems-level behaviors. When the expression
levels of thousands of genes change due to a growth-related stimulus, which
underlying regulatory parameters are responsible? In this paper, we identify quantitative aspects of the transcriptional regulatory
mechanisms underlying cellular growth in Saccharomyces cerevisiae,
and we develop a model to predict instantaneous growth rates of cellular cultures
based on gene expression data. The model enables the estimation of growth rates
under any conditions for which expression data is available, even on a very short
time scale, where standard experimental techniques cannot measure cellular growth
directly [1]. For example, a culture undergoing continuous growth
in a chemostat [2] can be perturbed from steady state by means of a brief
heat pulse, but the departure from and the return to steady state growth is too
brief to capture with optical density measurements. Our model allows such a decrease
(and subsequent resumption) of growth rate to be quantified under a variety of
conditions: batch or chemostat cultures, different microarray platforms, and under
any environmental stimulus for which gene expression can be assayed. Surprisingly,
this model also successfully predicts growth rates from Saccharomyces
bayanus and Schizosaccharomyces pombe expression data,
the latter of which is evolutionarily diverged from S. cerevisiae
by an estimated billion years [3]. Our findings suggest that the proposed statistical model of cellular growth provides
a broadly applicable biological characterization of the transcriptional regulatory
network underlying growth rate control. We have previously observed that the
expression of ~25% of the yeast genome responds to changes in
growth rate [4]. The response is functionally cohesive, with genes
up-regulated with increasing growth enriched for translational and ribosomal
functions, and with down-regulated genes enriched for oxidative metabolism and the
peroxisome. This functional portrait provides a rich environment in which to study
transcriptional regulation of growth; for example, production of new proteins at the
ribosome is vital to cellular proliferation, and yeast devotes some
~60% of its transcriptional throughput to ribosomal RNA [5].
Similarly, growth rate regulation is highly interconnected with a variety of other
cellular processes (e.g. the environmental stress response [6], metabolic cycling [7], and
the cell cycle [8]), and we discuss potential causative regulatory
signals from the Ras/PKA pathway [9] and growth-related transcription factors. Our recent analysis of gene expression measurements from a collection of S.
cerevisiae chemostat cultures across several nutrient limitations and
growth regimes [4] offered intriguing evidence for a notion of
instantaneous growth rate. In this paper, we develop a model to characterize such a
notion quantitatively, in a statistically principled fashion. We further assess the
robustness of the proposed characterization by presenting new computational evidence
on six additional published data sets and on four newly collected data sets. More in
detail, we demonstrate that the model can accurately predict relative growth rates
under a variety of conditions and is robust to the conditions of the originating
culture, the technological platform used to assay gene expression, and evolutionary
conservation to other organisms ( S. bayanus and S.
pombe). The model allows us to predict growth rates for published
genome-wide collections of expression data (e.g. the stress response [6] or gene
deletions [10]) and for four new data collections we have generated
for this work ( Tables S1, S2, S3, S4), providing biological insight into the growth
rate response at very short time scales—minutes, rather than the hours
necessary to experimentally assay doubling times. This biological validation of the
predictions is accompanied by an out-of-sample validation and outlier analysis to
assess the statistical accuracy of the model. We have made an implementation of this
model available to the public at http://function.princeton.edu/growthrate. Additional analyses offer biological insights that support and further substantiate
the empirically observed robustness of the predictions based on the newly
characterized growth-rate genes. Our insights rely, in part, on the quantitative
identification of binding motifs of known (and uncharacterized) transcription
factors associated with the genes responding to growth. Moreover, our model enables
a quantitative characterization of growth profiles underlying puzzling experimental
evidence that provides a first convincing explanation of observed cell death in
response to a perturbation in the Ras/cAMP/PKA pathway. More in detail, we apply our
model to study two important aspects of cell growth regulation: nutrient sensing and
the cell cycle. Artificial activation of the Ras/cAMP/PKA pathway has been
previously observed to recapitulate approximately 85% of the expression
response associated with increased growth in the presence of glucose [11]; here,
we show that the cell's regulatory state during this activation is
indicative of an up-regulated growth response, even in the absence of appropriate
nutrient availability. This conflict between internal regulatory state and the
external environment leads to rapid cell death. In contrast, analysis of growth rate
regulation during metabolic cycling [12] and synchronous cell cycles [8], [13] indicates that growth
rate regulation is not specific to cell cycle phases, but it is strongly limited to
the oxidative phase of the metabolic cycle. These observations, coupled with an
analysis of putative transcription factors mediating the growth response, establish
a substantial foundation on which to base further experimental work on the
systems-level control of cellular growth rate. Background: Measuring Growth Cellular growth is typically quantified in one of two experimental environments:
batch culture or the chemostat. In a batch culture, cells are provided with a
saturating amount of nutrient [1]. Growth is quantified by measuring the optical
density (OD) of the culture over time, X. illustrates three typical phases of
an OD growth curve: a slow initial phase (lag), a fast exponential growth phase
(exp), and a slow saturation phase (stationary). Solving the appropriate
differential equation leads to an exponential model of cellular growth,
X  = e μ·t. In
practice, the OD of a culture is sampled at discrete points over time, and the
growth rate parameter μ (in units of inverse hours
h −1) is estimated from an exponential fit to the OD
measurements. In the chemostat, a specific growth rate is maintained by limiting the
concentration of a controlling nutrient provided to the cells [14].
illustrates the
principle behind the chemostat. A limited concentration (S 0 in the
tank) of the controlling growth factor is provided in media flowing continuously
into a growth tube of limited capacity. Changes in density of the culture, X,
and in concentration of the controlling nutrient (S), in the growth tube, are
driven by Michaelis-Menten dynamics [15]. In this regime, the
growth rate is a function of the concentration of the controlling nutrient,
μ = μ(S). In particular,
dX/dt = [μ(S)−D]
X; at steady state, the density of the culture no longer changes,
dX/dt = 0, and the concentration of the
controlling growth factor also stabilizes,
dS/dt = 0. The growth rate then equals the flow
rate set by the experimenter,
μ(S*) = D. In a batch culture, the growth rate is generally not controlled; it is determined
by a complex interaction of environmental and genotypic states, and it is
maximal during the exponential phase of growth (μ max). Under
these conditions, the growth rate of the culture (the first derivative of the
curve in ) changes
with time. In a chemostat, the growth rate is controlled by setting the nutrient
flow rate D below an organism's maximum possible growth rate
μ max as estimated from batch culture. In either
experimental environment, the growth rate μ is directly related to the
doubling time, T d= ln(2)/μ. Our model is built on a collection of gene expression data from chemostats at
known growth rates, and it allows us to quantify a notion of instantaneous
growth rate in chemostat and batch cultures, even in cultures in which the
growth rate is changing rapidly over time. We fit a linear model to a collection of expression data drawn from S.
cerevisiae chemostat cultures over several growth rates and nutrient
limitations. Estimates of the parameters characterize each gene's response
to changes in growth rate, and provide insight into the transcription factors and
regulatory network responsible for yeast growth homeostasis. By applying this model
to new expression data sets, we are able to predict instantaneous growth rates for
any yeast culture. The model is robust to the biological and technical conditions of
the originating gene expression data and enables the prediction of growth rates at
instantaneous time scales inaccessible to standard experimental methods (e.g.
optical density). We have also successfully applied the model to the related
organisms S. bayanus and S. pombe. Data and tools
relating to this model are made available at http://function.princeton.edu/growthrate. Experimental Design and Data Our model is based on a collection of gene expression measurements from steady
state (chemostat) cultures limited across several nutrients and growth regimes.
Briefly, 36 CEN.PK derived S. cerevisiae chemostat cultures
were grown under six nutrient limitations: Glucose (G), Nitrogen (N), Phosphate
(P), Sulfur (S), Leucine (L), and Uracil (U). Six growth rates were used for
each nutrient, ranging by steps of 0.05 h −1 from 0.05
h −1 to 0.3 h −1. Agilent Yeast V2
microarrays were used to measure gene expression in the resulting 36 chemostats;
for details, see [4]. This experimental design provides the
opportunity to discover gene expression patterns correlated with growth rate,
independently of nutrient-specific responses. highlights the
sources of variability in the gene expression profiles that the experimental
design aims at capturing. The resulting data contain a number of characteristic
gene expression patterns, including genes with strong growth-specific
transcriptional regulation and negligible nutrient-specific response (). Other genes
include a growth-specific expression component but are also strongly up- or
down-regulated under specific nutrient limitations (). Finally, displays expression profiles that
show unsystematic or negligible responses under these conditions. The linear
model described below summarizes the variability in the expression profiles of
individual genes specifically due to changes in growth rate, which leads to a
characterization of growth-specific calibration genes such as
those shown in .
This growth-specific signature enables predictions of the instantaneous growth
rate of any cellular culture based on the relative expression values these
growth-specific genes. Table 1 summarizes the
collections of expression data analyzed in this study. Six collections were
previously published by others, one was published in our previous work [4], and
four are new to this study: 1. chemostats limited for different nitrogen
sources, 2. heat pulses inducing a temporary departure from steady state growth,
3. artificial activation of the Ras/PKA pathway, and 4. S.
bayanus diauxic shift and heat shock time courses. All gene expression
collections were pre-processed as in [16]. The gene
expression values for all growth-specific genes are provided in Tables S1,
S2,
S3,
S4,
respectively. | Table 1Overview of expression data analyzed in this study. |
Linear Models and Identification of Growth-Specific Signature We sought to identify a small set of genes providing a quantitative summary of
cellular growth rate regulation. Genome-wide expression measurements underlying
the 36 chemostat cultures provided us with the opportunity to determine which
genes were responding linearly to changes in growth rate, and not to differences
in nutrient limitation. To identify such gens in a statistically principled
fashion, we performed four steps, beginning by using maximum likelihood to fit a
linear model of each gene g's expression under all
training conditions ( Yg) based on the conditions' known growth rates ( Xc): This step yields two learned parameters per gene, a baseline expression level
αg and a growth rate response
βg. The model is fit to minimize the
residual error εg, which can capture either non-growth-related biological variability or
technical noise. We fit this model for the yeast genome using the expression
levels from our 36 chemostat conditions, recording each gene's
αg and
βg parameters and its goodness of fit (total
explained variability) R2g. We next used the bootstrap (i.e. a randomized re-sampling technique) to assess
the expected background distributions of these parameters in the absence of a
growth-related biological signal (i.e. the null distributions). We constructed
100,000 randomized expression vectors of length 36 by sampling each component
(with replacement) from the collection of gene expression values in the
corresponding condition, i.e., the same combination of growth rate and nutrient
limitation. For example, the first value randomly chosen for such a vector could
be drawn from any gene or nutrient limitation in our chemostat data at a flow
rate of 0.05 h−1, the second from any flow rate of 0.1
h−1, and so forth. Note that by re-sampling the
expression values of putative genes column-by-column, we do not wash away the
average transcriptional response that is expected to be associated with
nutrient-growth rate pairs. In this sense, the null distribution we derive
carries information about how genes respond to growth across nutrient
limitations, on average. As a consequence, the statistical significance of the
differential response we compute is biologically justified. In other words, this
sampling scheme maintains average nutrient specific and growth rate specific
information, and leads to an estimate of the null distribution in the absence of
gene-specific growth related and nutrient related gene expression. This process
yields null distributions for parameters αg,
βg, and the goodness of fit R2g. Third, from these null distributions, we assign false discovery rate corrected
p-values [17] to each gene's
αg,
βg, and R 2g values. Finally, a gene was deemed to have a significant expression
response to changes in growth rate if it fit this model well (R 2g p<0.05) and was up- or down-regulated significantly with growth
( βg p<0.05); this information
is available in [4]. We further characterized a specific set of
growth-specific calibration genes responding only and
significantly to changes in growth rate ( βg
p<10 −5 and R 2g p<10 −5) that we used to infer
instantaneous growth rates in new expression data ( Table S5
and Dataset
S1). Model-Based Prediction of Instantaneous Growth Rates from Expression Data The set of growth-specific genes identified with the four-step procedure above
represents a quantitative signature of a cellular culture's
transcriptional regulation of growth rate, i.e. the speed at which cells are
proliferating. By examining these genes' expression levels in a new
collection, we can predict the instantaneous growth rate of the cellular culture
the expression measurements correspond to. This notion of instantaneous growth
rate is comparable to the derivative of an optical density growth curve, but it
can be inferred robustly by our model on any time scale, e.g. minutes, from
expression data, without the need to measure one or more full doubling times of
a culture. Given expression data for a new experimental condition, we use an iterative
maximum likelihood approach to infer its growth rate using the parameters
captured by our linear model. Formally, consider a vector of expression
measurements for n growth-specifc genes,
Z1:n. As described above, the
expression of these growth-specific genes varies primarily in response to
changes in a condition's growth rate, which we model as the mean
μ of a Gaussian with variance
σ2. Using our previously calculated
maximum likelihood estimates of the calibration gene parameters
α1:n and
β1:n, the expected value of
a gene's expression is thus: Here, δ is a condition-specific parameter that captures
the condition's baseline gene expression, i.e. an average offset
between a new experimental condition and our training expression data. In
dual-channel data, this parameter may capture differences between a new
condition's reference channel and our training data's
reference channel; for a single-channel array, δ may
capture the absolute difference between the platform baseline and our training
data. In any event, the expected variability of a new measurement is: The likelihood of the expression measurements
Z1:n is thus a product of Gaussians: From this, we derive the maximum likelihood estimate of the condition's
growth rate μML: Similarly, the maximum likelihood estimate of the condition's baseline
δML is given by: Note that the estimate of δML depends on the
estimate of μML, and vice versa. To
calculate these estimates, we initialize
μML(0) assuming
δML(0)= 0
and iterate subsequent computations of
μML(t+1)
and
δML(t+1)
to convergence. In practice, individual growth-specific genes with residuals
outside the inner fences of all growth-specific gene residuals (more than 1.5
inter-quartile ranges from the lower or upper quartiles, [18]) are noted as
outliers and do not participate in that condition's growth rate
inference procedure. This allows outlier genes responding to non-growth related
stimuli (which are, in general, infrequent, e.g. six in one of our most variable
conditions as discussed below) to be noted for further investigation, while also
decreasing the cross-validated error of predicted growth rates. Extending Predictions to Additional Organisms In principle, this model of growth rate can be extended to study and predict
instantaneous growth in any organism for which appropriate homology to
growth-specific genes exists. To analyze growth rates in expression data from
S. bayanus and S. pombe, the S.
cerevisiae calibration genes were mapped to known orthologs. This
mapping was performed using the unambiguous pairings from [19] for S.
bayanus and the curated orthologous groups from [20]
for S. pombe. This resulted in 51 growth-specific genes for
S. bayanus and 74 for S. pombe, the
increase being due to one-to-many mappings; see Table
S5. Online Tool Availability The parameter estimates driving our predictions and tools allowing users to
predict growth rates in new data sets are available at http://function.princeton.edu/growthrate. Specifically, users
can upload S. cerevisiae expression data (single- or
dual-channel in standard PCL format) to receive estimates of relative growth
rate for each condition. If a reference with known growth rate is provided,
absolute rate estimates will be generated. This growth rate prediction tool has
been implemented in R and is also available for offline use, allowing further
customization (such as application to additional organisms). We apply our linear model of growth rate regulation to predict instantaneous growth
rates for a variety of expression data. This includes new chemostat cultures used to
assess prediction quality, publicly available stress response and gene deletion
microarrays from batch cultures, growth differences between metabolic cycling and
the cell cycle, several different microarray platforms, and an out-of-sample
validation to quantify model accuracy. We also observe good predictive performance
for growth rates in S. bayanus and S. pombe data
sets, the latter despite up to a billion years of evolutionary divergence from our
S. cerevisiae training data. This suggests that the
growth-related transcriptional regulation captured by our model is a key feature of
unicellular homeostasis, a feature we explore by examining nutrient sensing inputs
through the Ras/PKA pathway and potential growth rate transcription factors and
binding sites. Relative Growth Rate Prediction in Novel Experimental Settings Our model of the growth rate transcriptional response can be used to predict
relative instantaneous growth rates from any S. cerevisiae gene
expression data. For example, shows our predicted growth rates for a gene expression
time course sampled from a steady state culture exposed to a brief (<30
s) heat pulse ( Table S1). The predictions clearly show a departure from steady
state within five minutes of the heat pulse, followed by recovery within 15
minutes. Similar predictions over a range of chemostat flow rates ( Figure S1)
reveal that this cellular behavior is consistent, although there is some
variation in the degree of growth cessation during stress, in agreement with
tolerance and sensitization models of the yeast stress response [21].
Notably, standard experimental assays for growth rate (e.g. optical density)
would be incapable of monitoring such a response, while our model is able to
observe these growth changes on an instantaneous time scale. A similar application of our model to predict relative growth rates for the
stress response conditions of [6] is presented in (see Figure S2
for complete results). These data represent yeast batch cultures assayed using a
variety of different reference mRNA samples on a custom spotted microarray
platform, none of which differences from our training data impair the growth
rate estimation process. While there are no direct measurements of growth rate
in these non-steady-state conditions, our predictions are consistent with known
yeast biology and agree with expected growth behavior. Most shock time courses,
including all heat shocks, peroxide, diamide, and hyper-osmotic stress, provoke
an initial sharp decrease in growth rate followed by a return to initial or
near-initial rate; shorter shocks, such as DTT, menadione, and peroxide
responses, capture only the rate decrease. Batch growth proceeds at a fairly
constant rate until nutrients become depleted, at which point the rate decreases
sharply; this pattern is also seen in intentional nitrogen depletion. Growth
rates across varying temperatures peak as expected at 25 C [1], falling off at
lower and higher temperatures. Finally, response to varying carbon sources is
also as expected [22], with ethanol inducing the slowest growth and
fructose, sucrose, and glucose allowing the most rapid. Our model's
inference of growth rate from gene expression data alone allows both post hoc
growth analysis (e.g. years after the original experiment) and an estimation of
growth rates for cultures where direct growth measurements would be unfeasible,
difficult, or time consuming. When applied to expression data from yeast mutant strains, in which one or more
genes have been deleted, predicted growth rates can be used to quantify single
mutant fitness. We used our model to analyze the knockout collection assayed in
[10]; predictions on the complete data set are
available in Table S6. Direct fitness measurements for 199 of the ~300
mutants assayed via microarrays is available as supporting information [10].
Our predictions for these 199 growth rates correlate very strongly with the
direct fitness measurements (ρ = 0.473,
p<10 −11) and are derived solely from expression
data. In contrast, methods for experimentally estimating single mutant fitness
from high-throughput growth curves showed substantially less agreement
(ρ = 0.321,
p<10 −6
[23];
ρ = 0.108, p>0.2 [24])
with the original publication's direct fitness measurements. These
results represent a compelling argument as to the relevance of our growth rate
model for fitness estimation. Absolute Growth Rate Prediction with One Shared Reference With a small amount of additional information (i.e., a scalar) the relative
growth rates inferred by our model can be made absolute, in units of chemostat
flow rate (hr−1). Our model's predicted rates for
a collection of arrays are relative estimates, to one another. This is due to
the unknown quantitative effects of the reference mRNA in our dual-channel
training data; it is impossible to know a priori the relationship between this
reference channel and the relative (for dual-channel) or absolute (for
single-channel) expression levels in new microarray data. However, if the
absolute growth rate is known for some array in a given collection, our model
can make absolute rate predictions for other two-color arrays in the collection,
given that they all share the same reference channel. shows actual growth
rates (dotted, black lines) for a collection of chemostats at various flow rates
limited on one of several different nitrogen sources ( Table S2)
along with estimates of the relative instantaneous growth rates (red, dashed
lines) and of the absolute instantaneous growth rates (solid, blue lines).
Absolute growth rates are estimated by recording the growth rate in the Proline
limited chemostat at μ = 0.35
hr −1, and shifting all the estimates accordingly, since
the dual-channel microarrays in this study all share the same transcriptional
readout in the reference channel. We sought to evaluate the goodness of the
predictions in by
computing the statistical significance of their correlation with the actual
growth rates. To this end, we computed the correlation between the true growth
rates and the predicted instantaneous growth rates. The correlation is the same
for both absolute and relative predicted rates, as they differ by a constant,
and equals ρ = 0.956
(p-value≈0). This computation provides statistical support to the
goodness of the predictions produced with the proposed model. More in general,
on normalized dual-channel microarrays, the doubling of any gene's mRNA
level in these conditions results in the same increase in its expression
readout. Thus one unit of predicted relative rates to corresponds to one unit of
absolute chemostat flow rate. However, since the reference channel differs from
that of the arrays used to train the model, all rate predictions are typically
off by a corresponding constant factor. By normalizing to any one of the
N arrays' known growth rates, this shift can be
automatically corrected for the N-1 other arrays, employing the
same reference channel. Accuracy of the Predictions and Outlier Detection We assessed the quality of our growth rate predictions using 1,000 out-of-sample
experiments, according to a hybrid bootstrap/cross-validation setup, using the
data from [4]. Results are shown in . In each experiment, we randomly
withheld 12 of the 36 conditions for testing, fit our linear model on the
remaining 24, derived bootstrapped null distributions using only these data, and
determined growth-specific gene sets to use for growth rate inference on the
held-out conditions. This experimental setup leads to absolute growth rate
predictions directly, as all the dual-channel microarrays share the same
transcriptional readout in the reference channel. This out-of-sample validation
allowed us to assess the accuracy and variability of our predictions on
conditions with known growth rates not included in the model building procedure.
In addition to the performance indicated by , the out-of-sample experiments
demonstrated robustness of p-value cutoffs and number of growth-specific genes;
these ranged in number from ~50 to ~110 across the randomized
validations (of a total ~5,500 possible genes), and changes of this
magnitude in the final calibration gene set had little impact on predicted
growth rates. We further quantified a notion of reliability for each of the 72
growth-specific genes. Specifically, we computed the percentage,
P, of bootstrap experiments in which each individual gene was
selected as a member of the growth-specific gene set. The percentages provide an
expectation about whether each individual gene should be considered reliable in
a new study. We found that 69 genes were selected in more than half of the
experiments, P> 0.5. Full results are
reported in Table 2. | Table 2Reliability study for the 72 growth-specific genes. |
In the process of estimating growth rates and determining this confidence score,
growth-specific genes with outlying expression values are also detected. While
most conditions induce few outlying growth-specific genes, when they occur, they
are not indicative of the quality of growth rate predictions.
We have found that neither the number of outliers nor their variability
correlates with prediction error (data not shown), but they call out genes that
may be responding to non-growth stimuli under specific biological conditions.
Excluding outliers from the growth rate estimation process improves the accuracy
of the predictions, and these outliers can in turn be biologically informative:
an outlying growth-specific gene is likely responding specifically to a stimulus
other than change in growth rate. For example, some of the only outliers in the
mild heat shock time course from [6] occur towards the end of a shift from 29 C to
33 C (). These
include HSP26 and HSP78, both known heat shock
chaperones [25], [26]. Three genes of
unknown function ( YLR327C, MOH1 and the
neighboring dubious ORF YBL048W, and TMA10)
are also outliers in this condition, which is evidence that these genes may have
specific expression responses (and thus biological functions) during heat shock.
HSP26 and YLR327C are frequent outliers in
stress-related conditions, perhaps suggesting a more general stress response
function. Predicting Growth Rates in S. bayanus and S.
pombe While our growth rate model is based on a transcriptional growth signature in
S. cerevisiae, the model can be applied to any organism
with sufficiently orthologous transcriptional activity. This is likely to be the
case within the sensu stricto yeasts, separated by ~25
million years of evolution [27]. By finding the ~50 S.
bayanus genes orthologous to our ~70 S.
cerevisiae growth-specific calibration genes [19], we can apply our
model directly to S. bayanus expression data ( Table S4).
demonstrates
such a result for two S. bayanus time courses assaying the
diauxic shift and a response to heat shock. These results have comparable
profile to those from S. cerevisiae and are similarly
biologically compelling. For example, the diauxic shift in S.
bayanus results in a very similar growth pattern to the known response
in S. cerevisiae, with a near-cessation of growth during the
shift and subsequent rebound. Conversely, S. bayanus is less
resistant to high temperatures than S. cerevisiae
[28], and our growth rate inferences show a
corresponding failure in its ability to grow following severe heat shock. We have also extended our model to a significantly further diverged yeast,
specifically the yeast Schizosaccharomyces pombe, separated
from S. cerevisiae by an estimated one billion years of
evolution [3]. A mapping of our growth-specific calibration
genes to S. pombe using information from [20] results in
~75 genes due to one-to-many correspondences, but these provide
sufficient calibration information to make high quality predictions (). Calibration gene
outliers and expression cohesiveness are not substantially changed relative to
S. cerevisiae and S. bayanus, and the
inferred relative rates reflect various biological expectations. All cultures
(data from [29]) show an initial increase from low growth rates
due to stalled growth during synchronization. An expected decrease in growth
rate is predicted during increased exposure to hydroxyurea (HU), and a
rad3Δ deletion ( S. cerevisiae
ortholog MEC1) incurs a mild overall growth impairment as well
as exacerbating HU sensitivity. While MEC1 is essential in
S. cerevisiae, this sensitivity has previously been noted
for deletions sod1Δ and lys7Δ,
both members of the MEC1 pathway [30], which is necessary
for the cell cycle checkpoint function. The extent to which transcriptional regulation is conserved between S.
cerevisiae and S. pombe, which allows us to
successfully apply the model despite the evolutionary distance that separates
these species, is reflective of cellular growth's central role,
particularly in unicellular organisms. While this model would become less
meaningful in metazoans, where the growth of individual cells is subjugated to
the growth and differentiation of the organism as a whole, certain
transcriptional growth behavior is of necessity conserved in single celled
organisms [31]. This is particularly true of the ribosome, one
of the main contributors to our model's predictive power; rRNA
regulation is purely transcriptional, and ribosomal proteins must be expressed
stoichiometrically. Since any cellular growth requires translation, observation
of ribosomal transcription is a strong indicator of unicellular growth [5].
This is one aspect of the transcriptional growth response made quantitative by
our model. Insights into Growth Homeostasis To further investigate the biological basis of growth rate correlated gene
expression, we used our model to predict relative growth rates for two
interesting cases: the yeast metabolic cycle [12] and the mitotic cell
division cycle [8], [13]. The expression
data published by Tu et al. was obtained for cells grown at high density in a
glucose-limited chemostat. Under this regime, cells within the culture become
metabolically synchronized and undergo periodic consumption of oxygen (defined
as the oxidative phase of the metabolic cycle) followed by periods of
undetectable oxygen consumption (termed the reductive building and reductive
charging phases). The cell cycle data sets by Spellman et al and Pramila et al
were obtained from experiments in which cells were uniformly arrested in the
cell division cycle using a variety of methods and then released to undergo
synchronous cell division cycles. Growth rate prediction applied to the yeast metabolic cycle data revealed a
striking periodicity (). The cyclical pattern of growth rate variation occurs completely in
concert with the metabolic cycle as defined by Tu et al. Specifically, the
culture's growth rate is predicted to be at minima during the reductive
phase of the metabolic cycle, when oxygen consumption is at a minimum, and reach
maxima during the peak of the oxidative phases when oxygen consumption is
maximal. In contrast, growth rate prediction for the cell cycle () show
virtually no variation in predicted growth rate during the different stages of
cell division. These data support and extend our previous assertions [4] that the there is a
close connection between the metabolic cycle identified in [7] and [12] and the
association we identify between growth rate and gene expression levels. This
result is consistent with two possible explanations. The first is that there is
variation in the growth rate of cells throughout the metabolic cycle. [12] and
[32]
have shown that under their specific experimental conditions, DNA replication
and cell division is restricted to the reductive phases of the metabolic cycle.
It is conceivable that growth per se (i.e. the accumulation of biomass) is
paused during the reductive phases of the metabolic cycle so that the cell can
replicate and segregate DNA and complete the complex processes of cell division;
growth may then be restricted to the oxidative phase of the metabolic cycle.
Alternatively, it is possible that as any heterogeneous culture grows faster, a
greater fraction of cells are in the oxidative phase at any point in time. Thus,
the growth rate gene expression signature we detect might reflect the fraction
of cells in the oxidative and reductive phases of the metabolic cycle in a
metabolically unsynchronized population. The absence of growth rate differences during the cell division cycle () supports our
previous claim [4] that the growth rate expression signature is
unrelated to the cell cycle. Moreover, since the published cell cycle
experiments were performed in rich media using a fermentable carbon source, the
results suggest that rapidly growing cells (which are almost exclusively
fermenting) do not partition metabolic activity into discrete phases, as their
energetic requirements are met in a continuously reductive metabolic state. It
is only when slowed growth is imposed upon the cell, due to stress, nutrient
limitation, or other suboptimal environments, that the metabolic cycle is
required. We sought to distinguish whether nutrient availability directly determines the
transcriptional state related to growth rate or whether nutrient availability is
integrated through an internal signaling pathway that controls the appropriate
transcriptional state. To address this issue, we examined the regulatory circuit
responsible for transcriptional changes in response to glucose availability in
yeast. Glucose addition to cells growing on glycerol elicits a rapid and massive
change in the pattern of gene expression, with more than half of all genes
changing at least twofold in expression. Previous work has shown that the
Ras/cAMP/PKA pathway is the major source for eliciting this transcriptional
change in response to glucose addition [9], [11]. Activation of the
Ras/PKA pathway in the absence of environmental signals, through induction of an
activated allele of RAS2
( RAS2G19V), recapitulates in magnitude and direction
more than 85% of the changes observed by glucose addition, and
inhibition of PKA (concurrent with addition of glucose) blocks most of the
glucose induced transcriptional changes ( [11], Table S3).
This mutation thereby represents a useful model connecting S.
cerevisiae's glucose sensory signaling to its transcriptional
regulation of growth rate. We used a gal1Δ strain carrying the activated allele
RAS2G19V under control of the galactose
inducible GAL10 promoter. Addition of galactose activates the
Ras/PKA pathway, but since galactose cannot be metabolized by this strain, the
metabolic state of the cell remains unaltered [9]. When grown on
glycerol our model predicts a relative growth rate of ~0.2 for this
strain (), which
changes to ~0.6 within twenty minutes following glucose addition,
consistent with the change in doubling time from 5.8 hr to 2.6 hr. When we
performed the same experiment on glycerol media and induced the
RAS2G19V by means of galactose addition, we
detected a transcriptional response within sixty minutes. The predicted growth
rate of the RAS2G19V mutant strain was comparable to
the addition of glucose despite the fact that galactose addition does not yield
an increase in growth, as measured by optical density, since the cells are
unable to metabolize galactose. In fact, while the model's
summarization of gene expression state indicates that the culture is attempting
to increase growth, induction of the RAS2G19V allele
results in an immediate decrease in growth rate and complete cessation of growth
within four hours [33]. These results are consistent with the cell
setting its growth-specific transcription program on the basis of its
perception of nutrients present in the environment, rather
than on the direct availability of energy or metabolites produced from such
nutrients. The mechanism by which the cell integrates this external state in
order to set the appropriate growth rate expression state must be mediated, at
least in part, through the Ras/cAMP/PKA pathway. Potential Transcriptional Regulators of Growth Rate To investigate the regulatory basis of growth-associated gene expression, we
identified motifs enriched in the 3′ and 5′ regions of genes
with strong growth rate responses (). We assigned genes to clusters based on their growth rate
response parameter ( βg) using k-means
clustering with k = 10. Using the FIRE motif
identification program [34], we identified enriched motifs in seven of
the resulting ten clusters. Consistent with the functional enrichments of
negatively growth rate correlated genes [4], we identified known
binding sites associated with the stress responsive transcription factors Msn2p
and Msn4p in genes negatively correlated with growth rate. Conversely, genes
that increase in expression with increased growth rate are enriched for the
Rap1p consensus motif, which is commonly found upstream of genes encoding
protein components of the ribosome. We also found enrichment of the Ino4p binding site in genes upregulated with
increasing growth rate. Ino4p forms a heterodimer with Ino2p to activate genes
involved in phospholipid, fatty acid, and sterol biogenesis, all of which are
required in greater abundance with increased growth rates. Furthermore, Ino4p
has been proposed to have an inhibitory effect on a number of genes, including
those that encode the heat shock proteins (Hsp12p, Hsp26p) and catalase (Ctt1p)
[35]. We also identified two additional enriched
motifs in the 5′ UTR for which the binding factor is not known,
suggesting that additional activators of growth-related transcriptional programs
remain to be determined. In addition to 5′ upstream motifs, we identified five enriched
3′UTR motifs, which are potential binding site for proteins that
promote mRNA degradation. Only a small number of mRNA binding consensus
sequences are known in yeast, all of which belong to the Puf family of mRNA
binding proteins [36]. Our analysis identified five enriched motifs
in 3′UTRs. Two of these motifs, found in genes positively correlated
with growth rate, were identified by the FIRE program as being targets of Puf4p.
As an independent test, we compared the distribution of growth rate responses in
the known gene targets of the five Puf proteins with the overall distribution of
growth rate slopes. Targets of both Puf3p (220 genes) and Puf4p (205 genes) are
enriched for genes that are upregulated with increasing growth
(Wilcoxon-Mann-Whitney two sample p-values
9×10 −23 and
7.23×10 −16, respectively; Figure S3).
The consensus motifs of Puf3p and Puf4p are very similar; investigation of the
PUF4 motif identified by FIRE suggests that the enrichment signal for at least
one of the motifs denoted PUF4 is likely to result from a composite of Puf3p and
Puf4p target genes (). Overall, this analysis is consistent with tight transcriptional regulation
underlying the cellular growth program; it is likely that mRNAs involved in this
process are also subject to extensive post-transcriptional control.
Interestingly, since our growth-rate prediction method is sensitive to changes
in gene expression levels that occur within minutes of a perturbation, we expect
that post-transcriptional regulation (both mediated decay of and stabilization
of transcripts) is involved in this response. Experimental analyses of the
effects of perturbations within this regulatory network promise to shed further
light on its organization. We present a statistical model of the gene expression response to changes in growth
rate in S. cerevisiae. Developed on expression levels from a
variety of steady state growth rates and nutrient limitations, the model captures
information regarding each gene's linear response to growth rate. As
detailed in [4], approximately half of the genome shows a significant
transcriptional response to growth rate with strong functional cohesiveness; here,
we extend this model to show its robustness, applicability to new data, and ability
to provide insight into the biological systems driving cellular regulation of growth
rate. New experiments with more complex models (quadratic and hierarchical)
demonstrated that additional model parameters did not provide substantial
performance gains, in terms of growth rate prediction accuracy, particularly
relative to their added complexity (data not shown). Similarly, variations in the
definitions of responding genes or of growth-specific genes did not substantially
alter results. This stability is reflected in the out-of-sample validation results,
which quantify the model's accuracy in predicting relative growth rates
from gene expression data, and in Table 2, which suggest that growth-specific signal is localized to a
small number of genes consistently across experiments. The model can be applied to new gene expression data to estimate the instantaneous
growth rate of the originating cellular culture. The estimated instantaneous rate
represents a measurement of the transcriptional state of cellular growth rate
control, and it provides insight into the cell's growth rate at arbitrarily
short time scales inaccessible by experimental measurements (e.g. optical density).
Moreover, genes with unexpectedly high or low expression values can be detected
during growth rate inference, and may indicate biological responses to non-growth
stimuli. The predictions based on the proposed model are robust to changing
biological conditions, experimental methods, and technological platforms; they also
extend to the related yeast S. bayanus and the highly diverged
yeast S. pombe, suggesting that the transcriptional control of
growth rate captured by the model are a fundamental aspect of unicellular biology. Through further analysis, we discovered several putative transcription factor binding
sites enriched in growth-correlated genes, most notably the stress-responsive Msn2p
and Msn4p, the Rap1p ribosomal factor, and Ino4p. Importantly, we have identified a
likely role for post-transcriptional regulation in modulating transcriptional states
related to growth rates. This finding is consistent with our ability to measure
changes in growth rate over very short time scales using gene expression signatures.
The abundance of any messenger RNA is a function of both its rate of production and
of its rate of degradation; however, since transcription is relatively slow, changes
in mRNA abundance can be most rapidly instantiated by altering the stability of the
existent mRNA population. The Puf proteins have known roles in mediating mRNA
degradation [37] and in mediating the association of functionally
related transcripts [36]. It has recently been proposed that modulation of
mRNA stability is an important factor in metabolic regulation [38]. The association of
Puf protein binding domains in the 3′ UTRs of genes with increased
expression at higher growth rates suggests that modulating mRNA stability is also
important in the regulation of the growth response at short time scales. From a statistical perspective, it is notable that a simple linear model accurately
and robustly captures a specific biological phenomenon. The model represents a
concise, functionally cohesive set of expression profiles regarding the
genome's transcriptional response to growth. This functional interpretation
of the model agrees with known aspects of the growth response, such as the
transcription of ribosomal components, and provides insight as to the mechanistic
roles of internal feedback, environmental sensing, and the stress response as growth
rate varies. By monitoring a small ensemble of genes—with few parameters
per individual gene—the model is easily applicable to new conditions and
organisms and is robust to technical and biological sources of variation. These
features enable our model to serve both as a practical tool for growth rate
estimation (available at http://function.princeton.edu/growthrate) and as a mechanistic
building block in the pursuit of a systems-level understanding of cellular growth
processes. Dataset S1 An RData archive containing the complete collection of programs and results.
The archive includes a Table (named frmeGRParameters) with the growth rate
slope, goodness of fit, and other parameters based on our expression data
and linear model. The linear model assigns each gene a growth rate slope
(i.e. response to increased growth rate), baseline response, and goodness of
fit (i.e. linearity of response) based on our 36 expression arrays. The
statistical significance of these parameters was tested against a null
distribution based on 100,000 bootstrap samples. We have also indicated
whether each gene is in our positively or negatively growth correlated gene
sets, whether it is up- or down-regulated in the Environmental Stress
Response (ESR) [6], whether it was used as a growth-specific
gene for inferring instantaneous growth rates, and whether it was reliably
unresponsive to changes in growth rate. (10.88 MB ZIP) Figure S1 Growth rate predictions for chemostat cultures subjected to a brief heat
pulse at various flow rates. Expression time courses were taken for a
collection of chemostats at increasing growth rates, each subjected to a
brief (<30 s) heat pulse at time zero; see Supplemental Table
S1 for details. Predicted growth rates show an immediate departure
from steady state as the heat pulse is administered immediately before time
zero, followed by a gradual return to steady state and regulatory overshoot.
This behavior is consistent across growth rates, with the lowest growth
rates potentially showing a lesser shock response due to stress tolerance. (0.02 MB PDF) Figure S2 Growth rate predictions for all conditions in the stress response expression
arrays in [6]. These predictions are generally
consistent with known yeast biology and agree with expected growth behavior;
most shock time courses, including all heat shocks, peroxide, diamide, and
hyper-osmotic stress, provoke an initial sharp decrease in growth rate
followed by a return to initial or near-initial rate. Shorter shocks, such
as DTT, menadione, and peroxide responses, capture only the rate decrease.
Batch growth proceeds at a fairly constant rate until nutrients become
depleted, at which point the rate decreases sharply; this pattern is also
seen in intentional nitrogen depletion. Growth rates across varying
temperatures peak as expected at 25 C, falling off at lower and higher
temperatures. Response to varying carbon sources is also as expected, with
ethanol inducing the slowest growth and fructose, sucrose, and glucose
allowing the most rapid. The model's inference of growth rate from
expression data alone thus allows both post hoc growth analysis (e.g. years
after the original experiment) and an estimation of growth rates for
cultures where it would be difficult or time consuming to measure directly. (0.03 MB PDF) Figure S3 PUF3 and PUF4 targets are enriched for genes that respond positively to
growth. We plotted the distribution of PUF3 targets (220 genes; black line)
and PUF4 targets (205 genes; red line) identified in [36] on the
distribution of slopes reported in [4]. Targets of both
these mRNA-binding proteins are enriched for genes that are increased in
expression at higher growth rates. This is consistent with an important role
for post-transcriptional regulation in modulating the growth-related gene
expression program. (0.05 MB PDF) Table S1 Expression of growth-specific genes for chemostat cultures at increasing
growth rates exposed to a brief heat pulse. A collection of chemostats was
run at growth rates ranging from 0.05/hr to 0.25/hr. A brief (<30 s)
heat pulse was administered immediately before time zero, and expression
arrays were collected in a time course from before the pulse (pre.) to two
hours after the pulse using the 0.1/hr pre-pulse time point as a reference.
(Here we provide expression data for all the growth-specific genes. The
genome-wide collection of gene expression data will appear in a subsequent
publication.) (0.06 MB XLS) Table S2 Expression of growth-specific genes for chemostat cultures at increasing
growth rates limited on various nitrogen sources. A collection of chemostats
was run at growth rates from ~0.06/hr to ~0.21/hr limited on
one of several different nitrogen sources, including ammonium, allantoin,
glutamate, arginine, glutamine, urea, and proline. (Here we provide
expression data for all the growth-specific genes. The genome-wide
collection of gene expression data will appear in a subsequent publication.) (0.04 MB XLS) Table S3 Expression of growth-specific genes for batch cultures grown on glucose,
galactose, and galactose with a constitutively activated Ras/PKA pathway. We
constructed a gal1 deletion strain carrying the activated allele RAS2(G19V)
under control of the galactose inducible GAL10 promoter. Addition of
galactose activates the Ras/PKA pathway, but since galactose cannot be
metabolized by this strain, the metabolic state of the cell remains
unaltered. Gene expression was then assayed at 20, 40, 60, and 80 minutes
(relative to time 0) after nutrient exposure. (Here we provide expression
data for all the growth-specific genes. See [9] for additional
data.) (0.03 MB XLS) Table S4 Expression of growth-specific genes for Saccharomyces bayanus orthologs under
the diauxic shift and heat shock. Gene expression was measured for time
courses of S. bayanus undergoing the diauxic shift and for a culture heat
shocked by shifting from 25 to 37 C. (Here we provide expression data for
all the growth-specific genes. The genome-wide collection of gene expression
data will appear in a subsequent publication.) (0.03 MB XLS) Table S5 S. cerevisiae growth-specific genes used for growth rate prediction in this
study with S. bayanus and S. pombe orthologs. S. cerevisiae growth-specific
genes were defined to have a bootstrapped p-value of growth rate response
and linear fit less than 10−5. S. bayanus orthologs were drawn
from [19] and S. pombe orthologs from [20]. (0.00 MB XLS) Table S6 Predicted relative growth rates for expression data from the deletion
collection in [10]. Our predictions for the 199 mutants for
which Hughes et al directly measured growth rates show significant
correlation to the experimental gold standard
(rho = 0.473, p<10−11), in
contrast to other single mutant fitness estimates based on growth curve
analysis (e.g. [23] reports
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