HOW TO AVOID
MISINTERPRETING
MICROARRAY DATA
Sungchul Ji, Ph.D.
Department of Pharmacology and Toxicology
Rutgers University
Piscataway, N.J. 08855
sji@rci.rutgers.edu
(DIMACS Workshop on Machine Learning Techniques in Bioinformatics,
Center for Discrete Mathematics and Theoretical Computer Science, Rutgers University,
Piscataway, July 11-12, 2006)
The DNA Microarray Technology [1]
inaugurated a new era in cell biology in the mid-1990’s. The integration of measurement, analysis and
interpretation of genome-wide expression data is essential for the successful application of this
revolutionary technology to cell biology. Of these three aspects of the new technology, interpretation
has been least developed, as evidenced by misinterpretations (secondary to conflating transcription
rates with transcript levels) of DNA microarray data found in numerous publications.
Measurement
Interpretation
Analysis
DNA Microarray Technology
•
Output
Input
•
10
9
•
Gradients
5
6
Proteins
7
3
RNA
8
1
Genes
4
2
Amino
Acids
•
Ribonucleotides
•
Figure 1. The molecular model of the cell known as the
Bhopalator [2].
A theoretical model of the living cell is
deemed essential in interpret DNA
mciroarray data.
The Bhopalator model of the cell
proposed in 1985 may provide a useful
starting point [2].
Because mRNA is broken down into
nucleotides (step 2) as rapidly as it is
synthesized (step 1), its concentration at
any time is determined by both
transcription (step 1) and degradation
(step 2) rates.
It has been the common practice in the
field of microarray technology to assume
that mRNA levels are determined mainly
by transcription rate, but this assumption
remains to be substantiated. On the
contrary, simultaneous measurements of
transcript levels (TL) and transcription
rates (TR) from budding yeast subjected
to glucose-galactose shift [3] indicate that
TL is always controlled by the dual
actions of transcription and transcript
degradation, except occasionally by
degradation alone [3a].
Another error commonly committed in
the field is to conflate “gene expression”
which is a rate process (i.e.,
concentration change per unit time) with
“mRNA levels” which are just
concentrations.
How To Avoid Misinterpreting DNA Microarray Data:
Do Not Say
Gene Expression.
Say
mRNA Levels.
•
‘Gene expression’ interpreted as transcription is not the same as mRNA levels: The former is a
process and the latter a concentration.
•
Gene expression rate, dnS/dt, and mRNA levels, n, are mathematically related as follows:
•
dn/dt = dnS/dt – βdnD/dt, where  and β are constants, and dnD/dt is the rate of
mRNA degradation. By a suitable integration, we obtain the following equation:
Δn = dnS – βdnD,
where  is the integration over a given time period and dnS can be calculated as the
AUC (Area Under the Curve) of the function, dnS/dt = f(t).
Mathematically speaking, Δn is a functional of dnS/dt. That is, mRNA levels are
functionals, not a function, of rates of gene expression.
The Six Modules of RNA Metabolism Observed in
Budding Yeast Undergoing Glucose-Galactose Shift
•
The mechanisms of interactions between transcription and transcript degradation
induced by the glucose-galactose shift in budding yeast have been studied based on
the dual measurements of TL (transcript levels) and TR (transcription rates) by
Perez-Ortin and his coworkers [3, 3a].
•
Δni denotes the changes in the number of the ith mRNA molecule experienced by a
cell during a given time period, and nS,i and nD,i indicate the numbers of the ith
mRNA molecules per cell synthesized and degraded between two time points,
respectively. The subscript i is omitted below for convenience.
•
There are six, and only six, modules of mRNA level control observed in yeast
during glucose-galactose shift, labeled as A, B, C, D, E and F in the following table.
Each module is characterized by a unique numerical value of the degradation-totranscription ratio, nD/nS. This ratio was calculated from the equation relating the
changes in transcript abundances (Δn) due to transcript synthesis (nS) and
transcript degradation (nD) and the experimentally measured values of Δn and nS in
[3, 3a].
Δn = nS – nD
(1)
•
The above equation can be visualized as a 3-dimensional plane (hyperplane) as
shown in the figure next to the table in the following slide.
Δn
Relative
sizes of nS
and nD
nD/nS
Modules of
mRNA
Metabolism
n S > nD > 0
<1
(ascending state
with dual control)
A
=0
(ascending state
with transcriptional
control)
B
+
n S > nD = 0
n S = nD > 0
=1
(steady state)
C
0
n S = nD = 0
(mathematically
undefinable;
equilibrium state)
D
0 < nS < nD
>1
(descending state
with dual control)
E
Infinity
(descending state
with degradational
control)
F
0 = nS < nD
The Cell-Brain-Computer Relation
•
Ontologically, cells gave rise to the human brain (step 1) which in turn gave rise to
computers (step 2).
•
Epistemologically, the well-known properties of computers will facilitate our
understanding of the functioning of the brain (step 3) and the cell (step 5).
Knowing how our brain works can also help us understand how cell works (step 4),
as exemplified by the recent proposal that cells use a language whose principles
share commonalities with those of human language [4].
•
As an example of the computer science helping biologists to understand the
workings of the cell (step 5), one may cite the application of the SVM (support
vector machine) approaches [5] to analyzing DNA mciroarray data.
The Cell as the Smallest DNA-Based Molecular Computer
(S. Ji, BioSystems 52, 123-133, 1999)
Cells
2
1
Brains
4
Computers
3
DNA
5
Ontogeny = 1, 2
Epistemology = 3, 4, and
5
The Budding Yeast as the Hydrogen Atom of Cell Biology: The DNA microarray
technique may play the role of the atomic spectroscopic technique in physisics
which helped unravel the structure of the hydrogen atom [6].
The Cytoskeleton
(Mouse Embryonic 3T3 cell)
The Complementary (+/-) Relations among the various DNA
and RNA molecules involved in Microarray Experiments
RP
RT
H
(+) DNA ------ > (-) mRNA ------ > (+) DNA ------- > (-) DNA.
|
| DNA
| Polymerase
RP = RNA polymerase
|
or
RT = Reverse transcriptase
| Synthesizer
H = Hybridizes to; no enzymes needed
|
\/
(-) DNA
(Used to fabricate DNA microarrays)
DNA Microarrays [1]
•
There are two kinds of DNA microarrays
– cDNA or EST microarray and the Gene
Chips.
•
One microarray can measure 104 mRNA
levels simultaneously.
•
mRNA levels in the cell are determined
by mRNA synthesis (Vsyn) and mRNA
hydrolysis or degradation (Vhyd), because
the rate of change in mRNA levels (R)
inside the cell is always:
dR/dt = Vsyn - Vhyd
(2)
•
Only when certain kinetic conditions are
met (discussed below) can mRNA levels
measured with DNA microarrays can be
interpreted as reflecting rates of gene
expression [1, 3a].
•
Each square can recognize one kind of
mRNA molecules.
How DNA Microarray Experiments are Done [1]
1
1.
2.
3
2
3.
4.
4
5.
5
6.
6
5
Isolate mRNA from broken cells.
Synthesize fluorescently labeled cDNA
from mRNA using reverse transcriptase and
fluorescently labeled nucleotides.
Prepare a microarray either with EST
(Expressed Sequence Tag) or
oligonucleotides (synthesized right on the
microarray surface; see Affimetric,Inc.).
Pour the fluorescently labeled cDNA
preparations over the microarray surface to
effect hybridization. Wash off excess
debris.
Measure fluorescently labeled cDNA
hybridized to a microarray using a
computer-assisted microscope.
The final result is a table of numbers, each
number registering the fluorescent intensity
which is in turn proportional to the
concentration of cDNA (and hence
ultimately mRNA) located at row x and
column y, row indicating the identity of
genes, and y the conditions under which the
mRNA levels are measured.
Covalent and Noncovalent Interactions in Microarray Experiments
1)
2)
3)
4)
CTAATGT
(Original DNA)
1
2
5)
3
6)
3
Transcription inside the cell
Reverse transcription inside the test tube
Hybridization on the microarray surface
Probably millions of cDNA molecules are
attached on each square on a DNA
microarray.
To the extent that mRNA is stable, the
amount of mRNA formed during Step 1
can be estimated from the amount of
cDNA bound to microarray surface in
Step 3.
But mRNA molecules inside the cell are
unstable, because they are rapidly
hydrolyzed into ribonucleotides by
various ribonucleases. Therefore, it is
impossible to estimate how many mRNA
molecules are formed in Step 1 by
measuring only how many molecules of
cDNA are bound to microarray surface in
Step 3 (more on this later).
Cluster Analysis

The changes in mRNA levels of human fibroblasts (cells
of connective tissues that synthesize and secrete fibrillar
procollagen, fibronectin, and collagenase) measured with
DNA microarrays over a time period of 24 hours.

Green represents a decrease in mRNA levels, black no
change, and red an increase.

Each kind of mRNA molecule is represented by a single
row of colored boxes, and a measuring time point is
represented by a single column.

Notice that the mRNA molecules belonging to cluster A
started to decrease around 8 hours after beginning
experiment.

The mRNA molecules belonging to cluster E began to
increase at around 5 hours after the beginning of the
experiment.

The phrase “mRNA levels” in above statements is almost
always replaced by “gene expression” (which phenomenon
may be referred to as the “gene bias”), which is strictly
speaking logically fallacious and can lead to false positive
and false negative conclusions regarding the identities of
the genes responsible for mRNA level changes.
The Duality of Transcription and Transcript Degradation.
The Principle of Dual Control of mRNA Levels by Transcription and Transcript Degradation.
The decreases in mRNA levels measured with DNA microarrays cannot be accounted for without invoking the
mRNA degradation step. If there were no transcript degradation step, the mRNA levels inside the cell can only
increase or remain constant, but never decrease.
Simultaneous Measurements of Genome-Wide Transcript Levels
(TL) and Transcription Rates (TR) from the Yeast [3] (I)
•
•
•
•
Most DNA array measurements reported in the literature since the beginning of the
DNA array era [1] have involved only mRNA levels, except Fan et al [7] and
Garcia-Martinez et al [3], who measured both transcript levels (TL) and
transcription rates (TR), the latter using nuclear run-on methods.
Because TL is determined by a dynamic balance between transcription and
transcript degradation, TL is a function of both TR and transcript degradation
rates, TD:
TL = f(TR, TD)
(3)
Eq. (3) has three variables. Hence it cannot be solved without the input of the
numerical values of any two of these three variables.
Most workers in the field in effect have been trying to solve Eq. (3) for TR by
inputting just one of the two remaining numerical values, namely, TL, ignoring TD.
This is mathematically impossible, and logically indefensible. Ignoring TD in an
attempt to determine TR with TL measurements alone is tantamount to violating
the Principle of Insufficient Reason, according to which if there is no sufficient
reason for something's nonbeing, then it will exist.
The significance of the TL and TR data obtained by Fan et al [7] and GarciaMartinez et al [3] is that their data allowed TD in Eq. (3) to be determined genomewide for the first time.
Simultaneous Measurements of Genome-Wide Transcript Levels (TL)
and Transcription Rates (TR) from the Yeast [3] (II)
•
•
•
•
•
•
•
Garcia-Martinez et al [3] measured TL and TR from budding yeast at six time points, 0, 5, 120, 360,
450 and 850 minutes after replacing glucose with galactose.
Typical plots of TR vs. TL revealed nonlinear trajectories as evident in the next slide. Each trajectory
is divided into 5 directed segments (hence to be called TR-TL vectors). These vectors seem to
assume all possible directions.
A total of 5,725 genes of budding yeast were analyzed and the directions (measured as  shown in
the next slide) of their component vectors (5 for each trajectory and 28,625 vectors in total) were
calculated from their coordinates in the TR vs TL plane. The direction of these vectors were grouped
into 9 categories based on their measured angles as follows: 1 = -3 to +3; 2 = 3 to 87; 3 = 87 to 93; 4
= 93 to 177; 5 = 177 to 183; 6 = 183 to 267; 7 = 267 to 273; 8 = 273 to 357; and 9 = 0 or
undefineable. (I want to thank Dr. WonSsk Yoo for carrying out these calculations.)
The measured percentages of the vectors belonging to each category is as follows with the expected
percentages given in parenthesis: 1 = 2.94% (1.67); 2 = 26.07% (23.33); 3 = 1.91% (1.67); 4 =
29.73% (23.33); 5 = 1.80% (1.67); 6 = 24.91% (23.33); 7 = 2.38% (1.67a); 8 = 10.26% (23.33); 9 =
(not determined). These values are graphically represented as a histogram in the next slide.
Three conclusions can be drawn from these measurements:
(i) The TL-TR vectors are distributed non-randomly over the 7 out of the 8 categories of directions.
(ii) TL can increase even when TR decreases or undergo no change.
(iii) TL can decrease even when TR increases.
Therefore, TL and TR can vary independently of each other.
Before these measurements were made, most workers assumed that TL and TR were related linearly.
TR
a
YBL091C-A
3
160
TR
120
1
80
2
4
6
40
0
-40 0
5
10
15
20

TL
5
b
1
9
YNL162W
40
1
TR
30
6
6
20
7
10
0
0
c
100
TL
200
300
YLR084C
2
6
TR
1.5
1
1
0.5
0
0
d
20
40
TL
60
80
YHR029C
25
20
TR
15
10
1
5
0
-5 0
8
6
10
20
TL
30
40
TL
Kinetic Equations Needed for Analyzing mRNA Metabolism
•
TL = Transcript Level, arbitrary unit
•
TR = Transcription Rate, arbitrary unit/min
•
n = mRNA molecules per cell = f(TL) = aTL + b
where a and b are constants determined empirically.
•
dnS/dt = rate of mRNA synthesis, molecules/cell/min = g(TR) = a’TR + b’
a’ and b’ are constants determined empirically.
•
dnD/dt = rate of mRNA degradation, molecules/cell/min
•
dn/dt =  dnS/dt – β dnD/dt
where  and β are constants
•
Δn =  dnS – β dnD
(5)
•
Δn = nS – βnD + ε
where ε is a constant and nS and nD are the number of mRNA molecules
synthesized and degraded, respectively, between two time points. If it is assumed that
 and β are unity and ε is zero, the above equation reduces to:
(6)
Δn = nS – nD + ε
(4)
(7)
which is visualized as the mRNA hyperpalne in the following slide.
(I want to thank Drs. R. Miura, N. Fefferman & W. Chaovalitwongse for helpful suggestions in formulating these expressions)
The RNA Hyperplane:
A Geometric Representation of the Six Modules of mRNA Metabolism Regulating mRNA
Levels in Budding Yeast. The Symbols are defined in the table in a previous slide. Please
note that Δn can be +, - or zero but nS and nD are always positive. (The delta sign in front of
nS and nD seem unnecessary.)
Kinetics of Genome-Wide mRNA Level Changes Induced
by Glucose-Galactose Shift in Budding Yeast
•
•
•
•
•
The genome-wide average TL values are plotted against time in Slide #25.
During the first 5 minutes after replacing glucose with galactose, the average TL
value drops by about 30%, decreasing maximally by 60% during the next two
hours.
The TL level begins to rise at about 360 minutes reaching a maximal value of 70%
by 450 minutes. The level then decreases gain to 55% by 850 minutes.
The initial decline probably results from the decrease in ATP level in the cell due to
inhibition of glycolysis, the main metabolic pathway to generate ATP in the
presence of glucose (see Slide #28).
The abrupt rise in TL beginning at 360 minutes is most likely due to the induction
of enzymes needed for metabolizing galactose, in part forming glucose (see the lefthand side of Slide #28). One evidence for this conjecture is the induction of the
mRNA molecules (Gal 1, 2, 3, 7 & 10) coding for the proteins required for galactose
metabolism beginning at 120 minutes (see Slide #27) and their suppression at
around 360 minutes, probably due to the presence of glucose newly synthesized
from galactose (see hi Glu in Slide #29).
Genone-Wide Average mRNA
120
mRNA, arbitrary unit
100
80
60
40
20
0
-200
0
200
400
time, min
600
800
1000
Genome-Wide Average Transcription Rate
25
mRNA/Cell/min
20
15
10
5
0
-200
0
200
400
-5
time, min
600
800
1000
GAL1, 2, 3, 7 & 10
Average TR (arbitrary
unit)
25
20
15
10
5
0
-200
0
200
400
Time, min
600
800
1000
Kinetics of the Glycolytic and Respiratory (or Oxidative
Phosphorylation) mRNA Metabolism in Glucose-Derepressed
Budding Yeast
•
•
•
•
•
Between 5 and 360 minutes after replacing glucose with galactose, the
average mRNA levels of glycolytic and respiratory genes change in the
opposite directions (see Slide #31a).
Strikingly, the average TR values for these two groups of genes change in
a parallel manner as shown in Slide #31b.
Therefore, the opposite changes in the TL values of glycolytic and
respiratory genes must be attributed to the opposite changes in the rates
of their transcript degradation (TD).
The average degradation to transcription (D/T) ratios for glycolytic and
respiratory genes at the 5 time points (corresponding to the mid-points of
the 5 time segments, namely, 0-5, 5-120, 120-360, 360-450, & 450-850
minutes) were calculated using nD/nS = 1 – Δn/nS, derived from Equation
(1) in Slide #5. These ratios are plotted in Slide #31c.
Based on the D/T ratios, we can assign the following sets of labels to the
glycolytic and respiratory TL trajectories:
Glycolysis = ECCAC
Respiration = EAAAC
b
Average m RNA Levels
= Glycolysis;
= Oxphos
a
Tim e - v_S plots
= Glycolysis;
= Oxphos
1
v_S, molecules/cell/min
mRNA, molecules/cell
50
40
30
20
10
0.8
0.6
0.4
0.2
0
0
0
200
400
600
800
-200
1000
0
200
Tim e, m in
400
Tim e, m in
Degradation/Transacription (D/T) Ratios vs Tim e
= Glycolysis;
= Oxphos
c
3.5
3.0
D/T Ratios
-200
2.5
2.0
1.5
1.0
0.5
0.0
0
200
400
Tim e, m in
600
800
600
800
1000
How to Interpret DNA Microarray Data (I)

What we measure with DNA microarrays are changes in florescence intensities.

The changes in fluorescence intensities can be divided into two categories – artifactual and non-artifactual. The
present state of the development of the microarray technique is such that artifactual fluorescence intensity
changes probably account for about 50%. This is why it is a common practice to use the notion of “fold
changes” referring to fluorescence intensity changes that are greater than 100% (or one-fold change).

Only the non-artifactual fluorescence intensities can be related to mRNA levels.

mRNA levels measured with DNA microarrays can be divided into two categories – steady state and nonsteady state. The difference between these two categories of mRNA levels can be represented mathematically
as follows, where R is a mRNA level and t is time:
Steady state :
dR/dt = 0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8)
Non-steady state: dR/dt  0

The steady-state mRNA levels divide into two categories – dynamic and equilibrium.
The intracellular levels of mRNA molecules are always determined by two terms – the source term (i.e., the
rate of mRNA synthesis, denoted by dRS/dt) and the sink term (i.e., the rate of mRNA hydrolysis into smaller
fragments, denoted as dRD/dt ):
dR/dt = dRS/dt - dRD/dt
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)
There are two ways of making Eq. (9) = 0; when dRS/dt and dRD/dt are equal and non-zero, and when dRS/dt
and dRD/dt are both equalt to zero:
Dynamic steady state: dRS/dt = dRD/dt
Equilibrium steady state: dRS/dt = dRD/dt = 0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . (10)
. . . . . . . . . . . . . . . . . . . . . . . . . (11)
How to Interpret DNA Microarray Data (II)

The non-steady state mRNA levels divide into two categories:
On-the-way-up, or Ascending:
dR/dt > 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)
On-the-way-down or Descending: dR/dt < 0 . . .. . . . . . . . . . . . . . . . . . . . . . . . . (13)

It is probably safe to assume that dRS/dt always independent of R (i.e., gene expression is turned
on or off by factors other than intracellular levels of corresponding mRNA levels). But dR D/dt
may often (if not always) depend on R, leading to the conclusion that there are at least two
categories of dynamic steady states:
Zero-order dynamic steady state: dRD/dt = k (R)0 = k . . . . . . . . . . . . . . . . . . (14)
First-order dynamic steady state: dRD/dt = kR
. . . . . . . . . . . . . . . . . (15)

These results can be summarized as follows:

Combining Equations (10) and (15) leads to the following useful relation:
dRS/dt = dRD/dt = kR
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (16)
Equation (16) states that, under the conditions of a dynamic steady state, the mRNA levels, R,
measured with DNA microarrays are directly proportional to the rates of expression of their
corresponding genes, dRS/dt, since it is equal to dRD/dt, the rate of transcript degradation, under a
dynamic steady state.

An important corollary of Equation (16) is that, under all other conditions, there is no direct
proportionality relation between mRNA levels and the rates of expression of their corresponding
genes.
Dissipative and Equilibrium Networks in the Living Cell:
I. Prigogine (1917-2003) distinguished between two fundamental classes of structures in nature – equilibirum and
dissipative structures. The former can exist without any input of free energy whereas the latter exist if and only if a
continuous dissipation of free energy supports them. Similarly, it is proposed here that ‘dissipative networks’ in cells
(e.g., some protein-protein interaction networks) disappear upon cessation of free energy input, while ‘equilibrium
networks’ (e.g., Krebs cycle, glycolytic pathway, etc.) can persist without any dissipation of free energy. The protein
network is unique in that it is the only network that can tap free energy from chemical reactions. Dissipative
networks may also be referred to as the “Self-Organizing-Whenever-and-Wherever-Needed (SOWAWN) Machine”.
References:
[1] Watson, S. J., and Akil, U. (1999). Gene Chips and Arrays Revealed: A Primer on
Their Power and Their Uses. Biol. Psychiatry 45:533-543.
[2] Ji, S. (1985). The Bhopalator – A Molecular Model of the Living Cell Based on
the Concepts of Conformons and dissipative Structures. J. theoret. Biol. 116:399-426.
[3] Garcia-Martinez, J., Aranda, A., and Perez-Ortin, J. E. (2004). Genomic Run-On
Evaluates Transcription Rates for all Yeast Genes and Identifies Gene Regulatory Mechanisms.
Mol. Cell 15:303-313.
[3a] Ji, S., Chaovalitwongse, W., Fefferman, N., and Perez-Ortin, J. E. (2006). The Six
Modules of Transcript Control Revealed by Genome-Wide Expression Data from GlucoseDerepressed Saccharomyces cerevisiae. (in preparation).
[4] Ji, S. (2004). Molecular Information Theory: Solving the Mysteries of DNA. In:
Modeling in Molecular Biology (Ciobanu, G., and Rozenberg, G., eds.), Springer, Berlin. Pp. 141150.
[5] Cristianini, N., Shawe-Taylor, J. (2000). An Introduction to support Vector
Machines and other kernel-based learning methods. Cambridge University Press, Cambridge.
[6] Ji, S. (2005). Semiotics of Life: A Unified Theory of Molecular Machines, Cells,
the Mind, Peircean Signs and the Universe Based on the Principle of Information and Energy
Complementarity. Reports, Research Group on Mathematical Linguistics, Rovira i Virgili
University, Tarragona, Spain. See the section entitled An Analogy between Atomic Physics and
Cell Biology on pp. 58-61, available at http://www.grlmc.com, under Publications.
[7] Fan, J., Yang, X., Wang, W., Wood, W. H., Becjer, K. G., and Gorospec, M. (2002).
Global analysis of stress-regulated mRNA turnover by using cDNA arrays. Proc. Nat. Acad. Sci.
US 99(16):10611-10616.